ENCYCLOPEDIA OF PHYSICS
CHIEF EDITOR
S. FLUGGE
VOLUME VIa/2
MECHANICS OF SOLIDS II
EDITOR
C. TRUESDELL
WITH 15 FIGURES
SPRINGER-VERLAG
BERLIN • HEIDELBERG • NEWYORK
1972

HANDBUCH DER PHYSIK
HERAUSGEGEBEN VON
S. FLUGGE
BAND VIa/2
FESTKORPERMECHANIK II
BANDHERAUSGEBER
C.TRUESDELL
MIT 25 FIG U REN
SPRINGER-VERLAG
BERLIN • HEIDELBERG • NEW YORK
1972

ISBN 3-540-05535-5 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-05535-5 Springer-Verlag New York Heidelberg Berlin
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Contents.
The Linear Theory of Elasticity. By Morton E. Gurtin, Professor of Mathematics,
Carnegie-Mellon University, Pittsburgh, Pennsylvania (USA). (With 18 Figures) . l
A. Introduction 1
1. Background. Nature of this treatise 1
2. Terminology and general scheme of notation 2
B. Mathematical preliminaries 5
I. Tensor analysis 5
3. Points. Vectors. Second-order tensors 5
4. Scalar fields. Vector fields. Tensor fields 10
II. Elements of potential theory 12
5. The body B. The subsurfaces Sfx and ^ of 8B 12
6. The divergence theorem. Stokes' theorem 16
7. The fundamental lemma. Rellich's lemma 19
8. Harmonic and biharmonic fields 20
III. Functions of position and time 24
9. Class C^.^ 24
10. Convolutions 25
11. Space-time 27
C. Formulation of the linear theory of elasticity 28
I. Kinematics 28
\12. Finite deformations. Infinitesimal deformations 28
v 13. Properties of displacement fields. Strain 31
vl4. Compatibility 39
II. Balance of momentum. The equations of motion and equilibrium .... 42
15. Balance of momentum. Stress 42
16. Balance of momentum for finite motions 51
\l7. General solutions of the equations of equilibrium 53
V 18. Consequences of the equation of equilibrium 59
\ 19. Consequences of the equation of motion 64
III. The constitutive relation for linearly elastic materials 67
^0 . The elasticity tensor 67
\21 . Material symmetry 69
v 22. Isotropic materials 74
v 23. The constitutive assumption for finite elasticity 80
* 24. Work theorems. Stored energy 81
\25. Strong ellipticity 86
\j26. Anisotropic materials 87
D. Elastostatics 89
I. The fundamental field equations. Elastic states. Work and energy .... 89
27. The fundamental system of field equations 89
28. Elastic states. Work and energy 94
II. The reciprocal theorem. Mean strain theorems 96
29. Mean strain and mean stress theorems. Volume change 96
30. The reciprocal theorem 98

[
Contents.
III. Boundary-value problems. Uniqueness 102
31. The boundary-value problems of elastostatics 102
32. Uniqueness 104
33. Nonexistence 109
\ IV. The variational principles of elastostatics 110
34. Minimum principles 110
35- Some extensions of the fundamental lemma 115
36. Converses to the minimum principles 116
37- Maximum principles 120
38. Variational principles 122
39. Convergence of approximate solutions 125
V. The general boundary-value problem. The contact problem 129
40. Statement of the problem. Uniqueness 129
41. Extension of the minimum principles 130
VI. Homogeneous and isotropic bodies 131
42. Properties of elastic displacement fields 131
43. The mean value theorem 133
44. Complete solutions of the displacement equation of equilibrium .... 138
VII. The plane problem 150
45- The associated plane strain and generalized plane stress solutions . . .150
46. Plane elastic states 154
47- Airy's solution 1 56
VIII. Exterior domains 165
48. Representation of elastic displacement fields in a neighborhood of
infinity 165
49. Behavior of elastic states at infinity 167
50. Extension of the basic theorems in elastostatics to exterior domains . . 169
IX. Basic singular solutions. Concentrated loads. Green's functions 173
51. Basic singular solutions 173
52. Concentrated loads. The reciprocal theorem 179
53. Integral representation of solutions to concentrated-load problems . . 185
X. Saint-Venant's principle 190
54. The v. Mises-Sternberg version of Saint-Venant's principle 190
55. Toupin's version of Saint-Venant's principle 196
56. Knowles' version of Saint-Venant's principle 200
56a. The Zanaboni-Robinson version of Saint-Venant's principle 206
XI. Miscellaneous results 207
57- Some further results for homogeneous and isotropic bodies 207
58. Incompressible materials 210
E. Elastodynamics 212
I. The fundamental field equations. Elastic processes. Power and energy.
Reciprocity 212
59. The fundamental system of field equations 212
60. Elastic processes. Power and energy 21 5
61. Graffi's reciprocal theorem 218
II. Boundary-initial-value problems. Uniqueness 219
62. The boundary-initial-value problem of elastodynamics 219
63. Uniqueness 222
III. Variational principles 223
64. Some further extensions of the fundamental lemma 223
65. Variational principles 225
66. Minimum principles 230
IV. Homogeneous and isotropic bodies 232
67. Complete solutions of the field equations 232
68. Basic singular solutions 239
69. Love's integral identity 242

Contents. VII
V. Wave propagation 243
70. The acoustic tensor 243
71. Progressive waves 245
72. Propagating surfaces. Surfaces of discontinuity 248
73. Shock waves. Acceleration waves. Mild discontinuities 253
74. Domain of influence. Uniqueness for infinite regions 257
VI. The free vibration problem 261
75. Basic equations 261
76. Characteristic solutions. Minimum principles 262
77. The minimax principle and its consequences 268
78. Completeness of the characteristic solutions 270
References 273
Linear Thermoelasticity. By Professor Donald E. Carlson, Department of Theoretical
and Applied Mechanics, University of Illinois, Urbana, Illinois (USA) 297
A. Introduction 297
1. The nature of this article 297
2. Notation 297
B. The foundations of the linear theory of thermoelasticity 299
3. The basic laws of mechanics and thermodynamics 299
4. Elastic materials. Consequences of the second law 301
5. The principle of material frame-indifference 305
6. Consequences of the heat conduction inequality 307
7. Derivation of the linear theory 307
8. Isotropy 311
C. Equilibrium theory 312
9. Basic equations. Thermoelastic states 312
10. Mean strain and mean stress. Volume change 314
11. The body force analogy 316
12. Sptecial results for homogeneous and isotropic bodies 317
13. The theorem of work and energy. The reciprocal theorem 319
14. The boundary-value problems of the equilibrium theory. Uniqueness .... 320
15. Temperature fields that induce displacement free and stress free states . . . 322
16. Minimum principles 323
17. The uncoupled-quasi-static theory 325
D. Dynamic theory 326
18. Basic equations. Thermoelastic processes 326
19. Special results for homogeneous and isotropic bodies 327
20. Complete solutions of the field equations 329
21. The theorem of power and energy. The reciprocal theorem .: 331
22. The boundary-initial-value problems of the dynamic theory 335
23. Uniqueness 337
24. Variational principles 338
25. Progressive waves 342
List of works cited 343
Existence Theorems in Elasticity. By Professor Gaetano Fichbra, University of Rome,
Rome (Italy) 347
1. Prerequisites and notations • 348
0
2. The function spaces Hm and Hm 349
3. Elliptic linear systems. Interior regularity 355
4. Results preparatory to the regularization at the boundary 357
5. Strongly elliptic systems 365
6. General existence theorems 368
7. Propagation problems 371
8. Diffusion problems 373
9. Integro-differential equations 373
10. Classical boundary value problems for a scalar 2nd order elliptic operator .... 374
11. Equilibrium of a thin plate 377

VIII
Contents.
12. Boundary value problems of equilibrium in linear elasticity 380
13. Equilibrium problems for heterogeneous media 386
Bibliography 388
Boundary Value Problems of Elasticity with Unilateral Constraints. By Professor
Gaetano Fichera, University of Rome, Rome (Italy) 391
1. Abstract unilateral problems: the symmetric case 391
2. Abstract unilateral problems: the nonsymmetric case 395
3. Unilateral problems for elliptic operators 399
4. General definition for the convex set V 401
5. Unilateral problems for an elastic body 402
6. Other examples of unilateral problems 404
7. Existence theorem for the generalized Signorini problem 407
8. Regularization theorem: interior regularity 408
9. Regularization theorem: regularity near the boundary 411
10. Analysis of the Signorini problem 413
11. Historical and bibliographical remarks concerning Existence Theorems in
Elasticity 418
Bibliography 423
The Theory of Shells and Plates. By P. M. Naghdi, Professor of Engineering Science,
University of California, Berkeley, California (USA). (With 2 Figures) 425
A. Introduction 425
1. Preliminary remarks 425
2. Scope and contents 429
3. Notation and a list of symbols used 431
B. Kinematics of shells and plates 438
4. Coordinate systems. Definitions. Preliminary remarks 438
5. Kinematics of shells: I. Direct approach 449
a) General kinematical results 449
/S) Superposed rigid body motions 452
y) Additional kinematics 455
6. Kinematics of shells continued (linear theory): I. Direct approach 456
8) Linearized kinematics 456
e) A catalogue of linear kinematic measures 458
£) Additional linear kinematic formulae 461
rj) Compatibility equations 463
7. Kinematics of shells: II. Developments from the three-dimensional theory . . 466
a) General kinematical results 466
(}) Some results valid in a reference configuration 471
y) Linearized kinematics 473
<5) Approximate linearized kinematic measures 476
e) Other kinematic approximations in the linear theory 477
C. Basic principles for shells and plates 479
8. Basic principles for shells: I. Direct approach 479
a) Conservation laws 479
(}) Entropy production 483
y) Invariance conditions 484
8) An alternative statement of the conservation laws 487
e) Conservation laws in terms of field quantities in a reference state .... 490
9. Derivation of the basic field equations for shells: I. Direct approach .... 492
a) General field equations in vector forms 492
/S) Alternative forms of the field equations 498
y) Linearized field equations 500
8) The basic field equations in terms of a reference state 502
10. Derivation of the basic field equations of a restricted theory: I. Direct approach 503
11. Basic field equations for shells: II. Derivation from the three-dimensional
theory 508
a) Some preliminary results 508
/S) Stress-resultants, stress-couples and other resultants for shells 512
y) Developments from the energy equation. Entropy inequalities 515

Contents.
IX
12. Basic field equations for shells continued: II. Derivation from the three-
dimensional theory 519
<5) General field equations 519
e) An approximate system of equations of motion 522
£) Linearized field equations 523
rj) Relationship with results in the classical linear theory of thin shells and
plates 524
12 A. Appendix on the history of derivations of the equations of equilibrium for
shells 527
D. Elastic shells 528
13. Constitutive equations for elastic shells (nonlinear theory): I. Direct approach 528
a) General considerations. Thermodynamical results 529
/3) Reduction of the constitutive equations under superposed rigid body
motions 534
y) Material symmetry restrictions 537
<5) Alternative forms of the constitutive equations 540
14. The complete theory. Special results: I. Direct approach 544
a) The boundary-value problem in the general theory 544
fl) Constitutive equations in a mechanical theory 544
y) Some special results 546
8) Special theories 546
15. The complete restricted theory: I. Direct approach 549
16. Linear constitutive equations: I. Direct approach 553
a) General considerations 553
/3) Explicit results for linear constitutive equations 555
y) A restricted form of the constitutive equations for an isotropic material . 557
8) Constitutive equations of the restricted linear theory 560
17. The complete theory for thermoelastic shells: II. Derivation from the three-
dimensional theory 561
a) Constitutive equations in terms of two-dimensional variables.
Thermodynamical results 561
f}) Summary of the basic equations in a complete theory 565
18. Approximation for thin shells: II. Developments from the three-dimensional
theory 566
a) An approximation procedure 566
/3) Approximation in the linear theory 568
19. An alternative approximation procedure in the linear theory: II. Developments
from the three-dimensional theory 569
20. Explicit constitutive equations for approximate linear theories of plates and
shells: II. Developments from the three-dimensional theory 572
a) Approximate constitutive equations for plates 572
f}) The classical plate theory. Additional remarks 575
y) Approximate constitutive relations for thin shells 578
8) Classical shell theory. Additional remarks 580
21. Further remarks on the approximate linear and nonlinear theories developed
from the three-dimensional equations 585
21 A. Appendix on the history of the derivation of linear constitutive equations for
thin elastic shells 589
22. Relationship of results from the three-dimensional theory and the theory of
Cosserat surface 594
E. Linear theory of elastic plates and shells 595
23. The boundary-value problem in the linear theory 596
a) Elastic plates 596
/3) Elastic shells 597
24. Determination of the constitutive coefficients 598
a) The constitutive coefficients for plates 598
/3) The constitutive coefficients for shells 606
25. The boundary-value problem of the restricted linear theory 607
26. A uniqueness theorem. Remarks on the general theorems 610
I b Handbuch der Physik, Bd. VI a/2.

X
Contents.
F. Appendix: Geometry of a surface and related results 615
A. 1. Geometry of Euclidean space 615
A.2. Some results from the differential geometry of a surface 621
a) Definition of a surface. Preliminaries 621
/3) First and second fundamental forms 623
y) Covariant derivatives. The curvature tensor 624
6) Formulae of Weingarten and Gauss. Integrability conditions 625
e) Principal curvatures. Lines of curvature 627
A. 3. Geometry of a surface in a Euclidean space covered by normal coordinates . 628
A.4. Physical components of surface tensors in lines of curvature coordinates . . 631
References 633
The Theory of Rods. By Professor Stuart S. Antman, New York University, New York
(USA). (With 5 Figures) 641
A. Introduction 641
1. Definition and purpose of rod theories. Nature of this article 641
2. Notation 642
3. Background 643
B. Formation of rod theories 646
I. Approximation of three-dimensional equations 646
4. Nature of the approximation process 646
5. Representation of position and logarithmic temperature 647
6. Moments of the fundamental equations 649
7. Approximation of the fundamental equations 652
8. Constitutive relations 654
9. Thermo-elastic rods 656
10. Statement of the boundary value problems 658
11. Validity of the projection methods 660
12. History of the use of projection methods for the construction of rod
theories 663
13. Asymptotic methods 664
II. Director theories of rods 665
14. Definition of a Cosserat rod 665
15. Field equations 666
16. Constitutive equations 669
III. Planar problems 670
17. The governing equations 670
18. Boundary conditions 674
C. Problems for nonlinearly elastic rods 676
19. Existence 676
20. Variational formulation of the equilibrium problems 676
21. Statement of theorems 680
22. Proofs of the theorems 682
23. Straight and circular rods 690
24. Uniqueness theorems 692
25. Buckled states 694
26. Integrals of the equilibrium equations. Qualitative behavior of solutions . 696
27. Problems of design 698
28. Dynamical problems 699
References 700
Namenverzeichnis. — Author Index 705
Sachverzeichnis (Deutsch-Englisch) 711
Subject Index (English-German) 729

The Linear Theory of Elasticity.
By
Morton E. Gurtin.
With 18 Figures.
Dedicated, to Eli Sternberg.
A. Introduction.
1. Background. Nature of this treatise. Linear elasticity is one of the more
successful theories of mathematical physics. Its pragmatic success in describing
the small deformations of many materials is uncontested. The origins of the
three-dimensional theory go back to the beginning of the 19th century and the
derivation of the basic equations by Cauchy, Navier, and Poisson. The
theoretical development of the subject continued at a brisk pace until the early 20th
century with the work of Beltrami, Betti, Boussinesq, Kelvin, Kirchhoff,
Lame, Saint-Venant, Somigliana, Stokes, and others. These authors established
the basic theorems of the theory, namely compatibility, reciprocity, and
uniqueness, and deduced important general solutions of the underlying field equations.
In the 20th century the emphasis shifted to the solution of boundary-value
problems, and the theory itself remained relatively dormant until the middle of the
century when new results appeared concerning, among other things, Saint-Ve-
nant's principle, stress functions, variational principles, and uniqueness.
It is the purpose of this treatise to give an exhaustive presentation of the
linear theory of elasticity.1 Since this volume contains two articles by Fichera
concerning existence theorems, that subject will not be discussed here.
I have tried to maintain the level of rigor now customary in pure mathematics.
However, in order to ease the burden on the reader, many theorems are stated
with hypotheses more stringent than necessary.
Acknowledgement. I would like to acknowledge my debt to my friend and teacher, Eli
Sternberg, who showed me in his lectures2 that it is possible to present the linear theory in
a concise and rational form—a form palatable to both engineers and mathematicians. Portions
of this treatise are based on Sternberg's unpublished lecture notes; I have tried to indicate
when such is the case. I would like to express my deep gratitude to D. Carlson, G. Fichera,
R. Huilgol, E. Sternberg, and C. Truesdell for their valuable detailed criticisms of the
manuscript. I would also like to thank G. Benthien, W. A. Day, J. Ericksen, R. Knops,
M. Oliver, G. de La Penha, T. Ralston, L. Solomon, E. Walsh, L. Wheeler, and
W. Williams for valuable comments, and H. Ziegler for generously sending me a copy of
1 Specific applications are not taken up in this article. They will be treated in a sequel by
L. Solomon, Some Classic Problems of Elasticity, to appear in the Springer Tracts in Natural
Philosophy.
2 At Brown University in 1959-1961.
Handbuch der Physik, Bd. VI a/2. 1

2
M. E. Gurtin: The Linear Theory of Elasticity.
Sect. 2.
Prangb's 1916 Habilitation Dissertation. Most of the historical research for this treatise was
carried out at the Physical Sciences Library of Brown University; without the continued
support and hospitality of the staff of that great library this research would not have been
possible. Finally, let me express my gratitude to the U.S. National Science Foundation for their
support through a research grant to Carnegie-Mellon University.
2. Terminology and general scheme of notation. I have departed radically
from the customary notation in order to present the theory in what I believe to
be a form most easily understood by someone not prejudiced by a past acquaintance
with the subject. Direct notation, rather than cartesian or general coordinates, is
utilized throughout. I do not use what is commonly called "dyadic notation";
most of the notions used, e.g. vector, linear transformation, tensor product, can
be found in a modern text in linear algebra.
General scheme of notation.
Italic boldface minuscules a,b,u,v ...: vectors and vector fields; x, y, z, £, ...:
points of space.
Italic boldface majuscules A,B, ...: (second-order) tensors or tensor fields.
Italic lightface letters A, a, on, 0, ...: scalars or scalar fields.
C, K: fourth-order tensors.
Sans-serif boldface majuscules M, N ... (except C, K): four-tensors or fields with
such values.
Sans-serif boldface minuscules u,%,...: four-vectors or four-vector fields.
Italic lightface majuscules B, D, E, ...: regions in euclidean space.
^: a group of second order tensors.
Script majuscules <€,£?,... (except ^): surfaces in euclidean space.
Italic indices i, j, ...: tensorial indices with the range (1, 2, 3)-
Greek indices a, /3, ...: tensorial indices with the range (1,2).
Index of frequently used symbols. Only symbols used frequently are listed. It
has not been possible to adhere rigidly to these notations, so that sometimes
within a single section these same letters are used for quantities other than those
listed below.
Symbol Name Place of definition
or first occurrence
A(m)
A
B
C
Dt
E
E
E(B)
<?
<f«>
Acoustic tensor
Beltrami stress function
Body
Elasticity tensor
Set of points of application of system I of
concentrated loads
Strain tensor
Traceless part of strain tensor
Mean strain
Three-dimensional euclidean space
= <f x(—00, 00) = space-time

Sect. 2.
Terminology and general scheme of notation.
3
Symbol Name Place of definition
or first occurrence
K
K
M
0
P
Q
R
S
o
S(B)
&\t &2
U{E}
W
-r
irtA)
w
or
a
b
c
c
Cl
c2
ei
f
/
i
k
I
tn
n
P
P
Pc
p
r
s
s
4
«tfffl
4
<»'
0
ij,
Sjmimetry group for the material at x
Kinetic energy
Compliance tensor
Stress-momentum tensor
Origin, zero vector, zero tensor
Part of B
Orthogonal tensor
Plane region
Stress tensor
Traceless part of stress tensor
Mean stress
Complementary subsets of 8B
Strain energy
Total energy
Vector space associated with $
= "Tx{—00,00)
Rotation tensor
Singular surface
Amplitude of wave
Body force
Centroid of B
Speed of propagation, also the constant
• i67in{\ — v)
Irrotational velocity
Isochoric velocity
Orthonormal basis
Pseudo body force field
System of forces
]f— i, also function with values i(t) —t
Modulus of compression
System of concentrated loads
Direction of propagation
Outward unit normal vector on dB
Pressure
Position vector from the origin 0
Position vector from the centroid c
Admissible process, elastic process
=|«—o|
Surface traction
Prescribed surface traction
Admissible state, elastic state
Kelvin state corresponding to a concentrated
load I at y
Unit Kelvin state corresponding to the unit load
ei&ty
Unit doublet states at y
Center of compression at y
Center of rotation at y parallel to the #,-axis
1*

4 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 2.
Symbol Name Place of definition
or first occurrence
t Time
u Displacement vector
u Prescribed displacement on boundary
u0 Initial displacement
v0 Initial velocity
v(B) Volume of B
w Rigid displacement
x, y, z Points in space
xt Cartesian components of x
z Complex variable
0{i} Functional in Hellinger-Prange-Reissner principle
A{t} Functional in Hu-Washizu principle
-¾ (y) Open ball with radius d and center at y
0{s>} Functional in principle of minimum potential
energy
!P{<i} Functional in principle of minimum
complementary energy
/3 Young's modulus
dtj Kronecker's delta
dv(B) Volume change
e Internal energy density
ei}k Three-dimensional alternator
eap Two-dimensional alternator
A Lame modulus
fi Shear modulus
fiu Maximum elastic modulus
fim Minimum elastic modulus
v Poisson's ratio
% Point in space-time
q Density
q> Scalar field in Boussinesq-Papkovitch-Neuber
solution, Airy stress function, scalar field in
Lame solution
\j/ Vector field in Boussinesq-Papkovitch-Neuber
solution, vector field in Lam6 solution
(o Rotation vector
1 Unit tensor
sym Symmetric part of a tensor
skw Skew part of a tensor
tr Trace of a tensor
® Tensor product of two vectors
V Gradient
F(4) Gradient in space-time
V Symmetric gradient
curl Curl
div Divergence
div(4) Divergence in space-time
A Laplacian

Sect. 3.
Points. Vectors. Second-order tensors.
5
Symb<
d1 Name
□ i, D2 Wave operators
[ ] Jump in a function
da Element of area
dv Element of volume
* Convolution
# Tensor product convolution
Place of definition
or first occurrence
B. Mathematical preliminaries.
I. Tensor analysis.
3. Points. Vectors. Second-order tensors. The space under consideration is
always a three-dimensional euclidean point space <f. The term "point"
will be reserved for elements of <f, the term "vector" for elements of the
associated vector space 'f. The inner product of two vectors u and v will be
designated by u • v. A cartesian coordinate frame consists of an orthonormal
basis {ej —{e1,ei,e^ together with a point 0 called the origin. The symbol p
will always denote the position-vector field on <f defined by
p(x)=x — 0.
We assume once and for all that a single, fixed cartesian coordinate frame is
given. If u is a vector and a; is a point, then u{ and xt denote their (cartesian)
components:
«,- = M-e,., xi = (x — 0)-ei.
We will occasionally use indicial notation; thus subscripts are assumed to range
over the integers 1, 2, 3» and summation over repeated subscripts is implied:
3
u-v=uivi='Zuivi.
i=l
We denote the vector product of two vectors u and »byMX». In components
(UXV){ = 8iihUtVh,
where ei;ft is the alternator:
+ 1 if (i, j, k) is an even permutation of (1, 2, 3)
ei;-ft = • — 1 if (i, j, k) is an odd permutation of (1, 2, 3)
0 if (i, j, k) is not a permutation of (1, 2, 3) •
We will frequently use the identity
eijk eipq — °jp °kq—°iq °kp>
where (5,,- is the Kronecker delta:
\\ if i=j

6 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 3.
For convenience, we use the term "(second-order) tensor" as a synonym for
"linear transformation from T^*into "T". Thus a tensor S is a linear mapping that
assigns to each vector v a vector
u=Sv.
The (cartesian) components S,-, of S are defined by
so that u = Sv is equivalent to
«< = 5,-/w,-
We denote the identity tensor by 1 and the zero tensor by 0:
lv =v and 0» = 0 for every vector v.
Clearly, the components of 1 are (5,-,. The product ST of two tensors is defined
by composition:
(ST) (v) =S(T(v)) for every vector v;
hence
(ST)i}—Sik Tkj.
We write ST for the transpose of S; it is the unique tensor with the following
property:
Su ■ v =u ■ ST v for all vectors u and v.
We call S> symmet ric if S—ST,skew if S = —ST. Since Sit- = Sjj, it follows that
S is symmetric or skew according as 5i;- = S,,- or S,; = —S,,-. Every tensor S
admits the unique decomposition
S — sym S + skw S,
where sym S> is symmetric and skw S> is skew; in fact,
symS = i(S + ST), skwS = ^(S-S1).
We call sym S the symmetric part and skw S the skew part of S.
There is a one-to-one correspondence between vectors and skew tensors: given
any skew tensor W, there exists a unique vector at such that
Wu = o) X u for every vector u;
indeed,
<Oi = -^eijkWjk.
We call o) the axial vector corresponding to W. Conversely, given a vector at,
there exists a unique skew tensor FT such that the above relation holds; in fact,
WH = — eiikcok.
We write tr S for the trace and det S for the determinant of S. In terms of
the components S,-;- of S:
trS=Sii(
and det S is the determinant of the matrix
•Jll -^12 --^13
^21-^2 2 -^2 3 ■
.-^31 -^32 -^3 3.

Sect. 3. Points. Vectors. Second-order tensors. 7
Given any two tensors S> and T,
tr(ST)=tr(TS),
det (ST) = (det S) (det T).
A tensor Q is orthogonal provided
The set of all orthogonal tensors forms a group called the orthogonal group;
the set of all orthogonal tensors with positive determinant forms a subgroup
called the proper orthogonal group.
If {e,} is an orthonormal basis and
with Q orthogonal, then {ej} is orthonormal. Conversely, if {ej and {e^} are ortho-
normal, then there exists a unique orthogonal tensor Q such that ef = Qe{. If
such is the case, the components w'{ of a vector to with respect to {e^} obey the
following law of transformation of coordinates:
Here Qit- and wi are the components of Q and w with respect to {e^}. Likewise,
for a tensor S,
Sij—QkiQljSkl-
The tensor product a ® 6 of two vectors a and b is the tensor that assigns
to each vector u the vector a (b ■ u):
(a ® b) (w) =a(b ■ u) for every vector u.
In components
(«$&)!, = «<&,■.
and it follows that
tr(a(g)b) =a ■ b.
Further, we conclude from the above definition that the negative of the vector
product a X b is equal to twice the axial vector corresponding to the skew part
of a®b.
The inner product S ■ T of two second-order tensors is defined by
S.T=tr(STT)=SiiTii,
while
|s|=ys-s
is called the magnitude of S>.
Given any tensor S and any pair of vectors a and b,
a-Sb=S (a®6).
This identity implies that
(«i®«/) • (e*®e,) =¾ ■ [(e<®e,.)e,] =¾ • [e,(ey • e,)];
therefore, since {e,} is orthonormal,
(e<®e/)-(e*®ei)=*i**/i.

8 M. E. Gurtin : The Linear Theory of Elasticity,
and the nine tensors et®e;- are orthonormal. Moreover,
(Si}- e,-®e,-) v = e,Sif{v ■ e,-) = e< Sif v^Sv,
S=Sijei<g)ei.
Sect. 3.
and thus
Consequently, the nine tensors e^®^- span the set of all tensors, and we have
the following theorem:
(1) The set of all
tensors
symmetric tensors
skew tensors
is a vector space of dimension
form an orthonormal basis.
, and the
9| [ et®e,-
6 ■ tensors < j/2 sym e^ ® e;-
3 J [j/2 skw et ® e;-, i < j
We say that a scalar X is a principal or characteristic value of a tensor S
if there exists a unit vector n such that
Sn=An,
in which case we call n a.principal direction corresponding to A. The
characteristic space for S> corresponding to A is the subspace of ^consisting of all vectors
u that satisfy the equation
Su=hu.
We now record, without proof, the
(2) Spectral theorem.1 Let S be a symmetric tensor. Then there exists an
orthonormal basis w^n^ng and three (not necessarily distinct) principal values
2^, X2, A3 of S> such that
Sni—hini (no sum) (a)
s = 2^M<®»i-
•=i
(b)
Conversely, if S admits the representation (b) with {nj orthonormal, then (a) holds.
The relation (b) is called a spectral decomposition of S.
(i) If Xy, A2, and X% are distinct, then the characteristic spaces for S are the line
spanned by %, the line spanned by n2, and the line spanned by ns.
(ii) If A2 =^3, then (b) reduces to
S =A!Mi®«l+^2 (1—^¾¾^¾) .
(c)
Conversely, if (c) AoWs m% ^4=^; iAe« ^ a«(f A2 are the only distinct principal
values of S, and S has two distinct characteristic spaces: the line spanned by 1¾ and
the plane perpendicular to n^.
(iii) %i = A2 = A3 = A if and only if
s=n, (d)
in which case the entire vector space "fis the only characteristic space for S.
1 See, e.g., Halmos [1958, 9], § 79-

Sect. 3.
Points. Vectors. Second-order tensors.
9
We say that two tensor S and Q commute if
SQ = QS.
(3) Commutation theorem.1 A symmetric tensor S commutes with an
orthogonal tensor Q if and only if Q leaves each characteristic space of S invariant,
i.e. if and only if Q maps each characteristic space into itself.
A tensor P is a perpendicular projection if P is symmetric and
P*=P.
If P is a perpendicular projection, then
(1-P)(1-P)=1-2P+P2 = 1-P,
and therefore 1 — P is also a perpendicular projection. Since
(1 -P) P=P(1-P) =P-P2 = 0,
it follows that
Pu ■ (1 -P) v =u ■ P(l -P) v = 0;
therefore the range of P is orthogonal to the range of 1 —P.
Two simple examples of perpendicular projections are 1 and 0. Two other
examples are n ®n and 1 — n ®n, where n is a unit vector. The first associates
with any vector its component in the direction of n, the second associates with
any vector its component in the plane perpendicular to n. The next theorem
shows that every perpendicular projection is of one of the above four types.
(4) Structure of perpendicular projections. IfP is a perpendicular
projection, then P admits one of the following four representations:
1, 0, n®n, 1— n®n,
where n is a unit vector.
Proof. Let A be a principal value and n a principal direction of P. Then
Pn—hn, and hence P*n —APn, which implies Pn=XPn. Thus each principal
value of P is either zero or one. If all principal values are zero, thenP = 0; if all
are unity, then P= 1. If exactly one principal value is unity, then we conclude
from the spectral theorem that P=n®n with n a unit vector. If exactly two
are unity, then
P=m1®m1-\-mi®mi,
where m^ and m2 are the corresponding principal directions, and it is a simple
matter to verify that this representation is equivalent to
P=l—n®n,
where n =m1xm!. □
Notice that, in view of (4), the spectral decompositions (a)-(d) in (2) are
all expansions in terms of perpendicular projections.
We shall identify fourth-order tensors with linear transformations on the
space of all (second-order) tensors. Thus a. fourth-order tensor C[-] is a linear
mapping that assigns to each second-order tensor E a second-order tensor
S = C[E].
1 This theorem is a corollary to Theorem 2, § 43 and Theorem 3, § 79 of Halmos
[1958, 9].

10 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 4.
The components of C [ ■ ] are defined by
Cijki=ei'c[e*®e,] ej = (e,-®ef) ■ C[ek<g>e,],
so that the relation S = C [JS] becomes
■5j,- — Cijk i^ki-
If C^; are the components of C[-] with respect to a second orthonormal basis
{ei, 4. 63} with e\ = Qei, then
Ciikl—QmiQnjQpkQql Cmnpq-
The transpose CT of a fourth-order tensor C is the unique fourth-order
tensor with the following property:
E ■ C[F] = CT[E] ■ F for all (second-order) tensorsE and F.
In components C^-jj = CkHi. The magnitude \C\ of C is defined by
|C|=sup{|C[JS]|:|£;|=l}.
Clearly,
|c[fl]|<;|c||fl|.
We will occasionally have to deal separately with the components %, w2 in
the %, #2-plane of a vector w. For this reason, Greek subscripts will always range
over the integers 1, 2 and summation from 1 to 2 over repeated Greek subscripts
is implied, i.e.,
2
a=l
The two-dimensional alternator will be designated by ea/3; thus
4. Scalar fields. Vector fields. Tensor fields. Let D be an open set in <f. By
a scalar, vector, or tensor field on D we mean a function <p that assigns to
each point xiD, respectively, a scalar, vector, or tensor <p (x). We say that a
sco/ar field <p is differentiahle at as if there exists a vector to with the following
property:
q>(y) — ^(x)=w-(y—x)+o(\y—x\) as t/->#.
If such a vector exists, it is unique; we write w = V<p{x) and call V<p{x) the
gradient of 99 at a*. The partial derivatives of q> at x, written <pti{x), are
defined by
More generally, let T^be a finite-dimensional inner product space, and let W
be a mapping from D into #7 For example, 70^ may be the translation space "T
corresponding to <f or the space of all tensors. Then W is differentiable at xtD
if there exists a linear function £?[ • ] from "Tm\.o T^such that
V(s)-V(x)=S?\3i-x]+o(\y-x\) as y-+x.
If .S?exists, it is unique; we write £?= VxP(x) and call VW{x) the gradient of W
at x. It is not difficult to show that the linear transformation VWfa) can be
computed using the formula
VV(x)\v] = -faW(x + 0iv)\am0,
which holds for every vector vd'f.

Sect. 4.
Scalar fields. Vector fields. Tensor fields.
11
We say that W is of class C1 on D, or smooth on D, if W is differentiable at
every point of D and VW is continuous on D. The gradient of VW is denoted
by [7(2) W. Continuing in this manner, we say that W is of class CN on D if it is
class Cw_1 and its (2V— l)th gradient V^N~X)W is smooth. Clearly, the gradient of
a smooth scalar field is a vector field, while the gradient of a smooth vector field
is a tensor field. We say that W is of class CN onZJ if W is of class CN on Z) and
for each «e {0,1,..., N} F(n> !F has a continuous extension to D. (In this case we
also write V^W for the extended function.)
A field W is analytic on D if given any x m D,W can be represented by a
power series in some neighborhood of x. Of course, if W is analytic, then W is of
class C°°.
Let u be a vector field on D, and suppose that u is differentiable at a point
a?eZ). Then the divergence of w at as is the scalar
div m (x) = tr Fm (as),
and the partial derivatives of u are defined by
«<.,■(*) = -¾^^ • F«(*)e, = [7«(*)]„;
hence
div w (as) = wi)t(as).
The curl of u at as, denoted curl u (x), is defined to be twice the axial vector
corresponding to the skew part of Vu(x); i.e. curl u(x) is the unique vector with
the property:
[Vu(x) — Vu(x)T]a = (curl u(x)) x«
for every vector a. In components
[c"rl«(*)]< = eijkuk>j(x) -5 vtn 0^,.-^. JU w.
We write Vu(x) for the symmetric gradient of w: "Ac |vu<. _^
Fm(jj) =sym Ftt(a?) = i{Vu{x) + Ftt(a?)r}.
Let S be a tensor field on D, and suppose that S is differentiable at x. Then
the tensor field ST is also differentiable at x; the divergence of S at as, written
div S(x), is the unique vector with the property:
[div S (x)1 • a=div [ST (x) a]
for every fixed vector a. In the same manner, we define the curl of S at *,
written curl S(as), to be the unique tensor with the property:
[curl S(x)] a = curl [ST (x) a]
for every a. The partial derivatives of S> are given by
thus
[divS(x)]i=Siu(x),
[curl S (x)]if = eipkSfk>p (x).
Let <p be a differentiable scalar field, and suppose that Vq> is differentiable
at x. Then we define the Laplacian of <p at x by
A <p (x) — div V<p (x).

12 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 5.
We define the Lapladan of a vector field u with analogous properties in the
same manner:
Au(x) =div Vu(x).
Clearly,
A(p{x)=(pM(x),
{Au(x)\t=uiii1(x).
Finally, the Lapladan AS(x) of a sufficiently smooth tensor field S is the
unique tensor with the property:
[AS(x)]a=A[S{x)a];
in components
[A8(xy]ti = Siiihh{x).
Of future use are the following
(1) Identities. Let <pbe a scalar field, u a vector field, and S a tensor field, all
of class C2 on D. Then
curl Vq> = 0, (1)
div curl m = 0, (2)
curl curl m = F div m — Au, (3)
' curlFtt = 0, (4)
curl(VuT) = Vcmlu, (5)
Vu = -VuT =» Wu = 0, (6)
div curl S = curl div ST, (7)
div(curlS)r = 0, (8)
(curl curl S)T = curl curl ST, (9)
cur%l) = — [curl(9?l)]r, (10)
div(SrM) =u ■ div S +S ■ Vu. (11)
If S is symmetric,
tr(curlS)=0, (12)
curl curl S = - A S+2 V div S - V V (tr S) +1 [A (tr) S - div div S]. (13)
If S is symmetric and S = 6? — 1 tr 6f, then
curl cwlS = -AG + 2V div Gf-1 div div Gf. (14)
If S is skew and u is its axial vector, then
curl S = l div u — Vu. (15)
II. Elements of potential theory.
5. The body B. The subsurfaces Sf\ and 5^g of dB. Given a region D in S,
we write D for its closure, D for its interior, and 8D for its boundary. We will
consistently write Ze (y) for the open ball in S of radius q and center at y.

Sect. 5.
The body B. The subsurfaces S\ and SP% of 8B.
n
Let D be an open region in <f or in the xx, #2-plane. We say that D is simply-
connected if every closed curve in D can be continuously deformed to a point
without leaving D (i.e., if given any continuous function g: [0, i]-+D satisfying
g(0) =g{\) there exists a continuous function /: [0,1] X [0,1]->D and a point
yeDsuch that/(s, 0) =g{s) and/(s, i) =y for every se[0,1]).
The following definitions will enable us to define concisely the notion of a
regular region of space.1 A regular arc is a set -a in <f which, for some orientation
of the cartesian coordinate system, admits a representation
«={»: #2=/(¾). #3=g(*i), «^#i^6},
where a<6 and / and g are smooth on [a, b]. A regular curve is a set c in <f
consisting of a finite number of regular arcs arranged in order, and such that the
terminal point of each arc (other than the last) is the initial point of the following
arc. The arcs have no other points in common, except that the terminal point
of the last arc may be the initial point of the first arc, in which case c is a closed,
regular curve. A regular surface element is a (non-empty) set 0t<=.S which, for
some orientation of the cartesian coordinate system, admits a representation
0t={x: xs=f{x1, x2), (xlt x2)<eR},
where ^Js_acomgact connected region in the %, %-plane, 8R is a closed regular
curve, and / is smooth on R. The union of a finite number of regular surface
elements is called a regular surface provided:
(i) the intersection of any two of the elements is either empty, a single point
which is a vertex for both, or a single regular arc which is an edge for both;
(ii) the intersection of any collection of three or more elements consists at most
of vertices;
(iii) any two of the elements are the first and last of a chain such that each
has an edge in common with the next;
(iv) all elements having a vertex in common form a chain such that each has
an edge, terminating in that vertex, in common with the next; the last may, or
may not, have an edge in common with the first.
The term edge here refers to one of the (finite number of) regular arcs
comprising the boundary of a regular surface element, while a vertex is a point at
which two edges meet. If all the edges of a regular surface belong each to two
of its surface elements, the surface is a closed regular surface. Note that a
regular surface (and hence a closed regular surface) is necessarily both connected
and bounded. Thus, e.g., the boundary of the spherical shell
{aj:oc^|a5-a50|^/S} («</?)
is not a regular surface, but rather the union of two closed regular surfaces.
Let B be an open set in &. We say that B is a bounded regular region
if B is the interior of a closed and bounded region E in S whose boundary dB
is the union of a finite number of non-intersecting closed regular surfaces. Note
that the boundary of a bounded regular region may have corners and edges.
We will occasionally deal with infinite domains. We say that B is an unbounded
regular region if B is unbounded, but J3n27e(0) is a bounded regular region
1 All of the definitions contained in this paragraph are from Kellogg [1929, 2], Chap. IV,
§8.

14
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 5.
for all sufficiently large q. Note that the boundary of such a region may extend
to infinity. An unbounded regular region whose boundary is finite is called an
exterior domain.
Throughout this article, unless specified to the contrary, B will denote a
bounded regular region. We will refer to B as the body. A (not necessarily
proper) bounded regular subregion P of B will be referred to as a part of B.
We shall designate the outward unit normal to 8B at x^8B by n(x). If
the function n is continuous at *, then x will be referred to as a regular point
of 8B. We say that Sf c dB is a regular subsurface of 8B if the intersection of
£f with each of the closed regular surfaces comprising 8B is itself a regular
surface. The symbols _^ and Sf% will alwaysjdenote complementwyLregMlar
subsurfaces oidB 1 i.e. regular subsurfaces of 8B with the following property:
where 5^ is the (relative) interior of ^. Note that since Sf^ is regular, 5^=^,
where £fa is the (relative) closure of 5^.
Let B be bounded or unbounded. A partition for B is a finite collection
B1, B2, ..., BN of mutually disjoint regular subregions of B such that
N
B=\JEn.
A function W is piecewise continuous on B if there exists a partition
S1( J32, ..., BN of S such that for each B„ the restriction of W to J5„ is bounded
onF„ and continuous onI?n—D, where I) is a (possibly empty) finite subset of
Bn. W is piecewise smooth onB if VW exists almost everywhere on B and is
piecewise continuous on E.
Piecewise continuity on regular surfaces is defined in a strictly analogous
manner. We say that a function W on S?a is jpieceioise regular if ¥* is piece-
wise continuous on £f„. and each regular point of ya is a point of continuity
of W. Suppose that W and W are piecewise regular on SPa. Then we write W = W
on ,5^ if ^(as) = ¥*(») at each regular point xeS^a.
We say that B is properly regular1 provided the following hypotheses are
satisfied:
(i) B is a bounded regular region;
(ii) for each x0z8B there exists a neighborhood N of x0 and a piecewise
smooth homeomorphism h of NnE onto the solid hemisphere
{as: #3^0, *? + #! +*|^;i}
such that ft maps2Vn0J3 onto the set {x: xz—0, x\-\-x\<,\};
(iii) det Vh is positive and bounded away from zero.
We say that B is star-shaped if there exists a point x0e B such that the line
segment connecting x0 and any point xe8B intersects 8B only at x. We state
without proof the following property of star-shaped regions:
(1) Let B be star-shaped. Then there exists a point x0&B such that
(x—x0) -n(x)>0
at every regular point xe8B.
1 This definition is due to Fichera [1971, 1], § 2.

Sect. 5-
The body B. The subsurfaces 5^ and S?2 of SB.
15
Let (xv x2, x3) denote cartesian coordinates. We say that B is X(-convex
(i fixed) if every line segment parallel to the #raxis lies entirely in B whenever
its end points belong to B.
Consider the following projections of B:
and for each xs£Bs, let
Note that
B12 = {(x1,x2):xeB},
B3={x3:xeB},
B12(xs) = {{x1, x2): (xlt x2, x3)eB}.
B12= (J J312(*3).
x,€Bt
(2) Lemma. Let M 2gl and N 2gl be fixed integers, and consider the following
hypotheses on B:
(i) B is xs-convex.
(ii) B12(x3) is simply-connected for each xseBs.
(iii) There exist scalar fields x^ and x2 of class CM on Es and CN on Bs such that
the curve
® ==l(%> X2> XSI • % = -*i(^3)> X2 = X2\X3)> X3^^S)
is contained in B.
(iv) There exists a scalar field xs of class CM on B12 and CN on B12 such that
the surface
^*~\\%1> X2> X3) • XZ~XZ\X1i X2)i \x1> ^-2)^-^12/
is contained in B.
j
|
¢3
+
1
^
/ A
Bn
—^-
*\>*1
Fig. 1.
Let &{M, N) denote the set of all scalar fields of class CM on B and class CN on B.
Then the following two assertions are true:

16 M.E. Gurtin : The Linear Theory of Elasticity. Sect. 6.
(a) If (i) and (iv) hold, and if X^{M, N), then there exists a field coi^(M, N)
such that
co)3=A.
(b) If (ii) and (iii) hold, and if vae^(M, N) and satisfies
^a—»8,1=0,
then there exists a field (p^[M, N) such that
v* = q>,a.
Proof. In view of (i)-(iv), the functions
q>(x1, x2,xs) = / va{xi, Xz, x3) dx*.
**
°* \x\, #2 > #3) = J « (%, #2' %) ** % »
where /"(%, *a, #s) *s any smooth path in 5ia(*s) connecting the points
(^Wi^W) ajl^ (%' *a)> have the desired properties. D
Let S^C dB. We say that B is convex with respect to Sf if the line segment
connecting any pair of points x and x of £f intersects dB only at x and 2. Note
that this implies, in particular, that £f is convex, though B need not be.
6. The divergence theorem. Stokes' theorem.
(1) Divergence theorem. Let B be a bounded or unbounded regular region.
Let cp be a scalar field, u a vector field, and T a tensor field, and let q>, u and T be
continuous onE, differentiable almost everywhere on B, and of bounded support. Then
/ q>nda=JVq>dv,
dB
/u®nda=jVudv,
SB B
f u-nda =/div u dv,
0B B
/ n X u da =/curl u d v,
dB B
/ Tn da = f div Tdv,
dB B
whenever the integrand on the right is piecewise continuous on E.
Proof. The results concerning q> and u are classical and will not be verified
here.1 To establish the last relation, let 6 be a vector. Then
b- fTnda= fb-Tnda= f{TTb)-nda
8B dB dB
=/div {TTb) dv =/ (div T) • b dv =b -/div T dv,
B B B
which implies the desired result since 6 is arbitrary. □
1 See, e.g., Kellogg [1929,1].

Sect. 6. The divergence theorem. Stokes' theorem. i 7
(2) Stokes' theorem. Let ubea smooth vector field and T a smooth tensor field
on B. Then given any closed regular surface £f in E,
/(curltt) ■nda = 0,
f {curl T)Tn da =0.
se
Proof. The relation concerning u is well known.1 To estabhsh the second
result, let 6 be a vector, and let u = TTb. Then
6 -/(curl T)Tn da =/b • (curl T)Tn da =fn ■ (curl T)bda
Sf 9> Sf
= fn • curl (TTb) da =/ (curl u) • n da =0,
which implies the desired result, since 6 is arbitrary. □
(3) Theorem on irrotational fields. Assume that B is simply-connected.
(i) Let ube a class CN (iVSH) vector field on B that satisfies
curl u = 0.
Then there exists a scalar field <p of class CN+1 on B such that
u — V<p.
(ii) Let T be a class CN tensor field on B that satisfies
curlT = 0.
Then there exists a vector field u of class CN+1 on B such that
T = Vu.
(iii) Let T obey the hypotheses of (ii), and, in addition, assume that
trT = 0.
Then there exists a skew tensor field W of class CN+1 on B such that
T = cwlW.
Proof. Part (i) is well known.2 Indeed, the function <p defined by
X
q>{x) = fu(y)-dy
has the desired properties. Here x0 is a fixed point of B, and the integral is
taken along any curve in B connecting x0 and x.
To establish (ii) let
ti = TTei. (a)
Then
curl tf = curl [TTe>) = (curl T) e, = 0,
and by (i) there exist scalar fields ut such that
ti = Vu{. (b)
1 See, e.g., Courant [1957, 4], Chap. 5.
2 See, e.g., Phillips [1933, 3], § 45-
Handbuchdcr Physik, Bd. VIa/2. 2

18
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 6.
Let u^UfCf. Then, since ui=wei, it follows that
Vu. = (Vu)Tei> (c)
and (a)-(c) imply
T = Vu.
To prove (iii) let u be the vector field established in (ii), and let W be the
negative of the skew tensor field corresponding to u. Then
div u = 0,
since tr T = 0, and we conclude from (15) of (4.1) that
cvilW—Vu. a
(4) Theorem on solenoidal fields. Assume that 8B is of class C3. Let !F
denote the set of all fields of class C2 on B and class C3 on B.
(i)1 Let u& be a vector field, and suppose that
fu- nda=0
se
for every closed regular surface S?cB. Then there exists a vector field we& such that
u = curl tt>.
(ii) Let T& be a tensor field, and suppose that
fTTnda=0
se
for every closed regular surface S^cB. Then there exists a tensor field W& such that
T=cwlW.
Proof. We omit the proof of (i).2 To establish (ii) let
ti = Te{.
Then
/ t{- nda=ei • f TTnda=0,
and we conclude from (i) that there exist vector fields w{ such that
f< = curli«t-.
Let
W—e^Wi
and note that for any vector a,
WTa = aiwi.
Thus
Ta = «,• Te,- = curl («,• w{) = curl (WTa) = (curl W) a,
1 The usual statement of this theorem (see, e.g., Phillips [1933, 3], §49, Courant
[1957, 4], pp. 404-406), which asserts that div u = 0 implies u = curl w, is correct only when
dB consists of a single closed surface. In this instance div u = 0 is equivalent to the
assertion that
/ u-nda = o (*)
for every closed regular surface S?CB. However, if SB consists of two or more closed surfaces
(e.g. when B is a spherical shell) there exist divergence-free vector fields in &• that do not
satisfy (*). Note that by Stokes' theorem, (*) holds whenever u is the curl of a vector field.
2 See Lichtenstein [1929, 2], pp. 101-106.

Sect. 7. The fundamental lemma. Rellich's lemma. 19
and hence
T = curlW. □
(5) Properties of the Newtonian potential.1 Let q> be a scalar field that
is continuous on B and of class CN (N ^ 1) on B, and let
rv ' 4n J \x-y\ y
B
for every xeB. Then f is of class CN+1 on B and
Ay>=q>.
Further, an analogous result holds for vector and tensor fields.
(6) Properties of the logarithmic potential.2 Let R be a measurable
region in the %, x2-plane. Let q> be a scalar field that is continuous onR and of class
CN{N^i) on R, and let
for every (xlt x2)eR. Then y> is of class CN+1 on R and
Ay> = q>.
(7) Helmholtz's theorem. Let u be a vector field that is continuous on B
and of class CN (N^tfon B. Then there exists a scalar field q> and a vector field w,
both of class CN on B, such that
u = Vq> + c\xrlw,
div«j=0.
Proof. By (5) there exists a class CN+1 vector field v on B such that
u=Av.
By (3) of (4.1),
u — Fdiv v — curl curl v;
thus if we let
9? = divt>, w = — curlv,
the desired conclusion follows. □
7. The fundamental lemma. Rellich's lemma. For x0 a point of <f and h a
positive scalar, let 0h (x0) denote the set of all class C°° scalar fields <p on S with
the following properties:
(a) (p > 0 on the open ball Eh(a50) of radius h and center at x0;
(b) q> — 0 on & — Zh(x0).
For h > 0 the set <H>h (x0) is not empty. Indeed, the function q> defined by
xsZh{x0)
belongs to 0h{xo).
lo,
1 See e.g., Kellogg [1929, 1], p. 156; Courant [1962, 5], p. 249.
a See, e.g., Kellogg [1929, 1], p. 174; Courant [1962, 5], p. 250.
2*

20 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 8.
Let «9" be a subset of 8B. We say that a function / on B vanishes near
^provided there exists a neighborhood N of S? such that /=0 on NnE.
We are now in a position to establish a version of the fundamental lemma of
the calculus of variations that is sufficiently general for our use.
(1) Fundamental lemma. Let W be a finite-dimensional inner product
space. Let w: B-+W be continuous and satisfy
fw ■ vdv=0
B
for every class C°° function v: B-+W that vanishes near 8B. Then
w = 0 on E.
Proof. Let e1, e2,..., en be an orthonormal basis in 'W, and let
»
w = ^wiei.
»=i
Assume that for some x0eB and some integer k, wh(x0)=£0. Then there exists
an h>0 such that Zh(x0)cB and wh>0 on £h(x0) or wh<0 on Zh(x0). Assume,
without loss in generality, that the former holds. If we let
v = <peh,
where q>e 0h (x0), then v is of class C°° on B and vanishes near 8B, and by
properties (a) and (b) of <p,
fw-vdv — f<pw-ehdv = f <pwhdv>0.
But by hypothesis,
fw • v dv=0.
B
Thus w = 0 on B. But w is continuous on B. Thus w = 0 on B. □
We now state, without proof,
(2) Rellich's lemma.1 Let B be properly regular. Further, let {un} be a sequence
of continuous and piecewise smooth vector fields on B, and suppose that there exist
constants Mx and M2 such that
flu^dv^M^ f\Vu„\*^M2, «=1,2,....
B B
Then there exists a subsequence {mm/J such that
lim f\u^—uni\2dv=0.
j, *->oo £
8. Harmonic and biharmonic fields. Let D be an open region in S, and let
W be a scalar, vector, or tensor field on D. We say that W is harmonic on D
if W is of class C2 on D and
AW = 0.
We say that W is biharmonic on D if W is of class C4 on D and
AAW=0.
We now state, without proof, some well known theorems concerning harmonic
and biharmonic fields.
1 Reixich [1930, 4]. See also, Fichera [1971, 1], Theorem 2.IV.

Sect. 8. Harmonic and biharmonic fields. 24
(1) Every harmonic or biharmonic field is analytic.
(2) Harnack's convergence theorem.1 Let Ws be harmonic on D for each
<5(0<<5<<50), and let
W^W as S-+0
uniformly on each closed subregion of D. Then W is harmonic on D, and for each
fixed integer n,
(7(»)y/«^F(»)<F as S-+0
uniformly on every closed subregion of D.
(3) Mean value theorem. Let q> and ip be scalar, vector, or tensor fields on
D with q> harmonic andip biharmonic. Then given any ball Ee =2e(y) in D:
Vto)=4-*pf<P*i>' (2)
sP
v>(y)+e*-My) = j~*fv><i<*, (3)2
dSp
dSp
(4)2
Sp ' dSp
We now summarize some results concerning spherical harmonics.3 Let (r, 6, y)
be spherical coordinates, related to the cartesian coordinates (xlt x2, xs) through
the mapping
x1=r sin 6 cosy, x2—r sin dsiny, xs=r cos 6,
0<Lr<oo, 0<^0^7t, 0^y<2n.
Then the general scalar solid spherical harmonic of degree h admits the
representation
HW(r, 6, y) =SW(0. y) r* (k = 0, ± 1, ± 2,...),
where S(*> (6, y) is the general surface spherical harmonic of the same degree:
1*1
S« (0, y) = 2 [4"> cos n y + btf sin n y] P*<"> (Cos 0), b^ = 0.
»=o
Here PS(M) designates the associated Legendre function of the first kind, of degree k
and order n, while a$\ a^\ b^ (n — 1, 2,..., \k\), for fixed k, are 2k + i arbitrary
constants. If k and n are non-negative integers,
where Pft is the Legendre polynomial of degree k. Consequently,
^(o) ® = Pk (i). nn) (i) = o («> * ^ o).
In addition, we have the recursion relations
PinU (I) = ^(M) (I),
1 See, for example, Kellogg [1929, J], p. 249-
2 Nicolesco [1936, 3]. See also Diaz and Payne [1958, 6].
3 See, for example, Poincare [1899, 1], Kellogg [1929, 1], and Hobson [1931, 4].

22
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 8.
which are valid for unrestricted ft and n. As is evident from the above relations,
every surface harmonic of degree — 1 — ft is a surface harmonic of degree ft. Next,
we recall that any two surface harmonics of distinct degree are orthogonal over
the unit sphere r = 1:
[S« S«~>] =// S<*>(0, y) S<m> (0, y) sin 0 <*0 dy = 0 (ft + », ft + - m -1).
o o
A vector solid spherical harmonic ft(ft) of degree ft is a vector-valued
function with the property that given any vector e, the function e • ft(ft) is a
scalar solid spherical harmonic of degree ft.
The solid harmonics of degree ft are harmonic functions, which are
homogeneous of degree ft with respect to the cartesian coordinates (%, x2, 3¾); they are
homogeneous polynomials of degree ft in the xf if ft is a non-negative integer.
Further, if H(ft) is a scalar solid harmonic of degree ft, then VH(k) is a vector solid
harmonic of degree ft — 1 and
p.FH<*>=ftH<*>,
where p (x) =x — 0. Similarly,
(Fft<*>)p=ftft<*>
for a vector solid harmonic ft1*' of degree ft.
Turning from solid spherical harmonics to general harmonic functions, we
cite the following theorem, in which 27 (oo) is the following neighborhood of infinity:
27(oo) ={»: r0< \x — 0| < oo}.
Here r0> 0 is fixed.
(4) Representation theorem for harmonic fields in a neighborhood
of infinity. Let <p be a harmonic scalar field on 27(oo). Then:
(i) <p admits the representation
A——oo
where the H(ft) are uniquely determined solid spherical harmonics of degree ft, and the
infinite series is uniformly convergent in every closed subregion of 27( oo);
(ii) q> has derivatives of all orders, series representations of which may be
obtained by performing the corresponding termwise differentiations, the resulting
expansions being also uniformly convergent in every closed subregion of 27( oo) ;
(iii) if n is a fixed integer, the three statements
(a) cp{x)=0{r»-i),
(/8) <p(x)=o(r»),
(y) H<*>=0 for ft=«,
are equivalent and imply
(d) V<p(x)=0{rn-2).
An analogous result holds for harmonic vector fields on 27(oo).
We now state and prove a theorem which is the counterpart for biharmonic
functions of (4).
(5) Representation theorem for biharmonic fields in a neighborhood
of infinity.1 Let ip be a biharmonic scalar field on 27(oo). Then:
1 Picone [1936, 4]. See also Gurtin and Sternberg [1961, 11].

Sect. 8.
Harmonic and biharmonic fields.
23
(i) y admits the representation
roAwe G(ft) a«^ H(*> are solid spherical harmonics of degree k, and both infinite series
are uniformly convergent in every closed subregion of 27(00);
(ii) y> has derivatives of all orders, series representations of which may be obtained
by performing the corresponding termwise differentiations, the resulting expansions
being also uniformly convergent in every closed subregion of E{ 00);
(iii) if n is a fixed integer, the three statements
(a) v(x) =0(^-1).
(fl)W{x)=o{r»),
(y) G<*>=H<*-2>=0 for k^n,
are equivalent and imply
(<5) Vy>(x)=0(rn-2).
An analogous result holds for biharmonic vector fields on E{00).
Proof. We show first that ip admits Almansi's representation
xp = <p-\-ri 0,
where 99 and 0 are harmonic on £(00). A completeness proof for this
representation, applicable to a region that is star-shaped with respect to 0 in its interior,
is indicated by Frank and v. Mises.1 To establish the completeness of this
representation in the present circumstances, it is sufficient to exhibit a harmonic
function 0 that satisfies
A(ip — r2 0)=0,
or equivalently,
4r8^- + 60=Arp. (a)
Since, by hypothesis, Aip is harmonic, we know from (4) that it admits the
expansion
00
Ay>(r,d,y)= ZSV»(d,y)r*, (b)
a=—00
where the 5(ft)(0, y) are surface harmonics of degree k and the infinite series has
the convergence properties asserted in parts (i) and (ii) of (4). Now consider 0
defined by
0(r, 6, y) = §J5« (0, y) r\ S<*> = ^3),
Clearly, 0 is harmonic. Further, this function also satisfies (a), as is confirmed
with the aid of (b).
On applying (i) and (ii) of (4) to <p and 0, (i) and (ii) of the present theorem
follow at once. With a view toward establishing (iii), we observe that (/?) is
immediate from (a). We show next that (/?) implies (y). Let s(ft)(0, y) and S(ft) (0, y)
be the respective surface harmonics corresponding to the solid harmonics G(*'
and H(ft). We now multiply the right-hand member of the equation in (i) by
r_ns(B,)(6, y) (m = 0, ±1, ±2,...) and integrate termwise over 0^0^^,
1 [1943,1], p. 848.

24
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 9.
0^,y^2n. Bearing in mind the convergence properties asserted in (i) as well
as the orthogonality relations, we find that the integral
Imn (*) = r- f f v (r, 0, y) s<"> (0, y) sin Od 0 dy, (c)
o o
for all sufficiently large values of r, is given by
Imn (r) = rm~n [jW, s<m>] + r-»-«+i [st-"-1), si"1*]
I rm-n+2 rg(m) s(»)-| i y-«-m+l rjl-w-l^ s(m)"| _ I '
It follows from (/?) and (c) that /m„(r) tends to zero as r->-oo for fixed w and «;
consequently, the same is true of the right-hand member of (d). Noting that the
coefficients of the inner products appearing in (d) are four distinct powers of r,
we conclude that [s{m\ s(B,)] =0 if m — wS^O. Thus s(B,) = 0, which implies that
G(m)=0 for m^n. In a similar manner we verify that Z/(m)=0 for m^n—2.
Hence (/?) is a sufficient condition for (y).
We have to show further that (y) implies (a). This is readily seen to be true
by observing that each of the series in (i) represents a field that is harmonic on
27(oo), and by invoking part (iii) of (4). Finally, (y) implies (d) because of (ii) of the
present theorem and (iii) of (4). Theorem (5) has thus been proved in its
entirety. □
The following well-known proposition on divergence-free and curl-free vector
fields will be extremely useful.
(6) Let v be a smooth vector field on B that satisfies
div«=0, curl«=0.
Then v is harmonic.
Proof. Let D be an arbitrary open ball in B. By hypothesis and (i) of (6.3),
there exists a class C2 scalar field <p on D such that
v — Vtp.
Since v is divergence-free, <p is harmonic, and this, in turn, implies that v is
harmonic. □
III. Functions of position and time.
9. Class CM,N. By a time-interval T we mean an interval of the form ( —oo,/0),
(0, /0), or [0, /0), where /0>0 may be infinity. We frequently deal with functions
of position and time having as their domain of definition the cartesian product
D x T of a set D in $ and a time-interval T. In this physical context x will always
stand for a point of D and t will denote the time. If W is a function on D X T,
we write W{ •, t) for the subsidiary mapping of D obtained by holding t fixed and
W{x, •) for the mapping of T obtained by holding x fixed. We say that such a
function W is continuous (smooth) in time if W{x, •) is continuous (smooth)
on T for each xeD. If D X T is open, we write V^Wfo, t) for the n-thgradient
of ¥ with respect. toje holding t fixed and IF'"' (x, t) for the «j^jienvahye_ of
W with respect to t holding x fixed. Ordinarily, we write W, ¥ instead of !F(1), if(2).
Let M and N be non-negative integers. We say that W is of class CM'N on
B X (0, /0) if W is continuous on B X (0, /0) and the functions
pW >p(n)> ma{Q,\,...,M}, na{0,i,...,N}, m+n^max{M, N},

Sect. 10. Convolutions. 25
exist and are continuous on Bx(0,t0). We say that W is of class CM,N on
Bx[0,t0) if W is of class CM<N on Bx(0,t0) and for each me{0, i,..., M},
»€{0, 1, .... N}, V(m) W(fi) has a continuous extension to Bx[0, t0) (in this case
we also write F(m) if1"' for the extended function). Finally, we write C^for CN,N.
We say that a function W on ^, X [0, /0) is piecewise regular if if is piecewise
continuous on £?, x [0, /0) and W(-,t) is piecewise regular on Sfa for 0 =/<t0.
10. Convolutions. Let T=[0,t0) or (— oo, oo), and let (p and f be scalar fields
on D X T that are continuous in time. Then the convolution <p*rp of <p and yi
is the function on D x T defined by
t
<p*xp{x, t) = f <p{x, t — r) ip(x, t) dr.
o
_ Let if be a function on D X [0, oo). We say that W has a Laplace transform
W if there exists a real number rj0= 0 such that for every rje[rj0, oo) the integral
oo
W(x,r))=fe-i'W(x,t)dt
o
converges uniformly on D.
(1) Properties of the convolution. Let <p, ip, and a> be scalar fields on
D X [0, /0) that are continuous in time. Then
(i) <p*ip = y)*<p;
(ii) (<p*rp)*a> = <p*(rp*a>) = q>*ip*co;
(iii) (p*(yi + co) = (p*xp-\-<p*co;
(iv) for xeD, if <p(x, •) >0 on (0, t0), then
(p*ip(x,-) —0 => rp(x, «)=0;
(v) for xeD, if t0 — oo, then
<p*rp(x,-)=0 =» (p(x, .)=0 or rp(x, «)=0;
(vi) «'/ <p is smooth in time, then
(p*ip = (p*ip-{- <p(-, 0)ip;
(vii) if t0 — oo and <p,xp possess Laplace transforms, then so also does <p*ip, and
(p*ip = (pip
on Dx[%, oo) for some r)07>0.
Proof. The results (i)-(iii), (v), and (vii) are well known,1 while (vi) follows
directly upon differentiating (p*y. To establish (iv) let <p = <p{x, •), rp=y)(x, • ),
assume that y> does not vanish identically on [0, /0), and let
^=inf{^[0,g:vW4=0}.
Since ip is continuous, there exists a number t2 with tx< t2< t0 such that xp > 0 on
(^, t2) or f < 0 on (<j, t2). In either case, since <p > 0, it follows that
'.
/ 9>(*2— t)v(t) ^t4=0,
o
and we have a contradiction. □
1 Properties (i)-(iii) are established in Chap. I of Mikusinski [1959, 10]. Titchmarsh's
theorem (v) is proved in Chap. II of the same book.

26 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 10.
Let #"bea finite-dimensional inner product space, e.g. if = "T or 1^= the
set of all tensors, and let 0 and !P be T^-valued fields on DxT that are
continuous in time. Then we write (p**P for the 7F-valued field on DxT
defined by
<p* Y(x, t) = $<p(x, t-r)V (a?, t) It,
o
and 0* !P for the scalar field on DxT defined by
0* W(x, *) = / 0(x, t-r)- T{x, T) dr.
o
Thus if 9? is a scalar field, u and v vector fields, and S and T tensor fields, all
defined on DxT, then (p*u is a vector field, <p*S is a tensor field, and u*v
and S*T are scalar fields; in components
[(p*u\i = (p*ui, [(p*S]ij = (p*Sij,
u*v = ui*vi, S*T = Sij*Tij.
Finally, we write S*u for the vector field
S*u(x, t) = f S(x, t — r) u(x, t) dx;
o
then
[S*u]i = Si]-*uj.
Properties analogous to (i)-(iv) of (1) obviously hold for the convolutions defined
above.
Let I and k be continuous vector functions on (— oo, oo). Then l#k is
for the tensor function on this interval defined by
l#k(f)=fl{t-T)®k(T)dr.
o
In components
For convenience, we write la for the function on (— oo, oo) defined by
la(t) =!(*-«).
Further, we let ££ denote the set of all smooth vector functions on (— oo, oo) that
vanish on (—oo, 0).
(2) Let I, keSC and let a>0. Then
la#k=l#kx,
la#k=l#ka.
Proof. Since
J=fc = 0 on (—oo.O],
(a)
Ia = fca = Ia=fca=0 on (—°°. «3.
the above relations hold trivially on (—oo, 0]. Thus choose te(0, oo). Then
la#k(t)=f l(t — r—a.)®k(r)dr.

Sect. 11. Space-time. 27
Thus, letting t=A—a, we arrive at
la#k(t)= fai(t-l)®ka(X)dX,
a
and, in view of (a),
Ia#fc(0=/I(*-A)®fca(A)iA = I#fca(<).
o
Next,
ia#fcW=i#*«W=/i('-T)®fca(T)iT. (b)
0
Since
i(/-T) = -^I(*-T),
if we integrate (b) by parts, we arrive at
i«#fcW = -[I(*-T)®fca(T)]JIo + /l(*-T)®fca(T)iT. (C)
o
Because of (a), (c) yields the second relation in (2). □
11. Space-time. We call the four-dimensional point space
space-time; in our physical context a point % = (x, fye&W consists of a point x
of space and a time /. The translation space associated with $<4> is
rw=-rx (-00,00);
elements of 1^"<4> will be referred to as four-vectors. As is natural, we write
a ■ b=a ■ b+a/J
for the inner product of two four-vectors a = (a, a) and b = (6, /?).
A four-tensor M is a linear transformation from T^"l4> into -)Tl4'. Thus M
assigns to each four-vector (a, a) a four-vector
(&,|8)=M(a,a).
Given a four-tensor M, there exists a unique tensor M (on 1^"), unique vectors
m, mei^\ and a unique scalar X such that for every four-vector (a, a),
M(a, 0) = (Ma, m • a),
M(0, a) =(ma, Aa).
Thus
M(a, a) =(Ma+ma, m • a + Aa).
We call the array
M m\
m X (a)
the space-time partition of M.
As before, we define the transpose Mr of M to be the unique four-tensor
with the property:
Ma • b=a • MT b for all four-vectors a and b.

28
M.E. Gurtin : The Linear Theory of Elasticity.
Sect. 12.
It is not difficult to show that the space-time partition of MT is
MT m
m X\'
Let (p be a smooth scalar field on an open set D in <f4>. Then the gradient in
<f(4> of q> is the four-vector field
VW(p = {Vq>,q>).
Let u = (u, /a) be a smooth four-vector field and M a smooth four-tensor field on
D (i.e. the values of M are four-tensors). We define the divergences of u and M
by the relations:
div(4) u = div u-\-fi,
(div(4) M) • a =div(4) (Mro) for every o6^4).
For future use, we now record the following
(1) Identity. Let M be a smooth four-tensor field on an open set D in <f(4>,
and let (a) be the space-time partition of M. Then
div(4) M = (div M + in, div m + A).
Proof. Since
M3 o = (MT a +ma, m ■ a + Aa),
it follows that
div(4)(/Mro) =div(Mra) +a div m+a • m+aA
= (div M + m, div m + A) • (a, a),
which implies the desired identity. □
(2) Divergence theorem in space-time. Let D be a bounded regular
region in $(4). Let u and M be a four-vector field and a four-tensor field, respectively,
both continuous onD and smooth on D. Then
f u ■ n da = f div(4) u dv,
dD D
f Mn da — f div(4)M dv,
dD D
whenever the integrand on the right is continuous on D. Here n is the outward unit
normal to dD.
Proof. The first result is simply the classical divergence theorem in <f(4); the
second can be established using the procedure given in the proof of (6.1). □
C. Formulation of the linear theory of elasticity.
I. Kinematics.
12. Finite deformations. Infinitesimal deformations. In this section we motivate
some of the notions of the infinitesimal theory. When we present the linear
theory we will give the axioms and definitions that form its foundation; while
j these are motivated by viewing the linear theory as a first-order approximation to
. the finite theory, the linear theory itself is independent of these considerations and
stands on its own as a completely consistent mathematical theory.

Sect. 12.
Finite deformations. Infinitesimal deformations.
29
Consider a body1 identified with the region B it occupies in a fixed reference
configuration.2 A deformation* of B is a smooth homeomorphism x of £ onto
a region x (B) in <f with det Vx > 0. The point x (a?) is the place occupied by the
material point x in the deformation x, while
u(x)=x(x)—x (a)
is the displacement of a? (see Fig. 2). The tensor fields
F = Vx (b)
and Fit are called, respectively, the deformation gradient and the displacement
gradient. By (a) and (b),
Vu=F — 1. (c)
Fig. 2.
The importance of the concept of strain cannot be established by a study of
kinematics alone; its relevance becomes clear when one studies the restrictions
placed on constitutive assumptions by material-frame indifference.4 Many
different measures of strain appear in the literature; however all of the properly
invariant choices are, in a certain sense, equivalent.5 The most useful for our
purposes is the finite strain tensor D defined by
D=%(FTF-1). (d)6
Of importance in the linear theory is the infinitesimal strain tensor
E = %(Vu+VuT); (e)
by (c),
D=E + %VuTVu. (f)
The infinitesimal theory models physical situations in which the displacement
u and the displacement gradient Vu are, in some sense, small. In view of (e) and
(f), D and E can be considered functions of Vu. Writing
s = \Vu\, (g)
1 See, e.g., Truesdell and Noll [1965, 22], § IS.
2 See, e.g., Truesdell and Noll [1965, 22], § 21.
3 Truesdell and Toupin [i960, 17], §§ 13-58 give a thorough discussion of finite
deformations.
4 See, e.g., Truesdell and Noll [1965, 22], §§ 19, 29.
5 See, e.g., Truesdell and Toupin [i960,17], § 32.
6 The tensor C = FTF — 2D + 1 is the right Cauchy-Green strain tensor (Truesdell and
Noll [1965, 22], § 23).

30 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 12.
we conclude from (f) that to within an error of 0(s2) as e->-0, the finite strain tensor
D and the infinitesimal strain tensor E coincide.
A finite rigid deformation is a deformation of the form:
x{x)=y0 + Q[x-x0],
where x0 and y0 are fixed points of & and Q is an orthogonal tensor. In this instance
F=-Q and Vu = Q-l (h)
are constants and
u (x) =u0 + Vu [x — aj0], (i)
where u0=y0—x0. Moreover, since QTQ = 1, (d), (e), and (h), imply
D=0,
E=m+QT)-i- (])
Thus in a rigid deformation the finite strain tensor D vanishes, but E does not.
For this reason E is not used as a measure of strain in finite elasticity. By (f)
and (j),
E = -%VuTVu,
and we conclude from (g) that in a rigid deformation to within an error of 0(sz)
the infinitesimal strain tensor vanishes. Thus
Vu = -VuT+0{e*);
i.e., to within an error of 0(e2) the displacement gradient is skew. This motivates
our defining an infinitesimal rigid displacement to be a field of the form
u(x) =11^ + W[x —x0]
with W skew.
The volume change dV in the deformation x is given by
dV= f dv-fdv.
k(B) B
Since the Jacobian of the mapping h is det Fx=detf,
Jdv=fdetF dv;
k(B) b
thus
dV = f (det F-\)dv,
B
so that detf — 1 represents the volume change per unit volume in the deformation.
In view of (c), we can consider F a function of Vu. A simple analysis based on
(c) and (g) implies that as e ->-0
det F =det(l + Vu) =1 +tr Vu+0{e*),
or equivalently,
det J— l=divw+0(e2);
thus to within an error of 0(s2) the volume change per unit volume is equal to div u.
For this reason we call the number
dv = / div udv— Ju-nda
B dB
the infinitesimal volume change.

