Mark LWi I kins
Computer
Simulation
of Dynamic
Phenomena

Scientific Computation
Editorial Board
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Scientific Computation
A Computational Method in Plasma Physics
F. Bauer, O. Betancourt, P. Garabedian
Implementation of Finite Element Methods for Navier-Stokes Equations
F. Thomasset
Finite-Difference Techniques for Vectorized Fluid Dynamics Calculations
Edited by D. Book
Unsteady Viscous Flows D. P. Telionis
Computational Methods for Fluid Flow R. Peyret, T. D. Taylor
Computational Methods in Bifurcation Theory and Dissipative Structures
M.Kubicek,M.Marek
Optimal Shape Design for Elliptic Systems O. Pironneau
The Method of Differential Approximation Yu. I. Shokin
Computational Galerkin Methods C. A. J. Fletcher
Numerical Methods for Nonlinear Variational Problems
R. Glowinski
Numerical Methods in Fluid Dynamics Second Edition
M.Holt
Computer Studies of Phase Transitions and Critical Phenomena
O. G. Mouritsen
Finite Element Methods in Linear Ideal Magnetohydrodynamics
R. Gruber, J. Rappaz
Numerical Simulation of Plasmas Y. N. Dnestrovskii, D. P. Kostomarov
Computational Methods for Kinetic Models of Magnetically Confined Plasmas
J. Killeen, G. D. Kerbel, M. C. McCoy, A. A. Mirin
Spectral Methods in Fluid Dynamics Second Edition
C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang
Computational Techniques for Fluid Dynamics 1 Second Edition
Fundamental and General Techniques C. A. J. Fletcher
Computational Techniques for Fluid Dynamics 2 Second Edition
Specific Techniques for Different Flow Categories C. A. J. Fletcher
Methods for the Localization of Singularities in Numerical Solutions
of Gas Dynamics Problems E. V. Vorozhtsov, N. N. Yanenko
Classical Orthogonal Polynomials of a Discrete Variable
A. F. Nikiforov, S. K. Suslov, V. B. Uvarov
Flux Coordinates and Magnetic Field Structure:
A Guide to a Fundamental Tool of Plasma Theory
W. D. D'haeseleer, W. N. G. Hitchon, J. D. Callen, J. L. Shohet
Monte Carlo Methods in Boundary Value Problems
K.K.Sabelfeld
Computer Simulation of Dynamic Phenomena
M.L.Wilkins
The Least-Squares Finite Element Method
Theory and Applications in Computational Fluid Dynamics and Electromagnetics
Bo-nan Jiang

Mark L. Wilkins
Computer Simulation
of Dynamic Phenomena
With 130 Figures
Springer

MarkL.Wilkins
Lawrence Livermore National Laboratory
Box 808
Livermore, CA 94550
Mail Stop 017
USA
ISSN 1434-8322
ISBN 3-540-63070-8 Springer-Verlag Berlin Heidelberg New York
Library of Congress Cataloging-in-Publication Data
Wilkins, Mark L., 1922- Computer simulation of dynamic phenomena / by Mark L. Wilkins.
p. cm. - (Scientific computation) Includes bibliographical references. ISBN 3-540-63070-8
(alk. paper) 1. Hydrodynamics-Computer simulation. 2. Gas dynamics-Computer simulation.
3. Elastoplasticity-Computer simulation. I. Title. II. Series. QC151.W495 1999 532\5-ddc21
99-42088 CIP
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Preface
This text describes computer programs for simulating phenomena in hydro-
hydrodynamics, gas dynamics, and elastic plastic flow in one, two, and three dimen-
dimensions. Included in the two-dimensional program are Maxwell's equations and
thermal and radiation diffusion. The programs were developed by the author
during the years 1952-1985 at the Lawrence Livermore National Laboratory.
The largest main-frame computers available in the early 1950s were re-
required to solve hydrodynamic problems in one space dimension by using
forty mass points. Subsequently, numerical methods were developed for solv-
solving problems in two and three space dimensions, but application of these
methods had to wait until the main-frame computers were large enough to
tackle meaningful problems. At the present time, lap-top computers can use
these methods to solve problems in three space dimensions with the detail of
10 000 mass points.
The numerical procedures described in the text permit the exact con-
conservation of physical properties in the solutions of the fundamental laws of
mechanics: A) conservation of mass, B) conservation of momentum, C) con-
conservation of energy. The laws of mechanics are universal in their application.
Examples are given for the same computer simulation programs solving prob-
problems of penetration mechanics, surface waves from earthquakes, shock waves
in solids and gases, failure of materials.
Many important concepts in mathematics have two or more equivalent
definitions. A particularity of this book resides in the choice of the physical
solid rather than the mathematical point as the basic concept in the schema-
tization of the material system. The gradient of a scalar point function is
expressed in the form of a surface integral. The divergence and curl of vector
point functions are represented in a similar manner. The integral definition
of a derivative defines a control volume in a scalar or vector field. The dif-
difference in the flux in and out of the control volume is the induced change
in the quantities in the control volume. Thus, the numerical system itself
is in a conservation form with zero truncation error in the solution of the
partial differential equations. Lagrange coordinates are employed, which per-
permits the history of the behavior of a mass particle to be followed. The mass
particle with its corresponding control volume can be made arbitrarily small.
Proceeding with the mathematics in the form of a finite material system is

VI Preface
consistent with the observation that the components of a vector force can
only be measured on a surface.
The approach here, of describing the mathematics in terms of mechanics
instead of the reverse, can lead to slightly different results. For example, some
of the sacred tenets of plasticity theory, which is a mathematical theory, are
not necessary when a physics approach is applied to the problem.
The closure to the solutions of the three fundamental laws of mechanics
is the model for the material behavior. When the model or material equation
of state is known, the engineer can use the simulation program to predict the
consequences of a material structure to a given situation. The programs are
especially useful in studying structures submitted to failure conditions that
would be very expensive to investigate by experiment.
With the ability to solve the three laws of mechanics to any degree of
accuracy, the problem can be turned around for the materials scientist to test
a hypothesis. The simulation of an experiment that measures an observable,
which by itself may yield no information of single significance, can be used to
construct a model of material behavior. A mechanical equation of state can be
developed in this manner. If the theoretical model of material behavior based
on fundamental assumptions cannot produce the same result, the theory must
be revised.
In my work I have relied on the skills of many colleagues at the Lawrence
Livermore National Laboratory. D.E. Giroux assisted by T. Suyehiro devel-
developed the programming for the two-dimensional simulation program HEMP1.
The program was further developed to include multiple sliding by Alan Leibee
and Karen Warren. S.J. French programmed early versions of the three-
dimensional program HEMP 3D in the late 1960s.
Eugene Cronshagen developed the production version of HEMP 3D with
vector programming, grid generators, and 3D graphics. Robert Gulliford and
David Turner programmed the sliding interfaces and fracture models in the
HEMP 3D program. Robert Dickens programmed the magnetohydrodynamic
version of HEMP.
Material models play an essential role in the usefulness of a computer
simulation program. Here I would like to acknowledge the many contributions
made by Jack Reaugh and Michael Guinan.
Livermore,
December 1998 Mark L. Wilkins
1 HEMP is an acronym stemming from the words Hydrodymanic, Elastic, Mag-
Magneto, and Plastic.

Table of Contents
Elements of Fluid Mechanics 1
1.1 Fundamental Equations 2
1.1.1 Equation of Motion 2
1.1.2 Continuity Equation 2
1.1.3 Energy Equation 2
1.1.4 Equation of State 2
1.2 Solutions to the Fundamental Equations 3
1.3 Propagation of Discontinuities 4
1.3.1 Sound Speed 4
1.3.2 Speed of Discontinuity Propagation 5
1.3.3 Characteristics 6
1.3.4 Shock Waves 8
1.4 Derivation of the Hugoniot Relations 9
1.4.1 Conservation of Mass 9
1.4.2 Conservation of Momentum 10
1.4.3 Conservation of Energy 10
1.5 Rayleigh Line 11
1.6 Applications of Hugoniot Equations to a Perfect Gas 13
1.6.1 Calculation of Shock Speed 13
1.6.2 Calculation of Shock Pressure 14
1.6.3 Calculation of Volume Behind the Shock 14
1.6.4 Graphical Representation 15
1.6.5 Reflection of a Uniform Shock 15
1.6.6 Conditions Behind the First Reflected Shock
from a Fixed Boundary 16
1.7 Detonation Waves 17
1.8 Elastic-Plastic Waves 17
1.9 Units and Orders of Magnitude 20
1.10 Measurements to Obtain Equation of State Data 20
1.10.1 Experimental Methods 20
1.10.2 Relation of the Free Surface Velocity
to the Shock Particle Velocity in a Solid 22
1.10.3 Form of the Equation of State for Solids 23
1.10.4 Detonation Pressure Measurement 25

VIII Table of Contents
2. Numerical Techniques 27
2.1 Von Neumann Finite Difference Scheme 27
2.1.1 Time Centering 28
2.1.2 Space Centering 28
2.2 Artificial Viscosity 28
2.2.1 Generalized Artificial Viscosity 28
2.2.2 Applications of the Generalized Artificial Viscosity
in One Space Dimension 29
2.3 Stability Conditions 32
2.3.1 Courant Condition 32
2.3.2 Von Neumann Stability Analysis 32
2.4 Finite Difference Scheme in Two Dimensions 33
2.4.1 Integral Definition of a Derivative 33
2.4.2 Integration Paths 34
2.4.3 Properties of the Integration Scheme 34
2.4.4 Continuity Equation 34
2.5 Finite Difference Scheme in Three Dimensions 35
2.6 Finite Difference Scheme for Double Operators
in Two Dimensions 35
2.7 Grid Stabilization 36
3. Modeling the Behavior of Materials 37
3.1 Introduction 37
3.1.1 Hooke's Law 37
3.1.2 Rigid Body Rotation 39
3.2 Plastic Flow Region 39
3.2.1 Yield Strength 41
3.2.2 Von Mises Yield Condition 43
3.2.3 Plastic Strain 45
3.2.4 Tresca Yield Condition 46
3.3 Flow Stress 48
3.3.1 Strain Hardening 50
3.3.2 A General Form of Strain Hardening 50
3.4 Rate Dependent Yield Models 52
3.4.1 Maxwell Solid 52
3.4.2 Dislocation Theory 53
3.4.3 Flow Stress Measurements 57
3.5 Upper Yield Point -. . 59
3.6 Nonhomogeneous Properties 60
3.7 Hydrostatic Pressure Equation of State 60
3.8 Modeling Fracture 62
3.8.1 Fracture Toughness Testing 65
3.8.2 Spallation 67
3.8.3 Ductile Fracture 68
3.8.4 Strain Damage 68

Table of Contents IX
3.8.5 Damage in Elastic Regime 69
3.8.6 Computer Simulation of Fracture 70
3.8.7 Damage in Plastic Regime 71
3.9 Equation of State of Explosive Detonation Products 75
3.9.1 Numerical Calculation of a Detonation 79
4. Two-Dimensional Elastic-Plastic Flow 83
4.1 Fundamental Equations 83
4.1.1 Equation of Motion in x, y Coordinates with Cylindri-
Cylindrical Symmetry and Rotation About the x Axis 83
4.1.2 Conservation of Mass 84
4.1.3 First Law of Thermodynamics 84
4.1.4 Velocity Strains 84
4.1.5 Stress Deviator Tensor 85
4.1.6 Pressure Equation of State 85
4.1.7 Total Stresses 85
4.1.8 Artificial Viscosity 85
4.1.9 Von Mises Yield Condition 86
4.2 Finite Difference Equations 86
4.2.1 Mass Zoning 86
4.2.2 Equations of Motion 87
4.2.3 Conservation of Mass 88
4.2.4 Calculation of Incremental Strain 89
4.2.5 Calculation of Stresses 90
4.2.6 Von Mises Yield Condition 92
4.2.7 Equivalent Plastic Strain, sp 92
4.2.8 Artificial Viscosity for Calculating Shocks 93
4.2.9 Navier-Stokes Artificial Viscosity
for Stabilizing the Grid 94
4.2.10 Material Internal Energy 96
4.2.11 Calculation of Time Steps, Zitn+3/2 and Atn+l 97
4.2.12 Energy Summations (Edit Routine) 97
4.2.13 Principal Stresses (Edit Routine) 98
4.2.14 Calculation of Load, L, on a Given k Line
(Edit Routine) 98
4.3 Boundary Conditions 99
4.3.1 Fixed Boundary on the x Axis 99
4.3.2 Fixed Boundary on the y Axis 100
4.3.3 Corner Zone on the x Axis 100
4.3.4 Corner Zone on the y Axis 101
4.3.5 Free Surfaces 102
4.3.6 Discussion 102
4.4 Applications 103

X Table of Contents
5. Sliding Interfaces in Two Dimensions 113
5.1 Sliding Interfaces Between Quadrilateral Lagrange Zones .... 114
5.1.1 Location of Master Points Associated
with a Given Slave Point 115
5.1.2 Calculation of the Volume of Sliding Zones
Associated with the Slave Grid 115
5.1.3 Advancing a Slave Point / in Time 116
5.1.4 Location of Slave Points
Associated with a Given Master Point 120
5.1.5 Advancement in Time of Point j, k
on the Master Grid 121
5.1.6 Testing for Penetration of Grids 123
5.1.7 Adjusting the Velocities of All Void Closed Points
Where d < 0 and Where in the Previous Cycle
the Point Was Void Open 124
5.1.8 Relocating Slave Points onto the Master Surface
when d < 0 126
5.2 Intersecting Slide Lines 126
5.2.1 Acceleration of Points on the Intersection
of Two Slide Lines 126
5.2.2 Adjustment for Grid Penetration 127
5.2.3 Relocation of Points when a Void Has Opened 127
6. Elastic—Plastic Flow in Three Space Dimensions 129
6.1 Fundamental Equations 129
6.1.1 Equations of Motion 129
6.1.2 Conservation of Mass 129
6.1.3 First Law of Thermodynamics 129
6.1.4 Velocity Strains 130
6.1.5 Stress Deviator Tensor 130
6.1.6 Pressure Equation of State 130
6.1.7 Total Stresses 131
6.1.8 Artificial Viscosity for Calculating Shocks 131
6.1.9 Von Mises Yield Condition 131
6.2 Finite Difference Equations for HEMP 3D 131
6.2.1 Mass Zoning 131
6.2.2 Equations of Motion 133
6.2.3 Conservation of Mass 136
6.2.4 Calculation of Incremental Strains 136
6.2.5 Calculation of Stresses 139
6.2.6 Von Mises Yield Condition 140
6.2.7 Plastic Strain 140
6.2.8 Artificial Viscosity for Calculating Shocks 141
6.2.9 Tensor Artificial Viscosity for Stabilizing the Grid .... 142
6.2.10 Material Internal Energy 145

Table of Contents XI
6.2.11 Time Step Calculations 146
6.3 Boundary Conditions 146
6.4 Check Problems 146
6.4.1 Simple Harmonic Motion 146
6.4.2 Plasticity 149
7. Sliding Surfaces in Three Dimensions 151
7.1 Calculational Steps to Advance in Time Grid Points
on a Sliding Surface 153
7.2 Applications of Sliding Surface Routine 163
7.3 Zone Dimension Change and Subcycling 163
7.3.1 Zone Dimension Change at an Interface
in Two Dimensions 163
7.3.2 Zone Dimension Change of an Interface
in Three Dimensions 167
7.3.3 Subcycling with Zone Dimension Change
in Two Dimensions 169
7.3.4 Example for a Zone Size Change of Two to One 169
8. Magnetohydrodynamics of HEMP 171
8.1 Finite Difference Scheme for Double Operators 172
8.2 Fundamental Equations of Magnetohydrodynamics 174
8.2.1 Equation of Motion 174
8.2.2 Electromagnetic Field Equations 174
8.2.3 Energy Equation 175
8.2.4 Continuity Equation 176
8.2.5 Constitutive Relations 176
8.3 Difference Equations for Magnetohydrodynamics 176
8.3.1 Equations of Motion 176
8.3.2 Magnetic Diffusion 177
8.3.3 Energy Equations 179
8.3.4 Continuity Equation 182
8.3.5 Time-Step Control 182
8.3.6 Boundary Conditions 183
8.3.7 Sliding Interfaces 183
8.3.8 Check Problems 185
Appendices 189
A. Effect of a Second Shock on the Principal Hugoniot 189
B. Finite Difference Program for One Space Dimension
and Time 191
B.I Fundamental Equations 191
B.2 Finite Difference Equations 192
B.3 Boundary Conditions 194
B.4 Opening and Closing Voids 195

XII Table of Contents
C. A Method for Determining
the Plastic Work Hardening Function 197
C.I Application to 6061-T6 Aluminum 199
D. Detonation of a High Explosive for a 7-Law Equation
of State 202
E. Magnetic Flux Calculation 211
F. Thermal Diffusion Calculation 224
G. Backward Substitution Method for Solving a System
of Linear Equations of the Form
AiHi+i + BiHi + Ciffi-i = Di 238
References 241
Subject Index 245

Notation
p local density
po reference density
V relative volume = po/p
W velocity vector
X, Y components of velocity vector
E stress tensor
t time
Y \r space coordinate with cylindrical symmetry about the
a axis
6 direction perpendicular to the X-Y plane
H magnetic field
/imc? magnetic flux
/zm magnetic permeability
P material pressure + radiation pressure
q artificial viscosity
E internal energy (per original volume) = cvT + clrVT4
em material internal energy (per original volume) = cvT
i y change in internal energy by distortion and magnetic
pressure stresses
5 surface
ds element of surface
n outward normal to surface
dl line element of contour (C)
A zone area
M zone mass
?xy,?xx, ?yy 5 ?eo components of the strain rate tensor
Txy,sxx,$yy, see components of the deviatric stress tensor

XIV Notation
T
K
A
A
E
J
C
Cv
C
temperature
thermal conductivity
Rosseland mean free path
K + 4/3aR.cAT3 = transmissivity
electric field
current density
electrical conductivity
specific heat
Stefan-Boltzmann constant
velocity of light

Units
Symbol
eu
P
V
E
t
T
P
H
E
J
C
A
K
v
C
Quantity
energy unit
density
relative volume
internal energy per
original volume
time
space coordinates
velocity
temperature
pressure (and
stress)
magnetic field
electrical field
current density
electrical
conductivity
Rosseland mean
free path
coefficient of
thermal conduction
specific heat at
constant volume
Stefan-Boltzmann
radiation constant
velocity of light
Units
1012 ergs
gm/cm3
dimensionless
eu pu
w x Po = S*
[is = lO'6 s
cm
cm/u.s
K
Mbar = 1012 dynes/cm2
106 gauss
104 volts/cm
107 amps/cm2
103 mho/cm
cm
eu
(cm)(K)(|xs)
eu x p0 eu
(gm)(K) (K)(cm3)
7.56 x 10~27 eu
(cm3)(K4)
3 x 104 cm/[is
SI equivalent
100 kJ
103 kg/m3
—
100 GJ
us
10 mm
10 km/s
K
100 GPa
100 tesla
MV/m
100 GA/m2
100 kmho/m
10 mm
10 TJ/m • K • s
100 GJ/K • m3
756 aJ/K4 • m3
3 x 108 m/s

XVI Units
Symbol Quantity Units SI equivalent
magnetic -. -.
permeability
T = tera = 1012; G = giga = 109; M = mega = 106
k = kilo = 103; a = atto = 108

1. Elements of Fluid Mechanics
The term 'fluid' will be used here in the ordinary sense, i.e., a material
medium that is continuously deformable, has very little cohesion between
its different particles, and may be compressible or incompressible. The ma-
material may totally or partially take the shape of its container. The concept of
a continuum is also implied. This means that the pressure, temperature, and
density vary continuously from point to point. Hydrodynamics is the study
of such a system.
The hydrodynamic approach to a given problem therefore assumes that
the thermodynamic variables have a definite value no matter where one at-
attempts to measure them. The whole system need not be in equilibrium, but
any arbitrarily small region within the system must be in equilibrium. The
space considered cannot be too small, for example, of the order of a molecu-
molecular mean free path, because variables such as pressure and temperature then
lose their macroscopic meaning. Therefore hydrodynamics does not apply in
the absence of local equilibrium, i.e., when the thermodynamic variables are
changing over dimensions of the order of a molecular mean free path. Ex-
Examples of this situation are strong shock fronts and the reaction zones of
detonations. For such cases a kinetic theoretical approach to the problem
must be taken and the fluid variables are replaced by molecular distribution
functions.
However, the fact that hydrodynamics does not apply in the interior of
detonation and shock-wave fronts is not a limitation on the studies under
consideration here. Because these zones are very small they can be replaced
mathematically by a discontinuous surface, on either side of which the macro-
macroscopic model of hydrodynamics is again valid. For example, the reaction zones
of explosives are typically ~ 0.1cm thick. Shock widths are of the order of
a molecular mean free path which in the case of air, for example, is about
10~4 cm. The dimensions of the physical problems to be considered here are
many times larger. Except when the motion is discontinuous, heat conduction
and viscosity in the medium can be neglected.
Hydrodynamics as described above applies to a liquid or a gas. Many of
the results also apply to solid media, metals for example.

2 1. Elements of Fluid Mechanics
1.1 Fundamental Equations
1.1.1 Equation of Motion
As was stated above, viscosity and heat conduction are neglected, as are all
exterior forces. The fundamental equation of mechanics (i.e., Newton's second
law) applied to a fluid element leads to Euler's equation:
where p is the density, t the time, U the velocity vector, and P the pressure.
This equation expresses the fact that the momentum of a fluid element can be
changed only by the pressure that this element experiences from its neighbors.
1.1.2 Continuity Equation
For flow that does not have any sources or sinks, the principle of conservation
of mass applied to the fluid during the motion is expressed by
!^ + V-l/ = 0. A.2)
p at
1.1.3 Energy Equation
Neglecting viscosity and heat conduction is equivalent to assuming that the
internal energy of a fluid element can be changed only by the work done by
the pressure of neighboring elements:
where E is the internal energy per unit mass, and V the specific volume,
which is also equal to 1/p.
From the first law of thermodynamics &E 4- PdV = Tds, where T is the
temperature and s the entropy of unit mass.
Thus A.3) expresses the fact that the entropy of a fluid element does not
change. The assumption, therefore, is that the change in state of each fluid
element is adiabatic and reversible.
1.1.4 Equation of State
At each instant and at each point in the fluid there is a state of thermody-
namic equilibrium defined in terms of the pressure, P, the internal energy per
unit mass, E, the density, p, the entropy per unit mass, s, and the tempera-
temperature, T. From thermodynamics it is known that only two of these parameters
are independent. It is advantageous to choose E and p as the two independent
variables. The equation of state is then the equation that relates P to E:

1.2 Solutions to the Fundamental Equations 3
P = P(p,E). A.4)
As an example we will consider the equation of state for an ideal gas.
A.4a)
where the gas constant R is related to the specific heat at constant volume
Cy and the specific heat at constant pressure Cp by
R = Cp-Cv. A.4b)
For an ideal gas the internal energy E is a function of the temperature T
above. If, in particular, this function is
E = CVT A.4c)
the gas is called polytropic.
It is customary to designate the ratio of the specific heats by 7:
The equation of state A.4a) can now be rewritten as
P = G - \)pE
or
p = G _ i)^? where V = -. A.4e)
V p
It turns out that the equation of state given by A.4e) applies to a large
number of real situations. For example, with 7 = 3, it describes to a good ap-
approximation the product gases from the detonation of a high explosive. When
A.4e) is substituted into A.3) and the resultant expression is integrated, one
obtains the familiar formula
PV1 = constant.
1.2 Solutions to the Fundamental Equations
Equations A.1-4) describe the behavior of a hydrodynamic system (except
at discontinuities, which will be discussed later). They are nonlinear partial
differential equations and can be solved in closed form only for a limited
number of special cases. The fundamental work on these equations was done
over a hundred years ago by G. Monge, B. de Saint-Venant, Lord Rayleigh,
G. Stokes and H. Hugoniot, and others.
Except for the work of a few ingenious people in the field of mechanics,
this area of hydrodynamics has been dormant. This is probably due to a reluc-
reluctance of researchers to work in a field where the fundamental equations can-
cannot be solved analytically. Technological developments in recent years have

4 1. Elements of Fluid Mechanics
established a strong requirement for understanding nonlinear wave motion.
The field of hydrodynamics is attracting much greater interest, especially now
that powerfull computers can solve the fundamental equations. The chapters
that follow will describe finite difference methods for the solution in one, two,
and three space dimensions and time.
1.3 Propagation of Discontinuities
1.3.1 Sound Speed
If a perturbation is introduced at a point in a fluid it will propagate and mod-
modify the physical and kinetic characteristics of the fluid. This perturbation will
describe a wave front or wave surface that at any time t will separate the dis-
disturbed fluid, 1, from the undisturbed fluid 0. As an example, consider a fluid
initially at rest in a tube. At one end a piston is moved into the fluid, Fig. 1.1.
The fluid molecules adjacent to the piston will be set in motion, whereas
those further away will still be at rest. Thus two distinct states have been
created, separated by a wave front that moves in the fluid. If the piston has
been put into motion without a discontinuity in its velocity the wave produced
will travel at the speed of sound with respect to the fluid. The velocity U of the
fluid particles as well as the pressure P, density p, and specific internal energy
E will be continuous functions of the position coordinate X. Only the space
derivatives of these parameters will have a discontinuity at the wave front.
In the next section it is shown that if there is a discontinuity in the
space derivative of the velocity [case (a), dU/dX ^ 0], then there will be
concurrent discontinuities in the derivatives of the other parameters also [1.1].
The discontinuity will travel in the fluid at a velocity equal to U ± y/dP/dp.
The quantity ^dP/dp is called the sound speed.
yWave front
(a) Sound wave:
(b) Shock wave:
Discontinuity in
the first derivative
Discontinuity in
the variable
Fig. l.la,b. Piston moving into a fluid at rest [1.1]

1.3 Propagation of Discontinuities 5
1.3.2 Speed of Discontinuity Propagation
For motion in only one space direction X, the hydro dynamic parameters are
a function of X and t. In this case, since d( )/dt = d( )/dt [dt/dt + d( )/dX]
dX/dt, and dX/dt — U the hydrodynamic equations become:
Conservation of momentum
dU TTdU ldP n , rN
+u + 0> (L5)
conservation of mass
equation of state and isentropic assumption
P = P(p). A.7)
If the fluid is separated into two regions at the wave front (Fig. 1.1 a,b)
each of the states corresponding to (?/o,Po,Po) and (?/i,Pi,pi) must satisfy
A.5-7). At the wave front the following difference parameters are defined:
L = Ux - Uo = 0,
M = pi - po = 0,
AT = Px - Po = 0.
It is assumed that at least one space derivative of L, M, or N is not equal to
zero.
If S is the speed of the wave, the condition that L, M, and N are always
zero on 5 is describe by
where Z is L, M, or N.
If A.5) and A.6) are written for the system where [/, P, and p are C/i,
Pi, and p\ and again where t/, P, and p are C/o, Po? and po and then a
term-by-term subtraction is made, the result is
dL T8L 1 dN
9f dX p dX ( 9)
dM TTdM dL K ' }
lH+Ul)X+pdX=0'
Equations A.9) apply at the wave front where U = U\ = C/o> P = P\ — Po,
and p = p\ — po.
Starting with A.7) P = P(p), it follows that:
dP _ dP dp
d* ~ ~dp dt'
+ u = ( + u
at ax dp \dt dx

6 1. Elements of Fluid Mechanics
A term-by-term subtraction similar to the above gives
If the time derivatives of A.9, 10) are eliminated by using A.8) the following
system of equations results:
I™ + (C_S)^=0,
p oX oX
Equations A.11) are three linear equations in dL/dX, dM/dX and dN/dX
that cannot all be zero since the original hypothesis was that at least one
discontinuity existed.
Equations A.11) can be satisfied in two ways:
(a) 5 - U = 0. This is not really a wave since the disturbance is traveling
at the fluid velocity.
(b) None of the partial derivatives is zero. This is possible only if
Thus it is seen that A) if there is a discontinuity in the derivative of one
parameter there is a discontinuity in the derivatives of all the parameters;
and B) the the discontinuity travels in the fluid at the velocity of sound. The
sign ± indicates that the propagation can be in either direction.
1.3.3 Characteristics
Once again A.1, 2) are applied to motion is one space dimension X.
Conservation of momentum
dU TTdU 1 dP ,
lH+U8X + p8X=0> (L12)
conservation of mass
Equations A.3) and A.4) will be specified through the sound speed C.

1.3 Propagation of Discontinuities 7
At this point the Riemann [1.2] parameter a is introduced through the defi-
definition a = / Cdp/p.
The derivatives of p and P can be restated in terms of
dp dp da p da
= =
dX CdX1
dp c da
Substitution of A.14) into A.12) yields
dt dx dx
A15)
Equations A.15) can be rewritten as
dt dx A16)
Now for a variable Z that is a function of two parameters C and t the total
derivative is
dZ _dZ dt dZ_ dX
~dt ~ ~di~dt + ~dX ~df' ( ' }
If (U - a) and (U 4- cr) are used in place of the parameter Z, equation A.17)
becomes:
d(U-a) _ d(U-a) d(U - a) dX
d* " 9t + dX "dT' h 1C.
(l.lo)
a) dX
dt ~ dt + dX dt '
Comparison of A.18) and A.16) shows us that along curves where
dX/dt — U ± C, the quantity U ± a = constant. Curves with this prop-
property are called characteristics. Prom the results of the preceding section it
is seen that discontinuities propagate along characteristic curves. Until now
no distinction has been made in the direction of motion of the piston, i.e.,
whether it moves into the fluid and transmits a compression wave or moves
away from the fluid and transmits a decompression wave. In both cases the
discontinuities induced follow the characteristic curves discussed. However,
the subsequent results are very different. In general, the sound speed is an in-
increasing function of P (or of p). Thus in a simple compression pulse (Fig. 1.2),
the high pressure portion of the pulse will travel faster than the low pressure
portion.

1. Elements of Fluid Mechanics
Distance
Fig. 1.2. Shock forming at time ?3 from a compression wave at time t\. The com-
compression portion steepens while the expansion portion flattens
The compression wave becomes progressively steeper until it eventually
approaches a discontinuity. The wave front is now called a shock wave and the
parameters ?/, P, p, and E become discontinuous in space across the wave
front.
If, on the other hand, the fluid is already under pressure and a pressure
decrease is induced by pulling the piston away from the fluid, a rarefaction
is introduced into the fluid. In this case, since the high pressure portions of
the signal can travel faster than the low pressure portion the wave will tend
to flatten with time.
1.3.4 Shock Waves
In the previous section it was seen that the differential equations of hydro-
hydrodynamics allowed discontinuities in the derivatives of the variables to propa-
propagate. It was shown that these discontinuities propagated along characteristic
curves. However, the nonlinear hydrodynamic equations break down com-
completely when there is a discontinuity in the parameters themselves (shock
wave). This situation occurs from the gradual steepening of a compression
wave front. A shock wave may also be initiated by suddenly giving a velocity
to the piston of Fig. 1.1. The fluid in contact with it will jump discontinuously
from a zero to a nonzero velocity, Fig. 1.1b.
In the physical shock phenomenon irreversible thermodynamic processes
occur caused by friction and heat conduction taking place in the shock region.
The neglect of viscosity and heat conduction in the mathematical formulation
of the problem is the cause of the difficulties encountered when a shock forms.
However, the mathematics becomes overwhelmingly complicated when these
effects are included.
Fortunately, the real shock phenomenon usually takes place over a very
narrow region, as was discussed earlier. Outside this region the fluid flow
obeys the isentropic formulation given here. The smallness of this region sug-
suggests its replacement by a surface across which pressure, density, and velocity
change in a discontinuous manner. The values of P, p, and U on the two sides
of the shock must of course obey the laws of conservation of mass, momentum,

1.4 Derivation of the Hugoniot Relations 9
and energy. In this way the effect of viscosity and heat conduction, necessary
to describe the real irreversible process, can be incorporated without actu-
actually specifying them. The conservation equations connecting the shocked and
unshocked fluids were first given by Hugoniot [1.3].
1.4 Derivation of the Hugoniot Relations
Consider a fluid in an initial state Eo, p0, ^b, and Uq representing energy,
density, pressure, and material velocity, respectively. A shock with velocity
S (with respect to the gas velocity in front) starts from the end and travels
through the fluid, changing the state from E$, /?o, Po, Uo to E\, pi, Pi, U\.
In a time t a length L — St will have been swept out (Fig. 1.3a).
The velocity of the rear surface relative to the fluid is (U\ — Uo); therefore
the rear surface will have been displaced (Ui-Uo)t during the time in which
the shock travels the length L = St (Fig. 1.3b).
1.4.1 Conservation of Mass
For the length of material being considered, conservation of mass requires
that the mass before and after passage of the shock should be the same. The
cross section is considered to be unity.
5 =
where
- Uo)
Pi
Vo
Pi ~ Po
Vo
A.19)
Shock
front
(a)
Shock
front
(b) U
rear
surface
(UrU0)t
Fig. 1.3. Portion of fluid length L in (a) initial state, just as shock front strikes,
and (b) final state, just after shock front has swept through

10 1. Elements of Fluid Mechanics
0
Po
Here the volumes are referred to the density po, making the relative volume Vo
equal to 1. Vq is carried through the equations, even though it is 1, in order
to describe the general case where the volumes are referred to a reference
density (pref) that is not the density p0 ahead of the shock. In this case the
result is Vo = pref/Po and Vi = pref/Pi-
1.4.2 Conservation of Momentum
Conservation of momentum for the length L requires that the net force mul-
multiplied by time equal the change in momentum.
(Pi - P0)t= PoLUx - poLUo
- poStUo, A.20)
E/o).
Substituting (U\ - Uo) from A.19) we get
Using A.19) and A.20) to eliminate 5 a very useful relation is obtained:
- U0J = (Pi - PoWo - V^. A.20b)
1.4.3 Conservation of Energy
Conservation of energy requires that the net work on the mass be equal to
the change in kinetic and internal energy:
{PXU, - P0U0)t = Lpo V-{U2X - U2) + Ei- Eo] ,
where E\ and Eq represent internal energy per unit mass. We also have
Pitfi - PoUo= Sp0 \\{U, + UoKUi - Uo)] + SpoiEt - Eo]
from A.19) and A.20). Furthermore

1.5 Rayleigh Line
11
Isentrope Hugoniot
Fig. 1.4. P vs. V for a given
equation of state. For small
V2-V1 =dv, l/2(Pi+P2)«
Pi and E2 - Ei = dE =
Pdv, or the isentrope and the
Hugoniot coincide
P0U0= -(A - P0
- t/0) =
(E, - Eo)poVo = \{Pi + Po)(Vo - Vx),
or, including poVo in the energy units,
E!-Eo = |(Pi + Po)(^o - Vi), A-21)
where ?^ — poVoE (or -^ is in units of the volume of the reference density,
i.e., energy per gram times the reference density).
Equations A.19-21) are the Hugoniot relations expressing conservation
of mass, momentum, and energy.
Equations A.19, 20) give the relation between the dynamic and thermody-
namic variables. The third equation is a relation between the thermodynamic
quantities alone. For a given equation of state of the form P{— V, E) the en-
energy E can in principle be eliminated by A.21). The result will be a curve
in the P = P(V, E) plane, which is the locus of all P, V states that can be
attained by a shock from a given initial state Po, Vo. This is called a Hugoniot
curve (Fig. 1.4).
The Hugoniot curve makes a second-order contact with the isentrope
through Po, Vo. The change in entropy across a shock increases with increas-
increasing shock strength, but the entropy increase is only of third order compared
to the shock strength.
1.5 Rayleigh Line
Shock strength can be measured by a change in the pressure, Pi - Po, or
particle velocity, U\ — Uq of a shocked medium. Given the shock strength and

12
1. Elements of Fluid Mechanics
Volume, V
Fig. 1.5. Typical Hugoniot and isentrope for a solid material
the initial conditions of a medium all of the other quantities describing the
medium in its shocked condition can be readily calculated from the medium's
equation of state.
For strong shocks in solids initially at atmospheric pressure it is usual to
set Po = 0 since P\ is so much larger. Figure 1.5 shows the Hugoniot and the
isentrope for a typical solid material with initial conditions Po, Vq.
Equation A.20a) describes a straight line of slope {poSJ in the P-V
plane and volume V denned as 1/p. This line is called the Rayleigh line
and represents the locus of all permissible P, V, states consistent with a
particular shock velocity 5. The intersection of the Rayleigh line with the
Hugoniot curve gives the P, V point consistent with the Hugoniot curve.
This is an example of the fact that only one shock parameter is necessary
to determine the other parameters when the equation of state and initial
conditions are known.
The isentrope sound speed as defined earlier is C2 = dP/dp. This can be
written as
A.22a)
or
C2 =
(pc)
re
V =
---V2
2 _ _
1
P
dP
dV
dP
dV'
A.22b)
The Rayleigh line has the same form as A.22b) above except that the differ-
differential becomes a finite difference

1.6 Applications of Hugoniot Equations to a Perfect Gas 13
Po
Vo - Vi'
corresponding to A.20a) with V defined as 1/p.
It follows that
U + O S>C0-
A.23)
A.24)
This inequality expresses the fact that shocks are supersonic with respect
to the material ahead and subsonic with respect to the material behind.
From A.21) it is seen that the change in internal energy across a shock is
the area under the Rayleigh line (Fig. 1.5). From A.20b) it is seen that the
change in kinetic energy per unit mass, 1/2U2 also equals this area. Here we
have considered the initial state to be Po = 0 and Uq — 0. The difference in
the area under the Rayleigh line and the area under the isentrope (Fig. 1.5)
represents the noncoverable energy of the shock process.
1.6 Applications of Hugoniot Equations to a Perfect Gas
1.6.1 Calculation of Shock Speed
Consider a column of gas at rest. A piston at one end is suddenly given the
velocity U which is maintained constant. A shock S travels down the column
changing the gas from the state subscript zero to subscript one. Applica-
Application of A.19-21) and the equation of state (Fig. 1.6) provide the new state
parameters V and P and the shock speed 5:
U = S{l-Vi) A.25)
Pi - Po = (H>US A.26)
A.27)
Pi = Po +
2-(l-V
PoU2
A - Vy
A.28)
Fig. 1.6. Propagation of a uniform
shock into a perfect gas at rest. Equa-
Equation of state: P = G - l)f

14 1. Elements of Fluid Mechanics
Substituting A.27) into A.28) one obtains
= 0. A.29)
Replacing A - Vi) by U/S from A.25) one obtains
S2 - -G 4- l)US - — = 0. A.30)
2 po
This equation gives the shock speed S when the piston velocity and the
state ahead of the shock are known. The roots of the equation are always real,
one positive and the other negative corresponding to the two cases where the
gas is to the left and to the right of the piston that is compressing it.
The positive part of A.30) is
5 = iG + \)U + ^G + 1J?/2 + ^. A.31)
4 y 16 p0
The speed of sound ahead of the shock is Co = y/jPo/Po- Hence
5 = IG + l)U + \/^G + lJ^2 + Co-
4 V lo
The shock speed 5 increases with U. When U is zero S = C0- For large
?/\ where Co/U is negligible, 5 = (l/2)G + 1I/.
1.6.2 Calculation of Shock Pressure
From A.26) we have 5 = (Pi - Po)/poU. Replacing 5 by this value in A.30)
we get an expression for the pressure Pi:
(Pi - PoJ - ^G + l)pot/2(Pi - Po) - 7PoPo^2 = 0. A.32)
Solving A.32) for the positive root gives
Pi=P0 + \poU2 + poU^I 1G + 1J[/2 + Cl A.33)
When U is large enough that Co/U is negligible,
1.6.3 Calculation of Volume Behind the Shock
The relative volume V behind the shock is obtained by eliminating S from
A.25) and A.31):
V, = 1 - V = A.34)

1.6 Applications of Hugoniot Equations to a Perfect Gas
15
Pi
Hugoniot — =
p0
Pi
PO
Fig. 1.7. Hugoniot and isentrope
starting from same point
r1 -Y
Isentrope — = V-.
p0
and when Cq/U is negligible,
which is the minimum relative volume that a single shock can produce.
1.6.4 Graphical Representation
To get a clearer view of these relations we shall represent the Hugoniot and
isentrope graphically. Taking as coordinates P\/ Pq and V\ (remembering that
V\ is the relative volume behind the shock, and that the relative volume ahead
of the shock is 1) and using A.27) for the Hugoniot, we get the curves shown
in Fig. 1.7
The Hugoniot starting from point A has the asymptote 7 — I/7 4-1 while
the isentrope has the ordinate as an asymptote. By differentiating the Hugo-
Hugoniot and the isentrope two times with respect to Vi, one obtains:
dVi
A(Hugoniot)
dVi
A(isentrope)
dVf
AV?
= 7G
A (isentrope)
A(Hugoniot)
which shows that the Hugoniot and isentrope have at point A the same
tangent and curvature, as was pointed in Fig. 1.4.
1.6.5 Reflection of a Uniform Shock
In the preceding analysis the column of gas was considered to be infinite
in length. Here it is assumed that the gas column is terminated by a fixed
boundary where the velocity is always zero.
When the shock S from the piston, described by A.31), reaches the fixed
boundary a reflection is produced. That is, a new shock S\ is formed which

