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Inelastic wave propagation under combined stress states

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Inelastic wave propagation under combined stress states
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Myers, Charles Daniel, 1945-
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x, 236 leaves. : illus. ; 28 cm.

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Combined stress ( jstor )
Constitutive equations ( jstor )
Datasets ( jstor )
Inertia ( jstor )
Mathematical variables ( jstor )
Shear stress ( jstor )
Sine function ( jstor )
Stress waves ( jstor )
Trajectories ( jstor )
Velocity ( jstor )
Dissertations, Academic -- Engineering Sciences -- UF
Engineering Sciences thesis Ph. D
Strains and stresses ( lcsh )
Stress waves ( lcsh )
Wave-motion, Theory of ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 231-235.
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Typescript.
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Vita.

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INELASTIC WAVE PROPAGATION
UNDER COMBINED STRESS STATES













By

CHARLES DANIEL MYERS


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY





UNIVERSITY OF FLORIDA
1973




































TO PEGGY










2. CKh'NO WLEDC UENT S


I would like to thank Professor Martin A. Eisenberg, Chairman of

the Supervisory Committee, not only for his untiring efforts during

the development and preparation of the material contained in this

manuscript, but also for being a counselor, teacher, and friend during

both my undergraduate and graduate studies. I am also indebted to

Professors L. E. Malvern and E. K. Walsh for their helpful criticism

and encouragement during my doctoral studies. In addition, I would

like to express my appreciation to the other members of my Supervisory

Committee: Professors U. H. Kurzweg, C. A. Ross, and R. C. Fluck.

A special word of thanks is extended to Professor N. Cristescu

for his many helpful discussions during the development of this

dissertation.

I thank my wife, Peggy, for her encouragement, moral support, and

understanding during the course of my studies. I also thank Peggy for

typing and proofreading the drafts of this dissertation. I appreciate

the efforts of Mrs. Edna Larrick for the final typing of the manuscript

and Mrs. Helen Reed for the final preparation of figures.

I acknowledge financial support from the National Defense Education

Act, the National Science Foundation, and the University of Florida

which made myv studies possible.

I also acknowledge the Northeast Florida Regional Computing Center

for the use of its IBM 370 Model 165 digital computer without which

the scope of this work would have been greatly curtailed.















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . . ... ...... iii

LIST OF TABLES . . ... ... vi

LIST OF FIGURES . .. . vii

ABSTRACT . . ... ... ix

CHAPTER 1. INTRODUCTION ... . 1

CHAPTER 2. THEORETICAL DEVELOPMENT . .. 12

CHAPTER 3. D]IENSIONLESS EQUATIONS AND NUMERICAL PROCEDURES 29

3.1. Wave Speeds as a Function of the State of Stress .. 29
3.2. Characteristic Solution in Terms of
Dimensionless Variables . ... 46
3.3. Numerical Grid for Characteristic Solution .. 51
3.4. Finite Difference Eouations . ... 58
3.5. Solution to the Finite Difference Equations .. 69
3.6. Calculation of the Strains .... . 77

CHAPTER 4. RESULTS AND DISCUSSION . ... 80

4.1. Introduction . ... .. 80
4.2. Grid Size Effects ... . .. 81
4.3. Effects of Radial Inertia . ... .84
4.4. Effects of Strain-Rate Dependence . .. 110

CHAPTER 5. SUIIARY . . ... 118

APPENDIX A. CONSTITUTIVE EQUATIONS . ... 125

A.1. Comments on the Constitutive Equation ... 125
A.2. Rate Independent Incremental Plasticity Theory ... 127
A.3. Rate Dependent Plasticity Theory . .. 135
A.4. Dimensionless Expressions for the Functions
0(s,a) and .(s, . . 137









TABLE OF COIM'ENTS (Continued)


Page

APPENDIX B. CHARACTERISTICS AND EQUATIONS
ALONG THE CHARACTERISTICS . ... 140

B.1. Equations for the Characteristics . .. 140
B.2. Equations along the Characteristics ... 141
B.3. Reducing Equations to Simpler Case . .. 156
B.4. Uncoupled Waves . .... .. 159
B.5. Elastic Waves .... . . 162

APPENDIX C. PROGRAMS FOR DETERMINING THE PLASTIC
WAVE SPEEDS . ... ... 164

APPENDIX D. SOLUTION TO THE FINITE DIFFERENCE EQUATIONS
IN THE CHARACTERISTIC PLANE . ... 171

D.1. Equations for Fully Coupled Waves . .. 171
D.2. Equations for Uncoupled Waves . ... 176
D.3. Solution at a Regular Grid Point
for Fully Coupled Waves . ... 180
D.4. Solution at a Regular Grid Point
for Uncoupled Waves ................. 183
D.5. Solution at a Boundary Point (X= 0)
for Fully Coupled Waves . ... 185
D.6. Solution at a Boundary Point (X= 0)
for Uncoupled Waves . ... 191

APPENDIX E. COMPUTER PROGRAM FOR CHARACTERISTIC PLANE
SOLUTION . ... 194

E.I. General Description of the Program . ... 194
E.2. Initial Conditions . . 196
E.3. Calculation of A . .... .. 201
E.4. Input Data . . 202
E.5. Listing of the Program . ... 204

LIST OF REFERENCES . . 231

BIOGRAPHICAL SKETCH . .... 236




















LIST OF TABLES



Table Page


1 Normalized Longitudinal Stress --) . 36
s



2 Normalized Hoop Stress (-- ............. 38
s



3 Normalized Shear Stress .-) . 40
S














LIST OF FIGURES


Figure Page

2.1 Coordinate System for the Thin-Walled Tube ... 13

2.2 Stresses on an Element of the Tube . ... 14

3.1 Yield Surface Representation in Spherical Coordinates 31

3.2 Plastic Wave Speeds as Functions of $ and y for
Poisson's Ratio of 0.30 . ... 41

3.3 Values of $ at v= 0 for which c =c =c ...... 45
f s 2

3.4 Numerical Grid in the Characteristic Plane ... 53

3.5 Regular Element in Numerical Grid . ... 54

3.6 Boundary Element in Numerical Grid . ... 55

3.7 Location of the Characteristic Lines Passing Through P 57

3.8 Numerical Representation of the Characteristic Lines
in a Regular Element . . ... 59

3.9 Representation of the Characteristic Lines in
a Boundary Element ... . .. 60

4.1 Grid Size Effects on the Longitudinal Strain
at X = 3.75 . . ... 83

4.2 Grid Size Effects on the Longitudinal Velocity
at X = 3.75 . . ... 85

4.3 Grid Size Effects on the Stress Trajectories
at X = 3,75 . . ... 86

4.4 Longitudinal Strain Versus Time at X = 3.75
for Data Set 1 . . ... 88

4.5 Change in Shear Strain Versus Time at X = 3.75
for Data Set 1 . . .89










LIST OF FIGURES (Continued)


Figure

4.6 Transverse Velocity Versus Time for Data Set 1
Without Radial Inertia . .

4.7 Longitudinal Velocity Versus Time for Data Set 1
Without Radial Inertia . .

4.8 Longitudinal Strain Versus Time for Data Set 1 .

4.9 Maximum Radial Velocity Versus X for Data Set 1
With Radial Inertia . .

4.10 Change in Shear Strain Versus Time for Data Set 1
Without Radial Inertia . .

4.11 Longitudinal Strain Versus X for Data Set 1 .

4.12 Stress Trajectories for Data Set 1
Without Radial Inertia . .

4.13 Strain Trajectories for Data Set 1
Without Radial Inertia . .

4.14 Shear Stress Versus Longitudinal Stress for Data S
With Radial Inertia . .


Page


. 93

. 94



96



. 97

. 98



- 99


100


et 1


. 102


4.15 Stress Trajectories for Data Set 1 With Radial Inertia

4.16 Hoop Stress Versus Longitudinal Stress for Data Set 1
With Radial Inertia . . .

4.17 Stress Trajectories for Data Set 2
Without Radial Inertia . . .

4.18 Stress Trajectories for Data Set 2 With Radial Inertia

4.19 Hoop Stress Versus Longitudinal Stress for Data Set 2
With Radial Inertia . . .

4.20 Shear Strain Versus Time for Data Set 2 .

4.21 Change in Longitudinal Strain Versus Time forData Set 2

4.22 Longitudinal Strain Versus Time for Data Set 3 .

4.23 Stress Trajectory at X= 0 for Data Set 3 .

4.24 Stress Trajectory at X= .25 for Data Set 3 .


viii


104



105



107

108



109

111

112

114

116

117











Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy



INELASTIC WAVE PROPAGATION UNDER
COMBINED STRESS STATES

By

Charles Daniel Myers

August, 1973


Chairman: Dr. M. A. Eisenberg
Major Department: Engineering Science, Mechanics
and Aerospace Engineering


The purpose of this dissertation was to investigate the effects

of radial inertia and material strain-rate dependence on the propa-

gation of inelastic waves of combined stress along a thin-walled tube.

A general quasilinear constitutive equation for multiaxial stress (and

strain) states was introduced. The equations of motion and the

strain-displacement equations, along with the constitutive equations,

were written to form a set of nine simultaneous hyperbolic, quasilinear,

partial differential equations. This set of equations was reduced to

a set of six equations which was then used to determine the expres-

sions for the characteristic lines and the equations along the char-

acteristic lines.

For combined torsional and longitudinal loading, two distinct

wave speeds were found. The values of these two wave speeds were

found as functions of the state of stress. Including radial inertia

effect in the formulation of the problem was shown to significantly

increase the wave speeds for a given stress state. Also certain










critical combinations of Poisson's ratio and the "effective tangent

mocdulus" caused the two wave speeds to be equal when the shear stress

vanished.

The equations for the characteristics and the equations along the

characteristics were written in terms of dimensionless variables.

These equations were then written as first order finite difference

equations. A computer code was written in the Fortran IV language,

and several problems were solved using an IBM 370 model 165 digital

computer. In order to obtain these solutions two particular forms of

the constitutive equation were used; one form represented a strain-

rate independent material while the other form represented a strain-

rate dependent material.

The strain at the impact end was considerably larger when radial

inertia effects were included than when radial inertia effects were

not included in the problem. However, radial inertia effects were

found to have little influence on the solution more than two diameters

from the impact end. The strain at the impact end was lowered by

including strain-rate dependence of the material. For any particular

set of initial conditions and boundary conditions, the stress trajec-

tories behaved in the same manner, at least qualitatively, whether or

not radial inertia effects or strain-rate dependence were included.

The details of the stress trajectories were more complicated when

radial inertia effects were included since the trajectories were

three-dimensional.















CHAPTER 1

INTRODUCTION


Stress wave propagation is the mechanism by which forces and

displacements are transmitted from one part of a structure to another.

Stress waves arise when a transient force is applied to a structure,

and they propagate through the structure reflecting (at least partially)

back into the structure whenever they encounter a boundary. After

several reflections the amplitude of the stress waves diminishes and

the structure reaches a state of equilibrium. In many engineering

problems the time required to reach equilibrium is very short, and for

practical purposes the structure can be assumed to reach equilibrium

instantly. Problems in which the forces are applied slowly or in which

the state of stress is required a long time after the forces are applied

are examples of instances when wave propagation effects may be neglected.

However, in many cases, the forces are applied rapidly (such as during

impact loading or explosive loading), and failure is most likely to

occur in the structure almost immediately after the application of these

forces. In these cases when it is necessary to determine the state of

stress during and immediately following the loading, wave propagation

effects may be significant and should be included in the analysis of

the problem. In order to understand the development of the theory of

stress wave propagation and the application of this theory to modern

engineering problems, it is instructive to review briefly the history

of wave propagation research.









The first serious attempt (at least in this century) to understand

nonlinear wave propagation in solids was made by Donnell (1930). In

this paper, Donnell used energy principles and impulse-momentum expres-

sions to find the particle velocity and the elastic wave speed for

longitudinal waves. He also predicted that if a material with a bilin-

ear stress-strain curve were impacted at the end by a stress above the

yield stress, two stress waves would propagate with distinct velocities.

However, after the publication of this paper interest in wave propaga-

tion subsided until the early 1940's.

A more general theory of longitudinal stress wave propagation was

developed independently by Taylor (1940), von Karman (1942) and

Rakhmatulin (1945) by assuming that the material exhibited a nonlinear

stress-strain curve above the yield point. This stress-strain curve

was assumed to be independent of the rate of straining. Using this

theory the velocity of propagation of the longitudinal waves was found

to be given by

1 da



where c is the wave speed, p is the density of the material, C is the

da
stress, and e is the strain. Thus represents the slope of the stress-
de

strain curve or the tangent modulus. This theory also considered the

stress-strain curve of the material which was obtained for the static

case to be valid in the dynamic case. With this assumption, the stress

and strain followed a unique functional relationship as long as no

unloading occurred. Because of this, the tangent modulus could be

written as a function of the stress (or strain) only, so that the

velocity of propagation then became a function of the level of stress









(or strain). This immediately led to the conclusion that a given

level of stress (or strain) propagated at a specific speed, and the

stress wave changed shape as it propagated along a prismatic bar for

stresses in the nonlinear region of the stress-strain curve. For a

bilinear stress-strain curve, the results of Donnell (1930) were again

predicted. However, these theories did not account for the lateral

inertia effects in the bar or the dependence of the stress-strain curve

on the rate of strain, and so more complex theories and constitutive

equations were proposed to account for these phenomena.

By the late 1940's many investigators including Davis (1938),

Manjoine (1940), and Clark and Wood (1950) had experimentally observed

the effect of the rate of strain on the stress-strain curve for several

materials. In order to incorporate this strain-rate effect into the

constitutive equations used to study plastic wave propagation,

Sokolovsky (1948a, 1948b) and Malvern (1949, 1951a, 1951b) independ-

ently introduced one-dimensional constitutive equations in which the

stress was a function of the plastic strain and the plastic strain rate.

By selecting a particular form of this constitutive equation, Malvern

(1951a, 1951b) was able to obtain a numerical solution which predicted

several experimentally observed phenomena. However, his numerical solu-

tion did not apparently predict a region of constant strain near the

impact end such as had been observed by Duwez and Clark (1947) and

others. This new strain-rate dependent constitutive equation also

predicted that, if a bar were strained statically above the yield stress

and then impacted, the first increment of strain would propagate with

the elastic wave velocity and not the velocity given by the tangent








modulus.in the strain-rate independent theory. Since this prediction

was quite different from that of the strain-rate independent theory

several investigators tried to verify one or the other. Bell (1951)

published the results of his experiments with aluminum which showed

that, for a bar stressed above the yield point, the initial strain

pulse propagated with the elastic wave velocity. These results were

in accordance with the strain-rate dependent model of Malvern (1951a,

1951b) as were the experimental results of Sternglass and Stuart (1953)

which were obtained using copper, Alter and Curtis (1956) which were

obtained using lead, Bell and Stein (1962) which were obtained using

aluminum, and Bianchi (1964) which were obtained using copper.

Encouraged by these experimental results, many investigators

continued the development of more general constitutive equations to

describe material behavior. Perzyna (1963) generalized the semi-linear

constitutive equation of Malvern (1951a, 1951b) to multiaxial states of

stress. At about this same time Cristescu (1964) introduced full quasi-

linear constitutive relations for a one-dimensional problem. This

quasilinear equation was used immediately by Lubliner (1964) to show

that the strain-rate independent constitutive equation of Taylor (1940),

von Karman (1942), and Rakhmatulin (1945), and the strain-rate depend-

ent constitutive equation of Malvern (1951a, 1951b) and Sokolovsky

(1948a, 1948b) were both special cases of this more general constitu-

tive equation. Later Cristescu (1967a) gave a generalization for multi-

dimensional stress states of the quasilinear constitutive equation as

well as an extensive summary of the developments in dynamic plasticity

until that time. Lindholm (1967) developed a constitutive equation

for combined stress states of aluminum which included strain-rate










effects and temperature dependence. He also presented extensive data

for one-dimensional loading and combined stress loading at several

strain rates and temperatures which were used in empirically determin-

ing the constants used in his generalized constitutive equation.

While these more general constitutive equations were being developed,

it was shown by Malvern (1965), by Wood and Phillips (1967), and by

Efron and Malvern (1969) that the semi-linear equation of Malvern"

(1951a, 1951b) did indeed predict a region of constant or nearly con-

stant strain near the impact end if the solution was obtained long

enough after impact. Suliciu, Malvern, and Cristescu (1972) have shown

that a region of constant strain is not possible for the semi-linear

constitutive equation but may be approached asymptotically. They have

also shown that a region of constant strain is possible when the quasi-

linear constitutive equation is used. However, in the interpretation

of experimental results it has been difficult to differentiate between

a region of truly constant strain and a region in which the constant

strain is approached asymptotically.

The experiments of Sternglass and Stuart (1953), Alter and Curtis

(1956), and others were believed by many investigators to be proof of

the strain-rate dependence of some materials. This led to the exten-

sive development of constitutive equations just discussed. However,

other investigators sought to explain the experimentally observed phe-

nomena by including radial inertia in the formulation of the wave prop-

agation problem. Plass and Ripperger (1960) introduced radial inertia

effects into the problem of longitudinal impact and used the constitu-

tive equation of Malvern (1951a, 1951b). In order to find a character-

istic solution, all of the variables were averaged at each cross section





6

and these averaged variables were used. The :results of this work were

given by Tapley and Plass (1961) but were somewv.hat inconclusive. More

work including radial inertia effects was published by Hunter and

Johnson (1964), and a year later DeVault (1965) showed that, at least

qualitatively, many observations formerly attributed to a material

strain-rate effect could be explained by including radial inertia effects

in the formulation of the problem of longitudinal impact of a bar.

Shea (1968) obtained good agreement between theory and experiment for the

propagation of longitudinal waves in a lead bar. He used the strain-

rate dependent constitutive equation of Malvern (1951b) and the

"correction" for radial inertia proposed by DeVault (1965). Mok (1972)

used the same averaging technique for the variables as Plass and

Ripperger (1960) for the problem of longitudinal impact of a bar with

radial inertia effects included. He used the strain-rate independent

constitutive equations and agreed in essence with DeVault (1965) that

radial inertia effects could explain, at least qualitatively, those

experimental results usually attributed to strain-rate sensitivity

of the material. Since radial inertia is always present in an experi-

ment using longitudinal impact it seemed that the only way to conclu-

sively determine strain-rate effects in a material would be to perform

the experiments using a torsional wave.

In an effort to determine the strain-rate dependence of various

materials, several investigators have recently conducted theoretical

and experimental studies concerning the propagation of torsional waves.

Convery and Pugh (1968) gave the results of their experiments in which

a tube was stressed statically above the yield stress in torsion and










then subjected to a suddenly applied increme-ntal torsional load. The

strain caused by this inciremcntal load was lound to propagate with the

elastic shear wave velocity. This seemed to be proof that the strain-

rate dependent theory was correct, but Convery and Pugh (1968) cau-

tioned against that conclusion. For Bell (1960, 1963) and Bell and

Stein (1962) had asserted that (based on experimental results with

annealed aluminum), while an increment of strain may propagate with the

elastic wave velocity, the larger amplitude strains propagate with the

wave velocity predicted by the strain-rate independent theory.

Nicholas and Garey (1969) tested aluminum samples in torsion at high

strain rates and found very little strain-rate dependence. However,

Yew and Richardson (1969) were able to measure some strain-rate depen-

dence in copper.

Another problem which was encountered in wave propagation studies
r
was that of unloading. The two most common unloading cases were when

the applied load was reduced and when waves were reflected from a bound-

ary. Unloading was examined for longitudinal plastic wave propagation

by Lee (1953) using the strain-rate independent constitutive equation

and by Cristescu (1965), Lubliner and Valathur (1969), and Cristescu

(1972) using the quasilinear constitutive equation. In all of these

investigations, regions of unloading and boundaries between regions of

unloading and loading in the characteristic plane were predicted but

the results have not been verified experimentally.

Many investigators in recent years have become interested in the

behavior of materials under combined stress and, more specifically, the









propagation of waves of combined stress. One of the first discussions

of combined stress wave propagation was given by Ranhmatulin (1958).

In this paper he developed the equations which must be solved for

elastic-plastic wave propagation under combined stress. Strain-rate

independent constitutive equations were used and only the problem for

the elastic case was solved. He found that the shear wave did not

affect the longitudinal wave in the elastic case. A similar discussion

of combined stress wave propagation was presented by Cristescu (1959).

Until now nothing has been said about the plasticity theory used.

The two plastic strain theories were the total strain theory proposed

by IIencky (1924) and the incremental strain theory proposed by Prandtl

(1924) and Reuss (1930). These two plasticity theories along with many

other developments in plasticity theory were presented in detail by

Hill (1950). The different plasticity theories were not presented

earlier because in many cases both theories gave the same results.

For instance, when a strain-rate independent constitutive equation was

used, the two plasticity theories led to identical results when one-

dimensional (either longitudinal or torsional) stress wave propagation

was studied, when combined stresses were used if the loading was pro-

portional, or even when unloading occurred in one-dimensional problems.

However, when strain-rate dependent material behavior of nonpropor-

tional loading under combined stresses was considered, most investiga-

tors used the incremental strain theory. Shammamy and Sidebottom

(1967) showed that the incremental strain theory more accurately pre-

dicted the experimental results when various metal tubes were subjected

to nonproportional static loading in tension (compression) and torsion.










Interest in the propagation of waves of combined stress continued

and Cli ton (1966) presented the results of his study of combined longi-

tudinal and torsional plastic wave propagation in a thin-walled tube.

Strain-rate independent material behavior and incremental strain

theory were used while radial inertia effects were ignored. The thin-

walled tube allowed Clifton to eliminate any dependence on the radial

coordinate so that a solution could be obtained in the characteristic

plane. (Earlier, Plass and Ripperger (1960) had used a rod and averaged

the variables over the cross section in order to eliminate the dependence

on the radial coordinate.) The results of this investigation were

based on a simple wave solution which resulted from applying a step

velocity impact at the end of the tube. Clifton (1966) found that when

the tube was stressed into the plastic range, an impact at the end of

the tube caused waves with two different speeds to propagate. These

waves were called the fast wave and the slow wave, and each wave was

found to carry both longitudinal and torsional stresses. Two special

cases were examined. The first case involved statically prestressing

the tube above the yield stress in torsion and then applying a longi-

tudinal velocity at the end. In this case the fast wave caused almost

neutral loading, that is, as the fast wave passed a point on the tube,

the shear stress decreased and the longitudinal stress increased in such

a way that the stress state at that point remained close to the initial

loading surface. Then as the slow wave passed the same point, loading

occurred so that the stress path was normal to the initial loading sur-

face. The second case was for a tube with a static longitudinal plastic

prestress impacted by a torsional velocity at the end. In this case the

fast wave caused unloading along the longitudinal stress axis followed










by an increase in shear stress at a constant value of longitudinal

stress and then the slow wave caused loading such lhail the stress path

was normal to the initial loading surface. Clifton (1966) also found

that for a given initial loading surface, the two wave speeds depended

upon the particular stress state on the initial loading surface, and

that for one particular initial loading surface the fast and slow wave

speeds were equal when the shear stress vanished.

This work of Clifton (1966) was a significant step forward in the

investigation of waves of combined stress. An extension of this work

was presented by Clifton (1968) in which the simple wave solution was

used along with unloading at the impact end. In this way certain unload-

ing boundaries for combined stress states were determined. Two years

later Lipkin and Clifton (1970) published their experimental results

from combined stress wave propagation tests and compared these results

to the simple wave solution developed earlier. Agreement between the

simple wave theory and the experiments was fair.

Cristescu (1967b) formulated the problem of combined stress wave

propagation in a thin-walled tube using general quasilinear constitu-

tive equations but again ignoring radial inertia effects. The equa-

tions for the characteristic lines and the equations along these char-

axteristic lines were determined. No numerical results were given

but the two waves (fast wave and slow wave) were shown to be coupled

during loading. Again Cristescu (1971) showed that the coupling of the

waves of combined stress depended on the constitutive equations and

yield conditions used.










This concludes a brief survey of the history of the development

of plastic wave propagation theory. No attempt ,vas m3nd1" tc givO

a complete historical background. For more information the reader is

directed to Hopkins (1961), Kolsky (1963), Olszak, Mroz, and Perzyna

(1963), and Cristescu (1967a, 1968).

The remainder of this dissertation will be devoted to solving

the problem of combined stress wave propagation in a thin-walled tube

when radial inertia effects are included. A general quasilinear

constitutive equation for multiple states of stress will be presented,

and it will be shown to be a generalization of the constitutive equa-

tions of both Lipkin and Clifton (1970) and Cristescu (1972). But

first the wave propagation problem itself must be developed.













CHAPTER 2

THEORETICAL DEVELOPMENT



The specific problem to be considered here is that of the propagation

of inelastic waves of combined stress along a semi-infinite thin-walled

tube, with the effects of radial inertia included. The material consti-

tutive equation used is a generalization for multiple states of stress

of the quasilinear constitutive equation used by Cristescu (1972) for

a single stress component, and is a special case of the very general

quasilinear constitutive equation given by Cristescu (1967a). The coor-

dinate system used is shown in Figure 2.1, and the stresses on an ele-

ment of the tube are shown in Figure 2.2, where r is the mean radius of

the tube.

The problem is assumed to be axisymmetric so that there is no

dependence on e. Since the tube considered is thin-walled, the stresses

Sr, Tr, and Trx are assumed to be negligibly small as are the strains

e r and e The strain e is not included in the problem. Stability
re rx r

of the tube wall and thermal effects are not included in the formulation

of the problem, and only small strains are used. The strain rate is

assumed separable into elastic, plastic, and visco-plastic parts. The

radial displacement is very small compared to the tube radius, and

plane sections of the tube remain plane. The material is assumed to be

isotropic and homogeneous, to obey the von Mises yield condition, and

to be isotropically work-hardening. All unloading is assumed to be

elastic.

















































Figure 2.1 Coordinate System for the Thin-Walled Tube











































ax

rx










Figure 2.2 Stresses on an Element of the Tube






15


The equations of motion in the cylindrical coordinates shown in

Figure 2.1 are given by

1 1
C + T + + -T =
x,x rx,r r ex,e r rx x,tt


1 1
T + C + -+ ) = p Ur
rx,x r,r r r@e, r r r,tt


1 2
T + T + -- T = 0 U
6x,x re,r r e,e r re9 ,tt


which, under the assumptions given above, become


(xx =p uxtt (2.1)
X,x P xtt -


-- = u (2.2)
r r,tt
o

Texx = P utt (2.3)


where the subscripts following the comma represent partial differentia-

tion with respect to the variables x (the coordinate along the tube

axis) or t (time). The density of the material is p, and ux, u and

u are the displacements of any point in the x, r, and 9 direction,

respectively.

For the cylindrical coordinates of Figure 2.1, the strain-

displacement equations are given by

e =
r r,r


C (1 + (u )
r 9,0 r


e = U
x x,x


r 11 r -
r 2(r r,6 + Ur -









1
S= (u + u )



= (u -+ u )
Px 2 x r x,




and under the above restrictions, these reduce to the following three

equations

S= u (2.4)
x X,x

1
e = u (2.5)
o
1
S= u (2.6)
ex 2 e,x


Defining the velocities v v, and va as u t, ur,t and ue ,

respectively, equations (2.1) to (2.6) become


xx = x,t (2.7)


-r e = r,t v(2.8)
o

Tx,x= p V,t (2.9)


S = v (2.10)
x,t X,X

1
= -- v (2.11)


xt = vx (2.12)
ex,t 2 e,x


Under the assumptions used here, the variables no longer depend on r,

so that the problem becomes two-dimensional (the independent variables

are x and t) and can be solved by the method of characteristics.

The equations necessary for completion of the set of simultaneous

partial differential equations describing the behavior of the body are









the constitutive equations. Ci-rstescr. (1972) uses a full quasilinear

constitutive equation for ,a -in:.;le sot'itu diinal stress as


Ot- =E t+ (, ~ + (~ ) (2.13)


As a generalization of this equation to a constitutive equation

governing multiaxial states of stress and strain, the following equation

is used


+ 3+ 3 ij
S l+v + (s,A)s +- (s,) (2.14)
ij E ij E ijkk A)s- (2.14)
s


where the dot represents partial differentiation with respect to time,

s.. is the deviatoric stress, 6.. is the Kronecker delta, v is Poisson's
ij ij
ratio, E is Young's modulus, 0(s,iA) and '(s,A) are material response

functions as yet unspecified, and s and A are defined as


s / s..s.. (2.15)
S 2 ij ij


/2 .P "P s
A ij ij dt + (2.16)
v 3 / 13ij E

*P
and e. is the inelastic portion of the strain rate which, using
ij
equation (2.14) can be written as


P 3 "3 sii
P. 3 0(s,A)s + (s,A) (2.17)
3ij 2L I
s


when the elastic, plastic, and visco-plastic portions of the strain

rate are assumed to be separable. The constitutive equation (2.14)

is a special case of the equation


k1
e. = fk. 0 + gi
ij ij kl ij









given by Cristescu (1967a). The form of equation (2.14) was chosen as

Lhe general constitutive equation because it contains terms which may be

considered separately as elastic, plastic, and visco-plastic strain-

rate terms, because the inelastic strain-rate tensor is proportional

to the corresponding deviatoric stress tensor, and because it reduces

to the form of equation (2.13) when the only stress present is the longi-

tudinal stress. This simplification to the form of equation (2.13) is

shown in Appendix A.

The functions 0(s,A) and *(s,A) are functions which depend on the

particular material being studied. The function 0(s,A) is a measure of

the rate insensitive inelastic work-hardening, and the function r(s,A)

is a measure of the visco-plastic strain rate due to the strain-rate

sensitivity of the material. In the classical rate independent plastic--

ity theory, t(s,) vanishes. When s < 0 or when s < a (the current

"yield stress"), 0(s,4) is set equal to zero. The unloading conditions

when ((s,6) = 0 are stated in equation (A.3.1).

Two separate materials are modeled in the numerical work done.

One is a 3003-H 14 aluminum alloy used in the experimental work of

Lipkin and Clifton (1970). This material is assumed to be insensitive

to strain rate and the functions 0(s,A) and r(s,A) are obtained using

the classical Prandtl-Reuss incremental plasticity theory with iso-

tropic work-hardening and the stress-strain curve for uniaxial tension.

(See Appendix A.) The other material used is a commercially pure

aluminum dead annealed at 1100F. This material is assumed to be

strain-rate sensitive, and the functions 0(s,A) and '(s,A) are

obtained from the data given by Cristescu (1972). (See Appendix A.)