Sect. 13. Properties of displacement fields. Strain. 31
13. Properties of displacement fields. Strain. In this section we study properties
of (infinitesimal) displacement fields. Most of the definitions are motivated by
the results of the preceding section.
A displacement field u is a class C2 vector field over B; its value u(x)
at a point xeB is the displacement of x. The symmetric part
E = i(Vu+VuT)
of the displacement gradient, Vu, is the (infinitesimal) strain field, and the
above equation relating E to u is called the strain-displacement relation. The
(infinitesimal) rotation field W is the skew part of Vu, i.e.
W = \{Vu-VuT),
while
to — \ curl u
is the (infinitesimal) rotation vector. Thus
Vu^E + W,
and 0 is the axial vector of W; i.e. for any vector a,
Wa=mxa.
We call
divit=tr E
the dilatation. The (infinitesimal) volume change dv(P) of a part P oi B due
to a continuous displacement field u on 5 is defined by
dv(P) = /«• nda,
dp
and we say that u is isochoric if du(P) =0 for every P. By the divergence
theorem,
dv (P) = f div u dv = f tr E dv;
p p
thus u is isochoric if and only if
trE = 0
on B.
An (infinitesimal) rigid displacement field is a displacement field u of the
form
u(x)=u0 + W0[x-x0],
where x0 is a point, w0 is a vector, and Wj, is a skew tensor. In this_ instance
Vu{x) = W0 = -Wf and JB(*)=0
for every x in B. Of course, u may also be written in the form
u(x) =u0 + o0 X [a? — ae0J,
where o0 is the axial vector of W0.
(1) Characterization of rigid displacements. Let u be a displacement
field. Then the following three statements are equivalent.
(i) u is a rigid displacement field.
(ii) The strain field corresponding to u vanishes on B.
(iii) u has the projection property on B: for every pair of points x.yeB
[**(»)-u(y)]. [a-y]=0.

32 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 13.
Proof. We will show that (i) => (iii) => (ii) => (i). If u is rigid, then
u(x)-u(y)=W0[x-y]
with W0 skew; hence
(x-y) ■ [u(x) -u(y)] = (x -y) ■ W0[x -y] = 0
and (iii) holds. Next, assume that u has the projection property. Then
differentiating the relation (iii) with respect to x, we obtain
Vu{x)T [x—y]+u(x) —u(y) = 0.
Differentiating this equation with respect to y and evaluating the result at
y = x, we arrive at
-Vu(x)T-Vu(x)=0,
which implies (ii). Now assume that (ii) holds. By (6) of (4.1), Wu=0. Thus
and (ii) implies that W0 is skew. □
An immediate consequence of (1) is the following important result.
(2) Kirchhoff's theorem1. If two displacement fields u and u' correspond to
I.'- the same strain field, then
u=u' -\-w,
where w is a rigid displacement field.
The next theorem shows that (iii) in (1) implies (i), even if we drop the
assumption that u be of class C2, but, as we shall see, the proof is much more
difficult.
/*y (3)2 LetFbe a non-coplanar set of points in S, and let u be a vector field on F.
K--Then u is a rigid displacement field if and only if u has the projection property.
Proof. Clearly, if u is rigid, it has the projection property. To establish the
converse assertion we assume that
[u(x)-u(y)] -(x-1/)=0
for every pair of points x, yiF. Let ziF, and let Fs be the set of all vectors from
s to points of F:
Fe={v:v=x—s,xeF}.
Let g be the vector field on Fs defined by
g(v) —u(x) — u(z), v=x—z, xeF. (a)
Then g(0) =0, and g has the projection property:
[g{v) -g(w)} ■ {v-w) =0 (b)
for every pair of vectors v,w£Fs. Taking w =0 we conclude that
g(v) -v=0
1 [1859,11
2 Cf. Nielsen [1935, 5], Chap. 3-

Sect. 13. Properties of displacement fields. Strain. 33
for every veFm, and this fact when combined with (b) implies
g(v)-w = -g(w)-v (c)
for all v, weF^.
Next, since the points of F do not line in a plane, Fx spans the entire vector
space 'V. Let w^, w)2, ws be linearly independent vectors in Fs, and let g be the
function on ^defined by
g(v)=vig(wi), (d)
where v1 are the components of v relative to wit i.e. v=viwi. Then g is linear.
We now show that g is the restriction of g to Fs. Indeed, (c) and (d) imply that
for v, keFe
g(v) . k = vig{wt) ■ k = —u«g(k) ■wi = —v-g{k)=g{v) k,
and since this relation must hold for every ktFx,
§{v)=g(v) for all 156¾. (e)
Since g is linear, we may write
g(t>)=wt>, (f)
where Wis a tensor. Further, using (c) and (d), it is a simple matter to verify that
g(v) -w = —g(w) -v
for all v, wt'f. Thus W is skew. Finally, (a), (e), and (f) imply
u(x) =u(s) + W[x— s]
for every xeF. □
A homogeneous displacement field is a displacement field of the form
u(x)—u0 +A[x—x0],
where the point x0, the vector u^, and the tensor A are independent of x. Such
a field is determined, to within a rigid displacement, by the strain field E, which
is constant and equal to the symmetric part of A. If u0 = 0 and A is symmetric,
i.e. if
u(x) =E[x—x0],
then u is called a pure strain from x0.
(4) Let ube a homogeneous displacement field. Then u admits the decomposition
u=w-\-u,
where w is a rigid displacement field and u is a pure strain from an arbitrary point x0.
Proof. Let
u(x)=u0+A[x—y0],
E = %(A+AT), W=${A-AT).
Then
u=w+2,
where
w(x) =w0 +W[x —x0], w0=u0 +A[x0 -y0],
u{x)=E[x—x0\, □
Handbuch der Physik, Bd. VI a/2. 3

34
M.E. Gurtin: The Linear Theory of Elasticity.
Sect. 13.
Let x0 be a given point, and let
p0(x)=x-x0.
Some examples of pure strains from the point
(a) simple extension of amount e in the direction n, where \n\ —i:
u=e(n -p0) n,
E = en0n;
(b) uniform dilatation of amount e:
u=ep0
E=e\;
(c) simple shear of amount x with respect to the direction pair (m, n),
where m and n are perpendicular unit vectors:
u =»[(m • p0) n + (n • p0) m],
jE = 2xsym(m(g)n) = K[m®n+n<g)m].
•To
$/j»j» / e extension
\ l s
-*—0 -<-o • O-*- O™*
/ \
Uniterm dilatation
Simple shear
Fig. 3. Examples of pure strains from x0.

Sect. 13.
Properties of displacement fields. Strain.
35
The displacement fields corresponding to (a), (b), and (c) are shown in
Fig. 3. The matrix relative to an orthonormal basis {n, e2,e^ of E in the
simple extension given in (a) is
[E]:
and the matrix of E relative to an orthonormal basis {m, n, e3} in the simple
shear given in (c) is
e 0
0 0
0 0
0
0
0.
"0 K
K 0
0 0
0"
0
0.
[E]
Notice that the homogeneous displacement field
u = 2x(m -p0)n
has the same strain as the simple shear defined in (c) and hence differs from it by
a rigid displacement field.
For a pure strain the ratio of the volume change dv(B) of the body to its
total volume v (B) is tvE. Thus for a simple extension of amount e,
dv(B)
v{B)~
= e,
for a uniform dilatation of amount e,
8v{B)
and for any simple shear,
v(B)
8v(B)
= 3«.
v(B
We call the principal values e1, e2, ez of a strain tensor E principal strains.
By (3.2), E admits the spectral decomposition
3
E=^eini®ni>
«=1
where n, is a principal direction corresponding to et. We now use this
decomposition to show that every pure strain can be accomplished in two ways: by three
simple extensions in mutually perpendicular directions; by a uniform dilatation
followed by an isochoric pure strain.
(5) Decomposition theorem for pure strains. Let u be a pure strain
from x0. Then u admits the following two decompositions:
(i) M=tl1+M2+M3,
where 1¾.¾½. and 1¾ are simple extensions in mutually perpendicular directions
from x0;
(ii) u=ud+uc,
where ud is a uniform dilatation from x0, while uc is an isochoric pure strain from x0.
Proof. Let E be the strain tensor corresponding to u. It follows from the
spectral decomposition for E that

36 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 13.
where
Ei = eini®ni (no sum).
Thus if we let
M4=E,-p0, ¢=4, 2, 3,
where
p0{x)=x-x0,
the decomposition (i) follows. On the other hand, if we let
ud = %(trE)p0,
uc = [E-%(trE)l]p0,
then ud is a uniform dilatation, uc is an isochoric pure strain, and
ud + ue=Ep0=u. □
(6) Decomposition theorem for simple shears. Let u be a simple shear
of amount x with respect to the direction pair (m, n). Then u admits the decomposition
u=u+ -\-u~,
where m* is a simple extension of amount ±x in the direction
^-(»»±»)-
Proof. Let E be the strain tensor corresponding to u. Then
E =x (m (g) n -\-n®m),
and it follows that A is a principal value and a a principal vector of E if and only if
x(n ■ a) m-\-x(m • a) n = Xa.
It is clear from this relation that the principal values of E are 0, +k, —x, the
corresponding principal directions
mxn, -^(m+n), ^(m-n).
Thus we conclude from (i) of (5) and its proof that the decomposition given in (6)
holds. □
Trivially, every simple shear is isochoric. The next theorem asserts that every
isochoric pure strain is the sum of simple shears.
(7) Decomposition theorem for isochoric pure strains. Let u be an
isochoric pure strain from x0. Then there exists an orthonormal basis {nj.n2.n3}
such that u admits the decomposition
u=tt1+u2+u3,
where 1¾. u2, and M3 are simple shears from x0 with respect to the direction pairs
(n2,n^), (nj, tij), and (1^, n2), respectively.
Proof.1 Let E be the strain tensor corresponding ton. If E=0 the theorem
is trivial; thus assume E^O. Clearly, it suffices to show that there exists an
1 This proof was furnished by J. Lew (private communication) in 1968.

Sect. 13- Properties of displacement fields. Strain. 37
orthonormal basis {n^.n^.tig} such that the matrix of E relative to this basis
has zero entries on its diagonal:
n^ • En^—n^ • £n2=ng • JEng = 0.
To prove this it is enough to establish the existence of non-zero vectors a, B such
that
a-0=0, (a)
x Ex = B EB=0; (b)
for if (¾. n2, Jig} is an orthonormal basis with
then (b) yields
n.- -"- n - f-
% • En^^n^ • En2=0,
and the fact that tr E = 0 implies
ng • JEng = 0.
Let m^, m2, inz be principal directions for E, and let e1, —e2, —es be the
corresponding principal strains. Since trJE=0, we may assume, without loss in
generality, that
(if two of the principal strains are positive and one negative consider — E rather
than JE). We shall now seek vectors «, B which satisfy not only (a) and (b), but
also the condition
&=ai> &=a2> &=—«3> (c)
where a,- and /^ are the components of a and B relative to the orthonormal basis
m^, m2, m,. In view of (c), conditions (a) and (b) reduce to the system:
a?+a!-a| = 0
e1a.l—e2<4 — esa.l=0.
It is easily verified that this system has the solution
a| = "2 + «8>
02 = ¾ — es,
«l=el+e2.
and since E =)= 0, the vectors a and B so defined will be real and non-zero. □
We now consider displacement fields that are not necessarily homogeneous.
Given a continuous strain field JE onB, we call the symmetric tensor
B
the mean strain. Clearly,
-w-tr E(B),
thus the volume change can be computed once the mean strain is known.

38 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 13.
(8) Mean strain theorem. Let u be a displacement field, let E be the
corresponding strain field, and suppose that E and u are continuous on E. Then the mean
strain E(B) depends only on the boundary values of u and is given by
E(B) = -^= f sym u ® n da.
dB
Proof. By the divergence theorem (6.1),
f (u®n+n®u) da= f (Vu + VuT) dv = 2 f Edv. □
dB B B
The next theorem, which is due to Korn, gives restrictions under which the
Z.2 norm of the gradient of u (i.e. the Dirichlet norm of u) is bounded by a constant
times the £2 norm of the strain field.
(9) Horn's inequality.1 Let ubea class C2 displacement field on E, and assume
that one of the following two hypotheses hold:
(a) u = 0 on dB;
(/3) B is properly regular, and either u = 0 on a non-empty regular subsurface
of SB, or
JWdv=0,
B
where W is the rotation field corresponding to u.
f\Vu\*dv^Kf\E\*dv,
B B
where E is the strain field corresponding to u, and K is a constant depending only
on B.
Proof. We will establish Korn's inequality only for the case in which (a) holds.
The proof under (/3), which is quite difficult, can be found in the article by Fi-
chera. 2
Thus assume (a) holds. It follows from the definitions of E and W that
|.E|2 = f(|J7M|2 + J7M. J7Mr),
|W|2 = i(|J7tt|*-|7tt. VuT);
hence
\E\*-\W\* = Vu-VuT. (a)
Further, we have the identity
Vu- VuT = div [(Vu) u - (div u) u] + (div u)\ (b)
By (a), (b), the divergence theorem, and (a),
f\E\*dv-f\W\*dv=f{divu)*dv,
B B B
which implies
f\W\*dv^f\E\*dv. (c)
B B
1 Korn [1906, 4], [1908, i], [1909, 3]. Alternative proofs of Korn's inequality were given
by Friedrichs [1947, l), Emus [1951, 5] (see also Mikhlin [1952, 2], §§ 40-42), Gobert
[1962, 6], and Fichera [1971,1]. Numerical values of K for various types of regions can be
found in the papers by Bernstein and Toupin [i960, i], Payne and Weinberger [1961, 17~\,
and Dafermos [1968, 3].
2 [1971, I\, § 12.

Sect. 14. Compatibility. 39
Thus we conclude from (c) and the identity
||7u|2=|E|2 + |W|2
that
J\Vu\2dv^2j\E\2dv,
B B
therefore Korn's inequality holds with K = 2. □
Let u be a displacement field on B. Then u is plane if for some choice of the
coordinate frame1
«a =««(%» x2>> «8 = 0.
In this instance the domain of m can be identified with the following region R
in the xlt #2-plane:
R={(x1,x2):xeB}.
Clearly, the corresponding strain field E is a function of (¾, x2) only and satisfies
■Ea/J=&Kt,/J+%d>
•&18 ==^'28 = ^83 ==0.
It is a simple matter to verify that a plane displacement field is rigid if and only if
where x = (%, #2) and «a and co are constants. We call a two-dimensional field ua of
this form a plane rigid displacement and a complex function of the form
u(z)=a1-\-ia2—icoz, z = x1-\-ix2,
a complex rigid displacement. Clearly, a complex function u is a complex
rigid displacement if and only if its real and imaginary parts u^ and u2 are the
components of a plane rigid displacement.
14. Compatibility. Given an arbitrary strain field E, the strain displacement
relation
E=h(Vu + Vur)
constitutes a linear first-order partial differential equation for the displacement
field u. The uniqueness question appropriate to this equation was settled in the
last section; we proved that any two solutions differ at most by a rigid
displacement. The question of existence is far less trivial. We will show that a necessary
condition for the existence of a displacement field is that the strain field satisfy
a certain compatibility relation, and that this relation is also sufficient when the
body is simply-connected.
The following proposition, which is of interest in itself, supplies the first step
in the derivation of the equation of compatibility.
(1) The strain field E and the rotation vector co corresponding to a class C2
displacement field satisfy
curl E = Vm.
Proof. If we apply the curl operator to the strain-displacement relation and
use (4) and (5) of (4.1), we find that
curl E = icurl (Vu + VuT) = iFcurl u = Vca. □
1 Recall our agreement that Greek subscripts range over the integers 1, 2.

40 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 14.
If we take the curl of the relation in (1) and use (4) of (4.1), we are
immediately led to the conclusion that a necessary condition for the existence of a
displacement field is that E satisfy the following equation of compatibility:
curl curl £ = 0.
This result is summarized in the first portion of the next theorem; the second
portion asserts the sufficiency of the compatibility relation when the body is
simply-connected.
■ (2) Compatibility theorem^ The strain field E corresponding to a class C3
displacement field satisfies the equation of compatibility.
Conversely, let B be simply-connected, and let E be a class CN(N^2) symmetric
tensor field on B that satisfies the equation of compatibility. Then there exists a
displacement field u of class CN+1 on B such that E and u satisfy the
strain-displacement relation.
Proof. We have only to establish the converse assertion.2 Thus let B be
simply-connected, and assume that curl curl £=0. Let
A=curlE. (a)
Then
curl4 = 0, (b)
and since E is symmetric, (a) and (12) of (4.1) imply that
trA=0. (c)
By (b), (c), and (iii) of (6.3), there exists a class CN skew tensor field Wsuch that
.4 = -curl HP, (d)
and (a), (d) imply that
cml{E + W)=0;
thus, by (ii) of (6.3), there exists a class CN+1 vector field u on B such that
E + W=Vu,
and taking the symmetric part of both sides of this equation, we arrive at the
strain-displacement relation. □
In components the equation of compatibility takes the form
or equivalently,
2^12,12 ==^11,22 T^22,ll<
^11,23 = ( ^23,1 +^31,2+^12,3),1) e*C-
Under the hypotheses of (2) it is possible to give an explicit formula which
may be useful in computing a displacement field u corresponding to the strain
1 Some of the basic ideas underlying this theorem are due to Kirchhoff [1859,1], who
deduced three of the six equations of compatibility and indicated a procedure for determining
the displacement when the strain is known. The complete equation of compatibility was
first derived by Saint-Ven ant [1864, 1], who asserted its sufficiency. The first rigorous
proof of sufficiency was given by Beltrami [1886,1], [1889, 1]. Cf. Boussinesq [1871, 1],
Kirchhoff [1876, 1], Padova [1889, 2], E. and F. Cosserat [1896, 1], Abraham [1901, 1}.
Explicit forms of the equation of compatibility in curvilinear coordinates were given by Od-
qvist [1937, 4], Blinchikov [1938, 1], and Vlasov [1944, 3].
2 This portion of the proof is simply a coordinate-free version of Beltrami's [1886, 1]
argument.

Sect. 14. Compatibility. 41
field E. Let x0 be a fixed point in B. Then for each xeB the line integral
ac
u(x) = fU(y,x)dy,
where
Ui,-(y. x) =Eij{y) + {xk-yk) [Eif>k(y) -Ekjii{y)},
is independent of the path in B from x0 to x, and the function m so defined is a
displacement field corresponding to the strain field E.1
An alternative form of the equation of compatibility is given in the following
proposition.
(3) Let E be a class C2 symmetric tensor field on B. Then E satisfies the equation
of compatibility if and only if
, ZlJB+F|7(trE)-2FdivE=0.
Proof. The proof follows from the identities:
curl curl E = - AE - VV(tv E) + 2 Fdiv E + 1[A (tr E) - div div JB], (a)
tr[AE + VV{tvE) -2FdivE] =2[A (tr JB) -div div JB]. (b)
Indeed, let curl curl E = 0. Then, if we take the trace of (a), we conclude, with the
aid of (b), that
Zl(trE)-divdivJE = 0, (c)
and this result, in view of (a), implies the relation in (3). Conversely, if that
relation hoids, then (b) implies (c), and (c) implies curl curl JE = 0. □
The following proposition will be quite useful.
(4) Assume that B is simply-connected. Let u be a displacement field, and
assume that the corresponding strainjield E is of class C2 on B and of class CN (Af 2:1)
on B. Then u is of class CN+1 on B.
Proof. Let u' be the displacement field generated by E using the procedure
given in the proof of (2). By (13.2), u and u' differ by a rigid displacement
field; hence ;it suffices to prove that u' is of class CN+1 on B. Since curl JE is of
class CN~X on B, the function W in (d) in the proof of (2) is of class CN on B, and
since
E + W= Vu',
u' is of class CN+1 on B. □
(5) Compatibility theorem for plane displacements. Assume that R is
a simply-connected open region in the xlt x2-plane. Let ua be a class Cs field on R,
and let
Exp^viu^p+upj. (i)
Then
2-^12,12=-^11,22+-^22,11- (11)
Conversely, let Eap(=Ep0[) be a class CN(N^2) field on R that satisfies (ii). Then
there exists a class CN+1 field ua on R such that (i) holds.
1 This explicit solution is due to CesAro [1906, 2]. In this connection, see also Volterra
[1907, 4], Sokolnikoff [1956, 12], Boley and Weiner [I960, 3].

4U
M.fc.. uurtin: The Linear iheory ot iblasticity.
beet. 14.
If we take the curl of the relation in (1) and use (4) of (4.1), we are
immediately led to the conclusion that a necessary condition for the existence of a
displacement field is that E satisfy the following equation of compatibility:
curl curl 12 = 0.
This result is summarized in the first portion of the next theorem; the second
portion asserts the sufficiency of the compatibility relation when the body is
simply-connected.
' (2) Compatibility theorem.1 The strain field E corresponding to a class Cs
displacement field satisfies the equation of compatibility.
Conversely, let B be simply-connected, and let E be a class CN(N 2; 2) symmetric
tensor field on B that satisfies the equation of compatibility. Then there exists a
displacement field u of class CN+t on B such that E and u satisfy the
strain-displacement relation.
Proof. We have only to establish the converse assertion.2 Thus let B be
simply-connected, and assume that curl curl JE=0. Let
A = cax\E. (a)
Then
curL4=0, (b)
and since E is symmetric, (a) and (12) of (4.1) imply that
trJl=0. (c)
By (b), (c), and (iii) of (6.3), there exists a class CN skew tensor field W such that
A =- -curl W, (d)
and (a), (d) imply that
curl(E + W0=°;
thus, by (ii) of (6.3), there exists a class CN+1 vector field u on B such that
E + W = Vu,
and taking the symmetric part of both sides of this equation, we arrive at the
strain-displacement relation. Q
In components the equation of compatibility takes the form
or equivalently,
2^-12,12 = ^11,22 + £-22,11»
£-11,23 = ( £23,1+-^31,2+-^12,3),11 e^c-
Under the hypotheses of (2) it is possible to give an explicit formula which
may be useful in computing a displacement field u corresponding to the strain
1 Some of the basic ideas underlying this theorem are due to Kirchhofe [1859,1], who
deduced three of the six equations of compatibility and indicated a procedure for determining
the displacement when the strain is known. The complete equation of compatibility was
first derived by Saint-Venant [1864, 1], who asserted its sufficiency. The first rigorous
proof of sufficiency was.given by Beltrami [1886, T\, [1889, 1]. Cf. Boussinescj [1871, 1],
Kirchhoff [1876, 1], Padova [I889, 2], E. and F. Cosserat [1896, X], Abraham [1901, 1].
Explicit forms of the equation of compatibility in curvilinear coordinates were given by Od-
qvist [1937, 4], Blinchikov [1938, 1], and Vlasov [1944, 3].
2 This portion of the proof is simply a coordinate-free version of Beltrami's [1886, 1]
argument.

Sect. 15. Balance of momentum. Stress. 43
We assume given on B a continuous strictly positive function q called the
density; the mass of any part P of B is then
p ,-.,..( ,.f
Let (0, t0) denote a fixed interval of time. A motion of the body is a class
Ca vector field aonBx (0, t0). The vector u(x, t) is the displacement of* at time
t, while the fields u, u,
E=$(Vu+VuT),
and E are the velocity, acceleration, strain, and strain-rate. We say that a motion
is admissible if m, m, it, JS, and E are continuous on S X [0, t0). Given an
admissible motion u and a part P of B,
I(P)=/ueii»
p
is the linear momentum of P, and
h(P)=fpxuodv V«-2
p
is the angular momentum (about the origin 0) of P. Note that, for P fixed, I (P)
and fc (P) are smooth functions of time on [0, t0); in fact,
i(P)= f ugdv, h{P)= fpxiiodv.
p p
A system of forces / for the body is defined by assigning to each (x, t) 6 B x
[0, t0) a vector b(x, t) and, for each unit vector n, a vector s„(x, t) such that:
(i) sm is continuous on B x [0, t0) and of class C1,0 on B x (0, <0);
(ii) 6 is continuous on B X [0, t0).
We call s„(x, t) the stress vector at (ae, i). Let Sf be an oriented regular
surface in B with unit normal n (Fig. 4). Then sn^(x, t) is the force per unit
Fig. 4.
area at x exerted by the portion of B on the side of Sf toward which n(x) points
on the portion of B on the other side; thus
fsnda = Jsnix){x,t)dax

44
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 15.
and
fp xs„da = fp{x)xs„(a,)(x, t) dax
sr se
represent the total force and moment across £?. The same consideration also
applies when x is located on the boundary of B and n is the outward unit normal
to 8B at x; in this case sn(x, t) is called the surface traction at (x, t). The
vector 6 (x, t) is the body force at (x, t); it represents the force per unit volume
exerted on the point x by bodies exterior to B. The total force f(P) on a part
P is the total surface force exerted across dP plus the total body force exerted
on P by the external world:
f(P)=Jsnda+fbdv.
dp p
Analogously, the total moment m(P) on P (about 0) is given by
m(P) = fpxsnda+fpxbdv.
dp p
An ordered array [u, /], where u is an admissible motion and / a system of
forces, is called a dynamical process1 if it obeys the following postulate: for
every part P of B
f(P)=i(P) K)
and
m(P)=h(P). (m2)
These two relations constitute the laws of balance of linear and angular
momentum;2 (1¾) is the requirement that the total force on P be equal to the
rate of change of linear momentum, (ma) that the total moment be equal to the
rate of change of angular momentum. Note that by (1%), the relation (m2) holds
for every choice of the origin 0 provided it holds for one such choice. Clearly, (1%)
and (m2) can be written in the alternative forms:
fs„da + fbdv=fuQdv, (mi)
dP p p
fpX.snda+fpX.bdv=fpxiiodv. (m2)
dp p p
For future use, we note that (m2) is equivalent to the relation:
skw { /p®snda + f p<S)b dv\ =skw/p®«g dv. (mj)
lap p I p
The next theorem is one of the major results of continuum mechanics.
(1) Cauchy-Poisson theorem.9 Let u be an admissible motion and / a
system of forces. Then [«, /] is a dynamical process if and only if the following
two conditions are satisfied:
1 For the results of this section weaker definitions would suffice for an admissible motion,
a system of forces, and a dynamical process. The strong definitions given here allow us to
use these definitions without change in later sections.
2 Cf. the discussion given by Truesdeix and Toupin [i960, 17], § 196.
3 Cauchy [1823, 1], [1827, 1] proved that balance of linear and angular momentum
implies (i) and (ii) in (1), while Poisson [1829,2] established the converse assertion. In a sense,
the essential ideas are implied or presumed in memoirs written by Fresnel in 1822 [1868,1),
but his work rests heavily on the constitutive assumptions of linear elasticity. In a still more
limited sense the scalar counterpart of (i) is foreshadowed in a work on heat conduction
written by Fourier in 1814 [1822, 1], which is even more involved with special constitutive
relations. For a discussion of the history of this theorem, the reader is referred to Truesdeix
and Toupin [i960,17]. Gurtin, Mizel, and Williams [1968,6] established the existence of the
stress tensor under somewhat weaker hypotheses.

Sect. 15. Balance of momentum. Stress. 45
(i) there exists a class C1'0 symmetric tensor field S on Bx(0, t0), called the
stress field, such that for each unit vector n,
sn=Sn;
(ii) w, S, and b satisfy the equation of motion:
divS + 6=gii.
The proof of this theorem is based on two lemmas. The first is usually referred
to as the law of action and reaction.
(2) Cauchy's reciprocal theorem. Let [u, /] be a dynamical process. Then
given any unit vector n,
sn= s_m.
Proof. Since B is bounded, the properties of q, u, and 6 imply that
k(t) =sup \b(x, t) —q{x) U(x, t)\
xeB
is finite for 0^.t<t0. Thus we conclude from (m^) that
I fsnda\ ^kv(P) (a)
\dP I
on [0, t0), where v(P) is the volume of P.
Now choose a point x0eB, a time te(0, t0), and a unit vector m. For
convenience we shall suppress the argument t in what follows. Let Pe be a rectangular
parallelepiped contained in B with center at x0 and sides parallel to m (see Fig. 5).
Fig. 5.
Suppose further that the top and bottom faces !F* and ^s~ with exterior
normals m and — m are squares of length e, and that the height of Ps is a2. Let ^e
denote the union of the four side faces. Then
dP,=*+KJ*t-KjVt (b)
and
v(Pt)=e*. «(*?)=«», «(*?e)=4e3, (c)
where, for any surface £?, a{SP) denotes the area of SP. By (a) and (c),
— J snda->0 as e->0. (d)

46
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 15.
Further, since st is continuous on B for each fixed unit vector I, we conclude
from (c) that
72- / s±™<*«-*s±m(a>o) and *, -J s„da-+0 as e-*0. (e)
By (b), (d), and (e),
sm (3¾) + s -m (¾) = 0 >
which completes the proof, since m and x0 are arbitrary. □
(3) Lemma. Let S be a class C1,0 tensor field on Bx (0, t0) with S and div S
continuous on Bx [0, t0). Then given any part P of B,
Jpx(Sn) da= f pxdiv S dv+2[a dv
dp p p
on [0, t0), where a is the axial vector corresponding to the skew part of S.
Proof. Let
j= fpx(Sn) da.
d'P
Then given any vector e,
j-e= / (exp) ■ Snda= / n-ST(exp) da =f div{ST(exp)} dv,
dP dP P
and hence (11) of (4.1) implies
Next,
j■ e = /(exp) • divSdv + fS- V(exp) dv.
p p
(a)
(b)
(exp) • div S =e • (pxdiv S),
S- F(exp)=2e-a;
since e is arbitrary, (a) and (b) imply
j= fpx div S dv+2fadv. □
p p
Proof of (1). Assume first that (i) and (ii) of (1) hold. Then it follows from the
properties of u and sm that S and div S are continuous on B x [0, t0). Thus given
any part P of B, (ii) and the divergence theorem imply (m^). Next, since S is
Fig. 6. Tetrahedron for the case in which /¾ = ex, fc2 = — e2> ^3 — — es

Sect. 1 5. Balance of momentum. Stress. 47
symmetric, its axial vector a vanishes, and (3) together with (i) and (ii) of (1)
imply
fpXs„da = fpxdivSdv = fpx (qu — b) dv,
dP p p
which is (m^).
Conversely, assume that [u, f~\ is a dynamical process. Let m be a unit
vector and assume that m 4= ±«j for any base vector e{ of the orthonormal basis
{e1, e2, e3}. Choose a point xeB, and consider the tetrahedron P(h) in B whose
sides 71(h) and 7ti(h) have outward unit normals m and k{) where
fc< = — [sgn (et • in)] et (no sum),
and whose faces 7ii(h) intersect at x (see Fig. 6). Let the area of 71(h) be a(h),
so that the area of 7i{ (h) is —a(h)m- kt■. Since sm is continuous on B x (0, t0) for
each fixed n, we conclude that
and, by (2), that
^y/sm <*«-►«».(*)
'«"(*) j s»ida->~(m'ki) S|«(*) = — (»»• e,.) Se,(») (no sum) (a)
"((A)
as A->0, where we have suppressed the argument t. On the other hand, it follows
from the inequality (a) in the proof of (2) that
*k/s-'
,da->0 as &->0. (b)
'a-P(A)
Thus, since dP(h) is the union of 71(h) and all three 7t,(A), (a) and (b) imply
sm (x) = (m • e<) sei (x) = [sei (x) ® et] m,. (c)
Our derivation of (c) required that m 4= ± e, hold for every t. However, it follows
from (2) that the first equation of (c) is also valid when m =±e,-; thus (c) holds
for every unit vector m. Now let S be the tensor field on B X (0, t0) defined by
S = sC(®e,.
Then
s„=Sn (d)
for every unit vector n. Further, S is of class C1,0 on B X (0, t0) and continuous
on B x [0, i0), because s„ has these properties for every n.
Next, if we apply (mi) to an arbitrary part P (with PC-B) and use (d) and the
divergence theorem we conclude that
divS + 6=gM. (e)
Thus to complete the proof we have only to show that S is symmetric. By (m^),
(d), and (3),
fodv=0
p
for every part P with FcB. Thus a — 0 on B x(0, t0), which yields the symmetry
ofS. □

48 M. E. Gurtin: The Linear Theory of Elasticity. Sect. 15.
We now give an alternative proof of the existence of the stress tensor.1 Let
[u, f\ be a dynamical process, and let (x0,t)eBx [0, t0). It suffices to show that
the mapping n -> sm(ae0, t) is the restriction (to the set of unit vectors) of a linear
function on "P". For convenience, we write
s(x,n) = s„ (x,t);
then for any xeB we can extend the function s(x, •) to all of "K as follows:
a{x,v) = \v\ 8^,-^, »4=0,
s(x, 0)=0.
Let a be a scalar. If a > 0, then
s(a») = |«»|s(-|^)=«|»|s(]||)=as(»), (b)
where we have omitted the argument x. If a<0, then (b) and Cauchy's
reciprocal theorem (2) yield
s(a»)=s(|a| (—»)) =|a|s(— v) =«s(»).
Thus s(x,•) is homogeneous.
To show that s(x, • ) is additive we first note that
s (a;, w1 + w2) = s (ae, wj + s (ae, mj2)
whenever w1 and mj2 are linearly dependent. Suppose then that wt and w2 are
linearly independent. Fix e > 0 and consider jty, the plane through x0 with normal
m^; 7t2, the plane through x0 with normal w2; and 7t3, the plane through x0 + ew3
with normal ws, where
w3 = — (w1+w2). (c)
Fig. 7-
Consider the solid jtf=s/(e) bounded by these three planes and two planes
parallel to both w± and w2 and a distance <5 from x0 (see Fig. 7). Let e and d be
sufficiently small that s/ is a part of B. Then
t=i
1This proof was furnished by W. Noll (private communication) in 1967.

Sect. 15. Balance of momentum. Stress. 49
where Wt is contained in n{ (¢ = 1,2,3), and ^4 and "^5 are parallel faces.
Moreover,
HSl"3 ¢=1,2),
«3 = 0(e) as £-*0,
v(sf) = — |MJ3|a3 = 25a4 = 25 a5,
where at is the area of ^. Thus, by the continuity of s„,
and (a) implies
3
c= 2 s(x0, Wi)+o(i) as e-^0.
»=i
On the other hand, we conclude from estimate (a) in the proof of (2) that
c=0(e) as e->0.
The last two results yield
3
2 s(ae0, *»;)=();
{=1
since s(ae,;) is homogeneous, this relation and (c) imply that s(x0, •) is additive.
Thus s(x0,') is linear, and, since ae0eB was arbitrarily chosen, Noll's proof is
complete.
The field S of the Cauchy-Poisson theorem (1) has the following properties:
(I) S is a symmetric tensor field of class C1,0 on B x (0, t0);
(II) S and div S are continuous on Bx[0, i0).
The continuity of div S onEx [0, i0) follows from the equation of motion and
the continuity of q, U, and 6 on this set. A field S with properties (I) and (II) will
be called a time-dependent admissible stress field. In view of the Cauchy-
Poisson theorem, the specification of a dynamical process is equivalent to specifying
an ordered array [u, S, 6], where
(a) « is an admissible motion;
(/?) S is a time-dependent admissible stress field;
(y) 6 is a continuous vector field on B X [0, t0);
(<5) «, S, and 6 satisfy the equation of motion
div S + b=oii.
Henceforth, a dynamical process is an ordered array [u, S, b] with properties
(«M<5)-_ _
If m is independent of time, the equation of motion reduces to the equation of
equilibrium
div S +6=0.
Handbuch der Physik, Bd. VI a/2. 4

50
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 15.
Let S be a stress tensor at a point x of the body. If
Sn = sn,
then the scalar s is called a principal stress at x and the unit vector n is
called a principal direction (of stress). Since S is symmetric, there exist three
mutually perpendicular principal directions 1^,1½.¾ and three corresponding
principal stresses s1( s2, s3.
If a symmetric stress field S is constant, it automatically satisfies the equation
of equilibrium (with vanishing body forces). Examples of such stress fields are:
(a) pure tension (or compression) with tensile stress a in the direction n,
where \n\ = 1:
S=on<g>n;
(b) uniform pressure p:
8 =-pi;
(c) pure shear with shear stress r relative to the direction pair m, n, where
m and n are orthogonal unit vectors:
S = 2t sym (m® n).
Pure tension
Uniform pressure
Pure shear
Fig. 8.

Sect. 16.
Balance of momentum for finite motions.
51
The matrix of the pure tension (a) relative to an orthonormal basis {n, e2, e3} is
a 0 0"
[S]= 0 0 0 ,
.0 0 0.
and the matrix of the pure shear (c) relative to an orthonormal basis {m, n, e3} is
"0
T
0
T
0
0
0"
0
0,
[S]
Given a unit vector k, the stress vector sfc for each of the above is:
(a) sh=a(k-n) n,
(b) sk=-pk,
(c) sk = r[(m-k) n + (n-fe) m].
These three surface force fields are shown in Fig. 8.
y 16. Balance of momentum for finite motions.1 In this section we show that the
definitions and results of the previous section are, in a certain sense, consistent
wkhjhajinite theory jxovided the deformations are small. We again remind the
reader that this section is only to help motivate the infinitesimal theory; the
remainder of the article is independent of the considerations given here.
Consider a one-parameter family Kt, 0<t<t0, of deformations of B, the time t
being the parameter. The point ic(a;, t)=Kt(x) is the place occupied by x at time t,
while
u(x, t) =k{x, t)— x (a)
is the displacement of x at time t.
The linear momentum I (P) of a part P is the same in the finite theory as in
the linear theory; i.e.
l(P)=fuQdv, (b)
p
where q is the density in the reference configuration. The angular momentum,
however, is now equal to
h*(P)=f{K-0)xuedv. (c)
By (a), k—u; thus, since «xm=0,
h*(P)=f{K-0)xuQdv.
p
(d)
In the finite theory we can specify the surface forces and body forces per unit
| area and volume in the deformed configuration or per unit area and volume in
; the reference configuration. The former specification leads to the Cauchy stress
tensor; the latter, which is most convenient for our purposes, leads to the Piola-
Kirchhoff stress tensor. A system of forces / is defined exactly as in the
infinitesimal theory with the quantities sm and 6 now interpreted as follows: given
any closed regular surface £f in B [rather than k (B, t)] with unit outward
normal n at 056^, sn(x, t) is the force per unit area in the reference configuration
1 See, for example, Truesdeix and Toupin [1960,17], §§ 196, 200-210, and Truesdell
and Noll [1965, 22], § 43 A.
4*

52
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 16.
exerted at the material point as by the material outside of £f on the material
inside; 6 (as, t) is the force per unit volume in the reference configuration exerted
on the material point as by the external world. The force f(P) on a part P is the
same as in the linear theory:
f(P) = fsnda + fbdv; (e)
dp p
however, the moment m*(P) on P is now given by:
m*(P) = f(K-0)xsnda+f{tc-0)xbdv. (f)
dp p
In terms of these quantities, the laws of balance of linear and angular momentum
take the forms
f{p)=i(P),
m*(P)=h*(P),
or equivalently,
/ snda+fbdv=foudv, (g)
dp p p
/(k — 0) xs„ da+J(k —0) x6 dv = /(k — 0) xgii dv. (h)
dp p p
Now let p;— — ~~{
\e = sup lit (as, t)\. | (i)
By (a), L_J »
ic(as, 2) — 0=ti\as, t) +as—0=tt(as, <) +p(a>), (j)
and if we consider the field k as a function of the /ieW m, then (i) and (j) yield
K-0=2>+O(e) as e->0. (k)
Using (k) and (d), we can write (c), (f), and (h) as follows:
h*(P)=J[j?+0(e)]xuQdv,
p + (1)
m*(P) = f[p+0(e)] xs„ da+ftpfiO{e)] xb dv,
dp p
f[p+0(e)]xs„da+f[p+0(e)]xbdv =/[p +0 (e)] X Qii dv. (m)
dP P P
These relations reduce to the corresponding relations in the linear theory provided
we neglect the terms written 0(e).
We can summarize the results established thus far as follows: In terms of
surface forces and body forces measured per unit area and volume in the reference
configuration, the linear momentum of any part P and the force on P are the same
as in the linear theory, and (hence) so also is the law of balance of linear momentum.
The angular momentum and the moment reduce to the corresponding quantities in the
linear theory provided we neglect the terms written 0(e) in (1); therefore when these \
terms are neglected the law of balance of angular momentum reduces to the
corresponding law in the infinitesimal theory.
It is a simple matter to establish the consequences of balance of momentum
for the finite theory. In fact, the analog of (15.1) may be stated roughly as

Sect. 17. General solutions of the equations of equilibrium. 53
follows: There exists a tensor field S, called the Piola-Kirchhoff stress tensor,
such that
s„=Sn,
divS + b = qu, v '
SFT = FST, (0)
where
F = Vk
is the deformation gradient. The Eqs. (n) ajce identical to their counterparts in ,
the infinitesimal theory. However, in the linearytheory the stress field Sis sym- %^-;,
metric; in contrast, it is clear from (0) that fn the finite theory S will in general
be non-symmetric. If we let -j -1
8=\Vu\, \ (p)
then, since X— i.
F = l + Vu, (q)
(0) implies
S(l+0(d))=ST (1+0(6)).
Thus S is symmetric to within terms of 0(8) as <5-v0. This fact further
demonstrates the consistency of the linear theory within the finite theory.
The Cauchy stress tensor may be defined by the relation
T = (detF)-1SFT. (r)
It follows1 from this definition that for any part P
fSnda = f Tvda,
dP »ki (P)
where v is the outward unit normal to 8 Kt(P); here K((P) is the region of space
occupied by P in the deformation Kt, i.e.
Kt(P)={y\y = Kt(x),xtP}.
Thus Tv is the force per unit area in the deformed configuration. Clearly, (r)
and (0) imply that T is symmetric. Further, as
detF = 1+0(.5), <U~ <~
it follows from (p) and (q) that ,( j^
T=S(i+0(8)). :^
I Thus to within terms of 0(8) the Piola-Kirchhoff stress tensor and the Cauchy stress S T "*?"
tensor coincide. > ,v"
17. General solutions of the equations of equilibrium. In the absence of body
forces the equations of equilibrium take the form:
divS=0, S=ST.
If A is a field over B, and if J27 is a differential operator with values that are
symmetric tensor fields, then
S=&(A)
will be a solution of the equations of equilibrium provided
divJS?(4)=0.
1 See, for example, Truesdell and Noll [1965, 22], § 43 A.

54
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 17.
A field A with this property isc^ed^stressJuncUon. The first solution of the
equations of equiUbriumnrEerms of ajtressTunction was Airy's1 two-dimensional
solution in terms of a single scalar field. Three-dimensional generalizations of
Airy's stress function were obtained by Maxwell2 and Morera3, who established
two alternative solutions, each involving a triplet of scalar stress functions.
Beltrami observed that the solutions due to Maxwell and Morera may be
regarded as special cases of
(1) Beltrami's solution.* Let A be a class C3 symmetric tensor field on B,
and let
S=curl curl A.
Then
divS=0, S=ST.
Proof. The symmetry of S follows from (9) of (4.1) and the symmetry of A.
On the other hand, by (7) of (4.1),
div S = div curl curl A = curl div (curl A)T,
and thus we conclude from (8) of (4.1) that
divS=0. □
Consider now a fixed cartesian coordinate frame. Then Beltrami's solution
takes the form:
Further, letting [A] denote the matrix of A relative to this frame, we have for
the following special choices of A:
(a) Airy's solution.
ro 0 0'
[4]= 0 0 0 ,
0 0 <p_
^11^ V,22» ^22 = ^,111 ^12 == V,12' ^13 = 1->23 = ^33 =0J
(b) Maxwell's solution.
[41 =
Jlx —«2,33+a!
«! 0 0
0 «2 0
0 0 «0
•->12
_fl!3,i2> etc.;
1 Airy [1863, !]■
2 Maxwell [1868, 2], [1870,1].
3 Morera [1892, 4].
* Beltrami [1892, 1]. This solution ha_s_.be.eri-.erx^
berg [i960, 12], to Gwyther anifFiNzi by Truesdell [1959, 16] and Truesdell and
ToupiN [i960,17], and to Blokh by Lur'e [1955, 10]. Related contributions and additional
references can be found in the work of Morera [1892, 5], Klein andWiEGHARDT [1905,1],
Love [1906, 5], Gwyther [1912,2], [1913, 1], Finzi [1934,2], Sobrero [1935, 7]. Kuzmin
[1945,2], Weber [1948,7], Finzi and Pastori [1949,1], Krutkov [1949,5], Peretti
[1949, 8], Blokh [1950, 3], [1961, 2], Morinaga and N6no [1950, 9], Filonenko-Borodich
[1951, 7,«], [1957,6], Arzhanyh [1953,2], Schaefer [1953, IS], [1955,-72], [1959, 12],
Kroner [1954, 11], [1955, 7,8], Langhaar and Stippes [1954,12], Ornstein [1954, 17],
Gunther [1954, 7], Marguerre [1955, 11], Dorn and Schild [1956, 1], BrdiCka [1957,
3], Rieder [i960, 9], [1964, 17, 18, 19], Minagawa [1962, 11], [1965, 14], Gurtin [1963, 9],
Kawatate [1963, 16], Pastori [1963, 21], Inov and Vvedenskii [1964,10], N6no [1964,15],
Carlson [1966, 6], [1967, 4], Stippes [1966, 25], [1967, 14], Schuler and Fosdick [1967, 13].

Sect. 17. General solutions of the equations of equilibrium. 5 5
(c) Morera's solution.
[A] =
0 ms co2
co3 0 a>!
a>2 a>! 0
^11 = -2^,23. ^12 = (^1,1+^2,2-^3,3),3. etc.
If we define a tensor field G on B through
G=A-\\txA,
then
A = G — \txG,
and it follows from (14) of (4.1) that Beltrami's solution (1) can be written in
the alternative form1
S = -A G +2V div G -1 div div 6?.
By a self-equilibrated stress field we mean a smooth symmetric tensor
field S on B whose resultant force and moment vanish on each closed regular
surface if in B:
/Snda — f px(Sn) da=0.
y y
Clearly, every self-equilibrated stress field S satisfies
divS = 0.
On the other hand, if the boundary of B consists of a single closed surface, then
any closed regular surface if in B encloses only points of B, and it follows from
the divergence theorem that every divergence-free symmetric tensor field will be
self-equilibrated. The assumption that dB consist of a single closed surface is
necessary for the validity of this converse assertion. Indeed, let B be the spherical
shell
B={as:a<|as—x0\<fi}.
Then the stress field S defined on B by
S W = a\~lp- (x~xo) ® (as -as„) (a + 0)
satisfies
divS = 0, S^ST,
but its resultant force on any spherical surface
^={as:|as -as0| =y} (a<y<)3)
is not zero. In fact,
/ fin da = —
/
bnda = — a.
This result when combined with the next theorem shows that the Beltrami solution
and hence the solutions of Maxwell and Morera are, in general, not complete; i.e.
there exist stress fields that do not admit a representation as a Beltrami solution.
(2)2 Beltrami's solution (1) always yields a self-equilibrated stress field.
1 This version of Beltrami's solution is due to Schaefer [1953, 18],
2 Rieder [i960, 9]. See also Gurtin [1963, 9].

56 M.E. Gurtin : The Linear Theory of Elasticity. Sect. 17.
Proof. Let
/S = curl curl A
with 4 symmetric and of class C3 on B, and let
W=curl.4. (a)
Then by the symmetry of S— curl curl A together with Stokes' theorem (6.2),
fSnda = fSTnda=f (curl W)Tnda=0
se se se
for every closed regular surface S? in B. To see that the resultant moment also
vanishes on SP, consider the identity
px(curl W)T n = [cwl(PW)]T n + W*n — (tr W) n, (b)
which holds for any tensor field W and any vector n, where P is the skew tensor
corresponding to the position vector p. In view of the symmetry of A, (a) and
(12) of (4.1) imply that
tr W=0.
Hence (a), (b), and Stokes' theorem (6.2) yield
/px(Sn) da = fpx(STn) da= fpx(curl W)Tnda
sr sr se
= / [curl(PH0]rnda+f (curl A)Tnda=0. □
Thus sfo-ess /ieWs #&atf are not self-equilibrated cannot be represented as Beltrami
[ solutions. The next theorem shows that Beltrami's solutiojris^omBleie, provided
"7 ! we restrict our...attfiDtifin_t.o self-equilibrated-Stress fields.. In order_to state this
fSeorem concisely, let & denote the set of ail fields of class C2 on B and class C3
onB.
(3) Completeness of the Beltrami solution.1 Assume that the boundary
of B is of class C3. Let S&^ be a self-equilibrated stress field. Then there exists a
symmetric tensor field A& such that
S= curl curl A.
Proof. Since S is self-equilibrated (and hence symmetric),
fSTnda=0, (a)
fpx(STn)da=0, (b)
for every closed regular surface S? in B. By (a) and (ii) of (6.4), there exists a
tensor field We & such that
S= curl W. (c)
1 Gurtin [1963, 9]. Completeness proofs were given previously by Morinaga and N6no
[1950, 9], Ornstein [1954,17], Gunther [1954, 7], and Dorn and Schild [1956, 1].
However, since the necessary restriction that S be self-equilibrated is absent in all of these
investigations, they can at most be valid for a region whose boundary consists of a single closed
surface. Gurtin's [1963, 9] proof is based on a minor variant of Gunther's [1954, 7] argument
and is quite cumbersome. The elegant proof given here is due to Carlson [1967, 4] (see also
Carlson [1966, 6]).

Sect. 17. General solutions of the equations of equilibrium. 5 7
It follows from (b), (c), the identity (b) given in the proof of (2), and Stokes'
theorem (6.2) that
f[W-{trW)l]Tnda = 0
se
for every closed regular surface S? in B. Thus, by (ii) of (6.4), there exists a
tensor field Fe J5" such that
W - (tr 17)1= curl F;
hence (c) implies
S = curl curl V + curl [(tr W) 1]. (d)
If we let
A=sym V,
then (9) and (10) of (4.1) yield the relations
sym{curl curl V} = curl curl A,
sym{curl[(trH01]} = 0.
Thus, since Sis symmetric, if we take the symmetric part of both sides of (d), we
arrive at
S = curl curl A. □
If the boundary of B consists of a single closed surface, then every symmetric
divergence-free stress field is self-equilibrated, and we have the following direct
corollary of (3).
(4) Assume that 8B consists of a single jtass C3 closed regular surface. Let
Sz& be a symmetric tensor field that satisfies
divS=0.
Then there exists a symmetric tensor field A^ such that
S — curl curl A.
The next proposition, which is a direct consequence of the compatibility
theorem (14.2), shows that two tensor fields are Beltrami representations of the
same stress field if and only if they differ by a strain field.
(5)1 Let
S— curl curl A, (a)
where A is a class C2 tensor field on B. Further, let
h=a+e,
E = i(Vu + VuT), (b)
where u is a class C3 vector field on B. Then
S = curl curl II. (c)
Conversely, let B be sim^J^connecjeji, and let A and H be class C2 symmetric
tensor fields on B that satisfy (a) and (c). Then there exists a class C3 vector field u
on B such that (b) holds.
It is clear from (5) that to prove the completeness of Maxwell's solution for
situations in which Beltrami's solution is complete it suffices to establish the
1 Finzi [1934, 2].

58 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 17.
existence of a class C3 displacement field u that satisfies the three equations
*(«<,,- + uiii) = -Aij (»■+,-). (I)
On the other hand, the completeness of Morera's solution will follow if we can
find a class C3 solution u of the equations
uiii=Aii (no sum). (II)
Here A is a given class C2 symmetric tensor field on B. Under hypotheses (i)
and (iv) of (5.2) with M = \ and IV=3, as well as analogous hypotheses obtained
from (i) and (iv) by cyclic permutations of the integers (1, 2, 3), the system
< (II) has a solution. Therefore^ under these hypotheses a stress field S_ admits
r 11 a Morera representation whenever ii admits a Beltt■ami representation.
To the author's knowledge, there is, at present, no general existence theorem |
-½ ' appropriate to the system (I). I;
Theorems (2) and (3) assert that the Beltrami solution is complete if and only
if we restrict our attention to self-equilibrated stress fields. The next two results
show that it_is_jDossible_ to■__mpdify_^ellxarQi's representation so. as_ to_fQnn.a j
completely general, solution of the equation of equilibrium. \
(6) Beltrami-Schaefer solution.1 Let A be a class C3 symmetric tensor
field and h a harmonic field on B, and let
S = curlcurl4+2Ffc — ldivfc.
Then
divS = 0, S=ST.
Proof. The proof follows at once from (1) and the identity
div[Vh + {Vh)T-ldivh]^Ah. □
(7) Completeness of the Beltrami-Schaefer solution.2 Let S be a
symmetric tensor field that is continuous on B and class C2 on B, and suppose that
divS = 0.
Then there exists a class C3 symmetric tensor field A and a harmonic vector field h
on B such that ^
S = cUTlcurlA+2Vh — 1 divfc.
Proof. By (6.5) there exists a class C3 symmetric tensor field G on B such
that
S = -AG.
Thus
S = -A G + 2P"div G -1 div div G -2Fdiv G +1 div div G,
and letting
A^G-ltvG,
h=~6ivG,
1 Schaefer [1953, 18].
2 Gurtin [1963, 9] introduced the following solution:
S = curl curl^1 + 2/4 Vb-FPdivb, (*)
where A is symmetric and 6 biharmonic, and established its completeness. Shortly thereafter
Schaefer noticed that the completeness of (*) implies the completeness of the Beltrami-
Schaefer solution (private communication). The direct completeness proof given above
appears here in print for the first time.

Sect. 18. Consequences of the equation of equilibrium. 59
we see from (14) of (4.1) that
S=curlcurl.A + 2Ffc — 1 divfc.
Further, the fact that S is divergence-free implies that h is harmonic. □
Let S satisfy the hypotheses of (7), and let
Se = curl curl A,
Sh = 2Vh — ldivh,
where A and h are the fields established in (7). By (2) Se is self-equilibrated,
while (6) implies that Sh is divergence-free and symmetric. Further, since h is
harmonic, so is Sh. Thus we have the
(8) Decomposition theorem for stress fields.1 Let S satisfy the
hypotheses of (7). Then
S = Se -\- Sh,
where Se is a self-equilibrated stress field, and Sh is a harmonic tensor field that
satisfies
divS,=0, Sh=Sf.
Finally, it is clear from the proof of (6) that2
S = 2Vh-l divfc,
Ah^-b
furnishes a particular solution of the inhomogeneous equations of equilibrium
divS + 6=0, S = ST.
v 18. Consequences of the equation of equilibrium. In this section we study
properties of solutions to the equation of equilibrium
divS+ 6=0,
assuming throughout that a continuous body force field 6 is prescribed on B.
For convenience, we introduce the following two definitions. A vector field u
is an admissible displacement field provided
(i)3 u is of class C2 on B;
(ii) u and Vu are continuous on B.
On the other hand, an admissible stress field is a symmetric tensor field S
with the following properties:
(i) S is smooth on B;
(ii) S and div S are continuous on B.
The associated surface force field is then the vector field s defined at every
regular point x£dB by
s(x) =S(x) n(x),
where n (as) is the outward unit normal vector to dB at x.
1 Cf. Gurtin [1963, 9], Theorem 4.4.
2 Schaefer [1953, 18].
3 Actually, for the results established in this section, this requirement may be replaced
by the assumption that m be smooth on B.

60 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 18.
The following lemma will be extremely useful throughout this article.
(1) Lemma. Let S be an admissible stress field and u an admissible
displacement field. Then
f s -uda= fu- div S dv + f S ■ Vudv.
SB B B
Proof. By the divergence theorem, (11) of (4.1), and the symmetry of S,
f u ■ Snda= f (Su) ■ nda= f div(Su) dv — f u ■ div Sdv-\- f S ■ Vudv.
dB dB B B B
The symmetry of S also requires that
S-Vu=S-VuT=S-Vu,
and the proof is complete. □
If S is a solution of the equation of equilibrium, then div S = — 6, and
(1) has the following important corollary.
y (2) Theorem of work expended.1 Let S be an admissible stress field and u
an admissible displacement field, and suppose that S satisfies the equation of
equilibrium. Then
fs-uda+fb-udv = fS-Edv,
dB B B
where E is the strain field corresponding to u.
This theorem asserts that the work done2 by the surface and body forces over
the displacement field u is equal to the work done by the stress field over the
strain field corresponding to u. If u is a rigid displacement field, then Vu =0,
and the above relation reduces to "
fs-uda+fb-udv =0.
dB B
Thus the work expended over a rigid displacement field vanishes.
Our next theorem shows that, conversely, the balance laws for forces and
moments are implied by the requirement that the work done by the surface and
body forces be zero over every rigid displacement.
(3) Piola's theorem.3 Let s be an integrable vector field on dB. Then
f s ■ w da-\- / 6 -wdv=0
dB B
1Ci. Theorem (28.3).
2 We follow Love [1927, 3], § 120, in referring to, e.g.,
fs-uda (a)
dB
as the work done by the surface force s over the displacement u. In the dynamical theory
the work is given by
f f st-u(da dt, (b)
(, dB
where [t0, <x] is the interval of time during which the time-dependent surface force st acts,
and where iif is the velocity of the time-dependent displacement field ut. Formally, (b) reduces
to (a) if we assume that the spatial fields s and m are applied suddenly, i.e., if st{x)—
s(x) h(t—X) and ut(x)=u(x) h(t—X), where h is the Heaviside unit step function and
3 Piola [1833, 2], [1848, 1]. See also Truesdeix and Toupin [i960,17], § 232.

Sect. 18. Consequences of the equation of equilibrium. 61
for every rigid displacement w if and only if
f s da+ f bdv = 0,
dB B
fpxsda+fpxbdv = 0.
dB B
Proof. Let
w=u0 + W0p
with W0 skew. In view of the identity
a • W0 e = W0 • (a ® e),
which holds for all vectors a and e,
A(w) ^fs-wda+fb-wdv
dB B
= u0- j fsda + f bdv\ +W0-i f s®pda + / b%pdv\.
Ub b > Ub b >
Thus A (w) =0 for every rigid displacement w (or equivalently, for every vector
u0 and every skew tensor W0) if and only if s and 6 satisfy balance of forces and,
in addition,
skwj fp<g>sda + f p<g)bdv\ =0.
las b J
This completes the proof, since the above relation is equivalent to balance of
moments. □
The theorem of work expended was generalized by Signorini1 in the following
manner. •
(4) Signorini's theorem. Let S be an admissible stress field that satisfies the
equation of equilibrium, and let w be a smooth vector field on B. Then
fw<S)sda-\-fw<S)bdv— f (Vw) S dv.
dB B B
Proof. By the divergence theorem,
fw®sda = fw<g)(Sn) da= /w>®(div/S) dv+ / (Vw) ST dv.
dB dB B B
This completes the proof, since ST =S and div S — — b. □
Since
Pw = %{Vw + VwT),
the symmetry of S yields
tr[(Fw) S]=8 -Vw = S- Vw.
Thus if we take the trace of Signorini's relation, we arrive at the theorem of
work expended.
Given an admissible stress field S, we call the symmetric tensor
sw = vkfsdv
B
the mean stress. Here v (B) is the volume of B.
1 Signorini [1933, 4]. Actually, Signorini's theorem is slightly more general.

62 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 18.
(5) Mean stress theorem.1 The mean stress corresponding to an admissible
stress field that satisfies the equation of equilibrium depends only on the associated
surface traction and body force fields and is given by
S(B)= u(1B) [ Jp®sda + jp®bdv .
Ufl B
Proof. We simply take w in Signorini's theorem equal to p and use the
identity Vp=l. □
The next three theorems give alternative characterizations of the equations of
equilibrium and compatibility.
X (6) Let S be an admissible stress field, and suppose that
f S • Vudv — f b ■ udv
B B
i" ■
for every class '.C*'vector field uon B that vanishes near dB. Then
divS+b=0.
Proof. Let u be a class C°° vector field on B that vanishes near dB. By the
symmetry of S, S • Vu=S • Vu; thus we conclude from (1) that the integral
identity in (6) must reduce to
/(divS + b) -udv=0.
B
Since u was arbitrarily chosen, we conclude from the fundamental lemma (7.1)
that divS +6 = 0. □
(7) Let S be an admissible stress field, and suppose that
f S-Edv=0
B
for every class C°° symmetric tensor field E on B that vanishes near 8B and satisfies
the equation of compatibility. Then
"~ " ~ divS = 0.
Proof. Let u be a class C00 vector field on B that vanishes near dB, and let
E be the corresponding strain field. Then E satisfies the equation of compatibility
and vanishes near dB; hence, by hypothesis,
/S- Vudv=0,
B
and the proof follows from (6). □
^(8) Donati's theorem.2 Let E be a class C2 symmetric tensor field on B.
Further, suppose that3
f S-Edv=0
B
1 Chree [1892, 2], p. 336. See also Signorini [1932, 5].
2Donati [1890, i], [1894,2]. See also Cotterill [1865, i], Southwell [1936,5],
[1938,5], Locatelli [1940,3,4], Moriguti [1948,5], Klyushnikov [1954, 10], Washizu
[1958, 20], Blokh [1962, 2], StickforthJi964, 21], [1965, 17], and Tonti [1967, 16, 17].
3 Notice that S-E is continuous on B, since S vanishes near dB.

Sect. 18.
Consequences of the equation of equilibrium.
63
for every class C°° symmetric tensor field S on B that vanishes near dB and satisfies
divS = 0.
Then E satisfies the equation of compatibility.
Proof. Let A be a symmetric tensor field of class C°° on_B that vanishes near
dB, and let
S = curl curl A.
Then S vanishes near dB, and by (17.1), S is symmetric and divergence-free.
Thus
/ E ■ (curl curl A) dv = 0.
B
If we apply the divergence theorem twice and use the fact that A and all of its
derivatives vanish on dB, we find that
/ E • (curl curl A) dv = / (curl E)T ■ (curl A) dv = f (curl curl E)-Adv.
B B B
Thus
/ (curl curl E) ■ A dv = 0.
B
Since curl curl E is symmetric, and since the above relation must hold for every
such function A, it follows from the fundamental lemma (7.1) that E
satisfies the equation of compatibility. □
Let s be an integrable vector field whose domain is a regular subsurface S?
of dB, let b be an integrable vector field on B, and let I be a vector field whose
domain is a finite point set {&, £2, ■••. &v} of B. Further, let ln=l(^n). The
function s can be thought of as a surface traction field on Sf, b as a body force
field, and 1^ as a concentrated load applied at £M. We say that s, b, and I are in
equilibrium if N
fsda+fbdv+Zln=0,
(E)
fpxsda+ fpxbdv + Z(l;n-0)xln = 0.
Sf B n=l
A somewhat more stringent condition is that of astatic_jguilibriumi in which the
system of forces is required to_remain in equilibrium..when..xQtated.through an
arbitrary angle. More precisely, s, b and I are in astatic equilibrium if Qs,
Qb, and Ql are in equilibrium for every orthogonal tensor Q, i.e. if
JQsda + fQbdv+ZQln = 0,
(A)
JpxQsda + fpxQbdv + Z(l;n-0)xQln = 0,
Sf B n=l
for every orthogonal Q.
(9) Conditions for astatic equilibrium.1 Let s, b, and I be as defined above.
Then s, b, and I are in astatic equilibrium if and only if the following two conditions
hold:
(i) s, b, and I are in equilibrium;
(ii) fp®sda+ fj>®brff+2(?»-0)®«„ = 0.
sr B n=X
1 In classical statics, in which the loads are all concentrated, this result has long been
known (see, e.g., Routh [1908, 3], p. 313). The extension to distributed loads is due to Berg
[1969, i].

64 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 19.
Proof.1 Assume that s, b, and I are in astatic equiMbrium. Then (i) follows
from (A) with Q = l. Let T denote the left-hand side of (ii). By (E2), skw T = 0;
thus T is symmetric. Next, by (A2),
0 = skw\ f p®Qs da + f p®Qb dv + % ($n-0)®Ql};
VST B n-l J
thus, since a®Qc = (a®c)QT for all vectors a and c, skw(TQT)=0.
Consequently,
TQT = QT. (a)
Let
T=^Xini®ni (b)
»=i
be the spectral decomposition of T (in the sense of the spectral theorem (3.2)).
By (a) and (b),
2 a*ih®0»<=2 *<«?»<) ®»«. (c)
»=i »=i
and this relation must hold for every orthogonal tensor Q. Let Q be the
orthogonal tensor with the following properties:
Qn1=n2, Qn^=nz, J?n3=n1. (d)
If we operate with both sides of (c) on 1¾ and use (d) and the fact that n^ify = <5<;-,
we find that
^2 **2 = /3 Hj,
which implies that A2 = A3 = 0. Similarly, ^=0. Thus T = 0 and (ii) holds.
Assume now that (i) and (ii) hold. Let T be as before. Then T = 0 and
O^TQT=Jp®Qsda + fp®Qbdv+Z(L-0)®Qln. (e)
Se B n=l
If we take the skew part of (e), we are led, at once, to (A2). On the other hand,
(At) is an immediate consequence of (¾. □
In view of (9), the mean stress theorem (5) has the following interesting
consequence.
(10)2 The mean stress corresponding to an admissible stress field that satisfies
the equation of equilibrium vanishes if and only if the associated surface traction
and body force fields are in astatic equilibrium.
19. Consequences of the equation of motion. In this section we study properties
of solutions to the equation of motion
div S+b=QU
on a given time interval [0, t0), assuming throughout that a continuous and
strictly positive density field q is prescribed on B.
Given a dynamical process3 [tt, S, b], we call
K=% / Qti2dv
B
1 This proof differs from that given by Berg [1969, T\.
2 Signorini [1932, 5].
3 See p. 49-

Sect. 19. Consequences of the equation of motion. 65
the kinetic energy and
fS-Edv
B
the stress power; here E is the strain field associated with u.
(1) Theorem of power expended.1 Let [u^S.b] be a dynamical process.
Then
fs-iida+fb-iidv = fS- E dv+K.
dB B B
Proof. The proof follows from the theorem of work expended (18.2) with b
replaced by b — qu and u replaced by u. □
Theorem (1) asserts that the rate of working of the surface tractions and body
forces equals the stress power plus the rate of change of kinetic energy. If
m(.,0) = u(-,t),
then, by (1),
T T T .
/ f s • iida dt+ // b ■ iidv dt= / f S • E dv dt;
0 SB OB OB
i.e., when the initial and terminal velocity fields coincide, the work done by the
external forces is equal to the work done by the stress field.
As we shall see later, in a homogeneous deformation of a homogeneous body
the stress and strain fields S and E are independent of position. In this instance
b=Qti and (1) implies
S-.13=-7^- ( suda;
dB
i.e., the stress power density is equal to the rate of working of the surface
tractions per unit volume.2
The next .theorem will be of use when we discuss variational principles for
elastcdynamics; it shows that the equation of motion and the appropriate initial
conditions are together equivalent to a single integral-differential equation.
Let i be the function on [0, t0) defined by
i(t)=t.
(2)3 Let bbea continuous vector field on Bx [0, t0), let u be an admissible motion,
and let S be a time-dependent admissible stress field. Further, let u0 and v0 be vector
fields on B, and let f be defined on Bx [0, t0) by
f(x, t) =i*b(x, t) +q(x) [u0(x) +tv0(x)].
Then [u, S, b] is a dynamical process consistent with the initial conditions
tt(.,0)=tt0, tt(.,0)=w0
if and only if
i*div S+f — QU. (a)
1 Stokes [1851, 2], Umov [1874, 2]. Cf. also v. Mises [1909, 11, Truesdell and Toupin
[1960,i7], §217-
2 Cf. Truesdell and Noll [1965, 22], Eq. (18.7).
8 Gurtin [1964, 8], Theorem 3.1.
Handbuch der Physik, Bd. VI a/2. 5

66 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 19.
Proof. If the equation of motion and the initial conditions hold, then
t
i* [div S + b] (x, t) =q (x) / (t — r) it (x, r) dr
0
= q (x) u (x, t)—q (x) [tv0 (x) + u0 (as)],
which implies (a). The converse is easily established by reversing the above
argument. □
The next lemma, which is a counterpart of (18.1), will be extremely useful
in establishing variational principles for elastodynamics.
(3) Lemma. Let w be an admissible motion and S a time-dependent admissible
stress field. Then
{ s*wda— f w*divS dv-\- f S*Vw dv.
dB B B
Proof. By (18.1),
t
f s*w da= f / s(x,t — r) • w(x, r) dadr
d'B 6 dB
t ^
= // [div S(x,t — r)-w (x, r) + S(x, t — x) ■ Vw (x, r)] dv dr
0 B
= / divS*wdv-\- f S*Vwdv. □
B B
When u0 and v0 are the initial values of u and u, we call the function f defined
in (2) the pseudo body force field. Our next theorem gives an analog of the
theorem of work expended (18.2).
(4) Let [u, S, b] be a dynamical process, let f be the corresponding pseudo body
force field, and let w be an admissible motion. Then
i* / s*w da-\- / f*w dv=i* f S*Vio dv + / qu*w dv.
dB B B B
In particular, when w=u,
i*Js*uda->rJf*udv=i*jS*Edv-\- / qu*udv.
dB B B B
Proof. By (2) and (3),
i* / s*w da= f (i*div S)*w dv-\-i* / S*Vw dv
dB B B
= / {QU—f)*w dv-\-i* / S*Vw dv. □
B B
(5) Reciprocal theorem.1 Let [u, S, b] and [u, S, b] be dynamical processes,
and let f and f be the corresponding pseudo body force fields. Then
i* / s*uda-\- f f*udv —i* / S*E dv
dB B „ B (a)
= i* f s*u da -\- f f*u dv — i* f S*E dv,
dB B B
/ s*iida-\- / b*udv— / S*Edv-\- / q(u0 ■ u-\-v0 ■ ft) dv
dB B B B ftA
= f s*uda+ f b*udv — f S*Edv + f q(u0- ii + v0- u) dv.
dB B B B
1 This theorem, which appears for the first time here, is based on an idea of Graffi [1947, 2],
[1954, 6], [1963, 8] (see Sect. 61 of this article).

Sect. 20. The elasticity tensor. 67
Thus if u0=v0=u0 = v0— 0, then
/ s*uda-\- f b*udv — / S*E dv = / s*uda-\- fb*udv — f S*E dv. (c)
SB B B SB B B
Proof. The relation (a) follows from (4) and the fact that
fqu*udv = /qu*udv.
B B
To verify (b) we note first that (vi) of (10.1) implies
i*y)—y). (d)
Thus, in view of the definition of /,
f*u= b*u+ q(u0-u+v0- u). (e)
Therefore if we differentiate (a) twice with respect to t using (d), (e), and the
analog of (e) for f and u, we arrive at (b). □
Given a time-dependent admissible stress field S and an admissible motion u,
let M denote the four-tensor field with space-time partition
/S —QU
0 0 J'
we call M the associated stress-momentum field. The importance of this
field is clear from the following theorem.
(6) Balance of momentum in space-time. Let b be a continuous vector
field on Bx [0, t0). Further, let S be a time-dependent admissible stress field, let u
be an admissible motion, and let M be the associated stress-momentum field. Then
[u, S, b] is a dynamical process if and only if
f Mnda + fbdv=0
3D D
for every regular region D with D in Bx(0, t0). Here n is the outward unit normal
to 8D and
b = (b,0).
Proof. By the divergence theorem (11.2), the above relation is equivalent to
/ (divwM+b)dv=0.
D
This equation holds for all regular D if and only if
div(4)M+b=0,
or equivalently, by the identity (11.1),
divS — gii + b = 0. Q
III. The constitutive relation for linearly elastic materials.
20. The elasticity tensor. The laws of momentum balance hold in every
dynamical process, no matter of what material the body is composed. A specific
material is defined by a constitutive relation which places restrictions on the
class of processes the body may undergo. Here we will be concerned exclusively
5*

68 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 20.
with linearly elastic materials. For such materials the stress at any time and point
in a dynamical process is a linear function of the displacement gradient at the
same time and place. Moreover, while this function may depend on the point
under consideration, it is independent of the time.
Therefore we say that the body is linearly elastic if for each xeB there
exists a linear transformation Cx from the space of all tensors into the space of
all symmetric tensors such that1
S(x)=Cx[Vu(x)-\.
We call Cx the elasticity tensor for x and the function C on B with values Cx
the elasticity field. In general, Cx depends on x; if, however, Cx and the density
q (x) are independent of x, we say that B is homogeneous. We postulate that a
rigid displacement field results in zero stress at each point? Let W be an arbitrary
skew tensor. Then the field
u{x) =W[x — x0]
is a rigid displacement field with
Vu(x)=W;
thus we conclude from the above postulate that
Cx[W]=0.
Since this result must hold for every xeB and for every skew tensor W,
Cu[A]=Cm[symA\
for every tensor A. Therefore Cx is completely determined by its restriction to
the space of all symmetric tensors. In particular, our original constitutive relation
reduces to
8{x)=Cm[E{x)],
where E is the strain field corresponding to Vu. Since the space of all symmetric
tensors has dimension 6, the matrix (relative to any basis) of the restriction of
Cx to this space is 6x6.
For the moment let us write C = CX. Consider an orthonormal basis (¾. e2, eg},
and let CijH denote the components of C relative to this basis:
Qi*i = (e,®e;)'C[es®e,].
Then S=C[E] takes the form
Moreover, since the values of C are symmetric, and since the value of C on any
tensor A equals its value on the symmetric part of A,
C»jki = (sym e{ ® e;.) • C[sym eh ® e{\;
hence
We call the 36 numbers Ciikl elasticities.
1 Thus we have made the tacit assumption that the stress is zero when the displacement
gradient vanishes; or equivaleritly, that the residual stress in the body is zero. At first sight
it appears that such an assumption rules out elastic fluids, for which there is always a residual
pressure. However, if the residual stress is a constant pressure, and if we interpret S(x) as
the actual stress minus the residual stress, then all of our results are valid without change,
provided, of course, we interpret the corresponding surface tractions accordingly.
2 Poisson 1829 [1831, T\. See also Cauchy [1829, T\. This axiom is discussed in detail by
Truesdell and Noll [1965, 22], §§ 19, 19A.