16 1. Elements of Fluid Mechanics
travels back toward the piston and changes the gas velocity from U\ to that
of the fixed boundary or zero. This shock reaches the piston and a new shock
52 analogous to S, is formed, etc.
We wish to find the values of Pn and Vn behind a shock 5, where for odd
n the shock is traveling from the piston toward the fixed boundary and for
even n the shock is traveling from the fixed boundary toward the piston.
Sn = (Un ~ Un-l)yVn~\ A-35)
Pn - Pn_i = pn-iSn(Un - ?/n_i). A.36)
[Note that A.35) and A.36) are generalized versions of A.25) and A.26), re-
respectively] The quantity (Un - [/n-i) alternates from (+) the piston velocity
to ( —) the piston velocity U. For a shock process, Pn > Pn_i, so 5 changes
sign corresponding to its direction. Eliminating S from A.35) and A.36) and
taking account of the fact that po-i — Po/Ki-i> one obtains
- A-37)
A.38)
[A.38) is derived from A.33)].
The solution of the problem is complete since all the quantities of index
n can be obtained from the quantities of index n - 1.
1.6.6 Conditions Behind the First Reflected Shock
from a Fixed Boundary
Equations A.37) and A.38) determine the conditions behind an incoming
shock from a piston moving at velocity U into the gas with initial conditions
Vo = 1, Po = 0 and reference density po (Fig- 1-6):
V = 7~ 1 - El
7 + 1 Pi'
Substitution of these values back into A.37) and A.38) determines the con-
conditions behind the shock after its reflection from a fixed boundary

1.8 Elastic-Plastic Waves 17
1.7 Detonation Waves
In Sect. 1.5 it was seen that a shock is a dissipative process and that further-
furthermore a rarefaction from behind the shock wave will always overtake it since
signals travel at U + C and U + C > S. This means that shocks will ultimately
die out unless energy is continuously supplied from behind the shock wave.
If, however, the passage of the wave involves a release of chemical energy in
the medium the wave propagation can be self-sustaining. Such a wave does
indeed exist in a high explosive and is called a detonation wave.
The calculation of a detonation wave differs in two principal ways from a
shock wave.
1. The Hugoniot conditions across the wave front still apply but it is nec-
necessary to supply the chemical energy released at the front.
2. The wave propagation is not controlled by conditions behind the front as
in the example of the piston considered earlier.
The three Hugoniot equations A.19-21) are not sufficient to determine
the four unknowns Pp, [/, and D. (Here D is the detonation velocity which
replaces S in the Hugoniot equations). A supplementary condition is given
by the assumption that at the detonation front a small disturbance travels
at the same speed D as the front itself. This is called the Chapman-Jouguet
(CJ) hypothesis and is stated mathematically as
D = U + C. A.39)
It has not been possible to supply a rigorous demonstration of this hy-
hypothesis, but it does give results verifiable by experiments and has been the
basis of much fruitful work on high explosives.
In reality there is a reaction zone at the detonation wave front where
an irreversible decomposition of the explosive takes place. Thermodynamic
equilibrium is assumed to exist immediately behind the reaction zone and
if there are any further chemical reactions occurring in this region they will
not affect the detonation velocity. It is in the region immediately behind the
reaction zone that the CJ hypothesis is applied.
1.8 Elastic-Plastic Waves
In contrast to fluids, solids resist shear distortion and as a result the equa-
equations of motion and the thermodynamic description applied to solids are much
more complicated. It can be argued that at stress levels greatly in excess of
the shear strength the stress system is effectively isotropic and equivalent
to a hydrostatic pressure. With this assumption the hydrodynamic analysis
discussed here can also be applied to solids. However, in recent years it has
been found that the presence of a small shear stress component has a large
effect on the manner in which a pressure wave attenuates. Furthermore, it

18 1. Elements of Fluid Mechanics
has been observed experimentally that the shear strength of some solids in-
increases with increasing pressure. These facts have led to the development of
an elastic-plastic model instead of a fluid model to describe the behavior of
solids even at high pressures.
Some of the outstanding features of an elastic-plastic material can be
demonstrated by considering a one-dimensional compression wave. The
stress, <t, is considered to be composed of a hydrostatic pressure, P, and
a distortion stress, s. The hydrostatic pressure can be thought of in the same
sense as a fluid pressure and described by an equation of state. The distor-
distortion stress, following elasticity theory, is considered to be a linear function
of strain. There is an upper limit to the magnitude of the distortion stress
and this limit is stated by a yield condition. After the yield point has been
attained the material deforms plastically under additional loading. A mate-
material is said to be elastic when the stress is proportional to strain and plastic
when the stress is no longer proportional to strain.
For the one-dimensional strain considered here, the stress a is given by
a = P + s,
where
_4 Vo-V
5M V "
Here k is the bulk modulus, // the shear modulus, and V the specific vol-
volume. Note that for the purpose of simple demonstration, the pressure P and
the stresses a and s are taken here to be positive in compression and nega-
negative in tension. (The usual notation counts the stresses a and s negative in
compression and positive in tension.) These equations apply until s reaches
a maximum value s = 2/3V, where Y is the yield strength in simple ten-
tension. The result that the maximum compression stress is s = 2/3Y stems
from the von Mises yield condition and is described in later chapters. For
all subsequent compression the material is taken to deform plastically with
s remaining equal to its maximum value and P increasing. The parameter k
is initially constant but then increases with increasing pressure. In the P-V
plane the pressure curve will be concave upward similar to fluids. Thus the
sound speed increases with pressure and shocks can form.
The Hugoniot equations still apply, but now the pressure P is replaced
by the total stress a.
In Fig. 1.8 point A is where the distortion stress component, s, has reached
its maximum value and is referred to as the Hugoniot elastic limit. The dis-
discontinuous decrease in slope at point A will cause the stress wave to break
into two steps for stress levels that are between points A and B in Fig. 1.8. An
elastic precursor (Fig. 1.9) of stress level a a will travel at the elastic velocity
CE

1.8 Elastic-Plastic Waves
Volume, V
k + 4/3/i
19
a.
essure,
Q.
o
6
co
Stre:
pAV
\v
V
\\\
\ \
\\
\
2/3Y
a=P+s
/-Rayleigh line
/
\
Fig. 1.8. One-dimensional
stress-strain for an elastic-
plastic material
A.40)
This stress will be followed by a plastic wave traveling at the shock ve-
velocity, Sp, given by applying A.20a).
where
A.41)
crB,
0
55
x-Plastic Wave
°A^-Elastic Precursor
Distance
Fig. 1.9. Space profile of an elastic-plastic stress wave

20 1. Elements of Fluid Mechanics
For cr > Gb the plastic wave velocity, Sp, will be greater than the elastic
precursor velocity, Ce, and the stress will propagate as a single shock. This
follows from noting that the slope of the Rayleigh line is greater than the
slope (aa — o"o)/(Vq — V&) for stress points above B.
1.9 Units and Orders of Magnitude
Pressure is a force per unit area. It is convenient to use a system of units that
fits in with the cgs system. The pressure unit is the kilobar (kbar) = 103 bar
or the megabar (Mbar) = 106 bar.
Here 1 bar = 106 dyne/cm2.
1 atmosphere = 76 cm of mercury = 1.012 x 106 dyne/cm2.
Hence one bar is approximately equal to one atmosphere.
A consistent set of units is:
Pressure
Distance
Time
Velocity
Density
Energy
Mbar
cm
M.s = 10s
cm/|is
g/cm3
1012erg/g.
A common source of high pressures in experimental hydrodynamics work
is high explosives. Typical explosive detonation velocities are 0.8cm/|j.s and
detonation pressures are ~ 0.3 Mbar. A 0.3 Mbar detonation pressure can
induce pressures between 0.1 and 0.6 Mbar in a material placed in contact
with the high explosive, depending on the equation of state of the material.
By accelerating a metal plate with a high explosive and allowing the plate to
strike a target plate, pressures of up to 2 Mbar may be attained.
1.10 Measurements to Obtain Equation of State Data
1.10.1 Experimental Methods
The greatest uncertainty in applying hydrodynamic theory to physical sit-
situations lies in the equation of state of the materials. Shock wave theory,
however, provides an experimental method to obtain information about the
equations of state.
A shock propagating into a material with an unknown equation of state
is completely specified by measuring any two of the variables P, p, [/, E, S,
and the three Hugoniot relations. The variables S and U are the easiest to
obtain experimentally, and one of three methods may be employed.

1.10 Measurements to Obtain Equation of State Data 21
A) When a free-flying plate is allowed to strike a target plate the interface
will acquire a new velocity and the pressure on either side will be the same
due to the principle of action and reaction. If the two materials are the same,
the interface velocity will be one-half the free-flying plate velocity.
Optical techniques or electric probes at fixed positions can measure the
flying plate velocity and the shock wave transit time in the target plate. By
repeating these measurements for different flying plate velocities the P-V
curve of the material can be determined.
B) When a shock wave reaches a free surface the subsequent velocity of
the free surface will be the result of the contributions of (i) the shock particle
velocity U associated with the change of state from Pq to P, po to p, etc., and
(ii) the isentropic velocity when a rarefaction proceeds back into the material
decompressing it and reducing the pressure from P to the boundary pressure
at the free surface P — 0. It is shown later that these two velocities are nearly
equal for shocks when p/ po is less than about 1.4. The front surface velocity
of the target material can be readily measured; one-half this velocity will
then be the required particle velocity. The shock velocity can be measured
as before.
C) Once a complete P-U curve for one material is known it can be used
with a material whose P-U relation is not known and only the shock speed
S need be measured for the new material to determine its complete state.
Consider a high explosive in contact with material A whose P-U relation is
known; next to this material is placed material B whose P-U relation is to
be determined. A shock from the high explosive will traverse material A and
enter material B. A shock will always enter material B, but the wave reflected
into material A at the interface may be:
(a) A reflected shock if material B has the greater shock impedance.
(b) A reflected rarefaction if material B has the smaller shock impedance.
(c) Neither a reflected shock nor a reflected rarefaction. In this case materials
A and B have the same shock impedance.
Figure 1.10 shows the known P-U curve for material A. One-half the A
measured front surface velocity of a free surface of material A will give the
particle velocity U\ and pressure P\ that is present just before the shock
reaches the interface of materials A and B. It is assumed that the reflected
wave into material A still follows the Hugoniot equation A.20), i.e., AP —
poSAU. For case (a) the pressure increases with a decrease in particle velocity,
while for the case (b) the pressure decreases with an increase in the particle
velocity. Graphically these states are given by reflecting a mirror image of
the P-U curve of material A about the point P\, U\.
When the reference density p0 of material B is known and the transmitted
shock speed 5T has been measured then P/U — po^T is the locus of all points
that satisfy the Hugoniot equation A.20). The intersection of this line with
the A reflected P-U curve A' is the desired PT, UT state in material B. This
is called the impedance match method.

22
1. Elements of Fluid Mechanics
Particle velocity, U
Fig. 1.10. Pressure-particle velocity curves to illustrate the impedance match
method to obtain equation of state data
1.10.2 Relation of the Free Surface Velocity
to the Shock Particle Velocity in a Solid
For a Hooke's law equation of state, P = k(p/p0 - 1), it is easy to show that
the front surface velocity equals twice the particle velocity (C/fs = 2Up):
2 dp k
o = —— = —,
dp po
where A: is a constant, po the reference density, and C the speed of sound.
The Riemann invariant a [1.2] for the hydrodynamic equations of motion
is
Since C is a constant for this equation of state we have
J P \PoJ
where 77 = p/p0.
For a shock traveling through undisturbed material we have
We want to find the front surface velocity when the shock reaches it. Taking
the +c characteristics we have
Up -f (Tp = E/fs + CTfs
at the front surface a = 0. Hence

1.10 Measurements to Obtain Equation of State Data 23
Substituting the equation of state into the relation poUp = P(l - V) we get
Po V
where
C = Up v ,
ap = C In 77 = ?/p -r-^—- In rj
= Up, since —-—- In r\ ~ 1.
From
we have
In the above derivation the equation of state was assumed to be linear
in 77. A more critical analysis, using a nonlinear equation of state, will show,
that the preceding result is valid over a large range of pressure and compres-
compression. Consequently the application of the principal Hugoniot in solving shock
interaction problems is reasonable. Appendix A gives an analysis of the effect
of a second shock on the principal Hugoniot.
1.10.3 Form of the Equation of State for Solids
Many equations of state are expressed in terms of pressure, volume, and
temperature. The use of temperature as a variable requires data on the spe-
specific heat so that an expression for energy may be obtained for use with the
Hugoniot equation A.21). Since hydrodynamic applications do not require
temperature explicitly, an equation of state relating P, V, and E is much
more desirable. A form that has been very successful for describing metals at
high pressures is the Mie-Gruneisen equation of state.
P = PL + PTl A.42)
where
^ and PT = 1(E

24
1. Elements of Fluid Mechanics
Pi
CL
1
I
1
PL
EL ^^^X
Volume, V Vo
Fig. 1.11. Shock compressibility
The interpretation is that the total pressure P is the sum of the lattice
pressure Pl due to the lattice potential energy E\, at absolute zero and the
thermal pressure Px due to the lattice vibrational energy E - E^. Here 7 is
called the Gruneisen ratio and is assumed to be a function of volume only.
Equations of state are discussed in more detail in Chap. 3 on modeling the
behavior of materials.
The thermal energy Et due to a shock is shown schematically in Fig. 1.11
as the striped area. The corresponding energy EL due to the compressibility
at absolute zero is also shown. The total energy change due to a shock is the
sum of these two energies.
The behavior of a shocked material can be seen by substituting the Hugo-
niot equation A.21) into the equation of state A.24) where Po = 0, Eo = 0.
A.43)
Equation A.43) gives the locus of all P-V states reached by a single shock
where the initial state is Po, Vq. It is seen that the pressure P becomes infinite
when the relative compression n = Vo/V = 1 + 7/2. For metals, the limiting
compression is ~ 2, implying 7 = 2.
Experimental P, V data along the Hugoniot curve may be used with A.43)
and Pl = — dE\^/dV of A.42) to derive consistent values of the functions
El(V), P(V) and j(V). (No distinction is made here between the absolute
zero isotherm and the room temperature isotherm). When this is done A.20)
will describe all P, V, E states of the material. It must be realized that the
shock wave data gives P and V and from this a P, V, E relation is developed
i.e., a line has been used to generate a surface. For this reason it can be

1.10 Measurements to Obtain Equation of State Data
25
Particle velocity, U
Fig. 1.12. A, B and C are known curves. Curve H is the locus of P, U states where
the initial state is Pa U&
expected that the experimental equation of state will be valid only in regions
near the Hugoniot curve.
At pressures above lOOMbar the electronic shells of atoms are crushed
and lose their individual structure. The Thomas-Fermi-Dirac (TFD) statis-
statistical model of the atom can be used to describe the compressibility in this
region. The experimentally derived, El, Pl, and 7 relations can therefore be
extrapolated from the experimentally determined portion, to the TFD zero
temperature isotherms. This is a rather long extrapolation since experimental
data end at 2 Mbar and the TFD data begin at 100 Mbar.
1.10.4 Detonation Pressure Measurement
A charge of high explosives detonated in contact with a metal witness plate
will transmit a shock wave into the plate. If the Hugoniot curve for the metal
plate is already known, a measurement of the plate free surface velocity will
determine the pressure in the metal. Repeating the experiment for metals
with different known Hugoniot curves will determine additional pressures.
These pressures lie on the reflected Hugoniot of the detonation products
of the high explosive; i.e., they represent states reached by the detonation
products where the initial state had the detonation pressure P<j detonation
density p^ and detonation particle velocity U&. A measurement of the det-
detonation velocity and the original high explosive density together with the
Hugoniot equation A.19), P/U — PqD, determine a line in the P-U plane.
The intersection of this line with the curve traced out by the experiments
with different metals gives the detonation parameters P& and U& (Fig. 1.12).
In measuring the detonation pressure care must be taken that the ex-
experiment is done in one-dimensional geometry. Also, it is the pressure at
essentially zero plate thickness that is required experimentally, so a suitable
extrapolation must be made from the finite plate thicknesses used.

26 1. Elements of Fluid Mechanics
This series of experiments provides a means of checking the Chapman-
Jouguet hypothesis. The Hugoniot conservation of momentum equation
A.20) can be used to describe the shocked states of the detonation prod-
product gases where the initial state is the detonation point, Pd, Ud, and pd.
P-Pd = PdS(U - Ud)
or
?5 <144»
In the limit of shock states very close to the detonation state, Pd, Ud,
equation A.44) leads to:
%
% P*C, A.45)
Here the shock speed S has been replaced by the sound speed at the
detonation state Cd. Equation A.45) gives the slope of curve H at the point
Pd,?/d, Fig 1.12.
The CJ hypothesis states
D = Ud + Cd. A.46)
The Hugoniot equation A.19) is used to express conservation of mass
across the detonation front
^ - 2-
Combining A.46) and A.47), we have
CdPd = poD. A.48)
Therefore, according to CJ theory, A.45) becomes
dP
= poD. A.49)
When the slope of curve H (Fig. 1.12) is measure for solid explosives it is
in fact equal to —poD, thus supporting the CJ hypothesis.

2. Numerical Techniques
The finite difference equations presented here follow the format and nota-
notation used by von Neumann [2.1] for the solutions of the differential equations
that describe fluid dynamics in one space dimension. The material is divided
into a Lagrange grid that moves with the flow. The space between consec-
consecutive grid lines is referred to as a zone. For multidimensional problems a
zone is defined as the interior space of intersecting grid lines. The intersec-
intersections are called zone node points. Subscripts define the Lagrange coordinates
and superscripts the corresponding times. See Refs. [2.2, 3] for complete de-
descriptions of mesh generators. For a one-dimensional network, X™ represents
the X position of Lagrange coordinate j at time tn. Intermediate points are
given by X]+l/2 = 1/2(X;+1 + XJ1) and X»+1/2 = l/2(X?+i + X?). A dot
over a parameter represents a time derivative. Thus, X™ represents the
velocity of node point j at time ?n+1/2. The complete set of equations for
one-dimensional calculations in gas dynamics as well as elastic-plastic flow
are given in Appendix B.
2.1 Von Neumann Finite Difference Scheme
Position and velocity are evaluated at zone node points; thermodynamic pa-
parameters, e.g., pressure, volume, and energy are evaluated at zone centers.
The von Neumann finite difference equations are second order in terms of
Taylor's series:
Subtracting yields
/*+* _ /n-i = Atf + third and higher order terms.

28 2. Numerical Techniques
2.1.1 Time Centering
Two time steps, At71 and Atn+l/2 are used to advance in time the set of
finite difference equations. Time step Atn = (tn+1/2 - tn~1/2) is used with
the pressure field defined at time tn to advance the velocity of a Lagrange node
point from time tn~1/2 to time tn+l^2. The positions of the mesh points and
subsequently the thermodynamic properties of the zones are advanced from
time tn to time tn+1 with time step Atn+Xl2 = (tn+l -tn). Time step At71*1'2
is determined in advance for each time cycle from stability conditions with
Atn = l/2{Atn+1/2 + Atn~l/2). The time centering is upset when stability
conditions require a smaller time step or when the conditions of the problem
permit a larger time step. In the latter case the time step increase can be
made gradually to help preserve the time centering.
2.1.2 Space Centering
Space centering is achieved when the Lagrange grid is generated by main-
maintaining a constant zone size. This in general is not practical. A geometric
progression can be employed with the mesh generator to make the transi-
transition from small to large zones while minimizing the centering error between
consecutive zones. For two materials of different densities in juxtaposition,
the interface error can be minimized by using zone sizes that are density
weighted. The error is manifest by an incorrect compression and energy for
the zones on the interface when a shock wave traverses the interface. A sim-
similar error occurs when a shock reflects from a fixed boundary. However, the
equations of motion conserve momentum and the correct pressure is reached.
The magnitude of an interface error in compression or energy depends on the
direction of the shock wave.
The complete set of equations for one-dimensional calculations in gas
dynamics as well as elastic-plastic flow is given in Appendix B.
2.2 Artificial Viscosity
For the calculation of shock waves the von Neumann artificial viscosity idea
for one-dimensional calculations is generalized to two and three dimensions
[2.4]. In addition, a linear viscosity term is added to correct for the fine-scale
error waves of the order of the zone-to-zone fluctuations that appear with the
von Neumann method.
2.2.1 Generalized Artificial Viscosity
The artificial viscosity, q, given below is used in all of the one-, two- and
three-dimensional and time finite difference programs presented here

2.2 Artificial Viscosity 29
-f Cl p La —
dt
q = 0 for -/ > 0
dt ~
p = local density
L = characteristic grid length \^-*-)
ds
— = rate of strain in the direction of acceleration
dt
fp
a — \ — where P = local pressure
V P
Co - 2; CL = 1.
For flow in one space dimension X, ds/dt — dX/dX and L = AX] thus,
the first term of the q is identical to the quadratic von Neumann artificial
viscosity. The linear term that follows was determined by comparing the
equation for a shock in a perfect gas derived by Hugoniot [see A.33)] with
the von Neumann artificial viscosity. The parameter a in the linear portion
of B.1) is proportional to the sound speed in a perfect gas. It is used for all
materials, solid or gaseous. The advantage ©f using the parameter a instead
of the actual sound speed in the linear portion of the viscosity B.1) is that
it provides a zero diffusion coefficient for waves propagating into solids at
rest [2.4]. This property helps to minimize the undesirable diffusion associ-
associated with a linear viscosity term that employs the actual sound speed of a
solid which is finite at zero pressure. With the formulation shown the linear
viscosity becomes effective behind the shock front, where it is needed to damp
numerical overshoots.
2.2.2 Applications of the Generalized Artificial Viscosity
in One Space Dimension
Figure 2.1 shows the results of a calculation using the above generalized
artificial viscosity q for a shock wave into a perfect gas G = 1.4) from a
constant pressure applied to the left-hand boundary. The shock proceeds
from left to right and reflects from the fixed right-hand boundary. From the
Hugoniot equations presented in Chap. 1, the ratio of the reflected shock
pressure Pr to the incident shock pressure Pi is: Pr/P[Cj - l)/G — 1) = 8.
Figure 2.1 shows the correct reflected shock pressure Pr — 80kbar has been
reached with no overshoots or oscillations.
Table 2.1 shows the numerical output for this calculation for zones near
the shock front of the incident shock, taken as the position of the maxi-
maximum value of the artificial viscosity q, j — 37. A comparison of the shock
parameters in Table 2.1 with those calculated from the Hugoniot equations
(Chap. 1) shows agreement to the fifth significant figure. The literature [2.1]

30 2. Numerical Techniques
Table 2.1. Numerical output for the region near the shock front at t = 1.2 (xs for the
problem given in Fig. 2.1 The symbols have the following meanings: J — Lagrange
coordinate, Q = artificial viscosity (Mbar), x = position (cm), E = internal energy
A012 erg) per original volume, U = particle velocity (cm/us), P = pressure (Mbar)?
ETA= compression
TIME
1.20057084E*00
CYCLE
1646
DTO
6.49460435E-04
P-JMAX
1.00000000E-02
U
ETA
00104 1.575142E*O0
00103 1,569386E*00
00102 1.564031E+00
00101 1.556475E+00
00100 1.552919E*00
00099 1.547363E+00
00098 1.541808E+00
00097 1.536252E*00
00096 1.530696E*O0
00095 1.525141E*00
000941,519585E*00
00093 1.514030E+00
00092 1.508474E+00
00091 1.502919E*00
00090 1.497363E+00
00089 1.491808E+00
00088 1.486252E + 00
00087 1.480697E+00
00086 1.475141E+00
00085 1.469586E+00
00084 1.464031E*00
00083 1.458475E+00
00082 1.45292OE+O0
00081 1.447364E+00
00080 1.441809E+00
00079 1.436254E+00
00078 1.430698E+00
00077 1.425143E+00
00076 1.419588E+00
00075 1.414032E+00
00074 1.408477E+00
00073 1.402922E+00
00072 1.397366E+00
00071 1.391811E+O0
00070 1.386256E+00
00069 1.380700E+00
00068 1.375145E+00
00067 1.369590E+00
00066 1.364034E+00
00065 1.358479E*00
00064 1.352924E+00
00063 1.347369E*00
00062 1.341613E+00
00061 1.336258E+00
* 330703E+00
325148E*00
..319592E*00
00057 1.314037E+00
00056 1.308482E+00
302927E^O0
297372E+00
291616E*00
286261 E-^00
..280706E+00
00050 1.275151E*00
1.269596E+00
" 264040E^00
256485E+00
252930E+00
..247375E*00
00044 1.241820E1-00
00043 1.236264E+00
00042 1 .230709E + 00
00041 1.225150E+00
00040 1 .219562E+00
00039 1 .213719E+00
00038 1.206051E+00
191638E+00
165922E+00
133329E+00
. ..100000E+00
00033 1.066667E*00
00032 1.033333E+00
00031 1.000000E+00
00061 1 .
00060 1 .
00059 1.
00058 1.
00055 1.
00054 1.
00053 1
00052
00051
00048
00047 1i
00046 1.
00045 1.
00037
00036
00035 1
00034 1
-2.6353E*00
-2.6353E*00
-2.63S3E+00
-2.6353E*00
-2.6353E+00
-2.6353E+00
-2.6353E*00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E*00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E*00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E*00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E+00
-2.6353E*00
-2.6353E+00
-2.6353E*00
-2.6353E+00
-2.6353E*O0
-2.6353E*00
-2.6353E+00
-2.6353E^00
-2.6352E+00
-2.6353E*00
-2.6353E+00
-2.6345E+00
-2.6270E+00
-2.5638E+00
-2.1953E+00
-1.2525E+00
-2.2379E-01
-2.5208E-03
-1.0573E-07
0.
0.
0.
1.OOOOOE-02
1,OOOOOE-02
1.00001E-02
1.O0001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-O2
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-O2
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.OOOO1E-O2
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.OOOO1E-O2
1.00001E-02
1.00001E-02
1.00001E-02
1 .00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1 .00001E-02
1 .00001E-02
1 .00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00001E-02
1.00002E-02
1.00002E-02
1.00002E-02
1.00002E-02
1.00002E-02
1.00002E-02
1.00002E-02
1.00002E-02
1.00002E-02
1.OOOO2E-O2
1.00002E-02
1.00001E-02
1.OOOOOE-02
1.00001E-02
1.00002E-02
9.99914E-03
9.99075E-03
9.91921E-03
9.30963E-03
6.17430E-03
1.74692E-03
1.27930E-04
2.59051E-07
O.
0.
0.
O.
0.
0.
4.66666E-10
2.56009E-10
5.127OOE-1O
6.20126E-10
1.06905E-09
1.25104E-09
1.33998E-09
1.31490E-09
1.15726E-09
8.45528E-10
3.79493E-10
O.
O.
O.
0.
O.
0.
0.
O.
0.
O.
3.60449E-10
).72689E-10
4.66440E-10
9.52294E-10
1.37691E-09
1.63310E-09
1.66107E-09
1.43720E-09
9.69426E-10
2.68001E-10
O.
0.
0.
0.
O.
0.
0.
0.
0.
0.
0.
0.
0.
5.92514E-10
8.71760E-10
7.75447E-10
4.02165E-10
0.
O.
1.28065E-09
0.
0.
1.85173E-09
1.61639E-08
0.
O.
1.61981E-07
2.34420E-07
O.
0.
7.62955E-06
7.50939E-05
6.248OOE-O4
3.23187E-03
5.21667E-03
2.44207E-03
7.60112E-05
9.22704E-09
0.
8:
4.1670E-03
4.1669E-03
A.1669E-03
4.1669E-03
4.1668E-03
4.1668E-03
4.1668E-03
4.1668E-03
4.1668E-03
4.1667E-03
4.1667E-03
4.1667E-03
4.1667E-03
4.1667E-03
4.1667E-03
4.1666E-03
4.1666E-03
4.1666E-03
4.1666E-03
4.1666E-03
4.1666E-03
4.1666E-03
4.1666E-03
4.1666E-03
4.1666E-03
4.1666E-03
4.1666E-03
4.1666E-03
4.1665E-03
4.1665E-03
4.1666E-03
4.1665E-03
4.1665E-03
4.1665E-O3
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
A.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
A.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
4.1665E-03
A.1665E-03
4.1665E-03
4.1664E-03
A.1664E-03
4.1665E-03
A.1663E-03
4.1653E-03
4.1568E-03
4.0802E-03
3.5509E-03
1 .8883E-03
2.4673E-04
6.3325E-07
O.
O.
O.
O.
6.9996E*00
5.9996E+00
5.9997E*00
5.9997E*00
5.9998E+00
5.9998E+00
5.9999E*00
5.9999E*00
0.9999E+00
6.0000E*00
6.0000E*00
6.0000E+00
6.0000E*00
6.0000E*00
6.0001E*00
6.0001E*00
6.0001E+00
6.0001E*00
6.0001E+00
6.0001E*00
6.0001E*00
6.0002E*00
6.0002E+00
6.0002E*00
6.0002E*00
6.0002E*00
6.0002E+00
6.0002E+00
6.0002E+00
6.0002E+00
6.0002E+00
6.0002E+00
6.0003E+00
6.0003E*00
6.0003E*00
6.0003E+00
6.0003E+00
6. 0003E-»-00
6.0003E+00
6.0003E+00
6.0003E+00
6.0003E+00
6.0003E+00
6.0003E+00
6.0003E+00
6.0003E+00
6.0003E+00
6.0003E+00
6.0004E+00
6.0004E+00
6.0004E*00
6.0004E*00
6.0004E*00
6.0004E+00
6.0004E+00
6.0004E+00
6.0004E*00
6.0004E*00
6.0004E*00
6.0003E+00
6.0004E*00
6.0004E+00
6.0000E+00
5.9964E+00
5.9657E+00
5.7042E*00
4.3470E+00
2.3128E+00
1.2962E+00
1.0227E*00
1.0001E*00
1.OOOOE+00
1.OOOOE+00
1.OOOOE+00

2.2 Artificial Viscosity
31
XL
12
10
8
6<
4
2-
0
P=10kb
P=<Y-1)f
Y=1.4
Fixed
boundary-—
(a) time 0.94 jis
100
80
|60
40
20
0
P=10kb
-4 -3 -2
(c)time 1.65 us
-1 0
x(cm)
12
10
8
6
4
2
0
(b)time 1.50 us
120r
P=10kb
-4 -3 -2
(d)time 1.77 jus
-1 0
x(cm)
Fig. 2.1a—d. Calculation of a shock wave into a gas at rest. The parameter values
used are the following: constant pressure: P = lOkbar; applied to Lagrange grid:
j = 151; original position: X151 = -5cm; fixed boundary at: j = 1, X\ — 0; q
constants: Co = 2, Cl = 0.5
refers to the von Neumann viscosity term as first order and the finite differ-
difference scheme as second order. The fact that the viscosity term is labelled only
first order is of no consequence in view of the accuracy it produces in shock
wave calculations.
Figure 2.2 shows the results of an elastic-plastic analysis of a flying plate
striking a target plate. The calculation illustrates the fact that the interac-
interactions of shock fronts with release waves can be calculated with no numerical
overshoots.

32
2. Numerical Techniques
80
40
0
-40
-80
80
40
0
-40
-80
Flying Target
plate .
¦ I
-
(a)
r
J-
t
L
e
_
L
P-
1
l-
L_
2 us
i
-
-
i
— — Void opens _
(e) t = 7.6us
Flying Target
plate ,
"(b)
^-n -1
1
2.5 us
_
80
c40
CO
.Q
-40
-80
-
/ i
/
__/ i
(c) t = 4.3us
i
i
i i
—
-
-
-
(d)
t =
I
-^. r
\A
\T
5.7 us
i i
-
-
-2-1012
3-:
x (cm)
-
(f)
I
? -
0
LT\L
7\
9.8 us
i i i
1 2
-
-
2
Fig. 2.2a-f. Calculated
stress waves from the col-
collision of two aluminum
plates, e.p. = elastic pre-
precursor, L — loading wave,
U = unloading wave
2.3 Stability Conditions
2.3.1 Courant Condition
The first stability requirement for the difference equations is the Courant
condition which demands that the time step At is less than the time for a
sound signal with velocity a to traverse the grid spacing L,
~ < C. B.2)
The reduction factor C is referred to as the "Courant number".
2.3.2 Von Neumann Stability Analysis
The stability analysis of von Neumann [2.1] identifies additional reductions in
the time step due to the material compressibility and the artificial viscosity
coefficient.

2.4 Finite Difference Scheme in Two Dimensions 33
It is convenient to lump the stability requirements into the single state-
statement
il^L B.3)
where Ln is the grid spacing, a the local sound speed, and b and Atn are
given by
Atn = -
The multiplier in the b term depends on the material compressibility and
can be smaller for materials with a stiffer equation of state. Appendix B
describes a finite difference program for the solution of problems in one space
dimension and time.
2.4 Finite Difference Scheme in Two Dimensions
2.4.1 Integral Definition of a Derivative
The fundamental equations are organized so that changes of variables associ-
associated with a mass point can be interpreted as due to a flux through a surface
surrounding the mass point. The difference operators are organized in the
same way, i.e., in the spirit of the divergence theorem. Space derivatives in
two dimensions are defined as the summation of the normal components of
the flux around an enclosed area. Thus, for a vector, F, representing a flux
(e.g., magnetic, heat flow, velocity, etc.), the following integral definitions of
the partial derivatives are used (Fig. 2.3):
dF , f F(n-i)de
~*V = lim / ~a
?L - lim / F(n-j)de B'4)
dY A->oJ{c) A
where d? is an element of arc length, (C) represents the boundary of area A,
n is the vector normal to the boundary, and r tangent vector.
Fig. 2.3. Integration scheme

2. Numerical Techniques
Fig. 2.4a,b. Integration paths,
j, k = Lagrange coordinates
2.4.2 Integration Paths
The integration path for evaluating the partial derivatives is defined in two
ways. Figure 2.4a shows the path for advancing in time a component of the
material velocity of a mass point centered at j, k that is accelerated by
the stress field surrounding it. Figure 2.4b shows the path for evaluating
components of the continuity equation centered at j + 1/2, k + 1/2 from the
velocity field that surrounds the zone area A; see Chap. 4 for details.
This staggered scheme reduces to the von Neumann equation for one-
dimensional flow.
2.4.3 Properties of the Integration Scheme
The difference equations have the properties that they conserve angular mo-
momentum and transform in the same way as the differential operators. For
example, in two-dimensional Cartesian coordinates the terms in the differ-
difference equations that represent the partial derivatives on the left-hand side of
each equation listed below will collect and be exactly equal to a time centered
difference of the quantity on the right-hand side:
v-w= r^ + ^\ =4
Vx W =
B.5)
where W is the velocity vector, A the area of the zone, and u the angle
of rigid rotation. This exact equality leads to zero truncation error in the
definition of stress and strain used in the formulations of the elastic-plastic
problem.
2.4.4 Continuity Equation
The continuity equation in x, y coordinates with cylindrical symmetry about
the x axis can be written as:
V
B.6)

2.6 Finite Difference Scheme for Double Operators 35
Here V is the volume swept out when the area zone A is rotated about the x
axis (Fig. 2.4b). It is important to recognize that the partial derivative terms
in this equation are independent of the coordinate system (also for Newton's
Law) since there is no truncation error with the terms in the bracket, it is
possible to evaluate V/V directly from the coordinates and express Y/Y as
Y _ V fdX dt\
Y~V~[dX + dYj' B'7)
Chapter 4 gives the complete set of equations of the HEMP program for
calculating elastic-plastic flow in two dimensions.
2.5 Finite Difference Scheme in Three Dimensions
The difference equations in three dimensions follow the same format as the
two-dimensional problem, but now the area A is replaced by a volume V and
the line contour becomes a surface contour.
dX~v™oJ{s) V
OF y f F(n-j
— = krn^l
dF , f F(n-k)d^
—— = lim '
dZ v"ojis) V
where dA is an element of surface area, (S) the boundary of volume V, and
n the outward normal to the surface. The complete set of equations for cal-
calculations in three space dimensions is given in Chap. 6.
2.6 Finite Difference Scheme for Double Operators
in Two Dimensions
The computer program HEMP was originally formulated to include the equa-
equations appropriate to magnetohydrodynamics. HEMP is an acronym from the
words Hydrodynamics Elastic Magneto Plastic. In addition to the Lorentz
force and magnetic diffusion, thermal and radiation diffusions are included.
The problem requires evaluations of the double operator, V x V x V, where
V is a vector function and the double operator V • VV, where V is a scalar
function. The physical equations and the constitutive laws are arranged so
that the finite difference equations implicitly conserve the physical properties.
The procedure is the same as described before; quantities are changed due to
a flux through a control volume. Referring to Fig. 2.4, the first derivative is

36 2. Numerical Techniques
evaluated with path (a) and the second derivative with path (b). To main-
maintain time centering a forward difference scheme is used resulting in a series
of implicit equations. The complete details are given in Chap. 8 for solving
problems in magnetohydrodynamics including thermal and radiation flow.
2.7 Grid Stabilization
Fine-scale errors of the order of zone-to-zone fluctuations can appear in cen-
centered difference schemes under certain boundary conditions. These errors are
suppressed by the linear term included in the von Neumann artificial viscosity
for one-dimensional fluid dynamic calculations. However, in multidimensional
calculations non-physical numerical oscillations often occur when a shearing
action is introduced on the boundary of a Lagrange grid. These oscillations
usually grow at a slow linear rate, and are not real instabilities, but nonethe-
nonetheless can lead to large grid distortions. In practical calculations, it is useful to
be able to control the onset of grid distortion due to numerical sources and,
in some cases, even to damp grid distortions due to real physical phenomena
that are not of interest to the calculation.
The addition of an artificial viscosity is a convenient way to control grid
distortions, since it represents a physical process whose influence on a problem
can be readily understood. This is in contrast to methods of damping spurious
oscillations that are implicit in the finite difference scheme.
Fluid dynamics involves large changes in the volume and shape of materi-
materials. Quadrilateral grids in two dimensions and cubic grids in three dimensions
allow large deformations to follow the flow without introducing artificial stiff-
stiffness typical of triangular and tetrahedral grids.
To control unwanted numerical oscillations, a Navier-Stokes artificial vis-
viscosity can be added to the stresses [2.4]. The formulation of the viscosity is
given in Chap. 4 for the two-dimensional problem and in Chap. 6 for the
three-dimensional problem.

3. Modeling the Behavior of Materials
3.1 Introduction
The first requirement in the calculation of problems in mechanics is a formu-
formulation of the material behavior. The material description should include elas-
elastic, elastic-plastic, and hydrodynamic flow. Appropriate yield criteria must
be employed. The literature includes many complicated forms to describe ma-
material behavior, some of which have been developed to aid the mathematics
in the analytical solution of the equations of motion. However, since numer-
numerical techniques are considered here, the equations of motion are completely
independent of equations that describe material behavior, and any mathe-
mathematical form may be used. The objective of the material models is to provide
a theoretical description applicable to a wide class of practical problems, but
using simple idealizations of the outstanding features of the real phenomena.
The problem of greatest present interest pertains to metal plasticity.
Therefore, details for describing elastic-plastic material are presented. The
formulation of this problem provides the framework for more sophisticated
descriptions of material behavior. The mathematics has been organized so
that a departure can be made from the elastic perfectly plastic model with-
without any change to the basic program that solves the equations of mechanics.
Some of the material descriptions presented include dynamic yielding based
on dislocation theory, work hardening, pressure, and temperature effects on
material strength. Incremental plasticity is used so that large deformations
with rotation can be modeled.
3.1.1 Hooke's Law
Only media which have the same material properties in all directions are
considered here (isotropic media).
A perfectly elastic material is characterized by a linear correspondence
between stress and strain. Hooke's law is used to describe the stress at a
point resulting from a strain at this point. The strain itself results from a
force displacing particles in the media. Hooke's law in terms of an incremental
strain resulting in an incremental stress may be written as

38 3. Modeling the Behavior of Materials
V
^2 = A-+2/ie2, C.1)
V
<Js = A- + 2/^3-
Here A and ji are the Lame constants, and ?i, ?2 and ?3 are the strain rates
in the direction given by the subscripts which refer to the principal axes; V
is the volume.
The dot means a time derivative along a particle path. It must be noted
that the time derivative provides a desired ordered sequence for the incremen-
incremental stress-strain relationship, but this does not mean that a rate-dependent
stress-strain relationship has been introduced. Hooke's law used in this way
gives natural strain, which means that the strain of an element is referred to
the current configuration instead of the original configuration.
The stress behavior of a material can be thought of as being composed
of a stress associated with a uniform hydrostatic pressure (all three normal
stresses equal) plus a stress associated with the resistance of the material to
shear distortion.
cri= -P + si,
OT2=-P + 52, C.2)
The stress components due to shear distortion (stress deviators s\, S2, 53)
are defined so that they do not contribute to the mean pressure, P.
P= ^(oi+^+aa), C-3)
where
si + s2 + 53 = 0. C.4)
The usual notation is followed here where stress is counted positive in tension
and negative in compression which is just the opposite for pressure. In general,
a state of stress is described by six components.
Stress (Hooke's Law):
C.5a)
&u = -P + su C.5b)
&ij = s^ for i ^ j C.5c)
-P = K^ C.5d)

3.2 Plastic Flow Region 39
with i,j = 1,2,3. Here iij is the strain rate deviator, V the volume, fi the
shear modulus, and K the bulk modulus = A 4- B/3)/i.
In the x, y, z coordinate system used here the stress deviators are given
by:
/ 1 V\ • • .
sxx = 2ji\exx - -—\ + Sxx; Txy= fi{exy) + 5xy
IV
3
Syy - ~ — + Syy] TZX = /ji(€ ZX) + JZ2; C.6)
11/ .
szz= 2ptrjrezz - - — 1 -f bzz\ Tyz =
where, from the continuity equation,
V
V
_ =ixx+eyy + ezz. C.7)
3.1.2 Rigid Body Rotation
The terms 5 in C.6) are corrections for rigid body rotation [3.1]. See Chap. 4.
^i= —2uzTxy + 2ujyTzx,
byy— -\-2ujzTxy — 2ujxTyx,
&zz— +2uxTyz — 2ujyTzx = —Syy — 8XX,
C.8)
SXy= UZ(SXX - Syy) + UyTyZ ~ UXTZX,
SyZ= UX(Syy - SZZ) +UJZTZX -UyTXy,
SZX= Uy(SZZ - 8XX)+ UXTXy ~ U)ZTyZ,
where ux = l/2[(dz/dy) - (dy/dz)], uy = l/2[(dx/dz) - {dz/dx)], uz =
l/2[(dy/dx) - {dx/dy)].
3.2 Plastic Flow Region
Plastic flow at a point of a material occurs when a certain stress combination
at this point reaches its limiting value. The fundamental relations of plasticity
theory are most simply explained with reference to the principal stresses. In
plasticity theory it is usually assumed that plastic behavior is independent
of the pressure. Therefore, the condition for plastic flow is written in terms
of the stress deviators
*2, s3)=0, C.9)
where si, S2, and 53 are the principal stress deviators.