Since the stresses a, T and T are assumed to vanish,
r r rx
equation (2.141) as applied to the present problem r reduces to

1 v 1 (s,)s+(sA)
2s

v 1 1
xt E x,t E E ,t 2
2s
e-+-1 ( 0 -( )s+ (s,A)s + (s.A) (2.18)

+ S

-xt E +x,t (sA)s (s, )
2s

where the deviatoric stresses are

1
s = s (2a ()
x 11 3 x




1
s = s (+ a )
r =33 3 x + )

s = S = s1 = 2 Tx
sx 12 21 ex

sr = s = .
re rx

Using these deviatoric stresses, the expression for s becomes

1 1
ds = t L- sijij 2 i sklkl s L 'ijsi


1 3 1 3 a F-
s L2 sijsi=-- LssIl + s22s22+ s33s33+ sl2sl2+ s21s21


S 3 o 2 2 2 2
s (Cx + -~ ) + 2T
4s

i 1 r -1
s 1 (2 ) (20 )a + 6T7 (2.19)
x x,t x ,t x x
2s







and equations (2.18) become

(2c % ,7"2 .(s,Z) (2- -a7,) (a -2cr) -
L ix ,t l-J i-' j et

-6(20 -a )T (2c -c )
42 ex ex (S A)] + t (S,) (2.20)
+ -2 (s,A) Text +- ((s,A) (2.20)
4s 2s


(20 -a,)(a -2 j F- (a -2c
,t E -2 (s A) x,t L+ 2 ( c ,t
4s 4s
(6(2crQ )IxTx j 2crQ cr
-6 ( 2 ,7 C Y x ) T e x 2 a 9 C 7x
+ ((s,A) Txt + (s,A) (2.21)
L 4s 2s

r3T e(2x ) 3Te] 3r e (2e 1)
ex, t 2= L0( xt 2(A ,t
4s 4s
18T 3T
P1_+ Ox ex
+ -- 0(sA) T + --- (s,6) (2.22)
4s 2s

The equations (2.7) to (2.12) and equations (2.20) to (2.22) form

a set of nine simultaneous hyperbolic quasilinear partial differential

equations for the nine unknowns ax' cre T6 x' V Vx', V, e', e and

e6x. A special case of this system of equations is the set of equations

obtained by neglecting radial inertia effects. When radial inertia

effects are ignored, the variables a', E and vr are not included

directly in the problem formulation. This case can be incorporated

into the more general formulation by multiplying a~, e vr, and their

derivatives by the dummy variable "a," where "a" has the value of 1

when radial inertia effects are included and the value of 0 when radial

inertia effects are neglected. Also the equation of motion, the kine-

matic equation and the constitutive equation for motion in the radial

direction must be multiplied by "a." Doing this, and defining the

quantities







2
1 (2 a% )
A = + -(s,6)
1 E -2(
4s


2 L


(2j ac,)(c
x x
-2
4s


2a~ )


6Te (2y aOe )
A3 -2 (s,A)
4s


A=
4 E


(7 2ac )2
+ s(-2)
4s


6 T Gex(ax 2aca ) -
A = 02(, A)
4s 2

4s


(2.23)


the nine simultaneous equations (2.7) to (2.12) and equations (2.20) to

(2.22) can be written as

,x=vx,t (2.24)

a
r a = apvt (2.25)



ex,x ,t (2.26)


x,x


a
ae = -r v
e,t r r
o
1
ex,t = Ve,x



ex, = Ax, + +aA + A +
x,t 1 x,t 2 e,t 3 ex,t


as ,t= aA 2 + aAa ,t
9,t 2 x,t 4 e,t


2 aaa
x ,
(s,6)
2s


2aa a

'2s


3T
1 a 1 ex
, = A3x + A + T + -- (sA)
Ex,t 2 3 ,t 3 A5 ,t 2 56 6xt 2S
2s


(2.27)


(2.28)


(2.29)



(2.30)



(2.31)



(2.32)


x,t


1
3(5.1_. )_1









Eliminating the strain rates from the last six of these equations,

and defining


2a au,
x e
x- (,)
2s

2aa -a
= x (sA)
2s


(2.33)


the system of

equations for


3T
Ox 4(s,/) J
2s

nine equations reduces to the following system of six

the unknown variables ex, 1O, Tx' Vx, v and v

a x= xt
X,X X,t


a
- a7
r
o


ap vr,t
r,t


T6x,x = Pv,t


v = Aa + aA + A T + X
x,x 1 x,t 2 9,t 3 ,t x

a -
-- v = aA +aAo + aAT + a
r r aA2 x,t 4 9,t 5 9x,t a
o


O,x= A3 x,t + aAO 9,t + A5 x,t + 2ex



2- 2 2 2 1
s = (c ax 0 + a + 3T Ox


(2.34)


(2.35)


(2.36)


(2.37)


Since the equations (2.24) to (2.26) and (2.34) to (2.36) form

a system of hyperbolic equations, they can be solved by the method of

characteristics. To do this, first the equations for the characteristic

lines must be determined, and then the equations along these characteristic


where









A w, B b (2.38)

where


p 0 0 0 0 0

0 ap 0 0 0 0

0 0 p 0 0 0
A = (2.39)
0 0 0 A! aA A3

0 0 0 aA2 aA aA5

0 0 0 A3 aA5 A6




0 0 0 -1 0 0 -

0 0 0 0 0 0

0 0 0 0 0 -1
B = (2.40)
-1 0 0 0 0 0

0 0 0 0 0 0

0 0 -1 0 0 0


V
x
v
r

w = (2.41)
CT
x



LT:xJ
a,.


and









0



r
o

0

x
x
av

o

-2a
r @Xe


- 2te X


and if the slope of the characteristic line (or the wave speed ) is

denoted by c where


dx
dc t

then the equation for the characteristic lines is given by


cA -B =0

and from the calculations shown in Appendix B, equation (2.


[ 2 2 Cr -2,2 c
a pc J L(pc ) [a}


(2.43)





(2.44)


44) yields


-2 (
- (pc ) {b + A4j = 0 (2.45)


where


S-2 -2- -2-
a = A 4 A6 + 2aA2A 3 5 aA 1 A5 aA26 A3A


(2.46)


-2 -2
= A1 -aA2 + A A a. 5
1 4 2 4 6 5


Setting the first factor in (2.45) equal to zero,

dx
ac = a = 0 (twice)
dt
and, setting the second factor in (2.45) equal to zero,


r= I 1 -2 1il 2
c = L-- (b -4aA4) 4
2ap


(2.42)


(2.47)




(2.48)









If the wave speeds in equation (2.48) are denoted by


c = i -- -b (b (2. 49)
f -L9_ fj
2ap
and


c = ( 4aA ) (2.50)
2ap


where cf is the fast wave speed and cs is the slow wave speed, then

the slopes of the characteristic lines are given by

c = 0 (twice)

c = f

c = c
s

Equations (2.47) and (2.48) are the six equations for the character-

istic lines for the set of six simultaneous, hyperbolic, quasilinear

partial differential equations of (2.38). When radial inertia effects

are not included (that is, when a= 0), the equations (2.47) vanish

identically and the remaining four simultaneous equations of (2.38) have

the characteristics given by equations (2.48). For this case (a=0) and

when 0(s,A) is obtained from incremental, rate independent plasticity

theory with isotropic work-hardening, equations (2.48) reduce to those

given by Clifton (1966) as shown in Appendix B.

The equations along the characteristics can be obtained in two

different ways, both of which are discussed in Appendix B. The result-

ing equations along the characteristic lines of equation (2.48) are









0 =


5 -A A ]dTex
5 3 4iJ x


-2 {r
+ pc2)(A2 6 -35 2 1L
o


1
c
c


+ 2(pc )aA2A5-A3A4 ( dt
2s


( 2aaG)

2s


(s,A) dt


[(P c) (iA6-A5) -A d-v -- (pc 2) (AiA5-A2A3) -A dvg
c


S -2 6 -
S (P c ) (A115-A 23) 5 dTgx (p 2 ) (A2A6-A3A5 dex
pc pc


[ 3 ] [ (2Jxa a) )]dt
2 x (s,
+ (PC )(2 6-AA 5 -A- (s,) 2dt
2s


1
+ -.
-2
PC


2 -2- -2 (-2 AA)]
(pc ) (A16-A3) (pc )(A1 )+1 1
o


(a -2aa )
x 6
+2
2s


2 3ex3T
+ 2 (PC)(A 1 -A 2 -A35 (s,6)]dt
2s


2-3 ( C 2 A jidA -aA 2 d
O =p c aA2 -A34dvx- oc c)( 1 4-aA2)4 dv


+ F(pc2)
L


(A A4-aA2)-Aj dTx
1 4 2 4~


+ ( c2a2--3
+ (pc )LaA2A5-3 A4] dx


-2 2 ( x c
+ (pc2)2 aA2AA-A 3J -x (sA) dt
2s


1 2 2
[(p-2) (A"- -2 (A 4 dv, ~ 6 La 2



+ --2 [(PC (A4A6-aA)-AX d [(pc ) (A4 -
pc

[(2a ac7 )
(s,A) dt
2s


(2.51)


(2.52)


(s,A) dt









S-2r 9 _2 r'o7 (, 2ao ) ]
+ a(pc ) (PC ) 1(A.-A2A)-i I) dt
0


+ 2(pc(p pc) (A1A 4-aA2)- 4I L- ,) dt (2.53)
2s


These three equations each represent four equations, one equation

in differential form along each of the four charaxeristic lines of

equation (2.48). When the waves are coupled, equations (2.51), (2.52),

and (2.53) are identical. That is, by multiplying equation (2.52) by

the quantity


(pC ) (aA25 3A4)
24
(pc )(A1A A2A3) A5


and using equation (2.48), equation (2.51) is obtained; or by multiply-

ing equation (2.52) by the quantity

-2 -2
pc (pc 2)(A a2) A

-2
(pc )(A A A2A3) A5


and using equation (2.48), equation (2.53) is found. When the numerator

and denominator of these multiplying quantities do not vanish, equations

(2.51), (2.52), and (2.53) are identical. However, when the waves

become uncoupled, a phenomenon discussed in Appendix B, A3 and A5 vanish.

In this case the multiplying factors used above become undefined and

the equations (2.51), (2.52), and (2.53) are not the same. When

A =A =0 the equations (2.51) and (2.53) reduce to equations (B.4.4)

and (B.4.6), respectively. Under these conditions, equation (2.52)

also reduces to the form of equation (B.4.4).











The equations along the characteristics of equation (2.47) may be

obtained (since :nlonig Lhese characteristics there is no variation in x)

directly from equations (2.25) and (2.35) by multiplying each of these

equations by the increment of time, dt. These equations then yield

aco
S- dt = ap dv (2.54)
r r
0


av
r dt = aA 2dxa + aA4do + aA 5dTrx + a4 dt (2.55)


The equations for the six characteristics and the equations along

these characteristics, along with the appropriate initial conditions

and boundary conditions, represent a complete mathematical formulation

of the problem, and the solution to these equations is the solution to

the problem posed here. The solution to these equations will be

obtained by using a finite difference numerical technique which will

be discussed in the next chapter.














CHAPTER 3

DIMENSIONLESS EQUATIONS AND NUMERICAL PROCEDURES


3.1 Wave Speeds as a Function of
the State of Stress


In this chapter the numerical schemes used to find the solution to

the wave propagation problem of Chapter 2 will be presented. In this

first section the dependence of the wave speeds on the stress state

will be shown. The stresses a T, and have already been assumed
r r' rx
negligibly small so that the scalar representation of the stress state

is given by equation (A.1.2) as

(2 9 9x9 2 2 -
s = au T + a + 3T 2 (3.1.1)
x x 6 9 6


Next, the new variables a ae and Tx will be defined so that the
x 9 ex
surface s = constant can be represented in terms of these variables as

a sphere, and the stress state on this surface in terms of these new

variables can be described in terms of spherical coordinates. Now

defining,
/ 1
= 2( + aca)
x 2x G


a (ae ) (3.1.2)


*'Bx- = /3 T
x ex


equation (3.1.1) can be written as

12 + 2 2T] (3.1.3)
s = / + a/ + T (3.1.3)
x 9 9xJ









and defining the angles y and S6 as shown in Figure 3.1 these new

variables defined in equation (3.1.2) Co-n Ibe written as


x = s cos y cos 6


e, = s cos y sin 6


(3.1.4)


x = s sin y


The angle y is the complementary angle to the one normally used in

spherical coordinates. It is used here to facilitate comparison of

results obtained later on to already published results.

From equations (3.1.2) and (3.1.4),


9 s cos v sin 6
/ -
S s cos v cos 6
x


so that the cx and ae
x 9


a -axis : 0 =


( + a(') 7 + ac
tan =

S(aa -a) a x_
2 9 x

axes are located by


and tan 6 =- and 6 = -60



and tan 6 and 6 = +60.


In order for the equations (3.1.4) to reduce automatically to the

simpler case when radial inertia is not considered, the angle 6 is

defined as


6 = a6' + (a-1)600


(3.1.7)


so that when radial inertia effects are included, a=l, and 6= 6', and

when radial inertia effects are not included, a= 0, and 6 = -60, which

from equation (3.1.6) automatically causes (a to vanish as it should.
e


(3.1.5)


(3.1.6)










x
ex


Figure 3.1 Yield Surface Representation in Spherical Coordinates









Using the uniaxial s;rs --strain curve in the form of equation

(A.2.13), the universal str'ess-:strain curve can Ie r ..itten as


(3.1.8)


S- n
S= B )
y


and letting Et(s) be the tangent modulus of this curve, yields


1 dA 1 n-l
S -+ Bn (s -
Et(s) ds y


(3.1.9)


and from equation (A.2.18) this becomes


1 1
+ O(s)
Et(s) E


() 1 1
E (s)


(3.1.10)


Now g = n(s) is defined so that


where


Et(s) = 1(s) E



0 I $(s) < 1


and when 5=1, the material is elastic, and when P=0, the material

is perfectly plastic. Using equation (3.1.11) in equation (3.1.10),

0(s)can be written as


Inverting equations (3


1 1
(s) = (- -- 1) .


.1.2), the stresses a


x x 3

/ 1 ,
ao = 0 + -- Ca
ao8 O Fx


(3.1.12)


re given by


(3.1.13)


1 1
78x :

(3.1.11)








and using equation (3.1.4) th-se become


i 1
-S CO '- jlC,; S c .


1 1
aoe = s cos y|cos 6 + -- sin


T -- s sin y


Now substituting equations (3.1.12) and (3.1.14)


S(3.1.14)





into equation (2.23),


F 1 ) 2 2 1 21
I 1+ -1) (1)(cos Y) (2 cos 6 --- sin 6 -cos 6 --- sin 6)

1r 1 1 2 2 1
E -[ (- 1) )(cos y(2 cos --- sin 6 -cos 6 --- sin 6)


(cos 6 -- sin 6-2 cos 6 -2 sin 6)

1 1 6 1 1
E -- sin v) (cos v)(2 cos 6 --- sin 6 -cos 6 --- sin 6)
S3 I 1 2 2 2N
= 1 + 14 1) (cos 6 -- sin 6 -2 cos --- sin 6) cos
E1 i (1 1 1 2 ]
E- (-1( sin y)(cos y)(cos 6 --- sin 6 -2 cos 6 --- sin 6)

1F 36 1 1 2
[2(+1 +-- (--1) ( sin Y)
E 4 B3


1 14 11 2 2 2
S[1 + ( 1) (cos v)(cs6 2 sin 6 cos + 3 sin 6)

1 I 1 1 2 2 2
1- + ( -1)(cos 2y) (-cos2 6+3 sin 6)



) (--) (I -1) (sin y cos y) (cos 6 -,3 sin 6)
E 2 J (3.1.15)

1 1 1 2 2 2
1 +(-+ 1) cos y(cos 6 + 2A/3cos 6 sin 6+ 3 sin 26)


E v( 1) (sin y cos y)(-cos 6 -/3 sin 6)


S2(1+v) + 3(1-1) sin y


A3


A4


A5

A 6

or

A1


2


A3

A4


A
A5


A6









The elastic wave speeds arc defined from equations (B.5.4) as


c /-
o / c


c1 = (3.1.16)

(1- )
c2 ~ p

and the wave speeds from equation (2.48) can be written in dimension-

less form as


c = -L b Vb 4aj a (G/p)
c2 k2 ap L 4
C2


c b 4A4a
1 Lv j 24Aa]
Ea

or

12 lv F 2 3-
C 2 =- E E2 b (E2b )2 4(EA )(E3 a) (3.1.17)
3- 4
E a

By defining the dimensionless functions from equation (3.1.15) as


Ai = E A i = 1,2,...,6 (3.1.18)
1
and
3- 2 2 2
a = Ea = A1A4A + 2aA2A3A5 -A3A4- aA1A aA2A6
S(3.1.19)
b' = E = A1A aA2 + A4A- aA ,
1 4 2 46 5 ./

then the wave speeds in dimensionless form become

/2 l+v r; / i ~2 ~) / c \2
c -= Vb b 4A4a = (3.1.20)
a 2


A computer program was written to solve this equation for the two

positive wave speeds as functions of the angles y and 6 for specified









values of v and 8. This pDrol- i- i lin ised in Appendix C. This pro-

gram also calculated the viuecs of the normalizz:ed stresses a /s,

0 /s, and ex/s as functions of v and 6. Those results are given in

Tables 1, 2, and 3. The wave speeds are shown in Figure 3.2 for

the case when 6 =-60o, which corresponds to j = 0. Also plotted in

Figure 3.2 are the results given by Clifton (1966). It is obvious

that the results are not the same and that including radial inertia

effects in the formulation of the problem can have significant effects

on the wave speeds and that, for any given state of stress, the waves

are always faster when radial inertia effects are included. The

results plotted in Figure 3.2 do not correspond to the case when

a= That is, although a = 0 when 6 =-60 a does not necessarily

vanish for this case. Ahen a= 0, the results obtained were identical

to those of Clifton (1966) as they should be, since a= 0 corresponds

to the absence of radial inertia effects.

An interesting phenomenon can be observed by remembering that the

physical presence of radial inertia is due to the Poisson effect.

That is, the longitudinal (fast) wave speed would be expected to be

the same when a=0 (no radial inertia effects) as when v=0 (the cause

of the radial inertia effects vanishes). However, in the fomulation

of this problem it is tacitly assumed that Poisson's ratio for the

1
inelastic portion of the material behavior is -, or that the material

behavior in the inelastic range is incompressible. When the material

is elastic ( = 1), this Poisson effect can be studied directly.

Comparing equations (3.1.15) with (B.5.1) when 1 = 1, the wave speeds

are given by equation (B.5.4) as










\
TABLE 1 NORMALtZLD ,LONGI'L:.UD.NAL STRESS (--)
s


Gamma

Delta
00 100 200 300 400


-900 0.57735 0.56858 0.54253 0.50000 0.44228

-80 0.74223 0.73095 0.69747 0.64279 0.56858

-70 0.88455 0.87111 0.83121 0.76605 0.67761

-600 1.00000 0.98481 0.93969 0.86603 0.76604

-500 1.08506 1.06858 1.01963 0.93969 0.83121

-40 1.13716 1.11988 1.06858 0.98481 0.87111

-30 1.15470 1.13716 1.08506 1.00000 0.88455

-200 1.13716 1.11988 1.06858 0.98481 0.87111

-100 1.08506 1.06858 1.01963 0.93969 0.83121

00 1.00000 0.98481 0.93969 0.86602 0.76604

100 0.88455 0.87111 0.83121 0.76604 0.67761

200 0.74223 0.73095 0.69746 0.64279 0.56858

300 0.57735 0.56858 0.54253 0.50000 0.44228

400 0.39493 0.38893 0.37111 0.34202 0.30253

500 0.20051 0.19746 0.18842 0.17365 0.15360

600 0.00000 0.00000 0.00000 0.00000 0.00000

700 -0.20051 -0.19747 -0.18842 -0.17365 -0.15360

800 -0.39493 -0.38893 -0.37111 -0.34202 -0.30254


-0.56858 -0.54253


900 -0.57735


-0.50000 -0.44228











TABLE 1 (Continued)


Gamma
Delta
500 600 700 800 900


-900

-800

-70

-600

-500

-400

-300

-200

-100

00

100

200

300

400

500

600

70

800


0.37111

0.47710

0.56858

0.64279

0.69747

0.73095

0.74223

0.73095

0.69747

0.64279

0.56858

0.47709

0.37111

0.25386

0.12889

0.00000

-0.12889

-0.25386


0.28868

0.37111

0.44228

0.50000

0.54253

0.56858

0.57735

0.56858

0.54253

0.50000

0.44228

0.37111

0.28867

0.19747

0.10026

0.00000

-0.10026

-0.19747


0.19747

0.25386

0.30253

0.34202

0.37111

0.38893

0.39493

0.38893

0.37111

0.34202

0.30253

0.25386

0.19747

0.13507

0.06858

0.00000

-0.06858

-0.13507


0.10026

0.12889

0.15360

0.17365

0.18842

0.19747

0.20051

0.19747

0.18842

0.17365

0.15360

0.12889

0.10026

0.06858

0.03482

0.00000

-0.03482

-0.06858


0.00000

0:00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000


-0.28868 -0.19747


900 -0.37111


-0.10026 0.00000













sa,
TABLE, 2 NOWRALIZFD ITOOIP S TiESS
S


Gamma
Delta
00 100 200 300 400


-900

-800

-700

-600

-50c

-400

-300

-200

-100

00

100

200

300

40

500

60"

700

800

900


-0.57735

-0.39493

-0.20051

0.00000

0.20051

0.39493

0.57735

0.74223

0.88455

1.00000

1.08506

1.13716

1.15470

1.13716

1.08506

1.00000

0.88455

0.74223

0.57735


-0.56858

-0.38893

-0.19746

0.00000

0.19747

0.38893

0.56858

0.73095

0.87111

0.98481

1.06858

1.11988

1.13716

1.11988

1.06858

0.98481

0.87111

0.73095

0.56858


-0.54253

-0.37111

-0.18842

0.00000

0.18842

0.37111

0.54253

0.69747

0.83121

0.93969

1.01963

1.06858

1.08506

1.06858

1.01963

0.93969

0.83121

0.69746

0.54253


-0.50000

-0.34202

-0.17365

0.00000

0.17365

0.34202

0.50000

0.64279

0.76605

0.86603

0.93969

0.98481

1.00000

0.98481

0.93969

0.86602

0.76604

0.64279

0.50000


-0.44227

-0.30253

-0.15360

0.00000

0.15360

0.30254

0.44228

0.56858

0.67761

0.76604

0.83121

0.87111

0.88455

0.87111

0.83121

0.76604

0.67761

0.56858

0.44228











TABLE 2 (Continued)


Gamma

Delta
500 600 700 800 900


-900

-800

-700

-600

-500

-400

-300

-200

-100

00

100

200

300

400

500

600

700

800

900


-0.37111

-0.25386

-0.12889

0.00000

0.12889

0.25386

0.37111

0.47710

0.56858

0.64279

0.69747

0.73095

0.74223

0.73095

0.69747

0.64279

0.56858

0.47709

0.37111


-0.28867

-0.19746

-0.10026

0.00000

0.10026

0.19747

0.28868

0.37111

0.44228

0.50000

0.54253

0.56858

0.57735

0.56858

0.54253

0.50000

0.44228

0.37111.

0.28867


-0.19747

-0.13507

-0.06858

0.00000

0.06858

0.13507

0.19747

0.25386

0.30253

0.34202

0.37111

0.38893

0.39493

0.38893

0.37111

0.34202

0.30253

0.25386

0.19747


-0.10026

-0.06858

-0.03482

0.00000

0.03482

0.06858

0.10026

0.12889

0.15360

0.17565

0.18842

0.19747

0.20051

0.19747

0.18842

0.17365

0.15360

0.12889

0.10026


0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000










/T
TABLE 3 NORLALIZED SHEAR STRESS \


T x
Gamma Value of x- Gamma Value of Tx
s s


00 0.0 500 0.44228

100 0.10026 600 0.50000

200 0.19747 700 0.54253

300 0.28868 800 0.56858

400 0.37111 900 0.57735


_- E
c =
f2
p (1 av2)


S= E -
Cs p 2p (1 +) -2



It is now obvious that the fast wave speed is the same when a=0 and

v 0 as when a= 1 and v= O. The slow wave speed (and consequently c2)

is the same when a=1 as when a=0, although it does depend on v.

Because of this dependence of c2 on v, the dimensionless fast wave

speed of equation (3.1.20) will have values when a=0 and v O differ-

ent from those when a=1 and v =0.

Also in Figure 3.2 it can be seen that when a=0 and y=0 the

fast and slow wave speeds are the same for = .385 and = .30. There

is usually some value of $ for which the two wave speeds are equal

at y=0 for each combination of values of v and 6. The condition for

which this is true can be obtained from equation (3.1.20) and is

















---Without Radial Inertia
Clifton (1I66)

-- With Radial Inertia
and ca = (6 = -600); 6 0


S= .01


y, Degrees


Figure 3.2 Plastic Wave Speeds as Functions of $ and y for Poisson's Ratio of 0.30


0, 8












ti'hen =- 0



A1


2
A


A3

A4


A =
5

A6
6


b/2 4a A = 0 .


equations (3.1.18) and (3.1.19) are


1 + -1(--1)(cos 6 -2v sin 6 cos 6 + 3 sin26)
43

-v + (--1)(cos 6-3 sin25)


0

1 1 2 2
1 (- -1)(cos 6 +23 sin 6 cos 6+ 3 sin 6)
1+45-


(3.1.21)


(3.1.22)


0


2(


:1 + )


2 2
a = AIAA aAA6


b/ = A1A + AA aA2


and using the same manipulations as in Appendix B, Section 4,

equation (3.1.21) becomes


0 = A(A -A ) + aA2


where Al, A2, A4, and A6 are given by equation (3.1.22). Now


fl = () = f cos26 2 3 sin 6 cos 6 + 3 sin2 ]


f2 =f2(6) = cos26 + 2v sin 6 cos 6 + 3 sin26


f= f(6) = 1 cos26 -3 sin26]

1
z = z( ) 1
c c
c


(3.1.23)








(3.1.24)


defining







(3.1.25)









the critical value of $ is found from


0= L L c f L 2(1-,.,)- (I + zcl) I+ a -(v cf3

0,= z -fl 2+af3 + z ([2(1+l ) -13 f- + a2vf3


+ [2(1 +v) -1+ av (3.1.26)


The expressions for the critical value of 3 will now be found

for the two separate cases of a= and a=1. First, when a= 0,

equation (3.1.24) becomes


A6 = A

2(1+v) 1+ z f
c 1
1 + 2v (3.1.27)
z --
c f
1

c z +1
C j

and in this case 6= -60" from equation (3.1.7) and

fl = cos2(-60) -273 sin (-600) cos (-600) +3 sin2(-600)


f -24( )( ) + 3( )
1 4 4 2 2 2


f =1

and
1
c (1+ 2v) +1

1
S2 ) (3.1.28)
c 2(1+ v)









When a = 1,

2 1 9
fl 2 af3 6 (cos" -2 ,':3sin cos c + 3 sin- )
22 2 2 21]

(cos52 + 2/3 sin cos 6+3 sin6) -(3 sin6 -cos2)2



ff2 af = 0
1 2 3

and equation (3.1.26) becomes


-[2(l +v) -1+ V
c [(2)(1 +) -1] f, -f +2vf


(1+v)
z (3.1.29)
S 3 (1 +v) sin 6 cos + 3v sin26



A short computer program was written to calculate the critical

values of (using equation (3.1.28) when a=0 and equations (3.1.29)

and (3.1.27) when a= 1) for various values of v and 6 when y= 0.

This program is shown in Appendix C and the results are plotted in

Figure 3.3. The only values of 3 which are physically possible are

between 0 and 1 and therefore only values of $ in this range are

plotted in Figure 3.3. For all other values of 6, there is no phys-

ically possible critical value of S; that is, there is no value of

such that the fast and slow wave speeds are equal at y= 0.

For the case when a = 0 (6 =-60), for any value of v the critical

value of $ is smaller when radial inertia effects are included.











v = 0, a 0

v = .25,

=.30, a


Figure 3.3 Values of 3 at y=00 for which cf cs = c
s 2


.6


.4 _


0 -30 -60 -90
6, Degrees










3.2 Characterirsi.c Solution in Terms
ol' uL'i ; loni:iolon ?le s C:: vria bles


In order to make the numerical solution in the characteristic

plane more general, the equations for the characteristics and the

equations along the characteristics given in Chapter 2 will be written

in terms of dimensionless variables. The dimensionless variables

used are

c
x 1 s
X T = t s = -
2r 2r E
0 0


x ex
S S T
x E E E



V VV V
v v v
x e r
V v =- V =-
x 9 r -



Sc c
c -f s
c c c (3.2.1)
s
c c c


r


C1
,(Sf) 2o








0 (S
Y(s,A) = -0----
s


where c1 given by equation (3.1.16) is the elastic longitudinal wave

speed when radial inertia effects are included. In terms of these

variables, the functions defined by equations (3.1.18) can be written

from equations (2.23) as







(2S aS )2
x 9
A1 EAI = 1 9(st)
A1 = E2 1 2 O(s,6)
4s

(2S aS(S 2aS)
A2 = EA2 L + 4s (s,
4s

6T(2S aS )
A3 = E 3 2 2(s)
4s

(S 2aS )2
A4 = EA4 2 -(s,) +
4s


A = EA =
5 5.


A6 = EA6 =
6 6


6T(S 2aS )
4x


2
9T2
2(1+ v) + 2 @(s,A).
s


Using these and equations (3.1.19), the fast and slow wave speeds can

be written from equations (2.49) and (2.50) as


cl


c
s
s (
c1


S {b + (b2 4aA4 (1 -2
2a p


= [E { b (b'2 4a'A4 (1
2a p


/2s
f = -- b' + (b'2 4aAA4)2
2a


es b' (b2 4a'A4
L 2a'


(3.2.3)


and the wave speeds in dimensionless form are given by equations


(3.2.3) and (2.47) and (2.48) as


(3.2.2)






48


c = cf


c = c (3.2.4)


c = 0 (twice)


where the wave speeds were written in dimensionless form by dividing

the wave speeds by cl. This was done because cl is the fastest wave

speed possible in the problem considered here, and thus all the

dimensionless wae speeds have values in the range


-1 c < 1 .


,When radial inertia effects are not included, the maximum value of the

fast wave speed is

C -
Cf p=
max


so that, for a=0, the maximum value of the dimensionless wave speed is


Cf
I cfI < max E/

1 E

p (1 -v2)

Next, the equations along the characteristics for fully coupled

waves will be written in dimensionless form. Along the vertical char-

acteristics (c=0), the equations can be written from equations (2.33),

(2.54) and (2.55) as

-aS dT = a L 1 dV
2(1 -o2)
or

a [2(1 -v2)S] dT = adV (3.2.5)










.I2aS *S a
a(2V dT) = aA2dS -A!dS +AArC + -- (s,. j
r L 2 x 4 + 5 \ s 0 o


(3.2.6)


The equations along the nonvertical characteristics (c= cf, cs)

can be written in three different forms from equations (2.51), (2.52)

and (2.53). In dimensionless form these become, respectively,


O = I ---(A4A6 aA5) -j dx A2A5 -A3A dV
1 -v -v


2 2
1- c 2
S[aA2A5 -A3A4] dT + 2 ( 2(A4A6 5 -A4dS
c 1 -v

[(2 ) r92S aS]
+ C -2-(A4A6 _A2) -< s "0 o0(s,)dT


K 2 \ n r -S 2aS
+ a L( 22(A A(3 -A A ) A 2V+
2+ (A2 A6-A -A3A5) 2J L2Vr s( )dT

22 ]
L+ 6 V-2 2 L _A o(s')Jd


(3.2.7)


2
S= c 2 (A A -AA5 )-A dV -




c 1 \


+ -( 2,) c <2A
c 1 -v


+ (--. (A2A A3A5)
L \1 -v 2A


r- 2
L c (A5-A2A )-A dV
c L5 2-v 5


- A5 dT


- A3A5) A dSx


2Sx aS)dT
-A2] Lx o(s dT
-i 2 "0


(1- 2 r( e2 \2 3 2
+ ( ) [ 2) (A1A6 --) (A1 +A 1
c 1 -v 1 -V


and









F S 2aS -
2V + ( (sA) dT
r s 0o



+ 6 c 2 (A1A5 A2A3) (s,A)]dT (3.2.8)


and
3 2
0 c 2 aA2A A3A4 d c 2) (A1A4-aA2) -A dV
(1 -v 2) 1 -v -v


+ c2 (A1A4 -aA 2)-A4 dT+ L_ 2A5-A3A 4 dSx
1 -v 1 -V
2 2 r -2S a

+ ) LaA2A-A3A4 L s o(s, l dT
1 -i-


+ a )L-- (AlA5-A 2A3)-A 2Vr x s(s'A)j dT



+ 6 2) (A- aA ] )-AA tsi AJdT (3.2.9)
1-v 1 -v

WVhen the equations are uncoupled (A3 = A 0), the equations along the

characteristics are given by equations (B.4.7) and (B.4.8). In dimen-

sionless form, the equations (B.4.8) along the vertical characteristics

(c=0) become

a[2(a -v )S ] dT = adV (3.2.10)
er
and
2aS S
a[2V dT] = a[AdS + A4dS + -- (s,)dT] (3.2.11)


The equations (B.4.7) along the nonvertical characteristics (c= c )

for the fast waves are given by








1 1 -2 2S aS
_0 1 dVd + 2- d + ---- dTA4
f c

r S 2aS
+ 2 L2Vr + (- ) o dT (3.2.12)


and the equations (B.4.7) along the nonvertical characteristics

(c= c ) for the slow waves can be written as
s

2
c 6c2
0 = s dV + dT + () o(s,T)d (3.2.13)
L(1-v2) u (l-v2)


The equations (3.2.5), (3.2.6), and (3.2.7), or (3.2.8), or

(3.2.9) are the equations along the characteristics for the fully

coupled waves written in differential form in terms of the dimension-

less variables. The solution to this set of equations will be obtained

numerically by writing them in finite difference form, and then solving

the resulting set of algebraic equations simultaneously. When the

waves are uncoupled, the equations along the characteristics are given

by equations (3.2.10), (3.2.11), (3.2.12), and (3.2.13). These equa-

tions will also be written in finite difference form and solved (when

applicable) in the same manner as described for fully coupled waves.