Sect. 21. Material symmetry. 69
(1) Properties of the elasticity tensor. For every xeB:
(i) Cx [A] — Cx [sym A] for every tensor A;
(ii) D • Cx [A] = (sym D) ■ Cx [sym A] for every pair of tensors A and D;
(iii) Cijk, = (sym et ® e;) • Cx [sym eh ® e{\, so that
Proof. Properties (i) and (iii) have already been established. Property (ii)
follows from (i) and the fact that the values of Cx are symmetric tensors. □
If the elasticity tensor is invertible,1 then its inverse
K — C-1
is called the compliance tensor; it defines a relation
E(x)=Kx[S(x)]
between the strain E(x) and the stress S(x) at x.
The following definitions will be important in what follows. Consider a fixed
point x of the body, and let C = CX denote the elasticity tensor for x. We say
that C is symmetric^
. A _ c^ =fi _ c^
for every pair of symmetric tensors A and B, positwe^semi-definite if
A- C[A]^0
for every symmetric tensor A, and positive definite if
A-C[A]>0
for every non-zero symmetric tensor A. Clearly, C is symmetric if and only if
its components obey
so that in this instance C has 21, rather than 36, distinct elasticities. Note also
that C is invertible whenever it is positive definite.
For the remainder of the article we assume given an elasticity field C on B.
\21. Material symmetry. Assume that the material is linearly elastic in the sense
of the stress-strain relation
S = CX[E]
discussed in the previous section. In order to specify the symmetry properties of
the material, we introduce the notion of a symmetry group. Our first step will
be to motivate this concept.
Let x be a ftm? point in jg, let u and u* be smooth vector fields whose domain
is a neigE5brHood of x, and suppose that u* is related to u by an orthogonal
tensor Q:
u*(y*)=Qu(y) whenever y* = Q(y—x)+x. (a)
1 By this statement we mean: "if the restriction of the elasticity tensor to the space of all
symmetric tensors is invertible." The elasticity tensor, when regarded as a linear
transformation on the space of all tensors, cannot be invertible since its value on every skew
tensor is zero.

70
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 21.
Roughly speaking, (a) asserts that if the point y* is obtained from y by a
rotation Q about x, then the displacement u*(y*) is simply the displacement
u (y) rotated by Q. If for every such u and u* the corresponding stresses
satisfy
-.Cx [«(*)], S*=Cx[E*(x)]
S*=QSQT,
(b)
(c)
then we call Q a symmetry transformation for the material at x. Condition (c) is
slightly more transparent when phrased in terms of the surface force vectors
s„=Sn, s£*=/S*n"; indeed, (c) is equivalent to the requirement that
(see Fig. 9).
s»» = J?s„
whenever n*=Qn
By (a)
Fig. 9.
Vu*(x)=QVu(x)QT,
and hence the corresponding strain tensors are related by
E*(x)=QE(x)QT.
(d)
It follows from (b), (c), and (d) that a sufficient condition for an orthogonal
tensor Q to be a symmetry transformation is that
QCm[E]QT = Cm[QEQT]
(e)
for every symmetric tensor E. That this condition is also necessary can be easily
verified using displacement fields which are pure strains from x.
Thus we have motivated the following definition:1 the symmetry group
%. for the material at x is the set of all orthogonal tensors Q that obey (e) for
every symmetric tensor E.
1 Our definition is consistent with the notion of symmetry group (isotropy group) as
employed in general non-linear continuum mechanics (see, e.g.,TRUESDELL and Noll [1965,22],
§ 31). Since the symmetry group of a solid relative to an undistorted configuration is a
subgroup of the orthogonal group, our treatment is sufficiently general to include elastic solids
of arbitrary symmetry. While we do not rule out non-orthogonal symmetry toansformations,
WfijiCLJKlt^tudjutheiiu, The treatment of more general symmetry groups is essential to a
systematic study of fluid crystals (see, e.g., Truesdell and Noll [1965, 22], § 33 bis.).

Sect. 21.
Material symmetry.
71
To show that the above definition is consistent, i.e. that (SX really is a group,
it is sufficient to verify that
Q,P^X^QP^X.
Note first that (e) can be written in the form
CX[E]=QTCX[QEQT]Q. (f)
If 06¾. then, since
00r = l.
(f) applied to the symmetric tensor QTEQ yields
cm[QTBQi=QTc*[Q(QT'SQ)QT]Q = QTc*mQ-
Thus we conclude from (e) that QT = Q'1 belongs to %.. Next, if Q, P£@x, then
two applications of (f) yield
Cx [E] = PTCX [PEPT]P=PTQTCX [Q(PEPT) QT] QP
= (QP)TCX[(QP)E(QP)T]QP;
thus QPe@x. Therefore @x is, indeed, a group.
, (1) Let Cx be invertible, and let Kx be the corresponding compliance tensor. Then
QKX[S]QT=KX[QSQT]
for every 06¾ and every symmetric tensor S.
Proof. Choose S arbitrarily, and let E = KX[S\. Then S = Cx[E] and
QSQT = CX[QEQT],
therefore
QEQT = Km[QSQT],
or equivalently,
QKm[S]QT = Km[Q8Q^]. D
We say that the material at x is isotropic if the symmetry group <SX equals
the orthogonal group, anisotropic if ^x is a proper subgroup of the orthogonal
group. Clearly, &x always contains the two-element group { — 1,1} as a subgroup.
In fact, ^x is the direct product of this two-element group and a group ?SX which
consists only of proper orthogonal transformations; hence the syjnmetry of Jthe
material is completely characterized by_the_grqup 5¾.. In particular, the material
at x is isotropic or anisotropic according as <&x equals or is a proper subgroup of
the proper orthogonal group. Although there are an infinite number of subgroups
of the proper orthogonal group, twelve of them seem to exhaust the kinds of
symmetries occurring in theories proposed up to now as being appropriate to
describe the behavior of real anisotropic elastic materials.
The first eleven of these subgroups of the proper orthogonal group correspond
to the thirty-two crystal classes.1 We denote~Tfaese subgroups by ^, ...,¾^.
Table 1, which is due to Coleman and Noll,2 lists the generators of each of
these groups; in this table {fe, I, m} denotes a right-handed orthonormal basis,
q= j/f (fe + Z+m), and Rg is the orthogonal tensor corresponding to a right-
1 The thirty-two crystal classes are discussed by Schoenfliess [1891, 1], Voigt [1910, 1],
Flint [1948, 4], Dana and Hurlbut [1959, Z], Smith and Rivlin [1958, 16], Coleman and
Noll [1964, 5], Truesdell and Noll [1965, 22], Burckhardt [1966, 5].
2 [1964, 5].

72 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 21.
handed rotation through the angle cp, 0< cp<2n, about an axis in the direction
of the unit vector e.
The last type of anisotropy, called transverse isotropy (with respect to a
direction m), is characterized by the group ^12 consisting of 1 and the rotations
R$n, 0<cp<2n. Transverse isotropy is appropriate to real materials having a
laminated or a bundled structure.
A material is called orthotropic if its symmetry group contains reflections
on three mutually perpendicular planes. Such a triple of reflections is — R%,
—JRf, -¾.] Since RgRf =¾ and (Rly=R%, it follows that the groups
^3> ^6 > ^6 > and ^7 correspond to orthotropic materials.
Table 1.
Crystal class Group Generators of #„ Order of #„
Triclinic system all classes
Monoclinic system all classes
Rhombic system all classes
Tetragonal system
tetragonal-disphenoidal
tetragonal-pyramidal
tetragonal-dipyramidal
tetragonal-scalenohedral
di tetragonal-pyramidal
tetragonal-trapezohedral
ditetragonal-dipyramidal
Cubic system
tetartoidal
diploidal
hexatetrahedral
gyroidal
hexoctahedral
Hexagonal system
trigonal-pyramidal
rhombohedral
ditrigonal-pyramidal
trigonal-trapezohedral
hexagonal-scalenohedral
trigonal-dipyramidal
hexagonal-pyramidal
hexagonal-dipyramidal
ditrigonal-dipyramidal
dihexagonal-pyramidal
hexagonal-trapezohedral
dihexagonal-dipyramidal
We say that the material at x has ^n-symmetry if ^„ is a subgroup of %.. We
call two groups <€m and e6n equivalent if given any symmetric elasticity .tensor Cx,
*« = «.«**»= %■
( A group $fB is distinct if there is no c€m (m=^n) equivalent to e&n. It will be clear
—>| from the results of Sect. 26 that ^, ^, ^3, ^4, ^B, ^8, and ^9 are distinct; ^6
j and c€1 are equivalent; ^10, ^11( and $flg are equivalent. By definition, the sym-
' metry group ¾ is a maximal group of symmetry transformations; thus the groups
1
Run
Ru.Ri
R%*tl
R%. Ri, R*q
Bin nln jiln
r%,r\Z,
/lin
Rk,R%
12
24
12

Sect. 21.
Material symmetry.
73
^6 > ^7» ^io < ^ii > ^12 cannot be symmetry groups, since invaxiance with respect
to one of these, <£„ say, implies invariance with respect to another group which
contains elements not in c€n.
A unit vector e is called an axis of symmetry (for the material at x) if
Qe=e
for some Q<c%, with Q 4= 1. Thus e is an axis of symmetry if and only if one of the
symmetry transformations is a rotation about the axis spanned by e; i.e., if and
only if R%£% for some <p (0<<p<2ri).
Consider now the stress S due to a uniform dilatation E=el:
S = C[«1],
where C = Cx. Then
QSQT = C[eQlQT]=C[el] = S
or
QS=SQ
for every Q&%.', i.e. S commutes with every Q in ^,. But by the commutation
theorem (3.3), a symmetric tensor S commutes with an orthogonal tensor Q if
and only if Q leaves each of the characteristic spaces of S invariant. The_only
subspace of V" invariant, under. all.i3rtJfa>o^ is the full vector^
space^'ftself. Thus if the material at x is isotropic, then every vector in ~if is a
principal direction of S. Therefore it follows from the spectral theorem (3.2)
that the response to a uniform dilatation is a uniform pressure, i.e.
q>i]=ai.
Now assume: tha^the^material at x is anisotropic and again consider the stress
S = C[el] due to a uniform dilatation. If (p^n, the only spaces left invariant
by the rotation R% are the one-dimensional space spanned by e, the
two-dimensional space of all vectors perpendicular to e, and the entire space y. The rotation
JR™ leaves invariant, in addition, each one-dimensional space spanned by a vector
perpendicular to e. These facts, the spectral theorem (3.2), and the commutation
theorem (3.3) enable one to establish easily the results collected in the following
table.1
Table 2.
Type of symmetry
Triclinic system (¾)
Monoclinic system (¾^)
Rhombic system (¾)
Tetragonal system (#4, #5)
Hexagonal system (<?8, #9, #10, ^fjj)
Transverse isotropy C^la)
Cubic system (#6, #7)
Isotropy
Restriction on the stress S due
to a uniform dilatation E= e 1
no restriction
m is a principal direction of S
ft, I, m are principal directions of S
S = al +§m® m
S = al
1 Coleman and Noll [1964, 5].

74 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 22.
Of future use is the following
> ^X (2) Identity. Given any two vectors a, b, let
A{a, b) = %(a®b+b®a).
Th&yi
C[A(m, n)] • A(a, b) =C[A(Qm, Qn)] ■ A(Qa, Qb)
for all vectors a, b, m, n and every Q£@x. Here C = CX.
Proof. Since
Q(m®n)QT = (Qm)®{Qn),
it follows that
QA{m, n) QT =A{Qm, Qn),
and thus
QC[A(m, n)] QT = C[A(Qm, Qn)] (a)
for every 06¾. Next, in view of the identity T ■ (u®v)=u ■ Tv,
(QBQT) ■ (Qa®Qb)=Qa ■ QBQTQb=a ■ Bb = B ■ (a®b);
therefore if we let B = C[A(m, n)] and use (a) and (ii) of (20.1), we arrive at the
desired identity. □
In view of (iii) of (20.1),
Ciju=A-{eit e,-) • Cx[A(ek, e,)],
and we conclude from the identity (2) that for an isotropic material the
components of Cx are independent of the orthonormal basis {e1,e2, e3}.
22. Isotropic materials. In this section we shall determine the restrictions
placed on the constitutive relation
S=CX[E]
when the material at x is isotropic; i.e. when the symmetry group @x equals the
full orthogonal group. In a later section we will discuss the form of the elasticity
tensor for various anisotropic materials.
n ^s (1) Theorem on the response of isotropic materials. Assume that the
material at x is isotropic. Then:
(i) The response to a uniform dilatation is a uniform pressure, i.e. there exists
a constant k, called the modulus of compression, such that
c*[i]=3*i-
(ii) The response to a simple shear is a pure shear; i.e. there exists a constant p,
called the shear modulus,1 such that
Cx [sym (m ® n)] = 2/j, sym (m ® n)
whenever m and n are orthogonal unit vectors.
(iii) If E0 is a traceless symmetric tensor, then
Cx[E0] = 2fiE0.
Proof. Assertion (i) was established in the previous chapter (see the remarks
leading to Table 2).
v To establish (ii) let
^- A(m, n) =syia(tn®n),
1 The shear modulus is often denoted by G.

Sect. 22. Isotropic materials. 75
where m and n are orthogonal unit vectors, let e=mxn, and choose Q such
that
Qm = — m, Qn = —n, Qe — e,
i.e. Q is a rotation about e of 180°. Then, writing C = CX, we conclude from the
identity (21.2) with a=m and 6 =e that
C[i4(m, n)] • A(m, e) = — C[i4(m, n)] • A(m, e),
and hence
C[A(m, n)] • .4(m, e) =0. (a^
Similarly,
C[.4(m, n)] • A(n, e) =0. (a2)
Next, if we choose Q such that
Qm=m, Qn = —n, Qe=e,
and take a —b =m, n, or e, we conclude from (21.2) that
C[^l(m, n)] • .4 (a, o) =0 for a=m,n, ore. (ag)
Now let
(i = C[A(m, n)] • .4(m, n). (b)
In view of (3.1), the six tensors ]/2A(m, n), ^2A(n, e), ^2A{m, e), V2A(m, m),
y2A(n, n), y2A(e, e) form an orthonormal basis for the space of all symmetric
tensors. By (a) and (b), the only non-zero component of C[A(m, n)] relative to
this basis is the component j/2^ relative to ^2A{m, n); thus
C[A(m, n)] = 2fiA(m, n). (c)
Now let u and v be orthogonal unit vectors and choose Q such that
Qm—u, Qn—v.
Then (21.2) with a=m, b=n implies
C[A(m, n)] • 4(m, n) = C[A(u, »)] • 4(m, »),
and thus the scalar (i in (c) is independent of the choice of m and n.
To establish (iii) let E0 be a traceless symmetric tensor. By (13.7) there exist
scalars ^,¾.¾ and an orthonormal basis e^ e2, e3 such that
JS0 = ^ 4(¾. e2) +¾ 4(e2, eg) +xzA{ez, ej;
thus (ii) and the linearity of C imply
^[«01 = 21««,. D

76 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 22.
V- U- (2) Theorem on the elasticity tensor for isotropic materials. The
//■' material at x is isotropic if and only if the elasticity tensor C — Cx has the form1
C[JB] = 2|»B + A(trB)l
for every symmetric tensor E; the scalars /j,=/j,(x) and X=X(x) are called the
Lame moduli.2. Moreover, /jl is the shear modulus and X =k —§ jx, where k is the
modulus of compression.
Proof. We first assume that
C[E] = 2/*E+X(txE)l.
Let Q be orthogonal. Then
ti(QEQT)=tr(QTQE)=txE,
and therefore
QC[E]QT=2/,QEQT+l.(tiE)QlQT
= 2fiQEQT +X[ti(QEQT)]l = C[QEQT].
Thus the material at x is isotropic.
Conversely, assume that the material at x is isotropic. Let E be an arbitrary
symmetric tensor. Then E admits the decomposition
E=E0 + i(trE)l, tvE0 = 0,
and (i) and (iii) of (1) yield
C[E] = 2/*E0 + k{tiE)l.
But since E0=E — ^(trE)l, if we let X=k— %/x, we arrive at the desired
result. □
Note that the isotropic constitutive relation
S = 2f*E+X(trE)l
takes the form
Sij = 2fiEij + XEkkdij
when referred to cartesian coordinates.
When the material is isotropic, (1) implies that C maps multiples of the
identity into multiples of the identity and traceless tensors into traceless tensors.
The next proposition shows that those are the only two tensor spaces that are
invariant under C.
\" (3) Characteristic values of the elasticity tensor. Assume that the
material at x is isotropic with 2^=4=3^- Then the linear transformation* C = CX has
exactly two characteristic values: 3& and 2ft. The characteristic space corresponding
to 3 & is the set of all tensors of the form ocl, where a is a scalar: the characteristic space
corresponding to 2fi is the set of all traceless symmetric tensors.
Proof. By (i) and (iii) of (1), 2(i and 3^=2^ + 3^ are characteristic values
of C. Further, if C[E]=fiE, then (2) implies that
(2p-p)E + X(trE)l=0; (a)
thus (for E 4=0)
P=2fi^trE=0. (b)
1 The basic ideas underlying this constitutive relation were presented by Fresnel [1866, 1]
in 1829. This form of the relation with fi = X was derived from a molecnlar model by Navier
[1823, 2], [I827, 2] and Poisson [1829, 2]. The general isotropic relation is due to Cauchy
[1823, 1], [1828, 1], [1829, 1], [1830, 1]. Cf. Poisson [1831, 1].
2 Lame [18S2, 2].
3 Here C is considered as a linear transformation on the space of symmetric tensors.
h

Sect. 22.
Isotropic materials.
77
(Note that A 4=0, since 3^4=2^.) On the other hand, if tr 22=4=0, then (a) yields
|8 = 2^ + 3A<^JS = i(trB)l. (c)
Therefore the two spaces specified in (3) are characteristic spaces of C. Taking
the trace of (a), we see that
(2^t + 3A—j8)trJS = 0.
Thus either /9 = 2^+3A or tr E = 0. If trJ3=0, then (a) implies that /5 = 2/«.
Thus 2^+3^ and 2[i are the only characteristic values of. C. □
Note that if 2/* = 3k, so that X = 0, then every symmetric tensor is a
characteristic vector for C.
Assume now that the material at x is isotropic, so that
S = 2(iE + k{trE)l. (I)
C1Carly' trS = (2^ + 3A)trE.
Thus if we assume that /jl 4= 0, 2/jl + 3 X 4= 0, then
E:
-s-
(trS)l,
2,U '" 2^(2^ + 3¾
and we have the following result.
/,\ (4) If the material at x is isotropic, and if fi 4=0, 2//4-3^4=0, then C is in-
vertible, and the compliance tensor K = C"1 is given by
(2^ + 3¾
(trS)l
K[S] = -£-|s.
/or «/ery symmetric tensor S.
Let ej, e2, eg be an orthonormal basis and consider the following special
choices for S in the relation E = K[S]:
(a) Pure shear. Let the matrix [S] of S relative to this basis have the form
[S] =
0
T
.0
T 0
0 0
0 0.
Then
[E] =
"0
X
0
h
0
0
0"
0
0.
with
i.e. E is a simple shear of amount
x = —-- t;
2[t
2[t
with respect to the direction pair (e1, e2). Conversely, if E is a simple shear of
amount x, S is a pure shear with shear stress r=2fix.
(b) Uniform pressure. Let S correspond to a uniform pressure of amount p;
i.e. let
S = -pl.
Then E is a uniform dilatation of amount
-I*-*

78
M.E. Gurtin: The Linear Theory of Elasticity.
Sect. 22.
with
Clearly,
:/»+*.
p = -k(tiE).
Conversely, S is a uniform pressure whenever E is a uniform dilatation.
(c) Pure tension. Let S correspond to a pure tension with tensile stress or
in the e1 direction:
Then [U] has the form
with
and
[SI-
[E] =
e = ja,
"or
0
.0
0
0
0
0"
0
0.
~e
0
.0
0
I
0
0"
0
I
—pa=-ve,
^ li + l '
= A
" 2(fl + X) -
The modulus 6 is obtained by dividing the tensile stress or by the longitudinal
strain e produced by it. It is known as Young's modulus.1 The modulus v is the
ratio of the lateral contraction to the longitudinal strain of a bar under pure
tension. It is known as Poissons' ratio.2 Table 3 gives the relations between
the various moduli-
o o
If we write E and S for the traceless parts of E and S, i.e.
E=E — l(trE)l, S=S-i{trS)l,
then the isotropic constitutive relation (I) is equivalent to the following pair of
relations3
S = 2/iE,
trS=ktrE.
Another important form for the isotropic stress-strain relation is the one
taken by the inverted relation (4) when Young's modulus /S and Poisson's ratio v
are used:
E
= J-[(l+v)S-v(trS)l].
1 Young neither defined nor used the modulus named after him. In fact, it was introduced
and discussed by Euler in [1780, 1] (cf. Truesdell [I960, IS], [i960, 16], pp. 402-403) and,
according to Truesdell [1959,17], was used by Euler as early as 1727. In most texts Young's
modulus is designated by E.
2 Poisson [1829, 2]. Poisson's ratio is often denoted by a.
3 Stokes [l 845, 1] introduced the moduli fi and k and noted that the response of an
isotropic elastic body is completely determined by its response to shearing and to uniform
compression.

Sect. 22.
Isotropic materials.
Table 3-
79
A, ,u
X,p
A,v
A, ft
P.P
/X,V
/X, k
P,v
p,k
V, A
A
{2/i-pL,
2/tv
(1 — 2v)
3k-^2/1
3
(l+v)(l —2v)
3ft (3ft-0)
9k-p
3kv
i+v
/"
//
{P -3/.) +
V'(/9 —3A)2+8A/5
4
A(l-2v)
2v
3 (A-A)
2
/^
/^
^/^
2(1 +V)
3Pk
9k-p
3ft(l-2j>)
2 (!+»>)
P
/i(3X + 2/i)
A(1+v)(1-2v)
V
9ft (ft-A)
3k--).
2,u(l+v)
9ft,«
3k+/t
//
n /^
3k{\ — 2v)
V
A
2(A+/u)
-Q?+A) +
V(i?+~A)2+8A2
4 A
A
"3*-A"
P-2/*
2/1
//
1
2
3k —2/1
3k+/i
]
^/
1
2
\3k-p~l
l 3ft J
A
3A + 2/1
3
(3A+j?) +
f(3"A+i?)2-TA7
6
A (i+v)
3r
^/
3(3A«-/3)
2,u(l+v)
3(1- 2v)
^/
P_
3(l-2v)
Since an elastic sohd should increase its length when pulled, should decrease
its volume when acted on by a pressure, and should respond to a positive shearing
stress by a positive shearing strain, we would expect that
/?>0, k>0, [t>0.
Also, a pure tensile stress should produce a contraction in the direction
perpendicular to it; thus
r>0.
Even though the aboveJnegualities„axe„ph^sically well., motivated, we will not
assume that they hold, for in many circumstances other (somewhat weaker)

80
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 2js.
assumptions are_more natural. Further, these weaker hypotheses are important
to the study of infinitesimal deformations superposed on finite deformations.
Finally, we remark that Gazis, Tadjbakhsh, and Toupin1 have established
a method of determining the isotropic material whose elasticity tensor is nearest
(in a sense they make precise) to a given elasticity tensor of arbitrary symmetry.
-/■ 23. The constitutive assumption for finite elasticity. In this section we will
show that the linear constitutive assumption discussed previously is, in a certain
sense, consistent with the finite theory of elasticity provided we assume
infinitesimal deformations and a stress-free reference configuration.
In the finite theory the stress S at any given material point as is a function
of the deformation gradient F at x:
8= 8(F). (a)
The function S is the response function for the material point x.
In view of (o) in Sect. 16, we assume that
S{F)FT=FS(F)T (b)
for every invertible tensor F; this insures that the values of S are consistent with
the law of balance of angular momentum. In addition, we assume that S is
compatible with the principle of material frame indifference; this is the requirement
that the constitutive relation (a) be independent of the observer.2 During a change
in observer F transforms into QF and S into QS, where Q is the orthogonal
tensor corresponding to this change.3 Since S must be invariant under all such
changes, it must satisfy4
S(F)=QTS(QF) (c)
for every orthogonal tensor Q and every F. By the polar decomposition theorem,5
F admits the decomposition
F=RU, (d)
where R is orthogonal and
U={FTF)i. (e)
Choosing Q in (c) equal to RT, we arrive at the relation
S = RS{U),
or equivalently, using (d),
8=FU-18(V). (f)
If we define a new function S by the relation
S(D) = U*S(U),
where D is the finite strain tensor
D=Js(U*-l)=%(FTF-l),
•X1 [1963, 7].
2 See, e.g., Truesdell and Noll [1965, 22], §§ 19, 19A.
3 See, e.g., Truesdell and Noll [1965, 22], § 19. The transformation rule for F is an
immediate consequence of their Eq. (19.2^. The rule for the Piola-Kirchhoff stress S follows
from the law of transformation for the Cauchy stress T [their Eq. (19.2)2] and (r) of Sect. 16
relating S to T and F.
4 Cf. Truesdell and Noll [1965, 22], Eq. (143.9).
5 See, e.g., Halmos [1958, 9], § 83.

Sect. 24.
Work theorems. Stored energy.
81
then (f) reduces to
S=FS(D). (g)
On the other hand, it is not difficult to verify that the constitutive relation (g)
is consistent with (c). Further, (g) satisfies (b) if and only if
FS(1))FT=FS(D)TFT,
or, since F is invertible,
S(D) = S(Df, (h)
so that the values of S are symmetric tensors.
It is important to note that the concept of strain arises naturally in the
study of restrictions placed on constitutive relations by material
frame-indifference. It is possible to derive other constitutive relations involving other strain
measures; we chose one involving D because it simplifies the discussion of
infinitesimal deformations.
Let S be differentiable at 0. Then
S(D)=S(0)+C[D]+o(\D\)
as |D| ->0, where C is a linear transformation which, by (h), maps symmetric
tensors into symmetric tensors. When F = l, D = 0; thus S(0) represents the
residual stress in the reference configuration. We now assume that this residual
stress vanishes;1 hence
S(D)=C[D]+o(\D\). (i)
Let
d = \Vu\.
Then (c), (e), and (f) of Sect. 12 yield
F = 1+0(6),
D=E+0(62) =0(6)
as <5->0, and we conclude from (g) and (i) that
S=C[E] + o(d) as 6^0.
Therefore if we assume that the residual stress in the reference configuration vanishes,
then to within terms of o(6) as <5->-0 the stress tensor is a linear function of the
infinitesimal strain tensor. Further, since E = 0 when Vu is skew, to within terms of
o (6) the stress tensor vanishes in an infinitesimal rigid displacement.
These remarks should help to motivate the constitutive assumptions
underlying the infinitesimal theory.
24. Work theorems. Stored energy. In this section we establish certain
conditions, involving the notions of work and energy, which are necessary and
sufficient for the elasticity tensor to be symmetric and positive definite. Thus we
consider a fixed point x of the body, and let C=CX denote the elasticity tensor
for x.
1 This assumption, while basic to the theory presented here, is not satisfied, e.g., by an
elastic fluid.
Handbuch der Physik, Bd. VI a/2. 6

82 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 24.
By a strain path rE connecting E0 and Et we mean a one-parameter family
^(a)» Oo = a = ai» °f symmetric tensors such that E(») is continuous and piecewise
smooth and
E(a„)=E0, E(<x1)=E1.
We say that rB is closed if
Clearly, the strain path rE determines a one-parameter family of stress tensors
S(a) = C[E(a)], 0¾ ^ a ^ 0¾;
the work done along rB is defined to be the scalar
W(rE)=/s(a)-^-ia.
a.
In view of the discussion following (19.1), we have the following interpretation
of w (rE): Consider a homogeneous elastic body with elasticity tensor C; w(rE)
is the work done (per unit volume) by the surface tractions in a time-dependent
homogeneous deformation with strain E.
y By a stored energy function* we mean a smooth scalar-valued function e
on the space of symmetric tensors such that e(0) =0 and2
C[E]=VEe(E)
for every symmetric tensor E. An immediate consequence of this definition is
the following trivial proposition.
(1) If a stored energy function exists, it is unique, and
w (rE)=e(iy-e(ig
for every strain path rE connecting E0 and Et.
The next theorem supplies several alternative criteria for the existence of a
stored energy function.
Y (2) Existence of a stored energy function.* The following four statements
are equivalent.
(i) The work done along each closed strain path is non-negative.
(ii) The work done along each closed strain path vanishes.
(iii) C is symmetric.
(iv) The function s defined on the space of symmetric tensors by
e(E)=$E-C[E]
is a stored energy function.
Proof. We will show that (i) => (ii) => (iii) =½ (iv) =½ (i). Assume that (i) holds. Let rE
be an arbitrary closed path, and let rE* denote the closed path defined by
E*(a.)=E(oL0-\-ci1—a.), 0^^0.^0^.
1 The notion of a stored energy function is due to Green [1839, 1], [1841, J].
2 The gradient p£;e(jE) of s at E is defined in a manner strictly analogous to the definition
given on p. 10 of the gradient of a scalar field on S. Note that VEs(E) is a symmetric tensor;
in components
3 Green [1839, 1], [1841, i] proved that (iii) O (iv), Udeschini [1943, 3] that (i) O (iv).

Sect. 24. Work theorems. Stored energy. 83
Then a simple computation tells us that
w(rE) = -w(rE.).
But (i) implies that w(rE) and W(rE*) must both be non-negative. Hence
w(rE) = 0, which is (ii). To see that (ii) => (iii) let A and B be symmetric tensors,
and let rE be the closed strain path defined by
22(a) =(cosa — 1) ^ + (sina) B, 0^a^27i.
Then
C \E (a)] • —J£ - = (sin a - sin a cos a) A ■ C [A]
+ (cos2 a — cos a) B ■ C [A]
- (sin2 a) A ■ C [B] + (sin a cos a) B • C [JB]
and
»(ii) = / C [£(«)] • ^°0■ ia = »(B • C [Jl] -A ■ C [B]).
o
Hence (ii) => (iii). Next, define
e(E)=kE-C[E].
Since the gradient VEe(E) can be computed using the formula1
VBe(E).A = -£lB(E + XA)\M,
it follows that
VBe(E)'-A = ± ^{E ■ C[E] +XE ■ C[A] +XA ■ C[E] +VA ■ C[A]}X=0
= ±-{E.C[A]+A-C[E]}.
Thus (iii) implies
VEe{E)-A=A-C[E]
for every symmetric tensor A, which yields (iv). Finally, by (1), (iv) implies (i). □
If C is symmetric, a stored energy function e exists, and the constitutive
equation
S = C[E]
may be written in the form
S = VEe(E).
Let C be invertible, and let
e{A)=kA-K[A]
for every symmetric tensor A, where K is the compliance tensor. In view of the
symmetry of K (which follows from the symmetry of C),
rae{S)=K[S],
and the above constitutive relation takes the alternative form
E = Vae{S).
Moreover,
B{S)=kC[E].K[C[E]],
1 Cf. p. 10.
6*

84 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 24.
and, since K is the inverse of C,
e(S)=e(E).
The next two theorems give a physical motivation for some of our future
assumptions concerning the elasticity tensor.
(3) The work done along each strain path starting from E = 0 is non-negative
if and only if C is symmetric and positive semi-definite.
Proof. Assume that the work done along each strain path starting from 0
is non-negative. Let 7¾ be an arbitrary closed strain path starting from E0,
and let
E*(a) =E(a) —E0, 0,,^0^0¾.
Then 7¾. is a closed strain path starting from 0 and
0^/c[B*(«)].-^i«.
a0
Thus, in view of the linearity of C,
0 =g f C [£(«)] • -*JM. da _ c [E0] ■ {E («,) -£(«„)}.
But 12(0¾) =£(00), since 7¾ is closed. Thus the work done along each closed
strain path is non-negative. Thus we may conclude from (2) that C is symmetric
and, in addition, that there exists a stored energy function. Let A be a symmetric
tensor, and let TE be the strain path defined by
E(d)=a.A, O^a^l.
Then by (1) and (iv) of (2),
0^w(ii) =8(-4)-6(0) =M-C[4];
hence C is positive semi-definite. Conversely, if C is symmetric, then (1) and (2)
imply that given any path rE connecting 0 and Elt
Thus if C is positive semi-definite, w (7¾) ^ 0. □
(4) The work done along each non-closed strain path starting from E = 0 is
strictly positive if and only if C is symmetric and positive definite.
Proof. Assume that the former assertion holds. Let rE defined by E(cn.),
0¾ 550^0¾. be an arbitrary closed strain path with
B(oo)=B(oO=0-
If E(<x) = 0 for all a, then w (7¾) = 0. Assume E{a) 4= 0 for some a, and let
a = sup{a:E(a)4=0}.
Then
-(rE) = /c[E(a)].^ia = lim/c[E(a)] ■ *£U«.
OCq OCo

Sect. 24.
Work theorems. Stored energy.
85
Since E(a) (a05Sa5ja*) for a*<a but sufficiently close to a defines a non-closed
strain path starting from 0,
/c[£(a)].^ia>0;
a,
hence
Thus the work done along each strain path starting from 0 is non-negative, and
we may conclude from (3) that C is symmetric. The remainder of the argument
is the same as the proof of the analogous portions of (3). □
sX (5) If the material is isotropic, then C is symmetric,
e{E)=fjL\E\* + \-{txE)\
and the following five statements are equivalent:
(i) C is positive definite;
(ii) /li>0, 2/a+3/l>0;
(iii) fi>0, k>0;
(iv) /li>0, —\<v<\;
(v) /?>0, —\<v<k-
Proof. The symmetry of C and the above formula for e(E)=^E ■ C[E]
follow from (22.2). That (i)<=>(iii) is an immediate consequence of (22.3) and the
sentence following its proof. Finally, the equivalence of (ii)-(v) follows from the
relations given in Table 3 on p. 79- □
Let
E=E + i(trE)l (trE=0).
Then
|E|2 = |E|2 + i(trJS;)2
and
This alternative form for e yields a direct verification of the implication (i)<=>(iii)
in (5).
At the end of the last section we gave physically plausible arguments to
support the inequalities: /?>0, &>0, /a>0, v>0. It follows from (5) that these
inequalities are more than sufficient to insure J^sitiw^efuuteness. On the other
hand7Bfiaf Cis positive "defimteTmplies fl> 6,k>0,/i> 0, but does not imply v>0.
Assume that C is symmetric and positive definite. Then the characteristic
values of C (considered as a linear transformation on the six-dimensional space
of all symmetric tensors) are all strictly positive. We call the largest characteristic
value the maximum_ela^Hci tnod^JMS, the smallest the minimum elastic
modulus.1 The following trivial, but useful, proposition is a direct consequence
of this definition and (22.3).
vt y (6) Let C be symmetric and positive definite and denote the maximum and
minimum elastic moduli by /uM and (im, respectively. Then
frW^E ■ C[B] ^M|E|2
1 Cf. Toupin [1965, 21].

86 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 25.
for every symmetric tensor E. If, in addition, the material is isotropic wUh^strictly
positive Lame moduli /u and X, then
Pm = 2^+3^, fJ.m=2fJ,.
Of course, if /u>0 and X—0, then ^=^ = 2/«; if /i>0 and X<0 (but
2/a+3A>0 so that C is positive definite), then /j,m = 2/j, and ^ = 2^+3^-
25. Strong ellipticity. In this section we relate various restrictions on the
elasticity tensor C = CX at a given point xiB.
We say that C is strongly elliptic if
A-C[A]>0
whenever A is of the form
A=a®b, a=t=0, 6=(=0.
This condition is of importance in discussing uniqueness and also in the study of
wave propagation.
By (20.1)
A-C[A]= (symA) ■ C[sym A];
thus
Cpositive definite => C strongly elliptic.
The f§£Lth^jth^_cony£rse assertion is not,true is_apgarent from the_nextj>ro-
posijkm.
&" (1) Suppose that the material is isotropic. Then
C strongly elliptic 4=>/a>0, A + 2/a>0,
— C strongly elliptic <=>/j,<0, A + 2/a<0,
where (i and X are the Lame moduli corresponding to C. 7« ferns o/ Poisson's ratio v,
C strongly elliptic <s=> /j, > 0, i>€ [$, 1],
— C strongly elliptic «=> ^ <0, i>€ [J, 1],
provided ^+^=^0-
Proa/. By (i) of (20.1) and f22.2),
(a®6) • C[a®6] = (a®6) • C[syma®6]
= (a®6) • |>(a®6 + 6®a)+A(a-6) 1]
=/*a2 62 + (X +/li) (a • 6)2
=/i [a2 62 - (a • 6)2] + (X + 2/i) (a • 6)2.
Since (a • 6)2^a262, /u>0 and A + 2^t>0 imply that C is strongly elliptic.
Conversely, if C is strongly elliptic, then X + 2^ > 0 follows by taking a = 6,
^ > 0 by taking a orthogonal to 6. The assertion concerning — C follows in the
same manner. Finally, the last two results are consequences of the relation:

Sect. 26.
Anisotropic materials.
87
These results, as well as some of the results established in (24.5), are given
in the following table:
Table 4.
Condition Restrictions on the moduli
Positive definite /*>0, 2/i + 3A>0 or ^,^¾. =*> S^fikyK
M>0, ^6(-1,¾ or
0>O, ve(-i.i)
Strongly elliptic /i>0, A + 2/i>0 or
H>0, vUi, 1]
> 26. Anisotropic materials. In this section we state the restrictions that material
symmetry places on the elasticity tensor C = Cm at a given point x. Let {ej be
a given orthonormal basis with 63=^X63, and let Cijkl denote the components
of C relative to this basis:
Cijhi = Cjiki = 0^ih = (sym e{®e?) • C[sym ek® e,].
We assume throughout this.section that a stored energy function exists, so that
C is symmetric; thus
Ci)kI=Cklij-
For the triclinic system (^-symmetry) there are no restrictions placed on C
by material symmetry.
For the monoclinic system ($i-symm.ttvy) let the group #2 be generated
by fl*. By identity (21.2),
Ciiki=sym{Qet®Qef) ■ C[sym(<?eft<g)0e,)]
for every 06¾. If we take Q = —R%,, then
(?«! = %, 0e2 = e2, 0e3 = -e3,
and the above identity implies
^1128 ==^'1181=:^2228==^'2281==^8328==^3881==^2812=:^3112=::0•
In the same manner, results for the other symmetries discussed in Sect. 21
can be deduced.1 We now list these results, which are due to Voigt.2 For
convenience, the 21 elasticities will be tabulated as follows:
^1111
C-1122
^2222
^-1183
^2288
^-8888
r
^1128
C-2228
^3323
^2323
^-1131
t-2231
^3331
t-2331
^-3131
^1112
t-2212
^3312
^-2312
^3112
^1212
1 They can also be deduced using the results of Smith and Rivjlin [1958, 16], Sirotin
[i960, 11].
2 Voigt [1882,1], [1887,3], [1900,6], [1910,1], §287. See also Kirchhoff [1876, 1],
Minnigerode [1884, 1], Neumann [1885, 3], Love [1927, 3], Auerbach [1927, 1], Geckeler
[1928, 2]. A discussion of methods used to measure the elastic constants of anisotropic materials,
as well as a tabulation of experimental values of these constants for various materials, is
given by Hearmon [1946, 2].

88 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 26.
Monoclinic system (all classes) <^2 generated by ft£ (13 elasticities)
^1111 C1122 C-1133 0 0 ^1112
^2222 ^"2233 0 0 ^2212
^-3333 0 0 ^3312
^2323 ^2331 ^
C3131 0
Cl212
Rhombic system (all classes) ^ generated by ft* , ft£a (9 elasticities)
t-im c1122 c1133 0 0 0
^2222 ^2233 0 0 0
c3333 0 0 0
^2323 0 ^
C3131 0
^1212
Tetragonal system (tetragonal-disphenoidal, tetragonal-pyramidal, tetra-
gonal-dipyramidal) <gi generated by ft*'2 (7 elasticities)
^1111
^-1122
^-1111
^-1133
^-1133
C-3333
0
0
0
t-2323
0
0
0
0
C2323
ClH2
^-1112
0
0
0
^-1212
Tetragonal system (tetragonal-scalenohedral, ditetragonal-pyramidal, te-
tragonal-trapezohedral, ditetragonal-dipyramidal) <€6 generated by ft*'2, ft*
(6 elasticities)
t-llll ^-1122 ^-1133 0
^1111 ^-1133 0
^-3333 0
c
^2323
0
0
0
0
C2323
0
0
0
0
0
^-1212
Cubic system (tetartoidal, diploidal) ^6 generated by
ft*,ft*,ft^8, q=yi{€,+€,+
■«*);
(hexatetrahedral, gyroidal, hexoctahedral) % generated by ft*(2, IJ£(2, ft*{2 (3
elasticities)
t-1111 c1122 c1122 0
t-1111 ^1122 0
Qui 0
r
^2323
0
0
0
0
^2323
0
0
0
0
0
^-2328

Sect. 27. The fundamental system of field equations. 89
Hexagonal system (trigonal-pyramidal, rhombohedral) ^s generated by
R%"JS (7 elasticities)
£-1122
^1111
^-1133
^-1133
C3333
^-1123
^-1123
0
r
u2323
^-1131
^-1131
0
0
c
u2323
0
0
0
~~ ^1131
^-1123
^"(Cllll
'C1122)
Hexagonal system (ditrigonal-pyramidal, trigonal-trapezohedral, hexag-
onal-scalenohedral) ^9 generated by R%^3, iJ£ (6 elasticities)
r
u1122
t-1111
^-1133
C1133
^3333
^1123
— ^-1123
0
t-2323
0
0
0
0
^2323
0
0
0
0
^-1123
¢(£-1111-
t-1122)
0
0
0
0
C-2323
0
0
0
0
0
3"(^1111 ^1122)
Hexagonal system (trigonal-dipyramidal, hexagonal-pyramidal, hexago-
nal-dipyramidal) ^w generated by B%&; (ditrigonal-dipyramidal, dihexagonal-
pyramidal, hexagonal-trapezohedral, dihexagonal-dipyramidal) #u generated by
Re?> R%,'y and transverse isotropy e&12 generated by R%t, 0<<p<2n (5
elasticities)
' Cim c1122 c1133 0
£-1111 C-1133 0
£-3333 0
£-2323
In view of (21.1), if the elasticity tensor C is invertible, then the components
of the compliance tensor K = C_1 will suffer restrictions exactly analogous to
the ones tabulated above.
D. Elastostatics.
I. The fundamental field equations.
Elastic states. Work and energy.
27. The fundamental system of field equations. The fundamental system of
field equations for the time-independent behavior of a linear elastic body consists
of the strain-displacement relation
E = Vu = k{Vu + VuT),
the stress-strain relation
S = C[E],

90 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 27-
and the equation of equilibrium1
div S+ 6=0.
Here, u, E, S, and 6 are the displacement, strain, stress, and body force fields,
while C is the elasticity field.
By (i) of (20.1),
C[Vu]=C[Vu];
thus when the displacement field is sufficiently smooth, the above equations
imply the displacement equation of equilibrium
divC[F«]+6 = 0.
Conversely, if u satisfies the displacement equation of equilibrium, and if E
and S are defined by the strain-displacement and stress-strain relations, then the
stress equation of equilibrium is satisfied. In components the displacement
equation of equilibrium has the form
(C</*j«*.j),/ + &< = 0.
We assumefor Jhe^emainder of this section that the material is isotropic.
Then
S=2/iE + X(tiE) 1,
where (i and X are the Lame moduh. Of course, (i and 1 are functions of position x
when the body is inhomogeneous. Therefore, since
div (jj,Vu) —pA u + (Vu)V/j,,
div (jj, VuT) =^ V div u + (Vu)T V/j,,
div [X (div u) 1] = I Pdiv u + (div u) VI,
the displacement equation of equilibrium takes the form
/iAu + {X+/A Vdivu + 2(Vu) F7^ + (div«) VX + b = 0.
Suppose now that the body is homogeneous. Then Vfi = VX= 0, and the above
equation reduces to Navier's equation2
fiAu + (X+/j)Vdivu + b=0, (a)
or equivalently,
Au -1 V Vdiv u + -6=0,
1 — 2v n
in terms of Poisson's ratio
Next, by (3) of (4.1),
Au = Fdiv u—curl curl u,
1 Since the values of C&l'] are assumed to be symmetric tensors, the requirement that
S = ST is automatically satisfied.
2 This relation was first derived by Navier [1823, 2], [1827, 2] in 1821. Navier's work,
which is based on a molecular model, is limited to materials for which ft = A. The general
relation involving two elastic constants first appears in the work of Cauchy [1828, 1]. Cf.
Poisson [1829,2], Lame and Clapeyron [1833,-?], Stokes [1845, i]. Lame [1852,2],
§26. The analogous relation for cubic symmetry was given by Albrecht [1951, i].

Sect. 27. The fundamental system of field equations. 0,1
and thus (a) yields1
(A + 2[a) Fdivu—fj, curl curl u + 6 = 0.
Operating on (a), first with the divergence and then with the curl, we find
that
(A+2^)4 divu =-div 6,
pA curlu = —curl 6.
If (A -f 2/j) =)= 0, /j, =)= 0, and div 6 and curl 6 vanish, then
^divi« = 0, Acurlu = 0.
Thus, in this instance, the divergence and curl of u are harmonic fields. Trivially,
these relations can be written in the form
div^M=0, curl^u = 0; |-> CA*]^ " ffi > ^v^
4
and since a vector field with vanishing divergence and curl is harmonic, we have
the following important result:3 'Ji»
4-
div6=0, curl6=0 =>AAu = 0. a.[a
Thus if the body force field 6 is divergence-free and curl-free, the displacement field \
is biharmonic and hence analytic on B. To prove this we have tacitly assumed
that it is of class C4; in Sect. 42 we will show that these results remain valid when
u is assumed to be only of class C2.
Another interesting consequence of the displacement equation of equilibrium
may be derived as follows: Let
v =/liu + £(A +p) p div u, p(x)~x — Q.
Then
Av—pAu + (h+/j.){Vdivu-\-?pA divt*},
and (a) and (b) imply
A v = - 6 - ----,4-¾ v P div b.
2(/ + 2/1) r
Thus we arrive at the following result of Tedone: 4 if the body force field vanishes,
then
Av = 0.
Assume, for the time being, that the body force field 6 = 0. A field u that satisfies .
Navier's equation (a) for all values of p and X is said to be universal. Since &iield~.
of this type is independent of the elastic constants p and X, it is a possible
displacement field for all homogeneous and isotropic elastic materials. Clearly, u is
universal if and only if5 "^ "
Au=0,
Fdivi«=0;
1 Lame and Clapeyron [1833, i]. Lame [1852, 2], § 26.
2 The first of these relations is due to Cauchy [1828, I]. See also Lame and Clapeyron
[1833, 1], Lame [1852, 2], § 27.
8 Lame [1852, 2], p. 70.
4 [1903, 4], [1904, 4]. See also Lichtenstein [1924, 1].
5 Truesdeix [1966, 28], p. 117-

92 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 27.
i.e., if and only if u is harmonic and has constant dilatation. Important examples
of universal displacement fields are furnished by the Saint-Venant torsion
solution.1
If/a=|=0,j'=t=— 1, the stress-strain relation may be inverted to give
2fl
S_ ?--(trS)l
By (14.3) we can write the equation of compatibility in the form
S?(E)=AE + VV(trE)-2pdivE=0. (c)
Therefore
0 = jS?(2/M B) =J2?(S) - -^-jS?[(tr S) 1]. (d)
But
JS?[(tr S)1]=A (tr S) 1 + FT(tr S), (e)
and by (c) with E replaced by S in conjunction with the equation of equilibrium,
&{S)=AS + W{tiS)+2pb. (f)
Eqs. (d)-(f) imply that
AS + -[^VV(trS)-iv+vA(tTS)l+2Vb=0, (g)
and taking the trace of this equation, we conclude that
\-VvA(trS) = -diVb. (h)
Eqs. (g) and (h) yield the stress equation of compatibility:2
AS + —— FT(tr S) +2Vb + --- - (div 6) 1 =0,
which in the absence of body forces takes the simple form
4S+~-FF(trS)=0.
On the other hand, if S is a symmetric tensor field on B that satisfies the
equation of equilibrium and the stress equation of compatibility, and if E is
defined through the inverted stress-strain relation, then E satisfies the equation
of compatibility. Thus if B is simply-connected, there exists a displacement field
u that satisfies the strain-displacement relation.
In view of the stress-strain and strain-displacement relations,
S=fi{Vu + VuT) + A (div u) 1,
and the corresponding surface traction on the boundary 8B is given by
s = Sn = fi(Vu + VuT) n + A(divw) n.
1 See, e.g., Love [1927, 3], Chap. 14; Sokolnikoff [1956, 12], Chap. 4; Solomon
[1968, 12], Chap. 5.
2 This equation was obtained by Beltrami [1892, 1] for 6==0 and hy Donati [1894, 2]
and Michell [1900, 3] for the general case. It is usnally referred to as the Beltrami-Michell
equation of compatibility although Donati's paper appeared six years before Michell's. The
appropriate generalization of this equation for transversely isotropic materials is given by
Moisil [1952, 3], for cubic materials by Albrecht [1951, 1].

Sect. 27. The fundamental system of field equations. 93
Let du/dn be defined by
then
But
8-=(Vu)n;
s=2fi -.-- +/i(VuT — Vu) n + A(div w) n.
(Fm2" — Vu) n=nxcurlm;
thus we have the following interesting formula for the surface traction:1
s = 2fi g-—h/«wxcurlM+A(divM) n.
We close this section by recording the strain-displacement relations and the
equations of equilibrium in rectangular, cylindrical, and polar coordinates, using
physical components throughout.2
(a) Cartesian coordinates (x, y, z). The strain-displacement relations
reduce to
E - 8^-
£«~ 8z
F —
^xx
F =
F —
8ux
8x '
1 /8ux
2 \ 8y
1 1 8us
"2 \ 8x~
F —
Vyy
. 8Uy
8x~
4. 8V
"•" 8z ,
8Uy
8y
)•
!■
and the equations of equilibrium are given by
8SXX I 8S*V |_ 8Sxz
9* ^ 8y ^ 8z ^u* v'
8S*y 1 8Syy 1 8Sya \h—n
8x + 9;>>""~r~9T~ -r°y-u'
Is*.: + Ihu + i?». +6=0.
9* ^ 9;y ^ 9* ^ * u
(b) Cylindrical coordinates (r, 0, z). The strain-displacement relations have
the form
p _ 8ur p — J_ 8m0 1 ur p _ 8m*
£rr —-3--. ^ee- , -gT^T' r'"~'8z ■
P 1/1 9wf 9we we
'e ~ 2" It ~w + "a? r
p = J / 8ml _i_ 8M p _ 1 /8^°. 4- J _8m*^
" 2\'8z ~^~ 9*-/' e* 2^0* "•" >- 90 J'
while the equations of equilibrium reduce to
0?rr 1 J. 8SrO 1 8Srz 1 SJil~lM. Mh —fl
9»- "^ / 90 ^ 8z + r -T-"r-v>
8Srg , 1 8Sgg 8SBt ,2-, _
-9/' + T 90" + -07-+ T^e+6e-°>
9r ^ r 90 + 9* + r ^"^^ u>
1 Betti [1872, i], § 10, Eq. (53). See also Somigliana [1889, 3], p. 38, Korn [1927, 2],
p. 13, Kupradze [1963, 17, IS], p. 9.
2 See, e.g., Ericksen [i960, 6], §§ 11-13.

94 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 28.
(c) Spherical coordinates (r,d,y). The strain-displacement relations
become
F _ 8ur J? _ * 8u6 I Mr
^^ - 8y . *ee — -r fa -I--7 -
£. _ 1 9«y , «, , Me cot 0
+ ^f +
p 1 ( 1 9m, My . 8uy\
"v ~~ 2" iTsin 6 8y V + ~9^j '
F = 1 /1 gwf «e_ , _d«e
'e 2 U 96 >• "•" Sf"
p _ 1 /_1_ 9«y _ My COt 0 . i 9ttfl \
^ — 2~ 1.7" "90 P ■" 7sin"0 "g^T/ '
and the equations of equilibrium are given by
8Sfr 1 8Sry 1 dSrl) 2Srr-SyY-Sm + Srg cote ,
9>- lysine 8y"" "^ r 86 ^ r " ^ ' U'
8Sry 1 85yy , 1 8Syp 3 5fy + 2 Syfl cot 8 , _
0r ~r rsin0 3y +7 30 " "^ " r i-Oy—U2
dSry , 1 8Sy6 1 9Sp8 35,a + {Sge — Syy) cot 8 , _
9»-' "^ >-sin0 8y ^r 86 ^ " r -|-0<,—U.
28. Elastic states. Work and energy. Throughout this section we assume given
a continuous elasticity field C on B. We define the strain energy U{E}
corresponding to a continuous strain field E on E by
•U{E}=$fE-C[E]dv.
B
Note that when a stored energy function e exists, its integral over B gives the
strain energy:
U{E} = fe{E)dv.
B
For future use we now record the following
(1) Lemma. Let C be symmetric, and let E and E be continuous symmetric
tensor fields on B. Then
U{E +E} = U{E} + U{E} + JE-C[E] dv.
B
Proof. Since C is symmetric,
E-C[E]=E-C[E].
Thus
(E +E) ■ C [E +E] =E -C[E]+E-C [E] +2E-C[E],
which implies the desired result. □
By an admissible state we mean an ordered array a = [u, E, S] with the
following properties:
(i) m is an admissible displacement field;1
(ii) E is a continuous symmetric tensor field on B;
(iii) S is an admissible stress field.1
1 See p. 59.

Sect. 28.
Elastic states. Work and energy.
95
Note that the fields u, E, and S need not be related. Clearly, the set of all
admissible states is a vector space provided we define addition and scalar
multiplication in the natural manner:
[a, E, S] + [«, E, S] = [u + u, E + E, S +S],
a [m, E, S] = [aw, a.E, aS].
We say that <s = [u,E,S] is ar^_elastic_§taisL (on jg)LCorresrjondingJ;o the
body force field 6 if o is an admissible state and r
' "" "" ~E = i{Vu + VuT), \
S = C[E], \
divS+6=0. \
The corresponding surface traction s is then defined at every regular point
of dB by
s(x) =S(x) n(x),
where n (x) is the outward unit normal to dB at x. We call the pair [6, s] the
external force system for o.
When we discuss the basic singular solutions of elastostatics, we will
frequently deal with elastic states whose domain of definition is a set of the form
D=B—F, where F is a finite subset of B. Wjj_say_that t = \u, E, S] is an
elastic state on D corresponding to 6 if u, E,J^jndJtJ^e Junctions, on D and
given any closed regular subregion PcD, the restriction of d to P is an elastic
state on P corresponding to the restriction of 6 to P.
When we omit mention of the domain of definition of an elastic state, it will
always be understood to be E.
K (2) Principle of superposition for elastic states. If [u, E, S] and [u, E, S]
are elastic states corresponding to the external force systems [6, s] and [6, s],
respectively, and if a and x are scalars, then a [it, E, S] +t[m, E, S] is an elastic
state corresponding to the external force system a [6, s] +t[6, s], where
a [6, s] +t[6, s] =[<x&+t6, ocs + tS].
A direct corollary of (18.2) is the
X (3) Theorem of work and energy.2 Let [u, E, S] be an elastic state
corresponding to the external force system [b, s]. Then
fs-uda + fb-udv = 2U{E}. P*» CUv»i»<>">
dB B l
The quantity on the left-hand side of this equation is the work done by the
external forces; (3) asserts that this work is equal to twice the strain energy.
^ P" (4) Theorem of positive work. Let the elasticity field be positive definite.
" JT Then given any elastic state, the work done by the external forces is non-negative and
np vanishes only when the displacement field is rigid.
Proof. If C is positive definite, then U{E} 5:0, and the work done by the
external forces is non-negative. If U{E} = 0, then E • C[E] must vanish on B.
Since Cis positive definite, this implies Ee=Q; hence by (13.1) the corresponding
displacement field is rigid. □
1 This notion is due to Sternberg and Eubanks [1955, 13].
2 Lame [1852, 21.

96 M. E. Gortin: The Linear Theory of Elasticity. Sect. 29-
The following direct corollary to (4) will be used in Sect. }2 to establish an
important uniqueness theorem for elastostatics.
(5) Let the elasticity field be positive definite. Further, let [u, E, S] be an
elastic state corresponding to vanishing body forces, and suppose that the surface
traction s satisfies
s-u = 0 on 8B.
Then u is a rigid displacement field and E == S ==. 0.
The next theorem, which is extremely important, shows that the Z,2 norm of
the displacement gradient is bounded by the strain energy.
(6) Alternative form of Horn's inequality.1 Assume that C is symmetric,
positive definite, and continuous on B. Let u be a class C2 displacement field on B,
and assume that either (a) or (/3) of (13.9) holds. Then
f\Vu\*dv^K0U{E},
B
where E is the associated strain field and K0 is a constant depending only on C and B.
Proof. Let /im (x) denote the minimum elastic modulus for Cx .2 By hypothesis,
fim (x) is bounded from below on B. Hence
T = supi ,--: xeB> <oo,
and we conclude from (24.6) that
\E\^xE-Cx[E]
for every xaB and every symmetric tensor E. This inequality and (13.9) yield
the desired result. □
II. The reciprocal theorem. Mean strain theorems.
29. Mean strain and mean stress theorems. Volume change. In this section we
establish some results concerning the mean values
B B
of the stress and strain fields when the body B is homogeneous.
■j: (1) Second mean stress theorem. The mean stress S corresponding to an
elastic state [u, E, S] depends only on the boundary values of u and is given by
S(B)=^c\Ju®nda
L3B
Thus for an isotropic body,
S(B)= vjp- f{/i(u®n+n®u)+X(u-n)i}da.
dB
Proof. Since B is homogeneous, C is independent of x and
*w=ik!cwi*=c\ik!Biv
B L B
= C[E(B)].
1 See footnote 1 on p. 38. See also HlavA6ek and Ne6as [1970, 2].
2 See p. 85.

Sect. 29. Mean strain and mean stress theorems. Volume change. 97
The first of the relations in (1) is an immediate consequence of this result, the
mean strain theorem (13.8), and (i) of (20.1), while the second follows from
(22.2). □
The above theorem shows that the mean stress is zero when the surface
displacements vanish, independent of the value of the body force field.
-7 (2) Second mean strain theorem.1 Supjwse that the elasticity tensorjC_.is
invertihle with K = C'1. Let [u, E, S] be an elastic state corresponding to the external
foTce'syslem'J\U7s\'.~Then the mean strain depends only on the external force system
and is given by
E{B)= vX(B) K\ fp®sda+ fp®bdv .
Thus for an isotropic body,
V8S B ' '
Proof. The first result follows from the mean stress theorem (18.5), since
E(B)=K[S(B)];
the second result follows from (22.4). □
Note that balance of forces
/sda+ J bdv =0
dB B
insures that the above formulae are independent of the choice of the origin 0,
while balance of moments in the form
skw I [ p®sda-\- J p®b dv]=0
IdB b 1
idB
insures that the right-hand side of each of the expressions in (2) is symmetric.
(3) Volume change theorem.2 Suppose that the elasticity tensor C is
symmetric and invertihle. Let [u, E, S] be an elastic state corresponding to the external
force system \b, s]. Then the associated volume change dv(B) depends only on the
external force system and is given by
dv{B)=K[l]- { fp®sda+ (p®bdv\,
Us B )
where K = C1 is the compliance tensor. If, in addition, the bodyisisotro^ic, then
dv{B) = -*k-l f p-sda+ j'p-bdv\,
'as b '
where k is the modulus of compression.
1 Betti [1872, 1], § 6 for the case in which 6 = 0. The terms involving the body force were
added by Chree [1892, 2], who claimed that his results were derived independently of
Betti's. See also Love [1927, 3], § 123; Bland [1953, 5].
2 Betti [1872, 1], § 6; Chree [1892, 2].
Handbuchder I'hysik, Bd. VIa/2. 7

98 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 30.
Proof. Let
D= Jp0sda + /p®6 dv.
SB B
Given any tensor A,
tiA=l-A.
Thus the remark made in the paragraph preceding (13.8), (2), and the symmetry
of K imply that
dv{B)=v{B)tiE{B)=trK[D]=l-K[D] = K[l]'D,
which is the first result. For an isotropic material, (22.4) and Table 3 on p. 79
imply that
hence the second formula for dv (B) follows from the first and the identity
1 • (o(g)c) =tr(o®c) =o-c. □
Assume that the body is homogeneous and isotropic. Then, as is clear from (3),
a necessary and sufficient condition that there be no change in volume is that
fp-sda-\-fp-bdv=0.
SB B
oi { Thus a solid sphere or a spherical shell under the action of surface tractions that are
I tangential to the boundary will not undergo a change in volume.
Recall from Sect. 18 that the force system [6, s] is in astatic equilibrium if
JQsda+ fQbdv = 0,
SB B
/ pxQsda + / pxQbdv = 0,
dB B
for every orthogonal tensor Q. We have the following interesting consequence
of (2) and (18.10).
(4)x Assume that the elasticity tensor is invertible. Then a necessary and
sufficient condition that the mean strain corresponding to an elastic state vanish is that
the external force system be in astatic equilibrium.
It follows as a corollary of (4) that the volume change vanishes when the external
force system is in astatic equilibrium.
30. The reciprocal theorem. Betti's reciprocal theorem is one of the major
results of linear elastostatics. Ii^ssjenceJ.it,expressesjthe fact that the underlying
system of field equations is self-adjoint. Further, it is essential in establishing
integral representation theorems for elastostatics. In this section we will estabUsh
Betti's theorem and~discuss~some oTTts consequences. We assume throughout
that C is smooth on B.
(1) Betti's reciprocal theorem.2 Suppose that the elasticity field C is
symmetric. Let [u, E, S] and [it, E, S] be elastic states corresponding to external force
systems [6, s] and [6, s], respectively. Then
f s-uda-\- fb-udv=Js-uda + fb-udv=fS-Edv— f S-E dv.
dB B dB B B B
1 Berg [1969, 1].
2 Betti [1872, 1], § 6, [1874, 1]. See also L£vy [1888, 3]. An extension of Betti's theorem
to include the possibility of dislocations was established by Indenbom [I960, 8]. For
homogeneous and isotropic bodies, an interesting generalization of Betti's theorem was given by
Kupradze [1963, 17, 18], § 1.1. See also Beatty [1967, 2].

Sect. 30. The reciprocal theorem. 99
Proof. Since C is symmetric, we conclude from the stress-strain relation that
S-E = C[E]-E = C[E]-E=S-E.
Further, it follows from the theorem of work expended (18.2) that
fs-uda+fb-udv — f S- E dv,
SB B B
Js-uda+fb-udv = JS-Edv,
dB B B
and the proof is complete. □
Betti's theorem asserts that given two elastic states, the work done by the
external forces of the first over the displacements of the second equals the work
done by the external forces of the second over the displacements of the first. The
next theorem shows that the assumption that C be symmetric is necessary for
the validity of Betti's theorem.
(2)1 Betti's reciprocal theorem is false if C is not symmetric.
Proof. If C is not symmetric, then there exist symmetric tensors E, E such
that
E-C[E]+E-C[E].
Let
u=Ep, S = C[E], 6 = -divS,
ft=jEp, S = C[E], 6 = -divS.
Then [u, E, S] and [ft, E, S] are elastic states corresponding to the body force
fields 6 and 6 and
/ S-Edv + / S-Edv. □
The following elegant corollary of Betti's theorem is due to Shield and
Anderson.2
(3) Suppose that C is symmetric and positive semi-definite. Let [u, E, S] and
[ft, E, S] be elastic states corresponding to external force systems [6, s] and [6, s],
respectively. Then
U{E} ^ U{E}
provided
/s • (m —ft) da-\- J 6 • (u — ft) dv^O
dB B
or
J {s—s) -uda-\- / (6 — 6) -udvsSO.
dB B
Thus, if £^[ and £?% are complementary subsets of dB,
u—u
s = 0
6=0
ii=0
s = s
6=6
on Sfx
on y%
on B
on &[
on ,9"2
on B
v*-
■^U{E}^U{E},
■=>U{E}^U{E}.
1 Truesdell [1963, 24]. See also Burgatti [1931, 1], P- 152.
2 [1966, 23].
7*

100 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 30.
Proof. By the principle of superposition (28.2), [u — u,E — E,S—S] is an
elastic state corresponding to the external forces [6 — b, s — s]. By (28.1) with
E replaced by E — E,
U{E} = U{E} + U{E -E} + JE-(S-S)dv,
and by Betti's theorem,
B
f E-(S-S)dv= f (b-b)-udv + f (s-s)-uda
B B dB
= / 6 • (w —u) dv + / s • (it —u) da.
B dB
Thus, since U{E — E] ;> 0, we have the inequalities
U{E}^ U{E} + /(6 -6) • udv + f (s-s) • u da,
B dB
U{E}^U{E}+ fb-{u-u)dv + Js-{u-u)da.
B dB
The remainder of the proof follows from these inequalities. □
y I Theorem (3) may be called a least work principle,, since by (28.3)
(1^ ^,. I U{E} 5S U{E} if and only if the work done by the external forces corresponding
n,te f| to [it, E, S] is less than or equal to the work done by the external forces of
& , (4)1 Suppose that Cis symmetric and positive semi-definite. Let [it, E, S] and
[u, E, S] be elastic states corresponding to external force systems [6, s] and [6, s],
respectively. Let S^0, £flt ..., SPN be complementary regular subsurfaces of dB, and
su^oseJhaL4o^-eaclLM=\,2, ...,N, the restriction o£uto< £^ is a rigid_displacfi-
ment of £fn. Then
s = s on S?0
f (s—s) da= f px(s—s) da = 0
>U{E}^U{E}.
n = \,2,...,N
6=6 on B
Proof. By (3) it is sufficient to show that
/=/ (s—s)-uda+ / (6— 6) -udv=0.
dB B
Since s=s on 5P0 and b—b on B,
/=24. I„=f{s-s)-uda.
«-i srn
Further, as the restriction of u to each S?n, n =1, 2, ..., N, is rigid, there exist
vectors v1, v2, ...,vn and skew tensors Wj, W2,..., WN such that
u = Wnp+vn on Sfn.
Thus
In = / (s-s) • vn da + / (s-s) -Wnpda,
1 Shield and Anderson [1966, 23],

Sect. 30. The reciprocal theorem. 101
and since vn and Wn are constant and (s— s) ■Wnp = Wn- [(s — 5)®p], this
expression reduces to
In=vn- f {s—s)da + Wn- f (s—s)®pda. (a)
But
J (s — s) da = f px(s—s) da = o. (b)
The second relation in (b) implies that the tensor
A= J (s — s)<S>p da
is symmetric. Thus, since Wn is skew, Wn -A =0, and it follows from (a) and the
first of (b) that I„ — 0 for each n. Therefore /=0. □
Theorem (29.3) can also be established using Betti's theorem. Indeed, the
volume change dv(B) is given by
dv{B) = jtvEdv.
B
If we assume that B is homogeneous and define
S = l,
B = K[1],
u=Ep,
then [u, E, S] is an elastic state corresponding to vanishing body forces, and we
conclude from Betti's theorem (1) that
fS-Edv=Js-uda + Jb-udv,
B SB B
= fs-Ep da+ f b-Ep dv.
dB B
Since
S-E = l-E = trE,
a • Ep —E • (a0p) for any vector a,
and E = K[l] is constant and symmetric, it follows that
dv(B)=K\l\- {fp®sda+fp®bdv\.
dB B
The following generalization of Betti's theorem holds even when C is not
symmetric. " """ " ~ *
(5) Reciprocal theorem. Let [u,E,S] and [u,E,S] be elastic states
corresponding to external force systems [6, s] and [6, s], respectively, and let [«, E, S]
correspond to the elasticity field C, [u, E, S] to the elasticity field C = CT. Then
fs-uda+fb-udv==fs-uda+fb-udv={S-Edv=fS-Edv.
dB B dB B B B
Proof. We will prove that S • E=S- E; the remainder of the proof is identical
to the proof of Betti's theorem (1). Since C = CT,
S-E = C[E]-E=E-CT[E]=E-C[E]=E-S. Q

102
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 31.
III. Boundary-value problems. Uniqueness.
31. The boundary-value problems of elastostatics. Throughout the following
six sections we assume given an elasticity field C on B, body forces b on B, surface
displacements u on &\, and surface forces s on £f2, where ,¾ and_<%, are
complementary, regular subsurfaces oidB. Given the above data, the mixed problem
of elastostatics1 is to find an elastic state [u, E, S] that corresponds to b and satisfies
the displacement condition
M=fi on &[
and the traction condition
s=Sn =8 on £f2.
We will call such an elastic state a solution of the mixed problem.2 When Sf2
is empty, so that ^ = dB, the above boundary conditions reduce to
u=u on dB,
and the associated problem is called the displacement problem.3 If .5^ = dB,
the boundary conditions become
s=s on dB,
and we refer to the resulting problem as the traction problem.*
To avoid repeated regularity assumptions we assume that:
(i) C is smooth on B;
(ii) b is continuous on B;
(iii) it is continuous on £^;
(iv) s is piecewise regular on £^2.
1 There are problems of importance in elastostatics not included in this formulation; e.g.,
the contact problem, studied in Sect. 40, in which the normal component of the displacement
and the"tarigehtial component of the traction are prescribed over a portion of the boundary.
Another example is the Signorini problem [1959, 13] in which a portion £?3 of the boundary
rests on a rigid, frictionTess "surface; but is allowed to separate from this surface. Thus s is
perpendicular to £fz and at each point of ,5^ either
u-n = o, s-n^o,
or
u-n<o, s • n = o.
A detailed treatment of this difficult problem is given by Fichera [1963, 6] and in an article
following in this volume.
2 Our definition of a solution, since it is based on the notion of an elastic state (§ 28), does!
not cover certain important problems. Indeed, as is well known, there exist genuine mixed '
r problems (,5^ o y2 4= 0) and problems involving bodies with reentrant corners for which the ,.s
corresponding stress field isunbounded. For such problems the classical theorems of elastostatics, N
as presented here, are vacuous. Fichera [1971, 1], Theorem 12.IV has shown that in the case of .-
the genuine mixed problem with C00 boundary, C°° body force field, C00 positive definite and ^
symmetric elasticity field, and null boundary data the displacement field u belongs to H1 (B) a
and is C00 on B —^¾^¾ ■ \Hl (B) is the completion of C1 (B) with respect to the norm defined \ ,
by ||tt||2 = ||u|f+Ypttf§rwhere || • ||2 is the usual L2 norm.] It is possible to establish minor >
variants of most of our results assuming only that ueH1 (B), but this would require analytical
machinery much less elementary than that used here. (In this connection see the article by
Fichera [1971, 1].) Finally, Fichera [1951, 6], [1953, 11] has_shown that for the genuine
mixed problem u will not be smooth, on B unless the. boundary data obey certain hypotheses.
3 Sometimes called the first boundary-value problem.
4 Sometimes called the second boundary-value problem. '

Sect. 31-
The boundary-value problems of elastostatics.
103
Assumptions (ii)-(iv) are necessary for the existence of a solution to the mixed
problem. In the definition of a solution [u, E, S], the requirement that S be
admissible1 is redundant; indeed, the required properties of S follow from (i), (ii),
the admissibility of u, and the field equations.
By a displacement field corresponding to a solution of the mixed
problem we mean a vector field u with the property that there exist fields
E, S such that [m, E, S] is a solution of the mixed problem. We define a stress
field corresponding to a solution of the mixed problem analogously.
(1) Characterisation of the mixed problem in terms of
displacements. Let u be an admissible displacement field. Then u corresponds to a solution
of the mixed problem if and only if
divC[Fw]+b=0 on B,
u = u on £^,
C[Vu] n = s on £f2.
Further, when B is homogeneous and isotropic, these relations take the forms2
/iAu + (X+/i) FdivM + b=0 on B,
u — u on 6^,
fi(Vu + VuT) n+A(divti) n = s on £^.
Proof. That the above relations are necessary follows from the discussion given
at the beginning of Sect. 27. To establish sufficiency assume that u satisfies the
above relations. Define E through the strain-displacement relation and S through
the stress-strain relation. Then, since u is admissible, E is continuous-on B, and,
by assumption (i), S is continuous on B and smooth on B. In addition,
div S+6=0 on B, (a)
Sn=s on £f2.
Further, (a) and assumption (ii) imply that div S is continuous on B; thus S
is an admissible stress field. Therefore [u, E, S] meets all the requirements of a
solution to the mixed problem.
When B is homogeneous and isotropic, the last three relations are equivalent
to the first three, as is clear from Sect. 27. □
In view of the discussion given at the end of Sect. 27, the boundary condition
in (1) on .9^ for isotropic bodies may be replaced by
2/*-7j h [iny. curl w+ A (div u) n = s on ,9"2.
(2) Characterisation of the traction problem in terms of stresses.
Suppose that the elasticity field is invertible with class C2 inverse K on B. Lej,,B_be_
sim^ly^£onn£ciid, and let S be an admissible stress field of class C2 on B. Then S
corresponds to a solution of the traction problem if and only if
divS + b=0l
\ on B,
curlcurlK[S]=Oj
Sn = s on dB.
1 See p. 59.
2 Lodge [1955, 9] has given conditions under which the basic equations of elastostatics
for an anisotropic material can be transformed into equations formally identical with those
for an isotropic material. See also Kaczkowski [1955, 6].

104
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 32.
When B is homogeneous and isotropic, these relations are equivalent to
divS + b
AS+~ilv VV(trS)+2Vb + wv_v (div b) 1
Sn
Proof. Let
E = K[S].
Then E is of class C2 on B and
curl curl E — 0.
Thus, by the compatibility theorem (14.2), there exists a class C3 displacement
field uon B that satisfies the strain-displacement relation. Moreover, we conclude
from (14.4) that u is of class C3 on 5.
When B is homogeneous and isotropic, the last three relations are equivalent
to the first three, as is clear from the remarks given at the end of Sect. 27. □
32. Uniqueness.1 In this section we discuss the uniqueness question appropriate
to the fundamental boundary-value problems of elastostatics.
We say that two solutions [w, E, S] and [u, E, S] of the mixed problem are
equal modulo a rigid displacement provided
[u,E,S]=[u+w,E,S],
where w is a rigid displacement field.
av (1) Uniqueness theorem for the mixed problem.2 Suppose that the
elasticity jidd is positive definite. Then any two solutions of the mixed problem are
equal modulo a rigid displacement. Moreover, if ^ is non-empty, then the mixed
problem has at most one solution.
Proof. Let 6 = [u, E, S] and % = [u, E, S] be solutions of the mixed problem,
and let
[u,E,S]=[,E,S]-[u,E,S].
Then, by the principle of superposition (28.2), [it, E, S] is an elastic state
corresponding to zero body forces. Moreover,
it = 0 on 6^,
s=0 on £f2;
thus, since ^ and SP2 are complementary subsets of dB,
s • u = 0 on dB,
and we conclude from (28.5) that u is rigid. Thus E=S =0, and the two
solutions a and 5 are equal modulo a rigid displacement. &\, since it is a regular surface,
if non-empty must contain at least three non-collinear points. Thus, in this
instance, since u =0 on £flt u must vanish identically. □
It is clear from the proof of (1) that the smoothness assumptions tacit in the
definition of a solution are more stringent than required. For the purpose of this
theorem it is sufficient to assume that a solution [w, E, S] is a triplet of fields
1 This subject is discussed at great length in the tract by Knops and Payne [1971, 2].
2 Kirchhoff [1859, 11, [1876, 1]. See also Clebsch [1862, 7].
= 0
on B,
= 0J
= s on dB.