40 3. Modeling the Behavior of Materials
This expression states that in the principal stress space there is a bound-
boundary condition on the magnitude of the stresses. After this value has been
attained, plastic flow begins. On a loading path prior to reaching the bound-
boundary condition /(si, $2, ss) < 0> anc^ the material is in the elastic region.
Equation C.9) is called the yield condition and it must be independent of
the choice of coordinates.
By perfectly plastic flow we mean that the function, f(s\, 52, 53) = 0,
retains its form during the whole process of plastic flow, i.e., / = 0. This
means there is no strain hardening and that the material flows plastically
under a constant yield stress.
A considerable simplification is obtained by assuming that C.9) is in-
independent of a change in stress sign (absence of Bauschinger effect). The
material thus behaves similarly in tension and compression.
By itself the yield condition is not sufficient to characterize the mechanical
behavior of a perfectly plastic material. It must be supplemented by a stress-
strain relation for the plastic region. Plasticity theories assume that during
plastic flow the rate of plastic strain is at any instant proportional to the
instantaneous stress deviator. Stated mathematically:
e\ = Asi,
el - As2, C.10)
el = \s3.
Here si, 52, and S3 are the principal stress deviators; e\ ?3, ?3 are the cor-
corresponding components of the plastic strain rate deviators; and A is a scalar
plastic flow-rate parameter. This parameter is different for different positions
and different for the same position at different times. It is to be noted that the
stresses s\, S2, and S3 are not rate dependent by C.10) but are proportional
to a rate-dependent parameter through a rate-dependent constant.
This is in contrast to elasticity theory which states that the stress is
proportional to the strain so that stress and strain determine each other.
Here, the stress is proportional to the plastic strain rate so a state of plastic
strain does not correspond to a unique state of stress.
Most theories of plasticity follow the experimental observation that there
is no permanent change in volume due to plastic strain (plastic incompress-
ibility). This can be stated mathematically as
?? + 4 + ?p = 0. C.11)
The total strain, e, is considered to be the sum of the plastic strain, ?p, and
the elastic strain, ee:
e2=el + e%, C.12)
S3 ^^-f^.

3.2 Plastic Flow Region 41
From C.11, 12) it is seen that the plastic strain is already a deviator. The
elastic portion of the strain is recoverable, but the plastic portion is assumed
to be permanent.
Equations C.9-11) express the fundamental assumptions of plasticity the-
theory. These assumptions must, of course, be consistent with the equations that
describe the material behavior. While the yield condition limits the magni-
magnitude of the stresses, there is still an ambiguity as to the implied boundary
conditions between an elastic and a plastic state. At this point, we will in-
introduce the plastic flow rule first proposed by von Mises [3.2] and proved by
Drucker [3.3]. According to the von Mises theory, which allows for more gen-
general behavior than C.9), the principal plastic strain-rate vector ip associated
with a principal stress vector, s, is directed outwards along the normal to the
yield surface at the point cri, cr2, and G3. Thus, if /(si,S2>?3) = 0 denotes
the yield conditions then
e- a df
?P=A^- (if / = / = 0) C.13)
p- i df
where A is a constant as in C.10).
In the principal stress space, C.13) corresponds to the gradient of a scalar
point function resulting in a vector. Drucker has demonstrated that the yield
curve must be convex (a curve is convex if it always lies on one side of
the tangent at any point) and that the work done on the material during a
loading and unloading cycle must be positive if plastic changes occur, and
zero if purely elastic change take place.
It will be pointed out later that while the requirements for normality and
convex surfaces are sufficient conditions they are not necessary conditions.
Calculations where the above two conditions are not imposed give stable an-
answers satisfying the laws of physics that are very near to those obtained when
the conditions are strictly imposed. A much greater ease of calculation and
versatility in modeling the physics of material behavior is attained without
these requirements.
3.2.1 Yield Strength
The isotropic and perfectly plastic assumptions stated above are in direct
contradiction to experiments on metals. For example, most metals strain
harden under plastic flow. However, we will formulate the problem with these
idealizations, but allow for a departure so that nonideal properties may be
incorporated.
The yield condition of von Mises is used to describe the elastic limit. In
the principal stress space, the yield condition can be written as

42 3. Modeling the Behavior of Materials
(*i " <r2J 4- (<T2 - a3J + (G3 - (JiJ = 2(Y0J, C.14)
where Y° is a constant which is taken as the stress where yielding occurs
in the simple tension test. It is seen that this equation is independent of
the pressure, P, since P is an additive constant in all the terms on the left
of C.14). The left side of C.14) is proportional to the second invariant of
the stress tensor and, therefore, is independent of the system of coordinates
which is required for an admissible form C.9).
The left side of C.14) can be shown to be proportional to the energy
required to change shape as opposed to the energy that causes a volume
change. The expression states, therefore, that plastic flow begins when the
elastic distortion energy reaches a limiting value, (Y°J/6/i, and that this
energy remains constant during the plastic flow. Thus, by the term "elastic-
plastic" is meant the state whereby the strained material has been loaded,
following Hooke's law, up to a state where the material can no longer store
elastic energy. All subsequent distortion will produce plastic flow, and plastic
work will be done. When the material is loaded beyond the yield strength
and subsequently unloaded, only the elastic distortion energy is recovered.
The work done against the material while in the plastic state is not recovered.
Another way of stating this is that the loading and unloading paths are not
the same when the material has been loaded beyond the elastic limit.
The left side of C.14) can also be interpreted in terms of shear strength.
There are several ways of viewing C.14) but for an elastic, perfectly plastic
material the left side of the expression is equal to a constant obtained from
the simple tension test.
In the single tension test when the stress in the axial direction is a\ and
the boundary stresses are zero, at the elastic limit the stress state is
<r1 = -P + 51=y°,
cr2 = -P + 52 = 0,
cr3 = -P + 53 = 0.
The simple tension test implies two-dimensional flow normal to the loading
direction because in order for the radial stress in the material to match the
zero stress boundary conditions there will be strains in the directions of the
radial stress a2 and the hoop stress &%. The strains in these directions will
result in deviatoric components of stresses that just cancel the pressure P.
Below the yield point the ratio of the strain in direction 2 to the strain in
direction 1 is Poisson's ratio, z/, and the ratio of stress G\ to the corresponding
strain, e\, is Young's modulus, E. Here 1 and 2 are the axial and radial
directions, respectively.
The elastic constants v and E can be expressed in terms of the bulk
modulus K and the shear modulus ji.
__ 3 K - 2fi
" 6K 2'

Yield circle
3.2 Plastic Flow Region 43
i3 Plane: S-j + S2 + S3 = 0
Y0
Fig. 3.1. Geometrical representation of von Mises yield criterion in principal stress
space
Any two constants can be used to describe the elastic behavior of a material.
However, we are interested in describing the behavior of materials beyond
the range of linear elasticity. Nonlinear material behavior will be modeled by
way of the bulk modulus and shear modulus since they do not depend on
the geometry. (Young's modulus and Poisson's ratio are associated with a
particular two-dimensional loading condition).
In <7i, G2 , <^3 space, C.14) describes the surface of a straight circular
cylinder. The axis of the cylinder is equally inclined to the cr1? cr2, cr3 system
of coordinates as shown Fig 3.1.
3.2.2 Von Mises Yield Condition
The von Mises condition C.14) is written as
a — Y° A 1 ^
yj eq — 1 ? yo.ioj
where aeq is the equivalent stress defined as
(V3 ~
In the x, y: z coordinate system of the equations of motion aeq is calculated
from the second invariant of the deviatoric stress tensor J2

44 3. Modeling the Behavior of Materials
Yield surface
= equivalent deviatoric
elastic strain increment
Aep = equivalent deviatoric
plastic strain increment
Ae' = equivalent deviatoric
total strain increment
\i = shear modulus
Fig. 3.2. (a) Intersection of von Mises yield surface with plane si + S2 + S3 = 0. (b)
Schematic of the stress scaling procedure for elastic-plastic behavior. The stress at
(n + 1)* is calculated assuming an elastic strain increment from the stress at n.
The portion of the elastic stress from (n + 1)* to (n + 1) is set to zero and the
corresponding strain increment is counted as plastic strain. The stress difference
between (n+1) and n represents the actual increase in the elastic stress for the
total strain increment AsT
C.16)
(syyJ + (szzJ + 2[{Txyf + (TyzJ + (TzxJ}.
Using these definitions, Fig. 3.2a shows the von Mises yield surface in
the plane s\ + s2 + s3 = 0. In Fig. 3.2a the equivalent stress, calculated
from Hooke's law, is shown inside the yield surface, i.e., aeq < Y°. After an
incremental strain from n to (n + 1)* the components of stress have changed
so that the equivalent stress extends beyond the yield surface. The star on
(n + 1)* indicates a temporary condition of the stresses. Their magnitudes
will be changed by scaling so as to satisfy the yield condition. The stesses
are then renamed as (n + 1) for use in the next time step, see Fig. 3.2a. We
assume the total strain associated with this stress is composed of an elastic
and a plastic component. The plastic component does not contribute to the
stress which is relaxed to the yield surface. The relaxation is achieved by
multiplying each of the six components of the deviatoric stress tensor C.16)
by the scale factor m:
Y°

3.2 Plastic Flow Region 45
This scaling process does not change the direction of the stresses. The total
deviatoric strain increment AeT, is considered to be the sum of elastic, AeE,
and plastic, Aep, components,
AeT = AeE + Aep. C.18)
The stress/strain calculations are carried out in the coordinate system of the
equations of motion as given by C.6). However, to explain the procedure it
is convenient to use principal stress space.
3.2.3 Plastic Strain
The plastic strain increments corresponding to each component of the devia-
deviatoric stress tensor can be calculated by subtracting from the stress at (n + 1)*
the scaled stress at (n + 1) and Hooke's Law.
- s[n+iy - ^(n+1) i = 1,2,3. C.19)
The scaled stress s"+1 is given by
s^1 =msjn+1)\ C.20)
Substitution of C.20) into C J.9) gives
C.21)
Equation C.21) corresponds to the Prandtl-Reuss condition:
—^ = —^- + —^-. C.22)
Thus, implicit in the method is the result that an increment of plastic strain is
related to the corresponding deviatoric stress by a positive constant, namely
it- \-— ll where m < 1.
It is seen from C.21) that the plastic dilatation is zero since the sum of
the deviatoric stress deviators is zero:
Ae\ + Ae\ + Ae\ = 0. C.23)
The increments of plastic strain can be integrated to give the equivalent
plastic strain ep.
-yyj ~ y^yy °zz) ~ \°zz cxxJ v '
The incremental plastic work AWP is calculated from the product of com-
components of the deviatoric stress tensor with the corresponding components of
the incremental plastic strain tensor.

46 3. Modeling the Behavior of Materials
C.25)
where p is the local mass density.
The plastic work is always nonnegative since from C.21) the stress appears
squared for each component of C.25). Thus, the increment of plastic work is
zero or positive for a loading or unloading cycle.
When one does not require the components of plastic strain C.21), a sav-
saving in computation time is realized by calculating the equivalent plastic strain
?p directly. The ratio of an increment in equivalent stress to an increment in
the equivalent strain deviator is 3/x. Referring to Fig. 3.2b,
C.26b)
Equation C.26a) is obtained from C.18) with
(n+l)*
n
feq
4? ^ ^
3/i 3/i
The equivalent strain ep is obtained by summing the increments Aep.
The incremental plastic work can be determined from the product of the
incremental equivalent plastic strain with the equivalent stress.
3.2.4 Tresca Yield Condition
As an example of the versatility of the method consider the Tresca yield
condition
si - S3 = c° = constant. C.27)
The principal stress deviators Si are assumed to be strictly ordered,
51 > 52 > 53. C.28)
After advancing the strain tensor to state (n 4-1)* (Fig. 3.3), calculate
If c(n+1) < c°, the stresses are left as they are and tagged with the superscript
(n + 1). If c(n+1)* > c° scale all of the components of the deviatoric stress
tensor by m = c°/c^n+1^, i.e., s^+1 = ms^+1^ . The plastic strains are
calculated as before. The parameter c° = constant in C.27) corresponds to
an elastic perfectly plastic model. A more general description of material
behavior is obtained by replacing c° with the flow stress Y described in the
next section.

3.2 Plastic Flow Region 47
(n + 1)*
Fig. 3.3. Tresca and von Mises
yield assumptions in principal stress
space. Hexagon: Intersection of the
Tresca yield surface with the plane
5i + S2 + S3 — 0. Circle: Intersection
of the von Mises yield surface with
the plane s\ 4- 52 + 53 = 0
2/3 Y°
i
-S1
/
Tension
i Compression
A B
A2
Slope
D/3) H
(b)
i
P
i
0
E1
^-
Slope
-P
K
(c)
or
P
~CTi
A .
0
*—
2/3 Y
*?- P
Slope (A. 4
Fig. 3.4. (a) One-dimensional strain for an elastic perfectly plastic material. Point
A is the elastic limit; for strains beyond this point, plastic flow occurs. For strains
between points O and A, the loading and unloading paths are the same. For strains
beyond point A, the unloading path is along BC. (b) Pressure P versus strain E\.
(c) Total stress -T\ showing the yield point A and the offset, 2/3y°, from the
pressure

48 3. Modeling the Behavior of Materials
It is seen that A) the Drucker postulate can be satisfied without the
requirement for normality and B) when a nonassociated flow law is used,
any shaped yield surface, even those containing corners or cusps, can be
utilized as required by experiments.
For the special case of one-dimensional strain, the von Mises and Tresca
yield conditions give the same result, shown graphically in Fig. 3.4. Point A is
the elastic limit; for strains beyond this point, plastic flow occurs. For strains
between point O and A, the loading and unloading paths are the same. For
strains beyond point A, the unloading path is along BC.
3.3 Flow Stress
The concept of perfect plasticity with a constant yield point is a very useful
idealization of the behavior of metals. The formulation provides a framework
that can be used as a starting point to include material descriptions that more
closely follow observed behavior. For real materials the yield point does not
remain constant but may change with plastic work, temperature, pressure,
and time. Since the combined stress where plastic flow begins changes with
continued loading, it is customary to use the term flow stress rather than
yield stress.
The term constitutive relation will be used for equations that describe the
flow stress. The forms of the constitutive relations are based on judgment in-
induced from experiment. It is convenient to use forms that are consistent with
micro-mechanical models of material behavior. However, we are only trying
to describe the outstanding features of the real phenomena. The object of the
constitutive relations is to describe the behavior of the present experiment
and predict the results of the experiment not yet performed.
In principal stress space @*1, cr2, cr3) it has been assumed up to now with
the von Mises condition that the yield surface is a cylinder. The scaling
procedure does not change the ratio of the stress deviators and since the
scaling is along the radius of a cylinder the normality condition is always
satisfied. The result corresponds to an associated flow law, i.e., the plastic
strain increments are associated with the yield surface.
Strain hardening is included by expanding the yield surface as a function
of equivalent plastic strain, Y° = Y(ep). (We are only considering direction-
ally independent yield surfaces). When strain hardening occurs, the yield
surface is not constant during a given strain increment. The conical surface
representing a material that strain hardens is approximated by a cylinder
with an incrementally increasing diameter as shown in Fig. 3.5. The error
accrued has been found to be negligible [3.4] as would be expected since the
increments of strain are small. (The increments of strain are controlled by
the stability conditions for integrating the equations of motion).
It is noted that the yield surface is located unambiguously by the scaling
method. Any yield condition which can be expressed in terms of invariants of

3.3 Flow Stress 49
Approximation to conical
yield surface
Conical yield surface
for a strain hardening
material
Axis of cone in [111] direction in (a-j, cv>, CJ3) space
Fig. 3.5. True yield surface of a material that strain hardens and the approximate
yield surface used in the stress scaling computational scheme [pictured in {<j\, 02, C3)
space]
the stress tensor can be used. The yield surface may have any shape including
corners and cusps. With this radial scaling method the relationship between
the incremental plastic strains and the corresponding stresses C.21) remains
unchanged for any yield surface. Drucker's postulate is still satisfied. When
the yield surface is not normal to the radius vector, a nonassociated flow law
is implied by the scaling procedure. The above results become clear when
it is noted that the radial scaling method locates a von Mises surface that
intersects the actual yield surface. An implied plastic potential flow rule is
applied to the von Mises surface by the scaling procedure. The fact that
the increments of plastic strain are not normal to the yield surface is of no
consequence. There is no experimental evidence to support this requirement.
It is, of course, required that the plastic flow process be dissipative, which
introduces irreversible thermodynamics into the mathematical formulation of
plastic flow. This result follows by the manner the total strain is partitioned
into elastic and plastic components. The stress increments corresponding to
part of the elastic strain increments are set to zero by scaling the stresses.
The corresponding strains are counted as plastic strains C.18). The remain-
remaining stress increments correspond one to one with the elastic strain increments
from the linear stress-strain relationship. This process is obviously dissipa-
dissipative since the capacity to do work for part of the strain has been lost. The
dissipation is quantitatively described by C.25).

50
3. Modeling the Behavior of Materials
3.3.1 Strain Hardening
A power law form of strain hardening is suggested by experimental results
and has proven to be satisfactory for many materials of interest:
Y =
C.30)
An iterative procedure for determining the coefficients is described in Ap-
Appendix C.
Equation C.30) together with the von Mises condition aeq = Y describe
isotropic hardening. The yield surface expands uniformly as the equivalent
plastic strain increases independent of the loading path, i.e., the yield sur-
surface is directionally independent. Experimental evidence does not support
isotropic hardening since the yield surface does not expand uniformly. Some
materials after being strained in tension will start plastic flow at a much re-
reduced stress level when the load is reversed to compression. This anisotropy
of strain hardening is known as the Bauschinger effect.
3.3.2 A General Form of Strain Hardening
The flow stress of materials measured in simple tension is usually different
when measured in pure shear [3.5]. For most applications of materials, the
stress state is neither simple tension nor pure shear. For these applications
a strain hardening function that depends on the state of stress is required.
Defined in Fig 3.6 is a parameter, A, that characterizes the stress field. The
parameter provides a measure of the asymmetry of the stresses. The magni-
magnitude of A ranges from 1 to 0 as the stress changes from simple tension, termed
A = max -± , -?
j = principal stress deviators
Uniaxial tension
1
S2 = S3 = - -—- S-j
Torsion
S-j = - S3
S-, = 0
Stress state
Uniaxial compression
Si=S2 = --j-S3
Fig. 3.6. Definition of the parameter A that characterizes the asymmetry of the
stress field

3.3 Flow Stress
51
Simple
tension
Si
Fig. 3.7. Intersection of the yield sur-
surface implied by C.31) with the plane
si -f S2 + 53 = 0. The short arcs are
portions of von Mises surfaces
symmetric loading, to pure shear, termed asymmetric loading. A general form
for a strain hardening function that is convenient for numerical calculations
and accounts for the stress state [3.5] is
Y = YT(ep)Ax + Ys(ep)(l - Ax). C.31)
Here Yy is the strain hardening function determined in simple tension and
Ys the strain hardening function determined in pure shear. Equation C.31)
has been found to be an effective way of expressing experimental data and
extending the usefulness of the von Mises approach to plasticity. The param-
parameter A serves to interpolate the yield surface between experimental data from
simple tension and pure shear tests. The tension test of flat plates [3.5] can
be used to determine A since the stresses for this test are a combination of
tension and shear. Figure 3.7 shows the intersection of the yield surface given
by C.31) with the plane S\ + S2 + S3 = 0. Every position on the intersec-
intersection curve corresponds to a von Mises circle with radius provided by C.31).
The fact that cusps are present, Fig. 3.7 is of no particular consequence.
The method of satisfying the yield condition, as explained earlier, is always
dissipative. Referring to Fig. 3.7 it is seen that an incremental change in
strain has changed the stress state from radius an to crn+1. The structure of
the yield condition has been passed over and the stress state of crn+1 is as
though the yield surface was flat between an and <7n+1. This could represent
a loss of accuracy in the calculation. Actually, the stability conditions of the
finite difference equations prevent large increments of strain from occurring
between consecutive cycles. With small increments of strain the details of
the flow stress function can be followed. The range of validity of C.31) is, of
course, confined to the region of the fit to the experimental data. Physically
plausible results are predicted by C.31) for stress states outside the region.
These states should be verified by experiments if they are important to the
problem.

52 3. Modeling the Behavior of Materials
For many problems the loading is in one general radial direction in stress
space. This is true for dynamic processes such as impact or explosively driven
materials. Other examples where the load is in a radial direction are the
various engineering tests that study fracture. Computer simulation of ex-
experiments involving mostly radial loading requires less information for the
development of a flow stress function compared to applications where a large
portion of the yield surface is interrogated.
Strain softening where the yield surface contracts can occur as the tem-
temperature increases. It has been observed experimentally that the flow stress
can increase with pressure. The effects of plastic strain ep, temperature T
and pressure P, can be incorporated into a constitutive relation for the flow
stress Y. When the flow stress changes as a function of state variables the
shear modulus must also be considered.
Equation C.32) below gives the Steinberg-Guinan constitutive model [3.6]
for the flow stress. The relation provides power law strain hardening, a linear
increase of flow stress with pressure, P, and a linear decrease with tempera-
temperature, T. The flow stress is zero when the melting temperature Tm is attained.
Flow stress, Y
Y = [y°(l + f3sp)n] [1 + bPVi - h(T - 300)], C.32)
where ?p is the equivalent plastic strain and V the relative volume, with the
conditions:
y°(i + /3?p)"<ymax,
Y = 0 for T > Tm; Tm = TmOvi exp[27o(l - V)].
Shear modulus fi
/i = /io [1 + bPV* - h{T - 300)].
Here Y°, Ym3iX, /3, n, /i0, b, h, Tm0 and 70 are material constants.
Appendix C outlines a procedure for determining work hardening param-
parameters.
3.4 Rate Dependent Yield Models
3.4.1 Maxwell Solid
The above formulation of the plasticity problem provides the framework for
introducing easily other models of material behavior. A simple viscous mo.del
that has a direct physical interpretation is the Maxwell solid. In this model
the total strain rate iij is the sum of two components: the elastic component
i*j and the viscous component ?^ . We will assume that the viscous behavior
applies only to the deviatoric stresses Sij

3.4 Rate Dependent Yield Models 53
The strains are the deviatoric strains, \x the shear modulus as before, and 77
the coefficient of viscosity. A dot over a symbol represents a material time
deviative.
Rewriting C.33) gives C.34) which describes the stress relaxation of a
Maxwell solid:
s = 2/ie - -s. C.34)
V
The finite difference approximation is given by:
_ a
For clarity the tensor indices have been omitted. Equation C.34) states that
the stress increases as the elastic strain increases, but is relaxed by an amount
proportional to the current stress.
Equation C.34) is realized by way of the von Mises yield condition
aeq -ay<0. C.35)
Here <7eq is the same as previously defined and ay is the flow stress with
l " -At] . C.36)
Here At the time increment corresponding to the strain and stress increments.
When aeq > ay the components of the deviatoric stress tensor are scaled by
m = ^- = 1 - -At = 1 -,m > 0. C.37)
The physical interpretation is that ay represents the material strength. Maxi-
Maximum and minimum boundary values placed on can be interpreted as changes
in the coefficient of viscosity rj:
Ymin <°y< Ymax. C.38)
Stresses falling below ymin correspond to rj = oo, i.e., no relaxation with the
result that the material is described by Hooke's Law in this stress region.
Figure 3.8 shows calculated stress profiles resulting from the impact of two
plates. The calculations compare the results for an elastic perfectly plastic
and a Maxwell solid model used to describe the behavior of the plates.
3.4.2 Dislocation Theory
Following dislocation theory, the maximum stress the material may attain
is considered to be the actual applied stress minus the component of stress

54 3. Modeling the Behavior of Materials
CO
.a
x
X
b
M
M
50 ;;
u
**:
S.D.
Target
i
u
M
M
U
J
1.0
1.0
50 i
i
o 4
S.D. Target
1.0
1.0
Fig. 3.8. Calculated stress profiles from a shock driver, S.D., striking a target at
0.7km/s. Materials: aluminum. Top: Elastic perfectly plastic model, flow stress,
Y° = 3kbar. Bottom: Maxwell solid model. The following parameter values are
used: s = 2/xe - {^/r))s] rj — 0.05Mpoise for axx > 1 kbar; 77 = 00 for axx < 1 kbar;
Vmax = 3 kbar; bulk modulus: 0.73 Mbar; shear modulus: 0.248 Mbar; density:
2.7-**
corresponding to plastic flow. The stress relaxation according to dislocation
theory is
5 = 2/i? - 2/X7. C.39)
Here the stress increases as the elastic strain increases, but is relaxed by the
plastic strain rate 7, which is given by dislocation theory. Equation C.39) is
realized by defining the flow stress ay of C.35) as
'eq
At
The scale factor becomes
m= 1
At , m > 0.
C.40)
C.41)

3.4 Rate Dependent Yield Models
55
It is seen from C.35, 40) that the stress increment corresponding to the strain
increment jAt is set to zero by the scaling method. The plastic strain rate 7
can be defined as
j = bNW, C.42)
where b is the Burgers vector. The mobile dislocation density N and the
average dislocation velocity W are functions of the plastic strain and the
applied stress. Equations C.43, 44) give simple models for N and W due to
J.J. Gilman [3.7]:
N = No
C.43)
C.44)
Fig. 3.9. Calculated stress profiles from a 0.25 mm thick shock driver, S.D., striking
a target, velocity 0.16km/s. Materials: iron with a dislocation model, E.P.^elastic
precursor. The following parameter values are used: s = 2 fie - 2/ry; 7 = bNW;
N = 108 + 2 x 10n7; 6 = 2.5 x 10~8; W = 0.32exp(-D/Vd); D = 0.015Mbar;
<Jd = creq - crmin; crmin = 0; bulk modulus: 1.74 Mbar; shear modulus: 0.814 Mbar;
density: 7.85 g/cm3

56
3. Modeling the Behavior of Materials
where No, a, D, and C* are material constants. The dislocation driving stress,
crd, is taken as the equivalent stress in excess of a background stress amin:
<7d = <Teq " <7min; °d > 0; <7min = Y°A + 0-y)n. C.45)
Here Y°, /?, and n are material constants determined from tension test ex-
experiments (Appendix C).
In the above formulation 7 is the equivalent plastic strain ep.
In terms of dislocation theory, the dislocations in motion and hence the
plastic strain rate 7 are postulated to be zero until the applied stress exceeds
the background stress. Equation C.39) reduces to Hooke's law when aeq <
0"min-
Figure 3.9 shows calculated stress wave profiles for the impact of two
plates with material behavior described by dislocation theory. The amplitude
of the elastic precursor, which is a measure of the yield stress, decays as the
stress wave advances.
The observation of a rate effect requires measurements at two different
times. For the geometry corresponding to Fig. 3.9, this is accomplished by
detecting the plastic precursor amplitude at two different target thicknesses.
Time = 0.
Time = 20 microseconds
L°=3cm
Time = 40
-
microseconds
1111111111 r
(H)H
Illllllllll
Time = 60 microseconds
11111 If
Fig. 3.10. Calculated time sequence of the impact of a cylinder, described by a
dislocation model, on a rigid boundary. Cylinder length to diameter ratio 3. Original
length L° = 3 cm. Impact velocity 0.3km/s

3.4 Rate Dependent Yield Models 57
3.4.3 Flow Stress Measurements
The impact of a cylinder on a fixed barrier is a convenient way to assess
the flow stress of the cylinder by comparing the final length of the cylin-
cylinder to the original length, [3.8, 9]. For the same impact velocity the strain
rate is increased as the linear dimensions of the cylinder are decreased. Fig-
Figure 3.10 shows the calculated shape of a cylinder that impacts a rigid barrier
at 0.3km/s. The original cylinder length to diameter ratio is three with the
original length L° = 3 cm. The cylinder material behavior is described by the
dislocation model using the parameters given in Fig. 3.9 except strain hard-
hardening is introduced by way of crmin- In place of crmin = 0 as shown in Fig. 3.9,
0"min = A + 30(tepH 3 kbar. The results given in Fig. 3.10 are identical to a
similar calculation without the rate dependence provided by the dislocation
model, but which did include the same strain hardening. As the impact ve-
velocity increases the strain rates increase correspondingly. A high strain rate
causes a rapid decay of the flow stress to the background stress, am[n> For
the calculation given in Fig. 3.10 the decay time is short compared to the
time scale of the event, i.e., the time required to decelerate the L° = 3cm
cylinder. The net result of the dislocation model at high strain rates is strain
= 3 cm
; = 0.58
= 0.3 cm
: = 0.60
Fig. 3.11. Comparison of original length L° and final length Lf for cylinders with
the same conditions as Fig. 3.10 except for L°

58 3. Modeling the Behavior of Materials
Time = 0. Time = .0005 microsecond
Time = .0015 microsecond
Time = .0025 microsecond
I
Fig. 3.12. Calculated time sequence for the cylinder shown in Fig. 3.10, but with
original length L° = 3 x 10 cm
hardening. Thus, paradoxically, the higher the strain rate the less important
is the rate dependence of the material. (However, it should be recognized that
strain rate material behavior could be important in low strain rate loading).
Reducing the linear dimensions of the cylinders reduces the event time.
Figure 3.11 compares the final shape of the calculation of Fig. 3.10 with
calculations using different scaled dimensions. It is seen that the ratio of the
final to initial cylinder length, L{/L°, varies from 0.58 to 0.63 for a scale
change of 10~2. When the scale factor is reduced to 10~4 the cylinder does
not deform and rebounds as an elastic body, Fig. 3.12. For this scale factor
the time for the decay of the flow stress is longer than the time for the stresses
in the cylinder to be released by rarefaction waves. Thus, during the event
time the cylinder behaves as an elastic body with no yield stress. If it can be
assumed that the decay rate of the flow stress used in the calculation shown
in Fig. 3.9 is reasonable, then it will require an inordinately large reduction
in dimensions to see a strain rate effect for impact problems similar to the
problem shown in Fig. 3.10.

3.5 Upper Yield Point
59
3.5 Upper Yield Point
The upper yield point that is seen for some materials in tension tests can be
modeled very simply by adding the term L to the flow stress:
Y =
L,
C.46)
with
L = (a-bep),L>0.
The upper yield point becomes Y = (Y° + a) when the lower yield is Y°.
Figure 3.13 shows the calculated stress profiles for colliding plates where
the upper and lower yield points are 9kbar and 4.5kbar, respectively. The
parameter b in C.46) is 4.5 x 103. (The lower yield point is reached when the
plastic strain, ?p, reaches 0.1 %.). At the yield point for this one-dimensional
3
55
100
50
S.D.
Target
4.5
i
4.5 kbar
)kb \
ft
T
9 kb
1
t1-
100
Radius Radius
1K0S-5 Time = 0.4011 Cycle =139 1KOS-5 Time = 1.6033 Cycle = 529
3mm 12mm
-^.^ ^
3
?
55
50
S.D.
Target
Radius Radius
1KOS-6 Time = 0.4032 Cycle = 139 1KOS-6 Time = 1.6112 Cycle = 526
Fig. 3.13. Calculated stress profiles for a shock driver, S.D., striking a target plate,
velocity: 0.4km/s. Top: Steel with upper yield 9 kbar and lower yield 4.5 kbar.
Bottom: Same as top but with a standard deviation of the upper yield of 0.5,
Gaussian distribution. axx —; ayy

60 3. Modeling the Behavior of Materials
geometry the difference of the stress in the direction of motion, axx, and the
orthogonal stress, ayyi is the yield stress, i.e., oxx — ayy = sxx — syy = Y at
the yield point. It is seen that the deviatoric portion of the elastic precursor
proceeds at a stress corresponding to the elastic limit of the upper yield point.
Behind the precursor in the shock front the deviatoric stresses are at the lower
yield point. The wave is unloaded by the rarefaction from the rear and the
deviatoric stress sxx becomes tensile and the stress syy becomes compressive.
3.6 Nonhomogeneous Properties
Mechanical properties of metals such as the yield point and fracture mecha-
mechanisms are closely related to the heterogeneous deformation which occurs at
various scales. The yield point of crystals comprising a metal, for example,
may have different properties in different directions. The nonhomogeneous
character of the real material permits sites for localization of plastic flow.
Nonhomogeneity can be readily introduced into a calculation by assuming a
Gaussian distribution of some material parameter, the flow stress for exam-
example, and a random number generator to distribute the parameter throughout
the calculational grid.
Figure 3.13 (bottom) shows the same calculation as Fig. 3.13 (top) but the
upper yield point parameter a, C.46), has a standard deviation of 0.5. It is
noted that the shape of the elastic precursor is similar to the result obtained
using a viscous model, Fig. 3.8. In calculations of tension tests Luders' lines
can be simulated by introducing nonhomogeneity into the calculation.
3.7 Hydrostatic Pressure Equation of State
A pressure equation of state that has been very successful for describing the
solid phase of materials is the Griineisen model
P-Po = l(e-eo). C.47)
Here P is the pressure, V the specific volume, e the internal energy, and 7 the
Griineisen parameter. The zero-Kelvin states are designated as Po> eo- The
pressure P and energy e are functions of volume and temperature while Pq
and So are functions of volume alone. The zero-Kelvin curve can be evaluated
using the Hugoniot as a reference curve [3.10]
Pu - Po = l[eu - e0]. C.48)
Here Pu and ?h &re obtained from shock wave measurements. The parameter
7 can be derived from double shock experiments. For many materials a linear
relationship between the shock wave velocity, Us, and the particle velocity,
Up, has been found from shock wave measurements,

3.7 Hydrostatic Pressure Equation of State 61
Us=c + sUp. C.49)
With this relation and the Hugoniot equations [3.10] the pressure on the
principal Hugoniot can be written as:
where x = 1 — V/Vq. The corresponding energy is
en = ^-(V0-V) = ^. C.51)
A good approximation of the Grlineisen parameter is
Equation C.48) is now a differential equation for e0 [3.10]
7o , de0 poc2x / 70
? + + I1
C-53)
as follows from C.50-52) and the definition of pressure, Po = —deo/
A power series expansion in x can yield sufficient accuracy for pressures
up to several hundred kilobars for metals and other engineering materials:
C.54)
p°= ~w = v ^Ol + 2?o2X)
The temperature T can be obtained by assuming a constant heat capacity
Cv = SR for the solid,
?-e0 = I CvdT = 3RT,
C.56)
3^ '
With the initial condition e = 0 for F/Vb = 1 and T = 300 K, the first term
in C.54) is eoo = — 900/2 The remaining coefficients of x are obtained by
substituting C.54) into C.53) and collecting terms so that C.54) is satisfied
for any value of x [3.10]:
?oo = —Q00R R = gas constant/atomic weight,
,o2 (c
?03 = gD
604 = 24^

62 3. Modeling the Behavior of Materials
The Griineisen equation of state becomes
= poc2 [x + Bs - |) x
where ? = s/Vq is the energy per original volume, Vq = 1/po? x = 1 — V
(here V is the relative volume). The temperature is determined from
Tables 3.1 and 3.2 give equation of state (EOS) and constitutive model
parameters for several materials. The EOS experimental data were obtained
from the Los Alamos National Laboratory report LA 4167-MS, May 1, 1969.
The units in the tables are consistent with computer simulation programs
where the pressure P is in megabars (Mbar) and the internal energy E is
Mbar • cm3/original cm3.
3.8 Modeling Fracture
Most engineering structures contain flaws or cracks. Thus engineering design
often requires evaluation of the maximum flaw size and operating stress level
for safe operation. Large flaws and/or high stresses can lead to crack growth
and ultimately to unstable propagation and structural failure. Knowledge
of the fracture toughness of a material, a measure of its resistance to crack
growth, is required to design against unstable crack propagation.
Small-scale specimens can be used to determine the resistance of a ma-
material to crack propagation, but measurements taken at small scale do not
necessarily coincide with large-scale results. Structures that are large enough
fail by brittle fracture1. In the brittle fracture regime, the failure stress varies
inversely as the square root of the crack size, so that larger geometrically sim-
similar structures will fail by brittle fracture at a lower average stress. Smaller
similar structures will fail at a higher average stress, until a certain size is
reached. For further reduction in size the failure mode changes to ductile
fracture. The material itself may be ductile on a micro-scale but the struc-
structure exhibits ductile or brittle fracture behavior depending on the geometry
and loading conditions.
The main objectives when modeling fracture are to predict where fracture
will occur from small scale tests and to do destructive large scale testing on
1 Brittle fracture refers to plane strain fracture as predicted by linear elastic frac-
fracture mechanics. The micro-mechanism leading to fracture is assumed to be simple
rupture due to micro-void coalescence regardless of whether the macro-scale is
ductile or brittle.

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64
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3.8 Modeling Fracture 65
the computer and not the engineering structure. To do this the model must
be able to describe A) how cracks form and grow from increasing loads or
displacements, and B) how cracks initiate and grow from existing cracks and
notches. An example of the first case is the tension test where a material is
loaded to failure without the existence of an initial crack on a macro-scale.
The second case refers to initiation of cracks from an existing macro-crack
as in fracture toughness testing. The observed geometric size effect must be
included in the material model.
The major difficulty in past attempts to develop a model consistent with
the above requirements has been the lack of knowledge of the stress-strain
state of the material for arbitrary loading conditions and geometries. With
powerful computers and two- and three-dimensional simulation programs,
the stress-strain states at the locations where fracture is observed experi-
experimentally can be calculated. This information coupled with micro-mechanical
understanding of fracture can in principle be used to develop an overall model
to predict initiation and propagation of fracture. The model should be able to
describe fracture initiation and propagation that is independent of specimen
size and to describe the material behavior up to the limit of load-carrying
ability.
3.8.1 Fracture Toughness Testing
Plane strain fracture toughness testing, in which a deliberately introduced
crack is loaded in tension, attempts to evaluate fracture behavior when the
hydrostatic tension is large compared to the flow stress. This represents a
worse-case condition where catastrophic failure can occur at small strains.
When the material can be considered a linear elastic solid the stress profile
near the crack tip from an externally applied stress a^ has the form
K
°* = T*
Here x is the distance from the crack tip, ay is the stress perpendicular to
the plane of the crack and K is a constant. The constant, called the stress
intensity factor, k\ has the dimensions of stress times the square root of
length, here k\ is defined as
Linear elasticity theory can identify the magnitude of k\ from the externally
applied load a^ and the geometry of the specimen. For a plate with a central
crack [3.11]
w
The crack length is a and the width of the plate is w. Thus the ay stress
field at the crack tip can be determined by way of linear elasticity theory

66 3. Modeling the Behavior of Materials
from external measurements. The test load that corresponds to k\ must be
reached before the intense gradient of the stress ay at the crack tip leads to a
region of plastic strain, since the correlation with the external measurements
and conditions at the crack tip relies on the stress field being linear elastic.
It is seen from the form of the equation relating k\ with the crack length that
unstable crack growth will result when the external load is increased to the
point where fracture initiation occurs. The value of the stress intensity factor
at this point is called K\c- Fracture toughness tests require sufficiently large
specimens and introduced crack lengths to achieve the conditions for unstable
crack growth. Large thickness dimensions of specimens are important because
the normal stress at a free surface is zero. If az is the stress in the thickness
direction, then
oz = -P + szz = 0
at the lateral boundaries of the specimen, where P is the hydrostatic pressure
and szz is a stress deviator. At the elastic limit, szz is limited by the material
flow stress, Y°. As a consequence, the hydrostatic pressure is limited at the
lateral boundaries. As the load on the crack tip of the specimen is increased,
the fact that the hydrostatic pressure is limited at the boundaries affects
an increasingly larger portion of the load-bearing surface in the plane of
the crack. The purpose of plane strain testing is to evaluate the resistance
to fracture of a material under conditions in which the hydrostatic tension
is not limited. The specimen thickness must be chosen large enough that
plane strain conditions exist when fracture first begins. For the reasons given
above, materials with high fracture resistance and/or low flow stress require
large test specimens to achieve the desired fracture conditions. A convenient
parameter for correlating the dimensions of cracked bodies, due to Irwin [3.12]
(see also [3.13]), is
which has the dimensions of length. Specimens with thickness dimensions
and crack lengths that are sufficiently large compared to Lie can maintain
plane strain conditions up to the point of unstable crack growth (brittle
fracture). Conversely, specimens sufficiently small compared to Lie can reach
conditions of macro plastic strains before fracture initiates. In this latter case
fracture can proceed stably as the load is increased (ductile fracture). Thus,
depending on the size of the specimen, a material can behave in a brittle or
ductile manner.
Valid K\q tests require that the minimum thickness of a compact tension
specimen be greater than 2.5 Lie [3.14]. For low- and intermediate-strength
steels Lie ranges from 0.02 to 40 in [3.13]. The method is too conservative
when the required test specimen size is larger than the actual structure;
in this case the structure does not fail by brittle fracture. The problem of
structural design is to determine fracture resistance when both Lie and the

3.8 Modeling Fracture 67
structure are large, and when it is impractical to test at the actual size of the
structure.
The usefulness of fracture toughness testing results from linear elasticity
theory which links external measurements with conditions at a crack tip.
An analogous theory does not exist for fracture initiation and propagation
under conditions of plastic strain. However, computer simulation programs
can provide the link between external conditions and the stress strain field
at the crack tip. In principle the critical stress fields could be determined for
several characteristic types of crack and for the cases in which they have the
most dangerous arrangement with close coupled experiments and computer
simulations.
3.8.2 Spallation
A fracture process that occurs at conditions of high hydrostatic tension with-
without an initial crack is spallation resulting from the impact of two solids.
Compressive stress waves generated at impact reflect from boundaries and
produce tensile stresses within the solids, causing fracture at conditions of
plane strain. There is a geometric size effect similar to that found in frac-
fracture mechanics studies. In small scale experiments, fracture requires larger
tensile stresses, achieved by higher impact velocities, than require in geomet-
geometrically similar large scale experiments. Tuler and Butcher [3.15] showed that
spall experiments could be correlated by a cumulative damage parameter D:
Fracture occurs for
D= [(a- a°Jdt > Dcrit, C.60)
i.e., when the tensile stress a exceeds a threshold stress <7° for a sufficient
time.
A time-dependent material behavior is not necessarily implied by the
results of these experiments. Colliding-plate experiments can produce one-
dimensional strain states that cannot be reached statically because of the
motion of lateral boundaries in static experiments. The same correlation of
experimental dynamic fracture data is obtained when the incremental time
dt in C.60) is replaced by an incremental distance dr divided by an arbitrary
velocity. With this substitution C.60) takes the form a2r = constant or
or1/2 = constant. Thus, fracture from dynamic spall experiments correlates
in the same manner as in static fracture experiments, and a time-dependent
material behavior is not required to explain fracture by spallation. The spall
results and the K\q analysis have in common a one-parameter model that
satisfactorily correlates experimental data for fracture at high hydrostatic
tension. The parameters K\c and Dcrit are not material parameters, since
they serve only to correlate fracture data at high hydrostatic tensions of the
order of the flow stress.