The procedure for obtaining these finite difference solutions is out-

lined in the next sections.


3.3 Numerical Grid for Characteristic Solution

Since the slopes of the characteristic lines at any point in the

characteristic plane depend upon the state of stress at that point and

upon the history of the deformation at the corresponding location along

the axis of the tube, the equations for the characteristic lines cannot








be determined before the solution (in terms of stresses) is known.

Because of this, the slope of the characteristic lines and the solution

to the problem must be determined at each point simultaneously. This

is done by using the iterative numerical technique described below.

The numerical grid shown in Figure 3.4 will be used. There are

two types of elements in this grid: boundary elements and regular

elements. All of the regular elements are alike, and all the boundary

elements are like the right-hand side of a regular element. A detailed

picture of a single regular element is shown in Figure 3.5, and a

single boundary element is shown in Figure 3.6. The grid is defined

in terms of the dimensionless variables given in equations (3.2.1) and

(3.2.3). It is diamond shaped with the straight outer lines corre-

sponding to the characteristic lines for elastic longitudinal waves

with radial inertia effects included. These outer characteristic lines

have slopes of either c= +1 or c= -1, which can be seen from equations

(3.2.1) and (B.5.4). The vertical straight line corresponds to the

two vertical characteristic lines, and the straight inner nonvertical

lines correspond to the characteristic lines (through the point P) for

the elastic shear waves. For both types of elements, the problem reduces

to that of determining the values of the stresses and velocities at the

point P, when their values at the points B, R, and L are known.

This grid with all the elements constant in size simplifies the

writing of the finite difference equations. The diamond shape allows

the vertical characteristic lines to automatically connect point P (at

which the solution is desired) with point B (at which the solution is

known) and makes the finite difference equations along the vertical




53





T




































AFigX -3.4 Nu l Gd in te Chc



Figure 3.4 Numerical Grid in the Characteristic Plane


2AT


2AT


77













































Ic = -c = --





--
AI-


1c


B

-a


Figure 3.5 Regular Element in Numerical Grid


op






55





T






P










c = -c = -


C 1 I












c- c= c =1



B


---- AX ----- k



LX












Figure 3.6 Boundary Element in Numerical Grid









characteristic lines very easy to obtain. The boundary lines for each

element are c-= This is the smallest value of c which insures that

all characteristic lines passing through tne point P will intersect

the line L-B between the points L and D if the lines have a positive

slope at P or will intersect the line R-B between the points R and B

if they have a negative slope at P. This is true since all of the

waves considered here will propagate with a speed less than or equal

to the speed of an elastic longitudinal wave with radial inertia effects

included. A larger value of c could be used, but the element size

would increase (for a given distance along the T axis), and the solution

would be inherently less accurate.

The straight lines representing the elastic shear wave character-

jstic lines are added to the grid elements as a convenience. The

results of Section 3.1 show that the fast wave speed always occurs in

the range


c2 : cf 1 c1

and the slow wave speed always occurs in the range


0 O c s c2
s 2

Therefore, these characteristic lines c=c2 divide each element so

that a characteristic line through P lies in one of the upper triangles

(P -L -LB or P -R-RB) if it is for a fast wave and in one of the lower

triangles (P -B-LB or P -B-RB) if it is for a slow wave. This is

shown for the regular grid elements in Figure 3.7. These characteristic

lines for the fast and slow waves will not, in general, be straight.









P




e ---/I \ \=\-


c = c2 / \ -0C 2




/ c c = -cC

c = 0





B



-------------------------------------------------------------------------------------------------- ------- ^- x


Figure 3.7 Location of the Characteristic Lines Passing Through P









3.4 Finite Difference Equations

General Discussion

1hile the actual characteristic lines for the fast and slow waves

are seldom straight, they can be represented as straight lines within

each grid element without introducing significant errors if the grid

elements are small. From the discussion in Section 3.3, it is known

that the slope of the characteristic lines at any point cannot be

determined before the solution at that point is known. Because of

this the solution at the point P (Figures 3.5, 3.6, and 3.7) must be

obtained by an iterative technique. Within any grid element, the slope

of each characteristic line will be constant during each iteration

although the slope of each characteristic line will change from one

iteration to the next as the solution at P is approached. These

straight lines are used to represent the characteristic lines for

th
c= c and c = c during each iteration and are shown for the i iter-
f s

ation as c=cfi and c= cs in Figure 3.8 for a regular grid element

and in Figure 3.9 for a boundary grid element. The points LLB, LBB,

RBB, and RRB are the intersections of the lines shown in Figure 3.8.

Each element has its own coordinate system X -T which is also shown

in Figures 3.8 and 3.9, and the finite difference equations are written

in terms of this local coordinate system so that the finite difference

equations for each element are the same.

First order finite difference equations will be written along each

characteristic line. The coefficients of the dependent variables in

these equations will in general be functions of the stresses and (s,A).

Thus, in order to linearize the equations, the coefficients for each





































T 2




bx


Figure 3.8 Numerical Representation of the Characteristic Lines in a Regular Grid Element






60


I
ST












P


c = -C1
C- i=

C = -cf







R


RRBI

c=-c )\
s. \
1
AT
RBB
c=0 RBB




B

^- ---- ^X ---- =






X




Figure 3.9 Numerical Representation of the Characteristic Lines
in a Boundary Grid Element








iteration will be calculated using the solution obtained in the

previous iteration. In this way the coefficients are :llays known

quantities.

One other scheme will be used with the coefficients in the finite

difference equations in order to reduce the time required for compu-

tation. Normally each coefficient used is the average value of that

coefficient at the end points of the interval over which the finite

difference equations are written. As an example, consider the charac-

teristic line from point LLB to point P, and let one term in the finite

difference equation along this characteristic line be


U(S S ).
xP xLLB


As a rule the value of the coefficient is calculated as

1
U = (Up + ULLB)


If this method is used, the coefficients of each variable in the equa-

tions along the characteristics of positive slope will be different

from the coefficients of the corresponding variables in the equations

along the characteristics of negative slope. For instance, one term

in the equation along c=+cf. can be represented as


S(SxP xLLB

and the corresponding term in the equation along c=:-cfi as


UR(Sx SxRRB

where
1
U = (U + ULLB)

S= +
R 2 p RRB









and U is the value of the coefficient at point P calculated from the

solution from the previous iteration.

When the coefficients are calculated in this manner, the number of

equations which must be solved simultaneously cannot be conveniently

reduced below five. However, if the coefficients are calculated in

such a way that the coefficient of any variable in the equation along

c=+cfi is equal to the coefficient of that same variable in the

equation along c=-cfi (so that UL = UR, etc.), then by adding these

two equations and subtracting one from the other, two different equa-

tions can be obtained, each with fewer variables than the two original

equations. If this procedure is applied to the equations along c=Csi,

then the set of five simultaneous equations can be reduced to at most

a set of three simultaneous equations and a set of two simultaneous

equations. This is shown in Appendix D. Since this set (or sets) of

equations must be solved during each iteration, the savings in compu-

tation time is significant.

One way to make the coefficients of similar terms equal is to

calculate the coefficients from values of the variables obtained at

point P during the previous iteration such as


UL = R = U *

The coefficients will be calculated in a somewhat more accurate manner

by using a weighted average of the value of each coefficient between

point P and point B, that is


UL = UR = alUP + (1 )UB










This gives the value of the coefficients at a point nearer the center

of each grid element. For this work, the value of ai is chosen arbi-

trarily as .625, so that the point at which the coefficients are calcu-

lated is at approximately the same location along the T'-axis

(Figures 3.8 and 3.9) as the centers of the four characteristic lines

c= cfi, Csi.

The values of all quantities at the points LB, LLB, and LBB will

be obtained by linear interpolation between the points L and B.

Similarly, the values of all quantities at the points RB, RRB, and

RBB will be obtained by linear interpolation between the points R and

B. From Figure 3.8, the times T1, T2, and T3 can be written as

2c 2c 2c
s 2 f
T, = AT T c AT T +c- AT
1 1 +c 2 1 + c 3 1 +c
s 2 f


and the interpolation constants for the points LB and RB are

2 2c2
CLRB -
AT 1+ c2


T2 1 2
CLRBI = 1
AT 1 + c2


Using subscripts to denote the grid point, the values of any quantity F

at the points LB and RB are


FLB = CLRB*FL + CLRBI*FB
LB L B


FRB = CLRB*FR + CLRBI*FB
RB R B









The interpolation constants for the points LLB, LBB, RBB, and RRB are


T -T 2(c c)
CON1 -
AT T2 (1 + cf)( c2)


T1 Cs ( + c 2)
CON2 = -- s
T c2(1 + c )
2 2 s


CON3 = 1 CON1



CON4 = 1 CON2


so that the value of F at each of these points is


FLLB = CON1*FL + CON3*FLB


FBB = CON2FLB + CON4*FB


FRRB = CON1*R + CON3*FRB


FRBB = CON2*F + CON4*FB
RBB RB B

For Fully Coupled Waves

When the equations are fully coupled, the equations along the

nonvertical characteristics (c=c cs) are given by either

equation (3.2.7) (3.2.8), or (3.2.9). Equation (3.2.7) will be used,

and the values of AV, A2, A3, A4, A5, and A6 of equation (3.2.2) will
th
be calculated for the i iteration as described for U earlier in this

section and defined as Ali, A2i, A3i A4i A5i, and A6i, respectively.

th
The coefficients will then be defined for the i iteration as









2
IR c f (A .A
If \ 2/ 4i 6i
1 v

2
C
R2f = 2 (AA 6i
1 v

2
c
/ s
R is 2 (A 4iA6i
1 v

2

R2s 2) (-2i 6i
1 v


2
- aA.) -
5i


- A3iAi) A2i
3i 5i 2i


2
- aA ) -
5i


- Ai ) A2i
*3i 5i 2i


Rfs= aA iA i- Ai A
fs 2i 5i 3i 4i


Thus, the finite difference equations when the waves are fully coupled

can be written directly from equations (3.2.5), (3.2.6) and (3.2.7),

and using the last subscript to represent the point in the numerical

grids of Figures 3.8 and 3.9 the equation

along c = 0 is


2 [SP SB AT = a(V
-2a(l v 2AT = a(V2
L 2 rP


rB- )


(3.4.2)


along c = 0 is

rP rB A + 2B\
2a VB (2LT) = a 2P )(S S )
2 1 2 ) xP xB


A4P + A4B (S
+ ( 2 (SeP


S()A5P + A5B)( B
- S ) + (A( T )
OB \ 2 P B


1 P- xp + B SxB
+ 2 -- + 2aS (2LT)]
SoP/ s
P


(3.4.3)


(3.4.1)








along c = + c is


-Rf
0 f V
Cf xP


+ R f(T p
fs P


c Rfs
xLLB 2 VP
1 -v


- V )LLB
eLLB


(1 2)Rf
) + (S S )
LLB 2 xP xLLB
Cf


+ R [lf1 (2Sxp


- aS + 2SLL
GP xLLB


- aS ) 2AT 1
6LLB 1 + cf_


+ FV +V + '1 + (Sxp 2aS +S 2aS
2irP rLLB 1 2 xP xLLB GLLB J


[2 T f
* T
L1 + c j


6c Rfs1 l1
2 2 (Tp + P LLB
1 v


2[rAT
/+f


(3.4.4)


along c = cf is


Rlf
0 = (V
C xP
f


c R
f fs
- V ) + (V V )
xRRB 2 GP RRB
1 v


+ R (TP RR) +
fs P RRB


+ R f 1i


(1 v2)Rf
2 R if(S S )
2 xP xRRB
f


(2Sxp aS p + 2SRRB
xP 6P xRRB


- aS )L cfJ
6RRB 1 + c


+ aR2Vr +VrRRB + (SP -2aSgp +SxRB aSRR)

2
S2T 1 R fs 1 R 2T- (3.4.5)
26T* L (T (3.4.5)
1+ c 2 P RRB c
L1 -C






67

along c = + c is
s

1s s -fs
c xP xLBB 2 eP 6LBB
s l-v

(1 V2)R

s

+ Rs 1(2Sxp aS + BB (aS BB+



+ aR V + V + -(S -2aS +S -2aSL)
s rP rLBB 2 P P xLBB LBB
6c 2R



2 2T 6csRf 1 T ) j (3.4.6)
+ c 2 L2 TP + LBB Ll+ (3
s 1 s


along c = c is
s

R cR
Is s fs
0 =- (V -V ) + (V -V )
c xP xRBB 2 GP eRBB
s 1-v

(1 v2)R
+ (T TRBB) + 2 (S SRBB)
fs P RBB 2 xP xRBB
C
s

+ R11 [+(2Sx -aS +2S -aS )] [2
1S 1 xP GP xRBB GRBB 1 +c
s


+ aR2s [Vrp VrRBB + 1{(Sx -2aS p+ SxRBB -2aSRBB)}


c 2 R
+C + (T + T ) 1 (3.4.7)
L +c2 2 [ (TP + RBB 1 +c -RB J
S 1 v s

where the values of *P / and s / obtained at point P from the previous

iteration are used and
oP roB
S= al --+ (1 -a) -s (3.4.8)
sp/ B

with al defined earlier in this section.









For Uncoupled Waves

Then the waves become uncoupled as described ir Section B. l, the

equations along the characteristic lines have a simpler form given by

equations (3.2.10) to (3.2.13). Using the averaging technique already

described in this section for the coefficients, the equations along

the characteristic lines for uncoupled waves simplify. The equation

along c = 0 is

Sp + SB
2 QP GB
-2a(1 -v )( )(2AT) = a(Vp V ) (3.4.9)
2 rP rB

along c = 0 is

[(A +A A +A
a(V + V )(2T) = a (2 2B)(S S) + 4( )(S -s
rP rB 2 xP xB 2 OP GB

S(2aS -S )* o (2aS -S )P oB
+ a op + --9B- B (2LT) (3.4.10)
2L Sp, s B

along c = + c is

2
S(V -V ) + (S -S
S= x LLB 2 xP xLLB
f c

1 2,T
+ 1 {(2S aS + 2SLLB aS ALLB ] i
frP r P xLLB GLLB 1 C B 4i


+ aA V+V + l(S -2aS +iS -2aS ) 2/T
+ VrPVrLLB+ P G xlLB LLB I

(3.4.11)









along c = c is


0 = A (vP-x ) + 12 (Sx -S
4 c xP xRRB 2 xP xRRB
f Cf


+ 1{ (2SxP -aS e+2SxRRB-2aS eRB)} -i


+ aA v +V +'{(S 2aS SX 2aS, 1 26T.
2iL rP YrRRB 1 (SxP-2aS PxRRB- SRRB L

(3.4.12)

along c = + c is
s
c
C
0= (V p V LBB) + (T -T LBB
2 P LLBB P LBB


S- .(Tp + TLBB) 2 A (3.4.13)
2 L2P
l-v s


along c = c is
s
c
s
0 (V e- V ) + (T T )
-2 eP eRBB P RBB

2

+ 22 (P + RBB) (3.4+
1-v s


3.5 Solution to the Finite Difference Equations


The solution to the finite difference equations of Section 3.4 are

given here for any iteration. The solutions consist of expressions for

VxP V p, VrP, Sx, S and Tp in terms of known quantities, including

quantities calculated during a previous iteration. The solutions given

in this section are obtained using Cramer's rule as shown in Appendix D,

and the definitions of the variables used in Appendix D will not be


repeated here.









At a Regular Grid Point for Fully Coupled Waves

The solution to the finite difference equations along the charac-

teristic lines at a regular grid point in the case of fully coupled

waves is given here. The longitudinal and transverse velocities from

equations (D.3.5) and (D.3.6) are
1
V = (Ds RHSBA D2 RHSDC) (3.5.1)
1


V --(Df RHSDC Ds RHSBA) (3.5.2)


When radial inertia effects are included, the stresses at point P

are given by equations (D.3.13), (D.3.14), and (D.3.15). These stresses

are

T = RHSF(D2 D1D7s) + RHSG(D1D D2D4f)


+ RHSH(D4fD7s D4sD7f)] (3.5.3)


1 r
S RHSF(A D DD ) + RHSG(D2D AD )
xP 2L 5Q 7s 23s 2 3f 5Q 7f


+ RHSH(D3sDf D3fD7s) (3.5.4)



S HSF(D1D A D4s) + RHSG(A5QD D1D)
P =2 53s 5Q 4s Q 4f 3f


+ RHSH(D3fD4s D4fD3s) (3.5.5)


and the radial velocity of equation (D.1.3) is


aVrp = a(D3 Q3S p). (3.5.6)

When radial inertia effects are not included, the hoop stress,

S p, and the radial velocity, VrP, automatically vanish, and the shear
veocty V,





71


stress and the longitudinal stress are given by equations (D.3.18) and

(D.3.19), respectively, as

S= (D RHSF D4f RHSG) (3.5.7)
P 4s 4f
1
S = (D RHSG D RHSF). (3.5.8)
xP 3 3f 3s
3


At a Regular Grid Point for Uncoupled Waves

When the waves are uncoupled, the solution to the finite differ-

ence equations has a much simpler form. In this case, the shear stress,

the transverse velocity, and the longitudinal velocity of equations

(D.4.1), (D.4.2), and (D.4.3), respectively, are


T = (RHSCE + RHSDE) (3.5.9)
P 2F2
2s





1
V = f (RHSBEM RHSAEM) .(3.5.11)
xP 2F


When radial inertia effects are included, the longitudinal and

hoop stresses from equations (D.4.8) and (D .4.9) are

1
SP = (D2 RHS3 F5f RHSEEM) (3.5.12)
5
1
S = (F RHSEEM D RHS3) (3.5.13)
OP 6 2f2 1
5

and the radial velocity is again given by equation (3.5.6).

When radial inertia effects are not considered, both the hoop stress

and radial velocity vanish, and the longitudinal stress of equation (D.4.10)

is
RHS3
S S (3.5.14)
xP F22
2f2









At a Boundary Point (X=0) for Fully Coupled Waves

In a boundary element, there are only four characteristic lines

(cO, c= O, c =-c c= -c ) and consequently only four equations along

these characteristic lines. Since the equations along the character-

istic lines are written in terms of six unknown variables at point P,

the solution at a boundary point can be obtained only if two of these

variables are prescribed at each boundary point. The hoop stress and

the radial velocity do not enter the formulation of the problem when

radial inertia effects are omitted, and therefore these variables are

not specified at the boundary. Thus, the four remaining variables, two

of which may be specified at any boundary point, are the longitudinal

stress, the longitudinal velocity, the shear stress, and the transverse

velocity. From a purely physical standpoint, it is also reasonable to

specify the longitudinal and transverse variables at the boundary since

these are the quantities which are normally associated with the impact

at the end of the tube and which can be measured more readily than

radial velocity and hoop stress. Only two of the four variables Sx,

Vx, T and V can be specified at any one boundary point. Furthermore,

at a given boundary point V and S cannot both be specified since they
x x

are not independent. Also, both T and V6 cannot be given at the same

boundary point. Therefore, four combinations of variables to be speci-

fied on the boundary will be considered: for Case I, S and T will be
x
given at the boundary, for Case II, Vx and V will be given, for Case III,

S and Ve will be given, and for Case IV, Vx and T will be given. The

solution to the finite difference equations at a boundary point for each

of these four cases when the waves are fully coupled is given below.









Case I: Traction boundary conditions

ihen Sp amd are known, then from equations (D.5.1), (D.5.5)

and (D.5.6) the solution to the finite difference equations at P is

a
Sp D (RHSH DS A5QT p) (3.5.15)



Vp = (B2s RHS1 Bf RHS2) (3.5.16)
4


V = 1 (B RHS2 B RHS1) (3.5.17)
OP A if is

and Vrp is given by (3.5.6).


Case II: Kinematic boundary conditions

When Vxp and Vp are given, the solution at P is given by

equations (D.5.11), (D.5.12), and (D.5.13) when radial inertia effects

are included as


T RHS4(D42 D2 + B7s D1 RHS5(D D2+BfD


+ RHSH(D4s2B7f D4f2B7)] (3.5.18)


Sx [RHS4(D3s D + AD) + RS5(D D + B A)
xP L6 3s2 2 5Q 7s 3f2 2 7f5Q


-RHSH(D3s2Bf D3f2B7)] (3.5.19)



S RHS4(D D A D ) RHS5(D D A D
P = RS 3s2 1 A5Q4s2 3f2D1 -5Q4f2


+ RHSH(D3f2D4 D4f2D3s2)] (3.5.20)


and Vrp is given by equation (3.5.6). When radial inertia effects are

not included, Vrp and Sp are zero and the solution given by equations

(D.5.16) and (D.5.17) is









T = A- (D sRHS4 D RHS5) (3.5.21)
P 4 4s2 4f2
7

1
Sxp = (D3 RHS5 D s2RHS) (3.5.22)
xP A 3f2 3s2


Case III: Mixed boundary conditions

When Sxp and Vp are known, the solution when radial inertia

effects are included is given by equations (D.5.22), (D.5.23), and

(D.5.24) as


S RHS6(D D + AB) RS7(D D + A5Q B 7f)
xP L 3s2 2 5Q7s 3f22 5Q7f
8

+ RHS8(D3s2B7f D3f2B7s) (3.5.23)



Tp F- (D2RHS7 + B7sRHS8) B s(D2RHS6 + B RHS8) (3.5.24)
P A Li~f 8 2 2


S 1 Bf(D3s2RHS8 -A5QRHS7) -B (D3f2RHS8-A5RHS6)] (3.5.25)


and Vrp again is found from equation (3.5.6). When radial inertia

effects are not included, VrP= S= 0 and from equations (D.5.27)

and (D.5.28), the solution at P becomes


V = (D 3s2RHS6 D 3f2RHS7) (3.5.26)
xP A 3s2 3f2


p = 1 (B RHS7 B RHS6) (3.5.27)
9 if Is
9


Case IV: Mixed boundary conditions

When Vp and T are known at the boundary, the solution at P is

found from equations (D.5.33), (D.5.34), and (D.5.35) when radial

inertia effects are included to be









i F
LP 10 RHS9(Ds2D + D1B7s) RHS10(D4f2D2 + D1B7f)

-7
+ RHSll(D4s2B7s D4f2B7s)J (3.5.28)



SxP 0 B2f(D2RHS10 + B7sRHS11) B2s(D2RHS9 + BsRHS11) (3.5.29)



Sp = LB2f(D4s2RHS11 -D1RHS10) B2s(D4f2RHS11 -D1RHS9) (3.5.30)


where again Vrp is given by equation (3.5.6). When radial inertia

effects are not included, Vr and S vanish, and from equations (D.5.38)

and (D.5.39), the solution at P is found to be



11


SP = (B2fRHS10 B2sRHS9) (3.5.32)
11


At a Boundary Point (X= 0) for Uncoupled Waves

When the waves are uncoupled, the solutionsto the finite differ-

ence equations are obtained at the boundary points for the same four

cases outlined above. When radial inertia terms are included in the

formulation of the problem, the expression for VrP is given by

equation (3.5.6), and in all cases when radial inertia terms are not

included both Vrp and S vanish. In all four cases the solutions can

be found in Appendix D.


Case I: Traction boundary conditions

When T and Sxp are known at a boundary point, then from equations

(D.6.1), (D.6.2), and (D.6.3) at that point










V = (RISDE F Tp) (3.5.33)
GP z9 2s P

a
S = (RHSEEM D S ) (3.5.34)
2
1
VP (RHSBEM aFfSp F2f Sx) (3.5.35)
XP F1 5f P 2f xP



Case II: Kinematic boundary conditions

When Vxp and Vp are prescribed at a boundary point, then the

solution at that boundary point is given by equations (D.6.4), (D.6.9),

(D.6.10), and (D.6.11). When radial inertia is included the solution is


S 1- (D RHS12 Ff RHSEEM) (3.5.36)
12

S P 1 (F RHSEEM D RHS12) (3.5.37)
P 2f 1
12

and when no radial inertia effects are included the solution becomes

RHS12
S (3.5.38)
xP F2f


The shear stress in both cases is

1
T (RHSDE Z V ) (3.5.39)
P F2 2 OP


Case III: Mixed boundary conditions

When Sp and Vp are known at a boundary point, then from

equations (D.6.12), (D.6.13), and (D.6.14), the solution at that point

is
Tp (RHSDE Z2 V ) (3.5.40)
F2s
a
S = D2 (RHSEEM DSpS ) (3.5.41)


1
Vxp (RHSBEM F S aF Sep) (
P 2f xP f P). (3.5.42)
if










Case IV: Mixed boundary conditions

Vihen VXP and are given at a boundary point, then the solution

at that point is given by equations (D.6.9), (D.6.10), (D.6.11) and

(D.6.15) i.e.,

1
p = (RHSDE F T ) (3.5.43)
oP 2 2s P


and Sxp and Sp are given by equations (3.5.36), (3.5.37), and

(3.5.38).


3.6 Calculation of the Strains


At any grid point P, the solution is obtained by an iterative

technique. Once this is done, the values of Sx, Se, T Vx, Vr, and

V are known at P as well as at points L, B, and R (see Figures 3.5

and 3.6). The strains at point P can be computed very easily from

equations (2.27), (2.28), and (2.29). These equations can be written

in dimensionless form using equation (3.2.1) as


x x
-x x (3.6.1)



T 2 (3.6.2)


= 2Vr (3.6.3)


For a regular grid element, these equations can be written in

finite difference form as









e V V V
xP xB xR xL
2ZT 2t X

C. C V -V
exP OxB 1 (R L
2AT 2 2AX

OP B Vrp VrB
9P eB rP rB
= 2 ( -)
2AT 2


where the final subscript on each variable denotes the point in the

grid element where that variable is evaluated. Now, since the outer
dX
grid lines defined in Section 3.3 have slopes of c = 1, AX and

AT are equal so that the expressions for the strains at point P are


C = x+ V V (3.6.4)
xP xB xR XL

1
CxP = exB + (V V ) (3.6.5)


e = CeB + 2(Vrp + VrB) AT (3.6.6)


For a boundary grid element, equations (3.6.1), (3.6.2), and

(3.6.3) can be written in finite difference form as

1
e C V -1 (V + V )
xP xB xR 2 xP xB
2LT 4x

GxP OxB 1 R 2 (Vp + B)
2AT L AX


eP V + V
eP BA2 [frP 2 rB
2 T 2

and again since AX = AT, the strains at the boundary point P are given

by





79


S= xB + 2V V VxB (3.6.7)
xP xB xR xP xB

1
xP = eB + VR 2 (Ve + ) (3.6.8)


e" = 6eB + 2(VrP + VrB) AT (3.6.9)


where equations (3.6.6) and (3.6.9) are the same expression.
















CHAPTER 4

RESULTS AND DISCUSSION


4.1 Introduction


In Chapter 2 the problem of inelastic wave propagation was

formulated and the equations for this problem were found. In Chapter 3

these equations were written in finite difference form and from them

expressions for the stresses and the velocities at the points in the

numerical grid (Figure 3.4) were determined. Next a computer code

(shown in Appendix E) was written to facilitate the calculation of

the stresses, velocities, and strains at the grid points in the charac-

teristic plane. Now, in this chapter the results obtained by using

this computer code will be discussed for several different combinations

of initial conditions and boundary conditions.

The computer code is written so that the boundary conditions are

specified by reading in values of two variables at each grid point

along the boundary (X=0). By specifying the boundary conditions in

this manner, any variable given as one of the boundary conditions can

have any functional shape. All of the data presented in this chapter

were obtained using the kinematic boundary conditions (Case II), that

is, by assigning values to the longitudinal velocity (V ) and the trans-

verse velocity (V ) at the impact end of the tube. Furthermore, the

same functional form was chosen for the two velocities in each case.

This form consists of assuming that each of the velocities at the










boundary increases linearly up to its final value (denoted by Vxf

or VG ) during a period of time called the rise time (T ) and then

remains constant. That is

T
TV Vqf if 0 T < T
T R R

V@ (X = 0) =


Vef if T > TR



T Vxf if 0 < T < T
Tr R R
R

V (X = ) =


Vxf if T > TR


Now that the computer code is set up, it would be advantageous to

compare the results from it to data which have already been published.

This is done in the following section by using the data of Lipkin and

Clifton (1970), and some interesting effects of the size of the numer-

ical grid are noted. Then, finally, the effects of radial inertia and

strain-rate dependence on the propagation of inelastic stress waves are

discussed.


4.2 Effects of Numerical Grid Size


Lipkin and Clifton (1970) published the results of three different

experiments where a thin-walled tube was given an initial static shear

stress and then impacted longitudinally. In this section the initial

conditions and boundary conditions from one of these experiments will

be used and the results obtained from the computer code will be compared










with the experimental and theoretical results of Lipkin and Clifton

(1970). The data vhich will be used are


0
T O


0
x


= initial static shear stress = 3480 psi


= initial static longitudinal stress = 0


Xf = final longitudinal boundary velocity = 500 ips


vf = final transverse boundary velocity = 23 ips


= rise time = 9.6 p sec


which can be written in terms of the dimensionless quantities for input

to the computer code as


0
T
-o 6x
E


= .0003480


0
o0
S x 0
S O 0
x E


Cl
T
R 2r


No radial inertia effects or rate

this section.


v
xf
Xf
V .002404
x
f c1


ef
V .0001106
f c1



t = 4.00


dependence will be considered in


The results from three different computer runs will now be made.

Each computer run used these initial conditions and boundary conditions

but had different grid sizes. The three grid sizes used were

AX=AT = .25, AX=AT= .125, and AX=AT= .05. The longitudinal strain

versus time obtained by using the computer code in Appendix E is shown

in Figure 4.1 along with the experimental results and the simple wave

solution of Lipkin and Clifton (1970). From this it can be seen that
































Simple Wave Solution, Lipkin and Clifton (1970)
Experimental Results, Lipkin and Clifton (1970)


----- X .05


---- X .125

---- X= .25


Figure 4.1


u G60
Time, T

Grid Size Effects on the Longitudinal Strain at X = 3.75


-.012


-.008











-.00,4









for the small grid size the strain follows closely the strain obtained

by Lipkin and Clifton (1970) for a simple ;'.ave with ;n instantaneously

applied velocity at the boundary. For the larger grid sizes the strain-

versus-time curve is smoother and follows more closely the experimental

results of Lipkin and Clifton (1970). Apparently, the larger grid

sizes tend to smooth out the data and eliminate the distinction between

the fast and slow wave speeds. For instance, in Figure 4.1, the' simple

wave solution of Lipkin and Clifton (1970) exhibits a region where the

longitudinal strain has the constant value of 0.00085. The strain

remains at this constant value from just after the fast wave passes

until the arrival of the slow wave.