Sect. 32.
Uniqueness.
105
on B with the following properties: (i) u, E, and S are continuous on B; (ii) u and
/S are differentiable on B; (iii) u, E, and S satisfy the field equations and
boundary conditions.
The last theorem implies that uniqueness holds for the displacement and
traction problems provided the elasticity field is positive definite. When B is
homogeneous, we can prove a much stronger uniqueness theorem for the displacement
problem.
^ (2) Uniqueness theorem for the displacement problem.1 Let B be homo-
geneous, let C be symmetric, and assume that either C or — Cjs_$trongly.dlipiiG,. Then
theliisplacement problem has at most one solution with smooth displacements on B.
Proof. By the principle of superposition, it suffices to show that 6=0on5
and
u=0 on 8B (a)
imply u e= 0 on B. Thus assume that the body forces and surface displacements are
zero. Then by (28.3)
2U{E} = jE-C[E]dv=0, (b)
B
and, since E = Vu, (ii) of (20.1) implies
2U{E} = fVu-C[Vu]dv. (c)
B
Let us extend the definition of u from B to all of S by defining
w = 0 on S—B. ' (d)
Then by (a) u is continuous and piecewise smooth on S, and Vu has
discontinuities only on 8B. Thus u and Vu are both absolutely and square integrable over
$ and possess the three-dimensional Fourier transforms a and A defined on £
by
s
a (*)=(~2ys/eip[x) 'p(v) vu ^ dv» • <f)
s
where p(x) =x— 0. Using (d)-(f) and the divergence theorem, we obtain
A = —ia®p. (g)
1 E. and F. Cosserat [1898, 1, 2, 3] proved^ that^ strong^eHigticity implies uniqueness for
the dispiacenSnFpTo^fem m'aTMmogeneous and isotropic body. (See alsoTredholm [l9067 3],
B6«ST6~[T9077 J]Y aiid Sherman [1938, 2]'.) TKiFresult can" also be inferred from the earlier
work of Kelvin [1888, 4']. For a homogeneous but possibly anisotropic body, theorem (2)
follows from the work of Browder [1954, 4] and Morrey [1954, 14]. The proof given here,
witicfi^jBjbase5^m_the work of Van Hove [1947^ 9], was Jurnished by R. A. Toupin (private
communication in 1961). A slightiy more general theorem was established by Hill [1957, 8],
[1961, 12, 13], who noted that uniqueness holds provided J7{u}>0 for all class C2 fields u
that vanish on 8B but are not identically zero. Hill's result is not restricted to homogeneous
bodies. Hayes [1966, 9] established a uniqueness theorem for the displacement problem
under a slightly weaker hypothesis on C, which he calls moderate strong elhpticity. It is
not difficult to prove that moderate strong ellipticity and strong ellipticity are equivalent
notions when the body is isotropic. Additional papers concerned with uniqueness for the
displacement problem are Muskhelishvili [1933, 2], Misicu [1953, IS], Ericksen and
Toupin [1956,3], Ericksen [1957,5], Duffin and Noll [1958, S], Gurtin and
Sternberg [i960, 7], Gurtin [1963, 10], Hayes [1963, 12], Knops [1964, 13], [1965, 12, 13],
Zorski [1964, 24], Edelstein and Fosdick [1968, 4], Mikhlin [1966. 16].

106 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 32.
Since B is homogeneous, a fundamental theorem of Fourier analysis1 yields
JVu-C [Vu] dv = Cijkl J uitj ukil dv=CilHi f A^A^ dv = JA-C [A] dv, (h)
where A is the complex conjugate of A.
If we write
A^AR+iAt,
a=aR -\-iaJt
then (g) implies that
AK=ajiSp, AI = -aK^>p. (i)
On the other hand, we conclude from the symmetry of C that
A-C[A]=AR-C[AK]+ArC[Al]. (j)
By (c), (d), and (h)-(j),
2 U{E} = f (aR ®p) • C [aR ®p] dv + f (at ®p) • C [a, ®p] dv. (k)
s s
Assume that u does not vanish identically on B. Then by the uniqueness of
the Fourier transform, a cannot vanish identically on i, and we conclude from (k)
that if C is strongly elliptic,
U{E}>0,
which contradicts (b). We also arrive at a contradiction if — C is strongly elliptic.
Thus u = 0 on B. □
For homogeneous and isotropic bodies, a converse to the uniqueness
theorem (2) is furnished by the
if" (3) Cosserat-Ericksen-Toupin theorem.2 Suppose that the material is
hoimgeneous_anA.isQtrQpic, and thatjn^ther the elasticity^tensor nor it&..negativ.e is
tylty&y...?Mptic- Then there exists a hody.B~and.an elastic, state <>= [m, E, S] on J3
with the following properties: 6 corresponds to. zero..body forces and satisfies the.null
boundary condition
m_=_0 _on,j9Z?,
and u is not identically zero.
Proof. By (25.1) we have only three cases to consider:
ju=0, 2 + 2^=0, [X+2fj) /lc<0.
In view of (31.1), it is sufficient to exhibit a body B and a non-trivial class C2
displacement field n on B that satisfies
(iAu + {?l+ix) Fdivu = 0, (a)
u = 0 on dB. (b)
(i)3 Assume X + 2/lc = 0. Then (a) is satisfied by any displacement field of the
form u = V<p. Thus we have only to exhibit a body B and a scalar field <p of class C3
on B such that
F9> = 0 on dB.
1 See, e.g., Goldberg [1961, 9], Theorem 13E.
2 Cosserat [1898,2,3], Ericksen and Toupin [1956,3], Ericksen [1957,5]. See also
Mikhlin [1966, 16].
3 Ericksen and Toupin [1956, 3]. See also Hill [1961, 13].

Sect. 32.
Uniqueness.
107
We take B equal to the open ball of unit radius centered at the origin 0, and
cp(x)=r3(r — l)3, r = |a? —0|.
(ii)1 Assume f/,—0. Then it suffices to find a displacement field u that satisfies
div u = 0 (c)
and (b). The field u = curl w obeys (c). Thus it suffices to exhibit a vector field
w satisfying
curlM> = 0 on dB. (d)
Let B be as in (i) and take
w(x) —r3(r — i)3e,
where e is a unit vector. Then it is easily verified that w satisfies (d).
(iii)2 Assume (A + 2/lc) fi < 0. Let
% (x) = 2/i x\ — (A + 2fi) (x\ + x\) —/*,
u2(x) — u3(x) = 0.
Then it satisfies (a) everywhere and vanishes on an ellipsoid. □
y That the assumption of homogeneity is necessary for the validity of theorem (2)
is clear from the following counterexample of Edelstein and Fosdick. 3 They
take B to be the spherical shell
B={x: n<\x — 0| <2n}
composed of the inhomogeneous but isotropic material defined by
M') = ;~{> +ln ~}, A(r) =-^ -2^(r).
Since /lc>0 and k + 2ft>0, the elasticity field is strongly elliptic on B. They
show that the displacement field u defined in spherical coordinates by
u,(r) = sin r, ue—uY—0,
satisfies the displacement equation of equilibrium with zero body forces and
vanishes on the boundary of B (i.e. when r = n or r = 2n).
(4) Uniqueness theorem for the traction problem for star-shaped
bodies/ Suppose that the body is homogeneous, isotropic, and star-shaded, and let
the Lame moduli /j, and X satisfy
___ _ (A + 2/a)/a<0. ^dth^mi'1"'
1 Ericksen and Toupin [1956, 3]. See also Mills [1963, 20].
1 Cosserat [1898, 2]. This solution was discovered independently by Erick§ijn [1957, 5].
3 [1968,4]. -- '"" ' '
4 Bramble and Payne [1962, 4~\. In terms of Poisson's ratio v (v4=i), theorems (J) and
(4) imply tfitCTTor a star-Shaped body uniqueness holds for the surface force problem provided
/«4=0 and — \<v< 1. The question of whether or not one can drop the hypothesis that B
be star-shaped is open. A partial answer was supplied by Bramble and Payne [1962, 3], who
have shown that uniqueness holds if — 1< v < 1 — iK(i + K)'1, where K is a non-negative
constant that depends on the shape of the body. (See also Hill [1961, 13].) A
counterexample by E. and F. Cosserat [1901, 3] (see also Bramble and Payne [1961, 3]) establishes
lack of uniqueness for — 00 < v < — 1, while counterexamples by E. and F. Cosserat [1901,
2, 4] and Bramble and Payne [1961, 3] indicate non-uniqueness for a countable number of
discrete values of Poisson's ratio in the interval [1, 00). See also Mikhlin [1966, 16]. For
the plane traction problem Sherman [1938, 2] has shown that uniqueness holds if »»#= 1,
/a, 4= 0. That these conditions are also necessary follows from counterexamples of Hill
[1961, 13] and Knops [1965, 12].

108
M.E. Gurtin: The Linear Theory of Elasticity.
Sect. 32.
Then any two solutions of the traction problem with class C2 displacements on B
are equal modulo a rigid displacement.
Proof. It is sufficient to show that
6=0 on B, s=0 on 8B
(a)
imply JS = 0 on B. Thus assume (a) holds. Then (24.5) and the theorem of work
and energy (28.3) yield
U{E} = f[M\E\*+l(trE)*
dv = 0.
Recall Navier's equation
Au+yVdivu — 0,
(b)
(c)
Since B is star-shaped, it follows from (5.1) that there exists a point in B, which
we take to be the origin 0, such that
p-n>0
at every regular point of dB. Consider the identity
f [Au+yVdivu]-(Vu)pdv = - f s-(Vu)pda + 1 U{E}
dB
-yVn)[|JSp+y2-1-(divw)'
dB
(d)
(e)
da,
which can be established with the aid of the divergence theorem, the stress
equation of equilibrium div S =0, and the stress-strain relation in the form
S=2ju
By (a), (b), and (c), (e) reduces to
E + ^- 1 (div w) 1
/ (P •»)
dB
y-\
|^|2+— ---(div u)2
ia=0.
(f)
(g)
Since n • Sn =0 on dB, (f) implies
(y — i) div w = — 2n- En.
Thus, f or y =(= 1, (g) becomes
jVn)[|E|» + -f--(n-En)«
dB
The inequality (A + 2/lc) /lc<0 implies y < — 1; hence
da =0.
— f >0.
y-i
(h)
W
Therefore (h) may be rewritten as
f (p-n) E—U + l/-^--) (n-Enjn®
dB
n
' da=0,

Sect. 33.
Nonexistence.
109
and by (d) this implies
E={\ +1/-^~)(n-£n)n<8)n on dB. (j)
In view of (i) and (j),
n-En=0 on dB,
and hence by (j)
E=0 on dB.
This implies that div u = tr JS = 0 on dS. But div u is harmonic;* hence div u = 0
on S, and we conclude from (b) that J3 = 0. □
^ 33. Nonexistence. For the traction problem a necessary condition for the
^existence of a solution is that"the external forces be in equilibrium, i.e. that
J s da+ f b dv=0,
dB B
f pxsda + f pxb dv=0.
dB B
A deeper result was established by Ericksen2, who proved that, in general,
lack of uniqueness implies lack of existence, or equivalently, that existence implies
uniqueness.3
Suppose there were two solutions to a given mixed problem, and that these
solutions were not equal modulo a rigid displacement. Then their difference
[w, E, S] would have E^O, would satisfy
u = 0 on S^, s=Sn=0 on ,9%, (N)
and would correspond to vanishing body forces. We call an elastic state [w, E, S]
with the above properties a non-trivial solution of the mixed problem
ivith null data.
(1) Nonexistence theorem.* Let the elasticity field be symmetric. Assume
tha^ there exists a non-trivial solution of the mixed problem with\ null data. Then
there exists a continuous body force field b on B. of. class C2 on B with the following
properly: the mixed problem corresponding to this body force field and to the null
boundary condition (N}_has no solution. Further, if Sfx is empty, then b can be
chosen so as to satisfy
fbdv=0, J pxb dv=0.
B B
Proof. Let 3 = [u, E, S] be a non-trivial solution of the mixed problem with
null data. Then if 6 = [u, E, S) is any other elastic state with b the corresponding
bodyTorce field, Betti's theorem (30.1) implies
/ S ■ u da = J s ■ ii, da + / b • u dv,
yt & b
where we have used the fact that 1 corresponds to null data. Nowichoose b = w,
and suppose that a satisfies thejniU^mmdaxy__TOn^tions_(N). Then the above
relation implies
f\u\2dv = 0,
B
1 This will be proved in Sect. 42.
2 Ericksen [1963, 4], [1965, 7].
3 This result is a direct analog of the Fredholm alternative for symmetric linear operators
on an inner product space.
4 The basic idea behind this theorem is due to Ericksen [1963, 4], who established a
theorem similar to (1) for the traction problem and asserted an analogous result for the
displacement problem. Cf. Fichera [1971, 1], Theorem 6.IV (with A = 0).

110 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 34
which yields u = 0, and we have a contradiction. Thus there cannot exist an
elastic state <f with the above properties. ^ ^
Now assume that ^=0. Clearly, if [w, E, S] is a non-trivial solution of the
surface force problem with null data, then [u+to, E, S] also has this property
for every rigid displacement field w. Thus to complete the proof it suffices to
find a rigid displacement w such that
J (u-\-w) dv = 0, J j)X(u+w) dv =0. (a)
B B
Let c denote the centroid of B and I its centroidal inertia tensor; i.e. if
pc(x)=x~c,
then
/ pcdv = 0,
B
B
Let w denote the rigid displacement field defined by
w=a-\-<oxpc,
a = ~ I udv, (o = —I'x I pcxndv.
B B
Then a simple computation based on the identity
/pcx(coxpc) dv=I(o
B
yields
/ (w+to) dv = 0, f pcx(u+w) dv—0,
B B
which is equivalent to (a). □
The non-existence theorem (1) has the following immediate corollary: If the
mixed problem with nidi boundary data has a solution whenever the body force field is
sufficiently smooth, then there is at most one solution to the mixed problem.
IV. The variational principles of elastostatics.
34. Minimum principles. In this section we will establish the two classical
minimum principles of elastostatics: the principle of minimum potential energy
and the principle of minimum complementary energy.1 These principles
completely characterize the solution of the mixed problem discussed previously.
We assume throughout that the data has properties (i)-(iv) of Sect. 31, and,
in addition, that
(v) the elasticity field C is symmetric and positive definite on B.
Let E and S be continuous symmetric tensor fields on B. For convenience,
we now write UC{E}, rather than U{E), for the strain energy:
Uc{E}=%f E C[E]dv.
B
1 The treatment of the minimum principles (1) and (3), which is based on the notion of
an elastic state, was furnished by E. Sternberg (private communication) in 1959.

Sect. 34.
Minimum principles.
Ill
Further, we define the s'*^£J^*^</2/ %{S} by
UK{S}=%fS-K[S]dv,
B
where K = C~l is the compliance tensor. In view of the remarks made on p. 84,
UK{S} = UC{E}
provided S — C[E].
By a fe»*£*!H?^i?JL^J2???^^^^??^e we mean an admissible state that
satisfies the strain-displacement relation, the stress-strain relation, and the
displacement boundary condition. " '?<""<. r^o-J\ z*x*.
(1) Principle of minimum potential energy.1 Let s# denote the_ set of^all
kinematicall_Yadmissible_siaies, and let 0 be the functional on stf defined by
#{d} = UC{E} — f b -udv — f s-uda
B ^
for every 4 = [u, E, S] €jj/. Further, let 4 be a solution of the mixed problem. Then
0{d}^#{d}
for every Ht-stf, and equality holds only if 4=4 modulo a rigid displacement.
Proof. Let 4, 4 ts# and define
4'=4—4. (a)
Then 4' is an admissible state and
\ rwx no**
E'=%(Pu'+Pu'T),
S'=>C[E'],
u' — Q on £fx.
Moreover, S = C[E], since 4£s/; thus (28.1) and (c) imply
UC{E} - UC{E} = UC{E'} + JS-E'dv.
(b)
(c)
(d)
(e)
Next, if we apply (18.1) to S and u', we conclude, with the aid of (b) and (d),
that
f /S • E'dv— f s ■ u' da— f u'■ 6xv S dv.
b &, B
1 It is somewhat difficult to trace the history of this and other variational and minimum j
principles, since in the older work the nature of the allowed variations was often left to be 1
inferred from the result. The basic, ideas appear in the work of Green [1839, 1], pp. 253-256, !
Haughton [1849, 1], P- 152, Kirchhoff.[1850,1], §'i/Kelvin"[i8'6*3, 3], §§61-62, Donati
[1894, 2]. J^agrgyeitheoremitappears.tp„hayefirstbeen giyei.Ej:i.iiyE,[19Q6ll], § 119for the
displacemeivFprpblem, by Hadamard [-1903,3], §264, Colonnetti [1912,1 J, and Prange [1916,
1], PP- 5154 for the traction problem, and by Trefftz [1928, 3], § 18 for the mixed problem.
An extension to "exterior domains was given by'Gurtin and Sternberg [1961, 11],
Theorem 6.5. Variants of this theorem, which are useful in the determination of bounds on
effective elastic moduli for heterogeneous materials, are contained in the work of Hashin
and Shtrikman [1962, 9], Hill [1963, 13], Hashin [1967, 8], Beran and Molyneux
[1966, 3], and Rubenfeld and Keller [1969, 6].
For the displacement problem, the hypothesis that £ be positive definite, is not necessary I
for the validity of the principle of rmnmimrr"potential energy. This fact is clear from the 1^-
work of Hill [1957, 8], Gurtin and Sternberg [i960, 7], and Gurtin [1963, 10]. >
For various related references, see Prager and Synge [1947, 6], Synge [1957, 15],
Mikhlin [1957, 10], and Oravas and McLean [1966, 19].

112
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 34.
Thus
¢{3} - #{«} == UC{E'} - / (div 8 + 6) • W dv + / (s - s) ■ W da, (f)
b sez
and since <) is a solution of the mixed problem, (f) implies
&{Pf-&{4} = Uc{E'}.
Thus, since C is positive definite,
0{<)}^0{3},
0{o} = ¢{3} <=>£',= £-£=().
Moreover, since <) and 5 are kinematically admissible, E=E only when <j=3
modulo a rigid displacement. □
In words, the principle of minimum potential energy asserts that the
difference between the strain energy and the work done by the body forces and
prescribed surface forces assumes a smaller value for the solution of the mixed
problem than for any other kinematically admissible state.
The uniqueness theorem (32.1) for the mixed problem follows as a corollary
of the principle of minimum potential energy. Indeed, let o and 2 be two solutions.
Then this principle tells us that
0(4^ 0{3}, 0{3}^0{o}.
Thus
0{a} = 0{3},
and we conclude from (1) that o and Z must be equal modulo a rigid displacement.
By a kinematically admissible displacement field we mean an admissible
displacement field that satisfies the displacement boundary condition, and for
which div C[Ptt] is continuous on B. The corresponding strain field is then called
a kinematically admissible strain field. If [«, E, S] is a kinematically
admissible state, then u is a kinematically admissible displacement field.
Conversely, the latter assertion implies the former provided E and S are defined
through the strain-displacement and stress-strain relations. In view of these
equations, 0{<s} can be written as a functional &{u} of u:
0{u} = £ f Vu ■ C[Vu] dv—Jb-udv—fs-u da;
b is SP,
therefore we have the following partial restatement of (1)
(2) Let u correspond to a solution of the mixed problem. Then
0{m}^0{«}
for every kinematically admissible displacement field u.
By a statically admissible stress field we mean an admissible stress
field that satisfies the equation of equilibrium and the traction boundary condition.
(3) Principle of minimum complementary energy.1 Let srf denote the
set of all statically admissible stress fields, and let ¥ be the functional on s/ defined by
V{$) =UK{S} -fs-uda
1 The basic ideas appear in the work of Cotteriix [1865, 1] and Donati [1890, 1],
[1894, 2]. As a proved theorem it seems to have first been given by Colonnetti [1912,1]
and Prange [1916, T\ for the surface force problem and by Trefftz [1928, S], §19 for the
mixed problem. See also Domke [1915, T\.

Sect. 34. Minimum principles. 113
for every S^jtf. Let S be a stress field corresponding to a solution of the mixed
problem. Then
for every Sts/, and equality holds only if S=S.
Proof. Let [u, E, S] be a solution of the mixed problem, let S(^/, and define
S' = S-S. (a)
Then S' satisfies
divS'=0 on B,
s'=S'n==0 on ^.
(b)
Since E = K[<S], (a) and an obvious analog of (28.1) imply
UK{S}-UK{S} = UK{S'} + fS'-Edv. (c)
B
Further, in view of (b) and (18.1),
J S' ■ E dv = f s' -uda.
b se,
Thus, since u— u on ,9\,
¥{S}-V{S} = UK{S'}.
Therefore, since K is positive definite, ¥{S} ^ ¥{§}, and W{S} = lF{S} only if
The principle of minimum complementary energy asserts that of all statically
admissible .stress fields, the one belonging to a solution of the mixed problem
renders a minimum the difference between the stress energy and the work done
over the prescribed displacements.
We shall now use the foregoing minimum principles to establish
^/(4) Upper and lower bounds for the strain energy.1 Let U be the strain
energy associated with a solution of the displacement problem. Assume thatjhebody
forces vanish. Then
/ s • u da - UK{S} ^U£ UC{E},
dB
where E is a kinematically admissible strain field, S a statically admissible stress
field, and s the corresponding surface traction.
On the other hand, let U be the strain energy corresponding to a solution of the
traction problem. Then
Jb-udv+ fs-uda- UC{E} ^ U g UK{S},
B oB
where u is a kinematically admissible displacement field, E the corresponding strain
field, and S a statically admissible stress field.
Proof, Let 6 = [u, E, S] be a solution of the displacement problem with
b =0, and let U be the associated strain energy:
U = UC{E} = UK{S}.
1 Cf. Aymerich [1955, 1] who obtains upper and lower bounds on the strain energy by
embedding the body in a larger body.
Handbuch der Physik, Bd. VI a/2. 8

114 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 34.
Then, since Sf% — 0, we conclude from the principle of minimum potential energy
that
U^UC{E}^UC{E}.
On the other hand, since S^ = dB, it follows from the principle of work and
energy (28.3) that the last term in the expression for *F{S} in (3) is equal to
— 2*7. Thus
¥{$} = -U,
and (3) implies
U^ f s-uda-UK{S}.
8B
Next, let o be a solution of the surface force problem. Then 5? = 0, and the
principle of minimum complementary energy yields
U^UK{S}.
Moreover, since S^ = dB, we conclude from (28.3) that the last two terms in
the expression for 0{o} in (1) are equal to —2U. Thus
#{<.} = -C7,
and (1) implies
U^ fb-udv + fs-iida-Uc{E}. □
B dB
Using (18.2) we can write the inequalities in (4) in the following forms:
/ S • {E - %K[S]} dv ^ U ^ UC{E},
B
f E ■ {S- 1C[#]} dv^U^UK{S}.
B
By a statically admissible state we mean an admissible state [«, E, S]
with <S a statically admissible stress field. For convenience, we now define W on
f ^j the set of all statically admissible states o — [u, E, S] by writing !F{<j} =^(8}.
.« Clearly, an admissible state o is a solution of the mixed problem if and only
; = |Cf 3 • i if 6 is both kinematically and statically admissible. We shall tacitly use this fact
I in establishing the next theorem, which furnishes an interesting relation between
*the functionals 0 and W.
i \ (5) Let o be a solution of the mixed problem. Then
0{<i} + W{<s}=O.
Proof. In view of the definition of 0 and W,
0{o} + ¥{<>} = UC{E} + UK{S} -fb-udv-fs-uda-fs-uda
B *i SPX
= 2UC{E} — f b • u dv — f s ■ uda,
B 8B
and the desired result follows from the principle of work and energy (28.3). □
If we assume that the mixed problem has a solution, then the preceding
theorem has the following corollary.
£> (6) Let 4 and 4 be admissible states with Z kinematically admissible and a
statically admissible. Then
0{4}+W{t}^O.

Sect. 35- Some extensions of the fundamental lemma. 115
Proof. Let o be a solution of the mixed problem. Then the principles of
minimum potential and complementary energy imply that
Thus
0{3} + TO^W + TO=O. □
35. Some extensions of the fundamental lemma. In this section we shall
establish three lemmas whose role in elastostatics is analogous to the role of the
fundamental lemma in the calculus of variations. Let dB(x0, h) denote the
intersection of dB with the open ball Eh (x0) of radius h and center at x0, and let
<&h(x0) denote the set of functions defined in Sect. 7.
(1) Let W be a finite-dimensional inner product space. Let w: S/'^-^-'W be
piecewise regular and satisfy
J w -v da=0
for every class C°° function v: B ~>Wthat vanishes near ^. Then
w=0 on £f2.
Proof. Let e,, e2,..., eB be an orthonormal basis for W, and let
H
»=i
Assume that for some regular interior point x0 of S^ and some k, wk (x0) > 0
(say). Then there exists an h>0 such that dB(x0,h) is contained in S^2 and
wk>0 on dB(x0,h). Let v = q>ek with q>€&h(x0). Then v is of class C°° and
vanishes near S^. In addition,
f w -v da>0,
which is a contradiction. Thus w(x0) =0 at every regular interior point x0 of
y%. But since w is piecewise regular, this implies w=0 on £f2. □
(2) Let u be a piecewise regular vector field on £flt and suppose that
f (Sri) -uda=0
for every class C°° symmetric tensor field S on B that vanishes near £?%. Then
u = 0 on S^.
Proof. Since S is symmetric,
(Sri) u=S- (u®n)=S-T,
where T is the symmetric part of u®n:
T = £ (u <g) n +n <g) u).
Therefore
jS-Tda=0
whenever S satisfies the above hypotheses. Consequently, letting #"in (1) be
the space of all symmetric tensors, we conclude from (1) that
T = 0 on 9{,
8*

116 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 36.
and hence that
2Tn=w+n(u • n) =0 on yx.
Taking the inner product of this relation with n, we find that u ■ n = 0 on Sflt
and this fact and the above relation imply «=0on^, □
(3) Let ube a piecewise regular vector field on SP2 • an^ suppose that
f u ■ div S da =0
for every class C°° symmetric tensor field S on B. Then
«=0 on Sf%.
Proof. Let v be an arbitrary class C°° vector field on B which vanishes near
,¾. Then by (6.5) there exists a class C°° vector field g on B with the property
that
Ag =v.
Let1
S=Vg + FgT-ldivg.
Then <S is a class C°° symmetric tensor field on B, and hence
/ u ■ divSi«=0.
But a simple calculation yields
div<S = Ag =v.
Thus
/ u- v da=0
for every class C°° vector field v on B which vanishes near 6^, and the desired
result follows from (1). □
36. Converses to the minimum principles. In this section we shall use the
results just established to prove converses to the principles of minimum potential
and complementary energy. We continue to assume that hypotheses (i)-(v) of
Sects. 31 and 34 hold. Our first theorem shows that if a kinematically admissible
state minimizes the functional 0, then that state is a solution of the mixed
problem.
Lf- (1) Converse of the principle of minimum potential energy. Let 4 be
a kinematically admissible state, and suppose that
for every kinematically admissible state 3. Then 6 is a solution of the mixed problem.
Proof. Let u' be an arbitrary vector field of class C°° on B, and suppose that
h' vanishes near S^. Further, let 0' = [«', E', S'], where
E'=^(Fu' + Fu'T),
S'=C[E'].
Then 3 = 0 + 4' is kinematically admissible, and it is not difficult to verify that
(f) in the proof of (34.1) also holds in the present circumstances. Thus, since
0^Uc{E'}- f (div S + b) -u'dv + f(s-s) u' da.
1 Cf. the remarks following (17.8).

Sect. 36.
Converses to the minimum principles.
117
Clearly, this relation must hold with u' replaced by aw' and E' by «.E'; hence
0 ^«?UC{E'} -a / (div S +b) ■ u' dv + a / (s - s) • u' da
b y,
for every scalar a, which implies
— / (div S + 6) • u' dv + / (s — s) • w' rfa =0 (a)
for every C°° vector field u' that vanishes near S^. If, in addition, u' vanishes
near dB, then .,,. „ , ,.
/(divS + b) w' dv=0,
B
and we conclude from (7.1) with #"= y that
divS + b=0. (b)
By (a) and (b), .,
^.
for every C°° field w' that vanishes near <ylt and (35.1) with #^= y yields
s = s on S?2. (c)
Thus a is a kinematically admissible state that satisfies (b) and (c); hence a is a
solution of the mixed problem. □
Note that (1) does not presuppose the existence of a solution to the mixed
problem. If onelthows a priori that a solution ? exists,1 then it follows from (34.1)
tnat any kinematically admissible state o that minimizes 0 must be equal to
1, and hence must be a solution. Indeed, if ? minimizes 0, then
0{?}^0{a};
but (34.1) implies
0{<S}^0{4},
so that
0{«}=0{3},
and we conclude from (34.1) that d=7 (modulo a rigid displacement).
The next theorem yields a converse to the principle of minimum complemen-.
tary energy. Toj3roye„iMs„theo^
and convex with respect to S/\ (when ^4^3). In view of the discussion given in[
the preceding paragraph, if B and the boundary data are such that existence
holds for the corresponding mixed problem,1 then this converse follows trivially
without the above assumptions concerning B.
(2) Converse of the principle of minimum complementary energy.2
Assume that B is simply-connected and convex with respect to £/^ and that K is of
class C2 on B. Let S be a statically admissible stress field of class C2 on B, and
suppose that „
for every statically admissible stress field S. Then S_is.Q stress field corresponding
to a solution of the mixed problem.
Before proving this theorem we shall establish two subsidiary results; these
results are not only basic to the proof of (2), but are also of interest in themselves.
1 Existence theorems for the mixed problem are given, e.g., by Fichera [1971, 1], § 12.
2 The basic ideas behind this theorem are due_to_CP-TTERILL [186;;, 1], Donati [1890, 1],
[1894,2], Domke [1915,-Z], Southwell [193673]T[1938, 5], Locatelli [1940,3,4], Dorn
and Schild [1956, 1]. As a proved theorem it seems to have first been given by Sokolni-
koff [1956, 12] for the traction problem and by Gurtin [1963, 11] for the mixed problem.

118
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 36.
(3)1 Let w be a continuous vector field on a regular subsurface Sf of dB, and
let B be convex with respect to Sf'. Further, suppose that
f (Sri) ■ wda = 0
for every class C°° symmetric tensor field S on B that vanishes near dB — Sf and
satisfies
divS = 0.
Then w is a rigid displacement field.
Proof. Let x and x be regular points of Sf, and choose a cartesian coordinate
system such that the coordinates of x are (0, 0, 0) and those of x are (0, 0, xa).
Let De be the closed disc in the x1, %,-plane with radius e > 0 and center at (0, 0),
and let fe be a class C00 scalar field on the entire xlt %-plane with the following
properties:
(«) /e^o;
(/3) /,=0 outside D,;
(?) //.<*« = !•
Such a function is easily constructed using the procedure given in the first
paragraph of Sect. 7. Clearly, the symmetric tensor field S on E defined by
[«(*)]
0 0 0
0 0 0
0 0 /,(¾. *,)
has zero divergence. Let C, be the infinite solid circular cylinder whose axis
coincides with the and whose cross-section is De. Then the assumed con-
1 Gurtin [1963,11].
Fig. 10.

Sect. 36. Converses to the minimum principles. H9
vexity of B with respect to £f and the regularity of Sf imply that for all
sufficiently small e there exist disjoint subregions SPe and 5^ of Sf such that (see
Fig. 10)
xe^s, xa<$?e, Ct^dB = £yj&t.
Thus S vanishes near dB —,SP and
0=/ (Sn) w da = f few3n3da-\- f few3n3da. (a)
Next, by property (y) of fe,
ffen3da = ffeda=\, j,
& c° r
f fEn3dar=- f feda = -\.
Thus if we let e->0 in (a) and use property (a) of fe in conjunction with the mean-
value theorem of integral calculus, we conclude that
w3(x)-w3 (2)=0,
or equivalently that
[w(x) —w(x)] ■ [■*—*] =0.
Since x and x are_ arbitrary regular points of y, and since 145 is continuous on
^w has the projection property.on £f. Moreover, from our hypotheses it follows
that SP is a non-coplanar point set; hence we conclude from (13.3) that w is
rigid. Q ,
(4)1 Let it be a continuous vector field on a regular subsurface £P of dB, and
let B be simply-connected and convex with respect to £f. Let E be a symmetric tensor
field that is continuous on B and of class C2 on B. Further, suppose that
f (Sn) -uda = f S-Edv
ST B
for every class C°° symmetric tensor field S on B that vanishes near dB —£f and
satisfies
divS = 0.
Then there exists an admissible displacement field u such that
j E = %(Vu + VuT), ■
\u — u on Sf.
Proof. Let Q> denote the set of all class. C°° symmetric tensor fields on B that
satisfy divTT^T'By "hypothesis
fS-Edv=0
B
for every ScS> that vanishes near dB. Thus we conclude from Donati's theorem
(18.8) that E satisfies the equation of compatibility. In view of the compatibility
theorem (14.2), this result and the fact that B is simply-connected imply the
existence of an admissible disp^cemelrFfieTajir"such"that E and u' satisfy the
1 For the case in which Sf=dB this theorem is due to Dorn and Schild [1956, 1]. The
present more general case is due to Gurtin [1963, 11].

120 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 37.
g^-arn-Hjcplarprnpnt relation. Thus we conclude from the present hypotheses and
(18.1) that for every St@> which vanishes near dB — £f,
f (Sn) -u'da= f (Sn) u' da = f S ■ Edv = f (Sn) -uda.
if 8B b se
Therefore if we let
w = u — u',
then
J (Sn) -wda—0
sr
for every Se@ that vanishes near dB —SP. Thus we may conclude from (3) that
w is the restriction^to SP of a rigid displacement field w, and the displacement
field u defined on B through
u=u' -\-w
has all of the desired properties. □
We are now in a position to give the
Proof of (2). Let S'a@ vanish near Sf2- Then S=S-\-S' is a statically
admissible stress field, and
¥{&}£¥{§}. (a)
Let
E = K[S].
Then (c) in the proof of (34.3) holds in the present circumstances, and we
conclude from this result, (a), and the definition of IF given in (34.3) that
0^W{S}-W{S} = UK{S'}+ f S'-Edv- fs'-uda. (b)
B S^
The inequality (b) must hold for every S'e@ that vanishes near £f2. If we replace
S' in (b) by a<S', where a is a scalar, we find that
f S' -Edv=f s'-uda
B y,
for every such field S'. Thus we conclude from (4) that there exists a vector
field u such that \u, E, S] is a solution of the mixed problem. □
■QvJ/37. Maximum principles.1 The principles of minimum potential and
complementary energy can be used to compute upper bounds for the "energies" ¢{0} and
^{0} corresponding to a solution 4. In this section we shall establish two maximum
principles which allow one to compute lower bounds for ¢{0} and lF{o}.
We continue to assume that the hypotheses (i)-(v) of Sects. 31 and 34 hold.
As our first step, we extend the domain of the functionals 0 and W to the set
of all admissible states in the obvious manner:
¢{0} = UC{E} — / b -udv — f s -uda,
b st,
W{<i} = UK{S}-fs-uda
for every admissible state a = [u,E,S].
1 Tig_basic ideas underlying these principles are diifi_ta_TREFFTZ [1928, 4], who estab-
\ lished analogous results for boundary-value problems associated with the equations A u = 0
"">and AAu = 0. See also Sokolnikoff [1956,12], § 118 and Mikhlin [1957, 10], §§55, 57, 59-

Sect. 3 7. Maximum principles. 121
(1) Principle of maximum potential energy.1 Let 0 be a solution of the
mixed problem, and let 3 be an elastic state that satisfies
f s • (u—u) daT^O,
»\
s = s on £?%.
Then
and equality holds if and only if 0 = 3 modulo a rigid displacement.
Proof. Let
o' = <i — 3.
Then (28.1) and the fact that S = C[E] imply
UC{E} - UC{E} = UC{E'} +JS-E' dv.
B
Next, if we apply the theorem of work expended (18.2) to S and u', we find that
f S ■ E' dv — f b ■ u' dv-\- f s ■ u' da.
B B dB
Thus, since u = u on Sfx and s = s on £f2,
#{«} - ¢{3} = UC{E'} +fs-{u-u)da.
This completes the proof, since C is positive definite and the last term non-
negative. ' □
It is clear from this proof that (1) continues to hold even when 4 is required
only to be kinematically admissible.
Note that in the principle of minimum potential energy the admissible states
were required to satisfy, in essence, only the displacement boundary condition.
On the other hand, the "admissible state" 3 in the principle of maximum potential
energy is required to satisfy all of the field equations, the traction boundary
condition, and a weak form of the displacement boundary condition.
(2) Principle of maximum complementary energy. Let 0 be a solution
of the mixed problem, and let 3 be an elastic state that satisfies
u = u on S^,
f (s — s) ■ u, da'StO.
sr.
Then
«F{<i}^«F{3},
and, equality holds if and only if ¢ = 2 modulo a rigid displacement.
Proof. Let
<(' = <> — 3;
then 0' is an elastic state corresponding to zero body forces. As before,
UK{S} - UK{$} = UK{S'} +fS'-Edv,
B
1 COOPERMAN [1952, 1].

122
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 38.
and by (18.1) and the fact that div S' = 0,
f S' • Edv = f s' ■ uda.
B SB
Thus, since ii = u on S^_ and s = s on Sf_,
W{i}-W{i) = UK{S'} + f(s-s)-u da,
which, in view of our hypotheses, implies the desired result. Q
Note that (2) remains valid under the weaker hypothesis that 4 be statically
admissible.
^\ 38. Variational principles. The admissible states appropriate to the minimum
principles of elastostatics are required to meet certain of the field equations and
boundary conditions. Injwrneji£gliciitia^ use variational
principles in which the~aHrn5sible states satisfy as few constraints~as~possTBle.
IrTtKs'sectibn'we establish two such principles.*
We continue to assume that the data has properties (i)-(iv) of Sect. 31, but
in place of (v) of Sect. 34 we assume that
(v') the elasticity field C is symmetric.
The two variational principles will be concerned with scalar-valued functionals
whose domain of definition is a subset s4 of the set of all admissible states. Let A
be such a functional, let
4 and 3 be admissible states,
(a)
4-\-Xt e.s/ for every scalar A,
and formally define the notation
d7A{4} = d\A{4 + M}U=0.
Then we write
if 32A{4^xists_and equals zerojor every choice pLl-consistent-wit-h (a).
We begin with a variational principle in which the admissible states are not
required to meet any of the field equations, initial conditions, or boundary
conditions.
(1) Hu-Washteu principle.2 Let jj/ denote the set of all admissible states,
and let A be the functional on jj/ defined by
A{<s} = Uc{E}- JS-Edv- f (divS + b) -udv
B B
+ f s ■ uda-\- f (s — s) -uda
_____ s\ -si
1 The basic ideas behind_these.Ennciples are contained in. the. yrQric._fflLBQim [1906, 1],
pp. 91—97, who established analogous_results. for t.he„plane.elastica^.Born was cognizant of
the fact thatTus-results applied to elasticity theory, as is clear from bis statement (p. 96):
„Ich will hier bemerken, daS sich der Vorteil dieser Darstellung eigentlich erst zeigt, wenn
man sie auf die allgemeine Elastizitatstheorie anwendet." See also Oravas and McLean
[1966, 19], pp. 927-929 for a detailed study of the early historical development of these
principles.
2 Hu [1955, 5],Washizu [1955,-74], [1968,25], § 2.3. See also de Veubeke [1965,6],Tonti
[1967, 16, 17], and HlAvaCek [1967, 6, 7]. An extension valid for discontinuous displacement
and stress fields was given by Prager [1967, 11], and one in which the stress fields of the
admissible states are not required to be symmetric was given by Reissner [1965, 16].

Sect. 38. Variational principles. 123
for every 4 = [u, E, S] £s/. Then
(5/l{o} = 0
at an admissible state 4 if and only if 4 is a solution of the mixed, problem.
Proof. Let 4 = [u, E, S] and 5 = [u, E, S] be admissible states. Then 4 + X 4 is/
for every scalar X, and in view of (28.1) and the symmetry of C,
UC{E + IE) = C7C{£;} + A2 C/c{£} + A / B • C [E] dv.
Thus, since
d7A{4}=-^A{4+X~4}\^0,
it follows that
d7A{4}=J{(C[E]-S) ■ E-(div S + b) ■ u-S ■ E -u ■ div §} dv
B
-\- f s ■ u da -{- f {s • u + (s — s) ■ u} da.
If we apply (18.1) to S and u, we find that
/ u • divSdv= f u -sda— f £(Vu + VuT) ■ Sdv;
B dB B
thus
d~tA{4) = f {C[E] -S) -Edv-f (div S + b)-udv
B B
+ f{i(Vu + VuT)-E}-Sdv+f(u-u)-sda (a)
B S^
+ f (s — s) ■ uda.
If 4 is a solution to the mixed problem, then (a) yields
(5y/4{<f}=0 for every 3€j2/, (b)
which implies
(5/1(4=0. (c)
To prove the converse assertion assume that (c), and hence (b), holds. If we
choose 3 = [u, 0, 0] and let u vanish near dB, then it follows from (a) and (b) that
/(divS + b) -udv=0.
B
Since this relation must hold for every such u of class C1 on B, we conclude from
the fundamental lemma (7.1) that div-S + b=0. Next, let 3 = [u, 0,0], but
this time require only that u vanish near &[. Then (a) and (b) imply that
/ (s — s) • uda=0,
sr,
and we conclude from (35.1) that s —s on ,¾. Now, let 3 = [0, E, 0] and suppose
E vanishes near dB. Then by (a) and (b),
/ (C[E]-S)-Edv=0.
B
Thus, since C[E]—S and E are symmetric tensor fields, it follows from (7.1)
with W equal to the set of all symmetric tensors that S = C[E]. In the same

124 M. E. Gurtin: The Linear Theory of Elasticity. Sect. 38.
manner, choosing 2 = [0, 0, S], where S> vanishes near dB, we conclude that
JB= ^(Vu-\-VuT). Finally, if we drop the requirement that S vanish near dB,
we conclude from (a) and (b) that
f (u—u) ■ (Sn) da = 0
for every class C1 symmetric tensor field S on B, and it follows from (35.2) that
u=u on S^. Thus 4 = [u, E, S] is a solution of the mixed problem. □
(2) Hellinger-Prange-Reissner principle.1 Assume that the elasticity
field is invertihle and that its inverse K is smooth on B. I^e^_^_denotethesetof_all
admis0M£^totesJhgi,£gtisly_J}}e strain^dis^lacement relation, and let 0 be the
functional on s/ defined by ~~
©{4} = UK{S} — f S • Edv + f b -udv + f s ■ (u—u) da-\- fs -uda
b b srt srt
for every <s = [u, E, S] {.stf. Then
at ids/ if and only if <s is a solution of the mixed problem.
Proof. Let <s = [w, E, S] and 3 = [ft, E, S] be admissible states, and suppose
that <f + A3c.K/ for every scalar X, or equivalently that 4, i£s4. Then, in view
of the symmetry of C, K is symmetric and
d7 &{<>} = f {(K [S] -E)-S-S -E + b-u}dv
B
+ / {s • (u—u) +s • u} da-\- f s ■ uda.
If we apply (18.1) to S> and u and use the fact that E and u satisfy the strain-
displacement relation, we find that
fS-Edv = fs-uda— f u, • div S dv;
B dB B
thus
676{i) = f (K[S] -E)-Sdv+ f (div S + b) ■ u dv
B B
+ / (u—u) ■ s da + / (s — s) • u da. (a)
If 4 is a solution of the mixed problem, then, clearly,
<5<6>{<)}=0 for every ^Gk/, (b)
which implies 66(4) =0. On the other hand, (a), (b), (7.1), (35.1), (35.2), and
the fact that 4 satisfies the strain-displacement relation imply that 4 is a solution
of the mixed problem. □
Let 4 = [u, E, S] be kinematically admissible. Then we conclude from
(18.1) that
/ u • div Sdv = f s • uda-\- f s uda — f S • Edv.
B if\ £^2 B
1 The basic idea is contained in the work of Heixinger [1914, 1]. As a proved theorem
it was first given by Prange [1916, 1], pp. 54-57 for the traction problem and Reissner
[1950,20], [1958, 14], [1961, 18] for the mixed problem. See also RUdiger [1960,20],
[1961, 19], HlAvaCek [1967, 6, 7], Tonti [1967, 16,17], Horak [1968, 8], Solomon [1968,12].

Sect. 39.
Convergence of approximate solutions.
125
Thus, in this instance, A{i}, given by (1), reduces to 0{<s}, where 0 is the
functional of the principle of minimum potential energy (34.1). Also, when i is kine-
matically admissible,
tfS-Edv.-=Uc{E} = UK{S},
m
and 0{<s} given by (2) reduces to — 0{<f}. On the other hand, if <f is a statically
admissible state that satisfies the stress-strain relation, then A{o} reduces to
— ^{i}, where *P is the functional of the principle of minimum complementary
energy (34.3); if i is a statically admissible state that obeys the
strain-displacement relation, then 0{<s} reduces to ^(<f}.
Table 5 below compares the two variational princi£les_established here|_with_the
minimum principles discussed "previously.
Table 5.
Principle
Minimum potential energy
Minimum complementary energy
Maximum potential energy
Maximum complementary energy
Hu-Washizu
Hellinger-Prange-Reissner
Field equations satisfied
by the admissible states
Strain-displacement
stress-strain
Stress equation
of equilibrium
All
All
None
Strain-displacement
Boundary conditions
satisfied by the
admissible states
Displacement
Traction
Traction, weak form
of displacement
Displacement,
weak form of traction
None
None
^ 39. Convergence of approximate solutions. Recall that the functional of the
principle of minimum potential energy (34.2) has the form
where
0{u) = U{u} — / b -udv — f s -uda,
U{u} =½ f Vu ■ C[Vu] dv
(a)
is the strain energy written, for convenience, as a functional of the displacement
field.
The standard method of obtaining an approximate solution to the mixed
problem is to minimize the functional 0 over a restricted class of functions.1
That is, one assumes an approximate solution uN in the form
;«jv + ZX/»'
(b)
where /i,/2, •••,/# are given functions that vanish on £flt and uN is a function
that ^approximates the boundary data_w on 5^. Of course, the teimu^'is omitted
when ,¾ is empty. The constants a1( a2,..., a.N are then chosen so as to render
¢{^} a minimum. Indeed, if we write
0(^,0^,...,0^)=0(¾}.
1 This idea appears first in the work of Rayleigh (1877) [1945, 6] and Ritz [1908, 2]; the
method is usually referred to as the Rayleigh-Ritz method.

y~
-126 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 39.
where uN is given by (b), then (assuming that C is symmetric)
N N
0(^,0.2, ...,0CN)=a + l 2 Dmn<*-m<*-n + 2 4¾.
m, k=1 n=l
where
0-,.=/ FA.-ciy/jif,, (c)
a — U{uN} — / b -uNdv — f s • uN,
b £
dn=JVuN ■ C[Vfn] dv-fb-fndv-fs.f„da. (d)
B S Sf,
If ^ is empty, the above relations still hold, but with uN = 0, so that
a=0,
d„ = - fb-fndv- fs-f„da. (e)
B dB
If C is positive semi-definite, the matrix [Dmn] will be positive semi-definite, and
¢(0¾. a2,..., oln) will be a minimum at 0^ = 0¾. a2=a2, ..., v.s = v.N if and
only if 6¾. bt2,..., a a- is a solution of the following system of equations:
2A»»«. = -4. (m=H, 2,...,iV). (f)
»=1
We will now estabHsh conditions under which solutions of (f) exist, and under
which tHe~resuTting approximate solutions uN converge in energy to the actual
solution as N'->66.'
We assume for the remainder of this section that C is symmetric and positive
definite, and that hypotheses (i)-(iv) of Sect. "}i hold.
We write #„ for the set of all continuous and piecewise smooth vector fields
on B. ~=
(1) Existence of approximate solutions. Let ^j^be an ^-dimensional sub-
space of #„ with the property that each /&^jv vanishes on Sfx,
(i) If ^ 4= 0, let uN be a given function in #„, and let £FN be the set of all functions
f of the form
f=g+uN, g^N.
(ii) If ^ = 0, let #N = &N.
Then there exists a function uNd3FN such that
¢ {uN} sS ¢ {»} for every ve^N. (g)
If 5^4=0, then uN is unique; if ^=0, then any two solutions of (g) differ by a
rigid displacement. Finally, uN is OjpJimalinJhe following, sense: If u is the
displacement field corresponding to a solution of the mixed problem, then1
U{u — uN} = inf U{u — v}. (h)
Proof. Let/1,/2, ...,fN be a basis for 3FN, and let Dmn and dn be defined by
(c), (d), and (e). Then, clearly, to estabHsh the existence of a solution uN of (g) it
suffices to estabHsh the existence of a solution 0¾, a2,..., aN of (f).
1 Cf. Schultz [1969, 7], Theorem 2.4.

Sect. 39. Convergence of approximate solutions. 127
Since C is positive definite, we conclude from (c) that
N
2 A»»«,»a»^0 (i)
for any iV-tuple (¾. a2,..., a^). Assume that
A7
2 A»»oc»a» = 0, (j)
m, «=1
AT
/=20»/»- (k)
and let
n=l
Then it follows from (c) and (j) that
fVf.C[Vf]dv^O,
B
and since C is positive definite, this, in turn, implies that / is rigid. Assume first
that 5^4=0. Then /sO, since it is rigid and vanishes on 3^- Consequently, as
/i . /a» • • •. /w is a basis for 3PN, this implies that
ai = aa= -•• =ajv=0. (1)
Therefore, if ^4=0, then (j) implies (1), and we conclude from (i) that the matrix
[Dmn] is positive definite 'L hence in this instance"^has~a unique solution.
Assume next that S^ — 0. Suppose that
N
ZDmn*n = 0'> (m)
»=1
then (j) holds, and hence f defined by (k) is rigid. Thus any two solutions of (g)
(if they exist) differ by afigicl displacement. Next, by (e) and (k),
N
2^4=-/ b -fdv — f s -fda,
«=1 B dB
and it follows1 from (18.3) that
N
2«»^„=o.
»=1
Thus \^i,dz,..., dN) is orthogonal to every solution («x, a2, •••, a#) of the homo-
geneous_eci.Tja.tiQn (m),. and. we conchide from the Fredholm alternative that the
inriornogeneous equation (f) has a solution.
We have only to show that 1% is optimal. To facilitate the proof of this
assertion, we now establish the following
(2) Lemma.2 Let [u, E, S] be a solution of the mixed problem. Then
0{v} — <P{u} = U{v —u} + / s-{v—u)da
for every ve^0.
Proof. Let ^
u' = v— u, E'=Vu'.
1 Here we tacitly assume that (when £^ =0) the external force system [6, s] is in
equilibrium.
2 Cf. Tong and Pian [1967, 15], Eq. (2.16).

128
M. E. Gurtin: The Linear Theory of Elasticity.
Sect. 39.
Then by (28.1),
U{v} - U{u} = U{u'} + fS-E'dv.
B
Further, since div S + b = 0, we conclude from (18.2) that
f S ■ E' dv — f s u' da + / b ■ u' dv.
B SB B
The last three equations and (a) imply the desired result, since u=u on 5^ and
s = s on Sf2. □
It is of interest to note that the principle of minimum potential energy in the
form (34.2) follows as a direct corollary of this lemma. Indeed, if v is kinematically
admissible, then »=uon £fx, and (2) implies that 0{v}^. &{u).
We are now in a position to complete the proof of (1). Assume that 5jH=0.
By hypothesis, every ve^N satisfies
v =uN on SP-l-
Thus, since uNe^N, it follows from (2) and (g) that
U{uN — u} = &{uN) — 0{u} — f s ■ (uN — u) da
^0{v} — 0{u} — / s • (uN—u) da = U{v—u}
for every ve&N. Thus
U{uN — u} <L U{v — u},
which implies (h). The proof when S^ =0 is strictly analogous. □
We now establish the convergence of the approximate solutions established
in (1). Thus for each N=i, 2, ... let uN and 3FN satisfy the hypotheses of (1).
We assume that every sufficiently smooth function that satisfies the displacement
boundary condition can be approximated arbitrarily closely in energy by a
sequence of elements of &\, &2, More precisely, we assume that
(A) given any kinematically admissible displacement field u, there exists a
sequence {vN} with vNe&N such that
U{u— »jv}->-0 as iV-»-oo. (n)
(3) Approximation theorem.1 For each N= 1, 2,... let uN and SFN satisfy
the hypotheses of (1), and assume thatj^yjiqld.s. Let {uN} be ~a~sequence of
approximate solutions, i.e. solutions of (g), and let u be the displacement field
corresponding to a solution of the mixed problem. Then
U{u—uN}^~0 as JV^-oo. (o)
1 Most of the ideas underlying this theorem are contained in the following works: Courant
and Hilbert [1953, 8], pp. 175-176; Mikhlin [1957, 10], pp. 88-95; Friedrichs and Keller
[1965, 8]; Key [1966, 11]; Tong and Pian [1967, 15]. These studies contain general results
on the convergence of the Rayleigh-Ritz and finite element methods.
In the usual applications of the Rayleigh-Ritz procedure
&it<#it+i- (*)
The abstract formulation given here also includes the finite element method (see, e.g.,
Zienkiewicz and Cheung [1967, 19], Tong and Pian [1967, 15], Zienkiewicz [1970, 4]) for which
(*) is not necessarily satisfied.

Sect. 40.
Statement of the problem. Uniqueness.
129
Proof. By (A) there exists a sequence {vN} with vNiJ^N such that (n) holds.
On the other hand, (h) and the fact that C is positive definite imply
0 ^ U{u — uN} sg U{u — vN},
and the desired result is an immediate consequence of (n). □ v
V. The general boundary-value problem. The contact problem.
40. Statement of the problem. Uniqueness. In the contact problem of elasto-
statics1 the normal component of the displacement and the tangential component
of the surface traction are prescribed over a portion 5¾ of the boundary:
u-n=u and s — (s-n)n=s on £f3,
where u and s are prescribed with s tangent to £f3. These relations can also be
written in the alternative forms:
(n®n)u=un and (1—n®n)s=s on S^3.
We now generalize this boundary condition as follows. We assume given a
tensor field P on 8B whose value P(x) at any xtdB is a perpendicular projection.
The generalized boundary condition then takes the form:
Pu—u and (1— P)s=s on dB,
where u and s are prescribed vector fields. Since2
(1-P)P=P(1-P)=0,
u and s must satisfy the consistency condition
(1—P)u = 0 and Ps = 0 on dB.
At a point * for which P(x) =1 we have u(x) =u(x), and no restriction is
placed on s(x). Thus displacements are prescribed over the subset of dB on
which P = 1; similarly, surface tractions are prescribed over the subset for which
P = 0. The mixed problem therefore corresponds to
P = l on ,9^, P = 0 on £f2,
where &[ and 5P2 are complementary subsets of dB.
If P=n(g)n at a point, then the normal displacement and the tangential
traction are prescribed. Thus the contact problem corresponds to situations in
which
P = l on £^, P = 0 on Sfit P=n®n on Sf%,
where £^, £"2, and 6^ are complementary subsets of dB.
In view of the preceding discussion, the general problem may be stated as
follows: given an elasticity field C and a body force field b on B together with
P, u, and s on 8B; find an elastic state [u, E, S] that corresponds to b and
satisfies the generalized boundary condition. We call such a state a solution of
the general problem.
1 Sometimes called the mixed-mixed problem.
2 See p. 9-
Handbuch dcr Physik, Bd. VIa/2. 9

130 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 41.
(1) Uniqueness theorem for the general problem.1 Assume that the
elasticity field is positive definite. Then any two solutions of the general problem are
equal modulo a rigid displacement.
Proof. Let 3 and ? be solutions of the general problem, and let
<s = [u,E, S]=? —?.
Then <s is an elastic state corresponding to zero body forces. Moreover,
Pw = (l-P)s = 0 on dB;
thus on dB
s-u = [(l-P+P)u] • [(1-P+P)s] = [(l-P)u] -Ps = [P(l—P)u] • s = 0.
Therefore we conclude from (28.5) that u is rigid and E =S = 0. □
Of course, (1) yields, as a corollary, the uniqueness theorem (32.1) for the
mixed problem.
41. Extension of the minimum principles. The minimum principles established
in Sect. 34 are easily extended to the general problem stated in the previous
section.2 For example, if we define a kinematically admissible state to be an
admissible state [u, E, S] that satisfies the strain-displacement relation, the stress-
strain relation, and the boundary condition
Pu=u on dB, (a)
then the principle of minimum potential energy (34.1) remains valid for the
general problem provided we let
<&{4} = UC{E}— J b -udv — /s-uda.
B dB
In view of the consistency condition and the remarks on p. 9,
s-u = (l-P)s-u = (l-P)s- (1-P)u,
and the integral over 8B can also be written in the form:
f(l-P)s-(l-P)uda. (b)
dB
For the mixed problem (P = l on Sfx, P = 0 on £?2) the boundary condition (a)
reduces to u = u on ^, and the integral (b) reduces to an integral over y2. Thus
in this instance the extended minimum principle is nothing more than the
traditional theorem (34.1).
On the other hand, if we define a statically admissible stress field to be an
admissible stress field S that satisfies the equation of equilibrium and the boundary
condition
(l-P)Sn = s on dB,
then the principle of minimum complementary energy (34.3) remains valid for
the general problem3 provided we let
W{S} = UK{S}-fs- uda,
dB
1 Cf. Sternberg and Knowles [1966, 24]. Other uniqueness theorems which are not
special cases of (1) are given by Bramble and Payne [1961, 5] and Knops and Payne
[1971,2].
2 Cf. Rudiger [I960, 10], Prager [1967, 11], HlavaCek [1967, 7].
3 An extension of the principle of minimum complementary energy to the mixed-mixed
problem was given by Sternberg and Knowles [1966, 24]. See also Hlavacek [1967, 7].
The result stated here includes, as a special case, the theorem of Sternberg and Knowles.

Sect. 42. Properties of elastic displacement fields. 131
or equivalently,
>P{S} = UK{S} -fPs-Puda.
dB
In a similar manner, the maximum principles discussed in Sect. 37 and the
variational principles given in Sect. 38 can be extended to the general problem.
VI. Homogeneous and isotropic bodies.
"^ 42. Properties of elastic displacement fields. As we saw in Sect. 27, when the
divergence and the curl of the body force field vanish, the displacement field u is
biharmonic, while div u and curl u are harmonic. To prove this we tacitly
assumed that u is of class C4. We now show that this somewhat stringent
assumption is unnecessary; we will establish the above assertions under the assumption
that u is of class C2.
By an elastic displacement field corresponding to the body force
field b we mean a class C2 vector field u on B that satisfies the displacement
equation of equilibrium
Au-\ Fdivu+ —6=0.
1 — Iv /1
We assume that Poisson's ratio v is not equal to f or 1, and that ^4= 0.
£ (1) Analyticity of elastic displacement fields.1 Let u be an elastic
displacement field corresponding to a smooth body force field b on B that satisfies
divb=0, curlb = 0.
Then u is analytic.
Proof. 'The proof is based on the following notion, due to Sobolev2 and
Friedrichs.3 A system of mollifiers is a one-parameter family of functions q8, 5>0,
such that for each fixed <3:
(i) qs is a C°° scalar field on V;4
(ii) j^Oonf;
(iii) q8 (v) = 0 whenever |»| ^ d;
(iv) / q8dv = \.
■r
An example is furnished by
\v\<6
with Ag chosen such that (iv) is satisfied. Let D be a regular region, and let f
be a continuous scalar, vector, or tensor field on D. Then the system of
mollified functions f, 3 > 0, is defined on $ by the transformation
f{x) = fe>(x-y)f<u)dvw.
D
Clearly,
(v) each f is a class C°° function on S.
1 Friedrichs [1947, 1}. See also Duffin [1956, 2], whose proof we give here.
2 [1935, 6], [1950, U}.
3 [1939,2], [1944,2], [1947, I].
4 Recall that y"is the vector space associated with S.
9*

132 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 42.
By (ii), (iii), and (iv),
\f{x)-f{x)\ = \ J e*(x-y)[f(y)-f(x)]dvJ
^suV{\f(y)-f(x)\:ytZs(x)},
for sufficiently small d, where £a (a?) is the open ball withradius #3;nd centa_at x.
Hence ———
N(vi) f -> f as d -> 0 uniformly on every closed subregion of D.
Assume now that / is of class C1 on D and, for convenience, that / is scalar-
valued. Let
Ds={xeD:Ud{x)CD}.
In view of the divergence theorem,
Vmf (x) = fv*es(*-y)f(y)<iv«=-f W* es (» - »)] / to) <*%
Z) D
= -/ 0tf(*-0) /(») n(y) ^%+ / Qs{x-y) Vyf{y) dvy.
3D D
But (iii) implies that
os{x—y)=0 for x e Ds and ye 3D.
Thus we have the following result:
(vii) Vjfl={7fy on Ds. \
Of course, (vii) also holds when / is a vector or tensor field.
We are now in a position to complete the proof of the theorem. Let u be an
elastic displacement field; let D be a regular region with DcB; let qs (8> 0) be a
system of mollifiers; let us and bs (8> 0) be the systems of mollified functions
corresponding to u and 6:
us(x) = / e'(x -y) u(y) dvy,
D
b8(x)=fe8(x-y)b(y)dv„.
D
Then, by (v), us and bs are of class C°°, and, by (vii), us is an elastic displacement
field on Ds corresponding to bs. Further, bs is divergence-free and curl-free on Ds.
Hence we conclude from the results of Sect. 27 (which hold in the present
circumstances) that us is biharmonic. Next, let
h=Au.
Then (vii) implies that
hs = (Au)s =A{us)
on Ds. Therefore, since us is biharmonic,
Ahs = 0
on Ds, and we conclude from (vi) that
hs-+h
uniformly on every closed subregion of D. The last two results and Harnack's
convergence theorem (8.2) imply that h is harmonic. But
Ah=AAu;

Sect. 43. The mean value theorem. \XX
thus to prove that u is biharmonic we have only to show that u is of class C4.
But since Au is equal to a harmonic function, (6.5) and f8.1) imply that u
is of class C°° on Z). Therefore w is biharmonic and, by (8.1), analytic on D. But D
was chosen arbitrarily. Thus u is analytic on B. □
In view of the results given in Sect. 27, we have the following immediate
corollary of (1).
(2) Properties of elastic states. Let [u, E, S] be an elastic state
corresponding to a smooth body force field b on B that satisfies
div6=0, curl 6=0.
Then:
(i) u, E, and S are biharmonic;
(ii) div u, curl u, tr E, and tr S are harmonic.
The next proposition will be of future use.
(3) If u is an elastic displacement field corresponding to null body forces, then
so also is u h = dw\dxh.
Proof. Since u is an elastic displacement field, u is analytic; therefore
0 = (^ + tzW ?div ")* =A{u-k) + T~2v Fdiv(M-*) •
Thus u>k is an elastic displacement field. □
Under-certain restrictions on the boundary of B, Fichera1 has established
the following important result: If u is an elastic displacement field that is
continuous on B and corresponds to zero body forces, then
sup \u\^H sup \u\, Wv^i, M-^
B 8B
where the constant H depends only upon v and the properties of dB. We omit
the proof of this result, which is quite difficult. Fichera1 has also established the
inequality
J\s\da^Hf\b\dv,
dB B
where s is the surface_traction and 6 the body force field of a smooth elastic
displacement field on B that vanishes on dB. ~~
43. The mean value theorem.3 The mean value theorem for harmonic functions3
asserts that the value of a harmonic function at the center of a sphere is equal to
the arithmetic mean of its values on the surface of the sphere. In this section we
derivejimiiar.xesuitsJQr the displacement,,.gtrajn>.and stress iidds-belonging to
an dastic state^Jn particular, we show that the values of these fields at the center
of a sphere are. jqjial to certain weighted averages of the displacement..on the
surface of the sphera We also establish a similar result for the stress field in
terms of its surface tractions.
1 [1961, «]. See also Adlkr [1963, 11, [1964, J].
2 Our presentation follows that of Diaz and Paynb [1958, 61.
3 See (l) of (8.3).

134
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 43.
(1) Mean value theorem.1 Assume that the body is homogeneous and
isotropic with shear modulus /j, and Poisson's ratio v, and assume that
' Let [u, E, S] be an elastic state^ corresponding to zero body forces. Then given any
ballSe=Se\y)inB, ~~
"^ = 76^7/(2-3,)/^ (M " n) n + (1 ~4v) M} da' (1)
E(y)
8jre3(7
- -r- — lOv / (u®n+n®u) da
— iov) ./
+ 7 / (m -n){5n(g)n — 1} da
S(^ = T^77-7o7) -iO»j"(H®n+n®H)i«
+ / (u -n){35n(g)n— <xl}i«
0i'p
20V2 — 28v + 7
(2)
(3)
1 —2v
3
^ = -8^77+5^ /Oo^sfgmJ+yls-n^nfgin-l]}^,
(4)
s=Sn.
Proof.2 Since the body forces vanish, the displacement equation of equilibrium
has the form
Au-\-y Fdivu = 0, (a)
where
y = - — . (b)
' 1— 2j> v '
Let
r(x)=x—y, r = \r\,
and notice that
(c)
n-
on 3¾.
Then (a), (b), (c), and the divergence theorem imply that
0 = J r*[Au +yFdiv «] rfi>
= / 0a[C7") n + 7 (div u)n]da — 2f [(Vu) r +y (div u) r] rf». (d)
dSp Up
1 Relation (l) is due to Aquaro [1950, 1], Eq. (13) and Synge [1950, 12], Eq. (2.36); see
also Sbrana [1952,4], Mikhlin [1952, 2], pp. 206-207, Garibaldi [1957, 1» Diaz and
Payne [1958, 6, 7], Bramble and Payne [1964, 4]. Relations (2) and (3) are due to Diaz
and Payne [1958, 6], Eqs. (4.11), (4.16). Relation (4) is due to Synge [1950, 12]; see also
Diaz and Payne [1958, 6], Eq. (5.21). Diaz and Payne [1958, 6], [1963, 3] and Bramble
and Payne [1964, 4], [1965, 2] have obtained a number of mean value theorems giving the
displacement and stress at the center of a sphere in terms of integrals over the surface of
various combinations of normal and tangential displacements and tractions. A mean-value
theorem for plane stress was given by Foppl [1953, 12].
2 The proof given here is that of Diaz and Payne [1958, 6].

Sect. 43-
The mean value theorem.
135
The surface integral in (d) is zero, since
0a/ [{Vu)n+y(divu)ri] da=g2 f [Au+yVdivu] dv=0.
dZp
If we use the divergence theorem on the remaining integral in (d), we conclude,
with the aid of (c), that
6 I [u+y{u ■ n) n] da — / (3 -\-y) udv=0. (e)
dZp Zp
On the other hand, by (4) of the mean value theorem (8.3),
zp dip J
If we eliminate the integral of u over Ze between (e) and (f), and use (b), we arrive
at the first relation in (1).
Our next step is to derive (2). Since u is biharmonic, Vu is biharmonic, and
we conclude from (4) of (8.3) that
F«w = M5»fVuiv-i>fVuda]' {g)
L Zx dZx
where 0< A Sag. Using the divergence theorem, we can rewrite (g) as follows:
Vu{y) = -^\l3 ju®nda-± j Vudc
If we multiply this relation by A4 and integrate from A = 0 to A = q, we find that
e* Vu(y) = ~-\$ fu®rdv- f r* Vu dv . (h)
By the divergence theorem,
J>3 Vudv =g3/M(g)n da — 2 j u®r dv,
Xp dZp i'p
and (g), (h), and the strain-displacement relation imply that
e-E(y) = -|—|7 f (u®r+r®u) dv—g2 f (w (g) n + n (g) w) dc
(i)
(i)
Next, we will eliminate the integral in (j) over Ue. By (42.3), uh, for each
fixed k, satisfies the displacement equation of equilibrium. Thus (e) holds with u
replaced by u>k, and we are led to the relation
I Vudv = - I [\+yn®n]Vuda,
z*. szx
which can be written, with the aid of the divergence theorem, in the form
/ M®nd« = - — / [l+yn<gm] Vuda.
)z*. dzx

136
M.E. Gurtin: The Linear Theory of Elasticity.
Sect. 43.
If we multiply this relation by X, use the fact that n = r/X on 327,, and integrate
from A =0 to X = q, we arrive at
I u®r dv = — - [>3l+y r(g)r] Vudv.
sP sp
Applying the divergence theorem to the right side of (k), we are led to
/ u®rdv = - —(q2 I [u0n+y(u-n)n0n]da
— / [2M®r+yr(g)M+y(r • m) 1] dv\.
If we add this equation to its transpose and rearrange terms, the result is
/ (u®r +r(g>w) dv
= -- — (g3 / [u®n+n®u + 2y(u-n)n®ri]da — 2yl J (r -u) dv
^ dZp zP
Taking the trace of this equation, we obtain
r ■ u dv = ~ u ■ n da,
and the last two equations yield
/ (u®r + r®u) dv
(k)
J M(g)n+n(g)M + 2y(u-n)(n®n—^ l)
azp
5+2y
If we combine (j) and (1) and use (b), the result is (2).
Since div u is harmonic, (2) of (8.3) implies that
(1)
da.
div w(ty) =-—- / divudv= -—5- u • n da.
vi" An q3 J An q3 J
(m)
The relation (3) follows from (2), (m), and the stress-strain relation in the form
S = 2p E + --1^. (tr E) 1 = 2p B + -^- (div u) 1
Since [u, E, S] is an elastic state corresponding to zero body forces, the stress
field S satisfies the equation of equilibrium
divS=0
(n)

Sect. 43. The mean value theorem.
and the compatibility equation1
AS+r]VF(tvS)=0,
1
where
Jj:
i+v
137
(0)
(P)
Further, by (42.2), S is biharmonic and tr S harmonic. Thus we may conclude
from the mean value theorem (4) of (8.3) that
By the divergence theorem and (n),
fs(n®n) «*« = -* f S(n®r)da= £ fsdv,
(q)
M
8S*.
3¾
and (q), (r) imply that
S(y) = ~^~ JS(n®n) da- Ij JSda
dZx
dZx
If we multiply this equation by A4, integrate from A = 0 to A = g, and solve for
S(y), the result is
S(y)=-£^ jS(r®r) dv- fr*Sdv .
On the other hand, the divergence theorem and (n) imply
/ g3S(n®n)da = f g*S(n®r)da = f r*S dv + 2 f S{r®r) dv; (s)
dZ„
thus
Further, since
where
it follows that
S (y) = T_~ 5\7 j S (r ® r) dv - q* f S (n ® n) da
L 2P dip
S(n®n) = (Sn)®n = s®n,
s = Sn,
S(y)=-~eY\7f S(r®r) dv-g* js®nda
(t)
(u)
To complete the proof we have only to transform the first integral in (u) into
terms involving surface tractions. By the divergence theorem and (o),
/ [(PS) n +r) (Ftr S) ® n] da = 0,
and using this indentity, (o), and two applications of the divergence theorem, we
see that
0 = fr*[AS +r)PP(tr S)] dv = -2 J [{VS) r +rj(Ptr S) ®r] rf»
= -2fA/ [S+J?(trS)n®n] rf«- / [3S+?y(tr S) 1] <*»
1 See p. 92.

138 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 44.
Taking the trace of (r), we obtain
JtrSdv= A f S ■ (n®ri) da,
Si. dhx
and the last two equations with (r) yield
J[r2S+r](trS)r®r]da= f tfS(r®r) +r)(r ■ Sr) I] da.
If we integrate this relation from A = 0 to A = q, we arrive at
/ 03 S +j;(tr S) r®r] dv = / [3 S(r <g) »•) +jj(r • Sr) 1] iw.
A simple computation based on (n) and (0) shows that r ■ Sr is biharmonic;
thus (4) of (8.3) implies
e3
■ I r • Sr dv= I 8 -n da.
£p dSp
By the last two relations, (s), and (t),
5 / S(r®r) dv=g3 / \s®n— v-(s -n) 1 da+rj f (trS)r®r dv. (v)
In view of the identity
and the symmetry of S, the volume integral in the right side of (v) takes the form
/ (tr S) r®rdv = Q3J(s-n)n®nda~f[S(r®r)+(r®r)S]dv. (w)
Ep dSp i:p
On the other hand, if we multiply (r) by A3 and integrate with respect to A from
0 to g, we arrive at
fS(r®r)dv = /A/ SdvdX. (x)
Sp 0 2*.
Since S is symmetric, the left-hand side of (x) is symmetric. Thus the volume
integral in the right side of (w) is equal to twice the volume integral in the left side
of (v); hence (v) and (w) imply that
I S(r®r) dv = -<[— \s®n+rj(n®n—l~l)(s-ri)da. (y)
Sp dSp
Eqs. (y), (u), and (p) yield the desired result (4). □
4>J 44. Complete solutions of the displacement equation of equilibrium. In this
section we study certain general solutions of the displacement equation of
equilibrium
Au+ — - Vdivu+ 1 6=0
1-2j> fi
assuming throughout that the body force 6, the shear modulus /j,, and Poisson's
ratio v are prescribed with /j, 4= 0 and v 4= \, 1 • Recall that a solution u of c]a§5 C2
is referred to_as_.an elastic.displacement field corresponding to b.