68 3. Modeling the Behavior of Materials
A size effect is revealed by the fact that small specimens sustain higher
tensile stresses than do large geometrically similar specimens. These results
are consistent with the idea that stress or damage of sufficient magnitude
must extend over a definite minimum distance before fracture will begin.
3.8.3 Ductile Fracture
Ductile fracture refers to failure conditions where sufficient plastic flow has
occurred that the material response is no longer linear elastic. Ductile and
brittle fracture are generally treated independently, but they both involve
material separations that are similar on a microscopic scale. Spall recovery
experiments where samples are subjected to increasingly larger tensile stresses
show the existence of holes and hole growth. The same progression is seen
for interrupted tension tests. The fracture conditions for spall and fracture
toughness tests are hydrostatic tensions several times larger than the flow
stress and small strains. The conditions at fracture for the same material in
a simple tension test are much larger strains and hydrostatic tensions of the
order of the flow stress, see Appendix C.
3.8.4 Strain Damage
The hole growth theory and supporting experiments of McClintock [3.16] have
provided a basis for explaining ductile fracture as the result of strain damage
due to the initiation, growth and coalescence of voids. The McClintock theory
has been applied to explain spall as well as ductile fracture in general. Thus,
the hole growth concept can span fracture phenomena that occur from large
deformations at relatively low stresses as in the simple tension test to high
hydrostatic tensions and small deformations as in spall. The model presented
here assumes the initiation, growth, and linking up of holes as the mechanism
of fracture initiation. The actual small scale processes are not described since
the overall objective of the model is to predict fracture on an engineering scale
and not to follow micro-mechanical processes due to stress-strain conditions.
The philosophy here is to use the micro-mechanical descriptions as a basis
for selecting the parameters and form of the macro-scale model of fracture
initiation.
In the McClintock theory hydrostatic tension accounts for the growth of
holes in fracture by spalling in which the loading consists of large triaxial
stress and small strain. Less damage will occur even at large strains if the
accompanying hydrostatic tension field is small as in the tension test. Asym-
Asymmetric strain accounts for the observation that the elongation before failure
decreases as the shear load increases in fracture tests with combined shear
loads. This was noted by Mogi [3.17], who studied the effect of the interme-
intermediate stress on the fracture of rocks. Mogi's data show that tensile fracture
occurs at smaller strain elongations when confining stresses are present that

3.8 Modeling Fracture 69
produce asymmetric strains. To account for this result it is assumed that
holes can link up as a band if the subsequent loading after a hydrostatic ten-
tensile field has initiated holes is shear. The parameter A, described in Sect. 3.3,
is used to characterize the asymmetry of the stress/strain field.
The important parameter for establishing a model for strain damage lead-
leading to fracture is the tensile hydrostatic pressure. The observed size effect can
be incorporated by requiring a critical damage to extend over a critical dis-
distance before fracture initiates. Thus the conditions for fracture will depend
on the gradient as well as the magnitude of the damage function. The stress
intensity factor approach to fracture initiation is consistent with the concept
of a function of the stress spanning a distance without specifying specific
magnitudes.
3.8.5 Damage in Elastic Regime
A damage parameter, De that follows the above tenets is
fracture when De > D2«,
a\ — -P + si (maximum principal stress).
Equation C.61) can be examined in terms of plane strain fracture toughness
testing. With /(cry) = 1/2 dy and a stress field of the form ay = K/y/x, it
follows that:
Dl = (KlcJ/2n. C.62)
At fracture initiation the stress field near a crack tip is: Kic/y/^wx. Thus, a
damage function that can incorporate fracture toughness testing data without
the requirement for a geometry with an existing crack is
De = —A nr-, fracture when De > ?>S,
AgxIAx ~ E' C.63)
D\ — constant.
In applications where a size effect is not considered, the damage parameter
can be taken as the maximum principal stress.
De — <f\» fracture when De > 0"?»
a i — constant.
It will be assumed here that the thresholds for fracture initiation parameters,
D\ and o\, are greater for fracture initiation in a crack free region than in a
region adjacent to an existing fracture.

70 3. Modeling the Behavior of Materials
3.8.6 Computer Simulation of Fracture
The simulation proceeds according to the following steps:
Referring to the sketch, calculate the damage D& for zone "a".
bs
b7
b2
a
b6
b3
bi
h
The above scheme shows zone "a" which is to be tested for fracture,
surrounded by eight zones b^.
where |Grad<Ji| = Max
|Grad<7i|'
a bi
and abi is the distance between centers of zones a and bi.
The difference (<Ti)a - (&i)bi is not formed across the material boundary
or with zones that have fractured.
If (&\)a — @"i)&» < 0? omit bi from search for maximum.
A zone that has fractured is tagged with the time tf, where t{ is the time
the zone met the criteria for fracture. For the fracture of zone "a" above two
geometries are identified:
A) zone "a" is next to a zone bi that has met the fracture criteria as deter-
determined by a finite value of time U;
B) none of the zones bi surrounding zone "a" has met the fracture criteria.
Fracture zone "a" if all of the conditions listed for either A or B are
satisfied and store the time U with zone "a". The conditions for fracture are
listed below:
Geometry A
(i) (t"+l-tf)Vc>^
where U is the time of fracture for zone bz, and Vc an estimate of crack
velocity,
(il) ((Tl)a > <7e>
(hi) DE > D%.
Geometry B
3.
(i) (<7l)a
(ii) DE >
where M is the fracture initiation threshold constant, 1

3.8 Modeling Fracture 71
After the conditions have been met for initiation of fracture of zone "a",
the next requirement is a method to represent the crack in the calculational
scheme. The meaning of a crack or fracture is that two stress-free surfaces
have been created in a local region of the material. Thus the general require-
requirement for the computer program is to create two stress-free boundary condi-
conditions in the continuous calculational grid. This will provide the appropriate
boundary conditions for a tensile fracture, where cracks open, as opposed to
a shear fracture where there may be compression across the fracture plane.
A fraction F is calculated describing partial fracture of a zone, with F — 1
denoting complete fracture and F = 0 denoting integral material.
Calculate fraction F
F" = ^. C.64)
Vc
Here tf is the time of fracture for zone "a", A the zone area, and Vc an
estimate of crack velocity. Furthermore
0 < Fn < 1.
When the fraction F — 1 the equation of state is changed to
(i) Minimum P — 0
(ii) Yn = hPn < Ymax
h and Ym3iX are constants.
The parameter Vc is not critical and does not need to be the actual
crack velocity, which in general is not known in advance. The computed
crack velocity is the result of the load and parameters in the model. The
simplest way to introduce fracture in the grid is to multiply the flow stress
parameter Y and the pressure P when negative, by the factor A — F). This
procedure allows a fracture to run through the grid independent of zone size.
The requirement (i) for geometry A is to prevent fracture initiation from
spurious signals that propagate at grid speed from a zone that has just met
the fracture initiation criteria.
3.8.7 Damage in Plastic Regime
The plastic strain must be included in a damage function that models fracture
at large strains and relatively low hydrostatic tensions. A simple model that
incorporated features already discussed is [3.18]:
Dp = / wiw2d?p, fracture when DP > D°, C.65)
where e is the equivalent plastic strain, W\ a hydrostatic-pressure weighting
term given by
+aP

72 3. Modeling the Behavior of Materials
Set Dp = D° when P < —I/a. w2 is an asymmetric-strain weighting term
w2 = B - AH,
and
$2 $2 \
—, — I , S\ > S2 > 53.
Here P is the hydrostatic pressure; s\, s2, and S3 are the principal stress
deviators; and a, a, /?, and D° are positive constants. The parameter A
ranges from 0 to 1; we call the stress field symmetric when A — 1 and
asymmetric when ^4 = 0. These limits correspond to the loading conditions
for the simple tension and the torsion test, respectively. The procedure for
relaxing the pressure P and flow stress Y is the same as described above.
Figure 3.14 shows an application of the procedure for a plate pulled in
tension. A 5% variation in the parameter D° was introduced in the grid as
explained in Sect. 3.6. Without this nonhomogeneous property the fracture
will be perpendicular to the load instead of the slant fracture shown.
When the location where fracture will occur is known in advance, sliding
interface logic can be used to create two free surfaces. Figure 3.15a shows
four zones of a two-dimensional grid surrounding point P. When the failure
conditions have been reached at point P (one zone that includes point a must
reach the critical damage and a second zone at least half the critical damage)
the grid is split into two parts. The direction of the split is taken perpendic-
perpendicular to the maximum principal stress of one of the zones (Fig. 3.15b). The
points P and P' are accelerated with free surface boundary conditions. Each
— 38.1
38.1mm
(a)
101.6mm
mmt"
3.175mm THK
FLAT
PLATE
-25.4mm
(b)
:: (c) 11
tUzk
Bfffiffl
Fig. 3.14. Thin plate pulled in tension (plane stress); (a) initial geometry; (b, c)
calculational grid at later times. The short lines in the grid designate zones that
have fractured

3.8 Modeling Fracture
73
IV
(a)
(b)
Fig. 3.15. Grid for
representing fracture;
(a) before fracture
conditions are reached;
(b) grid split with
free structure bound-
boundary conditions applied
to points P and P'
acceleration is multiplied by the factor F. Prior to splitting the zones the
acceleration of point P is calculated in the usual manner.
Figure 3.16 shows an example of this technique for a steel projectile per-
perforating an aluminum target.
Figure 3.17 shows another example applied to the cutting tool problem.
When the exact location of the fracture is not known in advance, more than
one slide line can be used in the vicinity where fracture is expected.
Figure 3.18 shows a fracture toughness test modeled on the HEMP 3D
time-dependent simulation program. The specimen is loaded by applying a
-0.3
0.3 0.6 0.9
Fig. 3.16a—d. Simulation of a steel rod penetrating an aluminum plate. Impact
velocity: 0.08 cm/us

3. Modeling the Behavior of Materials
Fig. 3.17. Simulation of a machining operation with a damage model to describe
the fracture of the work piece. A: aluminum work piece; B: AI2O3 insert; C: steel
small velocity at the load pin position. When the damage criteria have been
met for points on the plane of symmetry, free surface boundary conditions
are applied. (For points on the plane of symmetry the normal component
of acceleration is zero). The resulting accelerations are multiplied by F as
described above. This procedure allows the crack to propagate smoothly along
the symmetry plane as the factor F changes from zero to one.
To validate a model that incorporates a size effect, scaled experiments
are required. Figure 3.19 shows several fracture toughness specimens that
are geometrically scaled so that fracture initiation spans the elastic to plastic
regimes. By scaling a single calculation can be done to simulate all of the
geometries. Fig. 3.18b shows the calculated conditions at fracture initiation
corresponding to a specimen thickness B — 0.5in.
For materials that are known to fracture in the linear elastic regime the
simple damage function D = a\ can be used by setting D^ = 0. Typical values
of a 1 for ceramics range from 1 —> 5 kbar. Figure 3.20 shows an application
of the fracture model for aluminum oxide with crj = 3 kbar.

3.9 Equation of State of Explosive Detonation Products
75
(a)
(b)
140
120
100
80
60
40
20
0
-20
-40
» i- . i I i i i « I . . . i
\\_ayy = -P+Syy
\\
-pj\
x = 0.382 in.^^
.... 1 ,...!....
1 ' ' ' '
-
-
Tension "
compression
0.1
0.2
x (in.)
0.3
0
1.0
0.8
0.6 la
O
0.4 ^
0.2 ^
0
-0.2
.4
Fig. 3.18a,b. Computer simulation of a compact tension test of aluminum 6061T6.
(a) Calculational grid and dimensions in inches. Specimen thickness, B = 0.5 in.
(b) Stress profiles at fracture initiation, x — distance from tip of fatigue crack
3.9 Equation of State of Explosive Detonation Products
In order to calculate the motion of systems that contain a high explosive
a knowledge of the equation of state of the products of detonation is re-
required. Ideally a thermodynamic approach that describes the chemical pro-
processes would provide the desired equation of state. However this approach has
led to equations of state, when used with calculations of explosively driven
systems, that deliver up to 20% more energy than observed experimentally.
In Ref [3.19] a technique is described in which experiments and computer

76 3. Modeling the Behavior of Materials
Fig. 3.19. Scaled aluminum fracture toughness specimens
simulations are used to locate the Chapman-Jouguet (CJ) adiabat that is
consistent with experiments of explosively driven plates. Spherical geometry
was employed to ensure one-dimensional flow and avoid uncertainties from
lateral effects.
The motion of metal spherical shells accelerated by explosives was accu-
accurately measured. Voids placed between the explosive and the metal permitted
the pressure-volume history to be sampled at different positions on the CJ
adiabat. With computer simulation of the experiments and an iteration pro-
procedure a CJ adiabat was constructed that was consistent with all the mea-
measurements. It was established that upon expanding from the CJ point the
pressure drops steeply and then eventually levels off as the volume continues
to expand. The equation of state that gave the best results was found [3.19]
to be
'-?-«¦* (i-&)«*-«">+?. CM)
V — —.
p
Figure 3.21, taken from [3.19] shows the correlation between calculation
and experiments for the high explosive PBX 9404-3. An equation of state
generated by this procedure can be considered a "mechanical equation of
state" as contrasted to an equation of state developed from thermodynamic
principles. Knowledge of the form the CJ adiabat must have can assist in
obtaining a better theoretical understanding. Jones [3.20] developed a the-
theoretical equation of state that contained a similar exponential term to rep-
represent an increased effect of repulsive forces on the internal energy at small

3.9 Equation of State of Explosive Detonation Products
Cycle = +167 Time = +2.652847 Cycle = +440 Time = +3.238295
77
0.32 in.
0.25 in.
Beginning of
fracture
conoid
.30 -cal
steel projectile
(velocity: 2500 ft/sec)
Beginning of
axial crack
Cycle = +693 Time =+4.151106 Cycle =+986 Time = +4.901122
Cycle = +1381 Time =+6.088165
Fig. 3.20. Simulation of a ceramic /aluminum target struck by a sharp steel pro-
projectile
volumes. However, the region of discrepancy between experiment and calcu-
calculation of plates accelerated by explosives was found to be at expansions from
the CJ point at relative volumes from 1 to 2. The form of C.66) was selected
so that the adiabat was integrable to yield the simplest possible equation of
state [3.19]. The exponential term in C.66) is arranged to provide a dip in the
pressure as the relative volume expands from 1 to 2. Figure 3.22a compares
the CJ adiabat of C.66) with the adiabat of a perfect gas.

78
3. Modeling the Behavior of Materials
cm Void
H.E.
Detonator
15.875cm
15.240cm
R
0.635cm
— = calculation
+ • = experiment
A= shot scaled by 4/3 E.06 cm Void)
i i I i
Fig. 3.21. Radius ver-
versus time curves for a 1/4-
inch-thick aluminum shell
accelerated by the high ex-
explosive PBX 9404-3 (Ta-
(Table 3.3). (The four curves
represent 0, 0.5, 2.0, and
3.81 cm voids between
the Al and the H.E.)
10
15
20
25
30
At large expansions from the CJ point the parameters in C.66) that oth-
otherwise fit the data very well yield pressures that are slightly high. A second
exponential term was added by Lee [3.21, 22] to improve the low pressure ac-
accuracy. The modified equation is referred to as the Jones-Wilkins-Lee (JWL)
equation and is given by
P=A 1
~RiV
exp(-^F)
C.67)
Figure 3.22b compares the CJ adiabat of C.67) with the adiabat of a perfect
gas.
With the knowledge of the general form of the CJ adiabat established
from the experiments with spheres, simpler experiments can be used to gen-
generate the coefficients. Measurements of the expansion rate of a metal cylinder
filled with an explosive detonated from one end together with computer sim-
simulations of the experiment provide a means to establish the equation of state
parameters [3.21]. The cylinder test does not necessarily provide a unique
equation of state nor can it yield an equation of state as accurate as can
be obtained from a series of sphere tests. Coefficients for C.67) for several
explosives are given in Table 3.3 [3.23]. Pressures in excess of the CJ pressure

3.9 Equation of State of Explosive Detonation Products
79
(a)
1.000
Fig. 3.22a,b. Comparisons of
CJ adiabats for PBX 9404-
3. (a) Equation C.66), coeffi-
coefficients from [3.19]; (b) Equa-
Equation C.67), coefficients from
Table 3.3
.0001
.1000 1.0 10.0
Relative volume
100.0
(b)
1.000
_ .1000
CO
CD
3 .0100
8
.0010
.0001
PVr = const.
equation C.67)
.1000 1.0
10.0 100.0
are outside the experimental data used to fit the CJ adiabat. For volumes less
than the CJ volume, a perfect gas equation with the CJ gamma is preferable
if other data are not available.
3.9.1 Numerical Calculation of a Detonation
The chemical energy to be releases through the equation of state of the high
explosive is stored in each high-explosive zone as an initial energy E°. The
time for the detonation front to reach a specific zone is calculated in advance
from the known detonation velocity, D, and the distance from the point of
detonation to the center of the zone. The detonation time, designated as t\> is
also stored with each zone. A burn fraction, F, is calculated so as to spread
the burn front over several zones analogous to the artificial viscosity "qn
that spreads a shock over several zones. The pressure is calculated from the
CJ adiabat using the zonal energy and volume and is multiplied by F. The
parameter F is arranged to vary from zero to one. When a large number of
zones is available, which can be the case for one-dimensional calculations, the

80 3. Modeling the Behavior of Materials
Table 3.3. Coefficients for the JWL equation of state C.67)
Explosive
Comp A-3
Comp B Grade A
Comp C-4
Cyclotol 77/23
HMX
LX-01
LX-04-1
LX-07
LX-09-1
LX-10-1
LX-11
LX-14-0
LX-17-0
Octol 78/22
PBX-9010
PBX-9011
PBX-9404-3
PBX-9407
Pentolite 50/50
PETN (Density=0.88)
PETN (Density=1.26)
PETN (Density=1.50)
PETN (Density = 1.77)
Tetryl
TNT
Nitromethane
BTF
DIPAM
EL-506A
EL-506C
Explosive D
FEFO
H-6
HNS (Density = 1.00)
HNS (Density = 1.40)
HNS (Density=1.65)
PBX-9501
PBX-9502
CJ
P0
[^]
1.6500
1.7170
1.6010
1.7540
1.8910
1.2300
1.8650
1.8650
1.8400
1.8650
1.8750
1.8350
1.9000
1.8210
1.7870
1.7770
1.8400
1.6000
1.7000
0.8800
1.2600
1.5000
1.7700
1.7300
1.6300
1.1280
1.8590
1.6500
1.4800
1.4800
1.4200
1.6900
1.7600
1.0000
1.4000
1.6500
1.8400
1.8950
Parameters
P
[Mbar]
0.3000
0.2950
0.2800
0.3200
0.4200
0.1550
0.3400
0.3550
0.3750
0.3750
0.3300
0.3700
0.3000
0.3420
0.3400
0.3400
0.3700
0.2650
0.2550
0.0620
0.1400
0.2200
0.3350
0.2850
0.2100
0.1250
0.3600
0.1800
0.2050
0.1950
0.1600
0.2500
0.2400
0.0750
0.1450
0.2150
0.3700
0.3020
D
[cm]
0.8300
0.7980
0.8193
0.8250
0.9110
0.6840
0.8470
0.8640
0.8840
0.8820
0.8320
0.8800
0.7600
0.8480
0.8390
0.8500
0.8800
0.7910
0.7530
0.5170
0.6540
0.7450
0.8300
0.7910
0.6930
0.6280
0.8480
0.6700
0.7200
0.7000
0.6500
0.7500
0.7470
0.6100
0.6340
0.7030
0.8800
0.7710
r
2.7900
2.7060
2.8380
2.7310
2.7400
2.7110
2.9350
2.9210
2.8340
2.8680
2.8680
2.8410
2.6580
2.8300
2.7000
2.7760
2.8510
2.5130
2.7800
2.6680
2.8310
2.7880
2.6400
2.7980
2.7270
2.5590
2.7170
2.8420
2.7520
2.7190
2.7500
2.5780
3.0920
2.4680
2.8810
2.8040
2.8510
2.6480
^0
["Mbarcm3 1
L cm3 J
0.0890
0.0850
0.0900
0.0920
0.1050
0.0610
0.0950
0.1000
0.1050
0.1040
0.0900
0.1020
0.0690
0.0960
0.0900
0.0890
0.1020
0.0860
0.0810
0.0502
0.0719
0.0856
0.1010
0.0820
0.0700
0.0510
0.1150
0.0620
0.0700
0.0620
0.0540
0.0800
0.1030
0.0410
0.0600
0.0745
0.1020
0.0707
Equation of State C
A
6.1130
5.2420
6.0980
6.0340
7.7830
3.1100
8.3640
8.4810
8.4810
8.8070
7.7910
8.2610
4.4600
7.4860
5.8140
6.3470
8.5240
5.7320
5.4090
3.4860
5.7310
6.2530
6.1700
5.8680
3.7120
2.0920
8.4070
4.2540
3.7380
3.4900
3.0070
3.8240
7.5810
1.6270
3.6650
4.6310
8.5240
4.6030
B
0.1065
0.0768
0.1295
0.0992
0.0707
0.0476
0.1298
0.1710
0.1710
0.1836
0.1067
0.1724
0.0134
0.1338
0.0680
0.0800
0.1802
0.1464
0.0937
0.1129
0.2016
0.2339
0.1693
0.1067
0.0323
0.0569
0.1496
0.0801
0.0365
0.0452
0.0394
0.0664
0.0851
0.1082
0.0675
0.0887
0.1802
0.0954
Ri
4.40
4.20
4.50
4.30
4.20
4.50
4.62
4.58
4.58
4.62
4.50
4.55
3.85
4.50
4.10
4.20
4.55
4.60
4.50
7.00
6.00
5.25
4.40
4.40
4.15
4.40
4.60
4.70
4.20
4.10
4.30
4.10
4.90
5.40
4.80
4.55
4.55
4.00
oeffic
R2
1.20
1.10
1.40
1.10
1.00
1.00
1.25
1.25
1.25
1.32
1.15
1.32
1.03
1.20
1.00
1.00
1.30
1.40
1.10
2.00
1.80
1.60
1.20
1.20
0.95
1.20
1.20
1.30
1.10
1.20
1.20
1.20
1.10
1.80
1.40
1.35
1.30
1.70
ients
w
0.32
0.34
0.25
0.35
0.30
0.35
0.42
0.40
0.40
0.38
0.30
0.38
0.46
0.38
0.35
0.30
0.38
0.32
0.35
0.24
0.28
0.28
0.25
0.28
0.30
0.30
0.30
0.39
0.30
0.30
0.35
0.38
0.20
0.25
0.32
0.35
0.38
0.48
burn fraction can be defined as F = A — V)/(l — Vqj). The burn calculation
is started by setting F = 1 in the zone that corresponds to the point of
detonation. The burn calculation will proceed to around three or four times
the number of zones that the artificial viscosity "g" is spread over before the
detonation velocity and pressure are correctly established. This amounts to
about 16 zones. In two- and three-dimensional calculations there is a limit to
the number of zones for a practical problem. It is usually necessary to have
the correct detonation velocity established immediately. (Very fine spatial
resolution is required to resolve the actual CJ pressure at the detonation front.
However, the equations of motion conserve mass energy, and momentum, and
the fact that the pressure does not attain the CJ value is of lesser importance).
A convenient way to do this is to start the burn calculation at the time the
detonation would reach a given zone as described above. To allow for the

3.9 Equation of State of Explosive Detonation Products
81
possibility of an overdriven detonation that may arise during the calculation
and result in a higher than normal detonation velocity, the burn fraction
F — A — V)/(l — Vcj) can be used in addition to the burn fraction that
is based on the known detonation velocity. The larger of the two is then
selected to multiply the pressure calculated from the equation of state of the
detonation products:
Burn fraction = Fxn+* = tn+l - th/AL.
Fort
n + 1
th
= o,
1 _
1-Vcj "
pn + \ _ maximum of
and F%
If Fn+1 >0.96, set Fn+1 = 1.
Here AL = rAx/D, t is the actual time, t\> the time for a zone to start
burning, Ax the grid spacing, D the detonation velocity, r is a constant
% 2.5, and Vcj is the Chapman-Jouguet relative volume.
Note: When the high explosive is in contact with a high sound speed
metal, it is possible for a signal to run ahead of the explosive. This can lead
to an incorrect ignition of the explosive. In this case the parameter /3 should
be set to zero in burn fraction F2.
Figure 3.23 shows applications to one-dimensional detonations using a
7-law equation of state for the detonation products. Comparison with the
method of characteristic solutions (Appendix D) shows agreement to within
a fraction of a percent. Reference [3.24] gives results comparing the method
of characteristics solution for the two-dimensional problem of a steady state
detonation of a cylindrical charge of explosive with the time-dependent tech-
technique described above. The two solutions agree to three significant figures.
x(cm)
Fig. 3.23. Calculation of a Chapman-Jouguet detonation with a 7-law equation of
state. Explosive originally between 0 < X < 1. Left: Detonation from a free surface
at X = 0. Right: Detonation from a fixed boundary at X = 0

4- Two-Dimensional Elastic-Plastic Flow
The equations listed below are used by the HEMP computer program to
solve problems in elasticity and plasticity in plane geometry or cylindrical
geometry including rotation about the axis of cylindrical symmetry.
The derivation of the equations can be found in Ref [4.1]. The problem is
formulated in Lagrange coordinates with sliding interfaces allowed between
adjacent regions. The equation of state is used in the same manner as de-
described in the preceding sections. There is, however, the additional complica-
complication that the stress-strain relationship must be independent of a rigid motion,
and hence the incremental stress-strain relationship must be corrected for a
rotation in the x, y coordinate system [4.1]. When a zone is displaced from an
initial state of stress, there may be a rotation through an angle uj as well as
a distortion. The rotation will not contribute to an increase in stress, but the
state of stress (s?x,, syy>i T™y) originally in the zone has been rotated through
the angle u. Since the equations of motion are referred to the fixed x-y co-
coordinate system, the totaled stresses must be recalculated in terms of the
coordinate system. The transformation equations [4.2] result in a correction
5 that is added to the stresses. The stresses can then be incremented by the
strain that occurred between time tn and tnJrl to give the stresses at time
\ rotation angle is given by
'dy dx
srnuo = —-— ( —
x dy
4.1 Fundamental Equations
4.1.1 Equation of Motion in x, y Coordinates with Cylindrical
Symmetry and Rotation About the x Axis
dt p I dx dy
D.u)
^ yi
dt p y dx dy y
dy _ 1 \dTxy dSyy Eyy - See] ^ ,2

84 4. Two-Dimensional Elastic-Plastic Flow
y dt p [ dx dy
where Q — y2uj and uj is the angular rotation also given by 6.
Note: For plane geometry omit Txy/y in D.1a) and (Eyy - Eee)/y and
cj2y in D.1b).
4.1.2 Conservation of Mass
dt -"
where M is a mass element.
4.1.3 First Law of Thermodynamics
E=-(P + q)V
+ V{Sxx^xx + syy?yy "+" s60^00 + J-xy^xy +
Here E1 is the internal energy per original volume, V the relative volume
= po/p, and p the actual density: po is the reference density of equation of
state.
4.1.4 Velocity Strains
dx
?xx - 7T-, D.2a)
D.2b)
D.2c)
D.2d)
D2e)
D-20
V
dy
dL
d(yu
dx
dx

4.1 Fundamental Equations 85
4.1.5 Stress Deviator Tensor
/ i v\
sxx = 2/i I ixx - - — 1 , D.3a)
Syy=2ll(iyy-±yJ1 D.3b)
/ 1 V\
O I • I (A O^^
tXy = fliXy, D.3d)
tyo = fiiyo, D.3e)
lex — H^ex, D.or;
where \i is the shear modulus.
4.1.6 Pressure Equation of Stat?.
p = a(r) — 1) + b(rj — IJ + 0G7— 1
where a, 6, c and d are equation of ${$\e constants.
4.1.7 Total Stresses
rM = -(P + q) 4- sxx,
Zee = ~(^ + 9) + 50^-
4.1.8 Artificial Viscosity
/ ds \ d s I
where Co and Cl are constants. Th quantity ds/dt is the rate of strain in
the direction of acceleration, L is a y^€&sure of grid size, a the local sound
speed, and p the local density.

86 4. Two-Dimensional Elastic-Plastic Flow
4.1.9 Von Mises Yield Condition
where Y is the plastic flow stress and 2 J the second invariant of the deviatoric
stress tensor.
4.2 Finite Difference Equations
4.2.1 Mass Zoning
The physical object is divided into zones denned by four points (Fig. 4.1).
The problem is formulated in x, y coordinates with cylindrical symmetry
about the x-axis, including rotation. Plane geometry is described by setting
to zero selected terms. See Refs. [4.3, 4.4] for details of generating problems.
Calculation of the volume of zone Q), vq. Refer here to Fig. 4.1.
(i) Cylindrical geometry
V(D= YaAa + YbAb
a=±[
Note: For plane geometry Ya — Yb — 1.
(Aa)(Q = area of Aa\ {Ab)® = area of Z\6,
i
{Aa)® =- [x2(ys - Va) + ^
(Ab)<x> =^2(y4 -y\)
(ii) Plane geometry
v® = Aa 4- Ab.
- 2/3)],
-2/4)].
j-1 j j + 1
-k + 1
¦k
•k-1
Fig. 4.1. Scheme for mass zoning

4.2 Finite Difference Equations 87
Calculation of the mass of zone Q, Mq.
where po is the reference density of equation of state, V° the initial relative
volume, and i/° the actual volume calculated from the x, y coordinates at
time t = 0.
4.2.2 Equations of Motion
Refer here to Fig. 4.2 for the notation.
i i /\tn r
™jk
5)(tfinv - y?)
D.4a)
- a;?,)
)©W,i - x?v)
- 2/inv)
- J/ii)
D.4b)
IV-
©
• k + 1
Ilk
k-1
- x Fig. 4.2. Scheme for equation of motion

4. Two-Dimensional Elastic-Plastic Flow
n®(y"v - 2/T)
S>(xPv - a;?)
where, at time t = 0, QQjk = [y2u}°jk
+ r" ^
D.4c)
B)
+ 3S ^r
xn+l _ n
a
X
Note: a, C and Q are zero for plane geometry.
4.2.3 Conservation of Mass
yn+l _ P0 yn+l

4.2 Finite Difference Equations 89
Fig. 4.3. Scheme for strains
4
11
3
X
12
4.2.4 Calculation of Incremental Strain
For velocity strains refer to Fig. 4.3 for notation.
u
[?
yy}®
—T [(^2 ~ ^4)(y3 ~ Vl) - B/2
- (x2 -
¦I®
; -2/i) - B/2 -2/-
- (x2 -x4)(x3
n+i D-5a)
2yll+^ '"
D.5b)
D.5c)
D.5d)
¦0
1
4(
0
D.5e)

90 4. Two-Dimensional Elastic-Plastic Flow
= +——r B/2^2 - 2/4^4)B/3 - yi) - B/2 -
94 +2 L
where
The incremental strains are
4.2.5 Calculation of Stresses
Stress deviators:
z\v j
3 \ V )
^it/)/) — I ——— I
®
0
D.5f)
!(D
D.6a)
D.6b)
D.6c)
D.6d)
D.6e)
D.6f)
D.7a)
D.7b)
D.7c)

4.2 Finite Difference Equations 91
® + Fxy)^, D.7d)
K D.7e)
^ l +?. D.7f)
Correction for stress rotation. If a mass element has rotated in the x-y
plane by an angle uj during the time interval /\n+1/2 _ fn+i _ ^ ^e stresses
must be recalculated so that they will be referred to the x, y coordinate
system in their new position.
The following transformation equations can be found in Ref. [4.5]:
Syy = Sxx
cos2 uj + s™y sin2 uj — 2Txy sin uj cos uj
2 ^ + 2Txy sin <*> COS U
Syy
s2o; sin2
D-8)
T^ = r^y(cos2o; - sin2 uj) + (s^x - s^y) cos uj sin uj .
The angle uj given by
9X ^
sinuj =
i (dy dx\
[- ¦»" I •
D 9)
Equations D.8) can be rewritten as:
s'xx = f^yf^ + -^^ cos 2a; - T?v sin 2a;,
s;y = !k±fk _ gSx-C cog 2w + Tny sin 2w>
sn _ ^n
Txy= ^?y COS 2UJ + ^ O V
In the incremental stress-strain relations,
f 1
s^1 = snxx 4- 2/1 [z^x - -
j J
etc.,
D.11)
the stresses 5^x,, 5^, and Txy must be replaced by s^, s'yy and T^y. In order
to preserve the form of D.11), it is convenient to introduce an additive term,
5, to the stresses such that:
t
r
]Aexx
6nxx = s'xx - snxx =
1 / Av\~\n*^
- [—J J + 8nxx, etc. D.12)
(cos2a; - 1) - Tx"y sin2a, D.13)

92 4. Two-Dimensional Elastic-Plastic Flow
%y = T'xy - Tx"y = l?B(cos2w - 1) + (^k^f^j sin2u}
ay
Total stresses
(Z,,)^1 = (s^1 - (Pn+1 + qn+?)®, D.14a)
1 ^ ' 4 D.14b)
D.14c)
where the pressure P is calculated from the equation of state and q is the
artificial viscosity.
4.2.6 Von Mises Yield Condition
o rn+1 / n+l\2 , (sn+l\2 . / ^ + 1
D.15a)
D.15b)
where y° is the flow stress, calculated from a constitutive equation that
describes the material behavior.
If: mn+1 < 1 multiply each of the stresses s?+\ 5?+\ s^\ T?+\ Ty6,
and T0X by mn+1.
If mn+1 > 1 use the stresses as they are for the next time step.
4.2.7 Equivalent Plastic Strain, ep
¦ D16a)
where Aep > 0 and \x is the shear modulus.
(?P)"+! = (?P)" + (Z\?P)"+1. D.16b)

4.2 Finite Difference Equations
93
4.2.8 Artificial Viscosity for Calculating Shocks
A shock process is assumed to occur when the volume of a zone is com-
compressed. The difficulty here is that in two and three space dimensions the
volume of a zone may be compressed due to convergence of the flow [4.6].
This is not a shock process. To describe a shock we wish to know the rate of
change of the volume due to one surface overtaking another surface which is
provided by dx/dx for flow in one space dimension. The equivalent result in
multi-dimensions can be accomplished by calculating the rate of strain in the
direction of the acceleration. Equation D.17) below gives the rate of strain
in the direction of the angle a for two space dimensions.
ds dx
dy
f dx dy
+ ?
— = — cos a + — sin a + -—h— I cos a sin a.
dt ox ox \oy o
D.17)
When the angle a is the direction of acceleration D.17) provides the two-
dimensional analog of the one-dimensional rate of strain dx/dx. If Ax and
Ay are the x and y components of acceleration of a zone, see Fig 4.4, then:
cos a =
sin a =
y
The components of acceleration are taken as the difference in velocity at
two consecutive times.
A characteristic grid length is obtained by calculating a length L in the
direction of acceleration, see Fig 4.5. The artificial viscosity q is given by
2
ds .
— + cLpLa
q =
-7T >0.
D.18)
< 1 i
4 > 4
\
Fig. 4.4. Scheme for calcu-
calculating the rate of strain ds/dt
of a zone defined by points 1,
2, 3, 4. Point c is the zone
center from average of coor-
coordinates 1, 2, 3, 4. Ax and Ay
are obtained by averaging the
respective components of ac-
acceleration of points 1, 2, 3, 4

94
4. Two-Dimensional Elastic-Plastic Flow
4?
Fig. 4.5. Scheme for obtaining a
characteristic grid length L. Here /
is the line through center c in the di-
direction of acceleration; di is the per-
perpendicular distance from point i to
line / (i = 1,2,3,4); and A is the
zone area defined by points 1, 2, 3
and 4
ds dx
Ki»2n , (d± , 92/
— = — cos a+— sin a + I ——h — cos a sin a;
dx
dy
dy dx
P and p are the local pressure and density respectively.
4.2.9 Navier—Stokes Artificial Viscosity for Stabilizing the Grid
In some types of problem it is possible to excite an "hour glass" pattern dis-
distortion in the quadrilateral grid as shown in Fig. 4.6. When the magnitude
of the grid velocities are equal and directed as shown in the figure, no com-
component of the strain tensor is activated. Hence, the artificial viscosity given
above will not damp this particular mode of distortion [4.7].
-X Fig. 4.6. "Hourglass" distortion

4.2 Finite Difference Equations
95
However, this pattern can be prevented from occurring by a q term that
recognizes the change in angle between grid lines.
A Navier-Stokes viscosity is formulated by using triangles to prevent grid
distortion and is referred to as the "triangle q". Figure 4.7 shows the grid for
calculating the acceleration of point j, k. A triangle, q, expressed as a stress
deviator, is formulated for each of the triangles in the four zones surrounding
point j, k. For example, the artificial viscosities for zone ® are given by
D.19)
Qxx — 2//0
Qyy — 2/^0
Qxy = M©4
where
2 .
36xx
2 .
1 .
1 .
and A is the area of triangle @, 1, 2); Cns is a constant w 10~3. Furthermore
po is the zone reference density and V the zone relative volume.
The velocity strains for triangle @, 1, 2) are:
dx
?yy- dy
dy
dy
dy
1 r
1 r
dx
- 2/1 )
4-
2B/1 - 2/2) 4- ?20B/2 - 2/o)j,
) 4 2/12B/1 ~ 2/2) 4 2/20B/2 - 2
?1 - X2) 4 ?20(^2 — xq)] >,
where
etc.,
\
V
k + 1
k-1
Fig. 4.7. Two-dimensional grid for acceler-
-X ating point j, A:

96 4. Two-Dimensional Elastic-Plastic Flow
and A is the area of triangle @, 1,2);
2
A triangle q for zones B), C), and (J) is formulated in the same manner.
The components of the viscosity are added to the corresponding stress in the
equation of motion for point j, k. As can be seen, q has been formulated for
plane geometry. The same q is used for cylindrical symmetry.
The viscous energy w dissipated by the Navier-Stokes viscosity is defined
at a node point j, k, Fig. 4.7. This energy is not included in the thermody-
namic equation of state.
(Aw)jik — ^ z^ vi^xx"xx ' wy^yy ' vxy^xyj^ - D 21)
w ¦
4.2.10 Material Internal Energy
Change in distortion energy, AZ\
D.22a)
4+" =\(sn+1+sn)(Di etc. D.22b)
Internal energy, per original volume, E:
En^ = [En - {Pn+q)(Vn+l - Vn) + AZn+l>}^, D.22c)
PJ+1 = P(^n+I,yn+1H. D.22d)
P(E, V) is the equation of state, and q = l/2(qn+1/2 -f ^n/2).
1 - ^n)]0. D.22e)
[E (P
When the equation of state, P(E, V), has the form P = A(V) + B(V)E, the
energy En+l can be calculated directly:
Pn] +q}{Vn+1 -Vn) + AZn+i D.22f)
1 -V") "

4.2 Finite Difference Equations
4.2.11 Calculation of Time Steps, Atn+3/2 and Atn+X
Ln+l
97
Atn+'i = 0.67-
min of all zones
If Atn+3'2 > l.lAtn+1/2 use Atn+V2 = l.lZ\tn+1/2. Here L is the minimum
zone thickness defined as the zone area divided by the longest diagonal; a is
the sound speed, and
d?
where Co2 and Cl are the quadratic and the linear q constants respectively
and ds/dt is the rate of strain used in the calculation of q.
1 / , 3 , j\
At — — \ At * 4- At * )
2 V / "
4.2.12 Energy Summations (Edit Routine)
Angular kinetic energy [<9KE]:
D.23a)
D.23b)
D.23c)
D.23d)
Linear kinetic energy [KE]:
Internal energy [IE]:
Energy dissipated by Navier-Stokes viscosity:
[NSE
n+l
= \W
™] k.
The total energy in the problem at any time is the sum of equations D.23a-d)
summed over all zones. For cylindrical geometry the mass parameter, M, is
multiplied by 2tt.