From these computer runs other quantities of interest can also be

plotted and the same grid size effect can be observed. This is shown

in Figure 4.2 for the longitudinal velocity versus time. The grid size

has a much smaller effect on the stress trajectory than on the time

history curves. The stress trajectory is shown in Figure 4.3.

Because the details of the solution depend on the size of the

numerical grid, all subsequent computer runs will be made using a small

grid. This small grid size necessitates a large amount of computer

time to obtain a solution more than 1.0 diameter from the impact end,

and most of the results given below are obtained near the end of the

tube.


4.3 Effects of Radial Inertia

In order to determine the effects of radial inertia, four separate

computer runs were made using the computer code in Section E.5. The

generalization of the uniaxial stress-strain curve of Lipkin and Clifton






































4/'

// -X= .05


,/I -*- X = .125


/ ----X = .25

-J/


20















10















0


Time, T


Figure 4.2 Grid Size Effects on the Longitudinal Velocity at X = 3.75


C
V
x

r4
o
X



o
>-,




-4(



o
'0
0
1-1
C3

C,


*H


4 --



















-.--- X =.05


*-- X = .125

----X = .25
















..... 5

|.rLr'


Yield Surface
after
Static Preload


-6 -9
Longitudinal Stress, S x '04
A


Figure 4.3 Grid Size Effects on the Stress Trajectories at X = 3.75


4











2











0









(1970) was used. This constitutive equation (shown in Appendix A) was

for strain-rate independent material behavior.

The first two computer runs (one including and one not including

radial inertia effects) were made using the initial conditions and the

boundary conditions which Lipkin and Clifton (1970) used in one of

their experiments. These input data used were


T = 4.00
R

AX = AT = .050


S= 0
x
Data Set 1
T = .0003480


V = .002404
x
-f

V = .0001106
1 f


These data represent a tube with an applied static pretorque

(above the yield stress) impacted longitudinally at one end. The time

history curves of the longitudinal strain and the change in shear

strain are shown in Figures 4.4 and 4.5, respectively, for the section

of the tube 3.75 diameters from the impact end. The simple wave solu-

tion and the experimental results of Lipkin and Clifton (1970) are

also shown in these figures. It can be seen in Figure 4.4 that the

longitudinal strain obtained in this work follows the experimental

results more closely than does the simple wave solution. Most of the

improvement over the simple wave solution is the result of using

a finite rise time (T = 4.0) for the impact velocity. The fast wave

has passed the point X= 3.75 at the time when the longitudinal strain













I- -




i/

/ /
/


- -=s-c.


Solution Without Radial Inertia

---- Solution With Radial Incrtia

---Simple Wave Solution Lipkin and Cliflton (i1 70)

--- Experimental Results Lipkin and Clifton (1.'70)


Figure 4.4 Longitudinal Strain Versus Time at X = 3.75 for Data Set 1


-. 012 -


-.008









-.004_


.7


- I I I
0 20 40 60
Time, T


~------
















.003


Solution Without Radial Inertia

-----Solution With Radial Inertia

-- Simple Wave Solution Lipkin and Clifton (1970)

- -Experimental Results Lipkin and Clifton (1970)


I
- I
4-















/ Time, T
-. I 1




-. 001


Figure 4.5 Change in Shear Strain Versus Time at X = 3.75 for Data Set 1









has reached the value of 0.00085. For the simple wave solution this

time is approximately T 6 Where for the finite rise time (T 41) this

time is approximately T=10. The difference in time when the fast wave

has passed can thus be accounted forby the finite rise time.

As the slow wave passes a point on the tube, the longitudinal com-

pressive strain begins to increase to values larger than 0.00085. The

higher levels of strain (e x -.008) occur later (in the results given
x
here) than in the simple wave solution. Again this can be accounted for

by the finite rise time.

The inclusion of a finite rise time in the theoretical solution

gives results which resemble the experimental data more closely than

the simple wave solution. It can also be seen that including radial

inertia effects in the formulation of the problem gives longitudinal

strains which are somewhat closer to the experimental data than the

corresponding strains when radial inertia effects are ignored.

The change in shear strain versus time curve in Figure 4.5

exhibits the same rise time effect as the longitudinal strain. The

results obtained here are much closer to the experimental data than

the results for the simple wave solution. The final value of the

shear strain appears to be low. Since this shear strain is calculated

from the values of the transverse velocity, it may be that the final

value of the transverse velocity should be larger.

This can be seen more easily by examining the transverse velocity

at several distance from the impact end as shown in Figure 4.6.

A transverse velocity is induced when the tube is impacted with a

longitudinal velocity, if the tube is statically preloaded in torsion.

From Figure 4.6 it can be seen that the transverse velocity induced




Full Text
233
Hunter, S. C, and Johnson, I. A. (1964), "The Propagation of Small
Amplitude Elastic-Plastic Waves in Prestressed Cylindrical Bars,"
Stress Waves in Anelastic Solids, I.U.T.A.M. Symposium, Brown
University (II. Kolsky and W. Prager, Eds.), Berlin: Springer-
Verlag, pp. 149-165.
Karman, T. von (1942), "On the Propagation of Plastic Deformation in
Solids, National Defense Research Commission Report Number A-29
(OSRD No. 365).
Kolsky, H. (1963), Stress Waves in Solids, New York: Dover Publications,
Inc.
Lee, E. H. (1953), "A Boundary Value Problem in the Theory of Plastic
Wave Propagation," Quarterly of Applied Mathematics, Vol. 10,
pp. 335-346.
Lindholm, U. S. (1967), "Some Experiments in Dynamic Plasticity under
Combined Stress," Mechanical Behavior of Materials under Dynamic
Loads (U.S. Lindholm, Ed.), New York: Springer-Verlag, pp. 77-95.
Lipkin, J. and Clifton, R. J. (1970), "Plastic Waves of Combined Stress
Due to Longitudinal Impact of a Pretorqued Tube," Journal of
Applied Mechanics, Vol. 37, pp. 1107-1120.
Lubliner, J. (1964), "A Generalized Theory of Strain-Rate-Dependent
Plastic Wave Propagation in Bars, Journal of the Mechanics and
Physics of Solids, Vol. 12, pp. 59-65.
Lubliner, J. and Valathur, M. (1969), "Some Wave Propagation Problems
in Plastic-Viscoplastic Materials, International Journal of
Solids and Structures, Vol. 5, pp. 1275-1298.
Malvern, L. E. (1949), "The Propagation of Longitudinal Waves of Plastic
Deformation in a Material Exhibiting a Strain-Rate Effect,
Brown University Dissertation.
Malvern, L. E. (1951a), "Plastic Wave Propagation in a Bar of Material
Exhibiting a Strain Rate Effect," Quarterly of Applied Mathematics,
Mathematics, Vol. 8, pp. 405-411.
Malvern, L. E. (1951b), "The Propagation of Longitudinal Waves of
Plastic Deformation in a Bar of Material Exhibiting a Strain-Rate
Effect," Journal of Applied Mechanics, Vol. 18, pp. 203-208.
Malvern, L. E. (1965), "Experimental Studies of Strain-Rate Effects and
Plastic Wave Propagation in Annealed Aluminum," Behavior of
Materials under Dynamic Loading, ASME, pp. 81-92.
Malvern, L. E. (1969), Introduction to the Mechanics of a Continuous
Medium, Englewood Cliffs, N.J.: Prentice-Hall, Inc.


161
However, along the characteristic lines for the slow wave speeds
(c = cg) equation (B.4.3) is identically satisfied by using equa
tion (B.4.2) and no information can be obtained from equation (B.4.3)
along the characteristic lines c = c Along the characteristic lines
for the fast waves (c= c^) given by equation (B.4.1), the equations
(B.4.3) become
0 =
(pcp6-lJ [ J [t idvx+ -A- dax+ dt]
Pc.
+ aj^(p c)Ag-l
(B.4.4)
In this case when the waves are uncoupled, equation (B.2.23) yields
no information about the variables vA and T. and by inspection, neither
9 9x
does equation (B.2.24). Therefore, we will next investigate the equa
tion along the characteristics of equation (B.2.25). This equation for
A_ =A_= 0 reduces to
O
0 = [(pc2)^^ aA2) T
<¡[
-2
^Cdve+ dTex + 2p hxdt
*]
(B.4.5)
However, along the characteristic lines for the fast wave speeds
(c = c^.) of equation (B.4.1), equation (B.4.5) is identically satisfied
Along the characteristic lines for the slow wave speeds (c = c ) given
by equation (B.4.2), equation (B.4.5) becomes
0 =
^2' '"'ij r
f sdve + dV + 2fcs't'exdt
]
(B.4.6)
Thus, in the case of uncoupled waves the nonvertical characteristic
lines are given by equations (B.4.1) and (B.4.2) and the equations along
these lines are given by (B.4.4) and (B.4.6) which may be simplified.


65
If
2f
Is
V 2y(A4iA6i
1 V
2
1 V
2
(A A -
2/ 2i 6i
1 v
2
(A,. A,,. -
2/ 4i 6i
aA .) A .
5i 4i
A A ) A .
3i 5i 2i
aA ) A
5i 4i
R
2s
1 v
2/(A2iA6i
A A ) A .
3i 5i 2i
R^ = aA .A A.A.
fs 2i 5i 3i 4i
(3.4.1)
Thus, the finite difference equations when the waves are fully coupled
can be written directly from equations (3.2.5), (3.2.6) and (3.2.7),
and using the last subscript to represent the point in the numerical
grids of Figures 3.8 and 3.9 the equation
along c = 0 is
2a(1 v2) [ SP 2 9B] 2AT = a(Vrp V^)
(3.4.2)
along c = 0 is
-V + V
2a a SXB,
A4P + N + A5B^,
2 /(S9P S0B + V 2 ) P 'b}
r2aS S 2aSn S i
+ 1 SP g toP 22_22 *J (T,]
§{-
S /
P
B
(3.4.3)


9
Interest in the propagation of waves of combined stress continued
and Clifton (1966) presented the results of his study of combined longi
tudinal and torsional plastic wave propagation in a thin-walled tube.
Strain-rate independent material behavior and incremental strain
theory were used while radial inertia effects were ignored. The thin-
walled tube allowed Clifton to eliminate any dependence on the radial
coordinate so that a solution could be obtained in the characteristic
plane. (Earlier, Plass and Ripperger (1960) had used a rod and averaged
the variables, over the cross section in order to eliminate the dependence
on the radial coordinate.) The results of this investigation were
based on a simple wave solution which resulted from applying a step
velocity impact at the end of the tube. Clifton (1966) found that when
the tube was stressed into the plastic range, an impact at the end of
the tube caused waves with two different speeds to propagate. These
waves were called the fast wave and the slow wave, and each wave was
found to carry both longitudinal and torsional stresses. Two special
cases were examined. The first case involved statically prestressing
the tube above the yield stress in torsion and then applying a longi
tudinal velocity at the end. In this case the fast wave caused almost
neutral loading, that is, as the fast wave passed a point on the tube,
the shear stress decreased and the longitudinal stress increased in such
a way that the stress state at that point remained close to the initial
loading surface. Then as the slow wave passed the same point, loading
occurred so that the stress path was normal to the initial loading sur
face. The second case was for a tube with a static longitudinal plastic
prestress impacted by a torsional velocity at the end. In this case the
fast wave caused unloading along the longitudinal stress axis followed


Transverse Velocity, VQx10
Figure 4.6 Transverse Velocity Versus Time for Data Set 1 Without Radial Inertia
CD


Shear Stress, T x10
Figure 4.15 Stress Trajectories for Data Set 1 With Radial Inertia
104


61
iteration will be calculated using the solution obtained in the
previous iteration. In this way the coefficients are always known
quantities.
One other scheme will be used with the coefficients in the finite
difference equations in order to reduce the time required for compu
tation. Normally each coefficient used is the average value of that
coefficient at the end points of the interval over which the finite
difference equations are written. As an example, consider the charac
teristic line from point LLB to point P, and let one term in the finite
difference equation along this characteristic line be
U(S
xP
- S ).
xLLB
As a rule the value of the coefficient is calculated as
P
+ U.
LLB
) .
If this method is used, the coefficients of each variable in the equa
tions along the characteristics of positive slope will be different
from the coefficients of the corresponding variables in the equations
along the characteristics of negative slope. For instance, one term
in the equation along c = +C£^ can be represented as
VSXP SxLLB>
and the corresponding term in the equation along c= -Cf as
UR(SxP SxRRB)
L
R
+ w
+ URRB)
where


68
For Uncoupled Waves
When the waves become uncoupled as described in Section B.4, the
equations along the characteristic lines have a simpler form given by
equations (3.2.10) to (3.2.13). Using the averaging technique already
described in this section for the coefficients, the equations along
the characteristic lines for uncoupled waves simplify. The equation
along c = 0 is
(3.4.9)
along c = 0 is
+
(2aSeB-sxBH
xB toB
(3. 4.10)
along c = + c^ is
+ ilr i (2S -aS + 2S aS^
Tll2 xP 0P xLLB 0LLB
+ aA V +V + ilf 1
2iL rP lLLB yll
>} ^ ] *4i
dLLB^Jj Ll + c^J
(3.4.11)


62
and Up is the value of the coefficient at point P calculated from the
solution from the previous iteration.
When the coefficients are calculated in this manner, the number of
equations which must be solved simultaneously cannot be conveniently
reduced below five. However, if the coefficients are calculated in
such a way that the coefficient of any variable in the equation along
c = +cf^ is equal to the coefficient of that same variable in the
equation along c = -c-f. (so that U = U etc.), then by adding these
two equations and subtracting one from the other, two different equa
tions can be obtained, each with fewer variables than the two original
equations. If this procedure is applied to the equations along c = cSi>
then the set of five simultaneous equations can be reduced to at most
a set of three simultaneous equations and a set of two simultaneous
equations. This is shown in Appendix D. Since this set (or sets) of
equations must be solved during each iteration, the savings in compu
tation time is significant.
One way to make the coefficients of similar terms equal is to
calculate the coefficients from values of the variables obtained at
point P during the previous iteration such as
U = U = U.
L R P
The coefficients will be calculated in a somewhat more accurate manner
by using a weighted average of the value of each coefficient between
point P and point B, that is
UL = UE = 1DP + (1 al)UB


TABLE 2 NORMALIZED HOOP STRESS
Gamma
Delta
0
H-1
O
o
O
O
CO
o
o
O
O
o
! o
CT>
-0.57735
-0.56858
-0.54253
-0.50000
-0.44227
o
O
CO
1
-0.39493
-0.38893
-0.37111
-0.34202
-0.30253
-70
-0.20051
-0.19746
-0.18842
-0.17365
-0.15360
-60
0.00000
0.00000
0.00000
0.00000
0.00000
o
O
in
i
0.20051
0.19747
0.18842
0.17365
0.15360
o
O
1
0.39493
0.38893
0.37111
0.34202
0.30254
1
CO
o
o
0.57735
0.56858
0.54253
0.50000
0.44228
1
to
o
o
0.74223
0.73095
0.69747
0.64279
0.56858
-10
0.88455
0.87111
0.83121
0.76605
0.67761
0
1.00000
0.98481
0.93969
0.86603
0.76604
h-1
o
o
1.08506
1.06858
1.01963
0.93969
0.83121
20
1.13716
1.11988
1.06858
0.98481
0.87111
CO
o
o
1.15470
1.13716
1.08506
1.00000
0.88455
o
o
1.13716
1.11988
1.06858
0.98481
0.87111
50
1.08506
1.06858
1.01963
0.93969
0.83121
60
1.00000
0.98481
0.93969
0.86602
0.76604
70
0.88455
0.87111
0.83121
0.76604
0.67761
00
o
o
0.74223
0.73095
0.69746
0.64279
0.56858
CO
o
o
0.57735
0.56858
0.54253
0.50000
0.44228


64
The
interpolation constants
CONI =
C0N2 =
for
the points LLB, LBB, RBB,
T2 2(Cf c2)
T2 = (1 + Cf)(1 C2>
Cs(1 + V
and RRB are
C0N3 = 1 CONI
C0N4 = 1 C0N2
so that the value of F at
F
LLB
F
LBB
F
RRB
F
RBB
each of these points is
= CONIF, + C0N3F
L LB
= C0N2F + CON4F
LB B
= CONI*I + C0N3F_
R RB
= C0N2F + C0N4F .
RB B
For Fully Coupled Waves
When the equations are fully coupled, the equations along the
nonvertical characteristics (c = c^, c ) are given by either
equation (3.2.7) (3.2.8), or (3.2.9). Equation (3.2.7) will be used,
and the values of A., A, A., A_, and A of equation (3.2.2) will
1 2 o 4 5 o
th
be calculated for the i iteration as described for U earlier in this
section and defined as A, , A., A., A.. , and A. respectively.
li 2i 3i 4i 5i 6i
th
The coefficients will then be defined for the i iteration as


78
e e v v
xP xB XR xL
2AT 2AX
e. s v v
0xP 0xB 1 0R 0L
2AT ~ 2 2AX '
e ~ v + v
0P 0B rP rB
2AT ( 2
where the final subscript on each variable denotes the point in the
grid element where that variable is evaluated. Now, since the outer
grid lines defined in Section 3.3 have slopes of c = ~ = 1, AX and
AT are equal so that the expressions for the strains at point P are
£^=e + V V
xP xB xR xL
£0xP G9xB + 2 (V9R ~ ^'0L)
CSP = 66B + 2 For a boundary grid element, equations (3. 6,1),
(3.6.3) can be written in finite difference form as
e -e v (V + V )
xP xB xR 2 xP xB
2AT
AX
e0xP £0xB 1
2AT 2
^V0R 2 (V0P + V0B)
AX
-]
(3.6.4)
(3.6.5)
(3.6.6)
(3.6.2), and
g9P ~ e6B
2AT
and again since AX = AT, the strains at the boundary point P are given
by


Y 1 = 2 5 PH I P
Y2 = Yl* F1
Y 3 = PH IP *F 3
YA = 1.5 Y3
A 1P = 1.0 + Y 2 F1
A 2 P = { NU + Y 2 F 2 )
A3P = Y A F1
AA P = 1.0 + Y1*F2**2
A5 P = -Y A *F 2
A6P = Q1 + 9.*Y3*F3
355 CONTINUE
!F( PHIFJ.LT. L.0E-O5.OR.PHIP.LT.1.0E-05JG TO 356
1 F( I.CT.2 ) GO TO 360
356 CONTINUE
All
-
Al P
A 2 I
A2P
A3 I
=
A3 P
AA I
A'+P
A 5 I
A 5 P
A 6 I
-
A6P
A2C
-
A2P
m A (.
=
AtP
A 5 C
=
A5P
CO
TO
365
CONTINUE
All
=
.625*A1P

A IRC
A2 I
=
.625*A2P
+
A2BC
A 3 I
-
.625*A3P
+
A3 OC
AA I
=
625*AAP
+
AA RC
A 5 I
,625*A5P
-f
A5BC
A 6 I
=
.625*A6P
+
A6BC
A2C
=
. 5 ( A 2 P
+
A2B )
AA 0
-
. 5 ( A A P
+
AAB )
A5G
=
. 5 ( A 5 P
+
A5B )
213


CD = CS(CELTAlJ))
SD = S IN ( DEL T A(J ) )
CD2 = CD *2
SD2 = S D 2
CSC = CD* SD
S X( I ,J) = C *(C D SD/b)
ST( I,J ) = C *(CD + SD/b)
A1 ( I J ) = 1. + .25*Z*C2*(CD2 2.*E*CS0 + 3.C-*SD2)
A2(I,J) = -NU .25*Z*C2*(3.*S02 CD2 )
A 3 ( I J ) = .5G*E7*CS*(CC E*SD)
A 4 ( I J ) = 1. .2WC2MCD R 2.£*CSD + 3.0*SD2)
A5(I,J)=.50*E*Z*CS(CD+E*SD)
A6 ( I J ) = 2.0 ( 1 * iMU) + 3 0 Z S 2
ABAR(I,J)=A1(I,J)*A4(I,J)*A6(I,J)-A3(I,J)*A3U,J)*A4 I(I,J)*A3( I,J)* A 5( I,J )-Al( I,J )*A5(I,J)*A5(1,J)-A2(I ,J)*A2( I ,J)*A6(I
2, J ) )
RBAR(I,J >-A4l I,J)*(A6(I,J)+Al(I,J))-A*{A2(I J)*A2(I ,J)+A5(I,J)*A5(
1 I J ) )
WAD2 = BBAR( I,J ) *BBARlI,J ) 4.* ABAR( I,J ) *A4(I,J )
IE(RAB2.GT.0.'-')G0 TU 13
RAD 3 = -RAD2
RAD4 = 110.*SQRT(RAD3)
I F(ORAR( I ,J) .LT.RAD4JG TO 13
RAD { I J ) = .i.O
GO TC 14
13 CONTINUE
RAD( I J ) = SQRT(RAD?)
14 CONTINUE
CFC2 ( I J ) =SORT( ( ( 1. 4-UU ) /ABAR( I J ) ) ( BBAR ( I J ) 4-RAD { I J) ) )
CSC2(I J)= S Q R T( ((l.+NU)/ABAR(I,J) )*(BBAR( I,J)-RAD(I J)) )
DELL= DELL+ DDEL
J = J + 1
IFIJ-20) 3,4,4
4 CONTINUE
166


Change in Shear Strain, y
Figure 4.10 Change in Shear Strain Versus Time for Data Set 1 Without Radial Inertia
CD
<1


60
/
f n rn
1 > a
Figure 3.9 Numerical Representation of the Characteristic Lines
in a Boundary Grid Element


47
(2S aS )
A1 = EA1 = 1 + S(s,A)
A = EA
2 2
-[
4s
(2S aS.)(S 2aS)
x G x 9
v + f(s
4s^
6t(2S aS )
A3 EA3 = $(s,A)
4s
(S ^ 2aS )
A4 = EA4 = - X 2 ?(s,A) + 1
4s
6T(S 2aS )
fl5 = s = 2 ~ 4 4s
- 9t
Ag = EAg = 2(1 + v) + $(s,A).
s
,&)]
y
(3.2.2)
Using these and equations (3.1.19), the fast and slow wave speeds can
be written from equations (2.49) and (2.50) as
Cf =
! 2a P
c =
s
C1 -2a p
<*'2 '*>*}
or
^ V
(b'2 4,,'a/1
- [7^ {' <*'2 '*>*} ]
\
(3.2.3)
J
and the wave speeds in dimensionless form are given by equations
(3.2.3) and (2.47) and (2.48) as


24
b =
a cr.
0
-
x
av
r 9
- 2i|r
6x
(2.42)
and if the slope of the characteristic line (or the wave speed ) is
denoted by c where
H v
(2.43)
c =
dt
then the equation for the characteristic lines is given by
c A B = 0
(2.44)
and from the calculations shown in Appendix B, equation (2.44) yields
|^a2p c^J j^(p c2) [a] (pc2) {b} + J = 0 (2.45)
where
s + SciA^A^Aj sA^A^.
^ A1A4 aA2 + A4A6 ^5
aAoAc A0A
2 o o 4
>>
V
Setting the first factor in (2.45) equal to zero,
dx
ac = a = 0 (twice)
dt
and, setting the second factor in (2.45) equal to zero,
i
c =
"2ap
f- -2 A")
jb (b 4aA4)e
IT .
(2.46)
(2.47)
(2.48)


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
MS A. Eisenberg, Chairman
^Associate Professor of Engineering Science,
Mechanics and Aerospace Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
L. E. Malvern
Professor of Engineering Science, Mechanics
and Aerospace Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
E. K. Walsh
Associate Professor of Engineering Science,
Mechanics and Aerospace Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
T4 Pfnv^w^cr
U. H. Kurzweg
Associate Professor of Engineering Science,
Mechanics and Aerospace Engineering


28
The equations along the characteristics of equation (2.47) may be
obtained (since along these characteristics there is no variation in x)
directly from equations (2.25) and (2.35) by multiplying each of these
equations by the increment of time, dt. These equations then yield
a8
- dt = a p dv (2.54)
r r
o
av
dt = aA^do + aA.don + aA^dT^ + ailr.dt (2.55)
r 2x 40 5 9xT0
o
The equations for the six characteristics and the equations along
these characteristics, along with the appropriate initial conditions
and boundary conditions, represent a complete mathematical formulation
of the problem, and the solution to these equations is the solution to
the problem posed here. The solution to these equations will be
obtained by using a finite difference numerical technique which will
be discussed in the next chapter.


67
along c = + c is
s
~ (AxP \xLBB) ~ ~2 0P 8LBb'^
s 1 v
(1 v2)R,
+ Rfs(TP TLBB > +
(s s )
xP xLBB
+ R. t, ^(2S ~ aS + 2S aS ) ,
lsYl [_2 xP 0P xLBB 9LBB J |_1 + c
>JM
+ aR2s [VrP + VrLBB + *l{s(S*P 2aSGP + SxLBB 2W) J
[rrT"J
6c2R \|r
s f s 1
s 1 v
2 L2(tp + tlbb}J Ll + c_J
][l
2AT "1
(3.4.6)
along c = c is
s
Ri c R*
0 = -ii(V -V ) + -S-f (V V )
c xP xRBB 2 GP 0RBb'
s 1 v
(1 v^)R
+ Rfs(Tp trbb) +
Is
(SxP SxRBB)
!is+i B
+ R, V. i (2S aS + 2S aS )
n xP 0P xRBB 0RBB
+ aR V +V + \I,t:(S 2aS + S 2aS )
2s L rP rRBB Yll2 xP 0P xRBB 8RBB
/I,
J
r bat i 6%RfS*i ri.. T ,ir 2at i
Ll + cJ+ 2 b P+ RBbJLi+cJ
S 1 v s
(3.4.7)
where the values of i|r / and s^/ obtained at point P from the previous
iteration are vised and
;1 1
*oP' N *oB
'I'-, = + (1 at)
V B
(3.4.8)
with a^ defined earlier in this section.


153
^ 3 5 r ______ o __ i) __ r)
_M^ ) = -a p x,_ p- |^A^A,A+2aAAA^-AA-aAH -aA.
46 235 34 15 26
]
4 2 2
3
dVQ
P ^ > p
Xr
d£
4 2
4
da
X
a p t,.
X£
d§
4 2 2
a p t,g
4
x, _
s
[-*:
4 2 2
3
dv
X
a p
XF
a
d§
4 ^ 3
2
da
X
p i
X§
d§
aA2A5_A3A4
J
r.
de I AlA4A6+2ail2A3A5_A3A4_aA2A6 &A
3
)(A2A6-A3A5)-2^0x(A2A5a-A3A4)
]
4 3 2 dT 0x
1 pt§ x§ dT L^Wa
J
5V^]+ a4pt,§ x,§ [^(r^eX] = -
Along the characteristic lines of equation (2.47), x,_ = 0 and this
equation is identically satisfied. Along the characteristic lines of
equation (2.48), t,F is never equal to zero since this would correspond
5
to an infinite wave speed. Therefore, dividing by t,P and noting that
dx/d£ dx
t, ~~ dt/d§ ~ dt
S
= c
(B.2.22)
the above equation becomes
dv
0 = aV2
iV^]
dv
d
]+ if hA-vJ
pc(aA2A5-A3A4) j + I aAA_-AA,
da
x
"dT
[p c2 (^46+2a235-2X4-a26-a142) (A^-^]
4/ig-drt3^ -4j c -r a \ ~ y g jj3 c ^2A6_A3A5^'
+ 2iojp£2(a52s5-y4)]t,5} .


58
3.4 Finite Difference Equations
General Discussion
While the actual characteristic lines for the fast and slow waves
are seldom straight, they can be represented as straight lines within
each grid element without introducing significant errors if the grid
elements are small. From the discussion in Section 3.3, it is known
that the slope of the. characteristic lines at any point cannot be
determined before the solution at that point is known. Because of
this the solution at the point P (Figures 3.5, 3.6, and 3.7) must be
obtained by an iterative technique. Within any grid element, the slope
of each characteristic line will be constant during each iteration
although the slope of each characteristic line will change from one
iteration to the next as the solution at P is approached. These
straight lines are used to represent the characteristic lines for
th
c = c^ and c = c^ during each iteration and are shown for the i iter
ation as c = c-f. and c = cs. in Figure 3.8 for a regular grid element
and in Figure 3.9 for a boundary grid element. The points LLB, LBB,
RBB, and RRB are the intersections of the lines shown in Figure 3.8.
/ /
Each element has its own coordinate system X T which is also shown
in Figures 3.8 and 3.9, and the finite difference equations are written
in terms of this local coordinate system so that the finite difference
equations for each element are the same.
First order finite difference equations will be written along each
characteristic line. The coefficients of the dependent variables in
these equations will in general be functions of the stresses and §(s,A).
Thus, in order to linearize the equations, the coefficients for each


CHAPTER 1
INTRODUCTION
Stress wave propagation is the mechanism by which forces and
displacements are transmitted from one part of a structure to another.
Stress waves arise when a transient force is applied to a structure,
and they propagate through the structure reflecting (at least partially)
back into the structure whenever they encounter a boundary. After
several reflections the amplitude of the stress waves diminishes and
the structure reaches a state of equilibrium. In many engineering
problems the time required to reach equilibrium is very short, and for
practical purposes the structure can be assumed to reach equilibrium
instantly. Problems in which the forces are applied slowly or in which
the state of stress is required a long time after the forces are applied
are examples of instances when wave propagation effects may be neglected.
However, in many cases, the forces are applied rapidly (such as during
impact loading or explosive loading), and failure is most likely to
occur in the structure almost immediately after the application of these
forces. In these cases when it is necessary to determine the state of
stress during and immediately following the loading, wave propagation
effects may be significant and should be included in the analysis of
the problem. In order to understand the development of the theory of
stress wave propagation and the application of this theory to modern
engineering problems, it is instructive to review briefly the history
of wave propagation research.
1


Longitudinal Strain,
Figure 4.11 Longitudinal Strain Versus X for Data Set 1


40
TABLE 3 NORMALIZED
SHEAR STRESS
(v)
S'
Gamma
-F Tx
Value of
s
Gamma
T
.. 9x
Value of
s
0
o
o
CJl
o
o
0.44228
O
O
tH
0.10026
60
0.50000
20
0.19747
70
0.54253
GO
o
0
0.28868
o
O
CO
0.56858
0
o
0.37111
90
0.57735
c =
p (1 av )
c
s
2p(l + v) C2
It is now obvious that the fast wave speed is the same when a=0 and
v^O as when a = 1 and v = 0. The slow wave speed (and consequently c^)
is the same when a=l as when a=0, although it does depend on v.
Because of this dependence of c^ on v, the dimensionless fast wave
speed of equation (3.1.20) will have values when a = 0 and v^O differ
ent from those when a =1 and v = 0.
Also in Figure 3.2 it can be seen that when a=0 and y = 0 the
fast and slow wave speeds are the same for 3= .385 and v= .30. There
is usually some value of |3 for which the two wave speeds are equal
at y = 0 for each combination of values of v and 6. The condition for
which this is true can be obtained from equation (3.1.20) and is


Longitudinal Strain,
Figure 4.4 Longitudinal Strain Versus Time at X = 3.75 for Data Set 1
oo
00


195
Once the initial conditions are known, the solution can be found
at each grid point along the line
T = X + 2AT
beginning with the point (0,2AT). Once this solution is known, the
solution can be found at the point (AX,3AT) and then at (2AX,4AT),
(3AX, 5AT) ,...,( (a^-l)AX, (aVl)AT) The solution is then found along
the line
T = X + 4AT
in the same manner, and then along succeeding lines until the solution
is found at the point (0,2a^AT).
At each grid point (except along the line X = T) the solution
must be obtained by an iterative technique using the equations from
Chapter 3 (or Appendix D). Since some of the coefficients in these
equations are determined from the values of some of the functions cal
culated at P for the previous iteration, these coefficients must be
specified for the first iteration. This is done in every case by assum
ing that the waves are elastic, that is, by assuming on the first iter
ation (i=1) that
$(s,A) = (s A) = 0
and
Ali = 1
A.. = 1
4i
A2i -
A = 0
3i
A = 0
5i
A = 2(1+ v) .
6i
Using these values, the solution can be found at P. The values
variables obtained at P in this manner are then used to compute
of the
the
values of the coefficients for the second iteration to obtain a second


Hoop Stress, SQxlO
105
Figure 4.16 Hoop Stress Versus Longitudinal Stress
for Data Set 1 With Radial Inertia


123
expected to affect significantly the theoretical results. The final
test of any theory (along with its assumptions) is how well it pre
dicts the real behavior of materials. This can only be determined by
comparison of the theoretical results with experimental results.
But experimentation in the field of dynamic plasticity under combined
stresses has only recently begun. Much work remains to be done before
enough experimental data have been gathered so that investigators can
determine a unified theory of inelastic wave propagation under combined
stress. At least a beginning has been made.