Sect, 44. Complete solutions of the displacement equation of equilibrium. 139
(1) Boussinesq-Papkovitch-Neuber solution.1 Let
u=w- 7(-/.:^- F(P-V> + <P)> (pi)2
where y and xp are class C3 fields on B that satisfy
AV = -[b, Acp^^p-b. (P2)
7¾½ m is an elastic displacement field corresponding to b.
Proof. Clearly,
A
Au + Fdiv u
1 1 — 2l>
=AV + -_1~FdivV- 4~-v) [/117^-^ + ^)+ -L-vVA{p-ip + cp) .
Since AV=VA and
zl(p -y) =p -A\p-\-2 divip,
it follows that
^M+ ^, ^divu=-dv+ i:^-FdivV- 2(T^jv) F(p-dv +zl<p + 2divy).
But
p-/ly = —zlq? and zl tp = — -6;
thus the above expression reduces to
-f6. D
1 Elements of this solution appear in the work of Bqussineso, [1878, 1], [1885, !]■ In fact,
the difference between the two solutions given by Boussinesq on p. 62 and p. 72 of [1885, 1]
yields the special form to which (Pj) reduces in the case of rotational symmetry. The
solution (Pj), (P2) with ¢) = 0 is due to Fontaneau [1890, 3], [1892, 3]. The complete solu-
\ tion (Pj), (P2) first appears in the work; of Papkovitch [1932, 3, 4]; it jyas later rediscovered by
<-A Neuber"'[19347!*]'. According to Papkovitch [1939, 4] (footnote on p. 136),(¾). (P2) should
be attributed to Grodski, who arrived at the solution in 1928 (unpublished research),
However, it is not clear from the statement of Papkovitch whether or not this work of Grodski
includes the scalar field q>; in fact, Papkovitch [1937, 5] claimed that it does not. The
complete solution (Pj), (P2) does appear in Grodski's [1935, 1] first published research on the
subject. Other references concerning this and related solutions and not mentioned elsewhere
in this section are: Boussinesq [1888, 1, 2]. Tedone [1904, 4], Serini [1919, 2], Burgatti
[1923, 1], [1926, 1], Timpe [1924, 2], [1948, 6], [1950, 13], [1951, U], Weber [1925, 1],
Trefftz [1928, 3], Margubrre [1933, 1], [1935, 3], [1955, H], Biezeno [1934, 1], Sobrero
[1934, 6], [1935, 8], Neuber [1935, 4], [1937, 3], Westergaard [1935, 9], Zanaboni [1936,
6], Slobodyansky [1938, 3, 4], [1959, 15], Platrier [1941, 1], Gutman [1947, 3], Shapiro
[1947, 7], Ter-Mkrtychan [1947, 8], Krutkov [1949, 5], Blokh [195o, 3], [1958, 2], [1961,
2], Sternberg, Eubanks, and Sadowski [1951, 9], Brdicka [1953, 6], [1957, 3], Churikov
[1953, 7], Dzhanelidze [1953,91, Hu [1953, 13], [1954,5], Schaefer [1953, 18], Trenin
[1953, 19], Eubanks and Sternberg [1954, 5], Kroner [1954, 11], Ling and Yang [1954,
13], Lur'e [1955, 10], Duffin [1956, 2], Solyanik-Krassa [1957. 13], Solomon [1957, 12],
Deev [1958, 5], [1959,3], Hijab [1959, 7], Truesdell [1959. IS], Bramble and Payne [1961,
4], Naghdi and Hsu [1961, 16], Alexsandrov and Solovev [1962, 1], [1964,2], Gurtin
[1962, 7], Tang and Sun [1963, 22], Volkov and Komissarova [1963, 25], Herrmank [1964,
9], Alexsandrov [1965,1], Stippes [1966, 25], [1967, 14], Solomon [1968, 12], deLaPenha
and Childs [1969, 3], Youngdahl [1969, 9],
2 Recall that p (as) = as — 0.

-140 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 44.
As a corollary of this theorem we have the following result.
(2) Assume that the body force b is continuous on B and of class CN (N 2_ 2)
on B. Then:
(i) there exists an elastic displacement field u of class CN on B that corresponds
to b;
(ii) every elastic displacement field that corresponds to b is of class CN on B.
Proof. Let
»(#) = — —— /
P(y)-b(y) ,
^)=4^/- -^
B
Then it is clear from the properties of the Newtonian potential (6.5) that <p
and \p are of class CN+1 on B and satisfy (P2); thus the field u defined by (P_) is
a class CN elastic displacement field corresponding to 6.
To prove (ii) let u be an arbitrary elastic displacement field corresponding to
6. By (i) there exists a class CN elastic displacement field u that also corresponds
to 6. Let
u=u— u.
It suffices to prove that u is of class CN. Clearly, ii is an elastic displacement
field corresponding to zero body forces; thus, by (42.1), u is of class C°°. □
For the case in which the body-force is the_jgr^iejol.oLa43olejitml, i.e.
6 = Fa,
it is possible to construct_jt.mucli_sim.plei- particular..-solution. Indeed, the field
u = V<p
satisfies the displacement equation of equilibrium provided
In the same manner, if
then
is a solution provided
__|m = _ a.
T X + 2/J,
6=curlr,
u = curl a
A
Am — T.
By Helmholtz's theorem (6.7) we can always decompose the body force as
follows:
6 = Fa + curlr.
Thus, combining the previous two displacement fields, we arrive at the solution:1
u = Vq>-\-cur\ m,
l l
Aw = —i-,-r--a, dfl) = T.
r A + 2/j, fi
1 Cf. Lam£ [1852, 2], p. 149; Bktti [1872, 1], § 5; Kelvin and Tait [1883, 1], p. 283;
Voigt [1900, 5], pp. 417, 432, 439; Tedonk and Timpe [1907, 3],

Sect. 44. Complete solutions of the displacement equation of equilibrium. 141
This solution, in general, is not complete. Indeed, consider the case in which
a = 0. Then
div «=zl 99=0;
thus, in this instance, the solution can represent at most divergence-free
displacement fields.
(3) Boussinesq-Somigliana-Galerkin solution.1 Let
u=Ag- 2(i1_y) Vdivg, (¾
where g is a class C4 vector field on B that satisfies
AAg = -ifib. (G3)
Then u is an elastic displacement field corresponding to b.
Proof. Clearly,
AuA Fdiv u
' \-2v
VdivA g -^- VA div g .
" 2(1— v) a\
Since A V—VA and div A =A div, the right-hand side of the above expression
reduces to A A g, which by hypothesis is equal to
Qu.rili5L^5EJ^A9..e.stab^sh the cpmpleteness of the Boussinesq-Papkoyitch-
Neuber^and Boussinesq-Somigliana-Galerkin representatiojis. The notion of
completeness is 6T"utmost importance to applications. Indeed, it is concerned with
the question: can any (solvable) problem be solved using a given
representation; or, in other words, does every sufficiently smooth solution admit such a
representation ?
(4) Completeness of the Boussinesq-Papkovitch-Neuber and
Boussinesq-Somigliana-Galerkin solutions.2 Let u be an elastic displacement field
corresponding to b, and assume that u is continuous on B and of class C4 on B. Then
there exists a field g of class C4 on B that satisfies (Gt), (G2); and fields <p, ip of
class C2 on B that satisfy (PJ, (P3).
Proof. We begin by establishing the completeness of the
Boussinesq-Somigliana-Galerkin solution. Clearly, (Gx) is equivalent to
Ag+ 1^Fdivsr+~6 = 0,
" 1 — 2v ft
1 This solution is due to Boussinesq [1885, T\, p. 281. It was later rediscovered by Somi-
gliana [1889, 3], [1894, 3] and Galkrkin [1930, 2, 3], [1931, 3] and is usually referred to as
Galkrkin's solution, although Tedonk and Timpe in their classic article [1907, 3], p. 155
attributed it to Boussinesq. See also Iacovachk [1949, 4], Ionescu [1954, 9], Moisil
[1949, 6], [1950, 8], Ionescu-Cazimir [1950, 7], Westkrgaard [1952, 7], Brdicka [1954, 3],
Bernikov [1966, 4],
2 Papkovitch [1932, 4] noticed that if a particular solution to the equation of equilibrium
exists for arbitrary values of Poisson's ratio and arbitrary body forces, then the Boussinesq-
Somigliana-Galerkin solution is complete. An alternative completeness proof for (Gj), (G2)
was furnished by Noll [1957, 11]. The proof given here, which is based on Papkovitch's
argument, is due to Stkrnberg and Gurtin [1962, 13].
=AAg-—^—-AVdivg+ -\
" 2(1 — v) ** ' \ — 2v

142
M.E. Gurxin: The Linear Theory of Elasticity.
Sect. 44.
where
-flU, v=^—v.
Thus the task of finding a class C4 field g that satisfies (Gj) is reducible to finding
a class C4 particular solution to the displacement equation of equilibrium
corresponding to given body forces that are continuous on B and of class C4 on B.
This latter objective is met by (i) of (2). Further, using steps identical to those
used in the proof of (3), it is easily verified that (Gx) implies
Au-\ — Vdivu=AAq.
Thus (Gx) and the displacement equation of equilibrium imply (G2), and we are
entitled to conclude the completeness of the Boussinesq-Somigliana-Galerkin
solution (Gx), (G2). Next, following Mindlin,1 we define
cp = 2divg— p-Ag, tp=Ag.
Then <p and \p are of class C2 on B, and (G2) implies (P2), (Gx) implies (Px). Thus
the Boussinesq-Papkovitch-Neuber solution is also complete. □
In cartesian coordinates the solutions (Px), (P2), and (Gx), (G2), as well as the
formulae for the corresponding stresses, have the following form:
Boussinesq-Papkovitch-Neuber solution
bt xj bt
V<M = -~> <P,ii = ^T
'*i = V* ~ Ji^vf fa" + *' Vi), i
Boussinesq-Somigliana-Galerkin solution
gi.jjkk — — p °i
__ 1
sa =f* \ii, jkk +gi, ikk + jzz^ (v&a Sk.h u — 8k,k i,) } •
In most applications of elastostatics the Boussinesq-Papkovitch-Neuber
solution deserves preference over the Boussinesq-Somigliana-Galerkin solution.2
Indeed, the transformation (Px) underlying the former is one degree lower than
that of (Gx); secondly, (in the absence of body forces) the stress functions involved
in the former are harmonic, while those of the latter are biharmonic; and, finally,
(Px), (P2) are conveniently transformed into general orthogonal curvilinear
coordinates, whereas (Gx), (G2) give rise to exceedingly cumbersome forms when referred
to curvilinear coordinates, with the exception of cylindrical coordinates.
1 [1936,2].
2 Cf. Sternberg and Gurtin [1962, 73].

Sect. 44. Complete solutions of the displacement equation of equilibrium. 143
Two other complete solutions of the displacement equation of equilibrium
(with 6=0) are:1
w=t— „,„ > Va.,
2(1 — v)
Ja=divT, Jt=0;
u=l— curl o),
\-2v
Aa)=curll, Al = 0.
Returning to the Boussinesq-Papkovitch-Neuber solution, we_note that__a
given equilibrium displacement field-does. not. .uniquely, determine its generating
stressTfunctionlf y and ig. In fact, it is just this lack of uniqueness which, in cer-
tain^OTctnnsrarices,' allows the elimination of y or one of the components of \p.
Thejiext_the2r£m, which_is duetto Eubanks and Sternberg,2 deals with the
possibility^ of setting <p = 0.3
(5) Elimination of the scalar potential.4' Let B be star-shaped with respect
to the origin 0. Let u satisfy the hypotheses of (4) with 6=0. Further, suppose that
—1 < v < \ with 4v not an integer. Then there exists a harmonic vector field \p on B
such that
Proof. By (4) and since 6=0, there exist harmonic fields <p and \p such that
u=w- 4(11_y) F(p-v + y).
Aip = 0, Aq>=0.
Thus we must show that there exists a harmonic field \p that satisfies
a.tp — V(p • ip) =a.ip — V(p -y> + (p),
where
«=4(l—i').
By hypothesis, we may confine a to the range
2<<x<8. (a)
1 The first solution, which is apparently due to Freiberger, [1949, 2, 3], is simply a
variant of the Boussinesq-Papkovitch-Neuber solution; indeed, take T = y/, 2a = q>-\-p ■ >//
in (Pj), (P2). The second solution is due to Horn [1915, 2], who used it t.ojrove existencefpr
fhp-ffl5E!2£^-rn^"t prp^'""1.
* [1956, 4]. Papkovitch [1932, 3, 4], through an erronoeus argument, was led to claim
that q> may be taken equal to zero without loss in completeness, a claim which is reflected
in the title of [1932, $}. In a subsequent note [1932, 2], Papkovitch refers to this argument as
"artificial". Neuber [1937, 3] asserted that q> may be set equal to zero without impairing
the generality of the solution. In his proof of this statement he takes for granted the existence
of a function satisfying a pair of simultaneous partial differential equations. Therefore, as
was pointed out by Sokolnikoff [1956, 72], p. 331, Neuber's argument is inconclusive.
Slobodyansky [1954, 79] proved that the stress function q> may be eliminated, but as was
observed by Sokolnikoff [1956,72], p. 331, Slobodyansky's proof is limited to a region
interior or exterior to a sphere.
3 The solution (Pi), (P2) with q> = 0 appears in the work of Fontaneau [1890, 3], [1892, 3],
and, according to Papkovitch [1937, 5], [1939, 4], p. 130, also in unpublished research of
Grodski.
4 This result has been extended by Stippes [1969, 8] to an exterior domain whose
complement is star-shaped, and to the region between two star-shaped surfaces.

144 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 44.
It is sufficient to establish the existence of a harmonic field 0 such that1
pV0 — a.0 = <p. (b)
Indeed, given such a field 0, the function
\p=\p + V0 (c)
has all of the desired properties.
Let (r, 6, y) be spherical coordinates. The function q>, by hypothesis, is
harmonic. Hence q> (r, 6, y) admits the following expansion in terms of solid spherical
harmonics
oo
<p{r,Q,y) = ZSn{Q,y)r\ (d)
»=o
which is uniformly convergent in any spherical neighborhood of the origin. The
functions S„ (6, y) are surface harmonics of degree n.
Next, define a function F through
[<x+l]
F(T, 6, y) =<p(r, 6,y)-Z Sn(6, y) r», (e)
»=o
where [a +1] denotes the largest integer not exceeding a +1. Clearly, F is defined
and harmonic on B. Further, by (d) and (e),
r-«-ip (r> 0( y) _> 0 as r .+ o. (f)
In spherical coordinates (b) has the form
r"d'r ~a-0=(P'
or by (e)
g, [<x+l]
^|(r«$)=F+2S„A (g)
Consider the function 0 defined by
nr,e,y)=G{r,e,y)+Z^^
»=0
G{r,6,y)=r*jF(£, 6, y) I""-1 if •
(h)
Since B is star-shaped with respect to the origin 0, and in view of (a) and (f),
the relations (h) define 0 on B. Moreover, 0 is of class C2 on B and satisfies (g),
and hence (b).
It remains to be shown that 0 is harmonic on B. To this end, it suffices to
show that G is harmonic. In spherical coordinates the Laplacian operator appears
as
^■HIKIm
ZlH,=cosec 6 -go-(sin6 -r-A + cosec2 6 ■ -,.
1 Neuber [1937, 3] assumed, without proof, the existence of such a ¢, in the absence of
any restrictions on B and v.

Sect. 44. Complete solutions of the displacement equation of equilibrium. 145
Clearly,
r
Ai,G=^JAlllF(i,d,r)ra-1di,
(i)
0
and, with the aid of two successive integrations by parts applied to the second
of (h), we reach the identity
a(« + l)G = -(« + 1)f-r-^-+f-/ £[p^^jr-1**. (k)
o
Substitution of (k) in (j), and use of (i), now yield
r*AG=r" f AF(£, 6, y) £---1 if.
o
But F is harmonic. Hence G and 0 are harmonic on B. □
(6)1 Zfeeorem (5) is false if the hypothesis that 4.v not be an integer is omitted /^.
Proof. Suppose that 4v is an integer. Then the modulus a =4(1 — v) is also
an integer with 2 < a < 8. Let B be a closed ball centered at 0, and let u be the
elastic displacement field (corresponding to 6=0) defined by
u = ~V<p, (a)
where <p is the solid spherical harmonic
V(r,e,y)=r«Sa(0,V). (b)
Assume that (5) remains valid. Then there exists a field xp that is harmonic on
B and satisfies
»=V- I V(p-ip). (c)
By (a) and (c),
V(p-xp)—a.xp = Vq>, (d)
and upon operating on (d) with the curl, we conclude that xp is irrotational.
Since B is star-shaped and hence simply-connected, there exists an analytic
field 0 on B such that xp = V0; thus we conclude from (d) that
p- V0 — a<P = 9P + c1, (e)
where cx is a constant. By (e) and (b),
_£ (,,-«#) =r-iSaM+Ci. (f)
The general solution of (f) is
0(r, 6, y)=raf(d,y)+ f(log r) Sa(6, y) + crr*+i. (g)
There is no choice of f (6, y) that leaves 0 without a singularity at the origin.
But 0 was required to be analytic on B. Hence (5) cannot be valid in the
present circumstances. □
1 Eubanks and Sternberg [1956, 4]. See also Slobodyansky [1954,79], Solomon
[1961, 20]. Even though the potential q> cannot be eliminated when 4i> is an integer, Stippes
[1969, 8] has shown that q> can be set equal to a solid spherical harmonic of degree 4(1 —v).
Handbuch der Physik, Bd. VI a/2. 10

146 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 44.
The next theorem, which is also due to Eubanks and Sternberg,1 deals with
the possibility of eliminating one of the components of tp.
(7) Elimination of a component of the vector potential. Assume that B
satisfies hypotheses (i) and (iv) of (5.2) with M=2 and N—^. Let u be an elastic
displacement field corresponding to 5=0, and assume that u is of class C4 on B.
Then u admits the representation (P2) with <p and xp harmonic on B and y)3 = 0.
Proof. By (4) there exist harmonic fields <p and xp such that
where
a=4(1— v).
Thus we must establish the existence of harmonic fields <p and \p such that
ccxp — V(p -tp + q)) — a.xp — V{p • ¢+^), (a)
¥-3=0. (b)
It suffices to find a harmonic field H that satisfies2
H,3=y>s, (c)
for then the functions
\p = \p — VH,
<p=<p-\-p-VH—aH
will be harmonic and will satisfy (a) and (b).
By tracing the_steps in the proof of (4), it is not difficult to verify that \p
is of class C2 on B, since u is of class C4 on B. By (a) of (5.2) with M = 2 and
iV = 3, there exists a scalar field J of class C2 on B and C3 on B such that
/,3=^3-
Then
(4/)j3 =Zl^3=0,
and there exists a continuous function h on B12 that is class C1 on B12 and satisfies
^J(x1,x2,xs)=h(x1,xi).
(Recall that B12 = {(x1, x2): ace 5}.) By (6.6) there exists a class C2 scalar field K
on B12 such that
AK = -h.
Thus if we define H on B through
Ji(Xi, x2, xs) ==J(x1, x2, x3) -j-K(xlt x2),
then H is harmonic and satisfies (c). □
Assume now that the body force 5 =0, and let ...ft Jbfi. a.giKen,.unit-KeGtoEr~If
in the Boussinesq-SornlgTIaniPGalerEiirsolution we take
0=ft.r,
1 [1956,4). ' ~~'
2 Nbuber [1934, 4], without restricting B, assumes the existence of such a function.

Sect. 44. Complete solutions of the displacement equation of equilibrium. -147
then (Gx), (G2) reduces to Love's solution1
u = kA3£- ■-,-■— Vlk ■ VX), (LJ
2(1— v) v v
J/l<r=0. (La)
On the other hand, if we take
^) = fcyi
in the Boussinesq-Papkovitch-Neuber solution (Px), (P2), we arrive at Boussi-
nesq's solution:2
u = kxp+ 4{ii_v)-V((P+zf), (Bj)
A<p = 0, Af = 0, (B2)
where z = k ■ (x — 0) is the coordinate in the direction of k. Both of these
solutions are extremely useful in applications involving torsionless rotational
symmetry.
It is clear from (LJ and (Bx) that these solutions can at most represent
displacement fields that satisfy3
fc-curlu = 0. (C)
The next theorem shows that under certain hypotheses on B, condition (C) is
also sufficient for the completeness of the above solutions.
(8) Completeness of Love's and Boussinesq's solutions.* Assume that
B satisfies hypotheses (i)-(iv) of (5.2) with M=2 and N—$. Let u be an elastic
displacement field corresponding to 6=0 that satisfies (C) with k — ez, and
suppose that u is of class C2 on B. Then there exist scalar fields 3C, <p, and %p on B that
satisfy (1^), (L2) and (Bj), (B2).
Proof. By hypothesis and (42.1), u is of class C°° on B. Since k = es, (C)
yields
and we conclude from (5.2) with M=2 and N=S that there exists a scalar
field t of class C2 on B and C5 on B such that
««=*,«• (a)
The displacement equation of equilibrium can be written in the form
Aua+yOia=0, (b)
Aus+y6>s=0, (c)
where
0=divu, (d)
y=-—■ . (e)
1 [1927, 3], p. 276. Cf. Michell [1900, 4].
2 [1885, -?], p. 62 and p. 72.
3 Noll [1957, 11].
4 The portion of this theorem concerning Love's solution is due to Noll [1957, H],
Theorem 2. Under the assumption of torsionless rotational symmetry, Love [1927, 3],
pp. 274-276 established the completeness of (LJ, (L2); under the same assumption Eubanks
and Sternberg [1956, 4] gave a completeness proof for (1¾). (B2) (see (9)).
10*

148 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 44.
Before proceeding any further, recall that
B12={(x1,xi):x^B}, B3={x3:xtB}.
By (a) and (b),
[Ar+yd]ta = 0;
thus
Ar+yd=X, (f)
where X = X (x3) is a class C3 function on B3 that is continuous on B3. Now let q
be a solution of
it --*■ w
and let
x(%, x2, x3) =t(^, *2, ^3) +e(*3). (h)
Then k is class C5 on 5 and class C2 on 5. By (h) and (a),
«a=*,«. (i)
and since
it follows from (f), (g), and (h) that
Ax+y6=0. (j)
If we differentiate (j) with respect to x3 and subtract the result from (c), we
arrive at
A{u3-x>3)=0. (k)
Next, by (d) and (i),
0 =M«,a + «3,3 —X,a<x + «3,3 =Ax + (U3 — X3)t3, (1)
and (j) and (1) imply
(i+y)Ax+y{u3-xt3):3=0. (m)
In view of (5.2), there exists a solution 0* of
<P*, = x (n)
that is class Ca on 5 and C5 on B; thus (m) takes the form
[(l+y)A$*+y{u3-xi3)]t3=0. (0)
Since B is %-convex, (0) implies that
g*=(i+y)A&*+y(u3-x_3) (p)
is independent of x3. Moreover, g* is continuous on B12 and class C3 on B12.
Thus, in view of the properties of the logarithmic potential (6.6), there exists a
class C4 function g on B12 such that
g,** = Y+J8*- (q)
If we let
0 (%, *2, x3) = 0* (xx ,x2,x3)—g (¾, x2), (r)
then 0 is of class C4 on 5 and (p), (q) imply
(i+y)A 0+y {u3-x_ 3)=0. (s)

Sect. 44. Complete solutions of the displacement equation of equilibrium. 149
Moreover, (n) and (r) yield
<P,8=*. W
Finally, using (i), (s), and (t)
u = uaea+uzez =x>aea+¾^ = Vx + (% —*,s) es
= V0,3-^A0e3,
and letting
y
we arrive, with the aid of (e), at (LJ. Since (Lj) and the displacement equation
of equilibrium imply (L2), the completeness of Love's solution is established.1
If we now define2
(p =^1 J, 3 %£ -C33,
%p = ./33,
then (Bx), (B2) follow at once from (LJ, (L2). □
Let (r, y, z) be cylindrical coordinates with the z-axis parallel to k. We say
that a displacement field u is torsionless and rotationally symmetric tcith
respect to the z-adds if
ut-=ur(r, z), uY = 0, uz — uz{r,z).
It is not difficult to show that such displacement fields satisfy (C), and hence,
by (8), admit the representation (BJ, (B2). An alternative proof of this assertion
is furnished by the following theorem due to Eubanks and Sternberg.3
(9) Suppose that dB is a surface of revolution with the z-axis the axis of
revolution. Let u bettor sionfess and~roiaiionally symmetric with respect to the z-axis and
satisfy the hypotheses of (4) with 6 = 0. Then there exist scalar fields (p = <p(r, z)
and y)=ip(r,z) on B such that (Bx), (B2) hold. Moreover, the functions q> and %p
are single-valued and analytic on any simply-connected plane cross-section of B
parallel to the z-axis.
Proof. By (4) there exist harmonic fields q>' and tp' such that (Px) holds. Let
t _ Sjh_ __ 8yt'r
lr dr dz '
Jz r Sr V Wr) -t- r Sy -+- Sz ■
Then an elementary computation based on (Px) and the fact that \p' is harmonic
and u torsionless and rotationally symmetric confirms that
dz dr
Thus there exists a field y> such that
'' dr' lz dz'
1 The proof up to this point is due to Noll [1957, 11~}.
2 Cf. the last equation in the proof of (4).
3 [1956, 4].

150
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 45.
In the same manner, let
Then
8lr — 8& -Q
"8*' 8f
and there exists a field a> such that
9o> dm
Now let
q> =a>—ztp;
then a trivial computation reveals that
U=ky)+ V{q>+zxp). □
VII. The plane problem.1
45. The associated plane strain and generalized plane stress solutions.
Throughout this section B is homogeneous and isotropic2 with Lame moduli jx and A.
Let (x1, x2, #3) be a rectangular coordinate system with origin at 0, and let B
be a (not necessarily circular) cylinder centered at J) with generators parallel to
Fig. 11.
1 Most of this subchapter was taken from the unpublished lecture notes of E. Sternberg
(1959). Extensive reviews of the subject are given by Teodorescu [1964, 22], and Musk-
helishvili [1965, IS]. See also the eighth chapter (by Barenblatx, Kalandya, and Mand-
zhavidze) of Muskhelishvili's book [1966, 17].
2 The plane problem for anisotropic media is discussed by Fridman [1950, 5] and Kupradze
[1963, 17, 18], Chap. VIII.

Sect. 45- The associated plane strain and generalized plane stress solutions. 151
the and with end faces at x3 = ± h (Fig. 11). Let £^ and S^ be complementary
subsets of the set
{xedB: x3^±h}.
We assume that Sfx and £"2 are independent of x3 in the sense that £fx n £"2
consists at most of line segments parallel to the #3-axis and of length 2h.
Given elastic constants fi and X, body forces 6 on B, surface displacements
« on ^, and surface tractions s on y2, with 6, u,_a^d.s_mdeQeadstd...aLxa and
paralleljto_the_j^,_^-J3lane, the plane problem of elastostatics consists in
finding an elastic state [«, E, S] on B that corresponds to the body force field 6
and satisfies the boundary conditions:
u — ii on &x, s=s on £fit
s = 0 when x3 = ± h.
Since the surface displacements, surface tractions, and body forces are all
independent of x3 and parallel to the xx, #2-plane, one might expect that the dis-
pl_acement_fidd_u_will.also have this property. With this in mind, we introduce
the following definition.
Let [u, E, S] be an elastic state. Then [u, E, S] is a state of plane strain
provided ~~^>
», = »«fe.*i). w3 = 0.
Here and in what follows Greek subscripts range over the integers (1,2). The
above restriction, in conjunction with the strain-displacement and stress-strain
relations, implies that E =E(x1, x%) and S =S(xlt x2). Further,
■^13 = -^83 = -^33 = 0.
Saf) = 2[i Eafj + A.dafl Eyy,
-^13 == ^23 =0, J33= Ahxx,
and the equation of equilibrium takes the form
S«/J,/J+*«=0-
A simple calculation based on the above relations yields the important relation
■->33 =*''->aa>
where v is Poisson's ratio:
A
Thus the complete system of field equations consists of
•Sa/J=/M(«a,/J+«/J,a)+^^«y,y.
supplemented by
Si3 = S23=w3 = 0, sss=vSaa; (Pf)
while the boundary conditions are
ua=ua on ^, S^^n^s,,, on ^, (P2)

152 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 45.
supplemented by
Si3=S23 = S33 = 0 when xs = ±h. (P2*)
Here ^ and ^2 are the intersections, respectively, of £^ and <S^2 with the xx, x2-
plane.
The system (P^, (P2) constitutes a two-dimensional boundary-value problem.
This problem has a unique solution which we call the plane strain solution
associated with the plane problem. As is clear from (P^) and (P*), this_,soluMgn
actwd]^£di)&sAe~Mm&..prsthkoLQnly..wheiL-iu^Q. or Su +S22 = q. However,
the plane strain solution is an actual solution of the modified problem that
results if we replace (Pf) by the mixed-mixed boundary condition:
S13 = S23 = % = 0 when xz = ±h.
For future use, we note here that in terms of jx and v the two-dimensional
stress-strain relation
s*() = 2/« Eafi + A<5a/! Eyv
can be inverted to give
2fiEafi=Safi—v8afiSYy.
We turn next to the associated generalized plane stress solution,1 which
characterizes the thickness averages of u, E, and S under the approximative
assumption :
S33 = 0. (A)
Thus let \u, E, S~] (no longer assumed to be a state of plane strain) be_a_solution
of the plane problem and assume that (A) holds. It follows from the symmetry
inherent in the plane problem that
Ua\Xl> X2> Xz) = Ua.\X\' X2> X3I' US\X1> X2> X3) = —%(%> X2> Xz) >
for otherwise the displacement field u* defined by
Ua (Xl> X2> X3I = Ua.\X\> X2' XZ) > US {Xli X2> X3) = — US\X1> X2> — XV
would generate a second solution of the plane problem, a situation ruled out by
the uniqueness theorem of elastostatics (at least when the elasticity tensor is
positive definite).
Given a function f on B, let f denote its thickness average:
+ h
f{x1,x2)=—^ I f(x1,x2,xs)ax3.
-h
Since the thickness average of an #3-odd function is zero, and since the %-deriv-
ative of an #3-even function is %-odd, it follows that
u3 =h13 =ii23 = ii3 ='->23 = 0.
Further,
^n=i(«M+"/i,«).
S«p = 2/U Eafi +A.dal!{EYY+ £33),
^33 = 2/M £33 + ^ {Eyy + -E33) •
1 Thickness averages of solutions to the plane problem under the assumption that S33 vanish
were first considered by Filon [1903, 2], who derived the results (a)-(c). See also Filon
[1930, i] and Coker and Filon [1931, 2]. The term "generalized plane stress" is due to
Love [1927, 3], § 94. The assumption S33 = 0 was studied by Michell [1900, 3].

Sect. 45. The associated plane strain and generalized plane stress solutions. 1 53
In view of (A), S33 = 0; thus the above relations imply that
^=/«(«a,P+«P,«)+^^"y,y. (a)
where
x=-j¥f-. v
Further, the averaged equation of equilibrium takes the form
Sap,p+K=o, (b)
while the boundary conditions become
ua=ua on ^, Safin^ = sa on ^. (c)
The relations (a), (b), and (c) constitute a two dimensional boundary-value
problem. Thisproblem has a unique solution called the generalized plane stress solution
associatedwith th¥pJane"'pTo^IelnTlT6^~tirat"tHe relations (a), (b), and (c), which
constitute~the generalized" plane"stress solution, are identical to the relations (Px)
and (P2) of the plane strain solution provided we replace the Lame modulus X by
It must be_emphasized_thal basic^Jo..the.d^xiv^ion^afJ^-is. the. approximative
assumption that S33 vanish.
The stress-strain relation for generalized plane stress is easily inverted; indeed,
where
- V
v =••
1+v
is the ^ppropjc^^ZE9i?IoT^I3^9ZA9L,K§Pi§T3^zP'^. plane stress. Fig. 12 show?
the graph of vjv as a function of v.
1.0
0J3
i*|* 0.8
0.7
0 0.1 0.2 0.3 Oi OS
V
Fig. 12.
Let us agree to call the elastic constants fi and v positive definite if
/n>0, -KKf.

154
M.E. Gurtin: The Linear Theory of Elasticity.
Sect. 46.
This definition is motivated by (iv) of (24.5). Note that
— Kv<i« — oo<v<£, (d)
so that /u, v positive definite does not imply //,, v positive definite. Note, however,
I that
|The result (d) will be of importance when we discuss the uniqueness issue appro-!:
- > priate to the generalized plane stress solution. '■',
46. Plane elastic states. In tlusjsection we study properties of the plane jBtrain
solution associated with the plane problem under the assumption that B is
homogeneous and isotropic with ^4=0, v=j=^, i)In addition, we assumejhroughout
this section and the next that the body force$~ttfe zero.
We write R for the intersection of B with the x1, #2-plane. Further, we let
^ and ^2 denote complementary regular subcurves1 of the boundary curve 8R
and write na for the components of the outward unit normal to dR.
In view of the discussion given in the previous section, to specify a state of
plane strain or generalized plane stress it suffices to specify the plane components
ua and Sap of the displacement and stress fields. This remark should motivate
the following definition. We say that [ua, Sap] is a plane elastic state
corresponding to the elastic constants /u and v if:
(i) ua is continuous on E and smooth on R;
(ii) Sap is continuous on R and smooth on R;
(iii) ua and Sap satisfy the field equations
^0,0 = 0,
Since every plane elastic state generates, with the aid of (P*) of Sect. 45, a
(three-dimensional) state of plane strain, we have the following direct corollary
of (42.1):
(1) Let [ua, Sap] be a plane elastic state corresponding to the elastic constants
fi and v. Then ua and Sa/8 are analytic on R.
Each state of plane strain is an actual solution of the modified plane problem
for which the boundary conditions at xs = ± h are replaced by the mixed-mixed
conditions
•^13 = ^23 = u3 = 0 when x3 = ± h.
Therefore, as a consequence of the uniqueness theorem (40.1), we have the
(2) Uniqueness theorem for the plane problem.2 Let [«„, SajS] and
[ua, Sap] be plane elastic states corresponding to the same elastic constants \i and v,
with fi and v positive definite, and let both states correspond to the same boundary
data, i.e.,
ua=ua on &! and Sapnp=Sapnp on<<f2.
1 <Sl and ^2 are assumed to have properties strictly analogous to those of S\ and £^2 given
on p. 14.
2 Trivially, this theorem is valid without change when the body forces do not vanish
provided, of course, that both states correspond to the same body forces. A computation of
the strain energy of a plane elastic state leads, at once, to the conclusion that uniqueness
holds under tteJ^g.restri<^yg^:y£oj^esesi/*^p, v < i. In view of (d), this extension is
important as"ft assures uniqueness when the plane states correspond to generalized plane stress
solutions and when the elastic constants fi and? =v/(1+v) are such that fi andi> are positive
definite.

Sect. 46.
Plane elastic states.
155
2
Then
5<x/i=Sa,8 on R,
and hence the two states are equal modulo a plane rigid displacement.
The next theorem shows that for plane states the stress equations of
compatibility take a particularly simple form.
(3) Characterisation of the stress field. Let [ua, Sai8] be a plane elastic
state corresponding to the elastic constants /u and v. Then Sa/8 satisfies the
compatibility relation1
ASaa=0. (a
Conversely, let Sai8 (=Spa) be of class C2 on R and smooth on R~, and suppose
that Sap satisfies (a) and the equation of equilibrium
5./,./, = 0. (b)
Assume, in addition, that R is simply-connected. Then there exists a field ua such
that [ua t Sap] is a plane elastic state corresponding to //, and v.
Proof. Let [ua, Sai8] be a plane elastic state. By (1) the strains Eap
corresponding to ua are analytic; thus we conclude from (14.5) that Eap satisfies the
compatibility relation
2^12,12 =-^11,22 +^22,11- (c)
The stress-strain relation, in conjunction with (c), yields
2^12,12 =Sll, 22 + ^22,11 ~vA3aa, (d)
while the equation of equilibrium implies
^12,12 = ^11,11)
°12,12— °22,22>
hence
2^12,12 = ^11,11 ^22,22- (e)
Eqs. (d) and (e) imply (a).
On the other hand, let Sai8 satisfy (a) and (b) and define Eap through the stress-
strain relation. Then by reversing the above steps, it is a simple matter to verify
that (c) holds. Thus (14.5) implies the existence of a class C2 displacement field
ua that satisfies the strain-displacement relation, and we conclude from (14.4)
that ua is continuous on E. Thus [ua, Sap] has all of the desired properties. □
Theorem (3) remains valid in the presence of body forces provided (a) and (b)
are replaced by
The next theorem asserts that the stresses corresponding to a solution, of the
tractionfroblem are"independent of the elastic constants3 (provided the body is
1 Levy [1898, 7].
2 Here A is the two-dimensional Laplacian:
_82 32
dx\ dx\'
3 This assertion was first made by Levy [1898, 7~\. However, Levy did not require that R
be simply-connected, and theoremjf4|Ja-laiaa.JL^?jrestaicMoa.i8_5anitted.._(cf. (47.5) and i
(47.6)). The first correct statement is due to Michell [1900, 3], p. 109. Reference should ,
also be made to a remark by Maxwell [1870, 1], p. 161, who appears to anticipate Levy's
claim, but I am unable to follow Maxwell's argument.

156 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 47.
simply-connected). Much of the experimental work in photoelasticity.theory is
based on this result. ~
(4) Levy's theorem. Assume that R is simply-connected. Let \ua, Sai8] and
[ua,Sap} be plane elastic states corresponding to positive definite elastic constants
fi, v and fx, v, respectively. Suppose, further, that Sa/8 and 5a/8 are smooth on R and
Safi»fi=Safinfi on dR.
Then
Proof. Clearly, Sai8 satisfies (a) and (b) of (3). Thus there exists a field ua
such that [«a,5ajS] is a plane elastic state corresponding to /j, and v. Therefore
[ua, Sap] and [ua,Sap] correspond to the same elastic constants and the same
surface forces. Thus we conclude from the uniqueness theorem (2) with ^=0
and <ra = dR that SaP =Safi on It. □
47. Airy's solution. In this section some of the functions we study will be
multi-valued. Therefore, for this section only, if a function is single-valued we will
say so; if nothing is said the function is allowed to be multi-valued. Thus the
assertion "<p is a function of class CN" means that "<p is a possibly multi-valued
function of class CN". However, we will always assume that the displacement
and stress fields corresponding to a plane elastic state are single-valued.
We continue to assume that the body forces vanish. Then, as we saw in the
previous section, a plane stress field is completely characterized by the following
pair of field equations:
Sa0,p=O, ASaa=0.
The next theorem shows that these equations can be solved with the aid of a
single biharmonic function.
(1) Airy's solution. Let <p be a scalar field of class Cs on R, and let
•->11 = <P,22< ^22 = 9^,11) S12— —<Pt12- (a)
Then
Sap.fi = 0. (b)
Further,
^5aa=0 (c)
if and only if <p is biharmonic.2
Conversely, let Sa/8 ( = Spa) be single-valued and of class CN (N~2z\) on R, and
suppose that (b) holds. Then there exists a class CN+2 function <ponR that satisfies (a).
Moreover, <p is single-valued if R is simply-connected.
Proof. Clearly, (a) implies (b), while cp biharmonic is equivalent to (c). To
prove the converse assertion assume that (b) holds. Then
(•->ii),i = ( 512)2.
( ^12),1 = (^22),2-
1 This solution is due to Airy [1863, 1]. See also Fox and Southwell [1945, i], Schaeper
[1956, 11], Teodorescu [1958, 19], [1963, 23], Strubecker [1962, 15], Schuler and
Fosdick [1967, 13]. A generalization for the dynamic plane problem is given by Radok
[1956, 10]. See also Sneddon [1952, 5], [1958,17], Sveklo [1961, 21].
2 Maxwell [1870, 1] was the first to notice that <p is biharmonic. See also Voigt [1882, 1],
Ibbetson [1886, 3], [1887, 2], pp. 358-359, Levy [1898, 7].

Sect. 47- Airy's solution. 157
and there exists class CN+1 functions g and h on R such that
•->ii=g,2> ^12 — S, 1» S2a = hilt 512= »>2. (d)
Hence
g,i=A,a.
and there exists a class 0^+2 function 97 such that
g = <P,i> h = <P,i- (e)
Eqs. (d) and (e) imply (a). □
Henceforth, by an Airy function we mean a (possibly multi-valued) bi-
harmonic function <p on R with the property that the associated stresses
^11 == 9?, 2 2 > ^22 = ^,11) '->12== ^,12
are_single-va,Lued. For future use we note that these relations can be written in
the concise form:1
where sai8 is the two-dimensional alternator.
By (a) of (1), any two Airy functions <p and <p generating the same stresses
differ by a linear function of position, i.e.
(p{x)-q>(x)=kaxa+c,
where ka and c are constants and x=(x1, x2). Conversely, we can always add a
linear function of position to an Airy function without changing the associated
stresses. Further, given a point y = [ylt y2) in R we can always choose the Unear
function so that the resulting function <p satisfies
9^)=^(^)=0.
The (two-dimensional) ffractjgn_problem\for a simply-connected region
consists in finding stresses Sa/8 that satisfy the field equations
Sa/U,=0, ^5aa=0,
and the boundary condition
Safinfi=sa on 8R,
where the sa are the prescribed surface tractions. In view of the last theorem, this
is equivalent to finding a scalar field <p such that
AA<p^=0,
e*ifyT<P,iTnp=s* on dR-
Thus the traction problem reduces to finding a biharmonic function q> that
satisfies the above boundary conditions. Our next theorem, which gives a physical
interpretation of the Airy function, allows us to write the boundary condition
for <p in a more familiar form. With a view toward a concise statement of this
theorem, let
r ^{x(a):0^a^a^
1 Finzi [1956, 5].

158
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 47.
■ be a piecewise smooth curve in R parametrized by its arc length a, and let na
! and ta designate the normal and tangent unit vectors along r, i.e.,
; where xa = dxjdo.
(2) Physical interpretation of the Airy function.1 Let cp be an A iry function
and write
<p(a)=<p(x(a)), -8l (a)=(p,a{x(a))na{a)
for the values if <p and its normal derivative along the curve r. Assume that
9>(0)=9>,«(0)=0.
Then
<p(a)=m{o),
2>)—'(*>.
where , , ac
m(°0 = I e*p[*«(<?') -xa(<?)] Sp{a') da', ^'' ^(,0,
* (*) = k W / ** (<?') iff'. w *^T*^
0
s<x (<r)=Safi(x{a))nfi(a).
Proof. By hypothesis
s«fi = Baliefir<piXr. (a)
Thus, in view of the identity
eaxeaT = dlr, (b)
the surface force field sa = Sai8 ^ can be written in the form
s<x =¾^ epx<piXT np = eaXcpjXtxr.
In view of (b),
<P, At x* — —EXrsr>
and thus
da(<P,x) = -eXlsr. (c)
Therefore, since <p x{0) =0,
<?>»=/», (d)
where
fxW) = -f BXrsT{a') da'. (e)
o
On the other hand, since cp (0) = 0,
(p(a) = fcp(a') da' = f <p x{p') xx(a') da'. (f)
o o
If we integrate (f) by parts and use (d), we arrive at
<p(<y)=hxx\ao-fh(o')xx(a')da'.
o
1 Michell [1900,3], p. 107 was the first to notice that q> and dtpjdn are essentially
determined by the surface forces along J1. The relations given here, which follow from a minor
modification of Michell's results, are due to Trepftz [1928, 3]. See also Phillips [1934, 5],
Sobrero [1935, 8], and Finzi [1956, 5].

Sect. 47.
Airy's solution.
159
By (e), fx(0) = 0 and fx — — sXrsr; therefore
a a
<p(e) = -f xx{a) sXr sr(a') da' + J xx{a') eXr sr{a') da',
o o
or equivalently
<p (a) — m (a).
Next, by definition,
dm
and (d), (e), and (b) imply
U («) = -^). D
If <p is of class C2 on R, then the results of (2) hold even when F is contained
in the boundary of R. If R is simply-connected, its boundary consists of a single
closed curve; thus taking; r^d^jjo^^X-^^-^^J^L^?'^^^^^^^^^^^
reduees,-to finding a biharmonic function.<p that satisfies.the "standard"
boundary conditions:1
<p = m and -„■ = — I on 32?,
where m and I are completely determined by the prescribed surface tractions
on dR.
If cp and <p a do not vanish at #(0), we can still apply (2) to the Airy function
y{x)=cp{x)~ka xa — c,
with the constants ka and c chosen so that
£(*(0))=£,«(*(0))=0.
Thus, in this instance, the relations for <p and dyjdn should be replaced by
q> = m+kaxa+c,
If R is multiply-connected, its boundary dR is the union of a finite number
of closed curves ^,¾. ■■■>^m- ^n this case it is not possible to adjust the Airy
function (without changing the stresses) to make <p and cp a vanish at one point
on each ^m. Thus relations of the form (*) will hold on each #m; of course, the
constants will, in general, be different on each #m:
<p=m+ti£xa + c» and |^-= —/ + &"»„ on <6m (m = i, 2, ..., M).
In the above relations &„ and cm can be set equal to zero on one of the curves.
^m,' as we shall see, the remaining constants can be evaluated using the
requirement that the displacements be single-valued. I
With a view toward deriving a relation for the displacement field in terms of
the Airy function, we now establish an important representation theorem for
the Airy function in terms of a pair of analytic functions of the complex variable
z = x1-}- i%z ■
1 Michell [1900, 3], p. 108.

160 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 47-
(3) Goursat's theorem.1 Let <p be biharmonic on R. Then there exist analytic
complex functions ip and % on R such that2
<p{x±, x2) = Re{zip(z) +%(*)},
or equivalently,
2<p(Xl, x2)=zip{z) +z^) + %[z) +%{*)■
Conversely, if ip and % are analytic functions and cp is defined on R by the above
relation, then cp is biharmonic on R.
Proof.3 Assume that cp is biharmonic on R. Since A<p is harmonic, there
exists an analytic function / such that A<p is the real part of /. Let ip be a solution
of the equation
dy) 1
and let
a,=V' (a)
?=*(&+*■&). (b)
where g1 and g2 are real. Then, since ga is harmonic,
A(q> — iga*a)=Aip-U*,a- (C)
By the Cauchy-Riemann equations,
gl,l=g2,2. (d)
and by (a) and (b),
gi.i=A<Pl (e)
thus (c) implies that
A(9-ig.*J=0. (f)
Let
qi = <p-kgax«- (g)
In view of (f), qr is harmonic and hence the real part of an analytic function %.
This fact, (g), and (b) imply that
<P(*i, %%) =Re{zy){z) +%{z)}. (h)
Conversely, let cp be given by (h) with ip and % analytic. Then, by reversing
the above steps, it is a simple matter to verify that cp is biharmonic. □
(4) Representation theorem for the displacement field.* Let [ua, Sai8]
be a plane elastic state corresponding to the elastic constants /u and v (with ,«4=0).
Further, let cp be an Airy function corresponding to Sap, and let
4f{z)=g1(x1,x2)+ig2{x1,x2),
where ip is the analytic complex function established in (3) and ga is real. Then
where wa is a plane rigid displacement.
Conversely, let cp be an Airy function of class C2 on R, let Sa/8 be the associated
stresses, and let ua, defined as above, besingle-yalued. Then [ua, Sap] is a plane
elastic state corresponding to /x and v.
1 Goursat [1898, 6].
2 Here Re{ } denotes the real part of the quantity { }, and a bar over a letter designates
the complex conjugate.
3 This proof is due to Muskhelishvili [1919, i], [1954, 16], § 31.
4 Love [1927, 3], § 145- See also Clebsch [1862, 1], § 39-

Sect. 47. Airy's solution. 161
Proof. Using (a) of (1) we can write the displacement-stress relations in the
form
2/x «M = — q>M + (i~v)A<p,
2/iu22 = — <p22 + (i—v)A<p, (a)
Thus we conclude from (d) and (e) in the proof of (3) that
2p Mjj = — <prll + (1 -v) glfl,
2^^2,2 = -^,22 + (^ -^)^2,2.-
hence
2/n*a = -(pia-\-(i~v)ga+2fiWa, (b)
where
^1,1=^2,2=0. (c)
Substituting (b) into the third of (a), we find, with the aid of the Cauchy-Riemann
equation
Si,z = £2, i>
that
^1,2+^2,1=0. (d).
By (c) and (d), wa is a plane rigid displacement; thus (b) is the desired
representation for ux.
To prove the converse assertion we simply reverse the steps (a)-(d). □
The next theorem yields conditions that must be satisfied by the Airy
function in order to insure that the corresponding displacements be single-valued.
(5) Theorem on single-valued displacements. Let R be
multiply-connected, and suppose that
M
m=l
where each ^m is a closed curve. Let <p be an Airy function of class C3 on R, and let
u«=tV ^- ^«+(1 ~") g<*] +w«'
where wa is rigid and ga is the function defined in (4). Then a necessary and
sufficient condition thai the displacement field ua be single-valued on R is that <p satisfy
the 3M —3 conditions:1
f
/[
Tom
1 These relations are due to Michell [1900, 3], p. 108. Prager [1946,. £| proved that for_a
4SSMZ^~'R§5i24jCSgiPitilese.r|?i^!0fl5.are the natural boundary conditions of a variational
principle for the Airy function.
~~HaSabubh aeirTHysiit, Bd. VIa/2."" 11

162 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 47.
for m = 2, 3,..., M, where1
and Sa g is the stress field generated by q>.
Proof. The field ua will be single-valued if and only if
J^«do=0 (m = 2,l,...,M),
or, in view of the definition of ua, if and only if
/l^^^/^W^ ^ = 2'3 M)- (a)
^m 'Km
It is a simple matter to verify that (c) in the proof of (2) is valid in the present
circumstances in which <p and <pa do not necessarily vanish at some point on
<&„. Thus (a) can be written in the form
/-^ = -1¾^ ^
where here and in what follows m is assumed to have the range (2, }, ..., M).
Clearly,
*£=&*.i>h (c)
on <Sm, where
x(a)=(x1(a),x2(a)), O^a^a^
is the parametrization for <Sm. Now let P1 and P2 denote the real and imaginary
parts of the complex function / established in the proof of (3):
f(z)=P1(xl, x2) + iP2{x1, x2).
By (a) and (b) in the proof of (3),
thus
gl,l=Pl> #2,1= A.
gl,2 = "2» 82,2="l>
and by (c) and (d) the relation (b) is equivalent to
(d)
f[Ptil + P1i9] da = -^^,
Win
J[P1i1-P^2]do = --1^vf,
(e)
,m)
2 •
Integrating the left-hand side of (e) by parts, we arrive at
(f)
1 We assume that the parametrization of the ^m are such that na = eap ia coincides with
the outward unit normal to 8R.

Sect. 47- Airy's solution. I63
It follows from the definition of the Airy function that P1=A<p is single-valued.
Thus for (f) to hold it is sufficient that
/l<
f dp2
J da
dP% dPx
'i~da +X2~da~
dP1 dP,]
1 da %2 da J
do=0,
1— V
da = --L~/j»>
i—v
(g)
Indeed, the first of (g) asserts that P2 is single-valued, and the second and third
of (g) are equivalent to (f) when both Px and P2 are single-valued. Next, by (d)
and the definition of ux,
^2=-73^ K,2-«2,i-«]. (h)
where co = w12—w2x is constant (since wa is rigid). To prove that (g) is also
necessary for (f) to hold it suffices to show that P2 is single-valued whenever ua
is single-valued. But this fact follows from (h).1 Finally, the Cauchy-Riemann
equations and the relation na = ea/5 x^ imply that
j ~ = °2,1 Xl + -^2,2 x2 = -^1,2% + -M.l x2
(i)
=^.,-,+3.^=-^=-^(^).
In view of (i) and since P1 = A<p, the relations (g) reduce to the desired 3M—3
conditions. □
The next result extends (46.4) to multiply-connected regions. It asserts
that the stresses corresponding to a solution of the traction problem with null
body forces are independent of the elastic constants if the resultant loading on each
boundary curve vanishes.
X(6) Michell's theorem.2 Let R be multiply-connected, and suppose that
M
dR={JVm,
m=l
where each ^m is a closed curve. Let [wa. Sa/3] and [wa,5a/3] be plane elastic states
corresponding to positive definite elastic constants (i, v and jx, v, respectively, and
assume that Sa/3 and 5a/5 are smooth on R. Then
Safj=Sx/) on R
provided both states correspond to the same surface tractions and the resultant force
on each ^m vanishes, i.e.,
Sa/3M/3=Sa/3M/5 = sa on dR
and
Jsada=0 (m = i, 2,...,M).
1 Since ua is analytic, uaa is single-valued whenever ua has this property.
2 [1900, 3], p. 109.
11*

164 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 47-
Proof. By (1) there exists an Airy function y generating 5a/3. Moreover,
<p is of class C3 on R, since 5a/5 is smooth there. Since [wa, 5a/3] is a plane elastic
state, we conclude from (4) that
with K>a rigid. Moreover, since ua is single-valued, and since the resultant force
on each ^, vanishes, it follows from (5) with /£"' =0 that
(a)
form = 2, 3, .
11^
j\Xl-ln{A<p) + x2^a{Acp)
j\Xl-l(A<p)-x2-l-(A<p)
da = 0,
da = 0,
da = 0,
., M. Now let
K= i-l-V.ai+ii—v)
&]•
By (a) and (5), ua is single-valued, and hence the converse assertion in (4)
implies that [ua, Sa/3] is a plane elastic state corresponding to /i and v. Therefore
[ua, Sa/}] and [ua, Sa/5] correspond to the same elastic constants and the same
surface forces. Thus we conclude from the uniqueness theorem (46.2) that
5a/; =3a/5 on K. □
(7) Kolosov's theorem.1 Let [ua, Sa/5] be a plane elastic state corresponding
to the elastic constants fi and v. Then there exist complex analytic functions tp and %
on R such that2
l
u1(xi,x2)+iu2(x1,x2)= 2fi [0-4v)f(z)-Zf'(z)—yjzJ]+w(z),
5u(%. x2)+S22(x1, x2) =4Re{y (*)},
5n(%. x2)-S22{xlt x2)+2iS12{x1, x2) = 2\zy>"(z) + %'(*)],
where w is a complex rigid displacement.
Conversely, let y> and % be complex analytic functions on R, and let ua and Sa/5
be defined by the above relations. In addition, assume that ua is single-valued and
ua and Sap continuous on R. Then [ua, Sa/5] is an elastic state corresponding to fi
andv.
Proof. Let q> be an Airy function generated by Sa/5. By Goursat's theorem (3),
there exist analytic complex functions f and % such that
<p(¾, x2) = Re{i f (z) + x(*)}■ (a)
A simple calculation shows that
q>,i(x1,x2)+i<pt2(x1,x2)=tp{z)+z^)+YWJ. (b)
1 Kolosov [1909, 2], [1914, 2], [1935, 2]. A related complex representation theorem was
established earlier by Filon [1903, 2]. See also Muskhelishvili [1932, 7], [1934, 3], [1954,26],
§ 32; Stevenson [1943, 2], [1945, ?]; Poritsky [1946, 3]; Sokolnikoff [1956, 22], Chap. 5;
Solomon [1968, 22], Chap. 6, § 7.
2 Here y>' = dipjdz.

Sect. 48.
Representation of elastic displacement fields.
165
Thus we conclude from (4) that the desired relation for ua holds. Further, by (a)
of (1),
Sn + iS12 = - t{q>A + *>>2)>2,
(c)
S22 — iS12=((pA + i(pf2)fl,
and (b), (c) imply the desired results concerning Sa/5. Conversely, let y> and % be
complex analytic functions on R, let ua and Sa/5 be defined by the relations in
(7), and let <p be defined by (a) so that (b) holds. Then it is not difficult to verify
that u satisfies the relation in (4), and that Sa/5 satisfies (c). Thus Sa/5 are the
stresses associated with <p, and we conclude from the converse assertion in (4)
that [ua, Sa/3] is an elastic state correspondign to /i and v. □
VIII. Exterior domains.
48. Representation of elastic displacement fields in a neighborhood of infinity.
The conventional proofs of the fundamental theorems of elastostatics are
confined in their validity to bounded domains. Of course, they can be trivially
generalized to exterior domains provided one is willing to lay down sufficiently
stringent restrictions on the behavior at infinity of the relevant fields, e.g., that
the displacement be 0 (r'1) and the stress 0 (r~2) as r-±oo. However, such a priori
assumptions are quite artificial: the rate at which these fields decay is an item
of information that one would expect to infer from the solution, rather than a
condition to be imposed on a solution in advance.
In this section we shall establish representation theorems for elastostatic
fields in the following neighborhood of infinity:
^(oo) = {x: r0< \x — 0| < oo},
where r0 > 0 is fixed. These representation theorems will be used in subsequent
sections to infer the rate at which elastic states decay at infinity, and this, in
turn, will be utilized to extend the fundamental theorems of elastostatics to
exterior domains.
We assume that the body B=Z(oo) is homogeneous and isotropic, and that
the elasticity tensor is positive definite; thus the Lame moduli satisfy
fi>0, 2(iJr'iX>0.
(1) Representation theorem.1 Let u be an elastic displacement field on
.S(oo) corresponding to zero body forces. Then u admits the representation
oo oo
u=2 0<*> + r«2 *w. (a)
k=—oo k = — oo
where </<*> and h^ are vector solid spherical harmonics of degree k, and
Both of the series in (a), as well as the series resulting from any finite number of
termwise differentiations, are uniformly convergent on every closed subregion ofZ(oo).
Proof. First of all, it is not difficult to verify that the denominator in (b)
cannot vanish, since the elasticity tensor is positive definite.
1 Kelvin [1863, 3]. See also Gurtin and Sternberg [1961, 11~\, Theorem 1.2. A
generalization of this result to anisotropic bodies is given by Bezier [1967, 3].

166 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 48.
In view of (42.2), u is biharmonic on Z(oo). Theorem (8.5) therefore assures
the existence of an expansion of the form (a) which has the asserted convergence
properties.
This leaves only (b) to be confirmed. A simple computation shows that
A {r* *W) = 4 (VhW) p + 6fc<*>,
and thus the identity
(p%(*>)p = ftft<*>, (c)
which is valid for any vector solid harmonic hw, implies
A(r2hW) = 2(2k + 3)hW.
Hence
Au = 2 2 (2* + 3)Aw. (d)
A=— 00
Next, by hypothesis and by (42.2),
divZlM=0, curl Au = Q;
thus (d) implies
00 00
2 (2£ + 3)divfc<*>=0, 2 (2& + 3)curlfc<*>=0. (e)
k=— 00 A=— 00
But divfc'*' is a solid harmonic of degree k — i. Hence the first of (e), by virtue
of the uniqueness of the expansion in (8.5), implies
divfc<*> = 0. (f)
Similarly,
curlfc(*> = 0,
or equivalently,
|7fcW = [|7fc(*)]r. (g)
Consequently, (a), (c), (f), and (g) imply
00
Vdivu= 2 [Vdivgw+2(k + i)hm]. (h)
k=—00
By hypothesis,
/jtAu+(X + /j,)Fdivu=Q, (i)
and substitution of (d) and (h) in (i) yields
00
2 {2[(A + l)(A + 3/«)+/«]^)+(A + ia)Fdiv9W} = 0,
which implies the desired result (b). □
The next theorem gives a particular solution of the equations of elastostatics
in a neighborhood of infinity.
(2)1 Let b be a smooth vector field on -S(oo), and assume that
div6=0, curl 6=0.
1 Gurtin and Sternberg [1961, 11], Theorem 4.3.

Sect. 49-
Behavior of elastic states at infinity.
167
Then:
(i) 6 admits the representation
00
6=2 &w,
A=— 00
where the b(*' are divergence-free and curl-free vector solid spherical harmonics of degree
k, and this infinite series, as well as the series resulting from any finite number of
termwise differentiations, is uniformly convergent in every closed subregion of 2 (00);
(ii) the same convergence properties apply to the series
tt=-— y 1 b<*>
2^^ + 1)(/1 + 3^)+^
and the field u so defined is an elastic displacement field on ^(00) corresponding to
the body force field b;
(iii) for any integer n,
b(x) = o(r") as r-±00,
implies
u(x) = 0(rn+1) as r-*-oo.
Proof. By (8.6) b is harmonic. Thus (8.4) implies part (i) of this theorem.
The uniform convergence of the series of (ii), and of the corresponding series
resulting from successive differentiations, is apparent from the convergence
properties of the series in (i). Thus u is of class C°° on -S(oo). That u satisfies the
displacement equation of equilibrium may be verified by direct substitution. This
establishes (ii).
To justify (iii) we note from (i) and (8.4) that b(x) =o(rn) implies bm(x) =0
for k^n. Bearing in mind (ii) and appealing once more to (8.4), we see that (iii)
holds true. □
49. Behavior of elastic states at infinity. We now use the representation
theorems developed in the preceding section to determine the behavior of elastic
fields at infinity.
We continue to assume that B =Z(oo) is homogeneous and isotropic and that
the Lame moduli obey the inequalities
fi>0, 2/i + }h>0.
In addition, we assume that the body force field 6 is smooth on .2 (00), and that
div 6 = 0, curlb=0.
(1) Behavior of elastic states with vanishing displacements at
infinity.1 Let [u, E, S] be an elastic state on £(00) corresponding to b, and assume
that
u(x)=o(\) t->
as r->oo. Then
u{x)=0{r-i), S(x)=0(r-*),
E (x) = 0 (r~2), b(x) = 0 (r~3)
as r-^00.
1 Fichera [1950, 4]. See also Gurtin and Sternberg [1961,11], Theorem 5.1. Kupradze
[1963,27,25], §111.3 established the estimate M=0(>-"1), but only with the additional
assumption that dujdr = 0 (y-1). Note that if u = 0 (1), then div u and curl u are both 0(r~2), a
conclusion which Duffin and Noll [1958, 8] reached by entirely different means.

168 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 49.
Proof. By (42.2) u is biharmonic on £(00). If we apply (8.5) to u, we
conclude that u=0(r'1). The remaining assertions then follow immediately from
(iii) of (8.5) in conjunction with the strain-displacement relation, the stress-
strain relation, and the equation of equilibrium. □
(2) Behavior of elastic states with vanishing stresses at infinity.1
Let [u, E, S] be an elastic state on 27 (00) corresponding to b, and assume that
S(x) = o(i), b{x) = o{r-*)
as r -^00. Then
u(x)=w(x)+0(r~1), S(x) = 0(r-Z),
E(x)=0(r-2), b(x)=0(r-3)
as r->oo, where w is a rigid displacement field.
Proof. In view of the assumed properties of 6, we infer from (48.2) the
existence of a particular solution u of the displacement equation of equilibrium
that is class C2 on I, (00) and satisfies u =0(r-1). Accordingly, by (1), the elastic
state associated with u (i.e., generated by the strain-displacement and stress-
strain relations) satisfies E=0(r'2) and S=0(r~2), while b must conform to
b—0(r'3). We are thus able to exhibit a particular elastic state corresponding
to 6 which possesses the requisite orders of magnitude at infinity. Therefore,
since the fundamental system of field equations satisfied by elastic states is
linear, it remains to be demonstrated that (2) is true when b = 0.
Suppose now that 6=0. Then, by hypothesis and by (48.1), u admits
the representation
u = u-\-u,
where
00 00
tt=I</<*>+>"2 2 Aw,
k=— 00 k=~ 00
and where the solid harmonics g^k\ h^ obey (b) of (48.1)
Next, define fields E and E through
E = ${Vu+FuT),
E = i(Vu+FuT),
so that
E = E + E.
The two infinite series entering the second of (a) each represent a harmonic field
on -S(oo). The second of (b) and part (iii) of (8.5) therefore imply
£ = 0(7--1). B = 0(r"2), (d)
while we conclude from the stress-strain relation that
S = 0(r-*).
Consequently, all we have left to show is that u is rigid; or equivalently that
E = 0. (e)
(b)
\
(c)
1 Gurtin and Sternberg [1961, 11], Theorem 5.2.

Sect. SO. Extension of the basic theorems. 169
To this end we conclude from (b) in (48.1) that
divfc<*> = 0,
and hence reach, with the aid of (a) and (b) above,
o oo oo
trB=2divg'*»+2 2 P'*(*'. (f)
where p(x) =x — Q. Next, since div</(*+2) is a solid harmonic of degree k + i,
p • Fdivsr<*+2> = (k + l) divg<*+2>, (g)
and since gm is a constant,
VgW=0. (h)
Thus if we eliminate fc'*' from (f), by recourse to (b) of (48.1), and use (g) and (h),
we arrive at
trE=y ?<2*-1>~_divoW (i)
By hypothesis, /S =o(l); thus the stress-strain relation implies
E = o(l),
and we conclude from (c) and the second of (d) that
E = o(\). (j)
Thus if we apply (8.4) to the harmonic function tr E given by (i), bearing in
mind that div </<*' is a solid harmonic of degree k — 1, we find that
divg(*' = 0 for k>\,
which, together with (b) of (48.1) and (h), yields
fc<*> = 0 for k^ — 2. (k)
Inserting (k) in the first of (a), and using (h) and the first of (b), we obtain
E = zvgm. (1)
Since Vg'*' is a solid harmonic of degree k — 1, we conclude from part (iii) of (8.4)
that (1) is consistent with (j) only if (e) holds. □
50. Extension of the basic theorems in elastostatics to exterior domains. We
now use the results established in the last section to extend the basic theorems of
elastostatics to exterior domains. Thus for this section we drop the requirement
that B be bounded and assume instead that B is an exterior domain. We shall
continue to assume that B is homogeneous and isotropic with Lame moduli fi and A.
Let 6= [u, E, S] be an elastic state on B corresponding to the body force
field 6, and suppose that the limits
/= lim / sda,
Q-+CO iSp
m-- lim / pxsda
e-*°o dSp

170 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 50.
exist, where s=Sn, n is the outward unit normal to dXe, and Ee is the ball of
radius q and center at 0. Then we call / and m the associated force and moment
at infinity. Since S and b satisfy balance of forces and moments in every
part of B,
fsda-irJbdv = — f s da,
SB B„ d£p
f pxsda+ f pxbdv = — f pxsda,
SB B„ d£p
for sufficiently large q, where BQ = BnZe} Thus, letting g-^-oo, we see that /
and m exist if and only if the integrals
fbdv, fpxbdv
B B
exist, in which case
/= — J sda— J b dv,
3B B
m = — Jpxsda— f pxb dv.
dB B
(1) Extension of Betti's reciprocal theorem.2 Assume that the Lame
moduli obey /i>0, 2/i + 3A>0. Let [u, E, S] and [u, E, S] be elastic states
corresponding to the external force systems [b, s] and [b, s], respectively, with b and b
divergence-free and curl-free and of class C2 on B. Further, assume that the associated
forces f, f and the associated moments m, rh at infinity exist. Finally, assume that
u(x) = w(x) + o(l), \
u(x) = w(x) + o(l),
as r-^-oo, where w and w are the rigid displacements:
w = uao + (oaoxp,
w = u00 + co00xp.
Then
fs-uda+ f b-udv + f-u^ + m-cox
dB B
= J s-uda+ J b -udv + f-u^ -f-m • (ox = JS-Edv=JS-Edv.
dB B B B
Proof. Applying Betti's theorem (30.1) to the region BQ=BnZt, with q
large enough so that dBCZe, we find that
/ s • uda-\- f b • udv~\- J s- uda
SB B„ SS„ , .
~ ~ (a)
= Js-uda + Jb-udv-{-Js-uda=JS-Edv = JS-Edv.
SB B„ SXP B„ B„
Clearly,
/ s • u da = / s • (it — w) da + / s • wda (b)
ssp ss„ ss„
1 Note that 8Be = dB \jdSe for q sufficiently large, since B is an exterior domain.
2 This result for the case in which w —w = 0 is due to Gurtin and Sternberg [1961, 11],
Theorem 6.1.

Sect. 50.
Extension of the basic theorems.
171
and
/ s-wda= f s- (un + co^xp)da = uai • f sda + co^- f pxsda. (c)
By hypothesis and by (49.1),
u = w + 0(r'1), u = w + 0 (r"1),
E = 0{r-*), S = 0(r~2), b = 0{r~s), (d)
E = 0(r~2), S = 0(r~2), b = 0{r~3).
Thus (b) and (c) imply
fs-uda^f-u^+m-con (e)
as g-s-oo. Similarly,
/ suda^fux+ma>x (f)
as q-^-oo. The estimates (d) also imply that the volume integrals over Be in (a)
are convergent as q-^-oo and tend to the corresponding integrals over B. Thus
if we let g-^-oo in (a) and use (e) and (f), we arrive at the desired conclusion. □
As a direct corollary of this result we have the
(2) Theorem of work and energy for exterior domains.1 Let [u, E, S],
[b, s], f, m, as well as /j, and A obey the same hypotheses as in (1). Then
f s-uda + f b-udv + /•tt00+m-oa) = 2UC{E}.
dB B
This relation differs from the corresponding result for finite regions by the
addition of the term fu^ + m • co^, which represents the work done by the
force and moment at infinity.
(3) Uniqueness theorem for exterior domains.2 Assume that
fi>0, 2j«+3A>0.
Let 4 = [u, E, S] and ~i = [u, E, S] be elastic states corresponding to the same body
force field and the same boundary data in the sense that
u = u on Sfx, s = s on £f2. (a)
Further, assume that either
u(x) = u{x)-\-o{\) as r-^oo, (b)
or both
S(x)=S(x)+o(\) as r-^oo (c)
and the corresponding traction fields are statically equivalent on dB, i.e.
J sda = f sda, J pxsda— J pxsda. (d)3
dB dB 8B dB
Then o and 1 are equal modulo a rigid displacement.
1 Gurtin and Sternberg [1961, 11], Theorem 6.2, for the case in which ic = 0.
2 Fichera [1950,4], Theorems II, V, and VI for case(b); Gurtin and Sternberg
[1961, 11] for case (c). See also Duffin and Noll [1958, 8] and Bezier [1967, 3]. Analogous
uniqueness theorems for two-dimensional elastostatics are given by Tiffen [1952, 6] and
Muskhelishvili [1954, 16], § 40.
3 Note that this condition is satisfied automatically when B is finite.

172 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 50.
Proof. Let <j'= <j— 3. It suffices to prove that
E' = 0. (e)
Clearly, o' is an elastic state corresponding to zero body forces and vanishing
boundary data. Assume (b) holds. Then
u'{x) = o{\) as r-+oo,
and (2) (with u'^ = to'^ = 0) yields
W} = o, (f)
which implies (e). Next assume (c) and (d) hold. Then
S' (a5) = o(l) as r->oo,
and we conclude from (49.2) that
u' (x) — w (x) + 0 (r'1) as r->oo
with w rigid. Further, by (d),
/'=— fs'da = 0, m' = - fpxs'da = 0,
dB dB
and (2) again implies (f) and hence (e). □
It should be noted that (3) is valid only for unbounded regions with
> finite boundaries. The uniqueness question associated with problems for general
domains whose boundaries extend to infinity is yet to be disposed of
satisfactorily.1
We emphasize that theorem (3), in contrast to (1) and (2), involves no >
explicit restrictions concerning the body force field. If this field is integrable '
. over B, then (d) can be replaced by the requirement that the forces and moments
at infinity coincide:
f=_f, m = m.
For the surface force problem, (d) may be omitted because it is implied by (a). On the
other hand, if £^ is not empty, (d) is an independent hypothesis. This hypothesis,
which requires the prescription of the resultant force and moment on dB (or
equivalently, the force and moment at infinity), appears at first sight artificial,
since the surface forces over at least a portion of the boundary are not known
beforehand. One is therefore led to ask whether (d) is necessary for the truth of
(3) when the regularity conditions are taken in the form (c). That this is indeed
the case is clear from the next theorem.
(4) Theorem (3) is false if either of the two hypotheses in (d) is omitted.
Proof. For the purpose at hand, let
B={a;:|a5-0|>1},
and assume that 8B =Sfx (6^ = 0) in (3)- It clearly suffices to exhibit two elastic
states, both corresponding to zero body forces, both having strains not identically
zero, both satisfying
m=0 on dB, (a)
S = o{\) as r->oo, (b)
1 For the special case of the displacement and surface force problems appropriate to the
half-space, this question was settled by Turteltaub and Sternberg [1967, IS]. See also
Knops [1965, IS].

Sect. 51- Basic singular solutions. 173
with one state obeying
f sda = 0, (c)
dB
the other
f pxsda=0. (d)
dB
To this end, let ^, =(=0 and to4= 0 be given vectors and consider the following
two displacement fields :1
« (-) = ^ + V(^ ™*(~^) " » ^ ®
Both of these displacement fields generate elastic states that correspond to zero
body forces and obey (a), (b). Moreover, for (e):
f= I sda = 24n/t--^- X+0, m= I pxsda = 0;
dB dB
while for (f):
/=0, m = 87tfi(o. □
The leading two terms in (e) represent the displacement induced in the
medium by a rigid translation A applied to the spherical boundary dB, the body
being constrained against displacements at infinity; the last term in (e)
corresponds to a rigid translation —A of the entire medium. On the other hand, the
leading two terms in (f) represent the displacement generated by a rotation co
of dB, while the last term corresponds to a rotation of the entire medium.
IX. Basic singular solutions.
Concentrated loads. Green's functions.
51. Basic singular solutions. In this section we study the basic singular solutions
of elastostatics, assuming throughout that the body is homogeneous and isotropic,
and that /* 4= 0, v 4= $» 1 • We begin with the Kelvin problem,2 which is concerned
with a concentrated load applied at a point of a body occupying the entire
space S. With a view toward giving a solution of this problem, we introduce the
following notation.
Throughout this section y is a fixed point of S,
r = x — y
is the position vector from y, and
r= \r\.
Further, we write 21, for the open ball Ev (y) with radius r\ centered at y.
1 Gurtin and Sternberg [1961,11] attribute these examples to R. T. Shield.
2 The solution to this problem was first given by Kelvin [1848, 3] (cf. Somigliana
[1885, 4], Boussinesq [1885, !])• It was derived by Kelvin and Tait [1883, 1], §§ 730, 731
through a limit process that was made explicit by Sternberg and Eubanks [1955, 13] (cf.
Love [1927, 3], § 130; Mindlin [1936, 1], [1953, 15]). The limit formulation presented here
follows closely that adopted by Sternberg and Al-Khozai [1964, 20] and Turteltaub
and Sternberg [1968, 14]. The solution for an anisotropic but homogeneous body was given
by Fredholm [1900, 2]. A precise statement of Fredholm's result is contained in the work
of SAenz [1953, 17]. See also Zeilon [1911, 1], Carrier [1944, 1], Lifshic and Rozencveig
[1947, 4], Elliot [1948, 2], Synge [1957, 15], Kroner [1953, 14], Basheleishvili [1957, 1],
Murtazaev [1962, 12], Bezier [1967, 3], Bross [1968, 1], Cakala, DomaiJski, and Milicer-
Gruzewska [1968, 2].