98 4. Two-Dimensional Elastic-Plastic Flow
4.2.13 Principal Stresses (Edit Routine)
n+l _
51 -
Qn+1
D-24b)
D.24c)
D.24d)
, D.24e)
with S2 the intermediate stress.
(In the plane x,y and cylindrical coordinate systems used here, s$$ is
already a principal stress).
Direction of maximum principal stress:
i ot1
t-1 xy D.25)
xy
The angle 0 is not calculated when
4.2.14 Calculation of Load, L, on a Given k Line (Edit Routine)
The scheme for this calculation is shown in Fig. 4.8.
crj+i = \sxxsvc?(ab) + Eyy cos2 (ab) - 2Txy cos(ab) sin(a6)l
L J
sin(a6) =
YaJ + (n - Ya
(Xb - Xa)
- xay + {xb - xa
cos(ab) —
-Xa)* + (Yb-Ya)*] k.
For plane geometry, k = 1 and for cylindrical geometry k = 27r[(ya 4- Yb)/2].
Jmax
Jmin
k-*X Fig. 4.8. Scheme for calculation the load on a /c-line

4.3 Boundary Conditions
99
4.3 Boundary Conditions
4.3.1 Fixed Boundary on the x Axis
Phantom zones are created by a mirror reflection across the boundary as
shown in Fig. 4.9
The point j, k can now be accelerated with the equation of motion for a
general point (Sect. 2.2) subject to the following conditions:
D.25a)
M(D = M®, M® = AfB).
The above procedure gives the desired acceleration along the boundary, but
it has the undesirable feature of not allowing for the situation when the point
j, A: is on a free surface since the point has the extra mass of the reflected
zones associated with it. It is more convenient to have the correct mass as-
associated with each point, determined once and for all when the problem grid
is generated, and use different acceleration routines for the case when the
boundary is fixed. Therefore, referring to Fig. 4.9, we calculate (jf-k as
alk =
1
4
1
— <
2
/ PoA7
\ Vn
fr <
1 \Tn
\[*v\
- I +
'0
fAn\]
— 1
\ v
+
0
in\
n s®_
r /
\Tn (
[Xy\
An
D.25b)
The acceleration equation for point j, k that gives the same results as the
equation of motion for a general point with the conditions D.25a) becomes
Wi -*Tu> - CC,)oWn -*fv)}
<4.25c)
+«
,*¦
1 ^^ Reflected zones
Fig. 4.9. Scheme for boundary conditions
on x axis

100 4. Two-Dimensional Elastic-Plastic Flow
LSi.
Fig. 4.10. Scheme for boundary conditions
on y axis
4.3.2 Fixed Boundary on the y Axis
Here we refer to the scheme shown in Fig. 4.10
D.26a)
Analogous to the fixed boundary on the x axis above, the effect of a
reflection about the y axis, subject to the conditions D.26a), is obtained
using the following equations for the acceleration of point j. k:
D.26b)
D.26c)
2^
Z<Pj,k
-(rxny)®B/F, - I/Si) -
4.3.3 Corner Zone on the x Axis
Here we refer to the scheme shown in Fig. 4.11.
P - 2/f,)}
+ i
IV
surface
J.k
Vixed boundary
Fig. 4.11. Scheme for corner zone on x axis

4.3 Boundary Conditions 101
Vj,k = 0,
1 = 0,
D.27a)
dx\ 1
4.3.4 Corner Zone on the y Axis
Here we refer to the scheme shown in Fig. 4.12.
Xj,k - 0,
xy)® = {Txy)(D = °'
D.27b)
D.27c)
D.28a)
4 Vn
D.28b)
3?fc. D.28c)
IVj
! (D
i
'Free surface
-X Fig. 4.12. Scheme for corner zone on y axis

102
y
4. Two-Dimensional Elastic-Plastic Flow
Fig. 4.13. Scheme for a free surface
4.3.5 Free Surfaces
For a free surface at j, k in Fig. 4.13, all quantities associated with the
phantom zones ® and ® are taken as zero. The equations of motion for a
general point can then be used, except that a™k and f3^k are calculated as
shown below:
For a corner free surface, the phantom zones in Fig. 4.14 are zones (J), B),
and
An
M
4.3.6 Discussion
The use of special acceleration routines for the different boundary condi-
conditions, as described above, allows the parameters 0, /?, and a to be calculated
the same for free surfaces and for fixed surfaces. This organization allows a
considerable simplification in the programming.
A good check of the program is to collapse a spherical shell by a pressure
field applied to the outside of the shell (Fig. 4.15). The problem can be set
IV
Free surface
Fig. 4.14. Scheme for a corner free surface

4.4 Applications 103
Fig. 4.15. Collapse of an aluminum spherical shell by an external O.lMbar pres-
pressure. Left: t — 0; right: t = 12 fis
up as a section of a shell in one of the four quadrants so that the x axis is
a line of cylindrical symmetry and the y axis a plane of symmetry. The shell
should collapse with perfect symmetry.
4.4 Applications
Figures 4.16-18 show applications of the computer simulation program to
problems in gas dynamics. Figure 4.16 shows results of a calculation that
simulates an explosion in air. The air is described by a 7 = 1.4 perfect gas
equation of state at an ambient pressure, P — 10~6 Mbar and density p0 =
12 x 10~4g/cm3. The explosion is simulated by assigning an energy density
to a spherical region corresponding to one kiloton of TNT. The shock wave
that propagates is shown graphically by plotting a symbol at the position of
the maximum value of the artificial viscosity, q. A Mach stem diverging shock
wave strikes the fixed boundary which represents the surface of the ground.
Figure 4.17 shows the shock waves that are formed when a slender cylin-
cylindrical body moves through air at Mach 2.34.
Figure 4.18 shows the shock patterns that form when the slender body
velocity is reduced to a velocity less than the sound speed of the ambient air.
Figures 4.19-21 show the effect of material strength on stabilizing the
growth of Taylor instability. This instability occurs on the surface of a dense
fluid that is accelerated by a less dense fluid. The initial conditions are simu-
simulated in Fig. 4.19 by applying a pressure boundary condition on the surface of
a copper plate described by a pressure equation of state only. A small initial
perturbation of amplitude 2 % of the plate thickness is seen to grow without
bound. Figures 4.20 and 4.21 show the effect of plate material strength in
reducing the instability growth for the same original conditions.
Figure 4.22 shows the simulation of a high explosive detonated in a copper
cylinder. This is a geometry used to develop an equation of state of the high

104 4. Two-Dimensional Elastic-Plastic Flow
HEMP MACH STEM3 CYCLE= +152. TIME= +2.00760 HEMP MACH STEM3 CYCLE= +847. TIME= +40.03914
Shock
Center of explosion
-Axis of cylindrical symmetry
HEMP MACH STEM3 CYCLE=+1209. TIME= +70.00103
HEMP MACH STEM3 CYCLE=+1518. TIME=+100.16033
c) d)
HEMP MACH STEM3 CYCLE= +1946. TIME= +136.05345 HEMP MACH STEM3 CYCLE= +2667. TIME= +200.05588
e) f)
HEMP MACH STEM3 CYCLE= +3244. TIME= +252.04984 HEMP MACH STEM3 CYCLE= +3789. TIME= +300.00459
Fig. 4.16a—g. Calculation of shock waves from a spherical explosion in air
explosive detonation products. The end of the copper cylinder actually melts
and forms a jet. The purpose of the calculations is to measure the turning
angle of the copper cylinder for comparison with experiment.
The effect of the distortion of the end of the cylinder is to reduce the time
step and increase the computational time of the problem. It is convenient in
this case to suppress the formulation of the jet by increasing the material
strength of the zones at the end of the copper cylinder. Figure 4.22d,e shows
the effect of adding strength to the corner edge of the copper cylinder. In
using material strength to reduce distortions that are not of interest to the

HEMP CIGAR I CYCLE= +217. TIME= +41.07336
4.4 Applications 105
HEMP CIGAR I CYCLE= +585. TIME= +90.09023
Mach 2.34
HEMP CIGAR I CYCLE=+681. TIME= +110.18223
HEMP CIGAR I CYCLE=+912. TIME=+160.14834
c)
HEMP CIGAR I CYCLE=+1088. TIME=+200.11361
HEMP CIGAR I CYCLE=+1088. TIME=+200.11361
e)
f) Same time as e) showing the
Lagrange grid
Fig. 4.17a-f. Slender body moving through air at supersonic velocity
purpose of a calculation it is advisable to provide a gradual transition to the
correct material strength. A large discontinuity in mechanical properties can
lead to additional distortions.
Figure 4.23 shows the simulation of a tension test of an elastic-plastic
material with a circular cut-out in the center. Using symmetrical boundary
conditions only one quarter of the geometry is required. Tension tests are
simulated by applying a pulling velocity at one end of the specimen and a
zero velocity in the pulling direction at the mid-plane. The velocity is chosen
so that approximately 4-5 round trips of a sound wave from the pulling
velocity to the specimen center are required to induce a 1 % strain. The
objective is to reach the desired strain with the minimum calculational time
while avoiding the introduction of excessive kinetic energy into the specimen.

106
4. Two-Dimensional Elastic-Plastic Flow
(a) HEMP SLENDER CYCLE=+923. TIME=+190.04838
(b) HEMP SLENDER CYCLE=+1930. TIME=+550.21891
Fig. 4.18a,b. Slender body
slowed from supersonic to
subsonic velocity, (a) Su-
Supersonic (b) Subsonic. Bow
shock has detached. Shocks
behind the body are from
bow shock reflections with
the cylindrical boundary
4 ::> :>6
L 0.1 cm
t = 0
= 0.8uS
t = 1.0 jis
Fig. 4.19a—d. Time sequence of the growth of Tay-
Taylor instability of a copper plate. Initial conditions: sine
wave of amplitude 10~3cm on the top surface and a
uniform 0.2 Mbar pressure boundary condition applied.
Yield strength Y° = 0
t = 1.0 us t = 1.2 us t = 1.4 us
Fig. 4.20a—c. Initial conditions same as Fig. 4.19 but yield strength Y = 0.01 Mbar
Fig. 4.21. Conditions same as Fig. 4.19 but yield strength
Y — 0.02 Mbar. Here the instability started to develop but
was arrested by the material strength
t = 5 jus

4.4 Applications 107
Various prescriptions have been used to damp the kinetic energy. However, it
has been found that there was no advantage in economizing on computational
time with these techniques and that it was even possible to converge to an
incorrect solution, depending on the damping coefficient used.
A typical pulling velocity is about 1.2 x 10~3 times the sound speed.
It is also helpful to use a velocity ramp as initial conditions for Lagrange
coordinates interior to the specimen.
Figure 4.24 shows a tension test of a plate with two notches. There is no
symmetry with this geometry and a pulling velocity is applied at both ends.
Figure 4.25 shows a comparison of a Rayleigh wave calculated by the
two dimensional HEMP program with an analytical solution [4.9, 10]. The
Rayleigh wave was generated by applying a pressure profile to a portion of
the surface of an elastic material.
Figure 4.26 shows a calculation simulating the origin of an earthquake.
The grid material represents rock with elastic properties, bulk modulus k =
0.4Mbar shear modulus \x — 0.15 Mbar and density po = 2.5g/cm3. A 0.5%
shear strain has been introduced into the material by slowly displacing the
right side of the grid in a downward direction and holding the left side of the
grid fixed. A fracture is allowed to occur in the center of the strained portion
of the material. Surface waves are seen to propagate in both directions from
the fracture lines.
Figure 4.27 shows a simulation of a torsion test of a solid aluminum cylin-
cylinder compared with experimental results. [4.10]
Figure 4.28 shows calculations representing the collisions of two aluminum
plates moving at 0.07cm/Vs. A supersonic closing velocity with respect to the
aluminum sound speed occurs for a 12° initial angle between the plates (top)
and a subsonic closing velocity for a 20° initial angle (bottom).
Figure 4.29 shows details at the collision positions for the two geometries.
For the supersonic geometry rarefactions from the lateral boundaries of the
plate cannot reach the shock front. A high pressure region, A, forms that
turns the material flow in the direction of separating the plates, region B.
For the subsonic velocity the lateral rarefactions can reach the shock front
and the material flow is turned parallel to the collision plane and the plates
maintain contact.
Figure 4.30 (top) shows the simulation of an esplosively formed copper
projectile. The copper disc is assumed to be held in place by a steel retaining
ring. Figure 4.30 (bottom) shows details of the edge of the copper disc and
the retaining ring.

108 4. Two-Dimensional Elastic-Plastic Flow
Shock In Cu
(b)
HEMP-- CYCLE TIME
LIP.00 53 20200000 5150379101
HEMP- CYCLE TIME
LIP.01 53 20200000 5150365133
(C)
(d)
HEMP- CYCLE TIME
LIP.02 53 20100000 51 50108226
Fig. 4.22a-e. Effect of material strength on the shape of the end of a copper
cylinder after interaction with a high-explosive detonation, (a, b) High explosive
detonated inside a copper cylinder, (c) Corner of copper cylinder at t = 5 ]xs with
copper strength Y° = 0. (d) Corner of copper cylinder at t = 5 ps Y° = 0.01 Mbar.
(e) y° = 0.02 Mbar Time, [is

4.4 Applications 109
Fig. 4.23a,b. Tension test of a plate with a circular cut out. (a) Grid at time 0.
(b) Plate after pulling. Lines show direction of principal stress for positions where
the plastic strain is greater than 15 %
(b)
Fig. 4.24a,b. Tension test of a plate with two notches, (a) Grid at time 0. (b)
Plate after pulling. Lines show direction of principal stress for positions where the
plastic strain in greater than 10 %
0.4
o
I °
CD
J-°-2
«-0.4
tr
> -0.6
-0.8
1 1 I 1 .
-
-
O i
O I
1 . I .
1 ¦ 1
/.
r
o
1 . 1
1 I • 1 . 1 .
t=100 ms
— Analytic solution
o Finite difference
solution.
. I.I.I.
100 200 300 400 500 600 700 800
—m
Fig. 4.25. Rayleigh wave calculated with the HEMP program compared to an
analytic solution [4.8,4.9]

110 4. Two-Dimensional Elastic-Plastic Flow
HEMP-- CYCLE TIME
EQ5 52 42000000 5145011090
Region of 1/2% shear strain
/Fracture line
HEMP-- CYCLE TIME
EQ5 52 55000000 51 60589538
HEMP-- CYCLE TIME
EQ5 52 80000000 51 90546541
.Surface waves
HEMP-- CYCLE TIME
EQ5 52 20100000 52 23553843
Surface waves (Rayleigh)
HEMP-- CYCLE TIME
EQ5 53 30100000 52 35536645
HEMP-- CYCLE TIME
EQ5 53 40100000 52 47519446
Fie 4 26a-f. Propagation of Rayleigh waves generated by a shear fractur. (a) Grid
with shear strain. All displacement from the original rectangular geometry are times
100 (b) Grid after fracture. Vertical boundaries on left and right are rigid, (e) bur-
face wave on left has reflected from the rigid boundary

4.4 Applications 111
Fig. 4.27. Torsion test of a solid aluminum 6061-T651 cylinder. A slow counter
rotating torque is applied to the two ends. Left: experiment showing scribe lines
originally parallel to the cylinder axis. Right: calculation showing lines formed from
the projection of the Lagrangian coordinates corresponding to the scribe lines of
the experiment
SUPERSONIC CLOSING VELOCITY @.67 cm/us)
Un
12°
T
1.2 jlls 7.2 fis 11.4 ns
SUPERSONIC CLOSING VELOCITY @.42 cm/usec)
20.0}is
ZJ
8.2 us 14.2 {is 18.2|is 20.0 jis
CALCULATIONS OF COLLISION OF ALUMINIUM PLATES
Fig. 4.28. HEMP calculations of collisions of aluminum plates. Time is measured
from the first contact of the plates. Only the main shock is shown, and the Lagrange
grid has been omitted for clarity. The plates are 1 cm thick and 8 cm long

112 4. Two-Dimensional Elastic-Plastic Flow
Fig. 4.29. Enlarged view of the calculations in Fig. 4.28 showing the directions of
material velocities in the colliding plates for each Lagrange coordinate. The vectors
are not plotted on the boundaries. Left: supersonic closing 8 ]is after contact. Right:
subsonic closing 10 f^s after contact. A = collapse point. B = beginning of rebound
w. ),. ...q.1-. i . i Li
Point
of
detonati*
Steel case
Steel
retaining
ring
Copper disc
i p • ¦ • i
Fig. 4.30. Calculation of an explosively formed copper projectile. Top: original
geometry and copper projectile at a later time. Bottom: enlarged view of the steel
retaining ring

5. Sliding Interfaces in Two Dimensions
When large relative displacements occur in fluid dynamic calculations that
are formulated in Lagrange coordinates, a decoupling of grid points must be
provided to allow slippage of one grid on the other. A sliding interface is a
line or a surface defined by a set of Lagrange coordinates. In two dimensions
the interface is a line defined by j, A: coordinates and iji three dimensions the
interface is a plane defined by z, j, k coordinates. In the method employed
here, a set of grid points associated with one side of the interface defines a line
or surface on which the grid points of the opposite side can slide. The process
is then reversed. Grid points associated with both sides of the interface are
tagged as either void open or void closed. The void open condition means
there is a gap or void between the given point and the surface opposite it.
The acceleration of void open points follows the same logic as a free surface
point. Void closed points are defined as points that are in contact with the in-
interface between the two grids. The acceleration of void closed points is broken
into components perpendicular and parallel to the interface. The perpendic-
perpendicular component of acceleration includes the mass of the materials from both
grids. The parallel component of acceleration only includes the mass of the
particular point being considered. The stresses of one grid provide boundary
conditions for the opposite grid. Both sets of points on the interface are ac-
accelerated by the same procedure, i.e., the motion of points on one side of the
interface includes the stress and mass from the opposite side and vice versa.
When penetration of a grid point through the surface of the opposite grid
occurs, the velocities of the interface grid points of both grids are adjusted
to conserve angular and linear momentum. The symmetry of the calculations
is especially useful in vector programming and in addition permits sliding to
occur simultaneously in more than one direction. After grid points on each
side of the interface have been advanced an integration time step, one set
of grid points is declared a slave grid and the other a master grid. All slave
points that have penetrated the master grid are set onto the master surface.
Provision is made for opening a void for a previously void closed point if the
acceleration is in the direction away from the opposite grid and the stresses
acting on the point from the opposite grid are tensile. When this condition
occurs, the point is tagged void open and free surface boundary conditions
are applied.

114 5. Sliding Interfaces in Two Dimensions
5.1 Sliding Interfaces
Between Quadrilateral Lagrange Zones
To describe the method, we will assume the sliding interface is a line of
constant Lagrange coordinate k as shown in Fig. 5.1. It will be convenient
to designate in advance the grid points on one side of the interface as slave
points, and the grid points associated with the opposite side of the interface
as master points. The steps to advance in time the points associated with the
master and slave grid points are given below:
Step 1. For a given void closed slave point, /, locate the portion of the
master line that contains point /. In general point / will be asso-
associated with a line segment formed by two master points, b and c;
see Fig. 5.1. (If the point / is tagged void open, this search is not
required).
Step 2. Calculate the volumes of void closed slave zones that take into
account any irregular shape of the master surface.
Step 3. Advance in time all void closed slave points with the sliding logic.
Void open points are advanced with free surface conditions. Check
for void opening of points previously tagged void closed.
Step 4. For a given void closed master point locate the slave points to be
associated with it.
Step 5. Advance in time all void closed master points with the sliding logic.
Check for void opening. Void open points are advanced with free
surface boundary conditions.
Step 6. Check all master and slave points for penetration into the opposite
grid.
k=slide line determined
by points h, a, b, and c
Fig. 5.1. Schematic grid for locating master points that lie to either side of a given
slave point. Master points are shown as closed circles and slave points as open
circles

5.1 Sliding Interfaces Between Quadrilateral Lagrange Zones 115
Step 7. If penetration has occurred and the void status has changed from
void open to void closed assign new velocities to the penetrated
point that satisfy conservation of angular and linear momentum
and tag the point void closed.
Step 8. Place slave points that have penetrated the master grid onto the
master grid interface.
Note: A slave grid point is checked for penetration into the master grid
and the master grid point is checked for penetration into the slave grid. New-
velocities are assigned to the points when the test is positive and the void
status has changed from open to closed. Only slave points are set back to the
master line. Points are left as they are if penetration has not occurred.
The subsequent sections give the details for the above eight steps.
5.1.1 Location of Master Points Associated
with a Given Slave Point
Consider a point (j,k) of the slave grid. Find points a and b on the master
grid that lie to either side of point (j, k). (Point (j, k) is shown as point / in
Fig. 5.1).
(a) Starting with the last master point on the slide line, (jmaX5fc), point c
in Fig. 5.1, calculate the area A of the triangle formed by points (j,k),
(j, k - 1) and point c. [It will be convenient to store with each point on
the sliding interface a point immediately inside the grid which is called
the "connector point". Here point (j, k — 1) is the connector point for the
interface point (j, k).]
2A = (xjfk-i - Xj,k) (yc - yjjk) - (y^k-i ~ Vj,k) (xc - xjyk)
(b) Repeat step (a) above using consecutive points on the slide line in place
of point c until a change in sign of the areas is found. The consecutive
points where the areas change sign will be points a and b in Fig. 5.1. If
the value of \2A\ < 10~4 the slave point is considered coincident with the
master slide line point.
It is only necessary to search the entire slide line when the grid is gener-
generated. Information on neighboring slide line points for a given sliding point is
carried from cycle to cycle and these points are tested first. If the points fail
to show a change in sign of the areas found these adjoining points on either
side are alternately tested.
5.1.2 Calculation of the Volume of Sliding Zones
Associated with the Slave Grid
A zone originally defined by four sides may have more than four sides when
it contains more than one line segment of the slide line. Referring to Fig. 5.2
we want to calculate the volume enclosed by P, /, 6, /' and G.

116 5. Sliding Interfaces in Two Dimensions
Fig. 5.2. Scheme for calculating the vol-
volume of a sliding zone
Here A\ is the area of triangle 1, etc.
2A = Xb~ Xp Vb ~ Vp
1 xf -xp yf -yp
= (xb - xp)(yf - yp) - (yb - yp)(xf - xp)
2A =
Xf' " Xp
Vv
2A,=
b -xP yb-yP
- xp)(yb - yp) - (yr - yp)(xb - xp)
xg -xp yg- yp
f - Xp yr - yp
= (x9 - xp)(yr - yp) - (yg - yp)(xr ~ xp)
(yP
[(yP
(yP
5.1.3 Advancing a Slave Point / in Time
Advance in time point / which has been found to be contained on line segment
ab; Fig. 5.3.
(a) Calculate the effective stress acting on the face ef of zone C) due to
material above the slide line, Fig. 5.3:
k+1
k-1 Fig. 5.3. Scheme for advancing
in time sliding point / on slide
line k

5.1 Sliding Interfaces Between Quadrilateral Lagrange Zones 117
IV
Fig. 5.4. Scheme for accelerating
point /
Eyycos2(ef)-2Txycos{ef)sin{ef)]^le
4- [?xxsin2(e/) 4- Syy cos2(e/) - 2Txy cos(e/) sin(e/)]0/a/},
where
sin(e/) -
E.1)
Vf ~
cos(ef) —
y/{xf -xeJ + (y/ -ye)
X* -X.
kf =
(ye -
etc.
It is seen that cr^ is a stress that acts perpendicular to the surface defined
by points e and / (Fig. 5.3).
Repeat the above procedure to get cre, the effective stress that acts on
the surface defined by points / and g.
Calculate the acceleration of point (j, k) (point /, Fig. 5.4).
The acceleration of point / is composed of components perpendicular and
parallel to the line formed by points a and b of the opposite grid, Fig. 5.3.
The perpendicular component includes the mass of the material of the
opposite grid while the parallel component does not. Refer to Fig. 5.4.
Gnj = [gcos(ab) +rsin(a&)]
gives the acceleration parallel to line ab and
R7} = - [ - g sin(afr) + r cos(ab)]
is the acceleration perpendicular to line ab.
E.2a)
E.2b)
sin(a6) =
- XaJ 4" (Vb ~ VaJ

118 5. Sliding Interfaces in Two Dimensions
cos(a6) = b a
V (xb - XaJ + {Vb - VaJ
E.2c)
j,k-
z<Pjfc L
E.2d)
E.2e)
E.2f)
Test for void opening (see Fig. 5.4). If g(x\ - x\\\) + r(y\ - yui) > 0 (the
acceleration of the slave point is away from the master grid and toward
the slave grid) and if ad + ae > 0 then point / is tagged a void open
point and the acceleration is recalculated as a free surface point. If the
test is negative, continue the calculation for a void closed point.
The factor z that appears in E.2b) above is obtained by associating with
the sliding point / (Fig. 5.3) the mass of the material above the slide line
between points e and g as well as the mass of zones C) and @:
z = 1
m
m—
lea (POA°
Ike + lea \ V°
Ifb
Igf
kg
la/ + hb \ V<
(d) Velocity components of point /
(p0A°\
Atn [Gcos(ab) -
J/'

5.1 Sliding Interfaces Between Quadrilateral Lagrange Zones 119
^+* = ^-\ + Atn[Gcos(ab) - Rcos(ab)]f.
(e) Displacement of point /
(f) Acceleration of point on the end of the sliding grid. When point / is at
the end of the grid (Fig. 5.5) the quantities ^, ctf and /3f are calculated
as follows:
(g)
The acceleration equations are similar to those given in (a) above with the
stresses for zone @ equal to zero (Fig. 5.4). The slide line k is extended
by connecting point c (end of slide line grid) with a point d that is outside
the grid. An alternate method of treating a slide line point / that has
moved off the end of the grid is to tag it a free surface point. It has been
found useful to have both methods available (see below).
Slide line extension. The coordinates of point d (Fig. 5.5) are determined
by extending the line segment be of the master grid.
od = ke (oc — ob) -f oc.
Here od, oc and ob are the vectors formed by the respective points d, c, 6,
and the origin. The parameter ke determines the extension in units of the
distance be of the master surface.
b . ,
k+1
I
j
r
Point
k-1
outside
•d
Free
the grid
surface
Fig. 5.5. Scheme for accelerating a sliding point that is also on a free surface

120 5. Sliding Interfaces in Two Dimensions
The coordinates of the extension point d are given by
xd = ke(xc-xb)+xci
yd = ke{yc -
yc-
If point / of the slave grid (Fig. 5.5) extends beyond point d then it is tagged
a free point. The position where a slave point / is treated as a free point is
controlled by the parameter ke. Normally ke is set equal to one so that point
/ must slide a distance equal to the grid spacing of the master surface before
it can become a free point.
5.1.4 Location of Slave Points
Associated with a Given Master Point
Given are the coordinates of point 1 and point a of the master grid. Find
points r and s of the slave grid that lie to either side of the line determined
by point 1 and point a (Fig. 5.6)
(a) Starting with the last point of the slave grid, calculate the area, A, formed
with this point and the line (xi,yi), {xa,ya)' (The point 1 is the "con-
"connector point" for point a on the interface, see step (a) of Sect. 5.1.1.)
2A = (xa - xi) (yt - yi) - (ya - yi) (xt - zi) •
Here (xt, yt) are the coordinates of the last slave point.
(b) Repeat step (a) above, using consecutive points of the slave grid in place
of point, t, until a change in sign of the areas is found. The consecutive
points where the areas change sign will be points r and s that lie on
either side of point a (Fig. 5.6).
(c) Determine the slave points that lie on either side of point b (Fig. 5.6) by
repeating step (a) and step (b) with points 6, 2, and consecutive slave
points.
k+1
Fig. 5.6. Scheme for locating
the slave material zones that
will be associated with the ac-
acceleration of a given master
point

5.1 Sliding Interfaces Between Quadrilateral Lagrange Zones 121
(d) The slave points can be set up in a numbered sequence. Steps (a) and (b)
will identify the number of the slave point directly to the right of point
b. The subtraction of these two numbers will identify the slave points
between points a and b.
5.1.5 Advancement in Time of Point jf, k on the Master Grid
Refer to point b of Fig. 5.6.
(a) Calculate the stresses that act on zones @ and B) due to the material
below the slide line.
ad= — | \SXX sin2(ab) 4- Eyy cos2(ab) - 2Txy cos(ab) sin(a6)](g)ios
4- [?Xx sin2(a6) 4- Eyy cos2(ab) — 2Txy cos(ab) sin(ab)]~lsb >;
sin(afr) =
cos(a6) =
Vb -ya
- xaJ + (yb - yaJ
xb -xa
Ls = V(xa -xsJ -f (ya -ysJ-
It is seen that ad is a stress that acts perpendicular to surface ab and is
obtained by mapping the components of the stresses in the zones below
surface ab that are perpendicular to ab (Fig. 5.7).
(b) Repeat the above procedure to get ae, the effective stress that acts on
surface 6c, Fig. 5.7.
(c) Advance point (j,k) (point 6, Fig. 5.7). The acceleration of this point is
composed of components perpendicular and parallel to the line formed by
points s and t of the opposite grid, Fig. 5.6. The perpendicular component
includes the mass of the material beneath the slide line while the parallel
components does not.
Referring to Fig. 5.6 calculate:
G —
-f
E.3a)
k+1
Fig. 5.7. Scheme for advancing in
time j, k on the slide line k

122 5. Sliding Interfaces in Two Dimensions
the acceleration parallel to line st and
R = -[- gs'm(st) -f rcos(st)],
the acceleration perpendicular to line st where
Vt -Vs
E.3b)
sin(st) =
cos(st) =
y/(xt - xsJ + {yt - ysJ
Xt -Xa
y/(xt - xsJ + (yt - ysJ
9=~:
1
)^A^ - tf) + (rxxM)(%n - yfi
E.3c)
y —_
n ~ xni)
~ x?v)
- if)
E.3d)
3,k —
E.3e)
E.3f)
Test for void opening (Fig. 5.7). If g(xm - x\) + r(ym - y\) > 0 (the
acceleration of the master point is away from the slave grid and toward
the master grid) and O& + ae > 0, the point b is tagged a void open point
and the acceleration if recalculated as a free surface point. If the test is
negative continue the calculation for a void closed point.
The factor z that appears in E.3b) above is obtained by associating
the slide line point j, /c, (point b) with the mass of the sliding material
between points a and c. See Fig. 5.8
m

5.1 Sliding Interfaces Between Quadrilateral Lagrange Zones 123
k+1
- k-1
Fig. 5.8. Scheme for
mapping mass of slid-
sliding material onto the
slide line k
m—
la
(
Ira + las \
°\ , U fPoA°\
~~ i i ;— 1 T rrt 1
(
lsb + lbt \ V° J
ltc (PqA°
( (
ht \ V* J® ltc + lcu \
(d) Velocity components of point j, k
±1+* = x^~* + Atn[Gcos(st) - Rcos{st)},
vTkh = vTkh + Atn[Gcos(st) - Rcos(st)}.
(e) Displacement of point j, k
xj,k
5.1.6 Testing for Penetration of Grids
Fig. 5.9 shows two grids, one defined by open circles and the other by closed
circles. Determine if a point / on one grid has penetrated the adjacent grid.
(a) Assume that point / has been determined previously from steps 1 or 4
(Sect. 5.1.1 or Sect. 5.1.4) to be between j and j' + l on line k correspond-
corresponding to Fig. 5.9. Calculate the direction numbers A, B of a vector through
point j, and perpendicular the segment j, j + 1 (shown as segment a, b
in Fig. 5.9)
A = (yb-ya),
B = -(xb~xa).
Check the direction of A, B.

124 5. Sliding Interfaces in Two Dimensions
Fig. 5.9. Schematic grids showing
point / of the current grid closest
to point j of the opposite grid
If (xj^k+i - %j,k)A 4- B/j,fc+i - Vj,k)B > 0, then the direction numbers A,
B, point into the grid, hence reverse the signs of A and B. Otherwise A,
B have the desired direction that points out of the grid,
(b) Calculate c/, the perpendicular distance from point / to line ab:
, A(xf-xa) + B(yf-ya)
VA2 + B2
Here point a is point j, k and point 6 is point j -f 1, k (Fig. 5.9).
If: 0 < d < + 5, leave point / as calculated with the same void status as
before.
If: d > 5, leave point / as calculated and tag as void open.
If: d < 0, point / has penetrated the segment ab and is tagged void closed.
If point / was closed the previous cycle proceed to step 8 (Sect. 5.1.8). If point
/ was open the previous cycle proceed to steps 7 and 8 (Sect. 5.1.7 and 5.1.8).
The parameter S is used to control the change in void status depending
on the relative displacement of a point with respect to the grid spacing. Here
S is 10 % of the grid spacing calculated as
5.1.7 Adjusting the Velocities of All Void Closed Points
Where d < 0 and Where in the Previous Cycle
the Point Was Void Open
(a) Calculate normal velocities for all master and slave points. Assume point
/ in Fig. 5.9 has penetrated line segment ab of the opposite grid and is
the void closed point being considered. Calculate the velocity components
Na, JVb, Nf, that are normal to the line segment ab.
Na = l%a + mya
Nb = lxb 4- myb

5.1 Sliding Interfaces Between Quadrilateral Lagrange Zones 125
= l±f +
A B
m =
Note: I and m are the direction cosines of the unit vector perpendicular
to line ab and directed outward toward the grid of point /. See step (b)
of Sect. 5.1.6.
(b) Calculate Nj~, the new velocity of point / that satisfies the conservation
of linear and angular momentum:
¦ + = A - a)MbPL - [A - a)Mh - aMa]Pu
f Mf [a2Ma -f Mb{\ - aJ] + MaMb
where
af • ab/\ab\
\ab\
)( - xa) + (yb - ya){yf - ya)
=
(xb - xaJ 4- (yb - yaJ
If a < 0, recalculate a using points a and b as points (j — 1) and j,
respectively. If a > 1, recalculate a using points a and b as points (j + 1)
and (j; + 2), respectively. See Fig. 5.9.
Pl = +(MaNa + M67V6 + MfNf),
Pu = +(aMfNf+MbNb),
where Ma, M^, and Mf are the respective masses at points a, 6, and /
(obtained by averaging the mass of the two zones in common with the
respective points).
Note: The new velocity of point /, iV^~ was obtained by the simultaneous
solution of the following three equations:
Conservation of linear momentum
Ma7Va + MhNb + MfNf = MaN2 + MbN+ + MfN+. E.4a)
Conservation of angular momentum about point a
abaNfMf + abNbMb = abaNJ'Mf + abN+Mb. E.4b)
E.4c)
where Nf is a linear interpolation of 7V+ and Nj}~.
(c) Calculate the x, y components of the new velocity of point /:
.71 +
Xr
2 _
y}+rn(ANf)

126 5. Sliding Interfaces in Two Dimensions
(here xj, yj refer to the velocity of point / before the velocity adjust-
adjustment), where
ANf = (JV+ - Nf).
5.1.8 Relocating Slave Points onto the Master Surface when d < 0
We have
Here / is a slave point while points a and b are points on the master surface.
The velocities of master and slave points are adjusted in the same manner
when penetration has occurred, i.e., when d < S. However, only slave point
positions are relocated. (The relocation of a slave point has been made per-
perpendicular to the master line segment associated with the slave point.)
5.2 Intersecting Slide Lines
Figure 5.10a shows schematically a slide line (s.l.I between regions A and
C and a second slide line (s.l.J between regions C and A and region B.
Figure 5.10b shows schematically a portion of the grid for the three regions
drawn separated for clarity. It is assumed that grid A is the master side of
slide line (s.\.)l and grid B the master side of slide line (s.l.J. Point a\ is a
master point with respect to slide line (s.l.^ and a slave point with respect
to slide line (s.l.J. Point C2 is a slave point with respect to both slide lines
(s.l^j and (s.l.J.
5.2.1 Acceleration of Points on the Intersection of Two Slide Lines
Assume that no voids are open so that points C2 and a\ occupy the same
position. Point C2 is accelerated with the z factor and ad stress obtained
from the zone of grid A for the component of acceleration perpendicular to
slide line (s.l.I. The acceleration component parallel to (s.l.I assumes point
C2 is a free surface point, i.e., the mass and stresses from grid B are not used.
Point ai is accelerated with the z factor and O& stress from the zones, of
grid C for the component of acceleration perpendicular to slide line (s.l.)^
The acceleration component parallel to (s.l.^ assumes point a\ is on a free
surface, i.e., the mass and stresses from grid B are not used.

5.2 Intersecting Slide Lines 127
(a)
1
c
B
A
(b)
b4
b2
C1
C4
C
c2
C3
a1
a4
a3
A
Ms..
Fig. 5.10a,b. Intersecting slide lines, (a) Regions C and A are separated from
region B by slide line (s.l.J. Regions C and A are separated from each other by
slide line (s.l.)r The circle shows the intersection of the two slide lines, (b) Grid
points for zones at the intersection position
5.2.2 Adjustment for Grid Penetration
If point c2 or point a\ has penetrated slide line (s.l.J the points are relocated
with respect to (s.l.J and the velocities adjusted to conserve momentum in
the manner already described.
If point c2 has penetrated slide line (s.l.I? it is relocated onto (s.l.)r The
velocity is adjusted to conserve momentum. When point a\ is involved in the
conservation of momentum equations for the velocity adjustment of point c2
the velocity of point a\ is used after having been adjusted for penetration of
(s.l.J.
If point c2 penetrates the extension of (s.l.I? it is given the velocity of
a\. The velocity of a\ is not adjusted if it has been used for the velocities
of slave extension points. It is recalled that when the penetration of slide
lines occurs the velocities of points are adjusted only when the void status
changes from open to closed. If the penetration of (s.l.I by c2 occurs in a line
segment containing point a\ the velocity of point a\ is adjusted even though it
may have been previously adjusted for penetration of (s.l.J. An exception is
made for any velocity adjustment required of grid points on (s.l.J associated
with grid B. The slide line intersection points c2 and a\ are not used in the
conservation of momentum equations for the velocity adjustment of grid B
points.
5.2.3 Relocation of Points when a Void Has Opened
An extension one zone thickness long is provided for intersecting slide lines.
When a void has opened the extension provides a continuation of the slide line
for sliding points associated with the slide line. When a point slides beyond
the extension it is considered a free point.

128 5. Sliding Interfaces in Two Dimensions
(a)
C1
i
e
'31
a4
:in+1 v"'"/2
a2
a3
(b)
Fig. 5.11a,b. Slide line extension for void open point, (a) Extension line a\e of
slide line (s.l.I intersects slide line (s.l.J. (b) Extension line a\e does not reach
slide line (s.l.J
Figure 5.11a shows point a\ as void open with respect to slide (s.l.J. The
slide line extension point e is calculated as follows:
xe — xa\ + \xa\ ~ xa^)^
Ve = yai +(yai -ya4).
Referring to Fig. 5.11a: If C\ penetrates (s.l.I in the time interval from tn to
tn+l then c\ is set to the intersection of (s.l.^ and line a\e when point a\
is void closed with respect to (s.l.J or relocated onto line a\e if point a\ is
void open with respect to (s.l.J. In both cases it is given the velocity of a\.
Referring to Fig. 5.11b: Points e and a\ are on the same side of slide line
(s.l.J. If point c\ lies between the extension point e and slide line (s.l.J it is
left as is.

6. Elastic-Plastic Flow
in Three Space Dimensions
Described here is the HEMP 3D computer simulation program for solving
problems in solid mechanics and gas dynamics in three dimensions.
The equations of motion, the conservation equations, and the constitutive
relations listed below are solved by finite difference methods following the
format of the HEMP computer simulation program formulated in two space
dimensions and time.
6.1 Fundamental Equations
6.1.1 Equations of Motion
P§=^+^+5k F,a)
dt ox oy oz
/l = ^y+^yy+^i F.lb)
dt ox oy oz
p§ = ^ + ^k + ^?i F.1c)
dt dx dy dz
6.1.2 Conservation of Mass
where M is a mass element.
6.1.3 First Law of Thermodynamics
E = -(P + q)V
Jt-v[Sxx?xx + syyeyy + szz&zz H~ Txy?xy + Tyz?yz -\- TZXSZX\. F.3)
Here E is the internal energy per original volume, V the relative volume
= po/p, in which p is the actual density and p0 the reference density of the
equation of state.

130 6. Elastic-Plastic Flow in Three Space Dimensions
6.1.4 Velocity Strains
exx = dfx, F.4a)
eyy = ^r, F-4b)
e,, =
dx dy
dy dx
(d±
= [Tz
6.1.5 Stress Deviator Tensor
( IV\
sxx = 2/i lexx - - — I , F.5a)
Syy = 2/i(eyy-~) , F.5b)
*« = 2mU**-~J, F.5c)
txy = n{sxy), F.5d)
rzx - /i(e2X), F.5e)
fyz=/i(iyzI F.5f)
where // is the shear modulus.
6.1.6 Pressure Equation of State
P = afa - 1) + 6(ry - IJ + c(r] - IK 4- drjE, F.6a)
V=^ = - F.6b)
where a, 6, c, and d are equation-of-state constants.

6.2 Finite Difference Equations for HEMP 3D 131
6.1.7 Total Stresses
Zxx = -(P + q) + sxx, F.7a)
Zyy = -(P + <l) + Syy, F.7b)
r22 = -(P + q) + szz. F.7c)
6.1.8 Artificial Viscosity for Calculating Shocks
s|- <6-8»
where Co and Cl are constants, ds/dt is the rate of strain in the direction
of acceleration, L a measure of grid size, a the local sound speed, and p the
local density.
6.1.9 Von Mises Yield Condition
F.9)
where Y is the plastic flow stress,
"v i V\ i p ^ ^
Here ?p is the equivalent plastic strain, 2 J the second invariant of the devi-
atoric stress tensor, and a, 6, and c are flow stress constants.
6.2 Finite Difference Equations for HEMP 3D
6.2.1 Mass Zoning
The physical object is divided into zones defined by eight grid points, Fig. 6.1.
The grid (i, j, k) moves with the material and the mass within a zone remains
constant. In the notation that follows a superscript refers to the time center-
centering of a parameter or equation and the subscript refers to the space centering.
See [6.1] for details of mesh generation.
Defining the Vectors. Three vectors are associated with each of the eight
grid points, g, shown in Fig. 6.1.
Vector Components
A: (a»)i = Z4-zi; (%)i = 2/4 - 2/ii (ak)i = z4 - Z\.