320 CONTINUE
I = 1
A1P = 1.0
A2 P = -NU
A 3 P = 0.0
A 4 P = 1.0
A5P = 0.0
A6P = Q1
PHIP = 0.0
CO TC 355
34t CONTINUO
1 = 1 + 1
I F { I.LE.MDGO TG 344
IF(JERROR.GT.0)GO TO 341
JERRCR = JERROR + 1
WRIT E(6 1033)
341 WRIT E(6110 3 2) X(MX ) ,T(MX ),DIFF,DENOM,ERROR
GU TC 430
344 CONTINUE
CALL PHI(PHIP,PH IQ,JRATE,JQuLL, 50PC,DEL TAP, SY,BtfAR,N, H XLAM, 3ETA, D
1Y,XM,XN,DZ,10)
I E ( S BPC. GT. SMAX3.UR I .GE .MJ )GU TO 351
PHIP = 0.0
351 CONTINUE
IE(SBPC.GT.1.CE-08)GO TO 353
FI = 0.0
F 2 = 0
F 3 = 0.0
GO TC 354
353 CONTINUE
FI = (2.*SX I A S T I J/SCPC
F 2 = (SXI I 2A*ST I ) /SBPC
F 3 = TAU1/S3PC
354 CONTINUE
212


02 S
=
2.* B2 S
06 S
=
2. B6S
DIF
=
2.*B1F
C2F
-
2 B2 F
C6F
=
2. B6F
C3S
2.*D3S2
C 3 F
=
2.D3F2
04 S
=
2. 04 S 2
C4F
=
2 C4 F 2
DBS
=
-2.BBS
DBF
RHSA
=
-2 BBF
= -BIF*VXLLB B2F*VTLLB + B3F TAULLB
1 A*(B5F*STLLB B6F*VRLL3)
RHSC = -B1S*VXLBB B2S VTL BB B3S*TAULBB
L A*(B5S*STLBB B6S*VRLB6)
OFL1 = O 1F*D2S D15*D2F
RHSBA = RHSB RHSA
RHi.CC = RHS 0 RHSC
VXI = (D 2 S R H S B A 02F*RHSDC)/DEL 1
VTI = (01 F*RFSDC D 1S*RHSBA)/DEL 1
C7F = DBF Q3*C6F
C7S = DBS Q3C6S
RHSF = RHSA + RFSB A D 3 D 6 F
RHSC = RHSC + RFSD A*D3*D6S
GO TO 3B 8
64F*SXLLB +
+ 34 S* SXLBB +
386 CONTINUE
RHS C E = -Z2*VTLSB + F3S*TAULBB
RHSAE = -F1F*VXLLB + F6F*SXLLB + A*(-F3F* STL LB + F4F*VRLLB)
FHSAEM = RHSAE + A*C3*F4F
TAUI = (RHSCE + RHSDE )/(2.*F2S)
VTI = (RHSDE RHSCF )/(2.*Z2)
VXI = (KHSBEM RHS A EM ) /(2.*FIF)
F2F2 = 2.*F2F
FBF2 = 2.F5F
221


Change in Shear Strain, Ay
Figure 4.5 Change in Shear Strain Versus Time at X = 3.75 for Data Set 1
oo
CD


The first serious attempt (at least in this century) to understand
nonlinear wave propagation in solids was made by Donnell (1930). In
this paper, Donnell used energy principles and impulse-momentum expres
sions to find the particle velocity and the elastic wave speed for
longitudinal waves. He also predicted that if a material with a bilin
ear stress-strain curve were impacted at the end by a stress above the
yield stress, two stress waves would propagate with distinct velocities
However, after the publication of this paper interest in wave propaga
tion subsided until the early 1940's.
A more general theory of longitudinal stress wave propagation was
developed independently by Taylor (1940) von Karman (1942) and
Rakhmatulin (1945) by assuming that the material exhibited a nonlinear
stress-strain curve above the yield point. This stress-strain curve
was assumed to be independent of the rate of straining. Using this
theory the velocity of propagation of the longitudinal waves was found
to be given by
'1 da
p de
c
where c is the wave speed, p is the density of the material, a is the
stress, and e is the strain. Thus represents the slope of the stress
de
strain curve or the tangent modulus. This theory also considered the
stress-strain curve of the material which was obtained for the static
case to be valid in the dynamic case. With this assumption, the stress
and strain followed a unique functional relationship as long as no
unloading occurred. Because of this, the tangent modulus could be
written as a function of the stress (or strain) only, so that the
velocity of propagation then became a function of the level of stress


o o r> o n n r- o o r¡ o r. r< n o o
DIMENSION SX ( 601 ) ST( 601 ) ,TAU( 6.' 1 ) VX( 61 ) VTI601 ) VR (601 ) ,trX(60l )
1 F;T ( CU ) FTX ( 6'U ) 0KLTA ( 601 ) HPLAS( 6'Jl ) ,CF ( oOl ) ,CS ( 6M ) t SB (6-n > ,
2E X P(6? 1 ) ,FTP(6 3 1 ) ,uTXP(601 ), PSI' (601),PHIU(601) ,1 TeK(601) ,Al (601 ) f
3 A2 ( 60 1 ) f A 3 I 6C l ) A4( 60 1 ) A5( 60 1 ) A6 ( 601 ) X ( 601 ) T ( 60 1) SMAX(6rl )
DIMENSION DELLHB(120),DELGAM(6'>1),DEX(601)
OI MANSION STRES(601)
DIMENSION SiP( 5? ),STRAIN( 50),VXP( 50) VTP( 50
PEAL N,riU
INTEGER A,UNCOUP
C
C UNCOUP = 1 WHEN THE TRANSVERSE ANO THE LONGITUDINAL WAVES.ARE UNCOUPLED
UNCOUP = O WHEN THE TRANSVERSE AND THE LONGITUDINAL WAVES ARE CUUPLED
A O POR NO RADIAL INERTIA EFFECTS
A = 1 FOR RADIAL INERTIA EFFECTS
JRATE = 0 FOP RATE INUEPENOENT CASE
JRATE = 1 FOR RATE DEPENDENT CASE
JDLL = C FOR CLIFTONS DATA
JBELL = 1 FOR BELLS DATA
IN NCNDIMENS 10NALIZINC THE PARAMETERS THE VALUES OF YOUNGS
MODULUS WHICH WERE USED WERE -
>z 1", 200, roo PSI FOR BELLS DATA
C = lO,v0O,''O0 PSI FUR CLIFTONS DATA
kL Afi (5,1C02) UNCOUP
Rf A0(5 1 Or1) A,JR AT 6,MXM AX,NU,H 1,H2,M3,U4,DEL X,PELT,N XKO
R t A D ( 5,1 0 C 2 ) JR ISE,XVFIN,TVFIN ,SXO,TAU j *SMALLBBARSY
REAL' ( 5,10f 2 ) KASE,SYS
PE All (5,10 02) JBELL,DZ,BETA, CHAT,XM,XN,H,XLAM
KEAD(5,lCn3) INCRX, I NCRT, I 0,MI,MJ,I PUNCH,I PUNI,IPUN2,INCRXP,INCRTP
W R I T E ( 6 ,1005) A,JRATE,DELX,CELT,MXMAX,SMALL
205


12 CONTINUE
WRITE(6109)
WRITE(6,110)(GAM( I), 1 = 1, I I)
WRITEI6, 111 ) (TAUl I ), 1 = 1, II )
101 F0RMAT(6F1G.4,15)
10? FORMAT('I',5X,'ALPHA = I 1 /6X 'BETA = ,F6.4/6X,NU = ',Fb.4)
103 FORMAT( / / / 5 O X ,FAST PLASTIC WAVE SPEEDS, CF/C2'/)
104 FORMAT(2X,'DELTA*,3X,'GAMMA = 10(F4.1,7X)/)
105 FORMAT(F8.24Xf10FI 1.5)
106 F0KMATI///5JX,SLOW PLASTIC WAVE SPEEDS, CS/C2'/)
107 FORMAT (' 1',50XNORMAL IZED AXIAL STRESS'/)
108 FORMAT (///5-jX,'NORMALIZED HOOP STRESS'/)
109 FORMAT(///50X,NORMALIZED SHEAR STRESS'/)
llu FORMAT(10X, 'GAMMA =,10 IF4.1,7X )/)
111 FORMAT(12X,10 FI 1.5)
STOP
END
168


196
solution at P. This is done until two successive solutions at P differ
by some value which is specified in the input. Once this convergence
is reached, this iterative technique is begun at the next grid point.
E.2. Initial Conditions
Initially the tube is assumed to be at rest, loaded by a state of
constant stress (and, therefore, constant strain). Designating the
initial state by a superscript o, the initial conditions are given in
terms of the dimensionless variables as
V = V = V = 0
x 0 r
and since the body is at rest
i|f = 0
o
and from equation (2.8),
s.o
It is assumed that initially the tube is in a state of static
prestress (T = T, S = S) and that this stress state is reached with-
x x
out any unloading so that the strains can be uniquely determined.
Furthermore, to simplify matters, it will be assumed that this initial
state of stress (and strain) is reached by proportional loading so that
if the initial stress state is given by a?, (or s. for the deviatoric
ij iJ
stress), then the stress state at any time during the static preloading
is given by
a. = C(\)o. .
iJ iJ
s = C(X)s.
ij ij
\
J
(E.2.1)


APPENDIX C
PROGRAMS FOR DETERMINING THE PLASTIC WAVE SPEEDS
In this appendix the listings are given for the two computer
programs used in the first section of Chapter 3. The first program
listed is the one which calculates the plastic wave speeds as functions
of v, 3, y, and 6 for the case when radial inertia is included or for
the case when radial inertia effects are not included. It is also used
to calculate the values of the normalized stresses as functions of y
and 6. The second program listed is the one which calculates the crit
ical value of 8 (as a function of v) at y = 0 for which c =c =c .
I s z
164


158
Using equation (B.3.3) in equations (2.46)
a = A.
rv1 ax
3Tt
Sx 2 2 2 \
(22 2\1
i
6T
at
+
S!
a
+
- H + a T H -
E x 0x /
(Ve* H )J
a = A.
r~
LEG
a2 3tB
x 0x
+ 3GH + ~E
= ]
- 1 x 1 2
b A4 |^ + -g-H + + 3T H
]
G 0x
and the equation (2.45) for the characteristics becomes after dividing
by A^ (which is always greater than zero)
(P^2)2 ih + 3^H+ ^H) (PS2)(i+^H+ ^+3T0xH) + 1 = O (B-3-4)
which is the characteristic equation given by Clifton (1966).
The equations along the characteristics can be reduced to the two
forms given by Clifton (1966). When a = 0, and ijj = |Q=fQ =0,
x 0 0x
equation (B.2.23) becomes
0=- \ [(pc2)X4X6-4]dvx+pc[3jdve-|]34]dTex+ 4(pc2)(/Ic)-jda,
4 6 4J x
or dividing by A
0 = [(p52)6-i][- i dvx + A_ dCTJ + a3 [P;cive-dT6x]
c pc
and using equation (B.3.3), this equation along the characteristic
becomes
[(I
+ 3T H
P c
J + [V0XH][P
ex-/-~r2j[dc7x~DcdvT H npdve-dT
6 9x
J (B.3.
5)
0


185
D.5. Solution at a Boundary Point (X=0)
for Fully Coupled Waves
At a boundary point, the solution when the waves are fully coupled
is obtained by specifying two of the variables at the point P and then
using equations (D.1.6), (D.1.19), and (D.1.21). Once again the
radial velocity is calculated using equation (D.1.3) after SQri has been
0P
determined for all cases considered. The solution will now be devel
oped for each of the four sets of boundary conditions discussed in
Chapter 3.
Case I: Traction boundary conditions
When S and T are specified, then the hoop stress can be deter-
xP P
mined directly from equation (D.1.6) and is
sep = 5-dfflSH - a5(jV .
(D.5.1)
Once this is known, two new quantities can be defined as
RHS1 SB1 D3f2Tp D4f2Sxp + aB7fSep
RHS2 = RHSDl D T D. S + aB S_
3s2 P 4s2 xP 7s 0P
(D. 5.2)
so that equations (D.1.19) and (D.1.21) can be written in matrix form as
Blf
B2f
VxP
RHS1
_ Bls
B2s
1
<
CD
'P
l
RHS2
(D.5.2)
With
*4 = (D.5.4)


84
for the small grid size the strain follows closely the strain obtained
by Lipkin and Clifton (1970) for a simple wave with an instantaneously
applied velocity at the boundary. For the larger grid sizes the strain-
versus-time curve is smoother and follows more closely the experimental
results of Lipkin and Clifton (1970). Apparently, the larger grid
sizes tend to smooth out the data and eliminate the distinction between
the fast and slow wave speeds. For instance, in Figure 4.1, the' simple
wave solution of Lipkin and Clifton (1970) exhibits a region where the
longitudinal strain has the constant value of 0.00085. The strain
remains at this constant value from just after the fast wave passes
until the arrival of the slow wave.
From these computer runs other quantities of interest can also be
plotted and the same grid size effect can be observed. This is shown
in Figure 4.2 for the longitudinal velocity versus time. The grid size
has a much smaller effect on the stress trajectory than on the time
history curves. The stress trajectory is shown in Figure 4.3.
Because the details of the solution depend on the size of the
numerical grid, all subsequent computer runs will be made using a small
grid. This small grid size necessitates a large amount of computer
time to obtain a solution more than 1.0 diameter from the impact end,
and most of the results given below are obtained near the end of the
tube.
4.3 Effects of Radial Inertia
In order to determine the effects of radial inertia, four separate
computer runs were made using the computer code in Section E.5. The
generalization of the uniaxial stress-strain curve of Lipkin and Clifton


76
Case II:
V0P = h,
S.0 = ~ (RHSEEM D S ) (3.5.34)
P D 1 xP
Cl
VXP = Flf <3'5'35>
Kinematic boundary conditions
When V^p and Vgp are prescribed at a boundary point, then the
solution at that boundary point is given by equations (D.6.4), (D.6.9),
(D.6.10) ,
and- (D.6.11). When radial inertia is included the solution is
S = (D RHS12 F RHSEEM) (3.5.36)
XP A12 3 5
= - (F RHSEEM D RHS12) (3.5.37)
0P A12 2f 1
and when no radial inertia effects are included the solution becomes
The shear
stress in both cases is
Tp = Y~ (RHSDE Z2 V0p) (3.5.39)
Case III:
Mixed boundary conditions
When S^p and Vgp are known at a boundary point, then from
equations
(D.6.12), (D.6.13), and (D.6.14), the solution at that point
is
Tp = (RHSDE Z2V p) (3.5.40)
2s
S0P = (RHSEEM DlSxP) (3.5.41)
2
VxP = Flf

55
T
Figure 3.6 Boundary Element in Numerical Grid


66
along c
0 :
along c
0
+ c is
If
(XxP VxLLB)
c _R
f fs
(V V )
2 0P 0LLB;
(1 v^)R
R_ (T_ T ) +
fs P LLB
If
(SxP SxLLB)
+ R-.-p'I'-, ^(2S aS+ 2S aS ) 1 ~j
If Y1 L2 xP @P xLLB 0LLB J Ll + cj
+ aR J V + V + ilr -¡^-(S 2aS+S 2aS )
2fL rP rLLB *1 12 xP eP xLLB 0LLB
}]
It7
2AT
+ c
-1 6efRfs*! fl ,
+ 2 I2 'p + 'LLB'
f-1
v
2AT "j
_1 + c J
(3.4.4)
c is
R-. CR
-1-f. (V V ) + --- (V V )
C xP xRRB 2 0P 0RRB
f 1 v
(1 v2)R
+ Rfs(Tp lRRB) +
If
2
(SxP SxRRB^
:*i B
+ R, .iL I (2S aS. + 2S aS )
lfyl L2 xP 0P xRRB 0RRB
][r¥y
+ aH2f[VrP + 'rERB + ltf }]
f 2AT 1 6cfRfs*l ri 1 r_2iTl
Ll + v2 L2 P RRB J Ll + c^J
(3.4.5)


39
TABLE 2 (Continued)
Delta
Gamma
o
O
60
70
00
o
o
90
1
CD
O
o
-0.37111
-0.28867
-0.19747
-0.10026
0.00000
1
00
o
o
-0.25386
-0.19746
-0.13507
-0.06858
0.00000
I
<1
o
o
-0.12889
-0.10026
-0.06858
-0.03482
0.00000
-60
0.00000
0.00000
0.00000
0.00000
0.00000
1
cn
O
o
0.12889
0.10026
0.06858
0.03482
0.00000
o
O
1
0.25386
0.19747
0.13507
0.06858
0.00000
o
O
CO
1
0.37111
0.28868
0.19747
0.10026
0.00000
-20
0.47710
0.37111
0.25386
0.12889
0.00000
o
O
H
1
0.56858
0.44228
0.30253
0.15360
0.00000
0
0.64279
0.50000
0.34202
0.17365
0.00000
O
o
0.69747
0.54253
0.37111
0.18842
0.00000
to
o
o
0.73095
0.56858
0.38893
0.19747
0.00000
CO
o
o
0.74223
0.57735
0.39493
0.20051
0.00000
o
o
0.73095
0.56858
0.38893
0.19747
0.00000
50
0.69747
0.54253
0.37111
0.18842
0.00000
o
o
CD
0.64279
0.50000
0.34202
0.17365
0.00000
o
O
0.56858
0.44228
0.30253
0.15360
0.00000
00
o
o
0.47709
0.37111
0.25386
0.12889
0.00000
90
0.37111
0.28867
0.19747
0.10026
0.00000


34
The elastic wave speeds are defined from equations (B.5.4)
; = /r ^
C1 =
p (1 v )
j
"2 Jo J
and the wave speeds from equation (2.48) can be written in
less form as
c'2 = ^ ^ {- [B ^~ 4V J )/
(G/p)
,2-yy [5 44;]
or
/2 1+vf Jr /*7T2 2 r 3 -
: = -33- I E b V (E
E a
b) 4(EA4)(E a)
J
By defining the dimensionless functions from equation (3.1.
A. = E A.,
1 1
i = 1,2,...,6
and
3 o 2 2
a' E a = A, A.A + 2aA AA,_ A'A. aA A aA A
146 235 34 15 26
b' = E)2£ = AA aA2 + A4A6 aA2 ,
then the wave speeds in dimensionless form become
/2 1 + v
7" _b' J b'2 ~ 4A4a' ] = (r
" \2
c \
as
(3.1.16)
dimension-
(3.1.17)
15) as
(3.1.18)
(3.1.19)
(3.1.20)
A computer program was written to solve this equation for the two
positive wave speeds as functions of the angles y and 6 for specified


Change in Shear Strain
100


73
Case I: Traction boundary conditions
When S^_p amd are known, then from equations (D.5.1)
and (D.5.6) the solution to the finite difference equations
Sep = (RHSH DlSxp A5QTp)
vxp = r4 (b2s RHS1 B2fRHS2)
vep = r (D.5.5),
at P is
(3.5.15)
(3.5.16)
(3.5.17)
and is given by (3.5.6).
Case II: Kinematic boundary conditions
When V^p and Vgp are given, the solution at P is given by
equations (D.5.11), (D.5.12), and (D.5.13) when radial inertia effects
are included as
= [^RHS4(D/1_ + B_ D.,) RHS5 (D^nD^+B^^D, )
P A,
4s2 2 7s V
4f2 2 7f 1J
A RHSH(D4s2B7f D4f2B?s)
]
(3.5.18)
i r
I R
'xP A L v3s22 1 *5Q7s' '3f2"2 ~7f"5Q'
S = I RHS4(D0_0D0 + A_DJ + RHS5(D0J,D0 + B^^A^)
A6
RHSH(D3s2B?f D3f2B'
*>]
(3.5.19)
^ = \ RHS4(D0 A^D,, ) RHS5 (DOJCOD., A^D^0)
0P A_ L
D
3s2 1 5Q 4s2
3f2 1 5Q 4f2
RI,SH (3. 5.20)
and V is given by equation (3.5.6). When radial inertia effects are
rP
not included, V and SD are zero and the solution given by equations
rP oP
(D.5.16) and (D.5.17) is


143
so it is apparent that the simultaneous equations (B.2.3) obtained from
equation (B.2.2) are not independent. Therefore, one of the unknowns
can be chosen arbitrarily, and the others can be solved in terms of it.
From equation (B.2.3) ,
12 = 0 (B.2.4)
and arbitrarily selecting
14 = 1 (B.2.5)
equation (B.2.3)^ requires that
1x= ~ ~ (B.2.6)
P c
Using equations (B.2.5) and (B.2.3) the last two of equations
(B.2.3) reduce to
which yield
acA,l + acA 1 = -acA
4 5 5 6 2
acA_l +
5 5
ft'S
cp
*6 = CA3
LA6 52]
A3A5 A2 l_ o c*
5 Z~TZ 1-I -2
A4LA6-J-^S
pc
(B.2.7)
and
X6 =
aA A A A.
D Z O 4
, aA2
4 e -21 5
pc
and equation (B.2.3) becomes

(B.2.8)
X3 =
pc
aA2A5 ~ A3A4
-A4 LA6 J ^5
pc
(B.2.9)


A dw
pdv
ap dv
pdv
Aldx+aA2da8+VT0x
| aA da + aA^da.+ aAdT.
j 2 x 4 9 5 0x
LA3dax+aA5dOe + VT0x
and using this and equation (2.42), the equations along the character
istics given by equation (B.2.1) become
[hVshVs]
p dv
ap dv
pdv
Aid,J*+aA2doe + A3dTe*
aA2dax+ aA4da9 + aA5dT0x
A3dax+aA5% + VTex
L
HVshVeJ
0
acr.
a-
2i|f
0x
or,


365 CONTINUE
366
367
RHSc = DLLTM2. *VRB-PSIR0*( 12A*5TB-SXB) ) + A2U SXB + A4Q* S TB + A50 TAU6
RHSH = RHSE 04
A 2 I 2 = A 2 I * 2
A512 = A 5 I *2
IF(UNCOUP.GT.0) GO TO 366
A3 = A4I(A1I*A6I-A3I**2)+A*(2.*A2I*A3I*A5I-A1I*A5I2-A6*A213)
BR = A4IMA11 + A61 } A*(A2I2 + A 51 2)
C02F = Q2/A3
ROOT = 03**2 4.*A4l*AB
RAC = SURTI ROOT)
CFI = SURTI COFFM BB + R A D ) )
CSI = SURTICUEFMBB RAO))
GO TC 367
CONTINUE
CFI = SURTIQ*A4I/I AlIA4I A A 2I2) )
CSI = SURTIQ/A6I)
CUNTINUc
CV 3 = 1. + CFI
CV4 = I. + CSI
CUN I = 2 IC FI C2 ) /IDV2*0V3)
C0N2 = 0V1*CSI/I C2*0V4)
CON 3 = L. CON 1
CON4 = 1. CON 2
TAURRO = CON I T AUR + C0N3*TAUR0
SXRRB = C ON I SX R + CN3*SXR0
STKRB = CON I*ST R + C0N3STRB
VXkRC = C ON1 V X R + CN3*VXR0
VTRRB = CON 1*V T R + C0N3*VTRB
VRRRB = CONI*VRR + C0N3*VRKB
TAURCB = C0N2*TAURB + C0N4*TAUB
SXRBB = C0N2*SXRB + C0N4*SXB
STREG = C0N2*STR8 + C0N4*STB
VXRBB = C CN2 *V X R B + C0N4*VXB
214


C fc L T A P = CELTAP + .5MPHIP + PHIO)*SBDIF
371 CUM I NUE
1F(SUPC.GT.1.0E-08)GO TU 372
RATICP 0.0
GU TO 373
372 CONTINUE
FAT ICP = PSIP/SBPC
3 73 CONTI.NUt
I F( I.GT.2 )G0 TO 375
RATIC = RAT I OP
GO TO 300
375 CONTINUE
RATIC = .625 *RATIOP + PSIBC
380 CONTINUE
77 = Z5*RATI0
ZR = Z6 K AT IU
79 6.*73*77
7 lu = 6.* Z4 Z 8
Cl = A20 OELT *RATIOP
L2 = AMJ + A*DELT2*RATI0P + 04
1F(RFSA.LT. 1 .0E-06.UR.UNCOUP.GT.O) GO TO 382
711
=
17* ( 2.* R1F
+ A R 2 F )
712
=
Z8*(2.*R1S
* A *R2 S )
CV7
=
UF/Z3
CV8
=
RIS/Z4
B3S
=
RF S *(1 -
Z 10 )
P3F
=
RF S*( 1 ~
79)
04 S
=
DVH Z 12
B 4 F
=
CV7 Zll
35 S
=
Z8*(RIS +
I 2 A R 2 S )
05 F
=
Z 7 ( R 1 F +
I2A*R2F )
0 7 S
=
05S + 03* B6S
B7F
=
B6F + G 3* B6F
D3S2
= R F S* ( 1. +
Z10)
216


o o n c~. o
THIS PROGRAM CALCULATES THE CRITICAL VALUES OF BETA FuR WHICH
CF/C2 = CS/C2 = l.C AT GAMMA = 0.0
THIS PROGRAM USES EQUATIONS (3.1.7), (3.1.27), (3.1.28), AND
(3.1.29)
DIMENSION DELTA(20) BETA(2u)
REAL NU
INTEGER A, AA(20)
1 REALMS.101) NU,CUTOFF
I F ( CUT Or F G E 1.0 ) GO TO 4
A = I
E = SORT(3.0)
PI = 3.141593
J = 1
CEL = -PI/2.
2 CFLTA(J) = A*DEL (A l)*(PI/3.0)
C = CCS( EL T A(J ) )
S = SINIDFLTA(J))
S 2 = S 2
CS = CS
CENOM = (l.+NU)*E*CS + 3.*NU*S?
IF(A.EQ.I)GO TO 5
Z = 1. + 2.Nil
CO TC 6
5 CONTINUE
C = ABS(DENOM)
IF(C.GT.1.0 E-4)CO TO 8
BETA(J) = 0.0
GO TO 7
8 CONTINUE
Z = -((1. NU ) *2 )/DENOM
6 CONTINUE
BETA!J) = l./( 1 .*Z )
169


70
At a Regular Grid Point for Fully Coupled Waves
The solution to the finite difference equations along the charac
teristic lines at a regular grid point in the case of fully coupled
waves is given here. The longitudinal and transverse velocities from
equations (D.3.5) and (D.3.6) are
VxP = ^ (D2s RHSBA D2f RHSDC)
(3.5.1)
vcr, = 7 RHSDC D RHSBA) .
0P A^ If Is
(3. 5. 2)
When radial inertia effects are included, the stresses at point P
are given by equations (D.3.13), (D.3.14), and (D.3.15). These stresses
are
TP A,
SF + RI1SG(DlD7f D2D4f)
* EHSH ]
(3.5.3)
SXP = T2 [SF(A5QD7s D2D3s> + RHSG + KHSH(D3sD7f D3f
*.>]
(3.5.4)
i r.
0P A0 LR v~l~3s 5Q"4s' "~'"5Q~4f ~l"3f'
RHSF(D D,
AD ) + RHSG(A D D D )
RHSH(D3fD4s D4f
D3s>]
(3.5.5)
and the radial velocity of equation (D.1.3) is
aVrP = a(D3 3S9P>-
(3.5.6)
When radial inertia effects are not included, the hoop stress,
Sgp, and the radial velocity, V^, automatically vanish, and the shear


critical combinations of Poisson's ratio and the "effective tangent
modulus" caused the two wave speeds to be equal when the shear stress
vanished.
The equations for the characteristics and the equations along the
characteristics were written in terms of dimensionless variables.
These equations were then written as first order finite difference
equations. A computer code was written in the Fortran IV language,
and several problems were solved using an IBM 370 model 165 digital
computer. In order to obtain these solutions two particular forms of
the constitutive equation were used; one form represented a strain-
rate independent material while the other form represented a strain-
rate dependent material.
The strain at the impact end was considerably larger when radial
inertia effects were included than when radial inertia effects were
not included in the problem. However, radial inertia effects were
found to have little influence on the solution more than two diameters
from the impact end. The strain at the impact end was lowered by
including strain-rate dependence of the material. For any particular
set of initial conditions and boundary conditions, the stress trajec
tories behaved in the same manner, at least qualitatively, whether or
not radial inertia effects or strain-rate dependence were included.
The details of the stress trajectories were more complicated when
radial inertia effects were included since the trajectories were
three-dimensional.
x


30 3
3 0 4
305
3 1C
IFCKASE.EQ.1) READ!5,10 C 4 ) SXI.TAUI
IFIKASE.EQ.2)
IF(K AS L .EQ.3)
IF(KASE.EQ.4)
GO TO 334
CONFINUE
IF(KASE.EQ.I)
IF(KASE.EQ.1)
IF(KASE.EQ.2)
IF(KASE.EQ.2)
IF(KASE.EQ.3)
IF(KASE.EQ.3)
IF(KASE.EQ.4)
IFIKASE.EQ.4)
R F AD( S, 100 4)
READ! 5 1004)
READ(5, 1004)
SXI = XVFIN
TAUI = TVFIN
VXI = XVFIN
VII = TVF1N
SXI = XVFIN
V r I = TVFIN
VXI = XVFIN
TAUI = TVFIN
CGNT I NUE
IF(MT.GT .2)GO TO 30 5
DEL TAP DELTAB
GO TO 32^
VXI,VTI
SXI,VTI
VXI,TAUI
DELTAP = 2.* DEL T AB DELLHB(MT-3)
CG 10 320
CONTINUE
sxl = sxirx-i)
STL = ST(MX-1)
VXL = VX(MX 1)
VTL = VT(MX-l)
VRL = VR(MX-l)
TAUL = TAU(MX-l)
CELTAP = DELTAIMX+1) + DELTA(MX-l)
TAULB = CLRB*TA'JL + CLRB I *T AUB
SXLB = CLRB*SXL + CLRBI*SXB
STLO = CLRB*STL + CLRBI*STB
VXLB = CLRB*VXL + CLRCI*VXR
VTL B = CLRB*VTL CLRBI *VTB
VRL B = CLRO*VRL + CLRBI*VRB
DELTAB