174 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 51-
By a sequence of body force fields tending to a concentrated load laty
we mean a sequence {bm} of class C2 vector fields on $ with the following
properties :
(i) 6OT = 0on«f-i;i/OT;
(ii) / bmdv-+l as m->oo;
s
(iii) the sequence |/|6m|<fw| is bounded.
8
(1) Limit definition of the solution to Kelvin's problem.1 Let {bm} be
a sequence of body force fields tending to a concentrated load I at y. Then:
(i) For each m there exists a unique elastic state 6m on $ that corresponds to the
body force bm and has uniformly vanishing displacements at infinity.
(ii) The sequence {6m} converges (uniformly on closed subsets ofS—{y}) to an
elastic state 6y [I] onS' — {y}.
(iii) The limit state 6y [I] is independent of the sequence {bm} tending to I and is
generated by the displacement field uy[l] with values
uw[l}{x) = -^- -^- + (3-4^)1
I,
c= i6n/j,(l — v).
We call 6y [I] the Kelvin state corresponding to a concentrated load I at y.
Proof. Let um be defined by
Um=Wm--^i_-}V{<PmJrr-Wm), \
where
Then, by (6.5) and (44.1), um is an equilibrium displacement field corresponding
to bm. Moreover, it is easy to verify that
um(x) = o(i) as \x — 0|-*-oo,
and, by the uniqueness theorem (50.3) for exterior domains, there is no other
equihbrium displacement field corresponding to bm with this property. This
establishes (i).
Next, a simple calculation shows that the functions defined on ^-{«/} by
<p(x) = 0, y>(x) = 1^3^-. (b)
generate, in the sense of the Boussinesq-Papkovitch-Neuber solution (44.1), the
displacement field uy[l] given in (iii). Thus to complete the proof we have only
to show that given any closed region R in ^~{y}, 4m-+4y[l] uniformly on E;
or equivalently, that q>m-*-<p and y>m-+ip uniformly on E, and that the first and
second gradients of <pm and \pm tend to the corresponding gradients of cp and \p
uniformly on R. Since the argument in each instance is strictly analogous, we
shall merely prove that \pm-+\p.
1 Sternberg and Eubanks [1955.^3], Theorem 4.2. An analogous theorem for
concentrated surface loads was given by Turteltaub and Sternberg [1968, lt\.

Sect. 51.
Basic singular solutions.
175
Let R be a closed subset of S— {y}. Then there exists an m0 > 0 such that
hence
-£i/». n R = !
for xc-R and
is finite. Thus
F-01
:W„
(c)
^ = ^1^-5)= ^^
F~»
i*-*r
3c-g|-|ac-y
Kmn
for ase.fi!, ^eZ1/m, and w>w0. Next, by (a) and (b),
|lP«.(»)-1P(»)| = |«
""-Jflla'-fl
I,
$bmdv-l
(d)
izj*
l/m
1
as —i/
consequently, (c) and (d) imply
1P»(*)-»(*)I ^- „—/|ft«.|<*" + «oH| $Kdv-l
•^llm
^llm
for a? 6 R and m>m0. This inequality, when combined with (ii) and (iii) in the
definition of a sequence ot body force fields tending to a concentrated load,
implies %pm-*%p uniformly on R. □
The need for condition (iii) in the definition of a sequence of body force fields
tending to a concentrated load was established by Sternberg and Eubanks,1
who showed by means of a counterexample that conclusions (ii) and (iii) in (1)
become invalid if this hypothesis is omitted.
We will consistently write
^[l] = [uu[llEy[l},Sy[l]]
for the Kelvin state corresponding to a concentrated load I at y. Clearly, the
mapping l>-+ oy[l] is linear on V.
In view of the stress-strain and strain-displacement relations, the stress field
Sy[l] has the form
Sw[t](») = ~%{}}!Ll£^r + (l-2v)[r® 1+ I®r- (r • I) 1]}.
Let S* [I] be the unique tensor field with the property that
Sw[v]l = S*[Z]Tv
for every vector v, and let
on 8B. We call Sy [I] the adjoint stress field and sy [I] the adjoint traction field
corresponding to oy[l]. A simple calculation shows that
w=-;p^;
Note that Sy [I] is not symmetric.
1 [1955, 13], Theorem 4.3-
+ (1 — 2v) [I® r - r® I + (r ■ I) 1]|.

176 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 51.
(2) Properties of the Kelvin state.1 The Kelvin state 6y [I] has the
following properties:
(i) 6y [l] is an elastic state on $— {y} corresponding to zero body forces.
(ii) uy [I] (a?) = 0 (r'1) and Sy [I] (as) = 0 (r'2) as r->0 and also as r-+oo.
(iii) For all r]>0,
f sy[l]da = l, frxsy[l]da = 0,
»■£„ as,
where
on 8ZV with n the inward unit normal.
(iv) For every vector v,
l ■ Uy[V] = V Uy[l] , l-Sy[V]=VS*[l],
where sy [v] is the traction field on dB corresponding to <sy [v] and Sy [I] is the
adjoint traction field on 8B corresponding to 4y[l].
Proof. Property (i) follows from (44.1) and the fact that the stress functions
<p and \p defined in (b) of the proof of (1) are harmonic on ^-{y}. Properties (ii)
and the first of (iv) are established by inspection of the fields uy[l] and Sy[l].
Property (iii) can be established by a direct computation based on the form
of Sy[l]. An alternative proof proceeds as follows. Let {<jm} and {bm} be as in (1).
Then balance of forces and moments imply that
-/ smda+ f bmdv = 0,
- f rxsmda + f rxbmdv = 0,
where sm is the traction field on that side of dZn facing the point y. If we let
m->oo in the above relations and use (ii) of (1) and properties (i)—(iii) of {bm},
we arrive at the desired relations in (iii).
Finally, to establish the second of (iv) note that, by the symmetry of Sy[l]
together with the definitions of Sy [I] and s^ [I],
I ■ sy[v] = I • (Sy[v] n) = (Sy[v] I) - u = (^ [l^v) ■ n
= v(S*[l]n)=vs*[l]. □
Sternberg and Eubanks 2 have shown that the formulation of Kelvin's
problem in terms of (i), (iii), and the portion of (ii) concerning the limit as r-+oo,
which appears in the literature,3 is incomplete in view of the existence of elastic
states on d>—{y) that possess self-equilibrated singularities at t/.4 In contrast,
properties (i), (ii), and the first of (iii) suffice to characterize the Kelvin state
uniquely.5
1 Properties (i)-(iii), in the precise form stated here, are taken from Sternberg and
Eubanks [1955, 13], Theorem 4.4. Assertion (iii) can be found in Love [1927, 3], § 131; Lur'e
[1955, 10], §2.1.
2 [1955, 13].
3 See, e.g., Timoshenko and Goodier [1951, 10], § 120.
4 E.g. the center of compression defined on p. 179.
5 As Sternberg and Eubanks [1955, 13] have remarked, Trefftz [1928, 3], § 32 and
Butty [1946, 1], § 350 approached the Kelvin problem on the basis of properties (i), (ii),
and the first of (iii) but made the erroneous assertion that (ii) is a consequence of the first of
(iii). That this is not true is apparent from the state Oy [I] + a%y, where a is a scalar and "ty
is the center of compression defined on p. 179. Indeed, this state obeys the first of (iii)
without conforming to (ii).

Sect. 51- Basic singular solutions.
Let uy[l] be defined on ^—{y} by
177
so that
tf,m(»)=-^«.[«](»).
tiy[l](x) =
r®ei+ei®r- 3J±^pr _(3_4v)rji
By (42.3), u%y\l\ is an elastic displacement field on $—{y}corresponding to null
body forces. Let 3y [I] be the elastic state on &—{y} generated by uy[l]. We call
<fy[l] the doublet state (corresponding to y, I and e^).1
Note that
Thus, since
it follows that
«iffl (») = Jim \ {uy [I] (x) - uy [I] (x -he{)}.
uy[l](x-a) = uy+a[l](x),
"»M (») = Um |«» I * (») + «»+*«,
or equivalently that
(*)}.
"»ffl=J™{«»
I
~T~ f^y+hei
I
Thus w^[7] is the limit as A-*-0 of the sum of two displacement fields. These
fields are associated with two Kelvin states: the first corresponds to a
concentrated load
-;- I applied at y;
the second, corresponds to a concentrated load
— v-l applied at yJrhei.
The two load systems are shown in Fig. 13. The above relation also yields the
important result
y+he,
Fig. 13-
1 These states were introduced by Love [1927, 3], § 132. See also Sternberg and Eubanks
[1955, 13].
Haadbuch der Physik, Bd. VI a/2. 12

178 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 51-
The next theorem is the analog of (2) for doublet states; its proof is strictly
analogous and can safely be omitted.
(3) Properties of the doublet states.1 The doublet state ~iy\l\ has the
following properties:
(i) ^ffl *s an Mastic s^e onS—{y} corresponding to zero body forces.
(ii) Tiy[l] (x) =0(r'2) and Sy[l] (as) =0(r~s) «se->0 and also as r-+oo.
(iii) For all rj>0,
fsy[l]da=0, frxsy[l]da = — etxl,
where
on dZn with n the inward unit normal.
(iv) For every vector v,
l-Uy[v] =V-Uy[l\.
(v) If 6y [I] is the Kelvin state corresponding to the concentrated load I at y, then
i.e.
^~uy[l] = -uy[l], -°y-Sy[l] = -Sy[l].
Property (iii) could be anticipated intuitively because of the physical meaning
attached to the doublet states (cf. Fig. 13). As is clear from (iii), the singularity
of 3y [I] is statically equivalent to a couple or to null according as
e^xZ^O or e,xJ = 0.
In contrast to properties (i)—(iii) of (2), which characterize the Kelvin state
uniquely, (i)-(iii) of (3), clearly, do not supply a unique characterization.2 Indeed,
*y [' + a e«•] (no sum) has the same properties as Zy [I].
We call
the unit Kelvin state corresponding to a unit load at y in the direction of e{.
On the other hand, the unit doublet states at y are defined by
By (3) the singularity of Z'J at y is statically equivalent to a couple or to null
depending on whether »4=7 or t = /. We refer to 3JJ, t'4=/» as the unit doublet state
with moment e,- X ej at y, and to dy (no sum) as the unit doublet state corresponding
to a force doublet at y parallel to the xraxis. Clearly,
0y[l] = h0y>
and the displacement fields belonging to 6y and 3y are given by
<(*) = cV[^ + (3-4v)e,](
«£ (») = ^3 [in r + U ei ~ -^- -(3- 4v) r, et
1 Sternberg and Eubanks [1955, 13], Theorem 5.2.
2 Sternberg and Eubanks [1955, 13], p. 152.

Sect. 52. Concentrated loads. The reciprocal theorem. 179
The state
S = ^4-3^4-333
°y °» ^ °y ^ °y
is called the center of compression at y, while
«w— 2eijk6y
is the center of rotation at y parallel to the %^-axis.1 Using the formula for m^,
it is a simple matter to verify that the displacement fields uy and uy associated
with ly and 6y are given by
0 , . 2(i — 2v)r
«»(«) = --—^3--
Moreover, it follows from (iii) of (3) that
/ syda = J rxsyda = 0,
327,, &i'„
/sl,<f« = 0, [ rxsyda = ei,
where
0 o- ° •
sy = Syn> Sy= Syn,
and n is the inward unit normal to dZ^.
52. Concentrated loads. The reciprocal theorem. By a system, of concentrated
loads we mean a vector-valued_ function 1! whose domain Dt is a finite set of
(boundary or interior) points of B; the vector l(£,) is to be interpreted as a concen-
trated load at £. If DtndB = 0, we say that the concentrated loads are internal.
Finally, we write I— I (£) when D{ is a singleton {£}.
Our next definition is motivated by (51.2) and the ensuing discussion. Let I
be a system of concentrated loads. We say that <s = [u, E, S] is a singular
elastic state corresponding to the external force system [6, s, I] if
(i) 6 is a (regular) elastic state on B —Dt corresponding to the external force
system [6, s];
(ii) for each ^Dt, u(x) =0(^1) and S(x) =0(r'2) as r = |a> —£|-*0;
(iii) for each %£Dt,
lim / Snda = l(Q,
where Sfe{£,) = Bnd2e(£) and n is the inward unit normal to dZe(^).
It follows from (51.2) that for £eS the Kelvin state corresponding to a
concentrated load 1! at £ is a singular elastic state on B corresponding to I and
to zero body forces.
By (ii) and (iii) we have:
(iv) for each £e Dt,
lim /pxSfi<*«=($-0)xI(fl.
1 These solutions were first introduced by Dougali, [1898, 4]. Ci. Love [1927, 3], § 132;
Lue'e [1955, 10], § 2.3; Sternberg and Eubanks [1955, 13].
12*

180 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 52.
Indeed, by (iii),
/ pxSnda-((-0)xl(Q = / (x-^)xS(x)n{x)daa. + o(l)
as (?->0, and by (ii) the integral on the right tends to zero in this limit.
The next theorem shows that for internal concentrated loads in a
homogeneous and isotropic body, every singular elastic state admits a representation
as a regular elastic state plus a sum of Kelvin states.
(1) Decomposition theorem. Assume that the body is homogeneous and
isotropic. Let I be a system of internal concentrated loads, and let 6 be a singular elastic
state corresponding to I and to zero body forces. Then 6 admits the decomposition
where <sR is a (regular) elastic state corresponding to vanishing body forces, and for
each £€.D|, o$( = <>g[J(£)]) *s ^e Kelvin state corresponding to the concentrated load
1(0 at I
Proof. For each t;£Dt let
i(0=2,«)-{«},
with q sufficiently small that
o
for all $<zDg. Then, on 2'(£), u admits the representation1
OO CO
u= 2 gf+r\ 2 hf, rt=\x-(\, (a)
ft=— oo h=— oo
where </j?> and h^ are vector solid spherical harmonics of degree k ({ being the
origin of the spherical coordinate system) and
Further, by (ii) in the definition of a singular elastic state,2
^) = ^-2) = 0, kS. — 2, (c)
and since g^\ k 2g0, is a homogeneous polynomial of degree k, (b) implies
^-2) = ^-1) = 0. (d)
For each ge-Djlet
us = g<fV + rlhl-V (e)
on $—{£); then a simple computation using (b) shows that the state <jg
corresponding to Ug (in the sense of the strain-displacement and stress-strain relations)
is an elastic state on<^—{£} corresponding to 6=0. Thus if we let
uR = u — 2 us (f)
1 It is clear that the representation theorem (48.1) remains valid when the deleted neigh-
o
borhood of infinity, 27(oo), is replaced by the deleted neighborhood of £, Z(%).
2 This follows from the analog of (8.5) for .£(£).

Sect. 52. Concentrated loads. The reciprocal theorem. 181
onB — Dlt then the associated state 6R is an elastic state on B—Dt also
corresponding to 6=0. It follows from (a) and (c)-(e) that for each tieDt,
00 00
^ = 2^+^2^-2^ fe)
O
on -£(ij); thus u£(x) can be continuously extended to a? = ij, and the resulting
extension is C°° on Ze (ij). Therefore 6R so extended is an elastic state on B
corresponding to zero body forces.
To complete the proof, it clearly suffices to show that <jg is the Kelvin state
corresponding to 1(1;). Since
9^^) = -,
where the vector a is a constant, (b) and (e) yield
us(x) = * + --.¾¾^ r\ Fdiv f "-). (h)
By (f) and (g), u—u^is C°° on i?e(¢); thus the stress field Ss corresponding to u%
satisfies (iii) in the definition of a singular elastic state:
lim / Ssnda =I(fl. (i)
An elementary computation based on (h) and (i) shows that
(2A + S/t) ,,-
hence we conclude from (h) and (51.1) that dg is the Kelvin state corresponding
to £(¢) at {. □
(2) Balance of forces and moments for singular states. Let I be a system
of concentrated loads, and let 6 be a singular elastic state corresponding to the external
force system [6, s, l] with b continuous on B. Then
J sda+fbdv + Z 1(0 = 0,
dB B gei>|
fpxsda+fpxbdv+ 2 (C-0)xI(0 = 0.
dB B {£ Dj
Here the surface integrals are to be interpreted as Cauchy principal values; i.e.
J s da= lim / sda,
dB e-*o *(«)
* (¢) = 35-(5 £„.(£,),
n=l
{&,...,^} = An95, c = (a,--.,ew).
Pra>/. Let Bl = Dlr\B,
B(d,Q)=B-{) Zd($)-{) Zen(U, (a)
gesj «=1
and choose d and q sufficiently small that the closed balls mentioned in (a) are
mutually disj oint. (Note that &,..., ¢^ are all points of dB; while each £ e Bt is an
interior point of 5.) Then 0 is a (regular) elastic state on 5 (5, q) ; hence
/ s<*«+ / 6<2i>=0. (b)
8B(<5, a) B(«,e)

182 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 52.
Since 6 is continuous on B,
f bdv^- f bdv as d-^0, <?->-0. (c)
B{d, S) B
Further, by (a), if d and q are sufficiently small,
/=/+1/+2/. (d)
BB{6,q) «"{«) n = l S^ejin) t&Bi^sit)
where
^,(aj) = 5nai:,(aj). (e)
Combining (b) and (d) and letting d->-0, g-»■(), we conclude, with the aid of (c)
and (iii) in the definition of a singular elastic state, that
f s da
dB
exists as a Cauchy principal value, and that the first relation in (2) holds. The
second relation is derived in exactly the same manner using (iv) in place of
(iii). □
We now give an extension of Betti's reciprocal theorem (30.1) which includes
concentrated loads. This result, which is due to Turteltaub and Sternberg,1
will be extremely useful in establishing integral representation theorems for
elastic states.
(3) Reciprocal theorem for singular states. Assume that the elasticity
field is symmetric and invertible. Let I and I be systems of concentrated loads with Dt
and Df disjoint. Further, let [u, E, S] and [u, E, S] be singular elastic states
corresponding to the external force systems [6 s, I] and [6, s, I], respectively, with b
and b continuous on E. Then
fs-uda+fb-udv+X «(£)' «(£)
dB b ssdj;
= fs-uda + fb-iidv+ £ *(£)'«(£)
dB B J6Dj
= fS-Edv = fS-Edv,
B B
where the surface integrals are to be interpreted as Cauchy principal values.
Proof. Let
Sf (x) =w (*) — w (£) (a)
for each x e B and ££Dt. Then (iii) in the definition of a singular elastic state
implies
f s-uda= f S'5*da + l(£) -m(£) +o(1) as q->-0 (b)
*ett) *e«)
for every {eZ)|. Since 4 is an elastic state on E—Dl and 3 an elastic state on
B—Df, and since Dlr\Dl =0, (a), (b), and (ii) in the definition of a singular
state yield
f s-uda^-l(£) •■&(!;), f s-uda^-0 (c)
felt) *e(«)
1 [1968,14].

Sect. 52. Concentrated loads. The reciprocal theorem. 183
as p-^-0 for each ^eDt. Similarly,
/ s -uda->-i(^) 'W(§), f s-uda->-0 (d)
as p-^-0 for each £eZ)j\
Now let
{& W = (Z),uZ)r)na£, tf = (a.-..,ew),
(e)
«6Bjj n=l
Then o and 3 are (regular) elastic states on B(8, q), and Betti's theorem (30.1)
implies
/ s-W(f«+ / b-udv = f s-uda-\- f b-udv
dB(d,Q) B(d,Q) 8B(6,Q) B(6,Q)
= / S-Edv = f S-Edv.
B(d,Q) B(S,Q)
(i)
(g)
Further, by (e), if 8 and q axe sufficiently small,
/ = / + 2 ,,/ + 2 /.
where we have used the notation given in (e) in the proof of (2).
Since C is invertible on B, we conclude from (ii) that E is 0 (r~2) as the
distance r from any of its singularities tends to zero. Thus, since DtnDi = 0, (ii)
implies that S • E is also 0 (f~2) near its singularities. Clearly, the same property
holds for S • E. Further, since 6 and 6 are continuous on B, b • it and 6 • u are
0 (r-1) near their singularities. Thus as 8 -*-0 and q -+0 all of the volume integrals
in (f) tend to the corresponding integrals over B. Therefore, if we pass to this
limit in (f) and use (g), (c), and (d), we arrive at the desired result. □
(4) Somigliana's theorem.1 Assume that the body is homogeneous and
isotropic with Lame moduli that satisfy /^,^=0,2(4,-^^^4=0. Let [u, E, S] be a singular
elastic state corresponding to the external force system [b, s, I]. Then2
«(!/) = / («, [«] - -¾ [«]) da + f uy [6] dv + 2 uy [I(fl] (£),
dB B J6Dj
or equivalently,
«<(») = / «-s-s^,-w)rf«+JX-6^+ 2 «4(0-'(C).
0B B_ J6D{
1 Somigliana [1885, 4], [1886, 4], [1889, <?J for a regular elastic state. See also Dougall
[1898, 4]. The extension to singular states is due to Turteltaub and Sternberg [1968, 14],
pp. 236-23 7. Somigliana's theorem was utilized to determine upper and lower bounds on elastic
states by Diaz and Greenberg [1948, 1], Synge [1950, 12], Washizu [1953, 20] and Bramble
and Payne [1961, 6]. Kanwal [1969, 4] used Somigliana's theorem to derive integral
equations analogous to those used to study Stokes flow problems in hydrodynamics. See also Maiti
and Makan [1971, 3].
2 The properties of the Kelvin state in conjunction with this result furnish an alternative
proof of the fact that u is of class C°° on B.

184 M. E. Gurtin: The Linear Theory of Elasticity. Sect. 52.
for yeB —Dt, where for any vector v,Uy[v] is the displacement field and s*[»]
the adjoint traction field corresponding to the Kelvin state1 4y[t>], while uy and s'y
are the displacement and traction fields for the unit Kelvin state2 oy.
Proof. Let v be an arbitrary vector. Then the reciprocal theorem (3) applied
to the states 6 = [w, E, S] and <,y [v] yields
u{y) •»== / (s • u [v] — u ■ sy[v]) da + f b • Uy[v] dv + £ l($) ' "»[>](£)•
BB B «£D|
In view of (iv) of (51.2), this relation implies the first formula in (4). On the
other hand, if we take v=et in the above relation, we are led to the second
formula. □
As a corollary of Somigliana's theorem we have the
(5) Integral identity for the displacement gradient.3 Assume that the
hypotheses of (4) hold. Then for every yeB—Dt,
uiJ(y) = -f(Uy<-s-sj>-u)da-fuyi-bdv-ZUyi(l;)-l(S),
SB B «6Dj
div u(y) = — / (Mj, • s — sy ■ u) da — J ky ■ b dv — £ %(£) • l(£),
dB B i&Dt
\[cwclu(y)]i= f (u„-s-Sy-u)da+fuy-bdv+Z «j,(£) • l($),
SB B S6D{
where u*J and s^ are the displacement and traction fields for the unit doublet state*
!$, uy and iy for the center of compression5 °6y, and 1¾ and sy for the center of
rotation e 4y.
Proof. If we differentiate the second relation in (4) with respect to yf, we
arrive at7
u.J^^f^^-s-^Sy-^da + f^-Uy-bdv+Z^.K^-m-
dB
«60l
In view of (v) of (51.3) and the definition of unit doublet states given on page 178,
this relation implies the first formula in (5). The second and third formulae are
immediate consequences of the first. □
It is a simple matter to extend the mean strain and volume change theorems
(29.2) and (29.3) to include concentrated loads. In particular, the formula for
the volume change in a homogeneous and isotropic body becomes
dv(B) = -*Tffp-8da + fp-bdv+2l((-0)-Ht\,
where k is the modulus of compression.
1 See p. 174.
2 Seep. 178.
3 The first identity is apparently due to Love [1927, <?], § 169- The formulae for the
divergence and curl of u are due to Betti [1872, 1], §§ 8, 9. See also Dougall [1898, 4],
Love [1927, 3], §§ 160, 162.
4 See p. 178.
5 See p. 179-
6 Seep. 179.
7 The differentiation under the integral sign is easily justified. Cf. the proof (Kellogg
[1929, 1], p. 151) of the differentiability under the integral sign of Newtonian potentials of
volume distributions.

Sect. 53. Integral representation of solutions to concentrated-load problems. 185
Suppose now that
s=0 on dB, 6=0 on B,
and that the system I consists of a pair of concentrated loads. If we let Dt = (¾, §2}>
then it follows from balance of forces and moments (2) that
*Ki) = -*(W = *e,
and there results the following elegant formula for the volume change:
Thus the volume change due to a pair of concentrated loads is independent of the
shape or volume of B; it depends only upon the magnitude of the loads, their
distance apart, and the modulus of compression of the material.
53. Integral representation of solutions to concentrated-load problems. In this
section we will use the reciprocal theorem for singular states to establish an
integral representation theorem for the mixed problem of elastostatics with
concentrated loads included. Thus we assume given a symmetric and invertihle
elasticity field C on B, surface displacements u on £fx, surface forces s on y2,
a continuous body force field 6 on B, and a system of concentrated loads I. The
generalized mixed problem is to find a singular elastic state [u, E, 8] that
corresponds to the body force field 6 and to the system of concentrated loads I, and
that satisfies the boundary conditions
m = m on S^—Di, s = s on 6^ — Dg.
We will call such a singular elastic state a solution of the generalised mixed
problem.
In order to introduce the notion of a Green's state we need the following
(1) Lemma. Let f and m be given vectors. Then there exists a unique rigid
displacement field w such that
f wda=f, f pexwda = m,
dB dB
where c is the centroid of dB and pc(x) =x—c. We call w the rigid field on dB
with force f and moment m.
Proof. Suppose such a field w exists and let
Since c is the centroid,
and thus
w = a-\-coxpc. (a)
Jpcda = 0, (b)
dB
J w da=aa, (c)
where a is the area of dB. Further, (a) and (b) also imply
/ pcxwda= f pcx((OXpc)da = I(o, (d)
dB dB

186 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 53.
where I is the centroidal inertia tensor:
I=S[\Pc\^-Pc®Pc\da. (e)
dB
Since I is invertible, (c) and (d) imply that if such a rigid field w exists, it is
unique and
a— -, (o — I^vn. (f)
Conversely, it is a simple matter to verify that w defined by (a) and (f) has all
of the desired properties. □
Let ytB. We call 6y = [iiy, Ey, Sy] (i = \, 2, 3) Green's states at y for the
mixed problem provided:
(i) ~4y is a singular elastic state corresponding to vanishing body forces and
to a concentrated load e4 at y;
(ii) if £f\_ is not empty, then
uy = 0 on ^, sy=0 on ^,
where sy is the surface traction field of ~6y;
(iii) if yx is empty, then
stt = wy on dB,
where wy is the rigid field with force — e, and moment — (t/ — c) xe,-.
The above boundary condition insures that balance of forces and moments are
satisfied when £fx is empty. Indeed, by (52.2) and (i), the Green's state ~<,y must
satisfy
fSyda+e — 0,
SB
f pcxs'yda+ (S-c)xe,. = 0;
BB
that ~4y is consistent with these results follows from (1) and (iii).
For a homogeneous and isotropic body, the Green's states iy may be
constructed as follows:
where 6y is the unit Kelvin state corresponding to a concentrated load e^ at y,
and 4y is a regular elastic state on B corresponding to zero body forces and chosen
so that the boundary conditions (ii) and (iii) for Zy are satisfied.
We say that an integrable vector field u on dB is normalised if
fuda = 0, f pexuda = 0.
dB BB
Given a solution u of the traction problem (^ = 0), the field u-\-w with w rigid
is also a solution. By (1) there exists a unique rigid w such that u-\-w is
normalized. Thus we may always assume, without loss in generality, that
solutions of the traction problem are normalized.

Sect. 53. Integral representation of solutions to concentrated-load problems. 187
(2) Integral representation theorem.1 Let [uy,Ey,Sy] (i = i,2, 3) be
Green's states at y<cB for the mixed problem. Further, let [u, E, S] be a solution of
the generalized mixed problem with u normalized if Sfx is empty. Then for y$Dt,
Ui(y) = ~fsy-uda + fuy-Sda+fiiy-bdv + Zuy($)-l(?).
Proof. Assume first that £fx is not empty. If we apply the reciprocal theorem
(52.3) to the states <, and Zy and use (i) and (ii) in the definition of 4y together with
the boundary conditions satisfied by <f, we find that
/ ~sy -uda+uM =fs-uyda+fb-uydv+Z '(«) •»%(«),
which implies the above expression for u^y).
If £^ is empty, we conclude from (iii) in the definition of Green's states that
there exist vectors a=a*y and a = (oy such that
sy = a-\- coxpc on dB.
Thus, since u is normalized,
f sy-uda = f (o+ coxpc) uda = a- f uda + at- f pcxuda = 0,
dB dB dB dB
and the reciprocal theorem again implies the desired result. □
(3) Symmetry of the Green's states. Let vOx and uy be displacement fields
corresponding to Green's states atx,yeB (x4= y) for the mixed problem. Further, if
£^ is empty, assume that w£, and uy are both normalized. Then
ul(y)-ei = uy(x)-er
Proof. We simply apply the integral representation theorem (2) to the state
[u, E, S] =¾ and use properties (i)-(iii) of Green's states. □
In the presence of sufficient smoothness one can deduce an integral
representation theorem for the displacement gradient (and hence also for the strain and
stress fields) by differentiating the relations in (2) under the integral sign. An
alternative approach is furnished by the
(4) Representation theorem for the displacement gradient. Assume
that B is homogeneous and isotropic. Choose y^B and let
1 A slightly different version of this theorem for the displacement and surface force
problems without concentrated loads was given by Lauricella [1895, 1], who attributed the
method to Volterra (cf. the earlier work of Somigliana discussed in footnote 1 on p. 183).
A precise statement of the Lauricella-Volterra theorem is contained in the work of
Sternberg and Eubanks [1955, 13]. In [1895, 1] and [1955, 13] the equilibration of the concentrated
load ei at y in the Green's state 3* when 5^ is empty is effected through the introduction of a
second internal singularity; the normalization of u is achieved by requiring the displacements
and rotations to vanish at the location of this added singularity. The method used here to
equilibrate the concentrated load and the normalization procedure for u is due to Bergman
and Schiffer [1953, 3], pp. 223-224 (cf. Turteltaub and Sternberg [1968, 14]). For the
surface force problem with concentrated loads this theorem is due to Turteltaub and
Sternberg [1968, 14]. See also Dougall [1898, 4], [1904, 1], § 32; Lichtenstein [1924, 1]; Korn
[1927, 2], § 14; Love [1927, 3], § 169; Trefftz [1928, 3], § 49; Arzhanyh [1950, 2], [1951, 2],
[1953, 1], [1954, 2]; Radzhabov [1966, 20]; Idenbom and Orlov [1968, 9].

188
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 53.
where:
(i) H'j} are the unit doublet states1 at y;
(ii) 4% are regular elastic states corresponding to zero body forces;
(iii) if £fx is not empty, then
ftj/=0 on yx, s*/ = 0 on ^,
where iij/ is the displacement field and sj/ the traction field of <i%;
(iv) if 6^ is empty, then
sj/ = w# on dB,
where w^ is a rigid field.
Let [u, E, S] be a solution of the generalized mixed problem with u normalized
if if^ is empty. Then for y$Dt,
«< ,(y) = -I ~s$-uda + jui;-sda+fu% -bdv+% «#'(«)• I«).
5» A B «£Dj
Proof. By (ii) and the reciprocal theorem (52.3),
/■(8-^-^-«)rfa+/ft-^rf» + 2l(«)-i^(«) = 0. (a)
dB B J6Dj
Since ?J» = <$ — 3y, if we add (a) to the right-hand side of the first relation in
(52.5), we find that
»*./(«) = / (» • «»' - 3g ■«) rfa + / ft • *# ^ + 2 I(«) • uj/(£). (b)
<JB B «£Dj
It follows from (iii) that when £fx is not empty (b) reduces to the required
expression for u{^(y). If if^ is empty, then an argument identical to that utilized in
the proof of (2) can be used to verify that
/ s*/ -uda = 0; (c)
dB
and (b) and (c) yield the desired relation for u{j(y). □
The rigid field w*J of (iv) is not arbitrary. Indeed, in view of (i) and (ii), w?y"
must obey balance of forces and moments on B—£e(y). If we apply these laws
and then let q -+0, we conclude, with the aid of (51.3), that
f sij da = 0,
dB
f rxs*? (f« = efxe-.
dB
Thus w*J must be the rigid field with force equal to zero and moment equal to et X e?-.
The proof of the next theorem, which we omit, is completely analogous to the
proof of (4) and is based on the relations for divw(t/) and curl w(t/) given in
(52.5).
(5) Representation theorem for the dilatation and rotation.* Assume
that B is homogeneous and isotropic. Choose yeB and let
~ * ° ~* ** °i
o„ = t„ — o„, "» = "» — "».
1 See p. 178.
2 Betti [1872, 1]. See also Love [1927, 3], §§ 161, 163.

Sect. 53. Integral representation of solutions to concentrated-load problems. 189
where:
(i) ly is the center of compression at y and ly the center of rotation at y parallel
to the Xf-axis;1
(ii) 6y and Oy are regular elastic states corresponding to zero body forces;
(iii) if £f{ is not empty, then
viy — Q on £fx, sw = 0 on £f2,
wj, = 0 on yx, s*, = 0 on ^,
where uy and Uy are the displacement fields and sy and sy the traction fields of 39
and 4y.
(iv) if £fx is empty, then
sy=0 on 8B,
Sy = wy on 8B,
where wy is a rigid field.
Let [u, E, S] be a solution of the generalized mixed problem with u normalized
if £^ is empty. Then for yiDt,
divw(t/) = — f sy-uda-\- f uy-sda + fuy- bdv + 2 ««(£)" ^(£)>
£ &% B J6Dj
J [curl u (t/)] ,• = / sy • u da — J uy • s da — f uy • b dv — £ uy ($) • I ($).
As before, the field wy of (iv) is not arbitrary; it follows from (i), (ii), and the
relations on page 179 that Wy must be the rigid field with force equal to zero and
moment equal to e,-.
In view of the last theorem, div u and curl u can be computed, at least in
principle, from the data. Further, since the displacement equation of equilibrium
is equivalent to the relation2
u+ \ (l+ ^Jpdivw
0,
the determination of w when div w is known is reduced to a problem in the
theory of harmonic functions. Moreover, since the surface force field on dB can
be written in the form3
s = 2ft +/unxcurlM+ A(divw)n,
when div u and curl u have been found, the surface value of du/dn is known
at points at which the surface force field is prescribed. This method of integration
is due to Betti.4
Ericksen's theorem (33.1) extends, at once, to problems involving
concentrated loads.
(6) Assume that there exists a non-trivial solution of the generalized mixed
problem with null data (u = 0, s = 0, 6=0,1=0). Then there exists a continuous
body force field b on B of class C2 on B with the following property: the regular
1 See p. 179.
2 See p. 91.
3 See p. 93.
4 [1872, l). See also Love [1927, 3], § 159-

190
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 54.
mixed problem corresponding to this body force and to null boundary data (it = 0, s = 0)
has no solution. If ,5¾ is empty, b can be chosen so as to satisfy
f bdv = f pxbdv = 0.
B B
In addition, there exists a system of concentrated loads I such that the generalized
mixed problem corresponding to I, vanishing body forces, and null boundary data
has no solution. If £fx is empty, I can be chosen so as to satisfy
2i«) = I«-o)xi«) = o.
The proof of this theorem is based on the reciprocal theorem (52.3); it is
strictly analogous to the proof of (33.1) and can safely be omitted.
As an immediate corollary of (6) we have the following theorem, due to
Turteltaub and Sternberg :1 If the regular mixed problem with null boundary data
has a solution whenever the body force field is sufficiently smooth, then there is at
most one solution to the generalized mixed problem.
In the case of internal concentrated loads, a uniqueness theorem can be
established that does not presuppose existence.
(7) Uniqueness theorem for problems involving internal
concentrated loads. Assume that the body is homogeneous and isotropic with positive
definite elasticity tensor. Then any two solutions of the generalized mixed problem
with internal concentrated loads are equal modulo a rigid displacement.
Proof. Let 6 denote the difference between two solutions. Then, by the
decomposition theorem (52.1), 6 is a regular elastic state corresponding to null data.
Thus the uniqueness theorem (32.1) implies that <j = [w>, 0, 0] with w rigid. □
X. Saint-Venant's principle.
54. The v. Mises-Sternberg version of Saint-Venant's principle. Saint-Venant,
in his great memoir2 on torsion and flexure, studied the deformation of a cylindrical
body loaded by surface forces distributed over its plane ends. In order to justify
ulHHg his results as an approximation in situations involving other end loadings,
SAiNT-V^NAN[rjnad^a_conjecture.which may roughly be worded as follows: If
two sets of loadings are statically equivalent at each end, then the difference in
stress fields is negligible, except possibly near the ends; or equivalently, a system
of loads having zero resultant force and moment at each end produces a stress
field that is negligible away from the ends. This idea has since been elevated to
the status of a general principle bearing the name of Saint-Venant. Its first
■A-^l-i general statement was given by Boussinesq :3 "An equilibrated system of external
\'~°*""Vforces applied to an elastic body, all of the points of application lying within a
ri'ic% given sphere, produces deformations of negligible magnitude at distances from
rxa*0 the sphere which are sufficiently large compared to its radius."
As was first pointed out by v. Mises,4 this formulation of Saint-Venant's
principle, which has since become conventional, is in need of clarification, since
1 [1968, 14], Theorem 5.2. Actually, the Turteltaub-Sternberg theorem is slightly
different from the one given above; the idea, however, is the same.
2 [1855,11
3 [1885, i], p. 298. "Des forces exterieures, qui se font equilibre sur un solide elastique
et dont les points d'application se trouvent tous a 1'interieur d'une sphere donnee, ne produisent
pas de deformations sensibles a des distances de cette sphere qui sont d'une certaine grandeur
par rapport a son rayon."
4 [1945, 5].

Sect. 54. The v. Mises-Sternberg version of Saint-Venant's principle. 191
the system of external forces must necessarily be in equilibrium, at least when
the body is finite. Further, as was noted by Sternberg,1 in the case of an un-
bounded bodyjoaded on_a finite portion joFits^surfice, The stresses are arbitrarily i
small at sufficiently large distances from the ioad region.2 I
The first of these criticisms led v. Mises3 to suggest the following
interpretation of the principle: "If the forces acting upon a body are restricted to several,
small parts of the surface, each included in a sphere of radius q, then the ... stresses':
produced in the interior of the body ... are smaller in order of magnitude when the
forces for each single part are in equilibrium than when they are not."
It is clear that this is the meaning intended by Boussinesq. Indeed, in order
to justify the principle, Boussinesq examined the strains at an interior point of
an elastic half-space subjected to normal concentrated loads upon its
boundary. Assuming the points of application of the loads to lie within a sphere of
radius q, he showed that the order of magnitude of the strains is q if the resultant
force is zero, andj)2when jthejresultant moment also vanishes. Various arguments
have since been given in support of the principle.*
v. Mises utilized two examples involving tangential as well as normal surface
loads to demonstrate that the above version of Saint-Venant's principle cannot
be valid without qualification. The two examples chosen by v. Mises are the
three-dimensional problem of the half-space and the plane problem of the circular
disk, each under concentrated surface loads.5 Guided by these examples, v. Mises e
conjectured a modified Saint-Venant principle-which he stated as follows:
" (a) If a system of loads on an adequately supported body, all applied at
surface points within a sphere of diameter q, have the vector sum zero, they
produce in an inner point y of the body a strain or stress value s of the order of
magnitude q.
(b) If the loads, in addition to having vector sum zero,... form an
equilibrium system within the sphere of diameter q, the s-value produced in the point
y will, in general, still be of the order of magnitude q.
(c) If the loads, in addition to being an equilibrium system, ... form a system
in astatic equilibrium, then the s-value produced in y will be of the order of
magnitude q2 or smaller. In particular, if loads applied to a small area are parallel
to each other and not tangential to the surface and if they form an equilibrium
system, they are also in astatic equilibrium and thus lead to an s of the order q2."
The conjectures of v. Mises concerning concentrated surface loads were
analyzed by Sternberg, whose results are also valid for continuously distributed
surface loads. In. order to give a concise statement of Sternberg's theorem, we
introduce the following definitions:
We say that !FQ, 0 < q gj q0 , is a family of load regions on dB contracting to
ztdBii:7
(i) &g = dB n S9 (0) for every q e (0, q0] ;
1 [1954, 20].
2 Toupin [1965, 20] gives several counterexamples to the conventional statement of
Saint-Venant's principle.
3 [1945, 5].
4 Southwell [1923, 2], Supino [1931, 5], Goodier [1937, 1, 2], [1942, 2], Zanaboni
[1937. 6, 7, 8], Locatelli [1940, 2], Dou [1966, 7].
6 Erim [1948, 3] has applied v. Mises' analysis to the half-plane under concentrated loads.
6 [1945, 5].
7 Cf. Sternberg and Al-Khozai [1964, 20], Def. 4.1.

192
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 54.
(ii) there exists a class C2 mapping x: ^->#"ffo, where & is an open region
in the plane containing the origin, such that
' 0¾ 8¾
!=0
Note that by (ii) the boundary of B possesses continuous curvatures near s;
thus this portion of the boundary is required to exhibit a higher degree of
smoothness than that automatically assured by the assumption that B be a
regular region.
For each &€{1, 2,..., K} let ^e(k), 0<q^q0, be a family of load regions on
dB contracting to a point 2k£dB. We say that ie=\ue, Ee, Se], 0<q<q0, is a
family of singular elastic states corresponding to loads on !FQ (k) if given
any 06(0, Qo)-1
(iii) ae is a singular elastic_state corresponding to zero body forces, to surface
tractions se, and to a system I of concentrated loads;" '
" ■ K
(iv) the traction field s8 vanishes on dB — (J !F (k);
A=l
K
(v) D^c^u^(*)j;
(vi) \se{x)\ ^.M1 for every xedB and |fff(£)| f^M2 for every £€£)¾» where the
constants M1 and M2 are independent of g;
(vii) ue and Fm? admit the representations
«»*. Ay) = /*,■*»'<*«+ z K (?) • fitf (?).
for every «/€5, where for each y£B the functions uy and 1¾ are continuous on
B-{y) and of class C2 on ^(fe) for £=1, 2,..., K.
Since D;g is the set of jjoints at which Jhe concentrated loads of the system
lQ are acting, (v)Ts"simply the requirement that there be no concentrated loads
applied'outside of the K load regions ^e(k).
In view of (iii), (v), and the integral representation theorem (53.2), the first
representation formula in (vii) is simply the requirement that the Green's states
for the surface force problem exist. For a homogeneous and isotropic body, it is
clear from the discussion on page 186 that such Green's states will exist if the
surface force problem for arbitrary (equilibrated) surface tractions has a solution
with sufficiently smooth displacements. In view of the integral representation
theorem (53.4), this also insures that the assertion in (vii) concerning VuQ hold.
For convenience, we adopt the following notation:
De(k)=DlQn3Pe(k),
me(k)= f pxseda+ Z «-0)x^(«),
Me(k)= f p®seda+ Z (¢-0)0^(0,
1 Cf. Sternberg and Al-Khozai [1964, 20], Dei. 4.2.

Sect. 54. The v. Mises-Sternberg version of Saint-Venant's principle. 193
so that fg (k) and me (k) denote the resultant force and resultant moment about
the origin of the surface tractions and concentrated loads on ^e(k). We say that
the force system is in equilibrium on jeacj^lgadregign^if
for k = 1, 2,..., K and 0<q<q0. Further, using the terminology introduced in
Sect. 18, we say that the force system is in astatic equilibrium on each
load region if given any orthogonal tensor Q
^JQseda+ £ <H(£) = 0,
JpxQseda+ £ («-0)xO^(0 = 0.
Taking Q = \, we see that astatic equilibrium implies equilibrium. In addition,
it follows from theorem (18.9) that the force system is in astatic equilibrium on
each load region if and only if
fe(k) = 0, Me(k) = 0,
for k = 1, 2, ..., K and 0 < q < Q0. Finally, we say that the forsesare parallel
and non-tangential, to. the boundary if there exist scalar functions <pe on dB
ancfy>e on Die such that for each k,
se(*) = %W «*. h(?) = fei^k
for all a?&^,(&), %eDe(k), and #e(0, q0), where ak is a unit vector that is not
tangent to 8B at zk.
(1) Sternberg's theorem.1 Let zk, k = \, 2, ..., K, be K distinct regular
points of dB, and for each k let ^e(k), 0<q^q0, be a family of load regions on
8B contracting to zk. Let oe = [ue, Eg, SJ, 0<q<q0, be a family of singular
elastic states corresponding to loads on ^(k). Then given y&B:
(a) ue(y)=0(es), Ee(y)=0(es), Se(y)=0(es) as q^O, where 6=0;
(b) <5 = 1 if fe(k) = 0 for k = \,2,...,K and 0<q<q0\
(c) 6=2 if on each load region the force system is in equilibrium with forces that
are parallel and non-tangential to the boundary;
(d) 6=2 if the force system is in astatic equilibrium on each load region.
Moreover, if the concentrated loads all vanish (i.e. lg = 0), which will be the case
if 6e is regular for each q, then:
(a') 6=2 (in general);
(b') <5 = 3 if /e(&) = 0 for £=1,2,..., K and 0<q<q0;
(c') <5=4 */ on each load region the force system is in equilibrium with forces
that are parallel and non-tangential to the boundary;
(d') <5=4 if the force system is in astatic equilibrium on each load region.
Proof. Fix yeB. In view of the strain-displacement and stress-strain relations,
it clearly suffices to establish the above order of magnitude estimates for uQ
and VuQ. Let ue denote either uei(y) or ueit1-(y). Then by (vii) uQ admits the
representation
ue = fse-gda+ £ le(®-g($), (e)
'[1954,20]. See also Schumann [1954, iS], Boley [1958,3], [1960,2], Sternberg
[i960,12], Sternberg and Al-Khozai [1964,20], Keller [1965, 11]. The last author
discusses the form which the principle might be expected to take in the case of thin bodies.
Handbuch der Physik, Bd. VI a/2. 13

194 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 54.
where g denotes either uly or «**. Since the zk are all distinct, there exists a
gj£(0, #0) such that the K sets ^e(k) are mutually disjoint for each #e(0, &).
Thus, in view of (iv) and (v),
«,= 2 »»(*). (f)
where
».(*)= /vs<*« + 2^(0-0(0-
We now fix the integer & and, for convenience, write
v9 = v9(k), &e = &e(k), De = De(k), z = zk,
so that
v0= Jse-gda+ZW-9(^)- (s)
&Q seoe
Let x: SP^-tF^ be the function specified in (ii). It follows from the properties
of x that there exists an open set 0 in 0 containing the origin, a number
#2£(0, &), and a mapping a = (¾.¾) of #"?J onto ^8 such that a is the inverse
of the restriction of x to ^. Further, it follows from the properties of x that
a (a?) = 0(\x — z\) as x^-z. (h)
Since the composite function gox (with values g(x(ot,))) is class C2 on ^, if we
expand this function in a Taylor series in a about a = 0 and evaluate the result
at a = a (x), we arrive at
g(x)=a0 + a1^1(x) + a2a2(x) + c(x) (i)
for a?e^"8j, where a0, (¾. and a2 are fixed vectors and, in view of (h), c is a class
C2 function on J^ that satisfies
c(x)=0(\x — z\2) as x^z. (j)
By (h)-(j) and (vi),
2
f st-gda = a0- / s8 d« + £ Op • /a^s^a + O^4),
&"q &q )3 = 1 ^)
= a0-/s^« + 0(e»), (k)
= 0(e*).
2 1,(0-9(0 = 00- 2 *ff(0 + 2<v 2 ^KHfO + ofe*).
?££<> $£De /9=1 $€D<,
=«o- 2 w+o(q), a)
= 0(1)
as g->0. Conclusions (a) and (a') follow from (f), (g), (k)3, and (1)3. Further, it
follows from the definition of fQ =fe(k) in conjunction with (f), (g), (k)2, and (1)2
that conclusions (b) and (b') are also valid.
To establish the remaining results we first expand the function x in a Taylor
series about a = 0:
x(a,)=s + d1La.1 + d2a.2 + h(<x), (m)
where
MW (/3 = 1,2), (n)
9a«
« = 0

Sect. 54. The v. Mises-Sternberg version of Saint-Venant's principle. 195
and h is a class C2 function on 3P that obeys
h(x)=0(\x\2) as a^O. (o)
Since p (x) — x — 0, we conclude from (m), (h), and (o) that
2
/ p <g) se da = (g — 0) 0 f se da + £ dp ® / (¾ s8 d« + 0(#4),
2 «-0)(8H(«) = (*-0)(8i 2 ^fl + ZH® I «/>(?) *8($) + 0 (<?2).
If we assume that the force system is in astatic equilibrium on each load region,
then
/e==0, Mj30, (q)
and it follows from (p) and the definition of /e and MQ that
/j=i (r)
= o(g*) if i^o,
where
/^/o^a+Z^KHtt). (s)
Each of the relations in (r) may be regarded as a system of inhomogeneous linear
equations in the two unknowns fpe{@ = i, 2). It follows from (n) and the
restrictions placed on x in (ii) that the coefficient matrix of this system has rank two.
Hence (r) implies
//,, = 0 (<?*). (t)
= 0(e*) if ^o,
and this result with (f), (g), (k)x, (1^, and (s) yields conclusions (d) and (d').
We have only to establish (c) and (c'). Thus suppose, in place of (q), that
/,==0, me^0. (u)
Clearly, (p) holds in the present circumstances with the operation " ® " replaced
by "x". It follows from this relation in conjunction with (u) and (s) that (r)
also holds with "®" replaced by "x". Thus if we assume that se(x) —<pe{x)a
and le(§) =ye(£) a on ^=^(fe) and De=De(k), respectively, then
£^xa{/^%rf«+2a^)%(£)} = 0(^),
/8=1 l^e g€D, >
= o(g*) if ¢,==0.
This, in turn, implies that
[(¾xd.) -a] { f^^da + £&,(?) fe(¢)} = 0(q*),
= o(e«) if 1,==0.
It follows from (n) and (ii) that d1x<f24=0. Therefore, since c^xefa is normal to
8B at g, the requirement that a not be tangent to 8B at g implies (e^ xd2) • a 4=0.
Thus, in this instance, (v) implies (t), and, as before, we see that (c) and (c') are
valid. □
On the basis of the traditional statement of Saint-Venant's principle
discussed at the beginning of this section, one would expect a reduction from d — i
13*

196 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 55.
to 8=2 in the order of magnitude estimates whenever the force system is in
equilibrium on each load region, even when the surface tractions are not of the
special form specified in (c), i.e. parallel and non-tangential to the boundary.
That this is not the case is apparent from counterexamples of v. Mises.1
The precedin^^onchj^ian has a counterpart in the. .theQry_j5f_hasic_sJngular
states discussed in Sect. 51. Thus consider the unit Kelvin state <>* and the unit
doublet state P1, with 0 the point of application in both cases. Both of these
states are regular on any infinite region S that does not contain 0. If the
boundary of B is bounded, then
/'=#0, m' = 0,
/*'>'= 0, m*'>'4=0 (*'4=/),
/*>' = 0, ml' = 0 (*==/),
where /', m' and /,;, m*; denote the resultant force and moment aoout the
origin of the surface forces on 8B corresponding to d' and ?*', respectively.
Consider the rate of decay of the corresponding stress field at infinity. It,is clear
from the results on p. 178 that
Sf (x) = 0 (r-*), S* (a?) 4= 0 (r~3)
S{> (x) = 0 (r -»), S* (x) 4= 0 (r "4)
as r =\x — 0| ->oo, regardless of whether i =/ or i4=/- Consequently, whereas the
stress field decays more rapidly when the resultant force on 8B vanishes than
when this is not the case, the additional vanishing of the resultant moment does
not produce a further reduction in the order of magnitude of the stresses as
f->-00.2
55. Toupin's version of Saint-Venant's principle. Two alternative versions of
Saint-Venant's principle were established by Toupin3 and K.NOWLES.4These authors
considered the solution corresponding to a single system of surface loads, and
both authors arrived at estimates for the strain energy U(l) of a portion of the
body beyond a distance I from the load region.8 Knowles' result, which we will
discuss in the next section, applies to the plane problem, while Toupin's result
is for a semi-infinite cylinder loaded on its end face.6
* (1) Toupin's theorem.1 Let B be a homogeneous semi-infinite cylindrical
body with end face Sf0, and assume that the elasticity tensor is symmetric and
positive definite. Let [u, E, S] be an elastic state corresponding to zero body forces and
to a surface force field s that satisfies:
(i) s=0 on dB — ^0,
(ii) / s da — fpxs da = 0,
(iii) lim / u- Snda=0,
1 [1945. 5].
2 The conclusions expressed in the last two paragraphs are due to Sternberg and
Al-Khozai [1964,20], pp. 144-145-
3 [1965, 21].
4 [1966, 12].
5 The idea of estimating the energy U(l) of that portion of a circular cylinder a distance I
from the loaded end appears in the work of Zanaboni [1937, 8].
6 Knowles and Sternberg [1966, 13] have established a Saint-Venant principle for the
torsion of solids of revolution. A result that is valid for more general regions was given by
Melnik [1963, 19].
7 [1965, 21]. See also Toupin [1965, 20].

Sect. 55.
Toupin's version of Saint-Venant's principle.
197
where S^\ is the intersection with B of a plane perpendicular to the axis e of the
cylinder and a distance I from 6^,. Let Oe^J,, let
B, = {xeB: (x-0)-e>l},
and let U(l) denote the strain energy of Bt:
%>c
-r
a *
Then
U(l) = ifE-C[E]dv.
Bl
1 It
U{l)£U(0)e J-W (l^t)
for any t > 0, where the decay length y (t) is given by
.»-0
PmW)
Here /xM and fim are the maximum and minimum elastic moduli, while X (t) is the
lowest (non-zero) characteristic value for the free vibrations of a disc with cross
section £f0 and thickness t, a disc that is composed of the same material and that has its
boundary traction-free.1
Then
Fig, 14.
Proof. Choose t, I > 0 and let
B{l,t)=Bl-Bl+t.
U{t) = \im\ JE-C[E]dv
t^oo B(l,t)
provided the limit exists. By (i) and the theorem of work and energy (28.3),
Ju-Snda+ f u-Snda= f E- C[E]dv.
r, srM B(i,t)
Thus if we let t ->- 00 and use (iii) we conclude that U(l) exists and obeys the formula
£/(/) = 1 fu-Snda. (a)
Equating to zero the total force and moment on the portion B (0,1) of B
between S"0 and S?i, we conclude, with the aid of (i) and (ii), that
J Snda= fpx(Sn)da = 0. (b)
1 Thus we make the tacit assumption that the characteristic value problem stated in
Sect. 76 has a solution, at least for the lowest non-zero characteristic value.

198 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 55.
Let s' be the vector field on 8B (0,1) defined by
s'=Sn on yu s'=0 on 35(0,1) -^,.
Then (b) implies
fs'da= Jpxs'da = 0,
dB (o, /) en (o, /)
and we conclude from Piola's theorem (18.3) with s = s' and 6 = 0 that
f w • Snda = 0
/
for every rigid displacement w. Therefore (a) yields
t/(/) = £ fu-Snda (c)
^,
provided
u = u + w. (d)
By (c) and the Schwarz inequality,
U(t)^i]/fu*da f \Sn\2da. te)
If we expand (a ]/#— ]/&)2, where a, b, and a are non-negative scalars with oc>0,
we arrive at the geometric-arithmetic mean inequality
' — 2
Applying this inequality to (e) yields
U{1) ^\U f\Sn\2da + ^ fu2da\. (f)
Since \n\ =1 and C is symmetric,
|Sn|2 ^ |S|2 = C[E] • C[E] = E ■ C2[E].
Let /<M and //,m denote the maximum and minimum elastic moduli for C; then
by (24.6)
^m\E\^E-C[E]^^M\E\K
Further, it follows from the definition of /iM and //,m that /x% and /i% are the
maximum and minimum moduli for C2; thus
^lEl^E-C^^El2.
The last three inequalities imply that
\Sn\*^//M\E\*^ii*E-C[E],
where
..* _ j"m .
thus (/) implies
C7(/) ^^ Lfi* fE-C[E]da + ~ fu2da\. . (g)

Sect. 55. Toupin's version of Saint-Venant's principle. 199
Next, choose t > 0 and let
i+t
0(U)=J-/ U{X)dX. (h)
If we integrate (g) between I and l-\-t, we arrive at
tQ(l, t) g A W [ E • C [E] rfo + *- J W dv\ . (i)
B (1,() B(l,t) '
Now let X{t) denote the lowest non-zero characteristic value corresponding
to the free vibrations of the cylindrical disc B(l, t). Then1
/ VvC[Vv]dv
X(t)<z.B<hlL
w — / K2 dv
B(l,t)
for every admissible displacement field v on B(l, t) that satisfies
fvzdv^=0, fvdv— / pcxvdv = 0.
B(l,t) B(l,t) B(l,t)
By use of the same procedure as that used to prove (53.1), it is not difficult
to show that the rigid^lisplacement w in (d) can be chosen so as to satisfy
fwdv = — f udv, fpcxwdv = — f pcxudv,
B(l,t) B (1,() B(l,t) B (1,()
where c is the centroid of B(l, t) and pc(x) =x — c. By (d) when this is the case
we have
J udv— f pcxudv = 0;
B (I, t) B (I, t)
thus
Ju'dv^-^y JE-C[E]dv, * (j)
B (1,() B(l,t)
and (i) and (j) yield
m,t)^{ (-^y+ «,**) JE-C[E}dv. (k)
B (I, t)
If we differentiate (h) with respect to I, we arrive at
--J!-=y ^+0-^01- _ w
On the other hand,
f E-C[E]dv = fE-C[E]dv- f E • C[E] dv = 2[U {1) -U{l + t)], (m)
B(l,t) B, B,+,
and (k)-(m) imply ~ ',;
y (0-^- + ^.0^0, (n)
where
1 See p. 264.

200 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 56.
Clearly, y (t) is a minimum when
a = •
Henceforth we assume that a has this value, so that
By(n)
and thus
■«>=^S
(t)
&M Q ft, t) - W® Qft, f)£0 ft ^y
or
e vW . (o)
Since C is positive definite, U(l) is a non-increasing function of I, and by (h)
@ (M)is the mean value of C7(/) in the interval [I, l-\-t~\; therefore
and (o) implies
Setting ^ = 0 and l2 = l — t in (p), we obtain the desired energy estimate. □
In view of (24.6), if the material is homogeneous and isotropic with strictly'
positive Lame moduli //, and X, then the characteristic decay length y (t) has the
iorm n , .
?W_ VaTTAM •
Toupin1 also derived an estimate for the magnitude of the strain tensor at
interior points of the cylinder. A somewhat stronger estimate was established by
Roseman 2 for an isotropic cylinder. Roseman proved that
\S{x)\*^.K\U{l — a) - U(l + a)]
for every a?e«$^, where if is a constant that depends on the Lame moduli and the
cross-section, while a is a constant that depends only on the cross-section. When
combined with Toupin's theorem, the above inequality yields pointwise exponential
decay for the stress throughout the cylinder.
56. Knowles' version of Saint-Venant's principle. Knowles' version3 of Saint-
Venant's principle is concerned with the plane problem of elastostatics for a
simply-connected region loaded over a portion of its boundary curve. To state
this result concisely we introduce the following notation.
Let R be a simply-connected regular region in the xx, #2-plane, so that its
boundary curve ^ is a piecewise smooth simple closed curve. As indicated in
Fig. 15, ~~
1 [1965, 21]. Toupin's estimate is based on (h) of Sect. 43, which is due to Diaz and Payne
[1958, 6].
2 [1966, 22]. The techniques used by Roseman are based on results of John [1963, 15],
[1965, 10].
3 [1966, 12].

Sect. 56.
Knowles' version of Saint-Venant's principle.
201
Fig. 15.
and we choose our coordinates so that the minimum value of x2 on "if0 is zero,
^^will represent that portion of the boundary curve along which surface tractions
are applied. Let
b = sup x2,
« = sup %,
Ri = {x€R: x1>l},
^Ci = {x€R: x1 = l).
We assume that for each le(0, a), the set ££l is the union of a finite number of
line segments; we regard J2?( as parametrized by x2.
Since R is simply-connected, (46.3) implies that the stress field Sa/8
corresponding to a plane elastic state is completely characterized by the field equations
Knowles' theorem gives a decay estimate for the norm
(a)
(b)
|S||;=1// ~SaPSaPda
of such a stress field over Rt.
(1) Knowles' theorem.1 Let Sa/8 (= S^) be smooth on E0 and of class C2
on R0. Suppose further that Sa/8 satisfies (a) on R0 and
(c)
Then for 0^/^ a
where
||S||;^]/2JS||0<r>~,
[1966, 12].

202 M. E. Guriin : The Linear Theory of Elasticity. Sect. 56.
With a view toward proving this theorem, we first establish the following
(2) Lemma.1 Let f be continuous on R0. Then
- I fda = — I fdx2.
dl
Ri Si
Proof. Consider the difference quotient
F(l + d)-F(l)
» J
for the function F defined by
F(l) = Jfda.
If ^i is connected, this difference quotient can be written, for sufficiently small d,
as an iterated integral, and the limit as <5->0 can be computed directly to
prove (2). If =2^ is the union of disjoint line segments, the difference quotient is
written as a sum of iterated integrals, and (2) again follows upon letting <5->-0. □
Proof of (1). By (47.1) there exists an Airy function cp for Sa/? on R0; thus
511 = 95,22' -^22 = ^,11- -^12 = -^,12. (d)
AA<p = 0. (e)
Since R0 is simply-connected, cp is single-valued; further, it follows from (d), (e),
and our hypotheses on Sa/? that <p is analytic on R0 and of class C3 on ^0.
Moreover, in view of (c), the relation <p(o) = m{o) in (47.2), (d) and (e) in the proof
of (47.2), and the discussion given in the second paragraph following (47.1),
we can assume, without loss in generality, that
<P = <P,« = 0 on <g0. (f)
Let
W®=fSafiSal,da, (g)
Ri
so that
||S||,=]/F(/J. (h)
Then (d) and (g) imply that
W(l) = J <pial>(pal>da. (i)
Ri
In view of (e), we can write (i) in the form
W{1) = / [95,195,11 — W.iii + 295,29^,12),1
Rt
+ (95,295,22- 95^,222- 295 95,112), i\ia>
where /6(0, a). Thus (f) and Green's theorem yield the relation
W(l) = - J (95,195,11 - 9595,111 + 295,2 95,12) d%s>
and by (2) this relation, in turn, implies that
W® = If I ^f'1 ^11 ~~ 95^.i" + 295,2 95,12) da.
r\
1 Knowles [1966, 12], Eq. (2.6).

Sect. 56. Knowles' version of Saint-Venant's principle. 203
Hence
a
fW(X) dX = - f {yA <ptll -<p<pini+2<P,2<P,i2) da> (J)
since the integral on the right vanishes when I = a. Finally, since
a
f W(X) dX = - J (?>?! + <p% - <p <ptll)tl da,
l Rt
the integral over Rt in (j) can be transformed as follows by again applying Green's
theorem and the boundary condition (f):
fW{X)dX = f{<p\ + <p%- <p<pA1)dx2. (k)
l S?i
Next, by (i) and (2),
dw(i) r ...
—di~ = ~ \ <P,*P<P,*fidxi- (])
Thus given any scalar x, the formulae (k) and (1) imply the following identity:
dW®+4x*fw(X)dX
i
J 0?n + 9^32+2^12- 4»2(9P?i + <f?t—<P<p,ii)] dx2.
dl
I
(m)
Letting
F(l) = W(I) + 2xf W(X) dX, (n)
i
we can write (m) in the form
-¾ + 2x F(l) = - f([Vt u + 2*V)2
js?« (°)
after a rearrangement of terms in the integrand on the right side of (m).
Our next objective is to determine a rx>sitive_yalue for x for which the right
sjde_of JpJ is non-positive. To do so we make use of the following inequality. Hip
is a smooth function on the closed interval [t0, /J, and if y>(t0) =^(4) =0, then
A^wA^- (p)i
where ip is the derivative of yi. This inequality reflects the fact that ^/½-¾)2
is the smallest characteristic value associated with the problem of minimizing
ff2dt,
1 For a discussion of inequalities of this type, reference may be made to § 0.6 of Temple
and Bickley [1956, 14]. A direct proof of (p) can be found in Hardy, Littlewood, and
Polya [1959, 6].

204 M. E. Guriin : The Linear Theory of Elasticity. Sect. 56.
with the minimum taken over all smooth functions f on [t0, /J that vanish at
the end points and have
*i
fy>2di = i.
h
We require a slight generalization of (p). If, for fixed I€(0, a), f is a smooth
function (of x2) on SPlt and if y> vanishes at the end points of each line segment
that constitutes .=2^, then
J f% d x 2 ^ |2- J y>2 d x2. (q)
This inequality follows upon decomposing the integral on the left in (q) into a
sum of integrals over the constituent line segments of 3?lt applying (p) to each
of these, and observing that the length of each line segment is at most b.
Fix /€(0, a). Since <ptl vanishes on ^0, it vanishes at the end points of the
line segments that form J?;. Thus, because of the smoothness of <p, we may take f
in (q) equal to <ptl with the result that
J <PAi dx2 ^ p" J <P*idx2
we can write (o) in the f
|2 + 2xF(l) ^- f [(-J- -4xj 9?! +tf,t-4x*cp%-4x*<p*
v2
In view of this inequality, we can write (o) in the form
dF(l)
dl
Alternatively, we can let f — <pt 2 in (q) with the result that
/ <pf22 dxz ^ ~ / <pfa dx2,
set sei
and this result when substituted into (r) yields
dF(l) /"!/■>■** \ . /»*
~dl
We now assume that
dx2. (r)
\f +2xF®?Z- f [(^-4^)^ + (½-4#)tf*-A*<fi
dx2. (s)
so that the coefficients of q?t and <p*2 in (s) are non-negative. Then we may take
y> = cp in (q) and infer from (s) that
2$. + 2xF(l) £ - [(-J -4**) £ - 4*] J> dx2.
The polynomial in x in the bracket has one positive zero, namely
k
where
"-t>
>-'(*-;-'?■
This value of x satisfies if). We now choose this value of x and conclude that
F(l) satisfies the differential inequality
^ + 2xF(l)^0, 0<l<a. (u)

Sect. 56.
Knowles' version of Saint-Venant's principle.
205
The inequality (u) implies that
F{l)^F{0)e-*Kl, 0<l<a.
Since x ^ 0, (i) and (n) imply that
W{1) ^F{0) e~2*>, O^l^a.
thus
(v)
(w)
It remains to compute an upper bound for F(0).
If we insert (n) into the left side of (v), we obtain an inequality which may be
written in the form
dl
1JW{X)
dk
<
F(0)e
-\xl
Integrating both sides of this inequality from /=0 to 1= a, we find, with the
aid of (n), that
« r u
$W{X) dX £ -¾ (1 - *-«"•) = ^ W(0) + 2xj W(X) dl
(1— e-ixa)
Therefore
lxfw{X)
dX:
1 — e-ixa
'X+e—***
W(0)£W{0),
(X)
and this result when inserted into (n) with 1=0 yields the inequality
F{0)^2W{0).
The inequalities (w) and (x) imply that
W(l)^2W(0)e-2*1, O^l^a;
in view of (i), this is the desired result. □
As was noted by Knowles,1 when R is a rectangular domain subject to self-
equilibrated tractions along one edge, the optimum choice of the constant k
in the estimate
|S||^1/2|S||0,
b
s approximately 4.2.a Thus the universal constant k in (1) appears, in this
particular case, to be conservative by a factor of approximately three.
As is clear from the relation on page 152, the strain energy U{t) of Rt is
given by
U(l) = {JSapE^dv
= V f[S«l>S°<l>-vSvv)dv-
1 [1966,12], p. 11.
2 See, e.g., Fadle [1940, 1], Horvay [1957, 9], and Babuska, Rektorys, and Vycichlo
[1955, £]• The discussion by Babuska, Rektorys, and Vycichlo of the exponential decay
of stresses in the rectangular case appears to be in error in that the roots of smallest modulus
of the equation sin2 ^-z2 = o are underestimated by a factor of two (cf. Fadle [1940,1]).3
3 Knowles [1966, 12], p. 11, footnote 1.

206 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 56 a.
If we assume that fi>0, 0< v < i, then the inequality
(1 — 2 v) Sa/? Sap ;£ Sa/? Sap — v Syy 5g Sa/? Sa/?
is easily derived using the same procedure as was used to establish (24.6). Thus
-*■■;? \S\,£U(DZ-±\Sl.
and the inequality in (1) can be written in the form M
U{l)^-^U(0)e-
b
Knowles1 also proved that under the hypotheses of (1) the stresses satisfy
the inequality
where 6 is the distance from the point (%, x2) to the boundary of ^¾ and k is
defined in (1). Note that this inequality is poor at points which are close to the
boundary. "" ~~
56 a. The Zanaboni-Robinson version of Saint-Venant's principle. (Added in
proof.) This version of the principle is concerned with a one-parameter family
Bv (0 < I < oo) of bodies (bounded regular regions) such that
BTCB, (r<l)
and
B(r,l) = Bl-Br (r<l)
has a regular interior. We assume given a continuous, symmetric, positive definite
elasticity field C on "
\FBl,
ie [o, oo)
a regular surface £f such that
tf'CdBj (0<l<oo),
and integrable surface tractions s on £f such that
J sda = fpxs da=0.
(1) Zanaboni-Robinson theorem? For each lC-[0, oo) let [ul,El,Sl]
be an elastic state on Bt corresponding to the elasticity field C (restricted to BJ, to
zero body forces, and to the following surface tractions:
S;=s on £?,
s,= 0 on dBt-y.
Let
U{T,I) = tfEfC[E,-\dv (t</).
B(r,0
1 [1966, 12], Theorem 2. See also Roseman [1967, 12].
2 The precise version stated here is due to Robinson [1966, 21], § 9.7; the underlying
ideas, however, are contained in the work of Zanaboni [1937, 6, 7, 8]. See also Segenreich
[1P7L £}• wn0 established interesting corollaries of this theorem.

Sect. 57. Some further results for homogeneous and isotropic bodies. 207
Then
lim U(r, 1) = 0.
r,l—*-oo
r<l
For the proof of this theorem we refer the reader to the book by Robinson.1
XI. Miscellaneous results.
57. Some further results for homogeneous and isotropic bodies. In this section
we shall establish some results appropriate to homogeneous and isotropic bodies.
Given a smooth vector field u on B and a closed regular surface if in B, we
define the vector field * (u) on £f by2
* (u) = (X +2/j) (div u) n -\-/n (curl u) x n,
where n is the unit outward normal to £f?
The first theorem, which is due to Fosdick2, shows that the vector field t(u)
when integrated over dB gives the actual load on the boundary.
(1) Let £f be a closed regular surface in B. Further, let u be a smooth vector
field on E, and let s be the corresponding surface traction field on if:
s=fi(Vu-\- VuT) n + X(divu) n.
Then
f t(u)da= f sda.
Proof. Since
(curl m)xm = (Vu — VuT) n,
it follows that
s — t (u) = 2fi [VuT — (div u) 1] n,
and thus that
/ sda— J t(u) da = 2/x f [VuT — (div u) 1] n da.
Letting T denote the skew tensor field whose axial vector is u, we conclude from
(15) of (4.1>that
/ s da— J t(u) da = — 2/j, / (curlT)Tnda,
and the desired result follows from Stokes' theorem (6.2). □
Note that since dB is the union of closed surfaces £fx, ..., £fN, if u is constant
on each £fn, then (1) implies
f t(u)'uda = f s-uda;
dB dB
thus, in this instance, "the work done by t(u)" is equal to the work done by s.
(2) Reciprocal theorem.* Let u and u be elastic displacement fields of
class C2 on B corresponding to body force fields b and b, respectively. Then
ft(u)-uda-\-fb-udv = ft(u),uda-\-fb-udv
dB B dB B
— I [(^ + 2/it) (divh) (divii) -\-/j,(cut\u) • (curlit)] dv.
B
1 [1966, 211 §9-7.
2 Fosdick [1968, 5].
3 Sometimes we will take SP=dB. It will be clear from the context when this is the case.
4 Weyl [1915, 3], p. 6; Fosdick [1968, 5],

208 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 57.
Proof Let I(u, u) denote the last term in the above relation. It follows from
the identity
/ (it X curl u) -nda= J (curl u) ■ (curl u) dv — / u • curl curl u dv
dB B B
in conjunction with the definition of t (u) that
J t(u) •iida= J [(X-\-2/j,)(divu) ii • n + /j,(uxcurlu) -n] da
dB dB
= I(u,u) -f f [(X + 2fi) Vdivu — fi curl curl u] -udv.
B
But since u is an elastic displacement field,
(A + 2/n) Pdiv u — fi curl curl u + b = 0.1
Therefore
/£(tt) •#<£«+ / b • udv = I(u, ii),
dB B
and, since I(u, u) —I(u, u), the proof is complete. □
If we let u = u and b = b in (2j, we arrive at the following result, which is
an analog of the theorem of work and energy (28.3).
(3) Kelvin's theorem.2 Let u be an elastic displacement field of class C2 on B
corresponding to the body force field b. Then
f t(u) -uda-j- f b •udv = 2G{u},
dB B
where
G{u} = ^J [{X + 2[j) (divu)2+^|curl m|2] dv.
B
In view of the remark made following the proof of (1), when u is constant
on each closed surface comprising dB, the left-hand sides of the relations in (3)
and (28.3) coincide; therefore, in this instance, G{u} — U{E}. In particular, it
follows that G{u} is equal to the strain energy when the displacement field vanishes
on the boundary.
Theorem (3) yields a much simpler proof3 for the uniqueness theorem (32.2)
when B is homogeneous and isotropic. Indeed, if u = 0 on dB and b=0 on B,
then Kelvin's theorem implies
G{u} = 0.
But if C or — C is strongly elliptic, we conclude from (25.1) that /j, (A + 2fj) > 0,
and hence that
divtt = 0, curltt = 0,
which implies A u — 0. But if u is harmonic and vanishes on dB, then u must also
vanish on B, and the proof is complete.
Since B is homogeneous and isotropic, the functional 0 of the principle of
minimum potential energy (34.2) has the form
0{u} = f [-£- IVu + VuT\2 + * (divtt)2 dv- fb-udv- fs-uda,
B B &
1 This identity is derived on pp. 90-91.
2 Kelvin [1888, 4]. See also Weyl [1915, 3], p. 6.
3 This argument is due to Ericksen and Toupin [1956, 3]. See also Duffin and Noll
[1958, S], Gurtin and Sternberg [i960, 7].