132 6. Elastic-Plastic Flow in Three Space Dimensions
(ij+1, k+1)
Fig. 6.1. Grid numbering scheme for zone 0
Vector Components
A: (a»J = zi-Z2; («jJ = 2/i - 2/2;
B: FiJ = X3-x2; F^J = 2/3-2/2; F^J = ^3 -
C: (C2J = X6 - X2\ (CjJ = V6 -2/2; (CfcJ = ^6 -
Vector
A: (a
Components
3; (ajK = y2 - y3;
3; (bjK = y4 - y3;
= z2 - z3.
= X7-X3;
|6
I
1
V
V- —
Vector
A: (ajL =x3 -
B : FiL = xi -
C : (c»L = x8 -
Vector
A: (aiM = x6-x5;
B : FiM = x8 - x5;
C: (ciM = X1-X5;
Components
(fljL = 2/3 - 2/4J
F^L = 2/1 - 2/45
(cjL = 2/8 - 2/4;
Components
(a7M = y6 - y5;
M = y8 - y5;
M = 2/1 - 2/55
(c/cL = ^8 -
(akM
(bkM
= z6 -

6.2 Finite Difference Equations for HEMP 3D 133
5 = 6
(
IB »
I /
9 = S
8 B J
5A.-
A
C I I
I I
1—/
Vector Components
A: (alN=x7-x6; (ajN = y7 - y6; (akN = z7 - z6.
B: (fe»N =X5-x6
C: (CiN=x2-x6
Vector
A : (oiO = x8 - x7;
B: (biO = x6 - x7]
C. (CiO = x3-x7;
Vector
A: (al)$ = X5-xs:
B: (bi)s = x7-x$;
C: (ci)8 = x4 - x8;
= 2/5 — 2/65
= 2/2 - 2/65
= zb - z6.
= z2 - z6.
Components
(a^y = ys - y7; KO = zs - z7.
(bjO = y6-y7; (bkO = z6- z7.
{cjO = y3 - y7; (ckO = z3 - z7.
Components
(aj)s = 2/5 - 2/8 5
Fj)s = 2/7 " 2/85
= z$ - z8.
= z7 - z$.
Calculation of the Volume of Zone
that
. Referring to Fig. 6.1 we see
Cjfe
This is repeated for g — 2 —> 8.
Calculation of the Mass of Zone 1,
where po is the reference density, V° the initial relative volume, and v° the
actual volume calculated from the coordinates at time t = 0.
Mass Associated with Point (i,j,
¦+¦
6.2.2 Equations of Motion
The following acceleration equations are applied to point z,j, k in Fig. 6.2.

134 6. Elastic-Plastic Flow in Three Space Dimensions
(a) OCTAHEDRON /
2
(b) CUBE
Fig. 6.2a,b. Grid for accelerating point. i,j, k — Lagrange coordinate
Motion in the x Direction.
/dx
where
'iasxx\n
dx
dZTT dTxy
' —^ i
dz
- yv)(z\v ~ zy)
~ Zy) - (zU - Zy)(yy\
- 2/v)]
~ Z\) -
Vi)(zy - zi) -
yiu)(ziv - zni)
ym)(zi -
(zn - zi)(yy - yi)]
- (zi - zm)(yiy - ym)]
- (zu -

6.2 Finite Difference Equations for HEMP 3D 135
To form A/p dTxy/dy)™jk, replace each Exx in the right side of the above
expression with Txy, every y with the corresponding 2, and each z with the
corresponding x.
To form A/p dTxx/dz)^-kl replace each Exx in the above expression with
T2X, every y with the corresponding x, and every z with the corresponding
y-
The x-direction velocity at n+1/2 and positions at times n+1 and n+1/2
are:
.n+1/2 _ .n-1/2 . /dx\n
xn+l = xn +x-fc4(n+1/2
n+1/2 _ I/Xn+1 , n \
Motion in the y Direction.
dtJij,k Pi,j,k L 9:r 92/ &
where
1 9TX2/\n same as A/p dExx/dx)"j k, defined above, except
p dx ) i ¦ k replace each Exx by the corresponding value of Txy.
1 dEyy\n _ same as A/p dTxy/dy)™jk, defined above, except
p dy ) • ¦ k ~ replace each Txy by the corresponding value of Eyy.
1 dTyz\n _ same as A/p dTzx/dz)™jk, defined above, except
p dz ) i ¦ k replace each Tzx by the corresponding value of Tyz.
The y-direction velocity at time n + 1 and positions at times n + 1 and
n + 1/2 are:
Motion in the z Direction.
d?\ 1 \dTzx ( dTyz t dEZ2
dt/z,j,k Pij,k L dx dy dz Ujk
where
1 dTzx \ n _ same as A/p dExx/dx)™jk, defined above, except
Kp dx J tj k replace each Exx by the corresponding value of Tzx.

136 6. Elastic-Plastic Flow in Three Space Dimensions
1 dTyz\n _ same as A/p dTxy/dy)^-k, defined above, except
P ty / i k replace each Txy by the corresponding value of Tyz.
1 dUzz\n _ same as A/p dTzx/dz)"jk, defined above, except
P dz ) i ¦ k replace each Tzx by the corresponding value of Ezz.
The z-direction velocity at time n + 1/2 and positions at times n + 1 and
n -f 1/2 are:
.n+l/2 _ .n-l/2 fdz\n
n+l/2 _ 1/rn+l , n
6.2.3 Conservation of Mass
where v^ is the volume at time t — n and Vq is the relative volume. Similarly,
T/n+l _ (P0\ n+l
where the volume vn+l is calculated from the coordinates at time n + l.
Vrn+l/2 A /-rrn-j-l T/n\
defines the relative volume at t = n + 1/2.
6.2.4 Calculation of Incremental Strains
The finite difference mapping procedure to calculate the surface integral of
zone Q, Fig. 6.1, covers the surface in units of triangles. The velocity asso-
associated with a given triangle is taken as the average of the velocities defined
at the triangle corners. The triangular surface area vectors are calculated
to point out of the zone surface. The dot product of the area vector with
the direction vector multiplied by the average velocity gives the velocity flux
through the surface in the given direction. The mapping procedure actually
covers the zone surface area, Fig. 6.1, two times. The difference equations
used to calculate dx/dx, dx/dy, and dx/dz are given explicitly below.
The remaining velocity derivatives required to calculate the components
of strain are calculated by replacing x in those equations by y and then by z
so as to complete the set:

6.2 Finite Difference Equations for HEMP 3D
dx dx dx
dx dy dz
dy_ dy_ dy_
dx dy dz
dz dz dz
dx dy dz
Velocity Derivatives Corresponding to Zone (J), Fig. 6.1.
137
dx
9=1
+xCa(C x A) • i 4- xBC{B x C) • i]
.in+l/2
where
(xca)9=1 = (X1+X4+X5),
(C x A • i)9=1 =
1 0 0
1 0 0
C-i Cj Ck
1 0 0
5=1
9=1
]
\g = V
\g=V
The above steps, written for g = 1, must be repeated for g = 2 —> 8.
n+l/2
?xC)-j]
where (&ab)q=\,{xca)q=\, and (iec)o=i are denned above.
(AxB.j)g=1 =
0 1 0
bi bj bk

138 6. Elastic-Plastic Flow in Three Space Dimensions
0 1 0
(BxC-i)g=l =
Ck
= [-{dak -ckai)}g=l.
9=1
0 1 0
hi bj bk
a cj ck
= [~{biCk -bkd)] .
9=1
The above steps, written for g = 1, must be repeated for g — 2 —>• 8.
j.xn+l/2 / 1 \ 8
dxY
9 = 1
+xCa{C x A) • k 4- xsc(? xC)-k]J
where (x^b)p=i, {xca)q=1i and (±#c)p=i are denned above.
0 0 1
«i dj dk
(AxB-k)g=1 =
(CxA-k)9=1 =
0 0
0 0
Cj
1
Ck
Vlg=V
Ci)]
The above steps, written for g = 1, must be repeated for g = 2 —> 8.
— = same as dx/dx except replace x by the corresponding y,
—— = same as dx/dx except replace x by the corresponding i,
ox
— = same as dx/dy except replace x by the corresponding ?/,
— — same as dx/dy except replace x by the corresponding i,
— = same as dx/dz except replace x by the corresponding y,
oz
— = same as dx/dz except replace x by the corresponding i.

6.2 Finite Difference Equations for HEMP 3D 139
Incremental Strains.
. \ n + 1/2
— I
±\n+1/2
d±\
Atn+1/2,
At"*1'2,
\ v J® XX 0
6.2.5 Calculation of Stresses
Stress Deviators.
\n + l _
{Aexx)
(Ae
(A?zz)
_ 1 (AV\
n+1/2
n+1/2
3V
+
+
Note: The terms S that have been added to the stress deviators are corrections
for zone rotations (see Chap. 4).
5nxx = -^
- 2ujnTn
rn rn
-dyy - dxx,

140 6. Elastic-Plastic Flow in Three Space Dimensions
rn , ,n/_n _,n \ , , nrpn , .nrpn
dxy = Uz(Sxx- Syy)+Uylyz-UJxlzx,
rn . ,n( on _n \ , / .nrpn . ,nrpn
dyz = ^xl5^ -S22) + ^>zizx -UJy1xy^
en , ,n / _n ^n \ . , .nrpn , .nrpn
hx = uy(szz- sxx)+ujxTxy-uzTyz,
where
n+1/2
Pressure Equation of State.
r>n+l _ /l/T/n+l\ , D/T/7i+l\ jpn+1
where A and B are functions of the volume V and E is the internal energy.
Total Stresses.
+
/ p \n + l _ / pn+l , n+l/2x
v-^yy/Q ~" v^® "^ ^® /
/r1 ^n + 1 — _/pn+l , n+l/2x
6.2.6 Von Mises Yield Condition
2
® = ^ ® ~ 3
If Kq~l < 0 use the stress deviators as denned above.
If K^1 > 0, then multiply each of the stresses
Ezz)n+1> (Tx )n+1, (T z)n+1, and {Tzx)n+l by
6.2.7 Plastic Strain
In the following definitions of plastic strain, the stress deviators at time n + 1
are taken as the values after the yield condition has been satisfied. If yielding
has not occurred, these equations are bypassed.

6.2 Finite Difference Equations for HEMP 3D 141
Components of Plastic Strain Rate.
1
P^n+l/2 __ ^n+1/2 _
P^n+1/2 _ -n+1/2
yy yy
P-n+1/2 -n+1/2 _
1
P-n+1/2 __ -n+1/2 _
^n+1/2
1
6xy
= e
P-n+1/2 _ ~n+l/2 _
P-n+1/2 __ ^n+1/2 _
*y Atn+1/2
1
= ?'
yz
-snxx-5xx , \Vn+l-Vn
on+\ _ cn
3 yn+i/2
3
s^-s«-6zz , IV
n+l
3 yn+l/2
p
xy
-T?x-*zx
-Tfz-Syz
The quantities eit etc. are the velocity strains in the calculation of the
stress deviators.
The equivalent plastic strain, ep is
v " \ (P • P a \2
- T"! V ?xx - ?yy)
(Pa Pa
\ Ezz ~ &X
1/2
The flow stress is
Here a, 6, and c are material constants, not to be confused with the vector
components ciij,k etc.
6.2.8 Artificial Viscosity for Calculating Shocks
An artificial viscosity is required to permit shocks to form in the grid. The
artificial viscosity, q, used here is composed of a quadratic and linear function
of the rate of strain. The quadratic portion is a generalization to three di-
dimensions of the one-dimensional von Neumann q for calculating shocks. The
linear portion provides damping for oscillations that can occur behind the
shock with the q method of calculating the shock front. The term ds/dt used
in the q calculations here is the rate of strain in the direction of acceleration
(see Chap. 4):
j\ +CLpLa

142 6. Elastic-Plastic Flow in Three Space Dimensions
q = 0 for ^ > 0,
where Ax, Ay and A2 are the x, ?/, z components of acceleration respectively,
L is a measure of the zone size taken here as: ZJ zone volume, and
a = \ —,
CL =
The g is added to the pressure P.
6.2.9 Tensor Artificial Viscosity for Stabilizing the Grid
For quasi-static problems in solid mechanics, nonphysical numerical oscil-
oscillations can occur in the grid under certain boundary conditions. A tensor
viscosity based on the rate of strain of volume elements formed by the zone
corners is used to damp this type oscillation. Referring to Fig. 6.2 it is seen
that surrounding point 0 there are eight tetrahedrons denned by the corners
of the eight zones. A Navier-Stokes type tensor viscosity based on the rates
of strain of the tetrahedron volumes is calculated for each tetrahedron that
contains 0, Fig. 6.2. The details for calculating the components of viscosity
for the tetrahedron in zone 0 are given below.
The tetrahedron corresponding to zone ® is shown in Fig. 6.3. The grid
numbering follows the scheme shown in Fig. 6.1. Here grid point 1 corresponds
to point 0 of Fig. 6.2. The finite difference integration mapping procedure is
applied to the four surfaces of the tetrahedron formed by vectors, A, J5, C,
of Fig. 6.3.
Volume vAbc formed by the vectors A, B, C, of Fig. 6.3 is:
1 = \{BxA).C
[
The notation for the components of the vectors is the same as used for the
vectors of the< volume of zone ®.

6.2 Finite Difference Equations for HEMP 3D 143
Fig. 6.3. Grid numbering
scheme for calculating the
tensor viscosity of the tetra-
tetrahedron associated with zone
©
Velocity Derivatives. The velocity derivatives corresponding to the tetra-
tetrahedron. Fig. 6.3, are
n + l/2
x B) • i + xCa(C x A) - i
dx)
1
6v
n+l/2
ABC ,
+xBC(B x C) - i + xED(E x D) • i]
.-, n+l/2
where
±5); X?D = (X2
and ^#c« = u^c- + v^J-;- This expression can be simplified by expressing
vectors D and E in terms of vectors A and B:
1+1/2
1
f (:ri - :/:2)(C X A) • i + (.'h - X
.-in+l/2
whern
(A x D -i) = {(ifbk - <ikbj): (C x A • i) = -(cjdk - ck(ij),
and
(BxC-i) =-

144 6. Elastic-Plastic Flow in Three Space Dimensions
+(ii - ±2)(C x A) • j + (±x - ±4)(B x C) • j]
where
(i4xB-j) = -(aibfc - akbi); (C x A • j) = -
and
(B xC -ft = -{bick-bka).
j]n+1/2
+(±i - ±2){C x A) • k + (±1 - ±4){B x C) • k]n+1/2,
where
(A x B ¦ k) = (aibj - ajb,); (C x A • k) = (c^- - Cjat),
and
(BxC-k) = FiCj -6jCi).
7— and 7— are calculated in the same way as dx/dx,
dx dx
but replacing x by y and then 2.
—- and 7— are calculated in the same way as dx/dy.
dy dy
but replacing x by y and then i.
7— and 7— are calculated in the same way as dx/dz,
dz dz
but replacing x by y and then z.
Components of the rate of strain of the tetrahedron defined by vectors A,
J3, C, Fig. 6.3, are
dx dy dz
dx1 yy dy' zz dz'
(dx dy\ . (dy dz\ . (dx dz"
?xy ~ \dy + dx) ; Syz ~ \dz + dy) ' ^zx " V^ ' dx J '
v _ dx dy dz
v dx dy dz'
Tensor artificial viscosity for tetrahedron A, B, C, Fig. 6.3, is

6.2 Finite Difference Equations for HEMP 3D 145
1 -171+1/2
3vJ
nr+l/2
^n+l/2 _ u in + 1/2. _ u in+1/2. _ |. jn+1/2
where
-in+l
— I
77) v^^BCl
V /0 J
and Cns is a constant w 10~2, po is the reference density of zone Q, and V
the relative volume of zone (J).
The above components of the tensor artificial viscosity are added to the
corresponding components of the stress tensor defined at time n + 1.
Increment of energy dissipated by the tensor artificial viscosity
Here i = 1 —¦ 8 are the eight nodes that define zone Q.
6.2.10 Material Internal Energy
Distortion Energy Increment.
+ TxyAsxy + TyzAsyz + TzxAezx] ^
where sxx, etc. and Aexx, etc. are the components of the stress tensor and
increments of strain respectively defined at the zone center.
Total Internal Energy per Original Volume.
f l) + Pn]+q}
CD
-Vn)
Note: It has been assumed here that the pressure equation of state has the
form P = A(V) + B(V)E.

146 6. Elastic-Plastic Flow in Three Space Dimensions
6.2.11 Time Step Calculations
rn+l
(At)n+3/2 = 0.67
jn of an Zones
and {At)n+^2 < l.l
L is the minimum zone thickness, defined as
where vn+l = volume of zone associated with point i,j,k at tn+1, and sj^+1
is the area of the largest side of the zone. Also, in this equation for At, a is
the sound speed calculated from the equation of state and
n+l/2
b - 8[O0 +CL\L I
where Cq and Cl are the quadratic and the linear q constants, respectively,
and ^| is the rate of strain used in the calculation of q.
Further,
(At)n+l = ~(Atn+3/2 -r Atn+1/2).
6.3 Boundary Conditions
Pseudo zones with zero mass are assumed to surround the grid that defines
the physical object. Thus points associated with the surface of the physical
object may be calculated without changing the logic. Normally a free surface
boundary condition is provided, i.e., the pseudo zone pressures are consid-
considered always equal to zero. Pressure boundary conditions may be applied by
entering the desired space-time values into the pseudo zones.
A reflection boundary condition is obtained by setting equal to zero the
normal component of accelerations of a surface point when it points into the
reflection surface.
6.4 Check Problems
6.4.1 Simple Harmonic Motion
The calculation of the motion of a vibrating plate, clamped at one end, pro-
provides a problem that can be readily checked by elasticity theory. Orienting
the plate at an arbitrary angle in three-dimensional space activates all six
components of the stress tensor.

6.4 Check Problems 147
(a) t = 275ns
Top plane
clamped
i
1
1
(c) t = 450 ms
i
1
w
0 200 400 600 800 1000
Time (fis)
Fig. 6.4a-d. Simulation of the motion of a vibrating elastic plate, (a) Position
of maximum positive displacement, t = 275 ms. (b) Position of maximum kinetic
energy, t = 360 ms. (c) Position of maximum negative displacement, t = 450 ms.
(d) Displacement history for a point in the geometric center of the bottom plane
In the calculations shown in Fig. 6.4 an elastic plate clamped at the top
is set into motion by applying a velocity v = 10 ms to the lower right
edge in the direction perpendicular to the edge for a time t = 50 jis. After
this time the applied velocity is released, but the lower portion of the plate
continues to move due to the kinetic energy. Actually upon release the end
of the plate initially moves faster than the applied velocity since this velocity
does not correspond to the natural frequency of the plate. Figure 6.4d is a
time-displacement plot for a position in the geometric center of the bottom
plane of the plate. It is easily verified that the calculation reproduces the
fundamental frequency of the plate.
Dimensions:
length: L = 52.5 mm,
width: W = 20.0 mm,
thickness: T — 10.0 mm.

148 6. Elastic-Plastic Flow in Three Space Dimensions
t = 0
L = 23.47 mm
D= 7.64 mm
V = 0.25 km/s
Fig. 6.5a,b. Simulation of the impact of a cylinder on a rigid wall. Constitutive
model: Pressure: P — 0.76(p/p0 — 1) Mbar; Density: po — 2.7gcm~~3; Shear mod-
modulus: fi = 0.248 Mbar; Flow stress: Y = 0.0046@.008 + ?pH1 Mbar, where ev is
the equivalent plastic strain, (a) Before and after views using the two-dimensional
HEMP program, (b) Two views using the HEMP 3D program
Elastic constants:
bulk modulus: k
shear modulus: \x
density: p0
= 1.88 Mbar,
= 0.814 Mbar,
= 7.72g/cm3.
The tensor artificial viscosity used in this calculation is Cns = 0 05, more
than enough to suppress the grid oscillations that would otherwise occur.
Figure 6.4d shows that the amplitude of the oscillation has not been damped
or affected by the artificial viscosity.

6.4 Check Problems 149
6.4.2 Plasticity
The impact of a right circular cylinder on a rigid boundary provides a cal-
calculation to test the plasticity aspect of the computer program. Since this
problem requires only two space dimensions it can be calculated with the
HEMP program. Figure 6.5a shows results of the HEMP calculation where
cylindrical symmetry is incorporated into the fundamental equations. Fig-
Figure 6.5b shows results of the same problem calculated with the HEMP 3D
program described here. It can be seen in Fig. 6.5b that the cylinder has
been discretized with three-dimensional zones. The calculated time to stop
the cylinder, 30 \is, and the final cylinder length, 19.28 mm, were the same for
both HEMP and HEMP 3D. Comparison of the cylinder profiles at t = 30 ^is
also showed almost identical results.

7. Sliding Surfaces in Three Dimensions
The sliding surface technique described here has evolved over several years of
applications. Very good results are obtained even for severely warped surfaces.
The implementation of sliding surfaces in a three-dimensional Lagrange
grid z, j, k follows similar procedures as slide lines in the two-dimensional
problem. However, instead of mapping stresses from one side of the interface
to the other side, the vector accelerations are added from one side of the
interface to the other. (This method can also be used in two dimensions but
there is no particular advantage.) Interfaces are defined in z, jf, k space that
separate two regions. The grid points at the interface of one region slide on
the surface provided by the grid points of the opposite region and vice versa.
The grid points associated with one side of the interface are designated in
advance as slave points while the grid points associated with the opposite side
of the interface are designated master points. The calculations are symmetric
in that the grid points of both regions at the interface are advanced in time
in the same manner. After the grid points associated with each region have
been advanced by the integration time step, the positions of slave points are
adjusted to lie on the surface defined by the master points when penetration
of one grid surface into the opposite grid surface occurs. It has been found
convenient to define a local surface at each grid point as the plane through
the grid point that is perpendicular to the normal vector defined at the point.
Thus, the interface between two regions is actually composed of a series of
local surfaces.
All grid points at the interface of the two regions are tagged as either void
open or void closed. Void open means there is a void between the point and
the opposite surface and void closed means the point is in contact with the
opposite surface. Void open points are advanced in time with the usual free
surface calculations.
At the interface between the two regions it will be convenient to refer
to the point that is currently being advanced as the "current" point. Pa-
Parameters associated with the other side of the interface that are required to
advance a current point are identified by the word "opposite". The roles are
then reversed after calculations have been completed for one side of the inter-
interface. The symmetry of the calculation permits sliding surfaces to be defined
simultaneously in more than one direction. However, for illustration of the

152 7. Sliding Surfaces in Three Dimensions
Fig. 7.1. Calculational grid
separated at a constant La-
grange coordinate j. View
of a current grid point / on
the interface
i,j,k = Lagrange coordinate
method, we will assume a sliding surface at a constant Lagrange coordinate,
j; see Fig. 7.1. The letter / will be used to designate a current point.
Free surface boundary conditions are used to calculate the acceleration of
point / in the x, y, z coordinate system. The components of acceleration are
transformed into a coordinate system where two components are in the plane
of the sliding surface interface at point /. The acceleration component normal
to the interface includes a contribution of mass from the opposite grid and,
in addition, the normal component of acceleration of the opposite grid. The
two acceleration components in the plane of the interface are unchanged by
the presence of the interface. The normal component of acceleration from the
opposite grid must include a contribution of mass from the present grid. Thus,
the symmetric treatment of the interface calculations requires preprocessing
each side of the interface. A final calculation is then made to advance in time
points associated with each side of the interface.

7.1 Time Grid Points on a Sliding Surface 153
7.1 Calculational Steps to Advance in Time Grid Points
on a Sliding Surface
Step I
1. Calculate the mass per unit area for all grid points on the sliding interface.
Referring to Fig. 7.1 assume point a is a point on the interface. The mass per
unit area, raa, is given by
m = 1
2
Mq is the mass of zone ®, etc. A® is the area of the triangle in zone ® that
is associated with opposite grid point a. The same applies to Aq, A®, and
A@. The parameter ma is seen to be the average of the masses per unit area
of the zones that share point a.
2. Calculate the acceleration of each point on the interface with free surface
boundary conditions.
For a given point z, j, k calculate the acceleration A*jfc (see point a,
Fig. 7.1):
. * dx. dy. dz^
(i) x direction
dTxy dTzxV
d
dx dy dz \.Jtk'
where
()
¦yv){ziv -zv)
- zv)tinv -
- yv)(zvi - zv)
-(zn - z\/)(yv\ -U
(ziv - zm)(yvi -yin)]
- yin)(zn ~ zni)
n — 2/111)] |;
To form [(l/p)dTxy/dy]\ k, replace each Exx in the right side of the
above expression with Txy, every y with the corresponding z, and each
z with the corresponding x.

154 7. Sliding Surfaces in Three Dimensions
To form [(l/p)dTzx/dz]™. k, replace each Exx in the above expression
with Tzx, every y with the corresponding x, and every z with the corre-
corresponding y.
(ii) y direction
'dy\n _ 1 \dTxy
where
(idrxy\n fidsxxy
I ——- = same as — ) denned above,
\P dx Ji,3,k \p dx Jijk
but with each Exx replaced by the corresponding value of Txy.
= same as I ——^ I defined above,
\p dy Ji,itk
but with each Txy replaced by the corresponding value of Eyy.
9Tyx \ (I dTzx \
—^— = same as — I denned above,
P ®z / i k \P &z / i k
but with each Tzx replaced by the corresponding value of Tyz.
(hi) z direction
dz\n _ i \&rzx dTyz dzzz
at) i k p^j k I ox oy oz
1 dTzx ~ -—XX
x — same as ( — ^ xx ) defined above,
(ldZxx\n
\p dx ),.
Kp dx
but with each Exx replaced by the corresponding value of Tzx.
— ^ — same as ( - ~~xy \ defined above,
but with each Txy replaced by the corresponding value of Tyz.
1 fir \n /I FTP \n
1 OZjzz \ I 1 Ol zx \ jcju
= same as I — denned above,
P dz Ji,j,k \P dz Ji,jtk
but with each Tzx replaced by the corresponding value of Ezz.
3. Determine an outward pointing unit vector normal to an element of surface
defined at each point on the sliding surface interface. Referring to Fig. 7.1
assume point a is any point on the sliding surface interface.

7.1 Time Grid Points on a Sliding Surface 155
(a) Calculate the normal vectors for each of the triangular surface areas
associated with point a. The normal vector corresponding to zone (T) is:
i j k
ra,v,iv = xy - xa yv - ya zy - za
XIV — Xa y\y — ya Z\y — Za
— ^4a,v,ivi 4- -Ba,v,ivj 4- Ca,v,ivk ,
where
Ai,V,IV = [(yv — ya){z\V — Za) — (zy — Za)(y\y — ya)] ,
Ba,V,lV = - [(XV ~ Xa)(ziv ~ Za) ~ (zy - Za)(x\y - Xa)] ,
Oq, v iv "" \Xy — Xdjyy\y — ya) — \2/V — ya)\X\V — *^a) •
The same applies to zones ©, C), and ®, Fig. 7.1.
Note: The vector cross products must be taken so that the normal vectors
point outward from the grid. A convenient way to assure that vector
ra,v,iv points outward from the grid is to take the dot product of roV,iv
with the vector formed by point a and its corresponding interior point.
The interior point is called a connector point. In Fig. 7.1 point VI is the
connector point for point a. If the product is positive, reverse the sign
ra,v,iv, otherwise the vector has the correct outward direction.
(b) Calculate ra, a unit vector obtained from the average of the vectors
perpendicular to the triangles that surround point a, i.e., in Fig. 7.1
triangles (a,V,IV), (a,IV,III), (a,III,II) and (a,II,V):
Cak ,
where Aa, Ba, and Ca are the direction numbers of vector ra and /, m
and n are the direction cosines,
Aa = (Ai,V,IV + Ai,IV,III + ^a,III,II + ^4a,II,v),
Ba = (Ba?v,IV + ^a,IV,III 4- Sa,III,II 4" #a,II,v),
Ca = (Ca,V,IV 4- Cajv,III + Ca,Ul,U + Ca,n,v)-
Step II
1. Locate the opposite surface points associated with each current void-closed
point /:
(a) Calculate d2^ the square of the distance from point / to successive points
i, k of the opposite grid.
d) - (xf - xhkJ 4- (yf - yl,k? + (zf - zhkJ.
(b) Let point a of the opposite grid be the point which has the shortest
distance from point /. See Fig. 7.2.

156 7. Sliding Surfaces in Three Dimensions
1
a) 3
Fig. 7.2a,b. Schematic grid to determine which of the four opposite grid zones
covers a current point /. (a) Quadrants surrounding opposite grid point a with
current grid point / in quadrant 3. (b) System of triangles shown for quadrant 3
(c) Project the points 1, 2, 3, 4, and / onto the local surface at point a,
defined by point a and the unit vector ra at point a. Designate as
(x*,y*,z*)i the coordinates of a point i and (x,y,z)i the coordinates
after projection onto the local surface at point a:
y% = Vi -mdi,
Zi = z* - ndl,
where d% = ra • ct, i = 1, 2, 3,4, and /. The vectors ct are formed by the
points i and the origin. Here I, m and n are the direction cosines of ra,
the unit vector at point a calculated in Step I.3(b).
(d) Point / can be in any one of the four quadrants formed by the projections
of opposite grid points surrounding point a onto the surface defined by
point a and the unit vector at point a, Fig. 7.2. The following procedure
is used to locate the quadrant that contains current point /.
(i) Referring to Fig. 7.2 calculate the area of triangles Z\a,2,3,a,/3, and
7 using the coordinates obtained in Step II.l(c) above.
(ii) Point / is contained within triangle Zia,2,3 if:
A>,2,3 - (a + /? + 7)
< 10~Ma,2,3. Here, A*,2,3, a, 0,7 refer to the areas of the triangles.
(iii) Repeat Step IIl(d)u for the remaining quadrants to locate the quad-
quadrant that contains /.
(iv) If all quadrant tests fail to locate point / then select the next closest
point a to point / and repeat Steps 1.1-1.3.
(v) Repeat 1.1-1.4 for all points / in the current plane.
Note: It is important that the search logic described above be
conducted with the grid points projected onto the local surface
(StepII.l(c)).

1
a
7.1 Time Grid Points on a Sliding Surface 157
Fig. 7.3. Schematic grid to determine whether a current
point / is outside the opposite grid
I
I
ie
t
3
(e) It is necessary to determine when a current point / is not covered by
the opposite grid as shown in Fig. 7.3 Assume current point / has been
determined closest to opposite grid point a and it is known that grid point
a is on the boundary of the opposite grid, Fig. 7.3. The four quadrants
for the search routine described above are formulated by extending the
opposite grid through point a. Referring to Fig. 7.3 an extension point 3
is established by calculating a vector extension:
(i)
= Va + ke(Va -
Here Va designates the vector formed by point a and the origin.
Similar for points 1 and 3. The parameter ke provides the dimension
of the extension. ke is taken as a large number, e.g., 1000, to assure
that point / is covered by the opposite grid for the search routine
that locates the quadrant,
(ii) Coordinates of extension point 3 (Fig. 7.3):
^3 = Xa + keixa ~ xl),
V3 = ya + h{ya -2/1),
Z3 = za + ke(za -zi).
(iii) When point / has been located in a quadrant that contains the
extension point it is outside the opposite grid. If the point is more
than one zone thickness off the opposite grid it is considered a free
point independent of the sliding interface. If the point is less than a
zone thickness off the opposite grid it is considered still on the sliding
interface. To make this distinction the extension point is recalculated
with a value of ke to provide an extension of the opposite grid of
approximately one zone thickness of the current grid. (The default
value is fce = 1, which assumes both grids are the same size.)
Point e in Fig. 7.3 is the new extension point. The following method
is used to locate the position of current point / with respect to the
extension surface. Calculate:
+
+

158 7. Sliding Surfaces in Three Dimensions
X'
y
\
X
X
\
y
y \
• \
Vil
V
current grid
opposite grid —
Fig. 7.4. Current point / is associated with the opposite surface formed at point
a.
Here Ae, Be and Ce are the direction numbers of vector Ce. S is
the vector from point a to current point / and Ce the vector from
point a to the extension point e.
If de > 1, point / is beyond the extension and is accelerated with
free surface boundary conditions.
If de < 1, point / is considered to be still on the sliding interface
and the calculation proceeds.
2. Calculate an interpolated mass per unit area, m/, at the position corre-
corresponding to point /. Figure 7.4 shows an overlay of the current grid containing
point / on the opposite grid. We wish to obtain the mass per unit area of the
opposite grid at the position of current point /. This mass per unit area will
then be used to increase the mass associated with point / for the acceleration
of point / in the direction normal to the sliding interface.
Referring to Fig. 7.5 the mass per unit area at point / is
maa -\- mb/3 4- ^ic7
see Step I.I for the calculation of raa, etc.
3. Calculate the mass weighting factor at point / (z factor):
(a) Let Mm be the mass due to the opposite surface that is to be included
with the mass of point /:
(see Fig. 7.4).

7.1 Time Grid Points on a Sliding Surface 159
Fig. 7.5. Weighting scheme for obtaining the value of
a parameter defined at points a, 6, c at position /. a =
area of Acbj, 0 — area of Acja, 7 = area of Ajba • mj =
maa + raj,/? + rac7
, raa. mb, mc are the mass per unit
a + 0 + 7
area of opposite grid points a, 6, c
Here A® is the scalar area of the triangle (/, V,IV) in zone (§), Fig. 7.6,
and is calculated as follows:
= yf(Af,v,ivJ + (?/,v,ivJ
where
<A/.v,iv = [(yy - Vf)(ziv - zf) - (zv - Zf){y\\ -
C/,V,IV — [(xy - Xf)(y\y - yf) -
Similar holds for 2A®, 2 A®, and
(b) Calculate the z factor
z = 1 + Mm
| (M(g) + M(g) + M@ -
Here M@ etc. are the masses associated with point /, see Fig. 7.6.
4. Calculate the acceleration of grid points on the sliding surface. The ac-
acceleration normal to the sliding surface includes the mass from the opposite
grid using the z factor determined from the preceding step.
(a) Calculate iVJ, the free surface acceleration of point / resolved in the
direction of the average normal, r/.
N} = (A} ¦ 17I7.
The free surface acceleration of point /, A**, was calculated in Step 1.2.
The average normal, 17, was calculated in Step I.3(b).
r~
1 d
vf-
1 d
IV
r --1
-f-
Fig. 7.6. Grid associated with current point /

160 7. Sliding Surfaces in Three Dimensions
(b) Calculate Nf, the acceleration of current point / in the direction of the
average normal that includes the mass of the opposite grid,
N*f
5. Repeat II.1-II.4 with the opposite grid as the current grid.
Note Nf is a partial acceleration normal to the surface defined at point
/ that includes the mass of the opposite grid. The total normal acceleration
of current point / must also include a contribution from the opposite grid
and is described in Step III that follows.
Step III
1. For the current void closed point /, locate the opposite grid points that
surround point /. See Step II. 1 and Fig. 7.4.
2. From the three opposite grid points a, b, c that surround point / (Fig. 7.5)
determine acceleration vector N®0. Vector N°G is an interpolated normal
component of acceleration from the opposite grid at the position of point /.
The interpolation method is shown in Fig. 7.5,
4- Nc7
where Na, AT6 and Nc are the accelerations of opposite grid points a, b and
c calculated in Step II.5.
Test for void opening. If Nf • Cf > 0 and NGG • Cf < 0 tag point / as
void open. Nf is calculated in Step II.4. The vector Cy is formed by point /
and the connector point associated with point /. If the above test is positive
the acceleration of point / is given by AJ, the free surface acceleration from
Step 1.2.
3. Calculate the total acceleration, A/, of point /,
Af = A} - N} + Nf + N^G = Ax\ + Ay] + A2k.
Note: Nf must be saved for use in the interpolation procedure when the
above process is reversed and the current grid becomes the opposite grid.
4. Calculate the x, y, z components of the velocity and the new coordinates
for the current point /.
(a) Velocity

7.1 Time Grid Points on a Sliding Surface 161
(b) New coordinates
xnf+1=xnf+±nf+1/2Atn+1'2,
5. Repeat III.1-III.4 for all interface grid points.
Step IV
1. Test to see if a point / has penetrated the opposite grid. Assume /, in
Fig. 7.2, is a point on a grid that is to be tested for penetration into the
opposite grid local surface at point a.
(a) Calculate d, the perpendicular distance from the point / to the local
surface at point a.
d = [/i 4- raj + rak] • [{xf - xa)i + (yf - ya)} + (zf - za)k]
or
d = l(xf - xa) + m(yf - ya) + n(zf - za).
Here /, m, n are the direction cosines of the unit vector defined at point
a; see Step I.3(b).
(b) If 0 < d < 5, point fn+l remains as calculated in Step III with the same
void status as before. Here S is a positive number equal to 0.1 times the
grid spacing calculated as:
S - O
(c) If d > 5, point fn+l remains as calculated in Step III and is tagged void
open.
(d) If d < 0 point /n+1 has penetrated the opposite grid and is tagged void
closed.
2. Adjust the velocities of all void-closed points.
(a) Calculate velocities normal to the interface. Assume point / in Fig. 7.4
has penetrated the local surface at point a of the opposite grid, calculate
the velocity components Nf, Na, JV&, and iVc that are normal to the
surface:
Na = lxa + rnya + nza,
Nb = lxb + myb + nzb,
Nc = lxc 4- myc + nic,
Nf = l±f -f mijf + nzf.

162 7. Sliding Surfaces in Three Dimensions
(b) Calculate Nt the velocity of point / from the conservation of linear
momentum:
'aMaNa + f3MbNb-, ,_c..c ._ AT
— + MfNf
a
h Mf
Ma + -[Af® + M^ + M® -f M0] = 2(<f>)a,
and similarly for M^ and Mc.
Note: The mass # associated with a point on the interface is calculated
in Step I.
Note also that to minimize the number of calculations, only the linear
momentum has been considered instead of including conservation of an-
angular and linear momentum as was done in the two-dimensional problem.
Actually it is the artificial viscosity, q, and the equations of motion that
accomplish the conservation of momentum. Adjusting the velocities at
the interface after a collision sets up the initial conditions for the artifi-
artificial viscosities on each side of the interface,
(c) Calculate the x, y, z components of the new velocity of point /.
xf
-+1/2
.n+1/2
where
= ±f-
= y}-
= z}-
\-mANf,
\-nANf,
ANf = (N+ - Nf).
Here x^,y^,i*/ refer to the velocity of point / before the adjustment
for conservation of momentum. The velocities of all void-closed points
are adjusted point by point. That is, only the velocity of the point under
consideration is adjusted, point / in the example described above. The
velocities of all mass points on one side of the grid are adjusted when
penetration has occurred. Subsequently the velocities of all points on the
opposite grid are adjusted. Thus, all of the old velocities and coordinates
must be retained until all of the velocities of both sets of sliding surface
points have been adjusted.
3. Declare one grid the slave grid and the grid opposite it the master grid.
Relocate slave points onto the master surface for slave points where d < — S. A
slave point / that has penetrated the master surface is set back to the master

7.3 Zone Dimension Change and Subcycling 163
surface by subtracting the length d from the position of the point. For the
sign convention used here d is a negative number. The direction cosines /, m,
n point outward from the grid, thus the new coordinates of point / are
z^1 = z}- nd.
Here x**,y*f, z^ refer to the coordinates of point / used to determine whether
d< -6.
When penetration occurs new velocities are calculated on both sides of
the interface, but only the positions of slave points are adjusted.
7.2 Applications of Sliding Surface Routine
The major difficulty with sliding surface routines arises from failure of the
search routines that must locate one grid with respect to the other. The
problem becomes aggravated with curved or warped surfaces with a search
routine that operates in three dimensions. The method used here projects the
grid point onto local two-dimensional surfaces to establish the orientation of
one grid with respect to the other. With this procedure the search technique
is robust even for distorted surfaces. Figure 7.7 shows an application with
two curved surfaces. Figure 7.8 shows the acceleration of a metal plate by
an explosive with a sliding surface between the two materials. The explosive
was detonated at nine equally spaced points on a line along the top surface
of the explosive [7.1].
7.3 Zone Dimension Change and Subcycling
It is useful to be able to change from coarse to fine zoning in a localized region
and to be able to join two independent grids at an interface. The latter being
especially important for constructing grids for three-dimensional problems.
The method is described first for a two-dimensional grid and subsequently
for a three-dimensional grid.
7.3.1 Zone Dimension Change at an Interface in Two Dimensions
Figure 7.9 shows schematically a zone change from large to small zones across
a Lagrange coordinate ks. The grid with the largest zone size is chosen as the
master grid and the small zone grid as the slave grid. The master grid defines
the interface ks. In Fig. 7.9 grid points associated with the master and slave
grids are shown as closed circles and open circles, respectively.
Refer to the master and slave points at the interface /cs, Fig. 7.9.