30
and defining the angles y and 5' as shown in Figure 3.1 these new
variables defined in equation (3.1.2) can be written as
q s cos v cos 6
x '
Cg = s cos y sin 6 \ (3.1.4)
V = Sin Y >
The angle y is the complementary angle to the one normally used in
spherical coordinates. It is used here to facilitate comparison of
results obtained later on to already published results.
From equations (3.1.2) and (3.1.4),
CT0 s cos y sin 6 ^ ^ g 2^x+aCT0^
<7^ S COS y cos 6 a/3
CTx+aCTe
vo, N y3 (aa. a )
r^e-V 9 x
(3.1.5)
so that the a and a axes are located by
x 0
c -axis : o = 0 and tan 5 =
x 0
ar-axis : ct = 0 and tan 6 =
0 x
¡3
V3
and 6 = -60
and 6 = +60 .
(3.1.6)
In order for the equations (3.1.4) to reduce automatically to the
simpler case when radial inertia is not considered, the angle 6 is
defined as
6 = a6' + (a-1)60 (3.1.7)
so that when radial inertia effects are included, a=l, and 6 = 6/, and
when radial inertia effects are not included, a=0, and 6 --60 which
from equation (3.1.6) automatically causes crQ to vanish as it should.
9


203
A = s+ E¡(s-s)n
y
(E.4.1)
in terms of the dimensionless variables, where
n = 1.923 ^
v= 33^si = 3 39 x1Q-4
y 10 psi
- n -10 7 1 923 4
B = BE =(6.1313 xlO ) (10 ) = 1.77236 x10 )
(E.4.2)
When using the data of Cristescu (1972), the generalized stress-
strain curve is found from equation (A.2.21) to be
i
-2ez nr
7 s \2 1
(s s ) + A s + xo
L q' y y J 2 L
P
& -SJ
(E.4.3)
where
and
s
y
a
ys
E
1068.24 psi
6
10.2 X 10 psi
1.047294 x 10
When the data of Cristescu (1972) are used, several dimensionless
quantities can be defined for Case XII. This case is for dynamic load
ing only, and the equations for §(s,A) and ^(SjA) are given in
Section A.4. Other dimensionless quantities for this case are


36
TABLE 1
/CTx\
NORMALIZED LONGITUDINAL STRESS (
\ >
s
Delta
Gamma
0o
o
O
rH
20
30
o
O
1
O
o
1
0.57735
0.56858
0.54253
0.50000
0.44228
1
GO
O
o
0.74223
0.73095
0.69747
0.64279
0.56858
o
O
I
0.88455
0.87111
0.83121
0.76605
0.67761
o
O
O
1.00000
0.98481
0.93969
0.86603
0.76604
-50
1.08506
1.06858
1.01963
0.93969
0.83121
O
O
1
1.13716
1.11988
1.06858
0.98481
0.87111
-30
1.15470
1.13716
1.08506
1.00000
0.88455
1
co
o
o
1.13716
1.11988
1.06858
0.98481
0.87111
-10
1.08506
1.06858
1.01963
0.93969
0.83121
0
1.00000
0.98481
0.93969
0.86602
0.76604
O
o
0.88455
0.87111
0.83121
0.76604
0.67761
20
0.74223
0.73095
0.69746
0.64279
0.56858
co
o
0
0.57735
0.56858
0.54253
0.50000
0.44228
o
o
0.39493
0.38893
0.37111
0.34202
0.30253
50
0.20051
0.19746
0.18842
0.17365
0.15360
60
0.00000
0.00000
0.00000
0.00000
0.00000
O
O
-0.20051
-0.19747
-0.18842
-0.17365
-0.15360
o
O
00
-0.39493
-0.38893
-0.37111
-0.34202
-0.30254
90
-0.57735
-0.56858
-0.54253
-0.50000
-0.44228


22
Eliminating the strain rates from the last six of these equations,
and defining
, 2ctx ~ acJe s
- t(s,A)
X 2s
2acfi a
ij/g = ^ (s ,A)
2s
3t
0x ,i,
Gx
?(s,A)
(2.33)
2s J
the system of nine equations reduces to the following system of six
equations for the unknown variables a aA, T. v vA, and v
x 0 Gx x 9 r
where
a =
X, X
4->
K
>
Q_
ii
CD
b
al I, O
a p v
r,
Gx,x
PV8,t
V =
X, X
V*,t
a
v =
r
o
a u
2 x,
ii
X
CD
>
Vx.t
s =
(cr2 .
X
aAT,
X
(2.
34)
+ avlr 0
(2.
35)
2\|f _
' 0x
(2.
36)
3tL>*
(2.37)
Since the equations (2.24) to (2.26) and (2.34) to (2.36) form
a system of hyperbolic equations, they can be solved by the method of
characteristics. To do this, first the equations for the characteristic
lines must be determined, and then the equations along these characteristic


95
impact end. Experimental results similar to these results were
presented by Bell (1960) for an aluminum bar under uniaxial compres
sive loading.
The distance from the impact end when radial inertia effects
become insignificant can be more easily determined from the radial
velocity. Since the radial inertia is proportional to the time rate
of change of the radial velocity, the maximum value of the radial
velocity at each tube location should give a reasonably accurate
comparison of the radial inertia at each of these locations. The
maximum value of the radial velocity is shown in Figure 4.9. From
this it can be seen that radial inertia effects are about four times
greater at the impact end than at 2.0 diameters from the impact end,
and that the effects of radial inertia decrease asymptotically toward
zero as the distance from the impact end increases.
The change in shear strain is shown at various locations in
Figure 4.10, and the longitudinal strain profile is shown at differ
ent times in Figure 4.11. In this last figure, the strain plateau has
developed near the impact for times greater than T= 20 when radial
inertia effects are not included. However, when radial inertia effects
are included, a strain plateau near the impact end is just beginning
to develop at the time T=20.
The stress trajectories and the strain trajectories are shown in
Figures 4.12 and 4.13 for the case when radial inertia effects are not
present. These trajectories are shown at several distances from the
impact end. The stress trajectory at distances greater than 1.0 diam
eter from the impact end behaves in essentially the same manner as the
ones given for the simple wave solution by Lipkin and Clifton (1970)


RHS2 = RHSDM + A*(B7S*STI R6S*D3)
CEL 4 = B 1 F B 2 S B1SB2F
VXI = (RHS1*82S RHS2*C2F)/DEL4
VTI = (RHS2*D1F RHS 1*BIS)/DEL4
GO TC 390
580 I F(A.EO.O )GU TO 585
IFIKASE.NE.2)GO TO 581
ZA = D4S2*D2 + C1*B7S
ZB = D3S 2 *02 + A5q* B7S
ZC = C35 2 *01 A50* D4S2
Cr L6 = D3F2*ZA D4F2*Z3 B7F*ZC
RHS4 = RHSB AD3*B6F B1F*VXI B2F*VTI
RHS5 = RHSD A D 3* B6S B1S*VXI B2S*VT1
TAUI = ( RHS4*Z A-RFS5* ( D4F 2 02 + 1* 3 7F ) -RHSH* {D4S2 3 7F-D4F2 *B7S ) ) /DL:L6
SXI=l-RHS4*ZB+RFS5*(03F2*D2+37F*A5)-RHSH*(03S2*B7F-D3F2*B7S))/DEL
16
ST1 = (RHS4*ZC-RHS5*(O3F2 *D1-4F2 *A5U)+RHSH*(D3F2U4S2-D4F2*D3S2) )/
1CEL6
GO TC 39 J
581 IFKASE.NE.3 ) GO TO 582
RHS6 = RHSB A*D3*B6F B?F*VTI D4F2*SXI
RHS7 = RHSD A*D3B6S B2S*VTI D4S2*SXI
RHS 8 = R H SH D1*SX1
ZA = C2*D3S2 + A 5Q B 7 S
ZB = D 2 D 3 F 2 + A 5 Q H 7 F
CtLB = 81F* Z A B 1 S Z B
VXI = {R H S6 Z A R H S 7 Z B + RHS8 *{D3S2*B7F 3F2*B7S) )/DEL8
TAUI = ( B1F*(RHS7*D2 + RHS8*B7S)-B1S*(RHS6*02 + RGS8*B7F) J/DEL8
ST I = (B1F*(D3S2*RHS8-A5Q*RHS7)-BlS*(D3F2*RHS8-A5Q*KHS6) )/EL8
GO TC 390
582 CONTINUE
RHS9 = RHSB A*D3*B6F B1F*VXI D3F2*TAUI
RHS10= RHSD A*D3*B6S B1S*VXI D3S2TAUI
RHS11= RHSH A 5Q*T AUI
to
oo


CHAPTER 3
DIMENSIONLESS EQUATIONS AND NUMERICAL PROCEDURES
3.1 Wave Speeds as a Function of
the State of Stress
In this chapter the numerical schemes used to find the solution to
the wave propagation problem of Chapter 2 will be presented. In this
first section the dependence of the wave speeds on the stress state
will be shown. The stresses a T ,, and T have already been assumed
r r9 rx
negligibly small so that the scalar representation of the stress state
is given by equation (A.1.2) as
s =
2
a
x
aa C- +
x b
(3.1.1)
Next, the new variables o', o', and T' will be defined so that the
X 0 0X
surface s = constant can be represented in terms of these variables as
a sphere, and the stress state on this surface in terms of these new
variables can be described in terms of spherical coordinates. Now
defining,
o
X
+
ace)
/ a/3 ,
On = (aff.
CTx)
y
(3.1.2)
equation (3.1.1) can be written as
s
i
/ 2 / 2 / 2~I 8
,CTx + CTe + Texj
(3.1.3)
29


179
and
RHSAE Fif^xLLB+ F6fSxLLB aF3fS0LLB + aF4fVrLLB
RHSBE = F V + F _S aF S + aF V
If xRRB 6f xRRB 3f 0RRB 4f rRRB
RHSCE = ZV + F T
2 0LBB 3s LBB
RHSDE = Z V. + F T
2 0RBB 3s RBB
and
RHSAEM = RHSAE + aD F
O I
RHSBEM = RHSBE + aD F
o ^xl
> (D.2.10)
y
(D.2.11)
and by substituting equation (D.2.1) into equations (D. 2.4) and (D. 2.5),
the equations (D.2.4), (D.2.5), (D.2.6) and (D.2.7) along the nonvertical
characteristics become as follows. The equation
along c = + c
is
-FlfVxP+F2fSxP+aF5fS0P=RHSAEM
(D.2.12)
along c = cp
is
FlfVxP+F2fSxP+aF5fS0P=RHSBEM
(D.2.13)
along c = + c
s
is
- Z2V0P + F2sTP = RHSCE
(D.2.14)
and along c =
- c is
s
Z2V0P + F2sTP=RHSDE *
(D.2.15)
Equations (D.2.3) and (D.2.12) to (D.2.15) form a set of five
simultaneous algebraic equations for the unknowns V^, Vgp, Sxp, sgp
and T Tlieir solution will be obtained at a regular grid point and
a boundary grid point in Sections D.4 and D.6, respectively. In all
cases the radial velocity, V will be found from equation (D.2.1)
after the hoop stress, Sa at the grid point is known.
0P


187
the expressions for the stresses become
Tp = i [rHS4(D2D4s2+ DiB7s) RHS5(D2D4f2+ W
6
+ RHSH(D4s2B7f-D4f2B,
*>]
(D.5.11)
xP _
b
- RI,S4 - RHSH ]
(D.5.12)
Sep = rr[RHS4 RHS5(DlD3f2 A5QD4f2^
4 RHSH(D3f2D4s2 04f2D3s2)j
(D.5.13)
When radial inertia effects are not included (a=0) the radial
velocity and hoop stress vanish as does the third equation of (D.5.8).
The remaining two equations of (D.5.8) become
D3f2
D4f2
1
o,
t-
L
RHS4
D3s2
4s2
1
CO
$
L
RHS5
(D.5.14)
and letting
A7 ~ D3f2D4s2 3s2D4f2
(D.5.15)
the two remaining stresses are given by
TP = ; (D.5.16)
SxP = A7(D3f2RHS5"D3s2RHS4)
(D.5.17)


LIST OF FIGURES (Continued)
Figure Page
4. 6 Transverse Velocity Versus Time for Data Set 1
Without Radial Inertia 91
4.7 Longitudinal Velocity Versus Time for Data Set 1
Without Radial Inertia 93
4. 3 Longitudinal Strain Versus Time for Data Set 1 94
4.9Maximum Radial Velocity Versus X for Data Set 1
With Radial Inertia 96
4.10 Change in Shear Strain Versus Time for Data Set 1
Without Radial Inertia 97
4.11 Longitudinal Strain Versus X for Data Set 1 98
4.12 Stress Trajectories for Data Set 1
Without Radial Inertia 99
4.13 Strain Trajectories for Data Set 1
Without Radial Inertia 100
4.14 Shear Stress Versus Longitudinal Stress for Data Set 1
With Radial Inertia 102
4.15 Stress Trajectories for Data Set 1 With Radial Inertia . 104
4.16 Hoop Stress Versus Longitudinal Stress for Data Set 1
With Radial Inertia 105
4.17 Stress Trajectories for Data Set 2
Without Radial Inertia 107
4.18 Stress Trajectories for Data Set 2 With Radial Inertia . 108
4.19 Hoop Stress Versus Longitudinal Stress for Data Set 2
With Radial Inertia 109
4.20 Shear Strain Versus Time for Data Set 2 Ill
4.21 Change in Longitudinal Strain Versus Time for Data Set 2 112
4.22 Longitudinal Strain Versus Time for Data Set 3 114
4.23 Stress Trajectory at X = 0 for Data Set 3 116
4.24 Stress Trajectory at X = 25 for Data Set 3 117
viii


174
along c = c is
BlfVxP + B2fV9P+ D3f2TP + D4f2SxP aB5fS0P + aB6fVrP
BlfVxRRB+ B2fVGRRB+ B3fTRRB+ B4fSxRRB
+ aB5fS0RRB aB6fVrRRB
(D.1.13)
along c = + cs is
" BlsVxp B2sV6P + D3s2TP + D4s2SxP aB5sS0P + aB6sVrP
"B1sVxLBB ~ B2sV6LBB+ B3sTLBB^ B4sSxLBB
+ aB5sS9LBB aB6s^rLBB
(D. 1.14)
and along c=-c is
s
B V + B V +D T + D S aB_ S. + aB V
Is xP 2s 0P 3s2 P 4s2 xP 5s 0P 6s rP
BlsVxRBB+ B2sV0RBB+ B3sTRBB+ B4sSxRBB
+ aB5sS0RBB_aB6sVrRBB '
(D.1.15)
Next, let
RHSA BlfVxLLB B2fV0LLB+ B3fTLLB+ B4fSxLLB+ aB5fS0LLB
aB6fVrLLB
RHSB BlfvxRRB+ B2fVeRRB+ B3fTRRB+ B4fSxRRB+ aB5fS0RRB
aB6fVrRRB
RHSC b1svxLBB B2sVeLBB+ B3sTLBB+ B4sSxLBB+ aB5sS0LBB
aB6sVrLBB
RHSD B2sV0RBB+ B3sTRBB+ B4sSxRBB+ B5sS0RBB
HB6s^ rRBB
J
(D.1.16)


15
The equations of motion in the cylindrical coordinates shown in
Figure 2.1 are given by
1 1
0* *f T 4- T 4. T p U
x,x rx,r r 0x,0 r rx r x,tt
T + u + T + ~(o o'.) = p u
rx,x r,r rr0,0 rr 0 H r,tt
T0x,x + Tr0,r + r CTe,e + r Tr0 ~ P U0,tt
which, under the assumptions given above, become
CT = P U 4-4-
x,x r x,tt
(2.1)
P u
r, tt
T0x,x p U0,tt
(2.2)
(2.3)
where the subscripts following the comma represent partial differentia
tion with respect to the variables x (the coordinate along the tube
axis) or t (time). The density of the material is p and u^, u^, and
u are the displacements of any point in the x, r, and 0 direction,
o
respectively.
For the cylindrical coordinates of Figure 2.1, the strain-
displacement equations are given by
e = u
r r,r
ee 3 r(u9,e + V
e = u
X x,x
, V1 1 ,
Sr9 = 2(7 Ur,0 + U0,r 7 V


120
The first numerical results were obtained for data given by
Lipkin and Clifton (1970) for one of their experiments. A rise time
of about 10 microseconds was used for the velocity impact. Several
grid sizes were used and the larger grid sizes were found to smooth
out the time histories of the variables. Because of this the strains
obtained using the larger grid sizes were very similar to the experi
mental results while the strains using smaller grid sizes were similar
to the simple wave solution. Since the larger grid sizes smoothed out
the numerical data, a small grid size was used for the remainder of the
computer runs.
Numerical data were first obtained using the generalization of
the Lipkin and Clifton (1970) strain-rate independent constitutive
equation. Two separate sets of initial conditions and boundary condi
tions were used. One was a tube prestressed statically above the yield
stress in torsion and impacted at the end by a predominantly longitu
dinal velocity pulse. The other was a tube prestressed statically
above the yield stress in compression and then impacted at the end by
a torsional velocity pulse. The rise time for the velocity pulse in
the first case was 10 microseconds and in the second case was 1 micro
second. Two computer runs were made with each data set to generate the
results both with and without radial inertia effects.
For the first case when the static prestress was torsional, the
stress trajectories were found. When radial inertia was not included
the stress trajectory at distances greater than 1.0 diameter from the
impact end behaved in exactly the manner predicted by Clifton (1966)
from his simple wave solution. That is, the stress path at any point


146
1 1
- dv
x -
c
'"aA2A5 A3A4
~i r
-2 -
c A A aA A /o c
4 6 5 4
2J dve + LAidcrx+ aA2dC7e
3' ej
+ AdT
K
AA_ -
A A +
3 5
2 6 -2
DC
A 4 -
-2 ^4
aA -
4 6
5 -2
pc -
f-
aA~A
- ia.
2 5
3 4
A
_
-2 4
A A -
aA^
L 4 6
5 -2_
p c

a" *

- 2
AA -
A A + -
3 5
2 6 -2
pc
[
aA d 2 x 4 0 5 0
J
A3dax+aA5% + VT0.x
J-
lit dt
Tx
- -2 4
A4A6 ^5 ~^2
Pc J
r
al ) ldt
o
0X
aA2A5 A3A4
- -2 A4
A,A aA
46 5-2.
pc
dt
r
and multiplying through by the term jp c A A aA
-2 A4
4 6 5 -2,
pc
A 1 >>
4]}.
this form
of the equations along the nonvertical characteristics becomes
-2
0 = -_-[(P c2) (A4A6 a2) A J dvx e|_ [aA^ 3 J dv
0
[<
+ I D
4 6 5
A,
dax+aA2do0 + A3dTe
J
2 i r -
+ I AA_ AA + 2 j aAdcf_ + aA^dcr^ + aA,_dT,
3 5 2 6
pc
2 x 4 0 5 0x
J[-]
[p 2J [ai2I5 -V j [Vx + aA5doe + A6dT0
j
[
+ I (pc2)(A4A6-aA2) -A4
'][
p c2 I I aA2A5 A3A4
1] +xdt {-
J [2,eJ
A3A5-A2A6 +
pc o


135
tys.A) = O
^0
if S <
0(s ,A) =<
2£2
z
i if cr < s < a /
E y z
(A.2.25)
2s 1 ..
- if a ^ s
~+2 E z
For this case the constitutive equation (A.1.1) during loading becomes
1+v
e. = o
ij E
x
Ks f 6ijk+ tiivr e} + 0
3 Sij
2 1+2
(A.2.26)
The expressions (A.2.18) and (A.2.25) are valid only when the
material is loading. During unloading, the constitutive equation
(A.1.1) is still valid, except that 0 (s ,A) = i|f (s ,A) = 0. This results
from assuming that the unloading is elastic.
A. 3. Rate Dependent Plasticity Theory
The constitutive equation (A.1.1) is shown in Section A.l to
reduce to the constitutive equation (2.13) when the only stress present
is a Therefore, the data from Cristescu (1972) Case XII, can be used
directly, and the expressions for 0(s,A) and \|r(s,A) can be written as
r
K(A)
>]
t(s,A) = <
s f (A) if s>f(A) and A^ A
if s s f (A) or A < A
and
V.
(A.3.1)
(A.3.2)


74
TP = h, (D4s2RHS4 D4f2RHS5) (35-21)
SxP r7 (D3f2RHS5 D3s2RHS4) (3-5'22)
Case III: Mixed boundary conditions
When S and V. are known, the solution when radial inertia
xP 0P
effects are included is given by equations (D.5.22), (D.5.23), and
(D.5.24) as
V
xP
h [RHS6 RIIS7
o
+ RHSS(D3s2B7f D3f2B7s)]
(3.5.23)
-.r,
Tp = ^Blf(D2RHS7 + B?sRHS8) Blg(D2RHS6 + B7fRHS8)J
8
GP
Blf
]
(3.5.24)
(3.5.25)
and V again is found from equation (3.5.6). When radial inertia
effects are not included, vrp=^Qp~ anc^ from equations (D.5.27)
and (D.5.28), the solution at P becomes
VxP = ^ T = (B7 RHS7 B RHS6) (3.5.27)
P If Is
Case IV: Mixed boundary conditions
When V and T are known at the boundary, the solution at P is
xP P J
found from equations (D.5.33), (D.5.34), and (D.5.35) when radial
inertia effects are included to be


3
(or strain). This immediately led to the conclusion that a given
level of stress (or strain) propagated at a specific speed, and the
stress wave changed shape as it propagated along a prismatic bar for
stresses in the nonlinear region of the stress-strain curve. For a
bilinear stress-strain curve, the results of Donnell (1930) were again
predicted. However, these theories did not account for the lateral
inertia effects in the bar or the dependence of the stress-strain curve
on the rate of strain, and so more complex theories and constitutive
equations were proposed to account for these phenomena.
By the late 1940's many investigators including Davis (1938),
Manjoine (1940) and Clark and Wood (1950) had experimentally observed
the effect of the rate of strain on the stress-strain curve for several
materials. In order to incorporate this strain-rate effect into the
constitutive equations used to study plastic wave propagation,
Sokolovsky (1948a, 1948b) and Malvern (1949, 1951a, 1951b) independ
ently introduced one-dimensional constitutive equations in which the
stress was a function of the plastic strain and the plastic strain rate.
By selecting a particular form of this constitutive equation, Malvern
(1951a, 1951b) was able to obtain a numerical solution which predicted
several experimentally observed phenomena. However, his numerical solu
tion did not apparently predict a region of constant strain near the
impact end such as had been observed by Duwez and Clark (1947) and
others. This new strain-rate dependent constitutive equation also
predicted that, if a bar were strained statically above the yield stress
and then impacted, the first increment of strain would propagate with
the elastic wave velocity and not the velocity given by the tangent


VTL8B = C0N2 *VT LB + CCN4*VTB
VKLBB = C0N2 *VRLB + C0N4*VRB
IF(RFSA.LT.1.0E-06.UR.UNC0UP.GT.0) GO TO 386
CIS = 2.*B1S
Cm
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U1
\ji
cn
CD
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sO
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CD
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73
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73
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XI
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a
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rr.
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73
XI
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73
X
**
n
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co
X
rx
ro
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m
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r^
3:

X
a
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x
rn
r*
r"
ro ro
ozz
VXI = (RHSBEM F 2 F SX I AF6FSTI )/FlF
IF(KASE.LQ.3)G0 TO 692
VTI = (RHSDE F 2 S T AUI )/Z2


130
Next, a expression for s, cr will be found as
kl kl
s,, = (a o 6, t a ) cr. ,
kl kl kl 3 kl nn kl
z 01
SklCTkl 3 f dt '
Using equations (A.2.11) in (A.2.10),
strain rate becomes
(A.2.11)
the expression for the plastic
P
e
ij
2f2 F' O'?)
s .
ij
(A. 2.12)
The function F/(W ) will be obtained by assuming that the material
obeys a "universal stress-strain law." The functions s and A are chosen
in such a form that if the only stress present is then s is equal to
a and A is equal to e as shown in Section A.l, and therefore the
x x
"universal stress-strain curve" is the same as the stress-strain curve
for simple tension with s replacing ct and A replacing e .
X X
Two forms of the uniaxial stress-strain curve will be considered.
The first form is the type used by Clifton (1966) and by Lipkin and
Clifton (1970). For this case the static and dynamic curves are both
given by
where
(A. 2.13)
r o if
a < ct
x y
(A. 2.14)
and a is the tensile stress,
x
yield stress, E is the elastic
e
(ct ct ) if cr
" x y x
is the tensile
x
tensile modulus,
> CT
strain, cr is the
y
and B and n are
constants used to adjust the curve to represent various materials.


173
and
Z7 = VZ5
Z8 VZ6
Z =
6*VZ7
Z10 = 6 Z4'Z8
Z11 = V(2,Rif + aR2f} }
z-.o = zc* (2 R + aR0 )
12 8 Is 2s y
and then
If
If
B2f = VRfs
B3f = Rfs*(1-V
R,
If
- Z*
4f Z3 11
B5f = V(Rlf+2aB2f> '
B61 = 2'VR2f
y
and
Is
Is c
B2s Z2*Rfs
B3s = Efs-(1-Z10>
lS
34s = ~Z~ ~ Z12
4
"N
B5s = Vis+2aR2s) (
B- = 2*ZC*ROc
6s 6 2s
y
and finally
D3f2 = BfS,(1 + Z9)
D3s2 = BfS-<1 + Z10>
Rlf
4f2 ~Z~+ Z11
R-
D = + Z
4s2 Z 12
4
The equations along the four nonvertical characteristic lines
(c = c^., cg) can now be written as given below. The equation
along c = + c^ is
BlfVxP B2fV0P + D3f2TP + D4f2SxP aB5fS0P + aB6fVrP
_BlfVxLLB "" B2fV0LLB + B3fTLLB+ B4fSxLLB
+ aB5fS0LLB aB6fVrLLB
(D.1.8)
(D.1.9)
(D.1.10)
(D.1.11)
(D.1.12)


115
highest rate of loading is at the impact end, and using the analogy
for longitudinal loading, the maximum value of the strain would be
expected to be lower here than at other positions along the tube where
the loading rate is lower.
The stress trajectories are shown for the four cases in
Figures 4.23 and 4.24. The behavior of these trajectories is similar
to the behavior already discussed. It can be seen that strain-rate
dependence does not significantly alter the stress trajectories except
at the impact end where the loading appears to be a little more severe
for a strain-rate dependent material.


32
Using the uniaxial stress-strain curve in the form of equation
(A.2.13), the universal stress-strain curve can be written as
| + B n
e y
(3.1.8)
and letting E^(s) be the tangent modulus of this curve, yields
dA
E^Cs) ds
1 /- xn-l
E + Bn
(3.1.9)
and from equation (A.2.18) this becomes
77 + 0(s)
E
Et(s)
or
0 (s)
Et(5)
1
E
(3.1.10)
Now 3 = ¡3(s) is defined so that
where
E (s) = 0(s) E
t
0 £ 3(s) 1
(3.1.11)
and when 3 = 1, the material is elastic, and when 3 = 0, the material
is perfectly plastic. Using equation (3.1.11) in equation (3.1.10),
0(s)can be written as
0(s) = \ ( 1) (3.1.12)
E 3(s)
Inverting equations (3.1.2), the stresses are given by
a = o' al ^
X X J3 6
ae CTx + CTe
1 /
0X yg. 0X
(3.1.13)


159
Also, under the conditions given here, equation (B.2.25) becomes
(p o (A.AJ-A, dv. +
2-3 r~-
i -r
0 = P c |_~A3A4
JdvxpcL
[(p52XA1A4)-A4JdTex-pi2 [vjda,
-2
and dividing by (p c ), this becomes
o=[h-^][d'
pc
P c dvft J + A3 |^p c dvx doj
6x ^ 0
and again using equation (B.3.3) this is
2
0 = [^2 ^iTH}][pSdv8-dTexJ+[v9xH][p3dvx-dCTJ- Equations (B.3.5) and (B.3.6) are the equations along the character
istics given by Clifton (1966) for the von Mises yield condition.
B.4. Uncoupled Waves
When the stress waves are fully coupled, the equations along the
characteristics given by equations (B.2.23), (B.2.24), and (B.2.25) are
equivalent. Each of these equations represents one equation along each
of the four nonvertical characteristics. The waves are fully coupled
when none of the coefficients of the variables in these equations vanish.
However, it can be seen that some of the coefficients vanish in these
equations when A =A =0. This condition can occur when loading is
o o
along the TQ axis (a =cto = 0), or when loading is perpendicular to the
OX X W
T- axis in the plane T. =0, or when loading occurs at stress levels
0x 0X
within the yield surface (0(s,A) =0), or when unloading occurs (0(s,A) = O)


402
WRITEl6, 1009) (T(L)*L=1,MX, INCRX)
WklTE(6,lCll)
WRITE(6, 1012)
WRITE(6, 1038 )
WRITE(6,1013)
W R I r E ( 6 10 1 4 )
W R I T F ( 6 ,1015)
WRIT6(6, 1016)
WRIT E{6 i 1Cl7)
W K1 F c(6 1^18 )
WRITEI6, 1019)
WRITE(6, 1020 )
WRITE6,1021 )
WRITE(6, 102 2 )
WRITE(6, in?3 )
WR I TE (6, K'24 )
WRITc(6,102 9)
WRITE16, 1^26)
W R 1 T E(o 1027)
WRITE-(6, 102 8 )
WRITE(6,1029)
WRIT F(6, 1036)
WRITlId,1037)
(SX(L ) ,1=1,MX,INCRX)
(ST(L ) ,L= 1,MX,INCRX)
(STRES(L),L=1,MX, I NCR X)
(TAU(L ) ,L = ltMX, INCRX )
(VX(L ) L= 1 .MX,INCRX)
(VT(L ) L = 1 MX.INCRX)
(V R(L ) L = 1 M Xi INCRX)
{CF(L ) L = 1,MX,INCRX)
(CSC L ) ,L l ,MX, INCRX)
C SB(L ) i L = 1 *MX, INCRX)
(EX(L),L=1,MX, INCRX )
( E T(L ) L=i,MX, INCRX)
( ETXtL )L=1MX, INCRX)
(EX P(L ) ,L = 1,MX,INCRX)
(E T P{L ),L = 1,MX, INCRX)
(ETXP(L ) t1 = 1,MX,INCRX)
( ITER(L),L = 1,MX,INCRX)
(CFL T A (l_ ) L = 1 MX, INCRX )
(CPLAS(L ) ,L = i,MX, INCRX)
(PH 10(L),L = l,MX, INCRX)
(LELCAMCL),L=1,MX,INCRX)
l f: F X ( L ),L = 1,MX, INCRX)
CONTINUE
IF( 1 PUNCH.E.0)C0 T0 404
LIN E P = LINIP + I
IFCLINEP.LT. INCRTPJGO TO 4U4
LINEP =
WRITt t 7, 1^3 5) (X(L ) ,L = 1,MX, INCRXP )
WRITEC7,1C 3 5 ) (T(L),L=1,MX,INCRXP)
WRITE(7, 1036) (SB(L ) L= 1,MX,I NCRXP)
WRITEC 7, 103 5) (CELTA(L),L = 1,MX, INCRXP)
IF( I PUNI.EQ.C)GO TO 40 3
WRITEC7, 1C35) {SXIL ) ,L= 1,MX,INCRXP)
225


Shear Stress, T x10
Figure 4.23 Stress Trajectory at X = 0 for Data Set 3
116


43
the critical value of 3 is found from
0 = 1 + vj [2(1+'') -(1+Vi>]+ a L-(v+zcVJ
'= Zc [flf2 + 0I1] + Zc [f2(1+',)-1}
J 2C1 + v) -1+ av2J .
(3.1
The expressions for the critical value of 3 will now be found
for the two separate cases of a=0 and a=l. First, when a=0,
equation (3.1.24) becomes
A6 = \
2(1 + v) = 1 + z f
c 1
1 + 2v
3. =
\
(3.1
c z +1
c
and in this case § = -60 from equation (3.1.7) and
f^ = i j^cos(-60) 2 */3 sin (-60) cos (-60) +3 sin2(-60)J
h =1
and
P = i
Hc (1 + 2v) + 1
1
3_ -
c 2(1 + v)
.26)
. 27)
(3.1.28)


46
3.2 Characteristic Solution in Terms
of Dimensionless Variables
In order to make the numerical solution in the characteristic
plane more general, the equations for the characteristics and the
equations along the characteristics given in Chapter 2 will be written
in terms of dimensionless variables. The dimensionless variables
used are
S =
x
cr
x
TT
v
X
V
X
c =
) (3.2.1)
(s,A) i|/(s,A)
Z1
§(s,A) = E0(s,A)
I (s,A)
Y(s,A) =
where c^ given by equation (3.1.16) is the elastic longitudinal wave
speed when radial inertia effects are included. In terms of these
variables, the functions defined by equations (3.1.18) can be written
from equations (2.23) as


234
Manjoine, M. J. (1944), "The Influence of Rate of Strain and Temper
ature on Yield Stresses in Mild Steel," Journal of Applied
Mechanics, Yol. 11, A211-A218.
Mok, C. H. (1972), "Effects of Lateral Inertia on the Propagation of
Elastic-Plastic Waves," BRL-CR-70, Contract No. DAAD05-70-C-0224,
Aberdeen Proving Ground, Md.
Nicholas, T. and Garey, G. F. (1969), "Torsion Testing of Aluminum at
High Rates of Strain," AFML-TR-69-172.
Olszak, W., Mroz, A. and Perzyna, P. (1963), Recent Trends in the
Development of the Theory of Plasticity, New York: Pergamon .Press.
Perzyna, P. (1963), "The Constitutive Equations for Rate Sensitive
Plastic Materials," Quarterly of Applied Mathematics, Vol. 20,
pp. 321-332.
Plass, H. J., Jr., and Ripperger, E. A. (1960), "Current Research on
Plastic Wave Propagation at the University of Texas," Plasticity
(E. H. Lee and P. S. Symonds, Eds.), New York: Pergamon Press, Inc.,
pp. 453-487.
Prandtl, L. (1924), "Spannungsverteilung in Plastischen Koerpern,"
Proceedings of the First International Congress for Applied
Mechanics, Delft, pp. 43-54.
Rakhmatulin, Kh. A. (1945), "Propagation of an Unloading Wave,"
Prikladnaia Matematika i Mekhanika, Vol. 9, pp. 91-100.
Rakhmatulin, Kh. A. (1958), "On the Propagation of Elastic-Plastic
Waves Owing to Combined Loading," Prikladnaia Matematika i
Mekhanika, Vol. 22, pp. 1079-1088.
Reuss, A. (1930), Beruecksichtigung der Elastischen Formaenderunger
in der Plastizitaetstheorie," Zeitschrift fur Angwandte Mathematik
und Mechanik, Vol. 10, pp. 266-274.
Shammamy, M. R. and Sidebottom, O. M. (1967), "incremental Versus Total-
Strain Theories for Proportionate and Nonproportionate Loading of
Torsion-Tension Members," Experimental Mechanics, Vol. 7,
pp. 497-505.
Shea, J. H. (1968), "Propagation of Plastic Strain Pulses in Cylindrical
Lead Bars," Journal of Applied Physics, Vol. 39, pp. 4004-4011.
Sokolovsky, V. V. (1948a), "Propagation of Elastic-Visco-Plastic Waves
in Bars," Doklady Akad. Nauk SSSR, Vol. 60, pp. 775-778.
Sokolovsky, V. V. (1948b), "Propagation of Elastic-Visco-Plastic Waves
in Bars," Prikladnaia Matematika i Mekhanika, Vol. 12, pp. 261-280.