Sect. 57- Some further results for homogeneous andjsotropicjbodies. 209
as is clear from (24.5). If we confine our attention to the displacement problem,
we can establish a minimum principle much simpler in nature than the principle
of minimum potential energy.
(4) Minimum principle for the displacement problem1. Consider the
displacement problem for which 6^ — dB. Let s/ be the set of all kinematically
admissible displacement fields, and let % be the functional on si defined by
x{u}=G{u}— J b -udv
B
for every ues/. Then given u£s#,
}{{u}^x{u}
for all u€ s/if and only if u is a displacement field corresponding to a solution of the
displacement problem.
The proof of (4) follows at once from (34.2) by virtue of the following
theorem, which establishes the connection between the functional 0 and x.
(5)2 Suppose that the hypotheses of (4) hold. Let si be the set of all kinematically
admissible displacement fields and let
X{u}=x{u} — 0{u}
for every ues/. Then
A{tt}=A{tt'}
for every u, u'£s/; i.e. A {it} is independent of u.
Proof. Let uesf. Since
\carl u\2 = Vu-(Vu-VuT),
| Vu + VuT\ 2 = 2 Vu • (Vu + VuT),
it follows that
A {it} —fi J {(divtt)2 — Vu • VuT} dv =/n f div[tt(divu) — (Vu) u] dv.
B B
Thus
X{u}=fi J tjda, tj = [u (div u) — (Vu)u]-n.
dB
We now complete the argument by showing that r\ depends solely on the
boundary displacement it, which is common to all members of si'. Let the origin 0
of the cartesian frame be an arbitrary regular point of dB, and choose the frame
so that the plane x3=0 coincides with the tangent plane of dB at 0, the
pointing in the direction of the inner normal of dB at 0. For this choice of
coordinates,
3«, , 3«, /3«, , 3«o\ , „ ,,
,=^+^_.._^__i_ + ._x) atO. (a)
In a neighborhood of 0 the boundary dB admits the parametrization
% = £i. *% = &%, *s =»*(©. 5=(¾.¾. (b)
where
■Jjr(°)=° («=i-2). (c)
Further, if we agree to write u(t-) =^(¾. |2< M£i> £2))» then, since uesf,
u(x) — ii(£,)
1 Love [1906, 5], § 116. See also Gurtin and Sternberg [i960, 7], Theorem 2.
2 Gurtin and Sternberg [I960, 7], Theorem 3.
Handbuch der Physik, Bd. VI a/2. 14

210 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 58.
in the neighborhood under consideration. Thus by (b)-(c)
■£-(0)=|j> (.=1.2).
and we conclude from (a) that
v=Ul^+U2.8^Ua^y + .dJL} at0.
Therefore rj at 0 depends exclusively on the boundary data u. But the point 0
was an arbitrarily chosen regular point of dB. Thus A{u} depends solely on the
boundary data u; hence A{u} — A{u'} for every u, u'es/. □
58. Incompressible materials. An incompressible body is one for which only
isochoric motions are possible, or equivalently only motions that satisfy
divtt = 0.
For a linearly elastic incompressible body the stress is specified by the
strain tensor only up to an arbitrary pressure. Thus
S+pl = C[E],
where
p = -±trS
and C = Cx is a linear transformation from the space of traceless symmetric tensors
into itself.1 The symmetry group ^x for the material at x is the set of all
orthogonal tensors Q such that
QC[E]Q*=C[QEQT]
for every traceless symmetric tensor E. This definition is consistent, since
tr (QEQT) = tr (QTQE) =trE = 0
when Q is orthogonal.
If the material is isotropic, then (iii) of (22.1) is still valid; therefore
C [E]=2ftE,
and hence
8+^1 = 2^.
Thus there is but one elastic constant for an isotropic and incompressible material—
the shear modulus /j,.
The fundamental field equations for the time-independent behavior of a
linearly elastic incompressible body are therefore
div tt=0,
E = k{Vu + VuT),
S+pi=C[E], p = -itr8, (a)
divS + b=0.
1 We could also begin by assuming that
S+p\ = C[Vu\;
the steps leading to the reduction C [Vu] — C [JEJ would then be exactly the same those given
in Sect. 20.

Sect. 58. Incompressible materials. 211
If we combine these relations, we are led to the displacement equation of
equilibrium
divC[Vu]-Vp+b=0,
which for a homogeneous and isotropic body reduces to
/j,Au-Vp+b=0. (b)
Since u is divergence-free,
Zl^>=divb; (c)
hence in the absence of body forces the pressure p is harmonic. In view of (a)x
and (3) of (4.1), we can also write (b) in the form ~
ficurl curlu-\-Vp — b = 0.
On the other hand, if b = 0, then (b) and (c) imply that
A curl it = 0,
AAu = 0.
Since tr E = 0, the compatibility equation (14.3) has the following form for
an incompressible body:
AE-VdivE-(VdivE)T = 0.
If the body is homogeneous and isotropic, then the same procedure used on
p. 92 to derive the stress equation of compatibility now yields
AS-2VVp + 2Vb + {div b) 1 = 0,
which in the absence of body forces has the simple form
AS-2VVp=0.
For an isotropic incompressible material the surface traction on the boundary
dB takes the form
s=Sn — —pn+fi(Vu-\-VuT) n.
Using the steps given on p. 93, we can also write this relation as follows:
s = —pn-\-2fi-g \-finxcuxlu.
We define, for an incompressible elastic body, the notion of an elastic state
in the same manner as before,1 except that the constitutive relation is replaced
by (a)3 and the incompressibility condition (a)x is added. If this is done, the
theorem of work and energy (28.3) remains valid for incompressible bodies,
where, as before,
U{E}=^fE-C[E]dv,
B
but now, of course, E is required to be traceless. The crucial step in proving this
result is to note that if S and E obey (a)3, then
S-E = -pl-E+E-C[E]=E-C[E],
since 1 • E =tr E =0. Using a similar result, it is a simple matter to verify that
Betti's reciprocal theorem (30.1) also remains valid.
^-95-
14*

212
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 59.
If we define the mixed problem as before,1 then steps exactly the same as
those used to establish (32.1) now yield the
(1) Uniqueness theorem for an incompressible body. Assume that the
body is incompressible and that the elasticity field C is positive definite. Let
6 = [u, E, S] and 3 = [u, E, S] be two solutions of the same mixed problem. Then
4 and 4 are equal modulo a rigid displacement and a uniform pressure, i.e.
[u, E,S] = [u + w,E,S+pl],
where w is a rigid displacement field and p is a scalar constant. Moreover, w = 0 if
<fx + 0, while p=Oif ^+ 0.
The principle of minimum potential energy (34.1) remains valid in the
present circumstances provided we require that the kinematically admissible states
satisfy div u = 0 and change the last few words to read: " only ifi — a modulo
a rigid displacement and a uniform pressure." On the other hand, the principle
of minimum complementary energy (34.3) remains valid if we replace &k{S}
by Uk{S-\~P 1}, where p = — J tr S and K is the inverse of C, and if we change
the last few words to read: "only ii S=S modulo a uniform pressure."
E. Elastodynamics.
I. The fundamental field equations. Elastic processes.
Power and energy. Reciprocity.
59. The fundamental system of field equations. The fundamental system of
field equations describing the motion of a linear elastic body consists of the
strain-displacement relation
E=j?(Vu + FuT),
the stress-strain relation
S = C[E],
and the equation of motion2
div S-\-b—Qii.
Here u, E, S, and 6 are the displacement, strain, stress, and body force fields
defined on B x (0, /0)> where (0, /0) is a given interval of time, while C and q are
the elasticity and density fields over B.
Since C[E]—C[Vu], if the displacement field is sufficiently smooth, these
equations combine to yield the displacement equation of motion:
div C[Vu\ -\-b=Qti,
which in components has the form
(CijkiUkil)tj + bi=Qui.
It is a simple matter to verify that, conversely, if u satisfies the displacement
equation of motion, and if E and S are defined by the strain-displacement and
stress-strain relations, then the equation of motion is satisfied.
lp. 102.
2 Since the values of C^ [•] are assumed to be symmetric tensors, we need not add the
requirement that S = ST.

Sect. 59- The fundamental system of field equations. 213
Assume for the moment that the body is homogeneous and isotropic. Then
the procedure used to derive (a) on p. 90 now yields Navier's equation of
motion1
[iAu-\-{X-\-[i) V6ivu-\-b =gu. (a)
By (3) of (4.1), this equation may be written in the alternative form8
(A + 2[i) V div u —fi curl curl u + b = qii.
If we assume that A + 2[i, [i, and q are positive, we can define constants q and
*-f¥-- c«-l/f; <b>
in terms of these constants the above equation becomes
c\ Vdivu — c\ curl curl u-\ =u. (c)
For convenience, we introduce the wave operators Qi and d2 defined by
Uj=clAf-'f (cc = l,2).
Then, since
—e--_Cl-c2,
Navier's equation (a) can be written in the form3
□2u+ (4-4) Vdiv u + -=0. (d)
Assume now that b = 0. It is clear from (c) and (3) of (4.1) that4
curl u = 0 => Di u = 0,
div w = 0=> 02^ = 0-
For this reason, we call cx the irrotational velocity, c2 the isochoric velocity.
Operating on Navier's equation, first with the divergence and then with the curl,
we find that5
□1 divw = 0,
(e)
□ 2CUI"1M = 0-
Thus the dilatation div u satisfies a wave equation with speed clt while the
rotation J curl u satisfies a wave equation with speed c2. Clearly,
1 = ][J+2JL = 1/2<1-V> .
1 This relation was first derived by Navier [1823, 2], [1827, 2] in 1821. Navier's theory,
which is based on a molecular model, is limited to situations for which fi = A. The general
relation involving two elastic constants first appears in the work of Cauchy [1828, 1]; cf.
Poisson [1829, 2], Lam£ and Clapeyron [1833, 1], Stokes [1845, 1], Lame [1852, 2], § 26.
a Lame and Clapeyron [1833, 1], Lame [1852, 2], § 26.
3 Cauchy [1840, 1], p. 120.
4 The first of these results is due to Lame [1852, 2], § 61-62; the second to Cauchy
[1840, 1], p. 137.
6 Cauchy [1828, 1], [1840,2], pp.120, 137- See also Lame and Clapeyron [1833,2],
Stokes [1851, 1], Lame [1852, 2], § 27.

214 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 59.
thus for most materials c1>c2, so that dilatational waves travel faster than
rotational waves.1 In fact, cx=2c2 when v—^.
Next, if we apply Qi to (a) (with 6=0) and use the first of (e), we arrive at
HilpAu — Qii) = 0,
or equivalently,2
DiD2m = 0. (f)
Eq. (f) and the strain-displacement and stress-strain relations imply that
Din2^=Din2s=o.
Eqs. (e) are counterparts of the harmonicity of div u and curl u in the static
theory; (f) is the counterpart of the biharmonicity of u.
We now drop the assumption that B be homogeneous and isotropic. The
quantity
represents the total energy per unit volume. If C is symmetric, then the stress-
strain and strain-displacement relations imply that
e=S • Vu-{-qU-u.
Thus, if we define
q = — Su,
we are led, with the aid of the equation of motion, to the following result:
e-\-div q = b-u. (g)
The quantity q is called the energy-flux vector, and (g) expresses a local form of
balance of energy.
Next, operating on the equation of motion with the symmetric gradient V
and using the strain-displacement relation, we find that
If we assume that C is invertible, so that
E = K[S],
where K is the compliance field, we arrive at the stress equation of motion3
If q is independent of x, this relation takes the form
V{div S + b)=gK[S].
If, in addition, the material is isotropic, then the formula on page 78 giving E
in terms of S yields
K[S]=|[(l+y)S-y(trS)l],
1 In seismology waves traveling with speed or are called primary waves (P-waves), those
with speed ca are called secondary waves (S-waves).
a Cauchy [1840,1], p. 137-
3 Ignaczak [1963, 14] generalizing results of Valcovici [1951, 12] for a homogeneous,
isotropic body and Iacovache [1950,6] for a homogeneous, isotropic, and incompressible
body. See also Ignaczak [1959, 8], Teodorescu [1965, 19].

Sect. 60.
Elastic processes. Power and energy.
215
where /5 is Young's modulus and v is Poisson's ratio; thus
p(Vdiv S + Vb)=Q[(l+v)S-v(trS)l].
Given initial values u0 and v0 for u and u, the stress equation of motion
implies the fundamental system in the following sense: let S satisfy this equation
and the initial conditions
S(-,0)=C[Vu0], S(.,0)=C[Vr0];
let E be defined by the inverted stress-strain relation; and let u be the unique
solution of the equation of motion with the initial values u0 and v0. Then the
strain-displacement relation holds, and hence [u, E, S] satisfies the fundamental
system of equations. The fact that in elastodynamics the fundamental system
of field equations is equivalent to a single equation for the stress tensor is most
interesting, especially when compared to the equilibrium theory where (for a
simply-connected body) the fundamental system is equivalent to a pair of
equations: the stress equations of compatibility and equilibrium.
60. Elastic processes. Power and energy. By an admissible process we mean
an ordered array p = [u, E, S] with the following properties:
(i) u is an admissible motion ;x
(ii) E is a continuous symmetric tensor field on B X [0, t0);
(iii) S is a time-dependent admissible stress field.2
Clearly, the set of all admissible processes is a vector space provided we define
addition and scalar multiplication in the natural manner; i.e. as we did for
admissible states.3
We say that p = [u, E, S] is an elastic process (on B) corresponding to the
body force 6 if p is an admissible process and
E = ±(Vu + VuT),
S = C[E],
div S+b=Qii.
As before, the surface force field s is then defined at every regular point of
95x[0,y by
s(x, t) = S(x,t) n(x),
and we call the pair [6, s] the external force system for p.
If
u(.,0)=u(-.0)=0 (I)
on B, we can extend p smoothly to B X (— oo, t0) by defining
w=0 and E = S=0 onBx(- oo,0).
When (I) holds we call the resulting extended process p an elastic process with
a quiescent past.* Further, to be consistent, we also extend the external force
system by assuming that 6 and s vanish for negative time.
1 Cf. p. 43-
a Cf. p. 49.
3 Cf. p. 95-
4 This terminology is due to Wheeler and Sternberg [1968, 16].

216 M. E. Gurtin: The Linear Theory of Elasticity. Sect. 60.
When we discuss the basic singular solutions of elastodynamics, we will be
forced to deal with processes whose domain is of the form $$ X [0, oo), where £
is a point of $ and
We say that p = [u, E, S] is an elastic process on $$ corresponding to 6 if u, E, S
and b are functions on $% x [0, oo) and given any closed regular subregion PC^j,
the restriction of p to P X [0, oo) is an elastic process on P corresponding to the
restriction of 6 to Px [0, oo).
When we omit mention of the domain of definition of an elastic process, it will
always be understood to be Bx [0, t0).
The following simple proposition will be extremely useful.
(1) If [u, E, S] is an elastic process corresponding to the body force field b,
then given any te(0,t0), [«(•,<), E {•,(), S(>,t)] is an elastic state corresponding
to the body force b—qii.
Given an elastic process, we call the function °tt on [0, t0) defined by
the total energy. Here K is the kinetic energy:
K — \ J qu2 dv,
B
U is the strain energy:
U = Uc{E}=%fE-C[E]dv.
B
If C is symmetric,
U = J / {E ■ C[E] +E • C[E]} dv = f E • C[E] dv = f S • E dv;
B B B
therefore the rate of change of total energy equals the rate of change of kinetic energy
plus the stress power.1 This fact and the theorem of power expended (19.1)
together yield the
(2) Theorem of power and energy. Assume that C is symmetric. Let
[u, E, S] be an elastic process corresponding to the external force system [6, s]. Then
fs-uda-\-fb-udv =<%,
BB B
where % is the total energy.
This theorem asserts that the rate at which work is done by the surface and
body forces equals the rate of change of total energy. The following proposition,
which is an immediate corollary of (2), will be useful in establishing uniqueness.
(3) Assume that C is symmetric. Let [u, E, S] be an elastic process corresponding
to the external force system [6, s], and suppose that
6 == 0 on B, s-u = 0 on 35.
Then the total energy is constant, i.e.
<&(t)=<&(0), 0^t<t0.
1 Cf. p. 65.

Sect. 60. Elastic processes. Power and energy. 217
Another interesting corollary of (2) may be deduced as follows. By (2)
/ / s-udadt + / / b-udvdt =<%{r) —%{0).
0 dB OB
The left-hand side of this expression is the work done in the time interval [0, r].
If the initial state [«(•, 0), £(•, 0), S(-, 0)] is an unstrained state of rest, i.e. if
£(.,0)=0, u(. ,0)=0,
then
#(0)=0.
If, in addition, the elasticity tensor is positive semi-definite and the density
positive, then #(r) Sg 0 and
r r
/ / s-iida dt+ // b -iidvdt^O;
0 dB 0B
in words, the work done starting from an unstrained rest state is always nonnegative.1
The next theorem, which is of interest in itself, forms the basis of a very strong
uniqueness theorem for elastodynamics. For the purpose of this theorem we
assume that the underlying time-interval [0, t0) is equal to [0, oo).
(4) Brun's theorem.2 Assume that C is symmetric. Let [u, E, S] be an
elastic process corresponding to the external force system [6, s], and suppose that
u(. ,0)=0, £(.,0)=0.
Let
P(a, /5) = / s (x, a) • u(x, /5) dax + / 6 (x, a) • u(x, /5) dvx
BB B
for all a, /Se [0, oo). Then
U(t) -K{t) =%f[P(t + X, t-X) -P(t-l t +X)] dX (0£t<oo),
o
where U is the strain energy and K the kinetic energy.
Proof. By (18.2) with 6 replaced by 6(., a) —gii(•, a), s by s (•,<*), u by
m(-,/S), andSby C[£(.,a)],
P(a, ft = / {C[B(«)J • £(/9) +<>« (a) •«(/?)} iw, (a)
where, for convenience, we have suppressed the argument x. Next, in view of
the symmetry of C and the initial vanishing of the strain field,
ff {C [E(t +X)]-E(t-X)-C [E{t - X)] -E(t+X)}dXdv
B0 (b)
= - ff -^- {C[E(t + X)]-E(t-X)}dXdv = 2U{t).
B0
Similarly,
// g{u(t+X) • ii(t-X) -ii(t-X) ■ ii(t + X)} dXdv = -2K(t), (c)
B0
and (a), (b), and (c) imply the desired result. □
1 Some related inequalities are given by Martin [1964, 14].
a Brun [1965, 4], Eq. (7b); [1969, 2].

218 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 61.
61. Graffi's reciprocal theorem. We shall assume throughout this section that
the elasticity field is symmetric.
Given an elastic process p = [w, E, S] corresponding to 6, we call the pair
«„=«(•, 0), »„ = «(•, 0)
the initial data for p and the function / defined by
f(x, t)=i*b(x, t)+g(x) [«„(x) +tv0(x)}
the associated pseudo body force field. Here, as before, i is the function on
[0, t0) defined by
i(t)=t.
Betti's reciprocal theorem has the following important counterpart in elasto-
dyn amies:
(1) Graffi's reciprocal theorem.1 Let [u, E, S] and [u, E, S] be elastic
processes corresponding to the external force systems [6, s] and [b, §]. Further, let
[u0, v0] and [u0, v0] be the corresponding initial data, f and f the corresponding
pseudo body force fields. Then
i* f s*uda-\- f f*udv=i* f s*uda-\- ff*udv, (a)
BB B BB B
J S*Edv= J~S*Edv, (b)
B B
J s*iida-\- f b*udv-{- J q(u0- u-\-v0- u) dv
BB B ^ B (C)
= J s*uda-{- J b*udvJr f q(u0-u-\-v0-u) dv.
»B B B
Thus if both processes correspond to null initial data, then
/ s*uda-\- / b*iidv= J s*uda-\- / b*udv. (d)
BB B BB B
Proof. In view of the reciprocal theorem (19.5), it suffices to show that
S*E=S*E.
By definition,
S*E(t) = J S(t-r)-E(r)dr,
o
where, for convenience, we have omitted the dependence of these functions on a?.
Since C is symmetric,
S(t-r)-E(r) =JB(t) ■ C[E(t-r)] =E{t — r) • C[E(r)] =E{t-r) • S(t).
Thus
S*E(t)=E*S(t)=S*E(t). n
1 The result (c) for homogeneous and isotropic bodies is due to Graffi [1939, 3], [1947, 2],
[1954, 6], whose proof is based on Betti's theorem and systematic use of the Laplace transform.
Later, Graffi [1963, 8] gave a direct proof utilizing the properties of the convolution. Hu [1958,
10] noticed that Graffi's results extend trivially to inhomogeneons and anisotropic bodies;
however, Hu gave his results in the Laplace transform domain rather than the time domain.
Wheeler and Sternberg [1968, 16], assuming homogeneity and isotropy, gave an extension
of Graffi's theorem valid for infinite regions; their result was extended to homogeneous but
anisotropic bodies by Wheeler [1970, 3]. Ainola [1966 2], showed that Graffi's theorem can
be derived from a variational principle. Some interesting applications of Graffi's theorem are
given by DiMaggio and Bleich [1959, 4] and Payton [1964, 16].

Sect. 62. The boundary-initial-value problem of elastodynamics. 219
Graffi1 also noticed that if both systems correspond to null external forces,
then (c) implies
J q(u0-u-\-v0-u) dv= f e(u0-ii-\~v0-u) dv;
B B
if, in addition, v0=v0 = 0, then
/ qu0-udv= f qu0- udv.
B B
This relation can be used to establish the orthogonality of the displacement fields
corresponding to distinct modes of free vibration.2
For the remainder of this section let t0 = oo. We say that two external force
systems [b, s] and [6, s] are synchronous if there exists a continuous scalar
function g on [0, oo) such that
b (x, t) = b0 (x) g(t), b (x, t) = b0 (x) g (t)
for all (x, t)eBx [0, oo) and
s (x, t) = s0 (x) g(t), s (x, t) = s0 (x) g (t)
for all (x,t)edB x[0, oo).
The next theorem shows that for synchronous external force systems a
reciprocal theorem holds having the same form as Betti's theorem in elastostatics.
(2) Reciprocal theorem for synchronous external force systems.3 Let
[u, E, S] and [u, E, S] be elastic processes corresponding to null initial data and to
synchronous external force systems [6, s] and [6, §]. Then
fs-uda-\-fb-udv= fs-uda-\- / b-udv.
dB B dB B
Proof. By hypothesis and (d) of (1), g*<p=o, where
<p = f s0-uda-j- f b0-u dv— fs0-uda—fb0-udv,
dB B dB B
and we conclude from (v) of (10.1) that either g = 0 or 95=0. If g = 0 the
theorem holds trivially. Thus assume <p = 0. Then gcp=0, and the desired result
follows. □
II. Boundary-initial-value problems. Uniqueness.
62. The boundary-initial-value problem of elastodynamics. Throughout the
following three sections we assume given the following data: a time interval [0, t0),
an elasticity field C on B, a density field q on B, body forces 6 on Bx(0, t0),
initial displacements u0 on B, initial velocities v0 on B, surface displacements u
on <$^x(0, /0), and surface tractions s on y2x(0, /0). Given the above data, the
mixed problem of elastodynamics consists in finding an elastic process [u, E, S] that
corresponds to 6 and satisfies the initial conditions
u(-,0) =u0 and u(-, 0)=v0 on B,
1 [1947,2]-
a Cf. {76. 3)
3 Graffi [1947, 2].

220 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 62.
the displacement condition
u=u on <S^X(0, /0),
and the traction condition
s=s on S^2X(0, t0).
We call such an elastic process a solution of the mixed problem.1 As before,
when 6^ — dB («$"2 = 0) we refer to the displacement problem, when
</>% = dB (£fx =0) we refer to the traction problem.
To avoid repeated hypotheses we assume once and for all that:
(i) C and q are smooth on B;
(ii) 6 is continuous on B x [0, /0);
(iii) u0 and v0 are continuous on B;
(iv) u is continuous on^X [0, t0);
(v) s is continuous in time and piecewise regular on £^x[0, t0).
The following proposition is an immediate consequence of the remarks made
in Sect. 59; its proof is completely analogous to the proof of (31.1) and can safely
be omitted.
(1) Characterisation of the mixed problem in terms of
displacements. Let u be an admissible motion. Then u corresponds to a solution of the
mixed problem if and only if
div C [Vu] -\-b = Qtt on B x (0, t0),
u(',0)=u0 and u(-,0)=v0 on B,
u=u on S^,X(0, t0),
C[Vu]n = s on S?2 X (0, t0).
Further, when the body is homogeneous and isotropic, the first and last of these
conditions may be replaced by
(iAu + (X+/i) Fdivw + b =qu on Bx(0,t0),
(j,(Vu + VuT)n-\-k(divu)n=s on S^X(0,t0).
For the remainder of this section we assume that
Q>0.
(2) Characterization of the traction problem in terms of stresses.2
Assume that:
(a) C is invertible on B;
(/9) q, u0, and v0 are of class C2 on B;
(y) b is of class C2>° on Bx [0, t0);
(3) S is a time-dependent admissible stress field3 of class C3>2 on Bx[0,(0)
with S continuous on Bx[0, t0).
1 See footnote 2 on p. 102.
2 Ignaczak [1963, 14] for the case in which u0 = «0=0.
3 See p. 49.

Sect. 62. The boundary-initial-value problem of elastodynamics. 221
Then S corresponds to a solution of the traction problem if and only if
pp!+6.)=K[S] on Bx(0,t0),
S(-,0)=C[Vu0] and S{-,0)=C[Vv0] on B,
s=s on dBx(0,t0).
Proof. The necessity of the above relations follows from the discussion given
at the end of Sect. 59. The sufficiency also follows from this discussion, but we
will give the argument in detail. Let S satisfy the above system of equations.
Then, since S is a time-dependent admissible stress field,
(1) S and div S are continuous onffx [0, /0); and, by (i) and (a),
(2) K= C'1 is continuous on E.
We now define u and E through the relations
tt=i.^S±/, e=K[S], (a)
where / is the pseudo body force field
f{x, t) = i*b (x, t) + g {x) [«0 (*) + tv0 (x)], (b)
i(t) = t.
It then follows that u, S, and b satisfy the equation of motion,1 and that u
satisfies the initial conditions. Next, by (ii), (iii), (/9), (y), (d), (1), (2), (a), and (b),
(3) u, u,.u, E, and E are continuous on ffx[0, t0), while u is of class C2 on
Bx[0,t0); _^_
(4) E is of class C°-2 on B X [0, t0), V{u) and Vu are continuous on B x [0, t0),
and V(u) = Vu.
To complete the proof we have only to show that the strain-displacement
relation holds (for then we could conclude from (3) and (4) that u is an admissible
motion). It follows from (4), (a)2, the first relation in (2), and the equation of
motion that
ij= K [S] = V (-^-) = V(u) = toi. (c)
On the other hand, since u satisfies the initial conditions, we conclude from (a)2
and the initial conditions satisfied by S that
E(..0)=Fu(.,0), E(.,0) = Pu{;0); (d)
thus, by (c) and (d), E = Vu. □
An interesting consequence of theorem (62.2) and the uniqueness theorem
(63.1) is that a body force field with Vlplg) skew induces zero stresses in the body;
i.e., if dB is traction-free, if uo=vo=0, and if F(6/g) is skew, then S=0 and
u—i*b\Q is a rigid motion.
The next two theorems give alternative characterizations of the mixed
problem in which the initial conditions are incorporated into the field equations.
These results will be extremely useful in the derivation of variational principles.
1 Cf. the proof of (19.2).

222
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 63.
(3)1 Let p — [u, E, S] be an admissible process. Then p is a solution of the
mixed problem if and only if
E = Pu,
S = C[E],
i*div S+f = QU
on Bx [0, t0), and
u = u on ,5¾ X (0, t0),
s=s on y2X(0, t0).
Proof. This theorem is a direct consequence of (19.2). □
(4) Characterisation of the mixed problem in terms of stresses.2
A ssume that hypotheses (a.), (/3), and (y) of (2) hold. Let S be a time-dependent admissible
stress field of class C3,° on Bx [0, tQ), and suppose that S is continuous on Ex [0, t0).
Then S corresponds to a solution of the mixed problem if and only if
P^J}^8+Lj=K[S] onBx[0,t0),
i*divS=QU—f on <S^x(0,£0),
s = s on y2X (0, t0).
Proof. Clearly, (l) and (2) in the proof of (2) hold. Assume that $ satisfies
the above relations. If we define
u=i*tor8+f Es=K[8]>
e
then (3) of (2) also holds, and it is not difficult to verify that Vu=E and u = u
on £fx x (0, t0). It therefore follows from (3) that p= [u, E, S] is a solution of the
mixed problem. Conversely, if ft is a solution of the mixed problem, then we conclude,
with the aid of (3), that the relations in (4) hold. □
63. Uniqueness. In this section we discuss the uniqueness question appropriate
to the fundamental initial-boundary-value problems discussed in the preceding
section. We assume throughout that the density q is continuous on B.
We will first establish the classical uniqueness theorem of Neumann. We will
then give a more general theorem due to Brun ; we present Neumann's theorem
separately because its hypotheses are sufficiently general to include most
applications, and because its proof is quite simple.
(1) Uniqueness theorem for the mixed problem.3 Suppose that the
density field is strictly positive and the elasticity field symmetric and positive semi-
definite. Then the mixed problem of elastodynamics has at most one solution.
Proof. Let p = [u, E, S] be the difference between two solutions. Then p
corresponds to vanishing body forces and satisfies
w(.,0)=m(.,0)=0 on B, (a)
m=0 on 5¾ x (0, t0), s=0 on 5¾ X (0, i!0). (b)
By (a)
E(., 0)=0, (c)
1 Gurtin [1964, 8], Theorem 3.2.
2 Gurtin [1964, 8], Theorem 3.4.
3 Neumann [1885, 3], § 61. An interesting generalization was given by Hill [1967, 9].

Sect. 64. Some further extensions of the fundamental lemma. 223
and (a) and (c) yield
«(0)=0, (d)
where $11 is the total energy.1 Moreover, in view of (60.3), (b) and (d) yield
<%{t)=0, 0<t<t0. (e)
By definition fy is the sum of the kinetic and strain energies, and by hypothesis
both of these energies are positive; therefore they must both vanish. In particular,
fQUzdv=0, (f)
B
and, since Q>0,
u = 0 on B x (0, t0).
This fact, when combined with the first of (a), yields
u = 0 on B x (0, t0),
and the proof is complete. □
Recently Brun has established the following major generalization of Neumann's
theorem (1); it is important to note that no assumptions other than symmetry
are placed on the elasticity field. We suppose for the purpose of this theorem that
;0=oo, so that the underlying time interval is [0, oo).
(2) Extended uniqueness theorem.2 Suppose that the density field is
strictly positive (or negative) and the elasticity field symmetric. Then the mixed
problem of elastodynamics has at most one solution.
Proof. As in the proof of (1), let p = [u, E, S] be the difference between two
solutions. Then p corresponds to vanishing body forces and satisfies (a), (b), and
(c) in the proof of (1). Thus we conclude from Bran's theorem (60.4) that
U{t)-K{t)=0, 0^t«x>. (a)
On the other hand, by (e) in the proof of (1), which also holds in the present
circumstances,
U(t)+K(t)=0, 0<t<oo. (b)
Thus i£(£) = 0 and the last few steps in (1) yield the desired conclusion that
u=o. a
III. Variational principles.
64. Some further extensions of the fundamental lemma. We now establish, for
functions of position and time, counterparts of the lemmas established in Sect. 35-
With this in mind, we introduce the following definition. We say that a function
/ on Bx[0,t0) vanishesnear SfCdB if there exists a neighborhood N of £f
such that / = 0 on (Nr« B)x[0, t0).
1 See p. 216.
2 Brun [1965, 4], [1969, 2]. This theorem was established independently by Knops and
Payne [1968,11], using an entirely different method. For the displacement problem appropriate
to a homogeneous and isotropic body, Gurtin and Sternberg [1961, 10] proved that
uniqueness holds provided both wave speeds are positive; i.e. provided q is positive and the
elasticity field strongly elliptic. This result was extended to anisotropic, but homogeneous,
bodies by Gurtin andTouriN [1965, 9]. Hayes and Knops [1968, 7] showed, without assuming
C symmetric, that uniqueness also holds when — C is strongly elliptic. Of course, the extended
uniqueness theorem (2) includes as special cases Neumann's theorem and all of the above
results except that of Hayes and Knops. See also Knops and Payne [1971, 2], § 8, for a
thorough discussion of uniqueness in elastodynamics.

224 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 64.
(1)1 Let W be a finite-dimensioned inner product space. Let w: Bx[0, Q->W
be continuous and satisfy
J w*v(x,t) dvx = 0 (0^t<t0)
B
for every class C°° function v. Bx[0, t0)->W that vanishes near 8B. Then
w = 0 on B x [0, t0).
Proof. Let /,, (t) denote the open interval (t— h, r+h), and for /a(t)C (0, t0)
let Wh (t) be the set of all C°° scalar functions y on [0, t0) with the properties:
(a) ^>0 on Ik{x),
(b) y=0 on [0,t0)-Ih(r).
It is clear from the discussion at the beginning of Sect. 7 that Wh (t) is not empty.
Let ej, e2,..., e„ be an orthonormal basis in W, and let
»
w= 2 wiei-
»■=1
As in the proof of the fundamental lemma (7.1), we assume that for some
(x0, t)gBx(0, t0) and some integer k, wk{x0, t)>0. Then there exists an h>0
such that £h(x0)cB, Ih(r)C(0, t0), and wk>0 on £k(x0) xlk(r). Let
v(x,s)=<p(x)f(s) ek,
where2 <pe 0h(x0),ye Wh(t — x), and t£ (r+h, t0). Then v is of class C°° on B x [0,t0),
vanishes near 8B, and
J w*v(x, t) dvx= J J w(x, t — s) -v(x, s) ds dvx
B BO
t
= /J'zMaM—s) <p(x)yi{s) ds dvx
B 0
t-r+h
= / / wk(x, t— s) cp{x)ip{s)dsdvx>0,
M*>) t—t-h
which is a contradiction. Thus w=Q on Bx\0,t0). □
The proofs of the next three lemmas can safely be omitted; they are completely
analogous, respectively, to the proofs of (35.1), (35.2), and (35.3) and utilize the
procedure given in the proof of (1).
(2)3 Let Wbe a finite-dimensional inner product space. Let w. y% x [0, t0) -*-W
be piecewise regular and continuous in time, and assume that
Jw*v(x, t) dax=0 (0^t<t0)
for every class C°° function v: Bx[0, t0) -*-W that vanishes near 3\. Then
w = 0 on ^x[0, t0).
1 Gurtin [1964, 8], Lemma 2.1.
2 <X>h (a;0) is defined in Sect. 7.
3 Gurtin [1964, 8], Lemma 2.2.

Sect. 65. Variational principles. 225
(3)1 Let ubea vector field on £^ x [0, t0) that is piecewise regular and continuous
in time, and suppose that
J (Sn) *u(x, t) dVn = 0 (0 ^t<t0)
for every class C°° symmetric tensor field S on Ex[0, t0) which vanishes near y2.
Then
« = 0 on ^x[0, t0).
(4)2 Let ubea vector field on y2 x [0, t0) that is piecewise regular and continuous
in time, and suppose that
/w*divS(ac, t) dvx = 0 (0^t<t0)
for every class C°° symmetric tensor field S on Bx [0, t0). Then
M = 0 on £?2x[0,t0).
65. Variational principles. In this section we will establish several variational
principles for elastodynamics. In addition to hypotheses (i)-(v)3 of Sect. 62 we
assume throughout that the elasticity field C is symmetric. We begin by giving
Kirchhoff's generalization of Hamilton's principle in particle mechanics.
By a kinematically admissible process we mean an admissible process
that satisfies the strain-displacement relation, the stress-strain relation, and the
displacement boundary condition.
(1) Ha-milton-Kirchhoff principle.4 Let «s/ denote the set of all kinematically
admissible processes p= [u, E, S] that have ii and E continuous on B X [0, t0] and
tt(-,0) = a, u(>,t0) = p on B,
where a and fi are prescribed vector fields. Let •?]{•} be the functional on s# defined by
rj{p}= f°U{E} dt- JJ (i Qitf+b • u) dv dt- f Js-udadt
0 0B 0 SP,
for every p= [u, E, S] e«s/. Then
dr]{p} = 0
at p£j/ if and only if p satisfies the equation of motion and the traction boundary
condition.
Proof. Let pes/, and let p= [u, E, S] be an admissible process with the
following property:
p+Apes/ for every scalar A. (a)
1 Gurtin [1964,5], Lemma 2.3. In 1965 HlavaCek (private communication) pointed
out that the proof given by Gurtin [1964, 8] is incorrect and furnished a valid proof.
2 Gurtin [1964, 8], Lemma 2.4.
3 For the validity of (1) we need the stronger assumption that (ii), (iv), and (v) hold with
the interval [0, t0) replaced by [0, t0].
4 Kirchhoff [1852, 2], [1859, 2], [1876, 2], § 11. See also Love [1927, 3], § 115; Loca-
telli [1945, 3, 4]; Washizu [1957, 18]; Kneshke and Rudiger [1962,10]; Chen [1964, 6];
Yu [1964,23]; Ben-Amoz [1966,2]; Karnopp [1966,20], [1968,20]; Komkov [1966,25];
Tonti [1966, 26].
Handbuch der Physik, Bd. VI a/2. 15

226 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 65.
Clearly, (a) holds if and only if p satisfies the strain-displacement relation, the
stress-strain relation, the boundary condition
m=0 on S^x[OJ0], (b)
and the end conditions
u(-,0) = u(-,t0) = 0 on B. (c)
Next,
to to (d)
= I j (S • E—qu- u—b ■ u) dv dt— I s -udadt,
OB o s?t
where we have used the symmetry of C and the stress-strain relation. By (c)
'° . • ^° .. ~
J qu ■ u dt= — J qu. • u dt, (e)
o o
and (d), (e), the fact that E and u satisfy the strain-displacement relation, (18.1),
and (b) imply
to (- t0
d^rj{p}= — J J (divS + 6 — qu) ■ udv+ f J (s — s) ■ udadt. (f)
OB o yt
Clearly, 8r}{ft}=0ii
divS+b = Qii on Bx[0,t0), s = s on S?2x[0,t0). (g)
On the other hand, if (f) vanishes for every p consistent with (a), then (f) vanishes
for every admissible motion u consistent with (b) and (c), and obvious extensions
of (7.1) and (35.1) yield (g). □
The Hamilton-Kirchhoff principle is concerned with the variation of a
functional over a set of processes that assume a given displacement distribution at
an initial as well as at a later time. This type of principle clearly does not characterize
the initial-value problem of elastodynamics, since it fails to take into account
the initial velocity distribution and presupposes the knowledge of the
displacements at a later time—an item of information not available in advance. We now
give several variational principles which do, in fact, characterize this initial-value
problem.
We assume for the remainder of this section that the density q is strictly
positive. Note that, in view of assumptions (i)-(iii) of Sect. 62, the pseudo body
force field
f(x, t) =i*b(x, t) +q{x) [u0(x) +tv0(x)], i(t) =t
is continuous on B X [0, t0).
We begin with a variational principle in which the admissible processes are
not required to meet any of the field equations, initial conditions, or boundary
conditions; this principle is a counterpart of the Hu-Washizu principle (38.1) in
elastostatics.
(2)1 Let j/ denote the set of all admissible processes, and for each td\0, t0)
define the functional /1,(-} on $0 by
At{p} = f {^i*E*C[E] +%QU*u — i*S*E — [i*divS+f]*u} (x, t) dv^
B
+ f {i*s*u}(x, t) dax-\- J fi*(s — s)*u\ da^
&x sr,
1 Gurtin [1964, 8], Theorem 4.1.

Sect. 65. Variational principles. 227
for every p = [u, E, S] e $¢- Then
6A,{fi} = 0 (0^t<t0)
at an admissible process p if and only if p is a solution of the mixed problem.
Proof. Let p = [u, E, S] and p = [u, E, S] be admissible processes. Then
p + Xp is an admissible process for every scalar X. Since C is symmetric,
E(t - t) • C [E{x)} = C [E(t - t)] • E{x),
and thus
E*C[E]=C[E]*E=E*C[E]. (a)
Let
Ut{E} = J / i*E*C[E] (x, t) dvx.
B
Then (a) imphes
Ut{E + XE}=Ut{E}+X2 Uf{E} +X f i*E*C[E] (x, t) dvx. (b)
B
Thus, since
d~At{p}= ddX At{p+Xp}\n,
it follows that
d~PAt{p}
= f {i#(C[E] —S)*E—i*S*E — (i*div S+f — Q u)*u — i*u*div S}(x, t) dvx
B
+ J {i*s*u}(x, t) dax-\- J i*{s*u-\-(s— s)*u}(x, t) dax.
If we apply (19.3) to S and u, we find that
i* Ju*div S dv =i* J s*uda—i* J S*Vu dv;
b m b
thus
d~pAt{p} = / ** (C[E] - S)*E(x, t) dvx
b
— J (i*div S+f — qu)*u(x, t) dvx
B
+ Ji*{Vu-E)*S{x, t) dvx (c)
B
+ /i*(u — u)*s(x, t) dax
+ f i*(s — s)*u(x,t) dax (0^t<t0).
If p is a solution of the mixed problem, then we conclude from (c) and (62.3) that
d~At{p}=0 (0^t<t0) for every pts/ (d)
which implies
dAt{p}=0 (0^t<t0). (e)
To prove the converse assertion assume that (e) and hence (d) holds. If we
choose p = [u, 0, 0] and let u vanish near 8B, then it follows from (c) and (d) that
/ (i*div S+f — qu)*u(x, t) dvx=0 (0^t<t0).
B
15*

228 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 65-
Since this relation must hold for every m of class C00 on Ex[0, t0) that vanishes
near dB, we conclude from (64.1) that i*div S+f = gu. Next, by choosing
jtistfvn. the forms [0, E, 0] and [0, 0, S], where E and S vanish near dB, we
conclude from (c), (d), and (64.1) with HP equal to the set of all symmetric tensors that
i*(C[E]—S)=0, i*(Pu—E)=0.
Thus, by (iv) of (10.1) with <p=i, ft satisfies the stress-strain and displacement-
strain relations. In view of (62.3), to complete the proof we have only to show
that p satisfies the boundary conditions. By (c), (d), (iv) of (10.1), and the
results established thus far,
/(« — «)* (Sn) (x, t) dax + f (s — s) * u (x, t) dax = 0 (0 ^ t < t0)
for every pzstf. Therefore, if we take pzstf equal to [«, 0, 0] and [0, 0, S] and
appeal to (64.2) and (64.3), we conclude that p satisfies the boundary
conditions. □
Our next theorem is an analog of the Hellinger-Prange-Reissner principle
(38.2).
(3)1 Assume that the elasticity field is invertible. Let s# denote the set of all
admissible processes that satisfy the strain-displacement relation, and for each
te(0, t0) define the functional ©<{"} on ^ by
0({p}= J {%i*S*K[S] —%QU*u — i*S*E+f*u} (x, t) dvx
B
+ / {i*s*(u —u)} (x, t) dax-\- f {i*s*«*} (x, t) dax
for every p = [u, E, S] e j</. Then
d@({p}=0 (0^t<t0)
at pzstfif and only if p is a solution of the mixed problem.
Proof. Let p and p be admissible processes, and suppose that p-\-hp£s& for
every scalar X, or equivalently, that p,p£s#. In view of the symmetry of C, K is
symmetric. Thus if we proceed as in the proof of (2), we conclude, with the aid
of (19.3), that2
d~6t{p) = / i*(K[S] -E) *S(x, t) dvx
B
+ / (i*div S +/ — qu) *u(x, t) dvx
B (a)
+ / i* (u — m) *s (x, t) dax
+ / i*(s — s)*u{x, t) dax (0^t<t0).
If p is a solution of the mixed problem, then (a), because of (62.3), yields
det{fi}=o (o^t<t0). (b)
On the other hand, (a), (b), (64.1), (64.2), (64.3), (iv) of (10.1), and (62.3)
imply that p is a solution of the mixed problem. □
1 Gurtin [1964, 8], Theorem 4.2.
2 Cf. the proof of (38£).

Sect. 65. Variational principles 229
If p = [tt, E, S] is a kinematically admissible process, then, by virtue of (19.3),
At{p} defined in (2) reduces to <&({p}, where
&t{p}=? f {i*S*E + gu*u — 2f*u} (x, t) dvx — f {i*s*u} (x, t) da^.
b sr,
Thus we are led to the following:
(4) Analog of the principle of minimum potential energy.1 For each
te[0, t0) let 0t{'} be the functioned defined on the set stf of all kinematically
admissible processes by the above relation. Then
at pes/ if and only if pis a solution of the mixed problem.
Proof. Let p = [«, E, S] be an admissible process, and suppose that
p + lpest for every scalar A. (a)
Condition (a) is equivalent to the requirement that p satisfy the
strain-displacement relation, the stress-strain relation, and the boundary condition
m = 0 on £fxx\0,t0). (b)
Next, since the restriction of At{-} of (2) to (the present) s4 is equal to $«{•},
we conclude from (c) in the proof of (2) and the properties of p and p that
h &t{p} = — I (*'*div S +/—qu) * u{x, t) dvx
B - (c)
+ fi*(s— s)*u(x, t) da^. {0^t<t0)
for every admissible motion it that satisfies (b). If p is a solution of the mixed
problem, then, by virtue of (62.3), 50,{p)==O (0^t<t0). Conversely, if the
variation of 0( vanishes at p for all t, then it follows from (c), (64.1), and
(64.2) that p is a solution of the mixed problem. □
It is a simple matter to prove counterparts of theorems (34.2) and (57.4)
characterizing the displacement field corresponding to a solution of the mixed
problem.2 Rather than do this we shall establish a variational principle which
involves only the stress field and which has no counterpart in elastostatics.
In addition to the assumptions made previously, we now assume that:3
(A) C is invertible onB; q, Uq, and v0 are of class C2 onB; 6 is of class C2'0 on
Bx[o,g.
(5) First variational principle for the stress field4. Let s£ denote the set
of all time-dependent admissible stress fields of class C3' 0onBx(0, t0) with S continuous
on Ex [0, rf0). For each te [0, t0) let ./]{•} be the functional on sf defined by
rt{S}=%J {i'*div S*div S +S*K[S] -2S* Vf'} (x, t) dvx
+ / {(/'—"**) *s} ix> 0 ^am + I {*'*(s —s)*div S} (x, f) dax,
where ^1 *"•
Q Q
1 Gurtin [1964, «], Theorem 4.3.
2 Cf. Gurtin [1964, «], § 5-
3 These assumptions are slightly more stringent than necessary.
4 Gurtin [1964, S], Theorem 6.1.

230 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 66.
c5/]{S}=0 (0<t<t0)
at Sestf if and only if S corresponds to a solution of the mixed problem.
Proof. Let Sc-s/, so that S + X Stsf for every scalar X. Then, since K is
symmetric, the divergence theorem and the properties of the convolution yield, after
some work,
ds rt{S} = -/ {V(i'*div S +/') -K[S]}*S{x, t) dvx
B
+ / (»'*div S — u-\-f')*s{x, t) dax (a)
+ f i'*(s — s)*divS(ac, t) dax (0^t<t0)
for every Sesf. If S corresponds to a solution of the mixed problem, then (62.4)
implies that
drt{S}=0 (0^t<t0). (b)
On the other hand, (a), (b), the symmetry of S, (64.1), (64.3), (64.4), (iv) of
(10.1), and (62.4) imply that S corresponds to a solution of the mixed
problem. □
Let <s/ be as defined in (5). By a dynamically admissible stress field
we mean a tensor field Sestf that satisfies the traction boundary condition. For
such a stress field, -T({S}, defined in (5), reduces to
Qt{S}=% J {i'*div S*div S+S*K[S]-2S*Pf'} (x, t) dvx
B
+ / {(/' — «) *s} ft, t) dax-
Thus we are led to the following theorem, the proof of which is strictly analogous
to that given for (5).
(6) Second variational principle for the stress field.J For each te [0, t0)
let Q,{'} be defined on the set of dynamically admissible stress fields by the above
relation. Then
6Qt{S}=0 (0<t<t0)
at a dynamically admissible stress field S if and only if S corresponds to a solution
of the mixed problem.
If we apply (19.3) to the term involving S * Vf', we can rewrite the expression
for Qt{S} m the form
Q({S} = J / {*''*div S*div S +S*K[S] +2 div S*f'} (x, t) dvx
— / {u*s} (x, t) dax — / {f'*s} (x, t) dax.
9*\ '^2
Of course, the last term may be omitted without destroying the validity of (6),
since it is the same for every dynamically admissible S.
66. Minimum principles. For the variational principles established in the
preceding section the relevant functionals were stationary, but not necessarily
a minimum, at a solution. On the other hand, the principles of minimum potential
1 Gurtin [1964, 8], Theorem 6.2.

Sect. 66. Minimum principles. 231
energy and minimum complementary energy are true minimum principles. This
difference is nontrivial; indeed, the lower bound on the energy provided by a
minimum principle is useful in establishing the convergence of numerical solutions.
In this section we establish minimum principles for the Laplace transforms of the
functionals associated with the variational principles (65.4) and (65.6).
We let t0 =oo, so that the underlying time interval is [0, oo). In addition to
hypotheses (i)-(v) of Sect. 62 we now assume that:
(vi) C is symmetric and positive definite;
(vii) q>0;
(viii) 6, u, and s possess Laplace transforms.
Let s£h denote the set of all kinematically admissible processes ft = [u, E, S]
such that1 ft and Vft possess Laplace transforms. Then for p&rfL the function
t^0t{ft}
of (65.4) has a Laplace transform
_ °°
<P„{*} = / «-" <P,{*} «tt (r?0^r?<oo)
0
for some rj0^0. In fact, since the Laplace transform of i(t) = t is \\rf, it follows
from (vii) of (10.1) and the transformed stress-strain relation S = C[E] that
*,{*} = y uc{E(;n)) + J{{ q(*) «(*.nY-/>. v) • «(*.v)}dv*
1 r- - V
— ^ J s{x,r))-u(x rj) dax (»?0^r?<oo).
It is interesting to note the similarity between &v{'} and the functional ¢{-} of
the principle of minimum potential energy (34.1). This similarity is emphasized
by the
(1) Theorem of minimum transformed energy.2 Assume that the
elasticity field is positive semi-definite. Let fa<zJ#L be a solution of the mixed, problem.
Then for each pd^L there exists an tjq^O such that
0n{ft}^0n{ft} (r?0^r?<oo).
Proof. Since ft, fi£<stfL, there exists an rfoJgO such that the Laplace
transforms of ft, p, Vp, Vft, and 6 exist on B X [%, oo). Let
ft'= p-ft. (a)
Then ft' is an admissible process, ft' and Vp' possess Laplace transforms, and
E'^{Vu' + Vu'T), (b)
S'=C[E], (c)
m' = 0 on ,9[. (d)
Moreover S = C[E], since p£s/L; thus (28.1) and (a) imply
UC{E} - UC{E} = UC{E'} +jS-E'dv. (e)
B
1 Here Vp = [Ft*, VE, VS].
2 Benthien and Gurtin [1970, 1].

232 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 67.
Similarly,
/ QtPdv— / QvPdv= / Qu'*dv-\-2 f QU-u'dv. (f)
B B B B
If we apply (18.1) to S and «', we conclude, with the aid of (b) and (d), that
fS-E'dv=fs-u'da-fu'-divSdv. (g)
B S% B
Further, since p€s#L is a solution of the mixed problem, if we take the Laplace
transform of the third and fifth relations in (62.3), we find that
ijf div S(*. V) + /(*. V) = Q (*)«(*' f) (h)
for every (ae, ^)6JBx[?;0, oo) and
s = s on ^gX[%,<»). (i)
By (L) and (e)-(i),
0„(p) - <?,(p) = jFUc{E'(.,r,)} + \ [q(x) u'(x, V)*dvm (% 5gV < oo);
B
this implies the desired result, since C is positive semi-definite and q positive. □
Note that, since q > 0, we can strengthen the inequality in (1) as follows:
$n{p) < $r,{p} whenever «(•,??)#=«(•,??).
We now assume that (A) on p. 229 holds, and, in addition, that Vb possesses
a Laplace transform. Let &L denote the set of all dynamically admissible stress
fields S such that S and VS possess Laplace transforms. Then for Se@L the
function tt-*Qt{S} of (65.6) has a Laplace transform £1^{S}(r)0<^r)<oo) for some
%>0. In fact,
_ /• r rjj„ 5/™ „\t2 _ _
dv„
_ I 2omr)"
B
B
and we are led to the
(2) Minimum principle for the stress field. Let Se3iL be the stress field
corresponding to a solution of the mixed problem. Then for each S£@L there exists an
rjo^O such that
The proof of this theorem is strictly analogous to that of (1) and can safely
be omitted.
IV. Homogeneous and isotropic bodies.
67. Complete solutions of the field equations. In this section we study certain
general solutions of the displacement equation of motion1
c\ Fdiv u — c\ curl curl u-\— = ii,
xtt. p. 213.

Sect. 67.
Complete solutions of the field equations.
233
assuming throughout that c|>c|>0 and that bjg admits the Helmholtz
decomposition1
— = — Vx— curly, divy = 0,
with x and y of class CM on B X (0, t0). A solution u of class Cz on B X (0, t0) will
be referred to as an elastic motion corresponding to 6.
(1) Green-Lame~ solution.2 Let
u — Vq> + curl \p, (LJ
severe 99 «m<2 y> are class C3 /ieWs on Bx (0, <0) ^«< satisfy3
Di <?> = :*, n2v=y- (L2)
TTtm it is an elastic motion corresponding to b.
Proof. By (L,)
c? Pdiv m — c| curl curl u — ii-\
e
= V{c\A <p—'(p — x} + curl{— c\ curl curl \p — y> — y}.
In view of (1) and (3) of (4.1),
curl curl curl ip = — curl A y>,
and the desired result follows from (L2). □
(2) Completeness of the Green-Lame~ solution.* Let u be an elastic
motion of class_C3 corresponding to b, and assume that u(>, r) and u[', r) are
continuous on B at some time t€(0, t0). Then there exist class Cz fields <p and ip
on Bx(0,t0) that satisfy (1^), (L2). Moreover,
divip = 0.
1 As is clear from the proof of (6.7), such an expansion is ensured if b/g is continuous on
B x (0, t0) and of class C2.° on B x (0, *0).
2 Lame [1852, 2], p. 149. The two-dimensional version was obtained earlier by Green
[1839,11- The solution u = V(p, Di <p = x appears in the work of Poisson [1829, 2], p. 404.
Related solutions are given by Bondarenko [i960, 4}, Kilchevskii and Levchuk [1967, Iff],
Chadwick and Trowbridge [1967, 5] show that
u = V<f — curl curl (p ip) — curl (pa>)
Di<p=o, n2v=o. n2«=o,
defines an elastic motion corresponding to b = 0. Further, they prove that this solution is
complete for regions bounded by concentric spheres.
8 The wave operators Da are defined on p. 213-
4 Clebsch [1863,2] was the first to assert that the Green-Lam6 solution is complete. Clebsch
also gave a proof which Sternberg [I960, 13] showed is open to serious objection. (Actually,
Clebsch's proof is valid when the boundary of the region consists of a single closed surface.)
Another inconclusive proof was given by Kelvin [1904, 5] in 1884. The first general
completeness proof was furnished by Somigliana [1892, 7]. Completeness proofs were also supplied
byTEDONE [1897, 1~\, Duhem [1898, 5], and Sternberg and Gurtin [1962, IS]. An explicit
version of Duhem's proof was given by Sternberg [i960, 13]. The work of Somigliana,
Tedone, and Duhem appears to be little known; in fact, Sneddon and Berry [1958, 18],
p. 109 state that the Green-Lame solution is incomplete if the region occupied by the medium
has a boundary. Further, as Sternberg [i960, 13] has remarked, the completeness proof
given by Pearson [1959, H] is open to objections.

234 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 67.
Proof. * If we integrate the equation of motion twice, we arrive at
u(t) = m(t) + m(t) (t - t) + V // [cfdiv tt(A) - k(A)] rfArfs
TT (a)
+ curl // [- c| curl m (A) - y (A)] rfA rfs,
T T
where, for convenience, we have omitted all mention of the variable x. If we
apply the Helmholtz resolution (6.7) to u(r) and u(t), we find that
u (t) + u (t) (* — t) = Fa (*) + curl /? (*), (b)
where a and /J are class C2 fields that satisfy
a = 0, /j = 0, div/J=0. (c)
Thus if we define functions <p and yj on £ x (0, £0) by
<p(t) = oL(t) + //[cf divw(A) -«(A)]rfAis, (d)
T T
v (*) = 0 (<) - // [4 curlw (A) + y (A)] df A rfs, (e)
T T
then
m= F95 + curly, (f)
and, since div y =0, (c), (d), and (e) imply
divip = 0. (g)
Moreover, by (c), (d), and (e),
<p = cfdivw — %, ip= — c|curlM — y, (h)
and (f), (g), and (1)-(3) of (4.1) yield
divu=A<p, curltt=— Atp. (i)
Finally, (h) and (i) imply the desired result (L2). □
It follows from (d), (e), and (i) that <p and tp are sufficiently smooth that
D2curl ip=curl D2V>
and (2) has the following corollary:
(3) Poisson's decomposition theorem.2 Let u satisfy the hypotheses of (2)
with body force field 6 = 0. Then u admits the decomposition
where u^ and u% are class C2 fields on Bx (0, t0) that satisfy
□iMi = 0, curltt^O,
□ 2^2 = 0, div «2=0.
1 Somigliana [-1892, 7]. See also Bishop [-1953, 4].
2 Poisson [1829, 3]. Although Poisson's results are based on a molecular model which
yields jx = X, his proof in no way depends on this constraint.

Sect. 67- Complete solutions of the field equations. 235
(4) Cauchy-Kovalevshi-Somigliana solution.1 Let
*=nig + ($-($rtivg, (Sx)
where g is a class C4 vector field on Bx (0, t0) that satisfies
n*ulg = -h-. (S2)
Then u is an elastic motion corresponding to b.
Proof. By (¾
n2M + (ci-c|)FdivM
= D2 Di 9 + (4 - cf) D2 Fdivg + (cf - ${{^ Fdiv g + (c| - cj) /I Pdiv g}.
In view of (S2) and the identity
D,/-Di/=(ci-c!)/l/,
the right-hand side of the above relation reduces to — bjq; thus we conclude
from (d) on p. 213 that u is an elastic motion corresponding to 6. □
If g is independent of time, then (SO, (S2) reduce to
u = clAg + (cl-cl) Vdivg,
which is, to within a multiplicative constant, the Boussinesq-Somigliana-Galerkin
solution (44.3).
If in the time-dependent Cauchy-Kovalevski-Somigliana solution we define2
V=Diff, 95 = 2c!divg— pxp,
then (S^, (S2) reduce to
M = V+ ~JyV{<p+ p-xp), \2tV = — —, ni9> = -p-DiV,
■»ci e
which may be regarded as a dynamic generalization of the Boussinesq-Papko-
vitch-Neuber solution (44.1). Since the potentials <p and \p obey a coupled
differential equation, this solution is of no practical interest in elastodynamics.3
We now establish the completeness of the Cauchy-Kovalevski-Somigliana
solution. For convenience, we shall assume that the body forces are zero.
(5) Completeness of the Cauchy-Kovalevski-Somigliana solution.*
Let u be a class C6 elastic motion corresponding to zero body forces, and suppose that
u(•, t) and u{-, t) are continuous on B at some t€ (0, t0). Then there exists a classC*
vector field g on Bx(0, t0) such that (S^, (S2) hold.
1 Cauchy [1840, 1], pp. 208-209; Kovalevsky [1885, 2], p. 269; Somigliana [1892, 7].
This solution was arrived at independently by Iacovache [1949, 4]; it was extended to
include body forces by Sternberg and Eubanks [1957, 14\ See also Bondarenko [1957, 2],
Predeleanu [1958, 13], Teodorescu [I960, 14].
2 Sternberg and Eubanks [1957, 14]. Cf. the remarks on p. 142.
3 Sternberg [i960, 13].
4 Somigliana [1892, 7]. This theorem was arrived at independently by Sternberg and
Eubanks [1957, 14] using an entirely different method of proof and without assuming that
b =0. A third completeness proof is contained in the work of Sternberg and Gurtin
[1962, 13].

236 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 67.
Proof. By hypothesis and (2),
cfFdivw — el curl curl it =u, (a)
w= V<p-\-c\xrly>, (b)
Di9» = 0, Di!V = 0- (c)
Further, it follows from the proof of (2) that q> and \p are of class C5.
Substituting (b) into (a),
u = c\A curl yj — Fdiv (4 curl ip — cf Vq>), (d)
where we have used the fact that div curl = 0. By (c)
c\A curl \p — curl ip = c\A curl \p — Di curl v,
and thus
cjM curl y = -j-^-j Di curl v • (e)
cl—c2
Eqs. (d), (e), and the first of (c) imply that
u = "2 *—» di(ci curl V — ci V<P) — Fdiv(c|curl vp— c\ Fq>),
c1 — c2
and thus
u{t) =v{x) +v(r)(t-r) + DiMO ~ (Ci-CD Fdiv*(0,
where
T T
*>=«+ „2—72" [cl curl v- ci v<p\ ■
cl c2
In the above equations we have omitted all mention of the variable x.
Next, since the Boussinesq-Somigliana-Galerkin solution (44.3) is complete,
we can expand v(r) — v(-, r) and v(r) = «(•, t) as follows
v(t) = clAg-L + (4 - c\) FdivfiT!,
v (t) = c\A g2 + (4 — c\) Fdiv g2,
where gx and g2 are of class C4 on B. Thus if we define
g(x, t)=h(x, t)+g1(x) + {t — x) g2(x),
then g is of class C4,
M = Di3 + (cl-c2)Fdivsr,
and (SJ holds. Finally, (Sj) and (a) imply (S2). Q
In attempting to judge the comparative merits of the Cauchy-Kovalevski-
Somigliana solution (SJ, (S2) and the Green-Lame solution (1^), (L2), the
following considerations would appear to be pertinent.1 The former remains complete
in the equilibrium case if the generating potential g [x, t) is taken to be a function
of position alone. In contrast, the Green-Lame solution has no equilibrium
counterpart in the foregoing sense. This circumstance reflects the relative economy
of the Green-Lam^ solution, which is simpler in structure than the available
complete solutions to the equations of equilibrium. Further, (SJ contains space
1 These observations are due to Sternberg [I960, 13].

Sect. 67- Complete solutions of the field equations. 237
derivatives of g up to the second order while only first-order derivatives of q>
and y> are seen to enter (L2). Finally, (1^), (L2) are conveniently transformed
into general orthogonal curvilinear coordinates, whereas (S^, (S2) give rise to
exceedingly cumbersome forms when referred to curvilinear coordinates, with the
exception of cyhndrical coordinates. For all of these reasons, the Green-Lame
solution deserves preference over the Cauchy-Kovalevski-Somigliana solution
in applications.
In the absence of body forces the Cauchy-Kovalevski-Somigliana stress function
satisfies the repeated wave equation
□2DiSr = 0.
The next result yields an important decomposition for solutions of this equation.
(6) Boggio's theorem.1 Let g be a vector field of class C4 on Ex (0, t0), and
suppose that
□2DiSr = 0.
Then
9 = 9i + 92,
where gt and </2 are class C2 fields on Bx (0, t0) that satisfy
Di»i = 0, n2»2 = o.
Proof.2 It suffices to establish the existence of a classC2 field ^onfix(0, t0)
with the following properties:
□ iSTi = 0, n2(sr-»i)=0. (a)
Indeed, once such a g1 has been found, we simply define
92 = 9-9i
Since
Ua9i=clAg1-g1 (<x = l,2),
it follows that if [Jj g1 = 0, then
c2 — c2
□2»i= V*'
Thus (a) is satisfied if
□i3i = 0, »!=/, (b)
where
/=-^-erD.»-
Notice that/is of class C2 on Ex (0, t0) and obeys
□i/=0. (c)
Thus to complete the proof it suffices to exhibit a field gt satisfying (b),
subject to (c).
Let t€ (0, t0) and consider the function g1 defined by
g-i.{x,t) = ///(*, X)dXds-\-h{ac) +tq(ac)
1 Boggio [1903, 1]. This theorem was established independently by Sternberg and
Eubanks [1957, 14].
2 Sternberg and Eubanks [1957, 1£\-

238 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 67.
for (x, t)eBx (0, t0), where h and q are as of yet unspecified. Then g1 meets the
second of (b). We now show that h and q can be chosen such that Q1g1=0.
Clearly,
□iffi(a>, t) = // 4Af(x, X) dlds -f{x, t) + c\Ah(x)+tc\Aq{x),
X X
and by (c)
clAf=f.
Thus
Di9i(x,t) = -/(a?, t) +T/(a?,T) -</(», t) +4Ah(x) +tc\Aq{x).
Since/is of class C2onSx(0, t0), /(•, t) and f(',r) are of class C1 on B; hence,
in view of (6.5), there exist functions h and q of class C2 on B such that
cf A h (x) =/(sb, t) — t/'(as, t) ,
c?Jqf(£c) =/(», t),
which yields nx £^=0. D
An interesting relation between the Cauchy-Kovalevski-Somigliana solution
and the Green-Lame solution was established by Sternberg.1 Assume that the
body forces vanish. Let g be the stress function in (Sj), (S2). Then by (S2) and
Boggio's theorem, there exist functions g1 and </2 sucn that
0=01 + 02, (a)
□i0i = O, D2»2 = 0. (b)
By the second of (b),
Di92:=(4-cl)Ag2; (c)
thus, using (a) and (b), we can write (SJ as
u = (4-4)[Ag2-V&iv(gi + g2)], (d)
and, since
curl curl g2 = V div g2 — A g2,
(d) can be written in the form
u = (4 — cf) [Fdiv g1 + curl curl g2].
If we define functions <p and \p by
99 = (c| - c2) div g1, xp = {c\- cf) curl g2, (e)
then (e) takes the form
u= V(p + cm\y), divip = 0, (f)
while (b) yields
□i9> = 0, D2V = 0. (g)
Eqs. (f) and (g) are identical to (L,) and (L2). Thus we have Sternberg's result:
The Cauchy-Kovalevski-Somigliana solution reduces to the Green-Lame solution
provided the field g is subjected to the transformation (a), (b), and (e). In view of
the completeness of the Cauchy-Kovalevski-Somigliana solution, this reduction
theorem furnishes an alternative proof of the completeness of the Green-Lame
solution.
1 [1960, 13].

Sect. 68. Basic singular solutions. 239
68. Basic singular solutions. In this section we will discuss the basic singular
solutions of elastodynamics, assuming throughout that the body is homogeneous
and isotropic with cx > 0, c2 > 0.
Throughout this section y is a fixed point of i,
r=x—y, r = \r\,
and Sn = %n (y) is the open ball with center at y and radius r\.
Recall1 that =Sf is the set of all smooth vector functions on (—oo, oo) that
vanish on (—oo, 0). For each le^Clet uy{l} be the vector field on (<? — {y})x
(— oo, oo) defined by
1/¾
Uy{l\ (X,t)= -— "
3r®r ^Jxi(t-Xr)dX
1/¾
Further, let E,{i) and £>,{&} be defined through the strain-displacement and
stress-strain relations, so that
S„{I}(».0 = ^>-[(r7i)-*®- -(»"/,) l-r®/,-/,®r
where
with
/X{1) (a?, 0 = -64 f XI {t-Xr) dX + i2 \l(t-r/cj --*-l{t -r/cj
1/¾
1/¾
/,{1} (as, Q=-64 [ M{t-Xr)dX+2l(t-rfa)+(i -4-J) I(*-r/e,)
1/¾
1/¾
/,{1} (a?, t) = -6c\jXl{t-Xr) dX+Wt-rlca)-?g Ip-r/c,)
1/¾
+ -"-^-r/Cl).
We write
^{1} = [«,(!}, E,(I},S,(I}]
and call #,{1} the Stokes process2 corresponding to a concentrated load I at y.
Clearly, the mapping Ii->^V(I} is linear on „SP.
1 See p. 26.
2 This solution is due to Stokes [1851, 1]. It was deduced by Love [1904, 3], [1927, 3],
§ 212 through a limit process based on a sequence of time-dependent body force fields that
approaches a concentrated load (cf. (51.1)). The form of Stokes' solution presented here is
taken from Wheeler and Sternberg [1968, 16]. A generalization of Stokes' solution, valid
for inhomogeneous media, is contained in the work of Mikhlin [1947, 5]. See also Babich
[1961, 1], Gutzwiller [1962, 8], Murtazaev [1962, 12], Nowak [1969, 5].

240 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 68.
(1) Let la&and define S*{1} on {&—{y}) X(—oo, oo) by
SJW (x, t) = -4 ~ \^\:^-r®U-h®r-(r-f3) l],
where f{ =/,-{!} (*, <) is defined above. Then
S„{k}*l=S*{l}T*k.
Proof. It follows from the definition of f{ and (10.2) that
hence
ft{l}* k = l*ft{k}.
Thus, in view of the definition of Sy{k),
4nr* S„{k}*l = (r •£{&}) • (r • I) -~ -(r -/2{fe}) *l -(/,{*}•!) r-(r • I)*/3{fc}
-®-) • (A{*}# I)]r-(l#/i{*}) *--(/3{fc}**) r-(/,{fc}# I)r
r-f r) • (*#*{!})]»•-(/.«# *) *--(fe*/aW)»— (fe#/3«) r
= 47rr3S*{t}r*fe. D
Let
on 8Bx( — oo, oo), where S*{Z} is the tensor field defined in (1). We call s*{l}
the adjoint traction field on SB corresponding to j&tf{l}.
(2) Properties of the Stokes process.1 Let leJ£. Then the Stokes process
fiu{l} has the following properties:
(i) If I is of class C2, then fiy{l} is an elastic process on <0—{y} with a quiescent
past, and #y{l} corresponds to zero body forces.
(ii) uy{l} (x, •) =0(r~x) and Sy{l} (x, •) =0(r~2) as r->0 and also as r-+oo,
uniformly on every interval of the form (— oo, t].
(iii) lim / sy{l}da=l, lim / rxsu{l}da=0,
where
on 327,, x(— oo, oo) with n the inward unit normal. Moreover, the above limits are
attained uniformly on every interval of the form (— oo, /].
(iv) For every &€-§?,
l*uy{k} = k*uy{l}, l*sy{k} = k*s^{l},
where sy{k} is the traction field on 8B corresponding to Py{k}, and s* {1} is the
adjoint traction field on 8B corresponding to py{l}.
Proof. Parts (i) and (ii) follow from the assumed properties of I and the
formulae for uy{l} and Sy{l}.
1 This theorem, with the exception of the second of (iv), is due to Wheeler and
Sternberg [1968, 16]. See also Love [1927, S], who asserts (iii).