164 7. Sliding Surfaces in Three Dimensions
Tungsten
Copper
1.2 km/s
t=0
t=6|is
(a)
(b)
Tungsten projectile at
later times
Point of impact
Experimental result
Fig. 7.7. (a) Simulation of a copper plate charge striking a tungsten projectile,
(b) Experimental result
(a) Calculate the partial acceleration of all master points j, k on /c-line ks.
j • \ master ., r
ax \ 1 I -i
G.1)
)UVU - Vfo) ~

Steel case
(b)
7.3 Zone Dimension Change and Subcycling 165
High explosive
t=0
t=50 us
t=70 us
t=90 us
t=125 us
Fig. 7.8a,b. Calculation of a three-dimensional implosion of a copper liner, (a)
Section view of geometry, (b) Time sequence of implosion
°A°
p°A
r\n — J \Tn
^ =
0 \Vn
G.2)

166 7. Sliding Surfaces in Three Dimensions
IV
Fig. 7.9. Schematic of
a zone dimension change
at /c-line ks. Master grid
points shown as closed cir-
circles and slave grid points as
open circles
The z factor that is found in G.1) above is obtained by mapping the
masses of the slave zones between master points II and IV onto the
master grid.
(b) Calculate the partial accelerations for all slave points on /c-line, ks. The
same procedure as above is used. The z factor now maps mass from the
master grid onto the slave grid.
For each master grid point on /c-line ks determine a slave grid partial
acceleration by interpolation. Referring to Fig. 7.9 the slave grid partial
acceleration corresponding to master grid point a is
(c)
slave
slave
slave
Here a — lat/ht where lat is the distance between points a and t and lst
the distance between points s and t.
Similar holds for
^ slave
(d) Calculate the total acceleration of all master points on /c-line ks. Referring
to Fig. 7.9 the total acceleration of master point a (point j, k Fig. 7.9) is
dx
Hi
i . \ slave
Similar holds for

7.3 Zone Dimension Change and Subcycling 167
(e) Calculate new velocities for all master points on fc-line /cs.
._ ^ri-l/2 A n+i/2
(f) Obtain new velocities for the slave grid points on fc-line ks by interpo-
interpolation so that the original spacing between consecutive master points is
maintained. Referring to Fig. 7.9 the new velocities for slave point t are
with
Here C is a constant calculated when the grid is generated. /^ is the
distance from point t to point b and lab the distance between points a
and b.
7.3.2 Zone Dimension Change of an Interface
in Three Dimensions
The same procedure is followed as described for the two-dimensional case.
Figure 7.10 shows two grids that are to be joined together without the inter-
interface grid points of both grids being necessarily coincident.
Refer to Fig. 7.10 where point a is considered a master grid point associ-
associated with master grid zones 0, ©, C), ®.
(a) Calculate the partial acceleration for all interface grid points associated
with the master grid.
Partial acceleration in x-direction.
i . \ master 1 r o r-i o/t-i o/t-i t n
dx\ 1 \dZxx dTxy dTzx\
where
dx
1
+(Zxx)®[(yvi - yin)(zu - zni) - (zvi - zUi)(yii - 2/ni)]}- G-4)

168 7. Sliding Surfaces in Three Dimensions
Master
Slave
Fig. 7.10. Joining of two independent grids
-f
G.5)
The factor z that appears in G.4) above is the weighting factor that maps
the mass of the opposite grid. The remaining terms in G.3) are composed
in the usual manner and include the z factor, as shown in G.4), the finite
difference equation for the first term in G.3). In a similar manner the y,
z components are calculated.
(b) Calculate the partial acceleration for all points associated with the slave
grid on the interface. The z factor now maps the mass from the master
surface onto the slave grid point.
(c) For each master grid point determine a slave grid partial acceleration by
interpolation. Referring to Fig. 7.11 assume master grid point a has been
found to be in the neighborhood of slave grid points /, g, h.
dx
slave n, (dx\
Here the subscripts /, g, and h denote the partial acceleration of the
respective slave grid points. Similar holds for the y, z components.

7.3 Zone Dimension Change and Subcycling 169
Fig. 7.11. Interpolation scheme for obtain-
obtaining at position a information defined at posi-
positions /, g, h where a = area of Aahg, 0 = area
of Aafh, 7 = area of Aagf
(d) Calculate the total acceleration for all master points on the interface. For
master point a (point i, j, fc, Fig. 7.10) the total acceleration is
/d±\ _ /dz\slave /dz\master
\dt)ijjc~{dt) +{dt)
and similarly for y, z components of acceleration.
(e) Calculate new velocities for all master points on the interface.
(f) Obtain new velocities for the slave grid points on the interface using the
interpolation scheme above.
7.3.3 Subcycling with Zone Dimension Change
in Two Dimensions
The time step for a given cycle is dictated by the zone with the smallest
zone dimension divided by the local sound speed. Rather than calculate the
entire grid with a given time step a saving in computer time can be obtained
by dividing the grid into different regions with a different time step for each
region. A region with small zones is calculated for several time steps until the
time step of a region that can use a larger time step is reached. The region
with the larger time step is then advanced with a single time step equal to the
sum of the time steps used in the region with the small zone. It is convenient
to calculate the region with the largest time step first. The time steps for
the regions with the smaller time step requirements are chosen so that an
integral number of equal times steps can be used to reach the time step used
for the largest grid.
7.3.4 Example for a Zone Size Change of Two to One
(a) With a time step At calculate new velocities for all grid points of the
largest grid using the stress boundary conditions provided by the small
grid. Advance all points of the large grid and calculate the new zonal
parameters. Save the old positions of the large grid for points on the
interface between the two grids.

170 7. Sliding Surfaces in Three Dimensions
(b) Find the new velocities of all small grid points that are on the interface
by interpolation. These velocities are the boundary conditions during the
subcycling of the small grid.
(c) With the old interface positions, so that all of the positions of the small
grid points are at the same time, calculate new velocities from the accel-
acceleration equations for all small grid points except those on the interface
using a time step At/2.
(d) Calculate new coordinates for all of the small grid points including those
on the interface. The points on the interface use velocities from step (b)
to obtain new coordinates and the remainder of the points of the small
grid use the velocities from step (c).
(e) Calculate the new zonal quantities for the small grid.
(f) Calculate new velocities again for all small grid points except those on
the interface using time step At/2.
(g) Calculate new coordinates for all grid points including those on the in-
interface.
(h) Calculate new zonal quantities for the small grid.
The procedure is similar for three dimensions.

8. Magnetohydrodynamics of HEMP
Using the continuum mechanics approach, details are presented for solving
the equations of magnetohydrodynamics in two space dimensions and time.
The problem considers cylindrical symmetry, in that only the He component
of a magnetic field is present. The problem is formulated so that the stress
contributions resulting from a magnetic field are incorporated into the stress
tensor of an elastic-plastic computer program. In addition to the Lorentz
force and magnetic diffusion, thermal and radiation diffusion are also treated.
Presented here is the magnetohydrodynamic portion of the HEMP program
including thermal and radiation diffusion. Details of the calculations are given
for the case where only the Hq component of an applied magnetic field, H,
is included. The problem consists of developing a finite difference approxima-
approximation to the double operator, VxVx V, where V is a vector function. The
mathematical problem is similar to that for approximating the double oper-
operator, V • VV, where V is a scalar function. For the problem at hand, V is
the magnetic field, H and V is the temperature, T. It is seen from the vector
identity
Vx Vxif - V(V • H) - V • Vif
that
Vx Vxif = -V-VH = -V2H,
since
V • H = 0.
In the calculation of thermal diffusion of the temperature, T, a simple ex-
exchange of if for T cannot be made in the difference equations. The reason is
that in cylindrical coordinates the Laplacian operation, V2, yields a different
result when applied to a vector than it does when applied to a scalar quantity.
For the case considered here, where if = He, the relationship between
the Laplacian applied to the vector, He and to a scalar, He is:
Here R is the radial space coordinate.

172 8. Magnetohydrodynamics of HEMP
An important consideration in calculating magnetic diffusion or thermal
diffusion is the manner in which the constitutive relations are introduced into
the difference equations. The constitutive relations describing the electrical
conductivity in the calculation of magnetic diffusion can be included in a
more physical way if the curl-curl or Vx Vx formulation is used. Thus for
the difference equations given here approximations are made to the operator
VxVx acting on the vector H$, and to V2 acting on the scalar, T.
8.1 Finite Difference Scheme for Double Operators
As explained earlier the changes of variables associated with a mass point are
interpreted as due to a flux through the surface surrounding the mass point.
Figure 8.1 shows the integration path for obtaining the first derivative defined
at a zone node (point 1) from parameters defined at zone centers. Figure 8.2
shows the integration path for obtaining the second derivative defined at
zone centers (position 0) from the first derivatives that were defined at zone
nodes.
Figure 8.1 shows the integration path for evaluating components of the
vector, VxH. Referring to Fig. 8.1, if F is a function (in this case, F = He)
defined at zone centers, then the difference equations for evaluating dF/dX
and dF/dY centered at point 1 are
8F_
dX
where
A dF
and w
f F(ni)dl = -
J(C)
and
IV
I
k+1
k
k-1
j-i j j+i
j,k Lagrange coordinates
—X
Fig. 8.1. Grid for calculating Vxif,
where H is given in a zone center. Dotted
line shows integration path for evaluating
components of Vxif, centered at zone
node point 1: similar integration paths
surround points 2, 3, and 4

8.1 Finite Difference Scheme for Double Operators 173
Fig. 8.2. Grid for calculating VxVxH,
where VxH is given at node points.
Arrows show integration path where
VxVxif is centered at j + 1/2, k + 1/2
(zone center)
F(n.j)d! =
(C)
- Xu)].
A is the area enclosed by the integration path, I, II, III, IV (Fig. 8.1), and is
taken as the average of the area contributions of the four zones surrounding
point 1 (Fig. 8.1).
Figure 8.2 shows the path for evaluating VxVx/f. If F is a function
(in this case a component of the vector VxH) defined at zone nodes, then
the difference equations for evaluating dF/dX and dF/dY', defined at a zone
center, are
dF
~dx
f,c)F(n-i)dl
0
Similarly,
dF
W
Here F23 = (F2 + F3)/2, etc., and A is the area of zone
1, 2, 3, and 4.
defined by points

174 8. Magnetohydrodynamics of HEMP
8.2 Fundamental Equations of Magnetohydrodynamics
The following equations are solved in Lagrange coordinates with 2D cylindri-
cylindrical symmetry, using the structure of the HEMP program.
8.2.1 Equation of Motion
dW
P^f
(8.1)
As shown in the following section, the term in braces [Lorentz force] is
incorporated into the total stress tensor, Z1, of the HEMP program.
8.2.2 Electromagnetic Field Equations
Faraday's law:
fim^- [ [ H nds= - [ Ed/. (8.2)
dt J Jo J(c)
Ampere's law:
VxH = 4ttJ. (8.3)
Ohm's law:
E=^. (8.4)
By Stoke's theorem and substitution of (8.3), (8.4), equation (8.2) can be
written as
/im^//ifnds = / E.<U = - /YvxE-nds
at J Js J(c) J Js
= -//sVx(^cVxH)nds- (85)
The above relations apply to a local region of the fluid that moves with the
fluid. Applying (8.5) to the Lagrange region and considering an He field only
gives the magnetic flux diffusion equation:
A, (8.6)
where /xm^ = nmHeA is the magnetic flux, H$ the theta component of mag-
magnetic field, if, and A is a surface element, defined later as the area of a
Lagrange zone. It is in the form of (8.6) that Faraday's law, Ampere's law,
and Ohm's law are used in the calculation. It is noted that the displacement
current has been omitted and that V • fimH = 0 is identically satisfied for
the case considered here with H = Hq only and symmetry in the 0 direction.

8.2 Fundamental Equations of Magnetohydrodynamics
8.2.3 Energy Equation
The total change in internal energy dE, in a given volume is
175
dE =
{P + q)dV + V
dZ,
K + -aRc\T3 ) VT
]}¦
72
+V—
G
(8.7)
where
(sxx -
xdt
+V
~ ^H2) eyydt + V (see + ^H2) ieedt.
The first term on the right in (8.7) is the change in internal energy from pres-
pressure forces; the second term is that from thermal conduction and radiation
flow; the third term is the energy change from ohmic heating; and the fourth
term that from distortion stresses.
The following definitions are used:
E = cvT + aaVT4 = material energy + radiation energy
dV
dE_
~df
dE
dV
v
dem
dT
dem
dV
dV
AaRVT3
(8.8)
(8.9)
(8.9a)
(8.9b)
here, em = material energy = cvT. Substitution of (8.9) into (8.8) gives:
dE
= - P
dE
dV
+(dtV)V ¦ {AVT) +
q)dV
VJ2
+ dZ,
(8.10)
where
A = K + -aRc\T3
o
and dZ is the change in internal energy from distortion stresses.
Rewriting (8.10):
df
(AVT)]dt + W,
(8.11)
where

176 8. Magnetohydrodynamics of HEMP
VJ2
+ q ) dV + -77-dt -f dZ,
P = material pressure 4- 1/3<2rT4.
It is in the form of (8.11) that changes in the internal energy are calculated.
Equation (8.11) is a diffusion equation for the temperature T.
8.2.4 Continuity Equation
8.2.5 Constitutive Relations
P
Pm
Pr
K
X
cv
C
dE
dT
dE
av
= Pm + PR
= Pm(T,V)
= K{T, V, H)
= \(T,V)
= cv(T,V)
= C(T,V)
= cvT
dem
v ®T
dem
dV
+ 4aRT3
V
T4
T1
(total pressure)
(material pressure)
(radiation pressure)
(thermal conductivity)
(Rosseland mean free path)
(specific heat)
(electrical conductivity)
(material internal energy)
(change in internal energy with respect
to temperature)
(change in internal energy with respect
to volume)
The above relationships must be given in advance in the form of explicit
functions or table look-ups.
8.3 Difference Equations for Magnetohydrodynamics
8.3.1 Equations of Motion
AW
(8.12)
where the Lorentz force (effect of the magnetic field on the fluid motion) is
4tt

8.3 Difference Equations for Magnetohydrodynamics 177
Using X, Y coordinates with cylindrical symmetry about the X axis, and
considering a field, H, in the theta direction Hq only, (8.12) becomes
dTxy Txy
dX dY
P
dX dY
Y
Eyy -
yy
dX
(_/
V
8tt
An Y
or
f ~dJT dY
dSyy
dY
dX dY
Eyy - See
Y
814)
where
~8nH '
It is seen that the magnetic force is like a pressure in the X and Y directions,
and like a tension in the theta direction. With the addition of the above
pressure-like terms from the magnetic field, there are no other changes to the
finite difference equations of motion of the HEMP program.
8.3.2 Magnetic Diffusion
We consider the calculation of d$/dt centered in space at j -f 1/2, k + 1/2,
(see Fig. 8.3) and in time at tn+l/2.
where H = He only, A is the zone area, C the conductivity, # = HA, and
/im is the magnetic permeability.
k+1
1 i
2:,,
t Axis of
-T~\ cylindrical Fig. 8.3. Lagrange zone defined by
* < symmetry points h 2, 3, and 4

178 8. Magnetohydrodynamics of HEMP
H
H
H
H
n+1
H
n+1
H
n+1
.n+1
H
H
H
H
•k+2
k+1
k
k-1
j+1 j+2
Fig. 8.4. Grid showing the time center-
centering of H used to evaluate the components
of V X if for calculating the flux change
in the /e-direction, where j and k are La-
grange coordinates
Procedure
(a) The vector VxH is evaluated at zone node points 1, 2, 3 and 4, using
the difference scheme of the preceding section, where H is considered defined
at a zone center (see Fig. 8.3).
(b) The required value of Vx (^Vxif) defined at j + 1/2, k + 1/2 (zone
center) is obtained by averaging the components of the vector VxH, so that
they are defined at the middle of the line connecting consecutive j, k points
and integrating around the area A, as discussed in the preceding section. A
representative value of the conductivity, C, for each leg of the integration
path is obtained by an averaging scheme based on current flow in parallel
circuits.
(c) A forward difference scheme is used which means that the desired value
of Hn+l at time, *n+1, is an implicit function of #n+1, i.e.,
For a given j-line, the value of Hn^,2 k+l,2 can be found by solving a set
of linear equations if we first evaluate Vxff at points 1, 2, 3 and 4, using
values of #n+1 for all /c-lines corresponding to the given j-line (Fig. 8.4). The
change of flux, /imd#, is then determined by using only sides 1-2 and 3-4 in
step (b) above. The conductivity, C, associated with a side is obtained from
the conductivity of the zones on either side of the line being considered. The
calculated value of /imd$ will be the change in flux in the ^-direction. Details
of the calculation are given in Appendix E.
(d) The calculation of VxH is repeated for points 1, 2, 3 and 4, but this time
using values of i/n+1 for all jf-lines corresponding to a given fc-line (Fig. 8.5).
The change in flux, /zmd# is determined as in step (c), but this time using
only sides 2-3 and 4-1. This will be the change in flux in the j-direction. See
Appendix E.
(e) The total change in flux /im$, for the zone j = 1/2, A: 4-1/2 is obtained by
adding together the flux change in the two directions; i.e., total flux change,

8.3 Difference Equations for Magnetohydrodynamics 179
n+1
H"
Hn+1
H"
Hn+1
h"
Hn+1
2h"
j-i
k+2
k+1
k
k-1
—- X
Fig. 8.5. Grid showing time center-
centering of H used to evaluate the compo-
components of V X if for calculating the flux
change in the j-direction
/imd# = /j,m(d<P)k + /jLm(d$)j, where (d^)* is obtained from step (c), and
(d$y is obtained from step (d).
(f) The desired value of Hn+l is obtained from step (e):
(HA)n+1 - (HA)n = [(HAO1*1 - (HA)n]k +
n)\
- (HA)n)\
[(HA)n]= [(HAT]3 = (HA)n,
Hn+l = (Hn+1)k
y _
An+l
When specified to one space dimension, the space and time-centering
of the difference equations correspond to the Crank-Nicholson differencing
scheme [8.1].
8.3.3 Energy Equations
Temperature diffusion
Rewriting (8.11) with the quantities at times n and n 4-1/2 considered to be
known, where the unknown quantity is Tn+1, gives:
(8.16)
where VT is calculated with Tn and 7in+1. The temperature, T, is centered
at j + 1/2, k + 1/2 (zone center).
(a) Equation (8.16) is solved in the same manner as the magnetic diffusion
equation (8.15) to obtain new temperatures (Tn+1)k and (Tn+l)j from the
temperature diffusion in the k and j directions, respectively. Details of these
calculations are given in Appendix F.
(b) The total energy change is the sum of contributions from the k plus
the j directions. The value of dE/dT is considered constant during the time
interval, tn —> tn+l. The new temperature for a zone is obtained as follows:

180 8. Magnetohydrodynamics of HEMP
rpn+l _ rpn (rpn+\\k _ rpn I /rTin+l\j _ rpn
thus
rjin+l /pn+\\k i (pn+\\j _ rpn
This temperature is considered to be the first approximation to the desired
temperature and will be designated as fn+1.
(c) The pressure, Pn+1, and (dE/dV)n+l are calculated from the given con-
constitutive relations using Tn+1 and Vn+1. The quantity, W, is then recalcu-
recalculated using Pn+1/2 anj (dE/dV)nJtl/2, with the other quantities remaining
the same, and where
pn+1/2 = i/
2V
1
dV } 2
as
(d) The quantity (dE/dTO1*1/2 is calculated from the given constitutive
relation. The time-centering of T is given by
rpn+l/2 _CT'n+l _|_ T171)
/rpn+l/2\3 _ \/rrin+l\3 , /jm+l\2/jm\ , /rpn+l /rrm\2 , /T^n>\3l
This definition of T3 assures that
dE = (cv + 4aRVrT3)dT,
if cv and V are constant in the time interval between n and n + 1.
(e) Steps (a) and (b) are repeated using up-dated values of dE/dT, A and
W obtained from the time-centering given in steps (c) and (d), and the con-
constitutive relations.
Space-centering of parameters at the interfaces of consecutive zones
In the integration scheme, representative values of parameters calculated from
the constitutive relations are required for each leg of the integration path.
The averaging technique used is given below.
Consider the leg 1-2 of the integration path 1, 2, 3, 4, 1 around zone ©
in Fig. 8.6. We then find the following:
Coefficient of thermal conduction, K
where A is the zone area.

8.3 Difference Equations for Magnetohydrodynamics 181
Fig. 8.6. Nodes 1, 2, 3, and 4 and zones ©, @,
®, and (§) surrounding zone ©
Rosseland mean free path A
A^\ -4- 4/FTs
A1-2 =
To insure the thermal flow is from high to low temperature zones, AjjU and
are calculated using the maximum temperature of the zones B) and © [8.2
Transmissivity, A
where
^1-2^1-2'
Electrical conductivity C
Cl-2 —
where M is the zone mass.
Similar expressions are written for legs 2-3, 3-4 and 4-1.
Ohmic heating, J2/C
c
16tt2C
1
(VxifJ
16tt2Cv" ' v h
where (Vx JT) = ai + 6j.
Referring to Fig. 8.6, the ohmic heating for zone © is given by
J2_
C
(8.17)

182 8. Magnetohydrodynamics of HEMP
where
( — ) =
2 2
C® \fa2+al
2
1
,
, >! + «? ,
2 J V C2_3
2
2 2
The superscripts /c and j refer to the /c and j sweeps described earlier.
8.3.4 Continuity Equation
Mass is conserved explicitly in the Lagrange formulation of the fundamental
equations.
The continuity equation
is used to describe components of the strain tensor and is explained in
Chap. 4.
8.3.5 Time-Step Control
The time step is taken as the minimum over the grid mesh of the mesh
size divided by the largest wave speed (Courrant condition). This choice
of time-step can insure stability of the difference equations, but does not
insure accuracy of the diffusion calculations. In the diffusion calculations a
much smaller time-step is required in order to provide communication of the
boundary conditions throughout the grid.
An extrapolation method devised by Julius Chang and Jim LeBlanc per-
permits a large time-step to be used, but still maintains the accuracy of a much
smaller time-step. The application specialized to one dimension is discussed
in Ref. [8.3]. This scheme provides considerable savings in computation time
in the solution of diffusion equations by the double sweep technique described
here.
With an equation of the form
l _ pn
this extrapolation method advances in time the parameter F from Fn to
Fn+1. Using the double sweep technique, one solves the implicit equation
three times, as follows:

F"
F"
- Fn
At
2
8.3 Difference Equations for Magnetohydrodynamics 183
- Fn
""' vn) (8.18a)
rii,Fn) (8.18b)
M =/(Fiii,Fii). (8.18c)
ThenFn+1 =2^"-^.
8.3.6 Boundary Conditions
The magnetic field, temperature, and pressure must be specified for all times
on the exterior boundaries of the material region being considered. Usually
the pressure is assumed zero on exterior surfaces. For problems where the
material occupies a region containing the axis of symmetry, the boundary
conditions for the axis of symmetry are
_dP _ dT _
where Y is the radial coordinate. For symmetry planes perpendicular to the
Xaxis:
8He _dP _ dT _
~dX ~"dX~ ~dX~ '
For the application of boundary conditions, it is convenient to provide phan-
phantom zones around the region of interest. The zero gradient conditions above
are achieved by reflecting the values of the interior parameters into the phan-
phantom zones.
8.3.7 Sliding Interfaces
For most problems of practical interest, it is important to allow for one ma-
material to slide upon another. The details of the computer logic for sliding
interfaces in the framework of elastic-plastic flow are given in Chap. 5. In
the solutions of the diffusion equations presented here, it is seen that the
zones must be continuous in the Lagrange zone directions, j and A:. This is
accomplished by subdividing the grid just before the diffusion calculations
are made.
Figure 8.7 shows a sliding interface that separates regions A and B. The
slide line logic of the HEMP program is used to develop in the computer the
geometry corresponding to Fig. 8.7. As shown by the dotted lines in Fig. 8.7
the grid of region B is extended through region A. Thus, region A is now
considered to be mapped by the original k lines of region A and the j lines of
region B. The new j grid of region A is generated so that it intersects the k

184 8. Magnetohydrodynamics of HEMP
Y
ia
bi
S
Sliding
Interface
> Region A
> Region B
¦e-
Fig. 8.7. Grid for calculating field and temperature diffusion across a sliding in-
interface. Solid lines — Lagrange grid at end of equation of motion calculation. An
ordered grid is created by the dotted lines for the diffusion calculations
lines in the same proportions to the original j lines as occurred on the slide
line. The new grid will overlay one or more zones of the old grid of region A.
A single value of the parameters H or T is obtained for each zone of the new
grid by an area weighting scheme. For example, if a zone of the new grid is
composed of parts of zone a and zone 6, see Fig. 8.7, then the appropriate
value of the parameter H for the new grid is
HgAg + HbAb
~ Aa + Ab
(and similarly for T). Here Aa and Ab refer to the area of the new grid that
overlay zones a and b respectively of the old grid.
Thus, the form of the original calculational scheme is established and the
same method can be used to solve the implicit equation for new values of the
magnetic field H and the temperature T. After the diffusion calculations are
completed, the updated parameters are reassigned to the solid line grid by
an area weighting scheme similar to the above.
In this example the zones of region B have been used to define zones in
region A. The same logic could have been applied using region A to define
zones in region B.
It is seen in Fig. 8.7 that region A extends beyond region B. The problem
requires that the boundary conditions are always known in advance of a time
step calculation. This information is carried in the computer logic by pseudo
zones that surround the entire calculational grid. The use of pseudo zones
permits parameters on the grid boundaries, including those of the overhang
of region A (Fig. 8.7) to be calculated in the same manner as any other point
in the grid. (A slide line extension is required if points associated with region
B on the slide line A:s move beyond the ends of region ^4).
The example shown here has considered a single sliding surface along a
k line. The same approach is used if more than one k line is a slide line.

8.3 Difference Equations for Magnetohydrodynamics 185
Axis of
symmetry
t = 0
- Air ^Fe High pressure gas
fr- Position reference
Axis of
symmetry
Later time
T
-Air <-Fe
h- Position reference
^-High pressure gas
Fig. 8.8. Application of slide lines in two directions. The calculation permits sliding
between each pair of adjacent materials
The region that is to control the zoning for the diffusion calculations must
be stated in advance.
Figure 8.8 shows a hydrodynamic calculation set up with slide lines in
two directions (j and k slide lines). This provision in the computer program
is very useful for permitting a fine-zoned region to be defined in the two-
dimensional grid. The same mapping scheme for calculating the temperature
and/or magnetic diffusion is used, the only difference being that additional
mapping is required so as to include all material regions surrounding a given
region, i.e., both the fc-lines and the j-lines are extended from the region that
controls the zoning.
8.3.8 Check Problems
The accuracy of the numerical technique that calculates the diffusion of T4
in one space dimension can be checked by a problem suggested by W. Schultz
of Lawrence Livermore National Laboratory. A material with the special
properties
106T2
K = 0, P = 0, Cv = p0T, and A =
is considered. An extreme density p0 = 105g/cm3, is used to insure that
the radiation energy density is negligible compared to the material energy
density.

186 8. Magnetohydrodynamics of HEMP
=t-H 16 zones
X— 16
Fig. 8.9. Calculational grid to test T4 dif-
diffusion, T — temperature, t — time
Specializing to one space dimension, the energy equation (8.7) with the
above material model can be written:
dT4-
dE
OX
Substituting for A, cv and po where E = cvT — poT2, V = 1 and c =
3 x 108 m/s gives
dt dX V
This equation has T4 = t — X as a solution.
A calculation was done using the zoning shown in Fig. 8.9 with boundary
conditions T4 = t at the left end and T = 0 at the right end. The results of
the calculation are shown in Fig. 8.10. The calculated quantity, T4, is plotted
at zone centers. It can be seen that all points of the calculated results fall on
the straight lines of the theoretical solution, T4 = t - X.
A test of the accuracy for solving the thermal diffusion equations in two
space dimensions and time can be obtained by imposing the first Fourier
harmonic to the general solution of the heat equations on a unit square,
as an initial condition. The theoretical result for this initial condition is an
exponential decay in temperature T as time t progresses. Material properties
are assumed such that the energy equation (8.7) becomes
dT/dt = V2T.
15
I i I
• ° Finite difference
— Theoretical
15
Fig. 8.10. Comparison of the finite dif-
difference calculation with a theoretical
calculation for the diffusion of T4

8.3 Difference Equations for Magnetohydrodynamics 187
(b)
t=0.025
t=0.075
(c)
(d)
0 0.5 1.0
X coordinate (along diagonal)
Fig. 8.11a-d. Calculation of thermal diffusion in two space dimensions and time.
(a, b, and c) Temperature distributions at t = 0 and at later times, (d) Compar-
Comparisons between the finite difference calculation (points) and the theoretical solution
(curves). Temperature is shown as a function of distance (X) along a diagonal of
the square
A calculation was done using a unit square with a 20 x 20 grid. Initial con-
conditions were T(X,Y,0) = sinGrX)sinGrF). Boundary conditions on all four
boundaries were T — 0 for all time.
The theoretical solution to this problem is
T(X,Y,t) = sinGrX)sinGry)
Figure 8.11a-c show temperature distributions for various times. Fig-
Figure 8.lid compares the calculated and theoretical solutions.
A check of the program for calculating thermal diffusion with different
grids is shown in Fig. 8.12. The initial conditions are T = 200° C for the
spherical region and T = 100° C for the region outside the sphere. Reflection

188 8. Magnetohydrodynamics of HEMP
o
II
T
^.
3
0
2(
V
30°
I I
T0=100
o
cv=P0=k=1
*^
3
0
I
2
i
c
0
>l
d
=1
e
0
li
0
n<
3
Fig. 8.12a—c. Calculated temperature contours at t = 1.02 ns for three calcula-
tional grids, (a) Undistorted grid, (b) Distorted grid, (c) Grid with slide line
boundary conditions are used for all four boundaries. It is seen that the
calculated temperature contours at t = 1.02 u.s are essentially the same.

Appendices
A. Effect of a Second Shock on the Principal Hugoniot
Given an equation of state
P = a/x 4- bfj? 4 c(l 4- /xJ5,
where
jjl — — — 1, V = relative volume.
We wish to find how much a second compression from a point on the principal
Hugoniot differs from the principal Hugoniot.
The principal Hugoniot is obtained by substituting in the equation of
state the relation E = Eo 4 1/2(P 4 Po)(V° - V) where EQ = 0, Po = 0, and
V° — 1. For the given equation of state this yields
P = — (principal Hugoniot). (A.I)
1 - c/i/2
For /i < 0.25 we may write
_ /, ac\ o 1 /, ac\ o /ax
P - a/i 4 (b + yJ yu2 4 -c [b 4 y) /i3. (A.2)
To calculate the Hugoniot starting from the point (Pi,Vi,iiq) on the
principal Hugoniot (Fig. A.I), we substitute in the equation of state the
relation E = Ex + 1/2(P 4 Pi)(Vi - V). Let Vi = 1 - 5, then Ex = l/2PiS.
We have
P = an + V2 + c(l + /x) [^ +

190 Appendices
Fig. A.I. Hugoniot from point (Pi, Vi) on
principal Hugoniot
principal Hugoniot.
The Hugoniot for points above the initial point (Pi, V\ — 1 — 6, E\ — ^- J
is
ajji -f 6/i2 -f §
(A.3)
Expanding, we get
ac
P = a/i + bfi 4- -Pi// -f —//
¦-"*i"-vJ + 7
ac E
be 2 6 c2 2 ^ c2
Rearranging the terms and replacing Pi by (A.2) we obtain
/, ac\ i \
+
ac\
y J
+
(A.4)
The underlined terms in (A.4) are the same as the principal Hugoniot.
Since y ~ [i\ and c is usually between 1 and 2, equation (A.4) is very nearly
equal to the principal Hugoniot except for high order terms.

B. Finite Difference Program for One Space Dimension and Time 191
B. Finite Difference Program
for One Space Dimension and Time
The partial differential equations and the corresponding finite difference equa-
equations are those used by the KO computer program. Time-dependent flows in
one space variable, r, are described for plane (d = 1), cylindrical (d = 2), and
spherical (d = 3) geometries.
B.I Fundamental Equations
1. Equation of motion
p0U dEr Er - Ee
—tit- = -^ h (a - 1) ,
V dr r
where U is the particle velocity.
2. Conservation of mass
with M a mass element.
3. First law of thermodynamics
E - V[siix + (d - l)s2e2] + (P + q)V - 0.
4. Velocity strains
dU
or
U
i<2 = 7'
5. Stress deviators
A, =
Note: Three stresses are identified here, even though they are not all re-
required in order to maintain an analogy with the two and three dimensional
programs.
6. Pressure equation of state
P = a(rj-1) + b(rj - IJ + c(rj - IK + drjE
with rj = l/V = p/po and where a, 6, c, and d are equation-of-state
constants.

192 Appendices
7. Total stresses
8. Artificial viscosity
where Cl and Co are constants, a — y/P/p, and Ar is the grid spacing.
9. Von Mises yield condition
with Y° the plastic flow stress.
B.2 Finite Difference Equations
1. Mass zoning
Vn
d
plane: d = 1
cylindrical: d = 2
spherical: d — 3
where po is the equation-of-state reference density and Vo the initial rela-
relative volume.
2. Equation of motion
_
'-1),
where
'-i \ Vn 1

B. Finite Difference Program for One Space Dimension and Time
3. Conservation of mass
193
PO
mj+i
4. Calculation of velocity strains
4 _ "+2
r?$ + u:+i
^2 = 0 for d = 1.
5. Calculation of stresses
(a) Stress deviators
+ 2fi
(b) Pressure equation of state
n+l __yn
6. Von Mises yield condition
If Kn+l < 0 the material is within the elastic limit. If Kn+l > 0 multiply
the stress deviators by ^2/3Y°/y/s\ 4- s^ 4- s§.
7. Artificial viscosity
and
Calculate only if
<
- V^_ i) < 0. Here a -
where P is the local pressure and Co = 2; Cl = 1.
8. Energy equations
The change in the internal energy, AE, is composed of a hydrodynamic
component and a distortion component:
AE = -(P + q)AV + AZ.

194 Appendices
The change in distortion energy, zAZ, is
where
The total internal energy, J5, is
where
This equation assumes that the equation of state has the form
P = A[ri) + B(r,)E.
9. Time steps
2 Arn+l
Atn+* = --
3Vo^TP
min over j
Arn+l = rn+l - rn+1
j'+l j
where a is the local sound speed and
v) •
6-0 if - > 0.
If Atn+i > (l.l)Atn+i, use Atn+i =
B.3 Boundary Conditions
At an outside regional boundary J (Fig. B.I)
1
TJ ~ TJ-l

Outside,
boundary
B. Finite Difference Program for One Space Dimension and Time 195
Fig. B.I. Grid boundary scheme
Increasing
Inside
boundary
i
At an inside regional boundary J
,„ 1
For a free surface at j = J, the stresses are set to zero at J + ^ for an outside
free surface or at J — ^ for an inside free surface.
B.4 Opening and Closing Voids
Many calculations require a routine that will permit a material to break
or spall. An additional requirement is a routine that will allow two materials
originally separated to join during the course of a calculation. Details of these
routines are given below.
(a) Opening of a void
Let
If Pjl+1 < Ps and Vj1*1 > Vs where PS,VS are material constants, then
introduce a new interface at j with the label V (Fig. B.2), with
rn+l = rn+l
In subsequent time steps, both j and V are treated as free surfaces where
V is an outside boundary and j an inside boundary (refer to Sect. B.3).
The criteria for the opening of a void given above are meant to serve as
an example. In general, the criteria for the calculation of spall involve other
parameters, stress gradients for example.

196 Appendices
r j+1 Fig. B.2. Scheme for void opening
(b) Closing of a void
At the beginning of each time step, the new positions of ry and rj are
calculated first, using a At that is 20% larger than the normal At for this
time step. If rv+1 < r"+1, calculate all grid points with the normal At. If
1 l
rv+1 >r"+l, solve for a new At as follows:
B = 2W + AAtn~^.
Note: In the calculation of 0 and /?, the subscript V refers to an outside
regional boundary and the subscript j to an inside regional boundary, see
Sect. B.3. Then
A{Atn^J + BAtn+* + 2R = 0.
To solve for Atn+i:
2AAU + B*
Start with Ati = 0 and iterate until (Ati — Ati+i) = 0. Solve equations of
motion for one time step with:
Atn+i = Ati+1
Remove the free surface boundary condition on j and set
rV ~
where *C/Jl 2 is the velocity of interface j when the void closed.
Note: no attempt has been made to conserve energy after setting the
velocity Uj to the value required to conserve momentum.

C. A Method for Determining the Plastic Work Hardening Function 197
C. A Method for Determining
the Plastic Work Hardening Function
The simple tension test of a cylindrical specimen offers a direct method for
relating the equivalent stress, <7eq, to the equivalent plastic strain, ep. For
this test the equivalent stress coincides with the uniaxial stress, azz, and the
equivalent strain coincides with the extension in the pulling direction. Very
large plastic strains can be made to occur in a local region when a ductile
cylinder is pulled in tension. A slight taper is used in the cylinder specimen to
control the position of the large strains; the smallest diameter being located
at the mid-section. The geometry lends itself to very easy measurements of
the equivalent stress and equivalent strain.
The uniaxial stress is taken as the load divided by the area at the mid-
section and is usually called the true stress, <7t- Thus the proposition is
This is only strictly true before the elastic limit has been reached and when
the axial stress azz is uniform across a section of the cylinder. As the plastic
strain increases, azz and creq become increasingly nonuniform.
It is instructive to review the definitions of the axial and equivalent
stresses. For the geometry of the tension test the mid-section is a plane of
symmetry and the coordinate axes are the principal axes. The second invari-
invariant of the deviatoric stress tensor, 2 J, can be evaluated from the principal
deviatoric stresses:
The radial and hoop strains are the same, thus srr — see- Since szz + srr +
see = 0> we have szz = —2srr. Thus
2J - -s2
and
The uniaxial stress is azz — — P -f szz. The radial stress arr must be zero at
the cylinder free surface, i.e., arr = — P + srr — 0. Thus
r = Srr = ~~SZZ
and gzz = -P + szz = \szz — aeq. This of course is not a result but rather,
the basis for the original proposition.
After the elastic limit is reached and the tension load on the cylinder
continues, the stresses depart from a uniform distribution. This is the region
of interest where plastic strain is occurring. Work hardening refers to the
increase in the flow stress as the plastic strain increases.

198 Appendices
The analysis of the experimental results of a tension test assumes that
the average uniaxial stress azz, and the average equivalent stress, aeq, equal
the true stress, or:
2 fr=zR _ 2 fr=R
= ~52 / GZz{r)rdr = aeq = —- / a
LOAD
<7T~ 7tR2 '
Here, R is the current outside radius of the cylinder.
The elastic limit is reached first and plastic strain begins at the mid-
section where the cross-sectional area, irR2, is smallest and hence the stresses
the largest. The flow stress, Y, at the mid-section increases when plastic
strain occurs. With continued loading positions adjacent to the mid-section
reach the elastic limit and the process continues until plastic flow extends
throughout the specimen length.
The magnitude of the plastic strain in the axial direction falls off as the
axial distance from the mid-section increases. A strain measurement must
be taken over a region where the strain is constant for the results to be
independent of the gage length and be of any value to the analysis. In the
tension test the strain in the radial direction remains fairly constant even for
large plastic deformations. The radial strain can be obtained by measuring
the change in diameter of the mid-section. Thus the diameter of the cylinder
serves as the gage length.
The simple analysis that follows shows the relationship between external
measurements and strains in a cylinder. External measurements of the radius,
R, and an arbitrary axial length, L, taken at the mid-plane of a cylinder can
be used to determine the average natural strains:
fL dL . / L \
ezz = — = In I —q I axial strain,
Jl° l \L /
fR dR . ( R \
?rr — I —- = In ( —- 1 radial strain,
Jr° R V-™ /
dRdO fR dR ( R
Here L° and R° are initial dimensions.
The strain sqq is the result of the change in length of a lineal element in
the 6 direction where the change in length is due to a displacement in the r
direction. The concept of natural strain compares the extension of an element
of length to the current length rather than the initial length.
The volumetric strain, / ^, is
dv
Hence, $ = -^ G^) where v0 is the initial volume.

C. A Method for Determining the Plastic Work Hardening Function 199
The strains ezz, err and see include both elastic and plastic components,
i.e., ezz = eezz + epz, etc. The elastic components are small compared to the
plastic components and will be neglected.
In the geometry considered here the plastic strains, epz, e^r an<^ ?^e are
also the principal plastic strains. As described earlier, the volume does not
change during plastic flow (plastic incompressibility) with the result that the
sum of the above three strains is zero. Since the radial and hoop strains are
equal, evzz = -2e*r = -2evee.
The equivalent plastic strain ep can be calculated and is equal to the axial
plastic strain e^z
C.I Application to 6061-T6 Aluminum
Figure C.I gives experimental results for two aluminum tension test exper-
experiments. The same data are shown as load and true stress, <7t, vs radial
strain err. When the data are plotted as lncrx vs ln?rr a straight line is
obtained. Thus, the data fit the form ctt = a{?Zz)n where ezz = — 2err =
-2\n(D/D°) and D is the cylinder diameter. The assumption here is the
average stress/strain data obtained experimentally can be used to suggest a
relation for a point function flow stress. A more convenient form for the flow
stress is Y = Y°(l+ Csp)n. Here Y° is the flow stress where plastic flow just
begins.
The experimental strain, ezz = —2\n(D/D°), includes the elastic strain
which of course will change as the load changes. Since these strains are small
they are ignored in the form of the flow stress given above. It can be easily
shown that with this form for the flow stress the exponent n corresponds to
the axial strain at maximum load. Figure C.I shows that this occurs at a
radial strain of —0.05, or an axial strain of 0.1. It is of course important that
fracture has not occurred in the region of the experimental data used to de-
develop the plasticity function. Fracture of these tensile specimens originates at
the center of the specimen. Examination of interrupted tests established that
fracture initiates after the peak load when the radial strain is approximately
0.26.
From the load vs radial strain curve the elastic limit is estimated to be
Y° — 0.0029 Mbar. This is also consistent with Hugoniot elastic limit results
that measure the elastic limit in compression. With the constants Y° and n
established, an iterative procedure can be used to select the parameter j3 by
computer simulation of the tension test. A Hooke's law material model was
used in the calculations. The best fit to the experimental data of Fig. C.I was
found with C — 125. The bulk modulus K = 0.79 Mbar and shear modulus
/i = 0.278 Mbar.