Internet Distribution Consent Agreement
In reference to the following dissertation:
AUTHOR: Myers, Charles
TITLE: Inelastic wave propagation under combined stress states, (record
number: 577578
PUBLICATION DATE: 1973
I, Charles D. Myers as copyright holder for the
aforementioned dissertation, hereby grant specific and limited archive and distribution rights to
the Board of Trustees of the University of Florida and its agents. I authorize the University of
Florida to digitize and distribute the dissertation described above for nonprofit, educational
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136
where
if A ^ A
(A) = l
8+ A
CJ + V A 2 (A-A ) if A SA SA
y 2 z y y :
V
, 1
3+Ae
if A ^ A
z
A =
z
3+A
/ 2 q+2
a -a/ a A P
y y y
K(A) = Kj_l exp (-A)J
a = m +
n\ZA~
X =
0 if s ^ f(A) + o' or < 0
1 if s > f(A) + o7' and > 0
ot
+
o + +
A = h + X
(A. 3.3)
(A.3.4)
(A. 3.5)
(A.3.6)
(A. 3.7)
(A. 3.8)
and A is the yield strain of the material in tension, y y
stress in tension, and ¡3' K A, m, n, h, and \ are constants of the
material. Because of the particular selection of the form of s and A,
the values of these constants, as determined experimentally by
Cristescu (1972) for uniaxially loaded 1100F aluminum, can be used in
the treatment of multiaxial states of stress for this same material.


157
and this function H(s) can be written in terms of 0(s,A) by using
equations (A.2.16) and (A.2.18) to get
H(s)
H(s) =
3Bn(s-a )
y
n-1
-2
s
30(s)
or
1 -2
0 (s ,A) = 0(s) = S H(s) .
(B.3.2)
Using equation (B.3.2) in equation (2.23), and remembering that
a = 0,
where
and
A5 2 CTxTGx H
V 5 + 3Te* H
(B.3.3)
2(1+y) = 1
E G
H = H (s) .


Longitudinal Strain,
Figure 4.8 Longitudinal Strain Versus Time for Data Set 1
<£>


189
] r n
Tp = [Blf(D2RHS7 + B7sRHS8) Bls(D2RHS6 + B7fRHS8) J
8 ~
S6P " [Blf o
(D.5.23)
(D.5.24)
When radial inertia effects are not included (a = 0), the last of
equations (D.5.19) vanishes, Vrp and Sgp are zero, and the first two
equations of (D.5.19) become
B_ ^
D
V
RHS6 j
If
3f 2
xP
B-,
Do o
T
RHS7
Is
3s2
P
1
-J
so that with
BlfD3s2 BlsD3f2
the solution is
V
xP
T
P
= ¡r
9
= t^-(B, RHS7 B, RHS6)
If Is
Case IV: Mixed boundary conditions
When V^p and Tp are specified at a boundary point,
RHS9 = BHSB1 BlfVxp D3f2Tp >
RHS10 RHSD1 BlsVxp D3s2Tp l
(D.5.25)
(D. 5.26)
(D. 5.27)
(D. 5. 28)
(D. 5. 29)
RHS11 = RHSH AT
o y p


101
and by Clifton (1966). That is, for a tube under a static torsion
load, a longitudinal impact creates a period of neutral loading as
the fast wave passes followed by loading normal to the yield surface
as the slow wave passes. During the neutral loading the shear stress
decreases while the longitudinal compressive stress increases. The
stress trajectory at the boundary (impact end) does not follow this
simple wave solution and changes character in the first diameter from
the end. This behavior is probably due to the particular selection of
the boundary conditions, coupled with the fact that near the impact end
both fast and slow waves pass the same point along the tube at almost
identical times. Therefore the tube is attempting to undergo both
neutral loading and loading simultaneously. The sharp corner on the
stress trajectory at X = 0 corresponds to the time T = T =4.0 which is
R
the peak of the input velocity ramp.
When radial inertia effects are included, there are two additional
variables: the hoop stress (Sg) and the radial velocity (V^). For
this case the stress trajectory will not necessarily be planar.
In spite of this, the stress trajectory (T versus S ) projected onto
the plane SQ=0 is shown in Figure 4.14. It appears at first that
0
the stress trajectory at X=1.0 represents loading followed by unload
ing and then by reloading. Recalling the definition of the general
ized state of stress from equation (A..1.2)when radial inertia is included
as
2 2 2 2
s = S + S S S. + 3T
x 6 x 9
the quantity S can be defined by
2 2 2
s = s + s s S-
N x 8 x 0
(4.3.1)


17
the constitutive equations. Cristescu (1972) uses a full quasilinear
constitutive equation for a single longitudinal stress as
3e
at
1 da
E 'St
+ 0(a,e)
3a
"3t
:(a,e)
(2.13)
As a generalization of this equation to a constitutive equation
governing multiaxial states of stress and strain, the following equation
is used
e
ij
1 + v
E
a. .
ij
v
E
6 a +
ij kk 2
0(s,A)s + \ji(s
,A)J
ij
s
(2.14)
where the dot represents partial differentiation with respect to time,
s is the deviatoric stress, 6 is the Kronecker delta, v is Poisson's
ij iJ
ratio, E is Young's modulus, 0(s,A) and i|f(s,A) are material response
functions as yet unspecified, and s and A are defined as
s = /- s. .s. (2.15)
V 2 xj ij
A
/I
V 3
J
P *P
e e
ij iJ
dt +
E
(2.16)
P
and e is the inelastic portion of the strain rate which, using
equation (2.14), can be written as
['
0(s,A)s + ^(s
IS..
(2.17)
when the elastic, plastic, and visco-plastic portions of the strain
rate are assumed to be separable. The constitutive equation (2.14)
is a special case of the equation
e
ij
_pk!
f. a + g. .
ij hi ij


7 CONTINUE
A A ( i ) = A
J J + I
CEL = DEL + PI/1H.0
IF(J.LT.2D)GO TU 2
A = u
IF(J.ED.20)GO TO 2
WRIT E(6,102)
CO 3 J=1,20
CELT A(J) = DELTA!J)*180./PI
WRITE(6103) DELTAJ ) ,AA(J), NU.BETAIJ)
3 CONTINUE
CC TC I
4 CONTINUE
Id FOkMAT( 2 F 10.3 )
102 FORMAT(' IFOR EACH VALUE F DELTA, THE APPROPRIATE VALUE OF Bn TA IS
1 THE ONE BETWEEN 0 AND +1'//' THE VALUc UF BETA GIVEN IS TH~ ON
AT WHICH CF/C2 = CS/C2 = 1.0 AT GAMMA = C//' A = ALPHA = 0 FO
3CLIFT0NS CASE'/' A = ALPHA 1 FOR RADIAL INERTIA EFFECTS////1
4, DELTA',7X,ALPHA*15X, 'NU13X, CRITICAL VALUE OF BETA /)
103 FORMAT IF 20.2,I 13,F2 > 3.F30.6)
STOP
END
o
uj a£ in


182
and with
42 = 3f + D7f
the stresses at a regular gird point are given by
TP = ^[RHSF(D2D46-D1D7S) + RHSG(DlD7f-D2D4f)
+ raSH(D4fDjs-D4sD7f)]
'xP h,
LrhSf + BiSE<3sD7f-V7S)]
6P A2 L
RHSF(JL D A D ) + RHSG(A D D, D )
1 3s 5Q 4s 5Q 4f 1 3f
+ RHSH(D3ffD4s-D4fD3s)
]
and the radial velocity, V^p, is given by equation (D.1.3).
When radial inertia effects are not included, that is, when
equation (D.1.6) vanishes, and the two remaining equations (D.3.
(D.3.10) for the stresses, can be written in matrix form as
D3f
I
Q
T
P
RHSF
D3s
I
w
Q
RHSG
and letting
a = D D D D
3 3f 4s 3s 4f
(D.3.12)
(D.3.13)
(D.3.14)
(D.3.15)
a = 0,
9) and
(D.3.16)
(D.3.17)


Longitudinal Velocity, V
Figure 4.2 Grid Size Effects on the Longitudinal Velocity at X = 3.75
CO
CJi


c
c
c
c
c
c
c
c
c
c
c
PROGRAM T CALCULATE; THE PLASTIC WAVE SPEEDS
THE EQUATIONS USED ARE (3.1.7), (3.1.18), (3.1.19), AND (3.1.20)
A = ALPHA = 0 FOR CLIFTON'S CASE
A = ALPHA = 1 FOR RADIAL INERTIA EFFECTS
0 E T A = SLOPE OF THE STRESS-STRA IN CURVE IN TENSION DIVIDED BY
YOUNGS MODULUS
NU = POISSONS RATIO
DIMENSION DELTA(20), GAM(IC), A1(10,2C), A2(10,20), A3(1C20)
DIMENSION A4(10,73), A5( 10,20), A6( 10,20), ABAR(10,20)
DIMENSION B B A R(ID 2 0) RAD(10,2?), CFC2(13.20) CSC2(10,20)
DIMENSION SX( 10 23), ST110.20), TAU(ll)
REAL NU
INTEGER A
1 REAC(StlOl) BETA,DEL GAM0,DGAM,DDFL,NU, A
I F ( NU. GT.C.5)GO TO 10
E = SORT(3.0 )
PI = 3.141593
1 = (l./BETA) 1.0
1 = 1
2 GAM(I) = GAMS
C = COS(GAM( I ) )
S = S I N ( G AM ( I ) )
C2 = C * 2
S 2 = S * 2
CS ^ C*S
TAU(I) = S/t
J = 1
CELL = DEL
3 DELTA(J) = ADELL+ (A l)*(PI/3.0)
165


110
The shear strain and the change in longitudinal strain are plotted
against time in Figures 4.20 and 4.21, respectively. Both of these
strains exhibit the period of constant strain between the passing of
the fast and slow waves. When radial inertia effects are not included
the shear strain at the boundary has a smaller value than at other
points along the tube. Physically this is difficult to realize, and
it may have been caused here by the manner in which the boundarycondi
tions are given since the strains are computed directly from the veloc
ities and the velocities are specified at the boundary. The change in
longitudinal strain (Figure 4.21) is interesting; initially the tube is
compressed so that the longitudinal strain is negative. As the fast
wave passes a point on the tube, the change in longitudinal strain is
positive, that is, the compressive strain decreases. This decrease in
compressive strain corresponds to the initial decrease in compressive
stress shown in Figure 4.17. Then as the slow wave passes a point on
the tube the change in longitudinal strain is negative. This repre
sents a further compressive loading of the tube and corresponds to
the region of loading normal to the yield surface shown in Figure 4.17 .
4.4 Effects of Strain-Rate Dependence
In order to investigate the effects of strain-rate dependence,
four computer runs were made using the computer code in Appendix E.
These four data sets were obtained using the generalization of the
stress-strain curve of Cristescu (1972). Two computer runs were made
without strain-rate dependence (one with and one without radial inertia
effects) and two with strain-rate dependence (one with and one without


180
D.3. Solution at a Regular Grid Point
for Fully Coupled Waves
For this case the solution will be obtained by solving equations
(D.1.6) and (D.1.18) to (D.1.21) simultaneously. These equations can
be reduced to a set of two simultaneous equations and a set of three
simultaneous equations as follows. Sutracting equation (D.1.18) from
equation (D.1.19) and subtracting equation (D.1.20) from equation.
(D.1.21) results in two equations involving only the unknowns V and
Vgp. They are
2BlfVxP + 2B2fV6P = RHSB1 RHSA1 (D.3.1)
2EL V + 2B = RHSD1 RHSCl (D.3.2)
Is xP 2s 0P
and if
II
tH
Q
2Blf
D2f "
2B2f
Dls =
2Bls
2s =
2B2s
RHSBA
= RHSBl
- RHSA1
= RHSB
- RHSA
RHSDC
= RHSD1
- RHSCl
= RHSD
- RHSC
(D. 3.3)
then these equations can be written in matrix form as


r .
Dlf
D2f
VxP
RHSBA
^Dls
D2s _
_V9P_
RHSDC
Solving this, the expressions for the longitudinal
velocities at a regular grid point are given by
(D.3.4)
and transverse
V = (D RHSBA D RHSDC) (D.3.5)
xP 2s 2f
V.n = -i- (D.. RHSDC D RHSBA)
6P A^ If Is
(D.3.6)


163
and the equations (B.4.1) and (B.4.2) for the characteristic lines
become
-2
Cf
(B.5.2)
and
-2 G
c =
S p
(B. 5.3)
which are
(B.5.4)
J
Also, for the elastic case we have iji = ij/ = i(r = 0 and the
x 0 0x
equations (B.4.7) and (B.4.8) along the characteristics further reduce.
The equation along c = c^ is
0 = I + dv + i da j i
I x -2 xl E
c_ P c
,v \
(-L)dt
E Vr /
o
(B.5.5)
along c
cs is 0 = + pcs dv0 + dT0x
(B.5.6)
along c = 0 is
aa
dt = ap dv
r r
(B.5.7)
and along c = 0 is
av
r ,, av a
dt = da + da
r E x E 8
where c^ and C£ are given by equation (B.5.4).
(B.5.8)


8
propagation of waves of combined stress. One of the first discussions
of combined stress wave propagation was given by Rakhmatulin (1958).
In this paper he developed the equations which must be solved for
elastic-plastic wave propagation under combined stress. Strain-rate
independent constitutive equations were used and only the problem for
the elastic case was solved. He found that the shear wave did not
affect the longitudinal wave in the elastic case. A similar discussion
of combined stress wave propagation was presented by Cristescu (1959).
Until now nothing has been said about the plasticity theory used.
ITie two plastic strain theories were the total strain theory proposed
by Hencky (1924) and the incremental strain theory proposed by Prandtl
(1924) and Reuss (1930). These two plasticity theories along with many
other developments in plasticity theory were presented in detail by
Hill (1950). The different plasticity theories were not presented
earlier because in many cases both theories gave the same results.
For instance, when a strain-rate independent constitutive equation was
used, the two plasticity theories led to identical results when one
dimensional (either longitudinal or torsional) stress wave propagation
was studied, when combined stresses were used if the loading was pro
portional, or even when unloading occurred in one-dimensional problems.
However, when strain-rate dependent material behavior of nonpropor
tional loading under combined stresses was considered, most investiga
tors used the incremental strain theory. Shammamy and Sidebottom
(1967) showed that the incremental strain theory more accurately pre
dicted the experimental results when various metal tubes were subjected
to nonproportional static loading in tension (compression) and torsion.


50
r
S 2aS.
I 2V + -i J
L r s
(s, A) IdT
+ 6
- 1 v
27 (A1A5 A2A3)
a5][} I."'1*]"
(3.2.8)
and
0 = -
2 2
(1 -v )
aA A -A A, dV
2 5 3 4J x 2
1 v
(-4) -AJdve
1 V
[(r4)-AJdT+ (r^s) [A2A5-A3AJdSx
1 V 1 V
2 % 2 p i -2S aS. -j
777) l^vvJL-t-- o 2 \ p / 2 v -t r S 2 aS A 1
7C) [i-" )(AlA5-A2A3>-A5][2Vr +^LT-? to'84]
i v 1 V
+ a
+ 6 (:
dT
-4) [(tAKv^-vI ¡} t0]dT
1 V 1 V
(3.2.9)
When the equations are uncoupled (A = A. =0), the equations along the
3 5
characteristics are given by equations (B.4.7) and (B.4.8). In dimen
sionless form, the equations (B.4.8) along the vertical characteristics
(c = 0) become
- a[2(a v2)Sd] dT = adV (3.2.10)
0 r
and
/2aSp S v
a[2VrdT] = a[A2dSx +A4dSQ+ { 1 -J *o(s,A)dT] .
(3.2.11)
The equations (B.4.7) along the nonvertical characteristics (c = c^)
for the fast waves are given by


4
modulus, in the strain-rate independent theory. Since this prediction
was quite different from that of the strain-rate independent theory
several investigators tried to verify one or the other. Bell (1951)
published the results of his experiments with aluminum which showed
that, for a bar stressed above the yield point, the initial strain
pulse propagated with the elastic wave velocity. These results were
in accordance with the strain-rate dependent model of Malvern (1951a,
1951b) as were the experimental results of Sternglass and Stuart (1953)
which were obtained using copper, Alter and Curtis (1956) which were
obtained using lead, Bell and Stein (1962) which were obtained using
aluminum, and Bianchi (1964) which were obtained using copper.
Encouraged by these experimental results, many investigators
continued the development of more general constitutive equations to
describe material behavior. Perzyna (1963) generalized the semi-linear
constitutive equation of Malvern (1951a, 1951b) to multiaxial states of
stress. At about this same time Cristescu (1964) introduced full quasi-
linear constitutive relations for a one-dimensional problem. This
quasilinear equation was used immediately by Lubliner (1964) to show
that the strain-rate independent constitutive equation of Taylor (1940),
von Karman (1942), and Rakhmatulin (1945), and the strain-rate depend
ent constitutive equation of Malvern (1951a, 1951b) and Sokolovsky
(194Sa, 1948b) were both special cases of this more general constitu
tive equation. Later Cristescu (1967a) gave a generalization for multi
dimensional stress states of the quasilinear constitutive equation as
well as an extensive summary of the developments in dynamic plasticity
until that time. Lindholm (1967) developed a constitutive equation
for combined stress states of aluminum which included strain-rate


56
characteristic lines very easy to obtain. The boundary lines for each
element are c = l This is the smallest value of c which insures that
all characteristic lines passing through the point P will intersect
the line L~B between the points L and B if the lines have a positive
slope at P or will intersect the line R-B between the points R and B
if they have a negative slope at P. This is true since all of the
waves considered here will propagate with a speed less than or equal
to the speed of an elastic longitudinal wave with radial inertia effects
included. A larger value of c could be used, but the element size
would increase (for a given distance along the T axis), and the solution
would be inherently less accurate.
The straight lines representing the elastic shear wave character
istic lines are added to the grid elements as a convenience. The
results of Section 3.1 show that the fast wave speed always occurs in
the range
C2 Cf C1
and the slow wave speed always occurs in the range
Therefore, these characteristic lines c = c divide each element so
that a characteristic line through P lies in one of the upper triangles
(P-L-LB or P R RB) if it is for a fast wave and in one of the lower
triangles (P B LB or P B RB) if it is for a slow wave. This is
shown for the regular grid elements in Figure 3.7. These characteristic
lines for the fast and slow waves will not, in general, be straight.


SYH = YS + H
DC = (SYH/BETA)*2
I F(CC.GT.DZ) GO TO 5
DC = 2.*H*DZ1/BETA + DY
5 CONTINUE
I F ( DGTCY)GO TG 6
F = YS
GO TO 8
b CONTINUE
IF(D.GT.DZ)GO TO 7
F = YS + EET A*(C 0Y)/(2.*DZI)
GO TO 8
7 CONTINUE
F = 01T A S O R T ( )
3 CONTINUE
CSTAR = H + XLAM*(D DC)
FM = F * OS TAR
IFIS.GT.FMJGO TC 9
P = 0.0
00 TC 10
0 CONTINUE
XA = XM + XN*SvjRT(D)
P = (3.*(C DY DSTAR + (XA/3.)**1.5)**(2./3.))/XA 1.0
10 CONTINUE
RETURN
END
229


35
values of v and 3. This program is listed in Appendix C. This pro
gram also calculated the values of the normalized stresses a /s,
x
a./s, and T /s as functions of v and 6. These results are given in
0 8x
Tables 1, 2, and 3. The wave speeds are shown in Figure 3.2 for
the case when 6 = -60, which corresponds to oa = 0. Also plotted in
Figure 3.2 are the results given by Clifton (1966). It is obvious
that the results are not the same and that including radial inertia
effects in the formulation of the problem can have significant effects
on the wave speeds and that, for any given state of stress, the waves
are always faster when radial inertia effects are included. The
results plotted in Figure 3.2 do not correspond to the case when
a = 0. That is, although crQ = 0 when 6 =-60 aQ does not necessarily
b 0
vanish for this case. When a = 0, the results obtained were identical
to those of Clifton (1866) as they should be, since a=0 corresponds
to the absence of radial inertia effects.
An interesting phenomenon can be observed by remembering that the
physical presence of radial inertia is due to the Poisson effect.
That is, the longitudinal (fast) wave speed would be expected to be
the same when a = 0 (no radial inertia effects) as when v = 0 (the cause
of the radial inertia effects vanishes). However, in the fomulation
of this problem it is tacitly assumed that Poisson's ratio for the
inelastic portion of the material behavior is i, or that the material
behavior in the inelastic range is incompressible. When the material
is elastic (3 = 1), this Poisson effect can be studied directly.
Comparing equations (3.1.15) with (B.5.1) when 3=1, the wave speeds
are given by equation (B.5.4) as


202
or
dA ,, ds ds
dT §(s,A) dT + 2ij.'o(s,A) + dT
and omitting the functional dependence, this is
dA _. ds
dT = <+1) di+ 2V
(E.3.2)
Along the vertical characteristics from point B to point P, this
equation can be written in finite difference form as
-| rs s
AP AB
2AT
-il
($p + 1) + ($B +
1} J [-W5]* I [2*op+2U
or,
, A r, ?p + -b
ap = ,sb + L1 +
0[Sp-SB] + [*oP+toB] W '
(E. 3. 3)
This expression is used to calculate the value of A at point P during
each iteration.
If A had been calculated from the plastic strain rates, it would
P *p
have been necessary to include expressions for e and e .
r e
E.4. Input Data
In each computer run, the grid size (AX and AT) must be specified
as well as Poisson's ratio (v), the rise time for the pulse at the
boundary (T = (JRISE 1)AT), the maximum number of grid points in the
X direction (MXMAX), and whether or not radial inertia effects and/or
strain-rate dependence is included. Other data which are used are
obtained directly from Lipkin and Clifton (1970) and Case XII of
Cristescu (1972).
When using the data of Lipkin and Clifton (1970), the generalized
stress-strain curv becomes


Longitudinal Velocity, V x10
Figure 4.7 Longitudinal Velocity Versus Time for Data Set 1 Without Radial Inertia
CD
CO


133
slightly different from the relaxation boundary, the relaxation boundary
will be used here for both the dynamic loading and the static preloading.
This relaxation boundary is defined as
r
Ee
if e s e
x y
1 _i
o = f (e ) = / a + g e 2 (e e ) if e s: s
x \ y 2 z xv V
x
x z
3 e2
V. x
if e e
z X
or as
r
X
-<
1
E CTx
l
2e2
(o a ) + e
,+ x y y
3^
V-
. 1 ,2
( O' )
p+ X
if O' =£ CT
x y
if a ^ a ^ o
y x z
if 0^0
Z X
(A. 2.20)
where o and a are stresses on the uniaxial stress-strain curve corre-
y z
sponding to the strains and respectively. The slope of this
curve is discontinuous at the point e = e This equation can be
x y
written in the form
e
x
1
E
o
x
+ X.
(a.
o ) + e
y y
i
E
(A.2.21)
where
Xr
if
o
£
a or
X
y
if
CT
<
q
A
Q
y
x :
if
o
<
a
X
z
if
o
a
x z


48
c =
c = c V (3.2.4)
S (
c = 0 (twice)
J
where the wave speeds were written in dimensionless form by dividing
the wave speeds by c^. This was done because c is the fastest wave
speed possible in the problem considered here, and thus all the
dimensionless wave speeds have values in the range
-l £ c <: l .
Mien radial inertia effects are not included, the maximum value of the
fast wave speed is
Cf
max
so that, for a = 0, the maximum value of the dimensionless wave speed is
cfl
max
V E/o
p (1 v )
Next, the equations along the characteristics for fully coupled
waves will be written in dimensionless form. Along the vertical char
acteristics (c = G), the equations can be written from equations (2.33),
(2.54) and (2.55) as
-aS0 dT
I
L
2(1
v2)
]
dV
r
or
- a [2(l-v2)S ] dT = adV
u r
(3. 2.5)


CT
r
Figure 2.2
Stresses on an Element of the Tube


CHAPTER 4
RESULTS AND DISCUSSION
4.1 Introduction
In Chapter 2 the problem of inelastic wave propagation was
formulated and the equations for this problem were found. In Chapter 3
these equations were written in finite difference form and from them
expressions for the stresses and the velocities at the points in the
numerical grid (Figure 3.4) were determined. Next a computer code
(shown in Appendix E) was written to facilitate the calculation of
the stresses, velocities, and strains at the grid points in the charac
teristic plane. Now, in this chapter the results obtained by using
this computer code will be discussed for several different combinations
of initial conditions and boundary conditions.
The computer code is written so that the boundary conditions are
specified by reading in values of two variables at each grid point
along the boundary (X=0). By specifying the boundary conditions in
this manner, any variable given as one of the boundary conditions can
have any functional shape. All of the data presented in this chapter
were obtained using the kinematic boundary conditions (Case II), that
is, by assigning values to the longitudinal velocity (V ) and the trans
verse velocity (VQ) at the impact end of the tube. Furthermore, the
same functional form was chosen for the two velocities in each case.
This form consists of assuming that each of the velocities at the
80


131
The slope of the curve defined in equation (A.2.13) is continuous for
n>l, and this equation becomes bilinear when n = l. The term may be
thought of as a "plastic modulus. From equation (A.2.13) the strains
are separable into the elastic and plastic portions
ct
e x
- T
P T^/ V11
e = B\ ct ct /
x X y
(A.2.15)
so that under uniaxial stress, the plastic work is given by
= J a. .deP. = f a deP = i s dP
ij xj J x x
ct* p
r*
= j a I Bn(a ct )n 1 da "1 = I sTb^s ct )
i XL X y xj o L.
v
n-1 dij
where o is the maximum stress on the "universal stress-strain curve"
which is reached during loading, and P .
W is zero. Thus
_* *
CT O ~
P p .n1 p n1
w = Bn s(s-o ds = ¡ Bn s(s a ) ds
ti y 0 y
OCT
y
and integrating this by parts
*
'"I-
W*3 =Hs(s-cr)n|_ B (s-CT__)nds
y CT
/ = a ,] B [j (J dy)ntlJ .
/ = bTct*(CT* CT..)n A- (CT*-CT_.)n+1l
n+1
J


Shear Stress, T x10
Figure 4.3 Grid Size Effects on the Stress Trajectories at X = 3.75
cc
Gi


198
Equation (2.16) defines the plastic part of the strain as
or
and from equation
ing becomes
s
(E.2.2), the plastic strain during the static preload-
P
e .
ij
s .
ij
(E. 2.4)


LIST OF FIGURES
Figure
2.1 Coordinate System for the Thin-Walled Tube
2.2 Stresses on an Element of the Tube ....
3.1 Yield Surface Representation in Spherical Coordinates .
3.2 Plastic Wave Speeds as Functions of [3 and y for
Poisson's Ratio of 0.30
3.3 Values of 8 at v= 0 for which c-c -c
Y f s 2
3.4 Numerical Grid in the Characteristic Plane
3.5 Regular Element in Numerical Grid
3.6 Boundary Element in Numerical Grid
3.7 Location of the Characteristic Lines Passing Through P .
3.8 Numerical Representation of the Characteristic Lines
in a Regular Element
3.9 Representation of the Characteristic Lines in
a Boundary Element
4.1 Grid Size Effects on the Longitudinal Strain
at X = 3.75
4.2 Grid Size Effects on the Longitudinal Velocity
at X = 3.75 .
4.3 Grid Size Effects on the Stress Trajectories
at X = 3 75
4.4 Longitudinal Strain Versus Time at X = 3.75
for Data Set 1 . .
4.5 Change in Shear Strain Versus Time at X = 3.75
for Data Set 1
Page
13
14
31
41
45
53
54
55
57
59
60
83
85
86
88
89
vii