Sect. 68. Basic singular solutions. 241
Consider now part (iii). A simple computation based on the formula for
Sy{l} leads to the result
J sy{l} (x, r) da3S= j I(t — ^/¾) + 21(t — ^/c2)
ssn
+ -V.-i(T-tilca + l?-i{T-rilcJ
which implies the first of (iii), since I€.§?. The second of (iii) is an immediate
consequence of the order of magnitude estimate for SU{1} in (ii).
The first of property (iv) is easily established with the aid of (10.2). To prove
the second, note that by (1)
l*su{k} = l*(Sy{k} n) =(Sy{k}*l) n
= {Sy{l}T*k)-n = k*{Sy{l}n)=k*sy{l}. □
Let Ie^? be of class C2, and let uy{l} be defined on (^ — {y}) X(— oo, oo) by
Uy{l}(X, t)=-*x-Uy{l}(X,t).
Then
r 1/¾
*r{0(*. 0 = -4^- 3(i^r--^l-e,®r-r®e<) fxi{t-kr)dX
+ (6':^_r.l_e.(g)r-r®e<)(^-l(<-r/Cl)-jrl(<-r/C2))
+ >(*(<-^2) + f i(*-r/c,))
c2 c2
We write
£{*} = ^(9.^(0.^(0].
where -Ej,{0 and Sj,{t} are defined through the strain-displacement and stress-
strain relations, and call py{l} the doublet process (corresponding to y, I,
and et).
(3) Properties of the doublet processes.1 Let lz££ be of class C2. Then
the doublet process ply{l} has the following properties:
(i) If I is of class C3, then fiy{l} is an elastic process on $' — {y} with a
quiescent past, and Py{l} corresponds to zero body forces.
(ii) Tiy{l} (x, •) =0(r~2) and Sy{l} (x, • ) =0(r'3) as r->0 and also as r-+oo,
uniformly on every interval of the form (— 00, f\.
(iii) Urn / ~sL{l}da=Q, lim / rx.sy{l}da = — etxl,
where
s<{l} = S»{I}n
on dSv with n the inward unit normal.
(iv) If keJi? is of class C2, then
k*uy{l} = l*Uy{k}.
1 Wheeler and Sternberg [1968, 26], Theorem 3.2.
Handbuch der Physik, Bd. VI a/2. 16

242
M. E. Gurtin : The Linear Theory of Elasticity.
Sect/69.
(v) If py{l} is the Stokes process corresponding to the concentrated load I at y, then
J-Py{l} = -py{l}.
69. Love's integral identity. In this section we will establish Love's theorem;
this theorem is a direct analog of Somigliana's theorem (52.4) in elastostatics.
We continue to assume that B is homogeneous and isotropic with c1 > 0, c2 > 0.
(1) Lemma.1 Let p = [u, E, S] be an elastic process with a quiescent past,
and let py{l} = [uy{I}, Ey{l}, Sy{l}] be the Stokes process corresponding to a
concentrated load leJ? at y&B. Then
lim / s*uy{l}da=0,
n-*> 82:„ (y)
where
lim / u*sy{l}da=l*u(y, • ),
s=Sn, sy{l}=Sy{l}n
on d£v(y) x(—oo, oo) with n the inward unit normal.
Proof. Since both states have quiescent pasts, the above relations hold
trivially on (— oo, 0]. Thus hold t>0 fixed for the remainder of the argument.
Let /8 be such that £p{y)CB, choose r\6(0,/8]. and write 2rl=2tj(y). Further,
let
v(x, r) =u{x, r) —u(y, r)
for (x, r)eBx [0, oo). Then
t
\I(rj)—l*u(y,t)\^ J f sy{l} (x,t — r)-v(x,r) drdaa
as. o
+
fu(y,t-
f sy{l} (x,r) daa, — l{x)
dx
(a)
(b)
where
I[rj) = J u*sy{l}(x, t) dax.
dZ„
The second term in the right-hand member of (b) tends to zero with r\, since the
limit may be taken under the time-integral2 and because of (iii) in (68.2).
Consequently,
\I(rj)—l*u(y,t)\^4nrj2tMs(rj)Mi(rj)+o(i) as »y->0, (c)
where
M3(rj) =sup{\v(x, t)| : (x, r)e8i7,X[0, t]},
M^r,) =sup {\sy{l} (x, t)| : (x, T)e027„x [0, (]},
for rj£(0,/8]. From (a) and the continuity of u on Ex[0, oo) follows
Af8(77)=0(1) as 77->0.
On the other hand, (ii) of (68.2) yields
Mi{v)=0(v~2) as i?->0.
The inequality (c), when combined with the above estimates, implies the second
result in (1). A strictly analogous argument yields the first result in (1). □
1 Wheeler and Sternberg [1968, Iff], Lemma 3.1.
2 See Mikusinski [1959, 10], p. 143-

Sect. 70.
The acoustic tensor.
243
(2) Love's integral identity.1 Let p = [u, E, S] be an elastic process on B
corresponding to the external force system \b, s], let p have a quiescent past, and
suppose that s(x, -)ejSf for every xedB, and that b(x, >)6JS?'for every xeB. Then
U(H>')= I luy{s} — «»{«}] da+ J uy{b} dv
dB B
for yeB, where, for any leJ?, uy{l} is the displacement field and s£{l} the adjoint
traction field corresponding to the Stokes process Py{l}.
Proof.2 By hypothesis, u is an admissible motion and
u(.,O)=u(.,O)=0 (a)
on B. Thus u(x, •)€ J? for every xeB. Therefore, since s(x, ')e3? for every
xedB, the above expression holds trivially on (— oo, 0].
We now establish its validity on (0, oo). Choose a>0 with 2a(y)CB, and set
B,=B-r,(») (b)
for rj€ (0, a]. Let le 3? be of class C2. By (a) and (i) of (68.2), p and py{l) both
correspond to null initial data, while py{l} corresponds to vanishing body forces.
Thus we are entitled to apply (d) of Graffi's reciprocal theorem (61.1) to p and
Py{l} on Bv with the result that
f s*uy{l}da-{- f b*uy{l} dv= f sy{l}*u da (c)
8B„ B„ 8B„
on (0, oo). If we let r\ ->0 in (c) and use (b) and (1), we arrive at the result
/ s*uy{l} da-\- J b*uy{l} dv= f sy{l}*uda-{-l*u(y, •). (d)
dB B dB
By (iv) of (68.2) and the properties of the convolution (10.1), (d) can be written
in the form
f / iuy{s}~ sy{u}] da-{- f uy{b} dv —u{y, -)1 * I = 0 (e)
IdB B J
on [0, oo). Since Ze jSf is an arbitrary class C2 function, (e) and (v) of (10.1) yield
the desired result. □
V. Wave propagation.
70. The acoustic tensor. As we will see in the next few sections, the notion of
an acoustic tensor is central to the study of wave propagation. In this section
we define this tensor and establish some of its properties.
Fix xeB and let C = Cx be the elasticity tensor and q — q (x) > 0 the density
at x. Let m be a unit vector. Since C is a linear transformation,
q'1 C [a (g) m] m
1 Love [1904, 3]. A similar result was deduced previously by Volterra [1894, 4], but
Volterra's result is confined to two-dimensional elastostatics. See also Somigliana [1906, 6],
De Hoop [1958, 4], Wheeler and Sternberg [1968, 16]. None of the above authors
utilizes the adjoint traction field, which considerably simplifies Love's identity. Related
integral identities are contained in the work of Arzhanyh [1950, 2], [1951, 2, 3, 4], [1954, 1, 2],
[1955, 2, 3], Knopoff [1956, 6], Doyle [1966, 8].
A similar integral identity for the stress field, involving linear combinations of doublet
processes, was established by Wheeler and Sternberg [1968, 16], Theorem 3.4.
2 The proof we give here is due to Wheeler and Sternberg [1968, 16], pp. 71, 72.
Love [1904, 3] based his proof on Betti's reciprocal theorem (30.1), treating the inertia
term as a body force.
16*

244 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 70.
is a linear, vector-valued function of the vector a and hence is represented by the
operation of a tensor A(m) acting on a:
A (m) a = q _1 C [a ® m] m for every vector a.
We call A(m) the acoustic tensor for the direction m. In components
Aik(m)=Q-lCiiklmiml.
(1) Properties of the acoustic tensor.
(i) A(m) is symmetric for every m if and only if C is symmetric.
(ii) A(m) is positive definite for every m if and only if C is strongly elliptic.
(iii) A(m) is positive definite for every m whenever C is positive definite.
(iv) Given any Q in the symmetry group ^,,
QA(m)QT =A(Qm)
for every m.
Proof. Let C=g_1C. Clearly,
b-A(m) a = b- C[a®m] m = (b®m) • C[a®m]. (a)
Properties (i) and (ii) are direct consequences of (a) and (ii) of (20.1). Further,
since C positive definite implies C strongly elliptic, (ii) implies (iii). Finally, since
(Qa)®(Qm)=Q(a®m)QT,
it follows from (f) on p. 7\ that
QTA(Qm) Qa = QTC[Qfi®Qrn\ Qm = QTC[Q(a®m) QT] Qm
= £\a®m~\ m=A(m) a,
which implies (iv). □ /
(2) Properties of the ajcoustic tensor when the material is isotropic.1
Assume that the material at £ is isotropic with Lame moduli /u and X. Then
A(mj) =c\ m®m + cl(l— m®m),
where cx and c2 are the wave speeds:
Further, c\ and c\ are the principal values of A(m); the line spanned by m is the
characteristic space for c\, the plane perpendicular to m the characteristic space for c\.
Proof. By (i) of (20.1) and (22.2),
C[a®m] =C[sym a®rri] =fi(a®m+m®a) -f-A(a-m) 1;
thus
C[a®m] m=fia-\-(X+fi) (a-m) m = [fjL \ + {X+p) m®m] a
~[f*(l— m®m) + (A + 2fi) m®m] a,
which yields the above formula for A(m). The remainder of the proof follows
from the spectral theorem (3.2) . □
1 The form 6f the acoustic tensor for various anisotropic materials was given by Sakadi
[1941,2] and Fedorov [1965,5], §19. Fbdorov [1963,5] has established a method of
computing the isotropic acoustic tensor that is closest (in a precise sense) to a given anisotropic
acoustic tensor.

Sect. 71. Progressive waves. 245
When we wish to make explicit the dependence of the acoustic tensor A(m)
on xzB, we write A(x, m), i.e.
A(x,m) a = g'1 (x) Cx[a®m\m.
In the following proposition the region B is completely arbitrary.
(3) Let C and q'1 be bounded on B with C symmetric and strongly elliptic.
For each xeB and unit vector m, let A (x, m) be the largest characteristic value of
the acoustic tensor A(x, m). Then the number %>0 defined by
c\[ =sup {A(x, m): xeB, \m\ = 1}
is finite, and
\G-Cx[G]\^eW<?M\G\*
for every xeE whenever G is of the form G=a®v. We call cM the maximum
speed of propagation corresponding to C and q.
Proof. By hypothesis and (i) and (ii) of (1), A(x, m) is always symmetric
and positive definite. Thus A (a?, m) >0 and
v-A(x, m) »5gA(as, m)|»|2 (a)
for every vector v. Thus if the supremuni of A (x, m) is finite, cM will be well
defined. Let e(x, m) with \e(x, m)\ = 1 be a characteristic vector corresponding
to A (a?, m). Then
A (x, m)—e (x, m) • A(x, m) e (x, m), (b)
and since
v-A(x, m) v=q~1(x) (v®m) ■ Cx[v®m]
^e-i(x)\Cx\ ^011^=^(05) |<y|e|« (C)
for every vector v, (b) implies that
A (a?, m) ^-1(32)10,1. (d)
Since q'1 and C are bounded on B, e_1|C| is bounded, and therefore cM is finite.
The final inequality in (3) follows from (a), (c), and the definition of cM. □
-^- 71. Progressive waves. In this section we study plane progressive wave solutionsx
to the displacement equation of motion. These simple solutions are important
in that they yield valuable information concerning the propagation characteristics
of elastic materials.
We assume that the body is homogeneous and the density strictly positive.
By a progressive wave we mean a function u on $ x (— oo, oo) of the form
u(x, t) =aq>(p-m — ct), (a)2
where:
(i) (p is a real-valued function of class C2 on (—oo, oo) with
(ii) a and m are unit vectors called, respectively, the direction of motion
and the direction of propagation;
1 For studies of steady waves, of which plane progressive waves are a special case, see
SAenz [1953, 11] and Stroh [1962, 14].
2 Recall that p (x) = x — 0.

246 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 71.
(iii) c is a scalar called the velocity of propagation.
We say that the wave is longitudinal if a and m are linearly dependent;
transverse if a and m are perpendicular; elastic if u satisfies the displacement
equation of motion
AivC[Vu]=QU (b)
on (?x(— oo, oo). Thus an elastic progressive wave is an elastic motion of the
form (a) that corresponds to zero body forces.
Let a denote any given constant, and let nt denote the plane defined by
nt={x: (a? —0) -m— ct=a}.
Then at any given time t the displacement field u is constant on nt. This plane
is perpendicular to m and, as a function of t, is moving with velocity c in the
direction m. For this reason u is sometimes referred to as a plane progressive
wave.
By (a)
Vu = q>' a®m, (c)
where
as 's=p-m-ct
r v ' as2 \s=p-m-ct
Thus
divu = <p' am,
curl u = q>' m xa,
and, since cp' ^0 (because <p" ^0), it follows that a progressive wave is:
longitudinal <=> curl w = 0,
transverse <=> div u = 0.
By(c)
C[Fw] =9>'C[a(g)»i],
and, as C is independent of x,
div C[Vu] =(p"C[a®m] m=Q<p"A(m)a, (d)
where A(m) is the acoustic tensor discussed in the previous section:
A(m) a = Q~1C[a®m] m.
Next,
Qti = Qc!i<p" a, (e)
and (d) and (e) yield the
(1) Propagation condition for progressive waves.1 A necessary and
sufficient condition that the progressive wave u defined in (a) be elastic is that the
Fresnel-Hadamard propagation condition
A(m) a=c2a
hold.
1 According to Truesdell and Toupin [1960, 17], § 300, the ideas behind this theorem
can be traced back to the work of Fresnel. See also Cauchy [1840, 1], pp. 42-50; Green
[1841, iy, Musgrave [1954, 15]; Miller and Musgrave [1956, 8]; Synge [1956, 13],
[1957, lay, Farnell [1961, 7]; Musgrave [1961, 15].

Sect. 71. Progressive wave. 247
Thus for a progressive wave to propagate in a direction m its amplitude must
be a principal vector of the acoustic tensor A(m) and the square of the velocity
of propagation must be the associated characteristic value. If the elasticity tensor
C is symmetric, then, by (i) of (70.1), A(m) is symmetric, and hence has at least
three orthogonal principal directions 0^,02,0^ and three associated principal
values c\, c\, c\ for every m. Further, if C is strongly elliptic, then A(m) will be
positive definite and %, c2, and c3 will be real. Thus, if C is symmetric and strongly
elliptic, there exist, for every direction m, three orthogonal directions of motion and
three associated velocities of propagation clt c2, c3 for progressive waves.
Theorem (1), when combined with (70.2), implies the
(2) Propagation condition for isotropic bodies.1 For an isotropic material
the progressive wave u defined in (a) will be elastic if and only if either
(i) c2 = c\ = — — and the wave is longitudinal; or
(ii) c2=c%=fiJQ and the wave is transverse.
Thus for an isotropic material there are but two types of progressive waves:
longitudinal and transverse. One can ask whether or not longitudinal and/or
transverse waves exist for anisotropic materials. This question is answered by the
(3) Fedorov-Stippes theorem.2 There exist longitudinal and transverse
elastic progressive waves provided the elasticity tensor is symmetric and strongly
elliptic.
Proof. By (1) there exists a longitudinal progressive wave in the direction m
provided
A(m)m = km (a)
for some scalar A>0. Since C is strongly elliptic, A(m) is positive definite;
thus, given any m,
m- A(m)m>0, A(m)m^0. (b)
Consider the function I defined on the surface of the unit sphere by
,. . A{m) m , .
I m) = -. .—'. -r . (c)
v ' \A(m) m\ v '
i By the first of (b), I is continuous, maps the surface of the unit sphere into itself, \
\ and maps no point into its antipode. By a fixed point theorem,3 any such map I
has a fixed point. Thus there exists a unit vector m^ such that l{m1) =m1, or
equivalently, by (c),
A(m1)m1 = A»w1, k = \A{m^m1\.
Thus there exists a longitudinal progressive wave in the direction m^. By the
spectral theorem (3.2), there exist unit vectors ma and m3 such that m^, m2, ms
are orthonormal and m2 and m3 are principal directions for A{m^. Since A(m^
1 Cauchy [1840, 1], pp. 137-142. See also Lam£ [1852, 2], § 59; Weyrauch [1884, 2],
§§ 98-99; Butty [1946, 1], §§ 60-62; Pailloux [1956, 9], p. 35-
2 This theorem was arrived at independently by Fedorov [1964, 7] and Stif-pes [1965, 18].
See also Truesdell [1966, 27], [1968, 13] and Fedorov [1965, S], § 18. Kolodner [1966, 14]
has extended the Fedorov-Stippes theorem by showing that there exist three distinct directions
along which longitudinal waves propagate. Sadaki [1941, 2] has calculated the actual
directions along which longitudinal and transverse waves propagate for various crystal classes.
See alsoBRUGGER [1965, 3]. Waterman [1959, 18] has established results for waves that are
nearly longitudinal or nearly transverse.
3 See, e.g., Bourgin [1963, 2], Theorem 8.7 on p. 132.

248
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 72.
is positive definite, the corresponding characteristic values c\ and c\ are positive
and hence are the squares of positive numbers. Thus there exist progressive
waves in the direction m^ whose directions of motion are equal to m2 and m^.
But m2 and wig are orthogonal to m^. Thus these progressive waves are
transverse. □
(4)1 Assume that the elasticity tensor is symmetric and strongly elliptic. Ldje
be an axis^oj_ syjfjmetiyJPLAlMAterial.2 Then there exist longitudinal and transverse
elastic progressive waves whose direction of propagation is e.
Proof. Clearly, we have only to show that e is a characteristic vector for
A(e). Since e is an axis of symmetry,
Qe=e (a)
for some ^=(= 1 in the symmetry group. Thus, by (iv) of (70.1),
QA(e)=A(e)Q,
and we conclude from the commutation theorem (3.3) that Q leaves invariant
each characteristic space of A(e). By (a) Q must be a proper rotation about e
through an angle •&. If ■& 4= n, then the only spaces left invariant by Q are the
one-dimensional space 91 spanned by e, the plane perpendicular to 91, and the
entire space "V. If # = n, then Q leaves invariant, in addition, each
one-dimensional space spanned by a vector perpendicular to e. In either event, it is clear that
at least one characteristic space of A(e) must contain e. □
It follows from the above proof that if #4=¾ elastic transverse waves can
propagate in the direction of e with any amplitude perpendicular to e, and all'
of these waves propagate with the same velocity.
72. Propagating surfaces. Surfaces of discontinuity. In this section we define
the notion of a propagating singular surface. Roughly speaking, this is a surface,
moving with time, across which some kinematic quantity suffers a jump
discontinuity.
Recall that
(fW =gx(-00,00), y<4> =y x (-00, 00).
By a smoothly propagating surface we mean a smooth_tln"ee-dimensional
manifold "W in Sw with the following property: given any (x, f)£iT, there exists
a normal" (ft, a^gY^4' to W at (x, t) with ft 4=0, « + 0. This assumption implies
the existence of a four-vector field
m = (m, — c)
on iT that is normal to "W and has
|m|=1, 00.
We call m the direction of propagation and c the speed of propagation.
For the remainder of this section let #"be a smoothly propagating surface
in Bx(0,t0).
Since iT is a smooth manifold, and since c is strictly positive, W has the
following property: given any point (x0, t^zW there exists a neighborhood £ of x0, a
neighborhood T of l0, and a smooth scalar field y> on Z such that
t=ip{x) for all (x,t)€Nr^W, N=ZxT.
1 Fedorov [1965, 5], pp. 92-94.
2 See p. 73.

Sect. 72. Propagating surfaces. Surfaces of discontinuity. 249
In terms of y>:
m(X, t) = -r~ ,T, clX, t) = 7^-r-rr-
V ' |Fy(a:)| I Py (*") I
for all (x,t)€N<^ir.
For t fixed,
is the smooth surface in B occupied by if at time t;1 it is not difficult to show
that the vector m (x, t) is normal to £ft at x.
Let r={y(t):ttT} be a smooth curve on W of the form y(t) = (y(t), t). We
say that J" is a ray if y{>) is a solution of the differential equation2
Thus i/ (•) is the path traversed by a particle on if whose trajectory at any time t
is perpendicular to Sft and whose speed is c. Assume, without loss in generality,
that the ray F passes through the neighborhood N in which the manifold if is
described by y>. Let a denote arc length along J":
t
<r= f\y(r)\dr.
o
Since \y\ never vanishes, a is an invertible function of t; we write t for the inverse
function, i.e.
t^t{a),
and we define
y{a)=y{t{a)),
c(o)^c(y(o)J(a)),
Vela) = |7a,c (x,w(x))\ =17(777^1-)
m(a)=m(y(a),'t(a)).
= m(<r) = C» VV{y(a)), (^.^=1^(^(a))|«,
Since
da
it follows that
Thus we are led to the following differential equation for the ray:
c* \daj\da ) c2
c da2
It follows from this equation that if the velocity c is constant on if, then every ray
is a straight line. """ ~
Let7 be a function on Bx (0, t0). We call if a singular surface of order
zero with respect to / if / is continuous on B x (0, t0) —if and / suffers a jump
disccaitinuity-^cross^. The jump in / across if is then the function [/] on if
1 In the literature it is customary to define a smoothly propagating surface to be a (smooth)
one-parameter family SPt, 0<t<tB, of smooth surfaces S?t in B. In this instance
•)T = {(aB, t):x£Srt, o<t<t„}.
2 Since y • m = (cm, 1) • (m, — c) = 0, this definition is consistent in the sense that the
tangent to P is also tangent to if.

250
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 72.
defined by1
[f] (x, t) = Hm {/(x,t+h)-f(x,t-h)}.
Thus the jump in / is the difference between the values of / just behind and
immediately in front of the wave. Since / is continuous on each side of if, we
can evaluate [/] by taking limits along the normal m:
W(§) = lim{/(§-Am)-/(§+Am)}
h-±0+
iox\={x,t)elf.
Let n S2 1 be a fixed integer. We call if a singular surface of order n
with respect to / if:2
(i) / is class C"-1 on B x (0, t0) and class Cn on Bx (0, t0)-if;
(ii) the derivatives of / of order n suffer jump discontinuities across if.
The next theorem shows that for a singular surface of order 1, the jumps
in the space and time derivatives of the associated function are related.
^.(1) Maxwell's compatibility theorem.3 Let cp be a scalar field, v a vector
field, and T a tensor field on Bx(0, t0), and suppose that if is a singular surface
of order \ with respect to cp, v, and T. Then
c[Vcp\ = — [y]m, (1)
c[Vv] = — [v]®m, (2)
c[divv]= — [v]m, (3)
c [curl v] = [v] x m, (4)
c[divT] = -[T]m. (5)
Proof. Choose a point \^if and a (non-zero) four-vector a such that
a -m(£)=0.
Since if has dimension 3, a is tangent to if at \. Further, since if is smooth,
there exists a smooth curve Q = {z (a): 0¾ ^ a ^ a2} on if through 2; whose tangent
at 2; is a; i.e. for some 0,,6(0¾. <x2)
*(oo)=?. *'K)=°> (a)
where
, dz
da
For ^¢^ and any function f on Bx(0, t0), let
/t(?) = lim/(?-Aiii).
ft—M)+
/,(0 = Iim/(?+*«•),
so that
_ W(S) =/i(S)-/,(?). (b)
1 This notation was introduced by Christoffel [1877, 1], § 6.
2 The notion of a singular surface of order n |ao is due to Duhem [1900, 1] and Hada-
mard [1901, 5], § 1, [1903, 3], § 75- See also Truesdell and Toupin [196O, 17], § 187.
3 Maxwell [1873, 1], § 78, [1881, 1], § 78a. Cf. also Weingarten [1901, 7], Hadamard
[1903, 3], § 73, Truesdell and Toupin [i960, 17], § 175, Hill [1961, 14].

Sect. 72. Propagating surfaces. Surfaces of discontinuity. 251
Then, by the smoothness properties of <p,
-/.- 9¾(*(«)) = -£ l™ 9» (z(«) -*•»)
= 11^1^(=0-^)
= lim ^(4) ?>(z(a) -Am) • *'(«) •
Thus (a) implies1
*->o+
da.
<Pl (* («)) !«=«„ = « • (P<4> <P)l (5) • (C)
A similar result holds also for cp^; if we subtract this result from (c) and use (b),
we arrive at the relation
-/.-[?>] (*(«))!«-«.=«• Ww¥\ (5)- (d)
On the other hand, since cp is continuous across #7
hence (d) implies
for every J-e^and every o tangent to if at $•. Thus [F^gp] (?) must be parallel
to m(g):
Wm<p] (9 = A(9 m(§); (e)
equivalently,
[[79?]= Am, [95] = —Ac,
which implies (1).
If we apply (1) to the scalar field v • a, where a is an arbitrarily chosen vector,
we find that
c [V (v ■ a)] = — [v ■ a] m.
Thus
c[VvT] a = —(m® [»]) a,
which yields (2).
If we take the trace of (2), we arrive at (3); if we take the skew part, we arrive
at (4).
Finally, (5) follows from (3) with v =TTa, where a is an arbitrary vector. □
We now sketch an alternative proof of the relation (1) in (1). By a four-
dimensional counterpart of Stokes' theorem,
/ skw {([7(4)cp) ®ri}da = Q
for every closed T^-regular2 hypersurface £f in Bx(0,t0). The same argument
used to derive (c) in the proof of (73.1) here yields
skw {[[7(4) cp] ® m} = 0.
Operating with this equation on m, we are led to the relation (e) in the proof
of (1).
The next theorem shows that for a vector field v with a discontinuity of order 1,
the jumgs^in Vv and v are completely determined by the jumps in the divergence
and curl of_».__
1 Hadamard [1903, 3], § 72. Cf. Lichtenstein [1929, 2], Chap. 1, § 9, and Truesdell
and Toupin [i960, 17], § 174.
2 This term is defined on p. 252.

252
M.E. Gurtin: The Linear Theory of Elasticity.
Sect. 72.
(2) Weingarten's theorem.1 Let ~W be a singular surface of order 1 with
respect to a vector field v on By.(0, t0), and let
a = [v\.
Then
a-m = o<=> [div»] =0, (1)
axm = 0<=> [curl»] =0, (2)
a = — cm [div v] + cm x [curl v]. (3)
Proof. The results (l) and (2) follow from (3) and (4) of (1). To establish (3)
we note that the identity
a = (a • m) m + (a<S>m) m — (m(g)a) m
in conjunction with (2) and (3) of (1) imply
a = —c[diw] m — c[Vv] m-\-c[VvT\ m.
This relation yields (3), since
[Vv — VvT] m = [curl v] xm. □
We will occasionally deal with integrals of the form
f Mn da,
where ,^ is a closed hypersurface with outward unit normal field n. If ~W is a
singular surface of order zero with respect to M, then M will suffer a jump
discontinuity across £fr\HT. To insure that the above integral exists and that the
value of M in the integrand at a point of discontinuity equals its limit from the
region interior to Zf, we give the following definitions.
Let ^bea closed regular hypersurface in J3x(0, t0), i.e. ,^ is the boundary
of a regular region D#> in B x (0, t0). A point £e £f is #"-regular if either %$ir or
"%<L-W and there exists an open ball Tin S(4) with center at £ such that {Wr^Z) C Sf
(see Fig. 16). We say that ,^ is if '-regular if the set of points of £f that are not
"^"-regular has zero area measure.
x
Fig. 16. %x and %% are l^-regular; i-3 is not nF-regular.
1 [1901, 7]. Cf. also Truesdell and Toupin [1960,17], § 175.

Sect. 73. Shock waves. Acceleration waves. Mild discontinuities. 253
\
I
Let iTbe a singular surface of order zero with respect to a four-tensor field M
on B x (0, *0). Then the limit
M(S = MmM(?)
exists for every regular "%^Sf. Further, the integral
J Mn da
ST
exists whenever if is a closed 1^-regular hypersurface in B x (0, t0). When such
is the case we define
r / Mnda= J Mn da.
73. Shock waves. Acceleration waves. Mild discontinuities. Throughout this
section -W is a smoothly propagating surface in B x (0, t0) with direction of
propagation m and speed of propagation c. Further, we assume that the elasticity
field C and the density field q are continuous on J5, that q is strictly positive, and
that the fo>^ /orce /»rirf vanishes.
Let «J2:1 be a fixed integer. By a wave of order n we mean an ordered
array [u, S] with properties (i)—(iii) below.
(i) u is a vector field and S a symmetric tensor field on B X (0, t0).
(ii) ~W is a singular surface of order n with respect to u and order « — 1 with
respect to S.
(iii) u and S satisfy balance of linear momentum in the following form:
given any closed ^-regular hypersurface £f in J5x (0, t0),
f Mn da=0,
where M is the stress^momentum field1 corresponding.t.QjSJ and u.
Requirement (iii) is motivated by (19.6). For n S: 2, (iii) may be replaced
by the assumption that
divS = e« on J5x(0, t0)— iT. (a)
For n = \ the discussion given at the end of the last section insures the
existence of the integral in (iii); in addition, the value of M in the integrand is equal
to its limit from the region interior to £f. When n—\, (iii) implies (a), but the
converse is not true. However, in this instance (iii) is equivalent to the following
requirement:
(iii') for every one-parameter family Pt (^ < t < ts) of parts of B,
j (Sn) (x, t) dax = -~ J (qu) (x, t) dvx
BP, Pt
whenever the integral on the left and the derivative on the right exist. Here the
stress Son 8Pt is to be interpreted as its limit from the interior of Pt.
Given a wave of order n, we call
r(»).
a = [u]
1 See p. 67.

254
M.E. Gurtin: The Linear Theory of Elasticity.
Sect. 73-
the amplitude, and we assume, for convenience, that a never vanishes. A wave
of"order"'{' is""sometimes referred to as a stress wave or shock wave; a wave
of order 2 is often called an acceleration waveTYof a'sEocK wave the amplitude
equals the jump in the velocity, while for an acceleration wave a is equal to the
jump in the acceleration. We call a wave longitudinal or transverse according
as a and m are parallel or perpendicular r For "a.wave"of order n, iT is a singular
surface of order 1 with respect to u ; thus Weingarten's theorem (72.2) implies
that a is completely determined by the jumps in the divergence and curl of u s :
a = — cm [div u ] + cm x [curl u ].
In addition, we conclude from (72.2) that the wave is:
(k-1)
longitudinals [curl u ]=0,
,.. (»-i),
transverse <=> [div tt]=0.
For a wave of order n the time-derivative of S of order w — 1 is allowed to
jump across the wave. The next result gives a relation between this jump and
the amplitude a.
(1) Balance of momentum at the wave.1 For a wave of order n Sg 1,
(»-D
[ S ] m= —Qca.
Proof. Let n = 1. Let £ be an open ball in B x (0, t0) with center on iT. For
£ sufficiently small, ~W divides £ into two regular regions £+ and £' with
"^-regular boundaries. Let £' be that portion of £ into which m points (see Fig. 17).
Fig. 17.
1 Although we have assumed that the body force field b = 0, the above result remains
valid if we assume instead that b is of class Cn—1 on B X (0, t0). For a shock wave (n = 1) the
relation in (2) has a long history. Hydrodynamical special cases for situations in which
S= —pi were obtained by Stokes [1848, 2], Eq. (3), Riemann [i860, 1], § 5, Rankine
[1870,2], §§3-5, Christoffel [1877, 1], §1, Hugoniot [1887, i], §140, and Jouguet
[1901, 6]; for general S but linearized acceleration, by Christoffel [1877, 2], § 4; for finite
elastic deformation by Zemplen [1905, 3], p. 448, [1905, 4], § 3, whose derivation is based
on Hamilton's principle. The general result (for n — 1) is due to Kotchine [1926, 2], Eq. (18).
Cf. also Truesdell and Toupin [1960, 17], § 205, Jeffrey [1964, 11].

Sect. 73. Shock waves. Acceleration waves. Mild discontinuities. 255
By (iii) ^
J Mnda= J Mnda = f Mnda—Q,
as 82+ 82-
where M is the stress momentum field; thus
fMnda-\- f Mnda— f Mnda=0. (a)
82+ 82- 8i'
The boundary of £ can be decomposed into a portion contained in d£* and a
portion contained in d£~; the remainder of d£+ and the remainder of d£~ are
equal, as sets, to £r\W; as surfaces,/Jfowever, they have opposite orientation.
Thus (a) reduces to
/ {lim M(5+) m(g) - lim M(%~) m©} da% = 0, (b)
Xr\nr K+-*% \~~-+%
where £* is a point of £*. In deriving (b) we have used the fact that the value of
M in the integral over d£* is equal to its limit from the interior of £*. Since
(b) must hold for every sufficiently small open ball £ with center on ~W, it follows
that
[M]m=0, (c)
or equivalently, using the definitions of M and m,1
[S] m = —Qc[u\ = —qca,
which is the desired relation for n = 1.2
Consider now a wave of order n2g2. Choose (cc, <) in Bx(0, /0) —#". Then
there exists an open ball £ in B centered at x and an open time interval TC (to>h)
containing £ such that #^ (27 xT) =0. Thus S is a time-dependent admissible
stress field and u is an admissible motion on £xT; (iii) and (19.6) therefore
imply
divS=@fi on £xT.
Since (x, t) in B x (0, i!0) — "#^ was arbitrarily chosen, the above relation holds on
B x (0, t0) — W. If we differentiate n — 2 times with respect to i! and take the
jump in the resulting equation, we arrive at
(«-a> («)
[div S ]=Q[u]=ga. (d)
(»-2)
Clearly, ~W is a singular surface of order 1 with respect to S ; thus (5) of (72.1)
yields
(»-2) (»-l)
c[div S ] = -[ S ]m. (e)
Eqs. (d) and (e) imply the desired result for n S2 2. □
We say that a wave [«, S] of order n 2g 1 is elastic if
S=C|Vu].
We now show that for an elastic wave of any order traveling in the direction
m^ the square of the speed.must.be a characteristic.„v.alue of the acoustic, tensor
A(ni)j..and the amplitude must lie in the corresponding characteristic space.
1 See pp. 67, 248.
2 The proof up to this point is due to Keller [1964, 12],

256
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 73.
(2) Propagation theorem for waves of order n.1 Elastic waves of all
orders n ^ 1 obey the Fresnel-Hadamard propagation condition
A{m) a = c2a.
Proof. Let [u, S] be an elastic wave of order n ;> 1. Since if is a singular
surface of order 1 with respect to v = u , we can apply (2) of (72.1) with the
result: (B_1) (M)
c[V u ] = — [m]®»h = — a®m. (a)
Next, (a) and the stress-strain relation imply
C["s ]=C[c[|7<"tt1,]]=-C[o®m], (b)
and since
A(m) a = g-1C[a(g)m] m,
(b) and (1) imply the desired result. □
More pedantically, the Fresnel-Hadamard condition reads
A(x, m) a (x, t) = c2 (x, t) a (x, t)
for every (x, t)^W.
Combining (2) and (70.2), we arrive at the
(3) Propagation theorem for isotropic bodies.2 If the body is isotropic
then a wave of order » 2:1 is either longitudinal, in which case
rl — A— 2t* + *
c -c1—e-,
or transverse, in which case
Comparing (2) and (3) to (71.1) and (71.2), we see that the laws of propagation,,,,
of waves of discontinuity are the same as those of progressive waves; thus the
discussion following (71.1) and (71.2) also holds in_the present circumstances.
| In particular, we note flialir^is^ymmetric" aii'ar'strongry~elliptic, there existTTor
I every direction m, three associated velocities of propagation for singular surf aces.
Note that the maximum speed of propagation cM defined in (70.3) is the
supremum over all possible velocities of propagation of elastic waves of any
order.
For a homogeneous and isotropic body the velocity of propagation c is
identically constant. Therefore (3) and the remark made on p. 249 yield the following
result.
1 This theorem is essentially due to Christoffel [1877. 2], who gave a proof for n = \.
Hadamard [1903, 3], §§ 260, 267-268, extended Christoffel's result to acceleration waves
(n = 2). That the Fresnel-Hadamard condition should hold for all n is clear from the results
of Ericksen[1953, Iff], §6, and Truesdeix[i 961,22], §3, in finite elasticity theory (cf. Trues-
dell and Noll [1965, 22], § 72). See also Hugoniot [1886, 2]; Duhem [1904, 2], Part 4,
Chap. 1, §5; Love [1927,3], §206-209; Finzi [1942, 1]; Pastori [1949, 7]; Petrashen
[1958,12]; Buchwald [1959.-7]; Truesdell andToupin [1960,17], § 301; Nariboli [1966,18].
2 Hugoniot [1886, 2]. Elastic waves of order zero (for which if is a singular surface of
order zero with respect to the displacement field) in an isotropic but inhomogeneous body are
discussed by Gvozdev [1959, 5] using the theory of weak solutions. Gvozdev shows that such
waves travel with a velocity equal to cx or c2. Other studies concerned with wave propagation
in isotropic elastic media are: Levin and Rytov [1956, 7], Thomas [1957, 17], Babich and
Alekseev [1958, 1], Skuridin and Gvozdev [1958, 16], Karal and Keller [1959,9].
Skuridin [1959, 14].

Sect. 74. Domain of influence. Uniqueness for infinite regions. 25 7
(4) Suppose that the body is homogeneous and isotropic. Then, for a wave^ of
any order nj=l,_thejJ?yj aZ? ^&!£!}£.M.?l$s-
74. Domain of influence. Uniqueness for infinite regions. In this section we
allow B to be a bounded or unbounded regular region. We assume that C is positive
definite and symmetric, that q is strictly positive, and that C and q'1 are continuous
and bounded on B. It then follows that the maximum speed of propagation cM
defined in (70.3) is finite.
Given an elastic process [u, E, S] and a given time te(0, t0), let Dt denote
the set of all xeB such that:
(i) if xeB, then
u(x, 0)=(=0 or u(x, 0)4=0 or b(x, t)=(=0 for some t€ [0, t];
(ii) if xedB, then
s (x, t) • ii (x, t) =)= 0 for some re [0, t].
Here [6, s] is the external force system associated with the process. Roughly
speaking, D( is the support of the initial and boundary data. Consider next the
set Dt of all points of B that can be reached by signals propagating from Dt with
speeds equal to or less than the maximum speed of propagation cM:
Dt={x^B:Dtr,Smt(x)^9).
We call Dt the domain of influence1 of the data at time t. The next theorem,
which is the main result of this section, shows that on [o, t] the data has no
effect on points outside of Dt.
(1) Domain of influence theorem.2 Let [u, E, S] be an elastic process, and
let Df be the domain of influence of its data at time t. Then
M=0, E=S=0 on (B— Dt)x[0, t].
The proof of this theorem is based on the following lemma. In the statement
and proof of this lemma e (x, G) is the strain energy density at x corresponding
to the (not necessarily symmetric) tensor G:
s{x,G)=\G-Cx[G}. (a)
(2) Wheeler-Sternberg lemma? Let [u, E, S] be an elastic state
corresponding to the external force system [6, s]. Further, letx:B-> [0, t0) be a continuous
1 Since Dt is based on the maximum speed of propagation, it is, actually, an upper bound
on the domain of influence. Cf. Duff [1960, 5].
2 This is a miuor modification of a theorem due to Wheeler and Sternberg [1968, Iff],
Lemma 2.2 and Wheeler [1970, 3], Theorem 1. These authors assume vanishing initial data
and do not utilize the notion of a domain of influence. The general idea is due to Zaremba
[1915, i], who established an analogous result for the wave equation. Zaremba's scheme was
rediscovered independently by Rubinowicz [1920, 2] and Friedrichs and Lewy [1928, 1].
See also Courant and Hilbert [1962, 5], pp. 659-661, Bers, John, and Schechter
[1964, 3], Duff [1960, 5].
3 Wheeler and Sternberg [1968, Iff], Lemma 2.1 for homogeneous and isotropic media;
Wheeler [1970, 3], Lemma 1, for anisotropic but homogeneous media. Both of the above
results assume vanishing initial data.
Handbuch der Physik, Bd. VI a/2. 17

B
— e
258 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 74.
and piecewise smooth field on B with {xeB: r(x) >0} finite. Then
t(sb) T(ac)
/ I s(x, t) • u{x, t) dt da + \ \ b(x,f) -u(x, t) dt dv
8B 0 BO
f {-%- [m2(*.t(*)) -w2(»,0)] -e{x,6[x)) +e(x, Vu(x,r(x)) + G(x)) (b)
(a>, Vu{x, 0))\dv
where
G{x)=ii(x, x{x)) <g> Vr(x). (c)
Proof of (2). For any function / on J3x[0, t0), let fT be the function on E
defined by
Mx)=f(x,T(x)). (d)
Further, let
r(aO
fe=Sti, «(«) = / k(x,t)dt. (e)
o
Then » is continuous on 5 and piecewise smooth on B, and we conclude, with the
aid of the equation of motion, that
t(sb)
div»(*)=fcT(*)-|7T(*)+ f\s- Vu + -e2-(u-u)—b-u (x,t)dt. (f)
0
Since
k=S-Vu,
(f) takes the form
div v (x) = kz (x) ■ Vr (x) + e (x, Vur (x))—e (x, Vu (x, 0))
t(sb)
+ e^- [uz(x) - wa(#, 0)] - f 6 •«*(«, *) rf*.
0
A simple calculation using the symmetry of C yields
kz(x) ■ Vr(x) = Vr(x) ■ (Sz(x) uz(x)) = Sz(x) -G(x)=G(x) ■ Cx[Vur(x)]
= s(x, Vuz(x)+G(x))-e(x, Vuz(x))—e(x, G(x));
thus
div t> (a?) =-^
«?(*)-u*(x, 0) -e(a>, G(*))+c(a!, |7Mr(a;)+G(cc))
T[X) (g)
-£ (¾. Ftt(a!, 0))-/6 •!*(*, <) <^.
Each of the terms on the right hand side of (g) is a piecewise continuous function
of x on B. Moreover, it is clear from (e) and the assumed properties of r that v
has bounded support. Thus div v is integrable on B, and the divergence theorem
together with (g) and the second of (e) imply (b). □
Proof of (1). Let (*, A) e (B - Dt) x (0, t) be fixed, let
Q = B^£XeM(z), (h)

Sect. 74.
Domain of influence. Uniqueness faMjnfinite regions.
vjjnfinit
259
Fig. 18.
and let r: B -> [0, t) be defined by
, > (A 1- \x—z\, xeQ,
t(*)={ cm ' ''
lo, »(f.Q.
(i)
Then t is continuous and piecewise smooth on B, and Vr has a weak point
discontinuity at x = z and a jump discontinuity across Br\dQ. Moreover,
IIZrHA- on£,
(i)
|7t = 0 on £-£.
Since A < t, we conclude from (h) and the definition of the domain of influence
Dt that
hence
By (i) and (k),
b=0 on Qx[0,t],
s-ti=0 on (Qr\dB)x[0,f],
u(-, 0) =m(., 0) =0 on Q.
(k)
(1)
r(x)
(m)
// b(x,rj) •u(x,if) drj dv=0,
b o
r(a>)
// s(£c,^) •«(», ?;) dtjda=0.
dB 0
Next, (i), (j), and (c) imply
u(x, r(x)) — u(x, 0), Fm(cc, t(cc)) = Vu(x, 0), C(cc)=0, #¢£. (n)
By (a), (n), and (1),
e(x, G{x))=0, XQ.Q,
it2 (x, t (x)) — m2 (as, 0) =
e [x, Vu(x, t(*)) + G(x)) — e (ae, Vu(x, 0)) =
(°)
u2(x, r(x)), xcQ,
0, cc$.Q,
e [x, Vu (x, x {x)) + G(x)), xeQ,
0,
x$Q.
17*

260 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 74.
Clearly, t satisfies the hypotheses of the Wheeler-Sternberg lemma (2). Therefore
we may conclude from (b), (m), and (o) that
f M&Hjiifa r(x)) -e(x, G(x)) +e(x, Vu{x, r(x))+G(x)) dv=0. (p)
a
Since
\G{x)\2 = \Vr{x)\2u2(x, r{x))
for xe(B— {«}), it follows from (a), (j) and the inequality in (70.3) that
e{x,G(x))^-e-{*±u*{x,T(x));
thus
^ it* (x, r (as)) - £ (x, G(x)) S: 0 (q)
for xe(B — {«}). Since C is positive definite, the function e is non-negative, and
(q) implies that the integrand in (p) is non-negative on Q — {«}. Thus, since this
integrand is also continuous on Q — {«}, we conclude from (p) that it must vanish
on this set. This fact, in conjunction with (q), implies
e (x, Vu(x, x (*)) + G{x)) = o, xe (Q — {s}).
Thus, since C is positive definite, (a) yields
E(x,r(x))+symG(x)=0, xe(Q-{s}). (r)
By (i)
In view of (i) and the continuity of E and u,
E(x,r(x))->E(s,X), u(x, t(x))->u(s, X) as x-^-z. (t)
If we let x—z-\-5e, where e is a fixed unit vector, and take the limit in (r) as
<5-»-0, we conclude, with the aid of (c), (s), and (t), that
E(z, A) + — sym {u(*. A) <g>e} = 0. (u)
CM
Since (u) must hold for every unit vector e, (u) must hold with e replaced by
— e; thus
E(», A)=0.
As (s, A) was arbitrarily chosen in (B—D() x(0, £), and since E is continuous,
£==0 on (E-Dt)x[0,f]. (v)
By the definition of D(,
6=0 on (5— D()x[0, (], tt(.,0)=ti(',O)=0 on 5— Dt; (w)
(v) and the first of (w) together with the stress-strain relation and the equation
of motion imply
S = 0, « = 0 on (B—D,)x[0,f]. (x)
The second of (x) and (w) yield
m = 0 on (B-Dt)x[0,t),
and the proof is complete. □

Sect. 75-
Basic equations.
261
We now use the domain of influence theorem to establish a uniqueness theorem
for elastodynamks thaHs valid for infinite domains. In "comparing: this theorem
to theUniqueness theorems of Neumann (63.1) and Brun (63.2) it must be
remembered that among the assumptions made at the beginning of this section
are the assumptions that Cbe_gositiye. definite and symmetricand .that q be strictly
positive.
(3) Uniqueness theorem.1 Under the assumptions made at the beginning
of this section the mixed problem of elastodynamics has at most one solution.
Proof. Let p = [u, E, S] be the difference between two solutions. Then p
corresponds to vanishing body forces and satisfies
tt(-,O)=M(-,0)=0 on B, s-m=0 on Bx(O,t0).
Choose t arbitrarily in (0, t0). Clearly, Dt is empty, and thus the domain of influence
Dt of the data at time t is empty. Therefore we conclude from (1) that p vanishes
on J5x [0, t\. Since t was arbitrarily chosen, the proof is complete. □
1 Itj^iould^be noted that Wheeler and Sternberg's uniqueness theorem (3)
I is valid for exterior domains and also for regions..whose.boundaries extend to
iriflnityT An elastodyhamic uniqueness theorem valid for infinite regions may
aTtefnatively be based on the classical energy identity following Neumann's
procedure (63.1), provided one introduces suitable restrictions on the orders of
magnitude of the velocity and stress fields at infinity. The advantage of (3) is
that it does nqt.i.nxQlv£.~sucli. artificial a priori assumptions. The analogous
uniqueness issue in elastostatics, where the equations are elliptic rather than
hyperbolic, is considerably more involved. For exterior domains elastostatic
uniqueness, theorems that avoid extraneous order prescriptions at infinity are
given in Sect. 50.
VI. The free vibration problem.
75. Basic equations. The free vibration problem2 is concerned with an elastic
body undergoing motions of the form
u(x, t)—u (x) sin (<ot+y),
E(x, t) =E(x) sin (cot +y), (a)
S (x, t)=S (x) sin (cot+y),
where a> is the frequency of vibration and y is the phase. If we assume null body
forces, then these fields will satisfy the fundamental field equations of
elastodynamics if and only if the amplitudes u(x), E(x), and S(x) obey the relations
E = $(Vu + VuT),
S=C[E],
divS+e<u2w=0.
Since
C[E]=C[Vu],
1 Wheeler and Sternberg [1968, 16] for the case in which B is homogeneous and
isotropic; Wheeler [197O, 3] for the case in which B is homogeneous, but anisotropic.
2 The vibration problem for infinite domains is treated in detail by Kupradze [1963,
17, 18].

262 M. E. Gurtin : The Linear Theory of Elasticity. Sect. 76.
the above relations yield the following equation for the displacement amplitude:
divC[Vu]+Q(oiu=0, (b)
which, for a homogeneous and isotropic body, takes the form
/j,A u + {X+[j) Vdivu->rQco2u — 0.
Also, a simple calculation, similar to that used to deduce (e) of Sect. 59, yields
the relations
c\ A divw+ft)2divw = 0,
c\ A curl u +co2 curl u = 0,
where c^ and c2 are the irrotational and isochoric velocities. Of course, these
formulae could also have been derived by substituting the relation for u (x, t)
in (a) into (e) on p. 213.
Returning to the general inhomogeneous and anisotropic body, we see that
the procedure given on p. 214. here yields the following relation for the stress
amplitude:
p(^_)+ft,aK[S]=0,
where K = C_1 is the compliance tensor.
76. Characteristic solutions. Minimum principles. We assume that the elasticity
field has the following properties:
(i) C is smooth on B;
(ii) C is positive definite and symmetric.
Let
3i = the set of all admissible displacement fields,1
and let £? be the operator on 3i defined by
SCu=divC[Vu].
For convenience, we introduce the following notation:
<m, v} = f u-v dv,
B
|u|=V<u,u>,
<u,v}c= f Vu- C[Vv] dv.
B
Note that, by (ii) of (20.1),
(u,v}c = fVu-C[Vv]dv.
B
The free vibration problem consists in finding combinations of the frequency to
and amplitude u that are possible when a portion £^ of the boundary is clamped
and the remainder ^ is free. This should motivate the following definition:
By a characteristic solution (for the free vibration problem) we mean an
ordered pair [A, u] such that A is a scalar, we^, and
.S?w+Aw = 0, ||w|| = l, w = 0 on Selt s=0 on ^2;
1 See p. 59.

Sect. 76. Characteristic solutions. Minimum principles. 263
here s is the traction field corresponding to u, i.e.
s = C[Vu] n on dB.
We call A a characteristic value, u an associated characteristic
displacement. Given A, the corresponding frequency of vibration is given by
» = ]/}:
as we will soon see, A is always non-negative. Finally, we say that the entire
boundary is clamped if Sfx = 8B (5"2 = 0), /ree if ,¾ = 8B (5^=0).
fl_) Lemma. Assume that:
(i) we^ re»i% Jzfw continuous on B;
(ii) « is continuous on B and piecewise smooth on B, while Vv is piecewise
continuous on B.
Then
<i?M, «> = — <m, w>c + / s • v da,
dB
where s is the traction field corresponding to u.
Proof. Let S=C[Fu], so that s=Sn on dB. Then
/ (Jzfw) -vdv=J (div S) • w rf v = / s-v da— f S- Vv dv
B B dB B
= / s • v da — (u, v}c- □
(2) Properties of characteristic values.1 Let [A, u] 6e a characteristic
solution. Then
A = <M,M>c,
so that
A^O.
Jm /ad, A = 0 if and only if u is rigid. Moreover, a characteristic solution [A, u]
mtt A=0 exists only if the entire boundary is free, in which case every rigid
displacement u with |tt|=l is a characteristic vector corresponding to A=0.
Proof. Since [A, u] is a characteristic solution,
J5?w + Aw = 0, H| = l, (a)
and
s-u=0 on dB, (b)
where s is the corresponding traction field. By (a)
<JSPu, m> = —A<m, m> = —A;
therefore we conclude from (b) and (1) that
A = <M, M>c.
Moreover, since C is positive definite,
<«, m>c = 0,
1 Clebsch [1862, T\, § 20, generalizing a result of Poisson [1829, 2] for the vibrations
of an elastic sphere. See also Ibbetson [1887, 2], § 256, CesA.ro [1894, 1], p. 58.
Azhmudinov and Mamatordiev [1967, 1] show that when the boundary is clamped A
is given by a certain integral over dB.

264 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 76.
with equality holding if and only if the associated strain field vanishes, that is, if
and only if u is rigid. Further, since u vanishes on 6^, a rigid u is possible only
if «9^=0. (Since 6^ is a regular surface, if non-empty it must contain at
least three non-collinear points.) On the other hand, if £fx — 0, it is a simple matter
to verify that [0, tt] is a characteristic solution whenever u is rigid and |tt| = 1. □
(3) Orthogonality of characteristic displacements.1 Let [A, u] and [A, u]
be characteristic solutions with A 4= A. Then
<u, w> = <M, m>c = <J£u, u> = 0.
Proof. By hypothesis
J5?w-|-Aw = 0( (a)
JSfu + iu = 0, (b)
s -14=8-^=0 on 3B, (c)
where s and s are the traction fields of u and u. By (a) and (b)
<.£?«,«> +A<w, m> = 0,
(d)
<j2?w, m> + A<w, w>=0.
On the other hand, (c), (1), and the symmetry of C imply that
<jg?H, M> = - <M, M>c = - <U, M>c = <.£?W, M>. (e)
By (d) and (e)
(A — l)<u, u)—0;
hence <u, ii>=0 if A 4= A. The remaining results follow from (d) and (e). □
Theorem (3) asserts that characteristic displacements corresponding to
distinct characteristic values are orthogonal. This is not necessarily true if the
characteristic values are equal. Indeed, if u^ ,1½,..., u„ are characteristic
displacements corresponding to the same characteristic value A, then it is a simple
matter to verify that any field u with flw J = 1 lying in the span of {%, u2,..., un}
is also a characteristic displacement corresponding to A.
Henceforth c€ denotes the set of all vector fields v with the following
properties^ ""'"
(i) v is continuous on B and piecewise smooth on B;
(ii) Vv is piecewise continuous on B;
(iii) H = l;
(iv) v==0 on £^.
(4) Minimum principle for the lowest characteristic value.2 Let
Uj^e^r,^" and assume that
(1¾. Wj>c5S <w, v}c for every veJf.
Define
A1 = <w1,e«1>c.
Then [^, wj is a characteristic solution, and ^ is the lowest characteristic value.
1 See footnote \ on p. 263. See also Poisson [1826, T\.
2 PoincarS [1890, 2], p. 211, [1892, 6], p. 103- See also Tedone [1907, 2], § l5d; Cou-
rant [1920, iy, Courant and Hilbert [1953, 8], p. 339; Hu [1958,11].

Sect. 76. Characteristic solutions. Minimum principles. 265
Proof. Let oc4=±l be an arbitrary real number and choose vejf. Then
1¾ + xv 14=0, and
||wi + at>| v 1 '
belongs to Jf; thus, by hypothesis,
1«! +at>
^ = <«!, M!>C ^ <M>, W>C = TuTXXiT» <"l +aW- "l +*®>C
and hence
If we expand both bilinear forms and use the relations ||t71| = [1^^11 =1 and
\ = <Mj, Mi>c» we arrive at
2a[A1<M1, v} — <«!, v}c] +a2[^ — <v, v}c] ^0.
Since this inequality must hold for every a 4= ± 1> we must have
A^m^i?) —<«!,w>c=0. (a)
As 17 = 0 on S?x, (1) implies that
<«i» «>c = — <-^«i» »> + / si" ® <*«.
where sx is the traction field corresponding to 1¾. In view of the last two relations,
/ (if Mj + h Uj) ■ v d v — / sx ■ v d a = 0
for <?wry continuous and piecewise smooth vector field v with v = 0 on Sfx and
|i?|| = l. Since this expression is linear in v, we can drop the requirement
that H| = l. Therefore it follows from (7.1) and (35.1) that
J?!*! + Ax 1¾ = 0 on B,
S! = 0 on ^2.
Thus, since u^X, [^, 1¾] is a characteristic solution.
If [A, u] is another characteristic solution, then (2) implies that
A = <W, W>c.
Thus we conclude from our present hypotheses that
and hence ^ is the lowest characteristic value. □
Theorem (4) asserts that the minimum (if it exists) of the functional
<». «>c (a)
under the supplementary condition
\\v\\ = \ (b)
is the lowest characteristic value. We now show that the determination of the
M-th characteristic value can be reduced to the variational problem of finding
a minimum of the functional (a) under the supplementary condition (b) and the
constraint
<i7,w,>=0, 7 = 1,2,...,» —1,

266 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 76.
where %, w2,..., un_1 are the characteristic vectors for the first n — \
characteristic values.
Let 1¾, w2, ...,un be piecewise continuous vector fields on B. Then we write
for the set of all piecewise continuous vector fields v on B such that
<v, w,>=0, 7 = 1,2,..., ».
f5_) Minimum principle.1 Let u^, u2,... belong to 2<-^$C, and let
*&• <&■
.^=,^^(11^1¾. ■.., M,-i}X, 7^2.
Assume that, for each »- S: 1, MreJ^ a«rf
<uf, ur}c 5i <w, » >c /or eaery w G Jfr.
Define A, by
A, = <w,,w,>c.
Then each [A,, ur] is a characteristic solution and
O^A^A^ ••• .
Proof. By (4), W, u{\ is a characteristic solution. We now proceed by
induction. Thus we assume that «3:2 and that [A,, u,] is a characteristic solution
for 1^7^» —1. Since M„e^,
<mb,m,>=0, 1 = 1,2,...,» — 1. (a)
On the other hand, our induction hypothesis and (1) imply that
0 = A,-<w„, u,-> = - <w„, .§?u,->
= <w„,w,->c— / s{-unda, t ==1, 2,..., m-1.
But m„=0 on ,¾ and 8^ = 0 on «5^; therefore
<«»»«i>c = 0, t = l,2,...,» —1. (b)
Now choose veJfn and let
t» = ii , ir («„ + aw),
where a=t=±l- Then weJf„, and steps directly analogous to those used to
establish (a) in the proof of (4) now lead to the result that
A„<w*,«>-<w„,«>c = 0 (c)
for every veJJT„.
Let vcJf be arbitrarily chosen. If v lies in the span of 1¾. w2, ..., uh_1, then
(a) and (b) imply that
*„<«». »>-<H„ «>c=0. (d)
Assume that » does not lie in the span of u^u^,..., un_x. Then there exist
scalars /3, ^, t2, ..., tn_1 such that
^=/3^+¾^) (e)
1 See footnote 2 on p. 264.

Sect. 76. Characteristic solutions. Minimum principles. 267
belongs to $Tn. Indeed, let
and (since »4= — 2 ^»u») choose (} such that ||w| = l. By (c) and (e),
_ _ n-l
*„<«». «>-<«„> »>c+ 2 k[K<un, «<> -<m„, u,>c] =0. (f)
t = l
Thus we conclude from (a) and (b) that (d) remains valid in the present
circumstances. Hence (d) holds for every v€Jf, and, as in the proof of (4), this implies
that [A„, un] is a characteristic solution. □
We now show that the minimum principle (5) characterizes all of the solutions
of the free vibration problem provided A^oo.1
(6) Completeness of the minimum principle. Let \, A2,... and 1¾, w2,...
obey the hypotheses of the minimum principle (5) and assume, in addition, that
A„->- 00 as n -> 00. Let [A, u] be a characteristic solution. Then X—X, for some integer
r, and if
Ar-i < Af = A,+1 = ■ • • = A,+M < A,+M+1,
then u is contained in the span of ur, ur.n,..., ur+n. In particular, if n =0 so that
Ar is distinct, then u=:t»r
Proof. Assume that A is not equal to any of the A,'s. By (4), X>XX. Thus,
since Xn->00, there exists anm^2 such that
Xm_x<X<Xm. (a)
Since [A,, wf] is a characteristic value for every r, we conclude from (3) that
_!*€{«!, 1¾. ..., MOT_1}-L.
Thus, as the hypotheses of (S) hold,
<Wm,W»,>C^<W.W>C.
or equivalently, by (2),
which contradicts (a). Thus X=Xr for some r.
Assume now that Af_1<Ar=Af+1= ••• =Af+M<Af+M+1, and let
W = {ur,ur+1,...,ur+n}.
Since A #= A?- for / < r, it follows from (3) that
«€{1^,1*2, ...,1^)-1-. (b)
We decompose u as follows:
M=M0 + M±, (c)
where w0 belongs to the span of °ll and u±e^L. Indeed, we simply take
;'=r
To complete the proof it suffices to show that wx=0. If w0=0, then u=uL,
and it follows from (b) and the fact that u^e'Bf-i- that
ue^.tta, ...,uf+B}-L. (d)
1 See (78.1).

268 M.E. Gurtin: The Linear Theory of Elasticity. Sect. 77.
Thus, as before,
K+n+l s£/>
and we have a contradiction. Thus M0=j=0. Assume now that wx=|=0 and let
- _ w„ - _ «1
"•-KT U±-Ju±\-
Since u0 is a linear combination of characteristic displacements corresponding
to the same characteristic value A, [A, u0] is a characteristic solution; this fact
and (c) imply that [A, HjJ is also a characteristic solution. Thus, by (3), u^ is
orthogonal to every u.<f<Bf, and hence (d) holds with u replaced by uL. Thus we
are again led to a contradiction; consequently
1*1=0. □
77. The minimax principle and its consequences. We now establish an
alternative characterization of the free vibration problem. We continue to assume
that C is symmetric, positive definite, and smooth on B. For convenience, we
introduce the following notation:
W = the set of all piecewise continuous vector fields on B,
5r=fX#"x-"Xf" (»times).
Recall that JT is the set of all vector fields on B with properties (i)-(iv) listed
on p. 264.
(1) Minimax principle.1 Let [A,,w,] satisfy the hypotheses of (76.5) for
1 ^ r 5g n +1. Define <p on Wn by
<p{w1,w2, ...,wn)=mi{(v,v}c: v^.Jf^{w1,wi,.... wn}±}.
Then
^»+i = ?>K.M2. ---,^) =sup{(p(wltw2,..., wn): {wltw2, ....w^aW").
Proof. By hypothesis,
A„+1 = 9?(M1,M2, ..., Uj= <MM+i, M„+i>c- (a)
Thus it suffices to prove that
A„+1^ cp (wltw2,...,wn) for every (w1,tot, ....wJeiT". (b)
To accomplish this we will show that for every (u>1> w2,.... w„)eWnthere exists
a ve.yT^{w1, w2,.... ic,,}-1 such that
<w,w>c^*»+i; (c)
for then (b) would follow from the inequality
q>(uj1,wt, ...,w„)^,<y,v}c^An+1.
Choose (w1,w2,..., wn)eW". Let
n+l
w=2«<«f. (d)
where the oc/s are as yet undetermined. Clearly, vejf^{w1, w2,.... w^L if
»+1
2«? = 1 (e)
1 Courant [1920, i]. See also Courant and Hilbert [1953, 5], p. 406.

Sect. 77. The minimax principle and its consequences. 269
and
n+1
2 *,■<«,-. «v>=o (7=1,2,...,»). (f)
t=l
The system (f) consists of n equations in # + 1 unknowns and can always be
solved for 0¾.¾. ...,ocM+1. Moreover, since the system is homogeneous, the
requirement (e) can also be met. We have only to show that the function v so
defined satisfies (c). Since1
<ui,ujyc = Xjdij (no sum) i, / = 1, 2, ..., » + 1,
(d) implies
<».»>e="2«?^- (g)
t=l
Further, (e) and the fact that
Af^^,+1, i^n + \
imply
"ZX ^ ^Vi. (h)
t = l
and (g), (h) yield the desired inequality (c). □
Let Ai, A2,... and 1¾. w2,... obey the hypotheses of the minimum principle
(76.5). We call the ordered array
{K> K> ■••}
the spectrum corresponding to C, B, S^, and £f2, and we say that u^, u2, ...
is an associated system of normal modes. By (76.6), (78.1), and Footnote 2 on
p. 270, the spectrum contains every characteristic value, and hence it is
independent of the choice of i^, w2,... (as long as this choice is consistent with (76.5)).
We now consider a second body B composed of an elastic material with
elasticity field C. We will discuss the vibration problem for such a body taking
as boundary conditions
w = 0 on £fx and s=0 on ^2,
where ,¾ and ^2 are complementary subsets of 8B. In particular, we compare
the spectrum corresponding to this problem with the spectrum for the problem
discussed previously (assuming, of course that both spectrums exist).
We assume that C is positive definite, symmetric, and smooth on B. We write
if
E-CX[E]^E-CX[E]
for every symmetric tensor E and every xeBr\B.
(2) Comparison theorem.2 Let {A^ A2, ...} be the spectrum corresponding
to C,B,Sfx, and S^', {A^ A2,...}, the spectrum corresponding to C,B,§\, and i^.
Assume that either
1 This follows from (d) and (e) in the proof of (16.3) and the fact that
<«,-, ufy = «5,?- (i, 7 = 1, 2, ..., n+ 1).
2 Cf. Courant and Hilbert [1953, 5], pp. 407-412.

270
M. E. Gurtin: The Linear Theory of Elasticity.
Sect. 78.
(i) B = B, §{<iSfx, C^C, or
(ii) the entire boundary of B is clamped, and
BCB, C^C.
Then
K^K. 7 = 1,2,...
Proof. Let X be as before, and let X be the corresponding set for B and 6^.
Assume that (i) holds. Then, since B = B and .¾ C .¾ >
&<£. (a)
Further, since C^C,
<v, w>c^<w,w>c- (b)
Let n ^ 1 be a fixed integer, let <p be as defined in (1), and let <p be defined on
•3T»by
£>(*»!, t»2,..., u>„) =inf {<v, w>g: ve^'^{w1, w2,..., w„}^-}.
Then by (a) and (b)
£=>.
and we conclude from the minimax principle (1) that
The proof under hypothesis (ii) is strictly analogous. □
78. Completeness of the characteristic solutions. In this section we show that
the characteristic values A„ become infinite as »^- oo. This implies, in particular,
that each characteristic value has only a finite multiplicity. More important,
however, is that the unboundedness of the characteristic values yields the
completeness of the characteristic displacement fields. Thus we assume that the
spectrum {Aj, A2,...} exists; in addition, we suppose that C is positive definite,
symmetric, and smooth on B.
(1) Infinite growth of the characteristic values.1 Assume that B is
properly regular and that ^4=0-2 Then the spectrum {^, A2,...} has the following
property:
lim Xn — oo.
n~>oo
Proof. Assume that there exists a real number M such that
K?iM (a)
for all n. Then by (76.2)
<un,u„yc^M,
where u^, w2,... is an associated system of normal modes. Thus, since
<«„. «„>c = 2 U{E„},
1 Weyl [1915, 3~\. See also Courant and Hilbert [1953,8], p. 412 and Fichera [1971, 1],
Theorem 6.V.
2 Note that we do not include the important case in which the entire boundary is free.
That the theorem also holds in that instance is clear from the work of Weyl [1915, 3] and
Fichera [1971, 1], Theorem 6.V. In fact, Weyl [1915, 3] proves that (for a homogeneous
and isotropic body) X„ goes to infinity as «*. Fichera (private communication, 1971) has
remarked that Weyl's result is easily extended to include inhomogeneous and anisotropic
bodies (with symmetric and positive definite C).

Sect. 78. Completeness of the characteristic solutions. 271
where En is the strain field corresponding to un, we conclude from Korn's
inequality (13.9) that there exists a constant K such that
f\Vun\*dv£K
B
for all n. Consequently, since
Kia = 1. (b)
we conclude from Rellich's lemma (7.2) that there exists a subsequence {u„t}
such that
lim |m„,—M„t|=0. (c)
;, A~>oo
But in view of the method of construction of 1¾. u2, ... in (76.5),
<«»/. «»t>=0, »,-#»*. (d)
By (b) and (d),
11««*-««* II2=\\un,f+K*|2=2,
which contradicts (c). Thus there cannot exist a number M that satisfies (a);
hence A„-> oo as m-> oo. [J
The next theorem shows that every vector field / with property (A) below
admits an expansion in terms of normal modes.
(A) / is continuous on B, piecewise smooth on B, and vanishes on yx; while
F/ is piecewise continuous on B.
(2) Completeness of the normal modes.1 Let u^.u^,... be a system of
normal modes corresponding to the spectrum {A1( A2,...}, and assume that
lim A» = oo.
Let f satisfy property (A). Then
lim 1/- £ cMwM| = 0,
JV-^-oo M=i
wfere iAe cM are 2Ae Fourier coefficients
cn = </,«„> •
Moreover, the cn satisfy the completeness relation
Proof. Let
with
Then, since
it follows that
AT
9n=1— 2c»m». (a)
n=l
C„ = </-"»> •
<"»,.«»> =^m». (b)
SM2 = |/II2-ZC»- (c)
1 Cf. Courant and Hilbert [1953, 8], p- 424.

272
M. E. Gurtin : The Linear Theory of Elasticity.
Sect. 78.
Thus the series 2 c\ converges and satisfies Bessel's inequality
«=i
By (a) and (c), to complete the proof it suffices to prove that
\\gN |->o as N->oo. (e)
Suppose that for some M,\\gM\\=0. Then (c) would imply that cM+n = 0,
n = \, 2,.... and hence that ||srM+n||=°. n = i,2 Thus, in this instance, (e)
holds trivially.
Assume now that ||sTiv[| never vanishes and let
9n
By (a), (b), and (f),
^ = Sl' N^'2>- (f)
<Srjv-M«>=0, « = 1,2, ..., AT, (g)
and we conclude from the properties of un and / that
9n£^n+i> (n)
where $TN+l is defined in (76.5). Hence
—. — 1
%n+i ^S (9n> 9n}c — T^jf (9n> 9n}c,
or equivalent] y,
\\9n
la < <gw. 9n}c
Since ^jvii-*"00 as N->oo, to complete the proof it suffices to show that
(9n> 9n}c remains bounded in this limit.
It follows from (76.1), the fact that [Xn, un] is a characteristic solution, (g), and
(h) that
<Srjv.M»>c=0, « = 1,2,..., N.
Thus, since
<un. um}c = K <5«„ (no sum),
we conclude from (a) that
N N
<f,f>C = <9N+ Z C»Un.9N+ Z C„M„>C
»=1 »=1
N
= <9n>9n>c+ Z h%-
»=1
Consequently, as An 5:0,
<9N,9N>C^<f,f>C;
hence (,gN, gN}c remains bounded as N->oo. □
Under reasonable assumptions on B and on S^lt the set of vector fields with
property (A) is dense (with respect to the norm | • |) in the space of continuous
functions on B. Thus, under such assumptions, (2) will hold for functions/ that are
continuous on B.

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1968 1. Bross, H.: Das Fundamentalintegral der Elastizitatstheorie far kubische Medien.
Z. Angew. Math. Phys. 19, 434-446. (51)
2. Cakala, S., Z. Domanski, and H. Milicer-Gruzewska: Construction of the
fundamental solution of the system of equations of the static theory of elasticity
and the solution of the first boundary-value problem for a half space [in Polish].
Zeszyty Nauk. Politech. Warszaw. Mat. No. 14, 151-164. (51)
3. Dafermos, C. M.: Some remarks on Korn's inequality. Z. Angew. Math. Phys. 19,
913-920. (13)
4. Edelstein, W. S., and R. L. Fosdick: A note on non-uniqueness in linear elasticity
theory. Z. Angew. Math. Phys. 19, 906-912. (32)
5. Fosdick, R. L.: On the displacement boundary-value problem of static linear
elasticity theory. Z. Angew. Math. Phys. 19, 219-233- (57)
6. Gurtin, M. E., V. J. Mizel, and W. O. Williams: A note on Cauchy's stress
theorem. J. Math. Anal. Appl. 22, 398-401. (15)
7. Hayes, M., and R. J. Knops: On the displacement boundary-value problem of
linear elastodynamics. Q. Appl. Math. 26, 291-293. (63)
8. Horak, V.: Inverse Variationsprinzipien der Mechanik fester Korper. Z. Angew.
Math. Mech. 48, 143-146. (38)
9. Indenbom, V. L., and S. S. Orlov: Construction of Green's function in terms of
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translated as J. Appl. Math. Mech. 414-420. (53)
10. Karnopp, B. H.: Reissner's variational principle and elastodynamics. Acta Mech.
6, 158-164. (65)
11. Knops, R. J., and L.E.Payne: Uniqueness in classical elastodynamics. Arch.
Rational Mech. Anal. 27, 349-355- (63)
12. Solomon, L.: Elasticity Line"aire. Paris: Masson. (27, 38, 44, 47)
13. Truesdell, C.: Comment on longitudinal waves. J. Acoust. Soc. Am. 43, 170. (71)

294
M. E. Gurtin : The Linear Theory of Elasticity.
14. Turteltaub, M. J., and E. Sternberg: On concentrated loads and Green's
functions in elastostatics. Arch. Rational Mech. Anal. 29, 193-240. (51, 52, 53)
15. Washizu, K.: Variational methods in elasticity and plasticity. New York: Perga-
mon. (38)
16. Wheeler, L. T., and E. Sternberg : Some theorems in classical elastodynamics.
Arch. Rational Mech. Anal. 31, 51-90. (60, 61, 68, 69, 74)
1969 1. Berg, C. A.: The physical meaning of astatic equilibrium in Saint Venant's
principle for linear elasticity. J. Appl. Mech. 36, 392-396. (18, 29)
2. Brun, L.: Methodes energe'tiques dans les systemes eVolutifs line'aires. J. Mecanique
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3. de La Penha, G., and S. B. Childs: A note on Love's stress function. ATAA J. 7,
540. (44)
4. Kanwal, R. P.: Integral equations formulation of classical elasticity. Q. Appl.
Math. 27, 57-65. (52)
5. Nowak, M.: Equations of motion of an elastic free-free body. Bull. Acad. Polon.
Sci., Ser. Sci. Tech. 17, 31-36. (68)
6. Rubenfeld, L. A., and J. B. Keller: Bounds on elastic moduli of composite
media. SI AM J. Appl. Math. 17, 495-510. (34)
7. Schultz, M. H.: Error bounds on the Rayleigh-Ritz-Galerkin method. J. Math.
Anal. Appl. 27, 524-533. (39)
8. Stippes, M.: Completeness of the Papkovitch potentials. Q. Appl. Math. 26, 477-
483. (44)
9. Youngdahl, C. K.: On the completeness of a set of stress functions appropriate to
the solution of elasticity problems in general cylindrical coordinates. Int. J. Eng.
Sci. 7, 61-79- (44)
1970 1. Benthien, G., and M. E. Gurtin: A principle of minimum transformed energy in
elastodynamics. J. Appl. Mech. 31, 1147-1149- (66)
2. HlavACek, I., and J. NeCas: On inequalities of Korn's type. I. Boundary-value
problems for elliptic systems of partial differential equations. II. Applications to
linear elasticity. Arch. Rational Mech. Anal. 36, 305-334. (28)
3. Wheeler, L. T.: Some results in the linear dynamical theory of anisotropic elastic
solids. Q. Appl. Math. 28, 91-101. (61, 74)
4. Zienkiewicz, O. C.: The finite element method; from intuition to generality. Appl.
Mech. Rev. 23, 249-256. (39)
1971 1. Fichera, G.: Existence theorems in elasticity. Below in Volume VIa/2 of the Hand -
buch derPhysik, edited by C. Truesdell. Berlin-Heidelberg-New York: Springer.
(5, 7, 13, 31, 33, 36, 78)
2. Knops, R. J., and L. E. Payne: Uniqueness theorems in linear elasticity. In:
Springer Tracts in Natural Philosophy (ed. B. D. Coleman), Vol. 19. Berlin-Heidelberg-
New York: Springer. (32, 40, 63)
3. Maiti, M., and G. R. Makan: On an integral equation approach to displacement
problems of classical elasticity. Q. Appl. Math. To appear. (52)
4. Segenreich, S. A.: On the strain energy distribution in self equilibrated cylinders.
5. Thesis. COPPE, Federal Univ. Rio de Janeiro.
Addendum.
The following additional papers would have been referred to, had I seen them in time.
1956 Teleman, S.: The method of orthogonal projection in the theory of elasticity. Rev.
Math. Pures Appl. 1, 49-66.
1957 Roseau, M.: Sur un theoreme d'unicit(5 applicable a certains problemes de diffraction
d'ondes elastiques. C.R. Acad. Sci., Paris 245, 1780-1782.
1959 Campanato, S.: Sui problemi al contorno per sistemi de equazioni differenziali lineari
del tipo dell'elasticita. Ann. Scuola Norm. Pisa (3) 13, 223-258, 275-302.
1961 Baikuziev, K.: Some methods of solving the Cauchy problem of the mathematical
theory of elasticity [in Russian]. Izv. Akad. Nauk USSR Ser. Fiz.-Mat. 3-12.
1962 Gegelia, T. G.: Some fundamental boundary-value problems in elasticity theory in
space [in Russian]. Akad. Nauk Gruzin. SSSR Trudy Tbiliss. Mat. Inst. Razmadze
28, 53-72.
Kupradze, V. D.: Singular integral equations and boundary-value problems of
elasticity theory [in Russian]. Tbiliss. Gos. Univ. Trudy Ser. Meh.-Mat. Nauk. 84,
63-75-
Rieder, G.: Iterationsverfahren und Operatorgleichungen in der Elastizitatstheorie.
Abhandl. Braunschweig. Wiss. Ges. 14, 109-343-

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