200 Appendices
14
13
(a) .-• 0
Experimental
• Specimen T-2
©Specimen T-3
65
Fig. C.I. (a) Load and (b)
true stress vs strain at mid-
section of a cylinder pulled
in tension, original diame-
diameter D° = 15.85 mm. Inset:
dimensions of cylinder
60
55
0.5
CO
CL
0.4&
03
0)
00
0.3
0 0.05 0.10 0.15 0.20 0.25
-In D/D°
The flow stress was described by
Y = 0.0029A + 125epH-1 Mbar.
Figure C.2 shows calculated profiles at the mid-section of the specimen when
R/R° = 0.772, the strain just prior to the observed fracture. The stress
profiles are not constant. However, it is noted in Fig. C.2 that the equivalent
plastic strain profile is fairly flat and that the strain calculated from the
external radius, -2\n(R/R°) = -21n@.772) = 0.52, is a good measure of
the average value of ep across the mid-section.
In Fig. C.2 it is seen that the axial stress, aZZ: which carries the load,
is quite different from the equivalent stress creq, shown as Y in Fig. C.2.
Prior to reaching the elastic limit they were equal with a constant value
across the radius. The true stress is obtained from the simulation program, by
summing zone by zone the product of the zone stress and zone area at the mid-
plane and dividing by the mid-plane area. The calculated value of the true
stress corresponding to Fig. C.2 is ctt = 4.6kbar which is not too different
from a mean value of the equivalent stress, aeq taken as 4.3kbar, where

C. A Method for Determining the Plastic Work Hardening Function 201
300
-10
0 0.2 0.4 0.6 0.8
R/R°
R/R
Fig. C.2. Calculated profiles at the cylinder mid-section at time of fracture,
R/Rq = 0.772. The flow stress Y is also the equivalent stress creq since the ma-
material is at the elastic limit
aeq = Y. Thus the external measurements on a tension test can give stress
strain information suitable for establishing a first guess for a constitutive
relation for the plastic work hardening function.
In Fig. C.2 it is seen that the hydrostatic stress, -P is responsible for the
nonuniform axial stress, azz.
For aluminum the power law form describes the real phenomena very
well and the constants were determined on the first try. For a metal with
a more complex work hardening behavior it would be expected that several
iterations of the computer simulation program would be necessary to develop
a satisfactory form to describe the flow stress.
Figures C.3 and C.4 show contours of the axial stress, aZZi and the hydro-
hydrostatic pressure P at the same radial strain as Fig. C.2. It is noted in Fig. C.4
that the pressure is compressive (it is recalled that here pressure is measured
positive in compression) in a region one to two radii away from the center.
This result is due to the free surface boundary conditions on the exterior of
the cylinder. The interior stress in the direction normal to the cylinder free

202 Appendices
Fig. C.3. Calculated contours of axial
stress gzz at time of fracture
Fig. C.4. Calculated contour of the hy-
hydrostatic stress — P at time of fracture
surface must be zero at the free surface. To maintain this stress-free condition
there is motion normal to the free surface as the cylinder is elongated. The
net effect is for material to move away from the center region similar to an
extrusion process by squeezing.
D. Detonation of a High Explosive
for a 7-Law Equation of State
The equation of state for the detonation products of the high explosive (HE)
is assumed to have the form:
P=G-l)f
The parameters at the detonation front can be calculated from the three
Hugoniot conservation equations and the Chapman-Jouquet (CJ) hypothesis:

D. Detonation of a High Explosive for a 7-Law Equation of State 203
Conservation of mass: = ———— = Vqj, (D.I)
Pcj V - Uo
Conservation of momentum: Pcj = PoDUqj, (D.2)
Conservation of energy: ?cj — eo — (^o ~ ^cj)> (D.3)
Definition of sound speed (s = entropy):
r? -
dJP
(D.4)
Chapman-Jouquet hypothesis: Uqj 4- ccj = D. (D.5)
The subscript 0 refers to the conditions ahead of the detonation where the
pressure and particle velocity are assumed to be zero. The subscript CJ refers
to conditions at the detonation front. The meanings of the symbols are the
following: D: detonation velocity, U: particle velocity, P: pressure, c: sound
speed, p: density, e: energy (volume units), and V: relative volume.
Equations (D.1)-(D.5) can be solved to give the conditions at the deto-
detonation front:
D (D.6)
(D.7)
(D.8)
(D.9)
<^CJ —
CCJ
Po
Pcj
Pcj =
7 +
7
7 + 1
CJ
A)
7 + :
- ^
¦D,
7
7
+ 1'
Equation (D.10) is obtained from (D.3) and the equation of state.
Calculation of the Parameters Behind the Detonation Front for a
7 = 3 HE Equation of State Using the Method of Characteristics.
Parameters are expressed as a function of the Lagrange coordinates h and t
where h is a parameter designating an element of volume which at time t = 0
has an Eulerian position of x — xq.
In general the problem of detonating HE from a surface involves two
regions. Region I is the rarefaction behind the detonation. Region II is the
rarefaction from the end of the HE. The characteristic line in separating the
two regions has the slope h/t = 4/9D and carries the information that at
t = 0, u = 0 (see below for proof). On crossing this line, if the boundary
condition where the detonation started was a plate of finite or zero mass,

204 Appendices
the particle velocity u will change signs. If the boundary condition is fixed
(infinite mass) then upon crossing to region II the velocity everywhere is zero.
The problem below considers the boundary condition to be a free surface
(plate of zero mass). In this case the solution for region I includes region II.
HE Detonating from a Free Surface. The characteristic equations in
Lagrange coordinates are:
H h no
— = — C+ characteristic along which U 4- <J = constant,
at p0
— = C characteristic along which U — a — constant.
at po
Here a is the Riemann function given by J cdp/p for 7 = 3, a = c.
The CJ hypothesis states: U + c = D along the detonation.
In the h, t plane ^ = D, a straight line for a steady detonation, and from
(D.7), (D.8)
Po
Thus
dh
along which U -\- c = D. For 7 = 3 these are just the requirements for a C+
characteristic. Since region I is adjacent to a line of constant state (U and
c each being constant along the detonation) it is a simple wave region. The
boundary condition requires the velocity to change instantaneously from a
rest velocity to a constant velocity. The family of C+ characteristics forming
the simple wave then degenerates into a pencil of lines through the origin.
(See Ref. [1.2], Chapter III, centered rarefaction waves.)
The equations for the C+ characteristics are
dh pc h
dt po t
(Different k for each C+. Along any C+, c is constant and U is constant.)
From the C characteristics,
U-c = k0
cC3 D.
7+1 7+I
(Same k for all C but c and U are different along a given C.)

D. Detonation of a High Explosive for a 7-Law Equation of State 205
^° > Solution of region I,
U-c=-~ J
where /? = ^ and p was eliminated by the equation of state as follows:
P p3 pc2
[by substituting the value of P from (D.4)],
c = P
ccj Pcj '
Pcj
p = c ,
„ Pcj ^PQ _G +1J
P — — r;—=r —
To determine the characteristic separating region I and region II as discussed
earlier, one proceeds as follows: The C characteristics emanating from the
detonation front carry the information that U — c — — y. As one approaches
the origin from the right and passes through it, the boundary condition U = 0,
at t = 0 is reached. See Fig. D.I.
Hence from U — c — —D/2, we have
c = D/2.
Fig. D.I. Detonation from a free surface
at h = 0

206 Appendices
The C+ characteristic that has this value of c is
t po
and thus
4 9D 9
t 9
The integration of the solutions in regions I and II yield x, [/, P, and p as
functions of h and ?:
T=c2/J
"
at
when
(D.ll)

D. Detonation of a High Explosive for a 7-Law Equation of State 207
Substituting the value of /3/2 in the equation for U above,
_ 3 [Dh D „ A
t
dh
dx
~dh
p(h,t)
13
equation of continuity
equation (D.ll)
p _ 4 jh h
~po ~ 3VZ>? > ~D'
Substituting (D.12) into c = U 4- f
_ 3 ADft ?> D
_ 3 Ada a
then substitute (D.13)
= I 1 equation of state
PCJ [3VI?<pcj
and from (D.8)
Po _ 7 _ 3
~7+1~4
3/2 L
for^-
equation of state.
DiJ
PV
?~t^T
Substituting (D.13), (D.15)
Pcj
7-1
3 )
h\Dt)
(D.12)
(D.13)
(D.14)
(D.15)
(D.16)

208 Appendices
Rewriting (D.11)-(D.16) in units of a given HE length, A (see Fig. D.2)
A = Dt
P
Pc~j
h_
~A
3 h
2~A'
(D.lla)
(D.12a)
(D.13a)
(D.14a)
(D.15a)
(D.16a)
Equation (D.lla) gives the Eulerian position of a particle originally at h after
the detonation has proceeded a distance A. The remaining equations give the
values of U, p, c, P, and e for a particle originally at h that finds itself at x
after the detonation has reached the distance A.
1.5
1.4
1.3
1.2
1.1
1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1
P
Po
/
- /
' /
v y
v/
Fig. D.2. High explosive, 7 = 3, detonated from a free surface, A — length burned,
h — Lagrange coordinate

D. Detonation of a High Explosive for a 7-Law Equation of State
Integration of the Kinetic and Potential Energy
to Find the Original Energy ?q
A — original length of HE
209
pA -t pA 1
Kinetic energy = / -U2{h)p0dh = / -p0
Jo z Jo z
1 2 f
2 Jo
1
IVZ + 4 \dh
Pcj = -A
Potential energy = f* e(h)dh = /^ g^^CJ = §Pcj^
Total energy in length A
, 3pcj A
16
Pcj
s0 = original energy per unit volume = ^-. Therefore for HE detonating
from a free surface 25% of the energy is in the form of kinetic energy and
75% in the form of potential energy.
Detonation from a Fixed Boundary at h = 0 (see Fig. D.3)
Region II
dh pc ,
— = — C characteristics.
dt po
Along C characteristics
U = 0 at the fixed boundary.
Thus c = D/2 everywhere in region II and
P =
from the equation of state.

210 Appendices
Fig. D.3. Detonation from a fixed
boundary at h = 0
= D
The C+ characteristics in region II become
dft _ pc __ PcjD2 7 + 1 1 D2 V7-Iix2
dt po
7 _i±l 4
' 7+1
7
p=^c =
CCJ
Z? _ 8
~2~ ~ 9;
P =
PCJp3
PV _ §?Pcj Po = ^
7 + 1 2 p 2 8'
? 6
For ft < |Z\, x = 0, t/ = 0, c = f, p = |po, P - ^Pcj, ^ = ^1-
For ft > |/i the solution is the same as for a detonation from a free
surface.

E. Magnetic Flux Calculation 211
E. Magnetic Flux Calculation
Calculation Scheme for fc-Sweep
First we calculate V x H to be used for the flux change in the &;-direction.
Subscripts in the equations refer to nodes and zones indicated in Fig. E.I.
(Note: Any quantity shown without a time index is centered at time tn+1/2.)
dH H\. dH.
dH
W
H
Y
= +-
2A\
(HA)
n+l/2
- X5)
-x2)]
4n+l/2
8H
~dX
i 2Anx
n+l/2
n+l/2
with
and
^+ /
, etc., as calculated in HEMP. (If k is the X-axis,
are zero.)
11
12
10
®
®
j+1 j+2
-k+2
-k+1
•k-1
Fig. E.I. Calculational grid for /c-sweep.
Plain integers designate a quantity at a
node (ji — k line intersection). Integers in
circles designate a quantity in the zone
center

212 Appendices
dY
H
Y
&H_
~dX
= +
"Tl / V V \ i Ljn+1/
V \ i LJTl ( V \r \\
— Aq) -+ IJ^{Aq — Aj)
AY2An2+l/2
H
W
H
Y
dH
3X
+
+
+
4y3^+i/2
dH
.n+1/2 _ 1 /jn+1/2 .n+1/2 .n+1/2 ,n+l/2
^ +A "+"/i +^

E. Magnetic Flux Calculation 213
dH
H
Y
= +-
~ XU)
n+l
dH
~dX
~ yio) + H®
.n-fl/2 _
.n+1/2 .n+1/2
1 ^^
Y* = l(y®
+l/2
Having computed V x if we now calculate the change in flux in the k direction
(i.e., across 1-2 and 3-4 in Fig. E.2).
—
where
V x H = ai + bj
'dH H
dH
X Fig. E.2. Calculational grid

214 Appendices
C is the electrical conductivity, and A the zone area. Hence
I 9i 1 da
where
——F3
C3_4
3_4
and
Therefore
4ir|un
[Hn+1An+1 - H"yln]
- y2)
3-4
where
(yn+i/2M.\ -
((jnyn+\/2M\ + (Qnyn
V po J (Si V
and
Rewriting the above, we have
4ir\±m —
+
*2 ~ -X-4
(Xj- X2
\ A
-4
~h —= ~h —=
H 7v—: h
A4 2Y3A3
- y4)
C3_
3_4

E. Magnetic Flux Calculation 215
- X2
- A6
71
2%A2
- Y4
Xq — X4
+ ^
C3-4
2Y3A3
YA-Yl2
Cl-2
-^3 ~ X4
C3-4
Y3 - Y4
C3-4
- X\
- X4
2Y4A4J V C3-4
Vio-yn^ (Y3-Y4\ (X10-Xn A^ \ fX3-X4
' V ^4 2Y4A4)\ C3-4
i7-^3\ /Ki-y5
2y
^2
C3-4
^2
X2-.
2Y2A2J
- X4
2Y2A2
C\-2
X\ - X2
Cl-2
A3
2y3^3
X3 — X4
(E.l)
Now we arrange the equations for HnJtl into a group of linear equations.
Referring to Fig. E.3, rewrite (E.I), which expresses the change in flux in the
/c-direction of zone j + 1/2, k + 1/2 (zone ©), as a linear equation in Hn+1.
__ TTTi+1 rrn+1
*+1 ~ n® - nj + l/
it _ rrn+1 _ jrn+l
1±l - ^© ~ /3j + l/

216
Y
Appendices
¦k+1
-k
k-1
Fig. E.3. Grid for calculating flux change in zone
E) from contributions in the /c-direction
j j+1
_
4 47T/Xn
Y9-Y4 Y3- Y10
- Y4
Xg - X4 t X3 -
A4 J V <?3-4
An+\ An+1 \
B% — Ar«
4 4?r/im
Y2-Y4
H-
A2
4-
v v
2 ~ 4
-h
2y2^2y
_ I
Xx-X2
Cl-2
2^3 ^ 2F4A4 I V ^3-
3-4
_ l
4
4 47T/im
A2
A*
\n+l
A2
2Y2A2
4 47T/im
"•
2f
"•
- X4
a,
10 Xn
V A4

E. Magnetic Flux Calculation 217
Yx-Y2
1-2
4.
+
2Y1A1
2Y2A2
Xi-X2\
Ci_2 /
X2 -X8 A® \
A YAJ
A3 2Y3A3J V <?3-
4-
1-2
+
"•
X\— X2
"•
+
A3 ' 2Y3A3/
The method of solution for a set of linear equations of this form is given in
Appendix G.
Calculation Scheme for jf-Sweep
Next, referring to Fig. E.4 for the numbering of nodes, zones, and values of
H, we execute an analogous calculation of V x H to be used for the flux
change in the j-direction.
dH H\. dH.
+ Ii
dH
W
H_
Y
X4)
X12)
- X5) + H^(X5 - X2)}
n+1/2

218 Appendices
Y
10 <
Hn
H_
2
.n+r-'
1 m
Hn
k+2
k+1
k
k-1
j-2 j-1 j
Fig. E.4. Calculational grid for j-sweep
dH
dX
- y5)
-Y2)]
dH
W
H
Y
dH_
dX
J
2
2A
- X6)
- X7)}
4Y2An2+1/2
^)(Y1 - Y6) + H&(Y9 - Y7)}

al =
E. Magnetic Flux Calculation 219
H\j , fdH\3
dH
y
¦l/V V \ i Z17n+1/'V
(A4 — A2J ~r ^B) V^-2 "~
^ + (if A)Js -f (ffi4)!L +
3 4F3A
n+1/2
ax
3 2A
f^(y8 - y9) + h&(y9 - y4) + Hn^\YA - y2)
H
dH_
dY
H
V
,n+l/2
4
-n)]
4y4A4
Y4An+1/2
~d~X
4 2A
n+1/2
4

220 Appendices
'dH H^j
? _ ^ fxrn+l/2
'
rn+1/2
"~
'©
Using this value of V x H we calculate the change in flux in the j-direction,
i.e. across 2-3 and 4-1 in Fig. E.5
±
Writing
V x H = a\ + 6j
we have
'dH H
(dH\
C is the electrical conductivity. Thus
db I da
where
and
i r i
C4_i
kt
X Fig. E.5. Calculational grid

E. Magnetic Flux Calculation 221
Therefore
Atn+l/2
-y3
(a{ + ai)(X4-
where
C2-3 =
and
(yn+\/2K\
©
.
C4-1 =
(yn+l/2M.\
V po/©
Rewriting the above gives
r+1 - (HA)*]®
I3 - il
+
C2-3
+
A4 Al
A
C4-1
An+l An+l
© ©
x1-x1 +
A3 ' 2Y3A2 2Y3A3)
~ A4 , ^(g)
[(
I2-I3 ( r2-y8^ ^y2-y3
¦"
- X3 X2 -
A ~l A
C2-3
^
H =
"9 - X4 ^®
1 a3 2r3^3
2-3
C2_3
2Y1A4 + 2Y1A1 I V C4-1
- X3
X4-X1

222 Appendices
fin
+ ~7%
' a2 2y2^2
/X12X5
+ ; r r^
-y7\ /r2-y3^
/X6X7
-3
X2 - Xs
Arranging the equation for Hn+l into a group of linear equations for a given
fc-line (see Fig. E.6):
TT __ i/n+1 Tjn + l
zr __ rrn+1 _ tjn+1
Rewriting again (E.2), the change in flux in the j-direction is
where
A l Atn+l/2 \(YlzIl Y2~YS\ fY2-Y3"
4 47r/im W A2 A3 J \
I X7 - X3 X2 - X$ ^0
( A2 A3 2Y2A2 2YSA3 I \ C2_

E. Magnetic Flux Calculation 223
Fig. E.6. Grid for calculating flux change in zone
© from contributions in the j-direction
k+1
-k
At"*1'2 \(Y3 - Yx + F4 - Y2\ (Y2 - Y3
4 47T/im
| (Yt-Yz | Y2-Y4\ (Y4-Yx
C4_i
/ v v v v >4 ^ 4
/A3-A1 A4-A2 ^© ^
+ :; 1 : V ~^ . +
fX2-X3\
V ^2-3 J
\n+\ An+1
H-
— X4 A/e\ A.
2y4^4 2Yi
C4-I
+
- A12
n+1
Ci-X
AU+1
n+1 \ /
v
A4 -
4 47r/im r"^ lv ^3 y v c2_3
+
C4-1
4 _^®_\ /X2-X3\
2^X3 A C2-3 y
C4-1
i —
+ -r=
C2_3 / V A,
" x 'X2-Xs
A2 2Y2A2J V C2_
2-3
4-1

224 Appendices
C4-1
Y2-Y3
C2-3
v An
— A7 -™-fa_
4- _ ^
A2
"• ^Vr
273^3
F. Thermal Diffusion Calculation
Calculation Scheme for fc-Sweep
First we must calculate VT to be used for the thermal flux change in the
/c-direction. Refer to Fig. F.I for numbering of nodes and zones.
dT. dT.
1 + i
(a) ™
dY
dT
~dX
+1
jn+l/2
dT
~dX
(b) ™
dY
+1
- X6)
-^7)]

101
11
12
6
j-1 j j+1 j+2
-k+2
^-k+1
-k
-k-1
F. Thermal Diffusion Calculation
Fig. F.I. Calculational scheme for /c-sweep
225
~dx
-i
*2
(c) —
' 0Y
dT
~dX
"¦ 2
+ 1
3 2A%
dX
(d) ?L
dY
dT
d~X
+1
8Y
2^+1/2
H-
4 2An4 + 1/2
rpn+l
(Y3 - Y10)
- X\\)
dX

226 Appendices
Having VT enables us to calculate the change in internal energy from
fluxes in the fc-direction (i.e. across 1-2 and 3-4 in Fig. F.2).
dT
~dX
dT
I =
m =
dY
(a, «"(?
T3A.
= VV • (AWT)dt + W
(F.I)
dV + V—dt + dZ
(Note: The factor 1/2 in the above expression for W is because we are going to
calculate the thermal diffusion energy change by adding the flux changes from
two directions to obtain the total change; hence, the W terms will appear
twice.)
(b)
VT = li + mj
(c) A^v = -
AxdYm
n
YdY
- Y2) + (AK^(lk3 + lk4)(Y3 - Y4)]
+mk2)(X1-X2)(Y1 +Y2)
mk4)(X3-X4)(Y3 + Y4)\.
X Fig. F.2. Calculational grid

F. Thermal Diffusion Calculation 227
Rewriting (F.I):
(a)
*©
+
+ 2
+2
(A1-A2) I
0
+ 2
Xi-X6\
+ —1 (A1-A2)
A2 J
+2
(A3-A4)
F3+I4
+ J2 + J3 + Jf4
+ 2
+ 2
I 1 j
\ Ai J
- A 4)
\ + Jr2 H- 13
Y3 4-I4
Yi 4- ^2 + J3
Y\ + Jr2 + r3 -h J4)
ri + Yo)
1 V^)l-2
0 V ~~A2—I ^ ~

228
Y
Appendices
k+1
•k
k-1
Fig. F.3. Grid for calculating temperature diffusion
from contributions in the /c-direction
j i+1
(Y3 -
i
— I (Yi ~ Y2){AI_2
2(X6-X7)(X1-X2)(Yl+Y2)
+ M (y1+y2 + y3 + y4)(/l)i-2
-X4){Y3 + Yl)i
where
See (8.17) for J2/C.
Referring to Fig. F.3 we see that for a given j-line
rp __ rpTi-\-\ rpTl+1
J-i+l — J-(^ — 1j+i/
rp __ rpn+1 rpn+1
T = Tn+1 nnn+1
J1 L 1
Rewriting again (F.2)
AiTi+i + BiTi + CiTi-i = Di,
where
l/2 2 r /y9 _ y•
1 =
®
+2
(F.2)

F. Thermal Diffusion Calculation 229
dE_
9T
Y4-Y2 , Fi - Y3
l
+2
X4-X2
Y2)
(X3-X4)(Y3
x + y2
y4
_^n+l/2
4*1+1/2
+2
—
+ -
+1/2
v/<5) 4A
n + 1/2
-^Hy3-
A4
0
4)
(y3 - Y4)(A)%_4
X3-X4)(Y3 + Y4)
+Y2
y4)
(
+
y2)
(X2-X8)(X3-X4)(Y3
~(^3-4]

230 Appendices
2(X6-X7)(X1-X2)(Y1+Y2),..n
1-2
+2
(XS-X9)(X3-X4)(Y3
and
c
(See D.22a) for the calculation of dZ.)
To calculate ohmic heating we expand J2/C as follows:
C L 2 VC L + V C
Cl-2
2
2 • 16tt2 V 2
"•
6|\ / 1
2 7 \C2.3
(F.3)
i
+ v
\ + a\ b\
2 '
The superscripts k and j refer to
^2 =
1
= +2lT
^©
+H^l(X5
^(X2-X4)
V \ \
¦n-'i) 1
(X7 - X3) +
(HA
Y ^ I
1 \
2 J
the /c
/l\n+l
and j
)(^4 —
V
sweeps, where
V \ i ZJn ( V
y\\2) "T ^0V^12
2YX
2y
2
/ tj a \n+l
-X6)
(HA)®

2^3
F. Thermal Diffusion Calculation 231
V \ i rrn+l/y Y\\ _t_ Hn^'1(YA Y \
8 — Ag) + ii^, (^9 ~~ ^4; ~r -"(g) V^M ~ ^-2j
= +
(H A)
2K,
%(Yi2 - Y6
- Y2)],
b =
— 1
n+l/2
(
(
%)(X6 - X7)

232 Appendices
1
a =
2^3
+l
- Xn)
v
A3
24?
- Y7)],
- y8)],
- V10)
- yn)
- Y3)].
Calculation Scheme for ./-Sweep
Calculation of VT to be used for the thermal flux change in the ^-direction
proceeds as follows. See Fig. F.4 for time and position number scheme.
dT. dT.
1 + i
(a) —
V ; 8Y
- X12)
- X6)
- X2)}

(b)
n
11
n+1
n+1
(9)
n
n+1
dr_J
~dX x
dT_
~dX
~3 _
dT
dT
dY
dT
- X6)
- X7)]
+T$>{Y1-Y6)+1%(Y6-Y7)
3
dX
j _ dT
l2~dY
(c) —
dY
dT
~dX
Fig. F.4. Calculational
grid for j-sweep
F. Thermal Diffusion Calculation 233
k+2
k+1
k
k-1
-y \
- X2)
- Xs)]
dT
mi =
dT
dY

234 Appendices
(d)
dT
dY
dT_
dX
2A\
dX
mi =
dT
With VT from the above we can calculate the change in internal energy from
fluxes in the j-direction (i.e. across 2-3 and 4-1 in Fig. F.5).
dT
= VV • (AWT)dt + W
(F.4)
dV + V^-c
dZ
(Note: The factor 1/2 is present because the total energy change by thermal
diffusion will be calculated by adding the flux changes from two directions.
Hence the W terms will appear twice.)
But
VT = l\ + mj
and
where
m =
dT
~dX
dY
T3A.
kt
-o-
Fig. F.5. Calculational grid

F. Thermal Diffusion Calculation 235
Therefore
A
8X 2A@
and
}2-3('J2 + IIW2 ~ Y3) + (AK.M + 1{)(Y4 - Y,)}
A 1 dYm 1
A--
Y 8Y A<S){Yl+Y2+Yz+Yi)
+(A)ti(m{ + m{)(X4-X1)(Y4
Rewriting (F.4)
v'<5)
+ I 1 +
Ll A4-A2 \ (
- + ^ ) {X2-^3)
+2 ( —^ +—j ] (A4-A!) I yi+y2 + n + y4 ] (%i
+2
+2
)
^i-^e\/v vx/ y2 + >3
{Y2 ~
+2 V—^—y

236
Appendices
X3 -
+2
Y\ + Yj
* 1 + Y2 + 13 + 14
— v
+2
X6-X7
_L9 I ^8 ~ ^9 \ (Y Y \
+2 I ) (A2 - A3)
® I V
+2
+ r2 + ^3 + ^4
n 1\
4-1 f
J J
where
/"n+l _ yn
See (F.3) for J2/C.
For a given /c-line, as shown in Fig. F.6:
nn+l
j+3/2,fc+l/2'
T7 /Ti71-f-l /TIT
(F.5)
k+1
k
j-1 j j+1 j+2
Fig. F.6. Grid for calculating temperature
flux changes in zone © from contributions
in the ^-direction

© i+l/2,Ar+l/2»
J~l/2,k+l/2'
. Thermal Diffusion Calculation 237
Rewriting again (F.5)
where
Y1~Y3 Y2~
+
+2
Y4-
+2 :
and
+2
-y.,M)J ,

238 Appendices
/ V* V \ / V I V \ 1
(A3 - rip ^ / r4 4- ri ^
+ 2 I 1 ) (*4 - Ai) I I (-AL-l
V ^4 / \>i 4- r2 4- Y3 + *4/ J
fXi2 X$\ / Y4 + Y1
Vi + ^2 4- Y3 4-
-)(y2-ir3)(iiM-3
+2 ( a ) (X2 - X3) \ v +y +v, , v / V^2-3
k Y1 4- r 2 4- Y 3 4- r 4
+2
G. Backward Substitution Method
for Solving a System of Linear Equations
of the Form AiHi+1 + B{Hi + CiH^ = D{
Given
AiHi+1 + BiHi + dH^ = A; (G.I)
with Him&x and Him[n being known quantities, define
Ei+\Hi 4- Fi+i = Hi+\. (G-2)
Substitute (G.2) into (G.I):
AiiEi+xHi -f Fi+i) + BiHi 4- C,E%.X = A,
or
Di — AiFt+i - CtHi-i
Rewrite (G.2) as
ffi = EM-! 4- Fi. (G.4)

G. Backward Substitution Method 239
Equate coefficients of (G.3) and (G.4)
F ~Ci tp A ~ Aj
for:
i = imax - 1 tO i = imin + 1,
where
?iniax =0 and FWx = Himax.
Store E'i and F^, then calculate Hi for i = zmin + 1 to i = imax — 1.
i/^ = EiHi-i -f F^
Example:
A2H3 + B2H2 + C2HX + 0 =D2 > (G.6)
AlH2 + BlH1+C1H0 =01. J
Given are #4, #0 and ^, J5», C^ for i = 1 —> i = 3. To calculate #3, H2,
and i/i, define the following recursive equations:
E3H2 + F3 =
\ (G.7)
Solving (G.6) in the form of (G.7) provides the following identificaitons:
?-3 = --p-; ^3 =
?>3
2 42?3 + B2' 2
1 4?? + B' *
or:
F A
Substitute Ei and F, into (G.7).

References
[1.1] J. Berger, J. Viard: Physique des Explosifs Solide (Dunod, Paris 1962)
[1.2] R. Courant, K.O. Friedrichs: Supersonic Flow and Shock Waves (In-
terscience, New York 1948)
[1.3] H. Hugoniot: Journal de l'Ecole Polytechnique Memoire, sur la Prop-
Propagation du Mouvement dans les corps 58, A889)
[2.1] J. von Neumann, R.D. Richtmyer: A Method for the Numerical Cal-
Calculation of Hydrodynamic Shocks, J. Appl. Phys. 21, 232-237 A950)
[2.2] D. Giroux: HEMP User's Manual, Lawrence Livermore National Lab-
Laboratory, UCRL-51079 Rev. 1 A973)
[2.3] K.H. Warren: HEMP DS User's Manual, UCID-18075 Rev. 1 A983)
[2.4] M.L. Wilkins: Use of Artificial Viscosity in Multi-dimensional Fluid
Dynamic Calculations, J. Comp. Phys. 36 281-303 A980)
[3.1] G. Maenchen, J. Nuckolls: Calculations of Underground Explosions,
Proceedings of the Geophysical Laboratory, Lawrence Radiation Lab-
Laboratory Cratering Symposium, Lawrence Radiation Laboratory, Re-
Report UCRL-6438, Part II A961)
[3.2] R. von Mises: Z. Angew Math. u. Mech. 8 161-185 A928)
[3.3] D.C. Drucker: A Definition of Stable Inelastic Material, J. Appl.
Mech., 26, 101-106 A959)
[3.4] W.L. Bradley: Strain Hardening in the HEMP Code, Lawrence Liv-
Livermore National Laboratory report UCID-16328 A973)
[3.5] M.L. Wilkins, J.E. Reaugh: Plasticity Under Combined Stress Load-
Loading, American Society of Mechanical Engineers Publication 80-C2/
PVP-106 (August 1980)
[3.6] D.J. Steinberg, S.G. Cochran, M.W. Guinan: J. Appl. Phys 51, 1498
A980)
[3.7] J.J. Gilman, W.G. Johnston: Dislocation and Mechanical Proper-
Properties of Crystals, ed. by J.G. Fisher, W.G. Johnston, R. Thompson,
T. Vreeland (Wiley, New York 1957)
[3.8] G.I. Taylor: Proc. R. Soc. A. 194, 289 A948)
[3.9] M.L. Wilkins, M.W. Guinan: J. Appl. Phys. 44, No. 3, 1200 A973)
[3.10] W. Gust: J. Appl. Phys. 53, No. 5, 3566 A982)

242 References
[3.11] G.C. Sih: Handbook of Stress-Intensity Factors, Institute of Frac-
Fracture and Solid Mechanics, Leigh University, Bethlehem, Pennsylvania
A973)
[3.12] G.R. Irwin: Trans ASME, Ser. D 82, 417 A960)
[3.13] V.M. Vainshelbaum, R.V. Goldshtein: On the Material Scale Length
as a Measure of the Fracture Toughness of Plastic Materials and its
Role in Fracture Mechanics, Institute of Problems of Mechanics, USSR
Academy of Sciences, Moscow A976)
[3.14] W.F. Brown, Jr., J.E. Srawley: Plane Strain Crack Toughness Test-
Testing of High Strength Metallic Materials, ASTM STP 410, American
Society for Testing and Materials, Philadelphia A966)
[3.15] F.R. Tuler, B.M. Butcher: A Criterion for the Time Dependence of
Dynamic Fracture, Int. J. Fracture Mechanics 4, No. 4, 431 A968)
[3.16] F.A. McClintock, A.S. Argon: Developments in Mechanics, Proceed-
Proceedings of the 11th Midwestern Mechanics Conference, Iowa State Univer-
University, Ames, Iowa A969). Also cited: Mechanical Behavior of Materials,
(Addison-Wesley, Reading, Massachusetts 1965), 524
[3.17] K. Mogi: Rock Fracture, in Annual Review of Earth and Planetary
Science, ed. F.A. Donath, (Annual Reviews, Palo Alto, CA), Vol. 1,
63-84 A973)
[3.18] M.L. Wilkins, R.D. Streit, J.E. Reaugh: Cumulative-Strain-Damage
Model of Ductile Fracture: Simulation and Prediction of Engineer-
Engineering Fracture Tests, Lawrence Livermore National Laboratory report
UCRL-53058 (October 3, 1980)
[3.19] M.L. Wilkins, B. Squier, B. Halperin: Equation of State for Detona-
Detonation Products of PBX 9404 and LX-04-01, Tenth Symposium (Inter-
(International) on Combustion. (The Combustion Institute 1965), 769-778
[3.20] R. Cole: Underwater Explosions, Princeton University Press Princeton
N.J. A948)
[3.21] J.W. Kury, H.C. Hornig, E.L. Lee, J.L. McDonnel, D.L. Ornellas,
M. Finger, F.M. Strange, M.L. Wilkins: Metal Acceleration by Chem-
Chemical Explosives, Fourth Symposium (International) on Detonation, Of-
Office of Naval Research, U.S. Naval Ordnance Laboratory, White Oak,
MD. ACR 126- Office of Naval Research/Department of the Navy,
(October 12-15, 1965)
[3.22] E.L. Lee, H.C. Hornig, J.W. Kury: Adiabatic Expansion of High Ex-
Explosive Detonation Products, Lawrence Livermore National Labora-
Laboratory report UCRL-50422 (May 2, 1968)
[3.23] B.M. Dobratz: Properties of Chemical Explosives and Explosive Simu-
Simulants, Lawrence Livermore National Laboratory Explosives Handbook
(March 16, 1981)
[3.24] M.L. Wilkins: The Use of One- and Two-Dimensional Hydrodynamic
Mechanics Calculations in High Explosive Research, Fourth Sympo-
Symposium (International) on Detonations, Office of Naval Research, U.S.

References 243
Naval Ordnance Laboratory, White Oak, MD. ACR 126- Office of
Naval Research/Department of the Navy (October 12-15 1965)
[4.1] M. Reiner: Twelve Lectures on Theoretical Rheology (North-Holland,
Amsterdam 1949)
[4.2] G. Maenchen, J. Nuckolls: Calculations of Underground Explosions,
Proceedings of the Geophysical Laboratory, Lawrence Radiation Lab-
Laboratory Cratering Symposium, Lawrence Radiation Laboratory, Re-
Report UCRL-6438, Part II A961)
[4.3] E.D. Giroux: HEMP User's Manual, Lawrence Livermore National
Laboratory, UCRL-51079 Rev. 1, A973)
[4.4] K.H. Warren: HEMP DS User's Manual, UCID-18075 Rev 1 (Feb.
1983)
[4.5] S. Timoshenko: Theory of Elasticity (McGraw-Hill, New York 1951)
[4.6] J. Von Neumann, R.D. Richtmyer: J. Appl. Phys. 21, 232 A950)
[4.7] M.L. Wilkins: Use of Artificial Viscosity in Multidimensional Fluid
Dynamic Calculations, J. Computational Phys. 36, No. 3, A980)
[4.8] J.A. Viecelli: Applications of the TENSOR Code to the Calculation
of Rayleigh Waves, Lawrence Livermore National Laboratory Report
UCRL 50992 (Jan. 1971)
[4.9] M.L. Wilkins: Calculation of Surface and Ground Waves from Above-
Ground and Underground Explosions, UCRL-93369 Rev. 1 (Nov.
1971). Prepared for the Proceedings of the Third International Collo-
Colloquium on Gasdynamics of Explosions and Reactive Systems, Interna-
International Academy of Astronautics, Marseille, France, 12-17 Sept. 1971
1971
[4.10] M.L. Wilkins, R.D. Streit, J.E. Reaugh: Cumulative-Strain-Damage
Model of Ductile Fracture: Simulation and Prediction of Engineering
Fracture Tests, UCRL-53058 (Oct. 1980)
[6.1] M. L. Wilkins, R. E. Blum, E. Cronshagen, P. Grantham: "A Method
for Computer Simulation of Problems in Solid and Gas Dynamics in
Three Dimensions and Time", UCRL-51574 Rev. 1, May 30, 1975
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November 3, 1982
[8.1] R. D. Richtmyer: Difference Methods for Initial Value Problems (In-
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[8.2] J. M. LeBlanc: Lawrence Livermore Laboratory, private communica-
communication, June 1972
[8.3] J. Chang: "An Introduction to Extrapolation Methods for Numerical
Solution of Differential Equations", Lawrence Livermore Laboratory,
Report UCID-15992 A972)

Subject Index
Artificial viscosity:
- for calculating shocks 28-32, 85,
93, 94, 131, 141, 142, 162, 192, 193
- for stabilizing the gried 94, 95,
142-145
Ampere's law 174
Behavior of materials 37
Boundary conditions 99-102, 146,
183, 195
Characteristics 6, 7, 206-208
Capman-Jouget (CJ) 17, 26, 76,
205-207
Conservation equations:
- Mass 9, 84, 87, 88, 136, 191, 193
- Momentum (equation of motion)
10, 83, 87, 133, 176, 191, 192
- Internal energy (first law of
thermodynamics) 10, 84, 96, 145,
191, 194
Courant condition 32
Damage:
- Strain damage 68, 69
- Damage in elastic regime 69, 70, 74
- Damage in plastic regime 71, 72
Detonation:
- Pressure measurement 25, 26
- Numerical calculation 79, 81
- Equation of state of detonation of
products 75, 79
- Detonation waves 17
Discontinuities 4-6
Dislocations 53-56
Double operators 35, 36, 171, 172
Drucker 41, 48, 49
Ductile fracture 68
Elastic-plastic waves 17-20
Electrical conductivity 183
Energy (internal) 2, 96, 129, 145, 175,
179, 180, 194
Energy summation 97
Equation of motion 2, 83, 87, 129,
174, 191
Equation of state 2, 3, 23-25, 205
Equation of state parameters 63, 64
Faraday's Law 174
Finite difference schemes:
- Von Neumann 27, 28
- One dimension 192, 197
- Two dimensions 33-35
- Three dimensions 35
- Double operators 35, 36, 171-173
First law of thermodynamics 2, 48,
129, 191
Flow stress 48, 49, 57-60
Fracture:
- Modeling fracture 62, 65
- Fracture toughness 65-67, 73
- Computer simulation of fracture
70, 71
- Spallation 67
Grid stabilization 36, 94-96, 142-145
Griineisen 23-25, 60-62
Harmonic motion 146, 147
Hooke's law 22, 37-39, 44
Hugoniot 3, 9, 11, 17, 18, 23-26, 29,
189, 190
Hydrodynamics 1
Ideal gas (perfect gas) 3, 13-16, 23,
202-210
Isentrope 11, 12
J-sweep:
- Magnetic-flux 217-224
- Thermal diffusion 232-238

246 Subject Index
K-sweep:
- Magnetic-flux 211-217
- Thermal diffusion 224-232
Load calculation 98
Magnetic diffusion 177-179
Magnetohydrodynamics 171-176
Mass zoning 86, 87, 131, 133
Maxwell solid 52-53
Measurements:
- Equation of state (EOS) 20, 21
- Detonation pressure 25, 26
- EOS of explosive detonation products
57-58
- Flow stress 57-58
Mie-Griineisen 23, 24, 60-62
Navier-Stokes 94-142
Nonhomogeneous properties 60
Ohmic heating 181
Ohm's law 174
Plasticity 149
Plastic flow 39-41
Plastic strain: 45, 46, 140, 141
- Equivalent plastic strain 45, 92,
141? 199
Prandl-Reuss 45
Pressure:
- Equation of state 60-62, 85, 130,
140, 193
- Radiation pressure 176
Principal stress 98
Radiation diffusion of T4 179, 180,
185, 186
Radiation pressure 176
Rate dependent yield models 52-57
Rayleigh 3, 11-13, 19
Reaction zone 1
Riemann 7, 22, 204
Rigid stress rotation 39, 83, 91, 139,
140
Shock pressure 14
Shock waves (shock speed) 8-16, 19,
20
Sliding interfaces 113, 151, 183
Sound speed 4, 6, 8, 12-16, 19, 22, 23,
26
Strain:
- Equivalent plastic strain 45
- Incremental strain 89, 90, 136-139
- Strain damage 68, 69
- Strain hardening 50-52
- Velocity strain 130, 141
Stress deviator tensor: 85, 130, 139
- Equivalent stress 43, 44
Temperature diffursion 179, 180
Time step calculations 146, 194
Transmissivity 181
Tresca 46, 47, 97
Upper yield point 59-60
Velocity strains 84
Von Mises 41, 43, 44, 47-51, 92, 131,
140, 193
Von Neumann 27-29, 31
Work hardening function 197
Yield strength 18, 41-43, 59
Z-factor 117, 118, 158-160

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Computer Simulation of Dynamic Phenomena
Preferred finite difference schemes in one, two, and three space
dimensions are described for solving the three fundamental
equations of mechanics (conservation of mass, conservation of
momentum, and conservation of energy). Models of the behav-
behavior of materials provide the closure to the three fundamentals
equations for applications to problems in compressible fluid
flow and solid mechanics.The use of Lagrange coordinates per-
permits the history of mass elements to be followed where the
integrated effects of plasticity and external loads change the
material physical properties. Models of fracture, including size
effects, are described.The detonation of explosives is modelled
following the Chapman-Jouget theory with equations of state
for the detonation products derived from experimenttAn equa-
tion-of-state library for solids and explosives is presented with
theoretical models that incorporate experimental data from the
open literature.The versatility of the simulation programs is
demonstrated by applications to the calculations of surface
waves from an earthquake to the shock waves from supersonic
flow and other examples.