197
where \ 'is a parameter along the static loading path and ij ij iJ
o
and s. are stresses which have not been made dimensionless. The incre-
ij
ment of plastic strain during this static loading is given by equation
(A.2.4) as
de. = s d\
ij ij
or
deP. = C(\)s?.d\
ij ij
and the plastic strain along the loading path becomes
P or1
e. = s. C(\)d\
ij iJ ^
where \ is a dummy variable, or
p = _H.
'ij C(\)
C(\)d\
and letting
U =
C(\)
C(\)d\
this plastic strain becomes
Now,
P
e. = cus .
13 ij
P P 2
e e = (ju s..s..
ij ij id ij
and from equation (2.15)
P P 2 2-2
e. e. = 77 ul) s
ij ij 3
or solving for (ju
P P
e. e. .
u = I
2 -2
s
(E. 2.2)
(E. 2.3)


186
the velocities at point P are given by
Vxp = (B2sRHS1 B2fRHS2)
4
V6P = A- 4
Case II: Kinematic boundary conditions
When V^p and V are known at a boundary point, by letting
and
RHS4 = RHSB1 -
RHS5 = RHSD1 EL V Bn V.
Is xP 2s 0P
the three equations for the stresses from equations (D.1.6), (D.
and (D.1.21) can be written as
D3f2TP + D4f2S*P B7fS6P = RHS4
+ D S aB S. = RHS5 \
3s2 P 4s2 xP 7s 0P /
aA T + aD S n + aD = aRHSH
5Q P 1 xP 2 0P J
When radial inertia effects are included (a=1) these equations
be written in matrix form as
D3f2
D4f2
B7f
r t n
p
" RHS4
D3s2
D4s2
"B7s
SXP
=
RHS5
1
>
Oi
D1
D2
_ s6p _
RHSH
_ _
so that with
A6 D3f2(D2D4s2 + D1B7s) D4f 2 (D2D3s2 + A5QB7s)
B7f(DlD3s2 A5QD4s2)
(D.5.5)
(D.5.6)
(D.5.7)
1.19),
(D. 5.8)
can
(D. 5. 9)
(D.5.10)


192
Case II: Kinematic boundary conditions
When V and V. are known, equation (D.2.15) becomes
xP SP
Tp = A-CRHSDE Z2V0p) (D.6.4)
If
RHS12 = RHSBEM .V _
If xP
then equation (D.2.13) is
F2fSxP + aF5fSGP = RHS12
(D.6.5)
(D.6.6)
and when radial inertia effects are included (a=l), equations
(D.6.6) and (D.2.3) can be written as
F
F_
S
RHS12
2f
5f
xP
D.
D
S.
RHSEEM
1
2
0P
(D.6.7)
By letting
J12
D F
2 2f
DlF5f
(D. 6. 8)
the expressions for the longitudinal and the hoop stresses can be
written as
S = -i-(D RHS12 F RHSEEM) (D.6.9)
xP A12 2 5f
S = ^(F-RHSEEM D..RHS12) (D.6.10)
6P 2f 1
When radial inertia effects are not included (a=0), equation
(D.2.3) vanishes along with S0p and V^p, and the expression for
is found from equation (D.6.6) to be
RHS12
(D.6.11)


H3F2 = R F S* ( 1. + 7.9)
G4S2 = 0V8 + 712
C4F2 = [)V 7 + 711
RHSB = 1 F V X RR B + B2F*VTRRG + B 3F* TAURK 4 B4F SXRRB +
1 A*(S5F* STRRB B6F *VRRRB )
RHS D = B 1SVXRBB 4 B2S* VTRBB 4 B 3S T AURBB 4 B4S*SXRBB +
1 A *(B5S*STRBB B6S VRR BB )
GO TC 333
382 CONTINUE
ARATIO = A2I/A4I
7 13 = 1 7Z3
714 = 7 7 ( 2 4 AARA TIU )
F2S = 1. 4 7 10
F3S = 1. 710
F1F =1./CFI
F2F = 713 4 714
F6F = 713 714
F3F = -7 7 *( I2A*ARATIU 4 1.)
F4F = -2. *75*ARATIO
F 5 F = F 3 F 4 Q 3 F 4 F
PHSBE = F1F*VXRRB 4 F6F SXRRB 4 A*{-F3F*STRRB 4 F4F *VRRRB)
RHSUE = 72*VTRBR 4 F 3S* T AURBB
KHSBEM = RHSBE 4 A*F4F*D3
PHSEt = (2.*VR8 PS I BB* l I 2A*STB SXB))*DELT 4 A2QSXB 4
RHSEeM = RHSEE 4 D4
383 CONTINUE
IF(MX.GT.1)CO TC 385
IF(RFSA.LT. 1.0E-06.OR.UNCOUP.GT.Q) GO TO 384
IFUASE.NE. 1 ) GU TU 500
RHSBM = RHSB C3F2*TAUI D4F2SXI
R FIS CM = RHSD C3S2*TAUI C4S2*SXI
R FIS EM = RHSH A5Q*TAUI D1*SXI
STI A*RHSEM/D2
RHS1 = RHSBM 4 A*(B7F*STI- B6F*D3)
A4Q*S TB
217


181
where
A1 = DlfD2s DlsD2f
(D.3.7)
Two equations involving only stresses can be obtained by adding
equations (D.1.18) and (D.1.19) and by adding equations (D.1.20) and
(D.1.21). Doing this and letting
D3f
2D3f2
D4f =
2D4f2
D7f =
2B7f
3s -
2D3s2
4s =
2D,
4s2
7s =
2B,
7s
RHSF = RHSA1 + RHSBl
RHSG = RHSC1 + RHSD1
these equations become
D3fTF + D4fSxP + aD7fS0P = RHSF
(D. 3.8)
(D. 3. 9)
D T + D S + aD = RHSG
3s P 4s xP 7s 0P
(D.3.10)
The third equation required to solve for the stresses is equation (D.1.6)
which is
aD S + aD S + aA T = aRHSH .
1 xP 2 0P 5Q P
When radial inertia effects are included, that is, when a=1,
these three equations can be written in the matrix form
D3f
D4f
D7f
T
P
-rhsf
D3s
4s
D7s
SxP
=
RHSG
_A5Q
D1
D2_
_SeF_
RHSH
(D.3.11)


Maximum Radial Velocity, V x10
3 6
Distance from Impact End, X
T
Figure 4.9 Maximum Radial Velocity Versus X for Data Set 1 With Radial Inertia
CD
05


Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy
INELASTIC WAVE PROPAGATION UNDER
COMBINED STRESS STATES
By
Charles Daniel Myers
August, 1973
Chairman: Dr. M. A. Eisenberg
Major Department: Engineering Science, Mechanics
and Aerospace Engineering
The purpose of this dissertation was to investigate the effects
of radial inertia and material strain-rate dependence on the propa
gation of inelastic waves of combined stress along a thin-walled tube.
A general quasilinear constitutive equation for multiaxial stress (and
strain) states was introduced. The equations of motion and the
strain-displacement equations, along with the constitutive equations,
were written to form a set of nine simultaneous hyperbolic, quasilinear,
partial differential equations. This set of equations was reduced to
a set of six equations which was then used to determine the expres
sions for the characteristic lines and the equations along the char
acteristic lines.
For combined torsional and longitudinal loading, two distinct
wave speeds were found. The values of these two wave speeds were
found as functions of the state of stress. Including radial inertia
effect in the formulation of the problem was shown to significantly
increase the wave speeds for a given stress state. Also certain
IX


I'-[ELASTIC WAVE PROPAGATION
UNDER COMBINED STRESS STATES
By
CHARLES DANIEL MYERS
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1973


200
so that
.Po
3
AP
(2
0-
o
<3
S
X
2
o
s
\3
o
s
X
Po
3
AP
/
(
1 s)
1
AP
0 "
2
o
s
V
3 x/
2
o
s
Po
:0x ~
3
2
APO
o
o
T
A
(E.2. 9)
J
and the total prestrains are then given by
o o Po
e = S + e
xxx
o o Po
e9 = vSx + Ee
o ... o Po
e0x <1 + V)T + e0x
>1
J
(E.2.10)
For the two stress-strain curves considered here, the expressions
for $(s,A) are given in equations (A.4.3) and (A.4.7). Therefore, the
initial values of §(s,A) are given by §(s) for these two cases as
x. o. / o vn-l
s(s) = Bn(s s )
and
(E.2.11)
i
(~2 i r- O |
-I**,
respectively.


183
the stresses at a regular grid point are given by
> = 4¡(D4sBHSF-D4fRHSG)
SxP = i¡ (D.3.18)
(D.3.19)
and the hoop stress, S ,-and the radial velocity, V are both zero.
H 0P J rP'
D.4. Solution at a Regular Grid Point
for Uncoupled Waves
For this case equations (D.2.3) and (D.2.12) to (D.2.15) must be
solved simultaneously. Once this is done V can be calculated from
equation (D.2.1). An expression for the shear stress is found by
adding equations (D.2.14) and (D.2.15) and dividing by to get
1 P
RHSCE + RHSDE
2F
(D.4.1)
2s
The tangential velocity is obtained by subtracting equation
(D.2.14) from equation (D.2.15) and dividing by 2Z^ to get
ep
RHSDE RHSCE
2Z
(D. 4.2)
By subtracting equation (D.2.12) from equation (D.2.13) and
dividing this by the expression for the longitudinal velocity
is found to be
V
RHSBEM RHSAEM
xP
2F
(D. 4. 3)
If


147
and regrouping the terms in this equation yields
r
-2. -
= rL(PC-)(A4A6-aA-) A
c
dv^ -pc
LaV5-vJdv
r -2 t t T t2 7 t 7 t2t t r r r2:
(pc)'
+ |_(pC) (A1A4A6-aA1A+aA2A3A5-aA2A6+aA2A3A5-;A4)-(14-a2)jdaj
+ [(Pi2)(W6-^5+W5-Â¥4V^5'V4V -(VVVV]adCTe
[(p;
+ ¡ (pc ) (A3A4A6-aA3A5+aA3A5-aA2A5A6+aA2A5A6-A3A4A6) (AgA4 aA2A5J d 0X
[<
+ | (p c2) (A4Ag-aA2) -aJ 4x dt a j^(p c) (A3A5-A2Ag)+A2 || l|r I dt
f (pc2)[aA2A5-A3A4][2^0J dt .
Using the characteristic equation (2.48), the above equation along these
nonvertical characteristics can be written in differential form as
0- r[-j
dvx-po|_aA2A5-3jdv0
-h [(pH2) (4 -a2)-jdax+ [a25-34JciTex
PC
[(pc2)(4A6-aX2)-4] [tlrjdt a[(Pc2)(35-26)+2] [--^Jdt
o
J[2*eJdt
-2
+ (p c )
aA2A5_A3A'
(B.2.11)
The second method for obtaining the equations along the nonvertical
characteristics may be shown as follows. Consider a curve in the x t
(characteristic) plane described parametrically by
x = x(§)
t = t(§)
(B.2.12)


71
stress and the longitudinal stress are given by equations (D.3.18) and
(D.3.19), respectively, as
TP = (D4s RHSF D4f RHSG) (3.5.7)
SxP = f At a Regular Grid Point for Uncoupled Waves
When the waves are uncoupled, the solution to the finite differ
ence equations has a much simpler form. In this case, the shear stress,
the transverse velocity, and the longitudinal velocity of equations
(D.4.1), (D.4.2), and (D.4.3), respectively, are
1
T -
P 2F
(RHSCE + RHSDE)
(3. 5. 9)
2s
V
6P 2Z
V
2
1
xP 2F
(RHSDE RHSCE)
(RHSBEM RHSAEM) .
(3.5.10)
(3.5.11)
If
When radial inertia effects are included, the longitudinal and
hoop stresses from equations (D.4.8) and (D.4.9) are
1
S = 7 (D RHS3 F RHSEEM)
xP A 2 5f 2
5
(3.5.12)
!ep r 5
(3.5.13)
and the radial velocity is again given by equation (3.5.6).
When radial inertia effects are not considered, both the hoop stress
and radial velocity vanish, and the longitudinal stress of equation (D.4.10)
is
S
xP
RHS3
F
2f 2
(3.5.14)


49
and
r- /2aSg S -j
a (2V dT) = a! AdS + A dS. + A dT + ( \¡J (s,A)dT .
r L 2 x 4 0 5 \s/ o J
(3.2.6)
The equations along the nonvertical characteristics (c = c_^, c^)
can be written in three different forms from equations (2.51), (2.52)
and (2.53). In dimensionless form these become, respectively,
2 i r* -1
0 =
c
1 V
x 2LaA2VA3\JdV9
1 V
+ [aA2A5"A3A4] dT+ ^2
c 1 v
5> -fllldSx
]
^ 1 -v
-| r-2S as
l_hJ vs'A,dT
/ 2 \ i p S 2aS
' ^)(A2A6 A3-V AJ [2Vr +
1 v
2
o(S,A)J
dT
+ 6
*)[
3A A A A
2 5 3
J[i o dT
(3.2.7)
and
0 =
I l(t5-2)(Vs-a3a5>-a2]dvx-1 [(r^KvAMj.
1 v 1 V
(A A A A ) A IdS
- aJ
, (AA AA
LA, 27 2 6 3 5
1 v
5> AJ J
-2S aSg
^c(s,A) |dT
,A)J'
(^) [(:
2' Lf (Aia6 A2) M-5; (a1 + a6) + i
1 v 1 V
]


188
Case III: Mixed boundary conditions
When S^p and VQp are known at a boundary point
RHS6 = RHSB1 B2fVep D4f2Sxp
RHS7 = RHSD1 B V D S
2s 0P 4s2 xP
RHS8 = RHSH D S
1 xP
(D.5.18)
so that equations (D.1.19), (D.1.21), and (D.1.6) can be written as
VxP + D3f2TP aB7fS0P = EHS6
B-> V + D T aB S_ = RHS7
Is xP 3s2 P 7s 6P
Vp + aD2sep = aRHSB y
CD. 5.19)
When radial inertia effects are included (a=l) these equations become
[hi
D3f2
_B7f
V
xP
RHS6
:
Bls
D3s2
"B7s
T
P
RHS7
L
A5Q
3 j
r
w
CD
"0
L
RHS8
(D.5.20)
and if
A8 Blf (D2D3s2 + A5QB7s) Bls('D2D3f2 + A5QB7f}
(D.5.21)
the expressions for the unknown variables become
V.. = j^RHS6(DD0_0 + A^BJ RHS7 (DnD0J,0 + A^B,,^)
xP A
8
2 3s2 5Q 7s
2 3f2 5Q 7i'
+ KHS8(D3s2B7f D312B,
*>]
(D.5.22)


90
has reached the value of 0.00085. For the simple wave solution this
time is approximately T-6 where for the finite rise time (T =4) this
time is approximately T=10. The difference in time when the fast wave
has passed can thus be accounted for by the finite rise time.
As the slow wave passes a point on the tube, the longitudinal com
pressive strain begins to increase to values larger than 0.00085. The
higher levels of strain (e ^ -.008) occur later (in the results given
here) than in the simple wave solution. Again this can be accounted for
by the finite rise time.
The inclusion of a finite rise time in the theoretical solution
gives results which resemble the experimental data more closely than
the simple wave solution. It can also be seen that including radial
inertia effects in the formulation of the problem gives longitudinal
strains which are somewhat closer to the experimental data than the
corresponding strains when radial inertia effects are ignored.
The change in shear strain versus time curve in Figure 4.5
exhibits the same rise time effect as the longitudinal strain. The
results obtained here are much closer to the experimental data than
the results for the simple wave solution. The final value of the
shear strain appears to be low. Since this shear strain is calculated
from the values of the transverse velocity, it may be that the final
value of the transverse velocity should be larger.
This can be seen more easily by examining the transverse velocity
at several distance from the impact end as shown in Figure 4.6.
A transverse velocity is induced when the tube is impacted with a
longitudinal velocity, if the tube is statically preloaded in torsion.
From Figure 4.6 it can be seen that the transverse velocity induced


CHAPTER 2
THEORETICAL DEVELOPMENT
The specific problem to be considered here is that of the propagation
of inelastic waves of combined stress along a semi-infinite thin-walled
tube, with the effects of radial inertia included. The material consti
tutive equation used is a generalization for multiple states of stress
of the quasilinear constitutive equation used by Cristescu (1972) for
a single stress component, and is a special case of the very general
quasilinear constitutive equation given by Cristescu (1967a). The coor
dinate system used is shown in Figure 2.1, and the stresses on an ele
ment of the tube are shown in Figure 2.2, where r^ is the mean radius of
the tube.
The problem is assumed to be axisymmetric so that there is no
dependence on 9. Since the tube considered is thin-walled, the stresses
a T and T are assumed to be negligibly small as are the strains
r r0 rx
and e The strain e is not included in the problem. Stability
r 0 rx r
of the tube wall and thermal effects are not included in the formulation
of the problem, and only small strains are used. The strain rate is
assumed separable into elastic, plastic, and visco-plastic parts. The
radial displacement is very small compared to the tube radius, and
plane sections of the tube remain plane. The material is assumed to be
isotropic and homogeneous, to obey the von Mises yield condition, and
to be isotropically work-hardening. All unloading is assumed to be
elastic.
12


177
terms together. Using equation (D.1.1), equation (3.4.9) along the
vertical characteristic (c=0), becomes
aVrP = a(D3-W which is the same as equation (D.1.3). Next, using equations
(D.1.1) and (D.1.5), defining
\
S 2aS. -i
L2 '
S 2aS
RHSEE = AT| 2V + ill
rB s_ toB.
B
J
A2QSxB + A4QS0B
and
(D. 2.2)
RHSEEM= RHSEE + 2T D
J J
and using equation (D.2.1) in equation (3.4.10), the other equation along
the vertical characteristic (c = 0) is
aD S + aD S = aRHSEEM (D.2.3)
1 xP 2 0P .
The finite difference equations along the nonvertical character
istic lines (c = c^, c^) will be simplified by using equations
(D.1.7) and (D.1.8) in equations (3.4.11) to (3.4.14) and noting that
A. is not zero so that the four equations reduce to
4i
v*p+[
1 + 2Z + aZ n
Z3 7 7 A4i-'
xP
aZv[1 + 2aS¡i]
A
S. + 2aZ_ -- V =
GP 5 A . rP
4i
V +
c xLLB
f z (2 + a S + aZ j 1 + 2a -Hi 1
LZ3 7 \ A4. /J xLLB 7L A4iJ
0LLB
- 2aZ i V
5 A . rLLB
4i
(D.2.4)


82
with the experimental and theoretical results of Lipkin and Clifton
(1970). The data which will be used are
o
T = initial static shear stress = 3480 psi
o
a = initial static longitudinal stress = 0
x
vx = final longitudinal boundary velocity = 500 ips
v6f
final transverse boundary velocity = 23 ips
t
R
rise time = 9.6 p sec
which can be written in terms of the dimensionless quantities for input
to the computer code as
6x
= .0003480
= .002404
= 0
V.
= .0001106
T = t
R 2r R
= 4.00 .
o
No radial inertia effects or rate dependence will be considered in
this section.
The results from three different computer runs will now be made.
Each computer run used these initial conditions and boundary conditions
but had different grid sizes. The three grid sizes used were
AX = AT = .25, AX = AT = .125, and AX=AT= .05. The longitudinal strain
versus time obtained by using the computer code in Appendix E is shown
in Figure 4.1 along with the experimental results and the simple wave
solution of Lipkin and Clifton (1970). From this it can be seen that


155
- | [(pE2)(26-A35)-2]dvj!4 [<1VV3>-5]dve
> A,A !
-2
P c
i r ~2 -
l''V'lf3' 5j u 9x + ~=2 L(p C ) A2A6" VV _A2J dCTx
(pe2) (1 -A AJ-.4 I d
[(PE2)(A2A6-A3A5)-A2]?xdt+ 2 [(DE2)(A1A5-A2A3)-A5]^0xdt
~2 j^(p c')2(AiA6-2) (pe2) (A. T
pe
i+V+1] (/^e)dt (B-2-24)
and,
2-3
O = pc~ [j^2A5-A3A4J dvx pe ^(pc2)(A1
Jdve
[^(p C2) (A1A4-a2) - J dTfa + -2
5-vJ
r
(PC >LaA2A3-A3A4J^x
J
-2 2
+ (p c )
,aA2A5
t dt + 2(p c)
[(pc2)(A1A4-aA2)-A4]1J/exdt
+ ap^^p^XA^-^-j (^-*e)dt
(B.2.25)
Equations (B.2.23), (B.2.24), and (B.2.25) are all forms of the same
equation when the waves are fully coupled. Uncoupled waves will be
discussed in Section B.4.
The equations along the vertical characteristics (2.47) can be
written directly from equations (2.25) and (2.35), since along these
characteristics only t varies (x is constant) and partial differential
equations with all partial differentiation with respect to t become
ordinary differential equations. Thus, along the vertical character
istics of equation (2.47), equations (2.25) and (2.35) may be written as
a ov
and
'a dv
0 r
= St
aVr dCTx dCTfi dTPx
= aA + aA - + aA
r 2 dt 4 dt
o
s dt + a*e


191
B2f
4f2
" V
RHS9
_ B2s
D4s2
s
xP
RHS10
so that with
11 B2fD4s2
B2sD4f2
the solution is given by
V
S
0P
xP
4s2
RHS9
2f
RHS10
D4f2RHS10)
B RHS9)
2s
(D. 5.36)
(D. 5.37)
(D. 5. 38)
(D. 5.39)
D.6. Solution at a Boundary Point (X=0)
for Uncoupled Waves
When the waves are uncoupled, the solution at a boundary grid point
is obtained by specifying two of the variables at the point and then
solving simultaneously the three equations (D.2.3), (D.2.13), and
(D.2.15). This will be done below for the four cases discussed in
Chapter 3. In all cases the radial velocity is calculated from equa
tion (D.2.1) after the hoop stress is found.
Case I: Traction boundary conditions
When S^p and are known, equations (D.2.3) and (D.2.15) yield
the expressions
S6P = ^-(RHSEEM ~ Vxp) (D.6.1)
V0p = ^-(RHSDE F2sV (D.6.2)
2
and once is known, the longitudinal velocity is found from equa
tion (D.2.13) to be
V = (RHSBEM 2F S ^ aF S0,_) .
xP F 2 xP 5f 9P
(D.6.3)


18
given by Cristescu (1967a). The form of equation (2.14) was chosen as
the general constitutive equation because it contains terms which may be
considered separately as elastic, plastic, and visco-plastic strain-
rate terms, because the inelastic strain-rate tensor is proportional
to the corresponding deviatoric stress tensor, and because it reduces
to the form of equation (2.13) when the only stress present is the longi
tudinal stress. This simplification to the form of equation (2.13) is
shown in Appendix A.
The functions 0(s,A) and i(r(s,A) are functions which depend on the
particular material being studied. The function 0(s,A) is a measure of
the rate insensitive inelastic work-hardening, and the function iji(s,A)
is a measure of the visco-plastic strain rate due to the strain-rate
sensitivity of the material. In the classical rate independent plastic-
- - *
ity theory, i|i(s,A) vanishes. When s < 0 or when s < u (the current
"yield stress"), 0(s,) is set equal to zero. The unloading conditions
when i|r(s,A) = 0 are stated in equation (A. 3.1).
Two separate materials are modeled in the numerical work done.
One is a 3003-H 14 aluminum alloy used in the experimental work of
Lipkin and Clifton (1970). This material is assumed to be insensitive
to strain rate and the functions 0(s,A) and i)/(s,A) are obtained using
the classical Prandtl-Reuss incremental plasticity theory with iso
tropic work-hardening and the stress-strain curve for uniaxial tension.
(See Appendix A.) The other material used is a commercially pure
aluminum dead annealed at 1100F. This material is assumed to be
strain-rate sensitive, and the functions 0(s,A) and jf(s,A) are
obtained from the data given by Cristescu (1972). (See Appendix A.)


156
of in differential form these become
e
dt = ap dv
r
r
o
(B.2.26)
B.3. Reducing Equations to Simpler Case
When radial inertia effects are not included and when the functions
0(s,A) and \|f(s,) are obtained from the rate independent incremental
plasticity theory, then the equation (B.1.2) for the characteristics
and the equations (B.2.23) and (B.2.25) along the characteristics
reduce to those given by Clifton (1966) for the von Mises yield condi
tion as will be shown below.
Under these conditions, equation (A.1.2) becomes
-2
(B.3.1)
s
and Clifton's expression for k can be written as
Now Clifton defined the function H(k) for the von Mises yield condition
as
H (k)
1
3


16
e
rx
e
ex
= i(u +
u
)
2 x, r
I
>x
= x-(u. +
1
u
2 0,x
r
X
CD
)
and under the above restrictions, these reduce to the following three
equations
Defining
respectively,
0 =
X
u
x,x
(2.4)
e0 =
1
u
r r
o
(2.5)
0x =
1

2 0,x
(2.6)
the velocities v ,
X
v and v. as u u ,
r 0 x,t r,t
and U0,t
equations
(2.1) to
(2.6) become
o =
X X
p v ,
x, t
(2.7)
1!
CD
t>
-u
l
P Vr,t
(2.8)
T0X,X
P V0,t
(2.9)
£X, t
V
X, X
(2.10)
e0,t =
1
v
r r
o
(2.11)
60x,t
1
V
2 0, x

(2.12)
Under the assumptions used here, the variables no longer depend on r,
so that the problem becomes two-dimensional (the independent variables
are x and t) and can be solved by the method of characteristics.
The equations necessary for completion of the set of simultaneous
partial differential equations describing the behavior of the body are


53
T
Figure 3.4 Numerical Grid in the Characteristic Plane


21
1 (2ox aCTfi)2
AX = E + =2 0(S'A)
4s
A2 = ~ Le +
(2ax aCT0> (ax 2acG)
1
0(s,A) J
- 6Tex<2ox V
A3 = 0(s,)
4s
1 (CTX 2aCTfi)2
A4 = E + ^ 0(sA)
4s
r6Tex(CTx ~ 2aCT8)
4s2
0 ( S
,A)J
- 2(1 +v) 36T0x .x
A6 = + ~=2 0(SA)
4s
the nine simultaneous equations (2.7) to (2.12) and equations
(2.22) can be written as
cr = p v
X, X r X, t
T- CTe = ^r.t
O
6x,x ^V0,t
e = v
x,t x,x
ae
9, t
a
v
r r
o
0X, t
1
V
2 8, x
'x, t
2x aCTe
+ Vex.t +
ae
8, t
2ao CTX
V*,t + Ve.t + Vex.t + a T
' r\
ox, t
2 A3x,t + 2 A5CT8,t 2 A6'8x,t
3t
6x
+ i|Ks>A)
(2.23)
(2.20) to
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
(2.31)
(2.32)
2s


235
Sternglass, E. J. and Stuart, E. J. (1953), '"An Experimental Study of
the Propagation of Transient Longitudinal Deformations in Elasto-
plastic Media," Journal of Applied Mechanics, Vol. 20, pp. 427-434.
Suliciu, I., Malvern, L. E. and Cristescu, N. (1972), "Remarks Concern
ing the 'Plateau' in Dynamic Plasticity," Archiwum Mechaniki
Stosowanej, Vol. 24, pp. 999-1011.
Tapley, B. D. and Plass, H. J., Jr. (1961), "The Propagation of Plastic
Waves in a Semi-Infinite Cylinder of a Strain-Rate-Dependent
Material," Developments in Mechanics, Vol. 1 (J. E. Lay and
L. E. Malvern, Eds.), Amsterdam: North-Holland, pp. 256-267.
Taylor, G. I, (1940), "Propagation of Earth Waves from an Explosion,"
British Official Report, R. C. 70.
Wood, E. R. and Phillips, A. (1967), "On the Theory of Plastic Wave
Propagation in a Bar," Journal of the Mechanics and Physics of
Solids, Vol. 15, pp. 241-254.
Yew, C. H. and Richardson, H. A., Jr. (1969), "The Strain-Rate Effect
in the Incremental Plastic Wave in Copper," Experimental Mechanics,
Vol. 9, pp. 366-373.


72
At a Boundary Point (X=0) for Fully Coupled Waves
In a boundary element, there are only four characteristic lines
(c = 0, c = 0, c = -c c = -c ) and consequently only four equations along
S X '
these characteristic lines. Since the equations along the character
istic lines are written in terms of six unknown variables at point P,
the solution at a boundary point can be obtained only if two of these
variables are prescribed at each boundary point. The hoop stress and
the radial velocity do not enter the formulation of the problem when
radial inertia effects are omitted, and therefore these variables are
not specified at the boundary. Thus, the four remaining variables, two
of which may be specified at any boundary point, are the longitudinal
stress, the longitudinal velocity, the shear stress, and the transverse
velocity. From a purely physical standpoint, it is also reasonable to
specify the longitudinal and transverse variables at the boundary since
these are the quantities which are normally associated with the impact
at the end of the tube and which can be measured more readily than
radial velocity and hoop stress. Only two of the four variables S ,
V T and V can be specified at any one boundary point. Furthermore,
X 0
at a given boundary point V and S cannot both be specified since they
are not independent. Also, both T and V cannot be given at the same
o
boundary point. Therefore, four combinations of variables to be speci
fied on the boundary will be considered: for Case I, and T will be
given at the boundary, for Case II, V and VQ will be given, for Case III,
S and V will be given, and for Case IV, V and T will be given. The
X 0 X
solution to the finite difference equations at a boundary point for each
of these four cases when the waves are fully coupled is given below.


S BO = SORT (SXO* 2 + 3.*TAUC**2)
CALL PH II PHIP,PHIQtOJBELL, SBOt DCLTAO SYS,BBAR,N,H,XLAM,BETA,DY,XM
1fXNtDZ,0)
IF(S BO.GT .1. E-0 8 ) GO TO 60
FI = 0.J
F 2 = 0.0
GO TO 65
60 CUNUNUL
Fl = sxo/suo
F 2 = TAUO/SBC
66 CONTINUE
G1 = PHIP*F1**2
G2 PHIP *F 2 * 2
C 3 = PHIPF1*F2
A10 = I. + GL
A 2 0 = -(NO 5C*G1 )
A 30 = 3 G3
A 40 = I. + 2 6 G1
A50 = -1.5 G 3
A6'j = Q l + 9 G 2
A 2 0 2 = A 2 C * 2
A 5 0 2 = A 5 ~ 2
IHUNCOOP.GT .0) GO TO 67
AO = A40(A10*Ao''-A30**2)+A*(2.*A20*A30*A5C-Ain*A502-A6,,*A2 0?)
P. = A 4 3 ( A 1 0 + A 60 ) A A 5 C 2 + A2G2)
COE F = 02/AO
PCCr = BC * 2 4. A 4 G A 0
RAO = SORT!ROOT)
CFO = SORT{CEF *(BO + RAO))
CSJ = SORT! COEFM BO RAD))
GO TO 6B
67 CONTINUE
CFO S0RT(0A4?/(A10*A40 A*A202)>
CSO = SORT!0/A60 )
207


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
LIST OF TABLES vi
LIST OF FIGURES vii
ABSTRACT ix
CHAPTER 1. INTRODUCTION 1
CHAPTER 2. THEORETICAL DEVELOPMENT 12
CHAPTER 3. DIMENSIONLESS EQUATIONS AND NUMERICAL PROCEDURES . 29
3.1. Wave Speeds as a Function of the State of Stress .... 29
3.2. Characteristic Solution in Terms of
Dimensionless Variables 46
3.3. Numerical Grid for Characteristic Solution 51
3.4. Finite Difference Equations ... 58
3.5. Solution to the Finite Difference Equations 69
3.6. Calculation of the Strains ..... 77
CHAPTER 4. RESULTS AND DISCUSSION 80
4.1. Introduction 80
4.2. Grid Size Effects 81
4.3. Effects of Radial Inertia 84
4.4. Effects of Strain-Rate Dependence 110
CHAPTER 5. SUMMARY 118
APPENDIX A. CONSTITUTIVE EQUATIONS 125
A. 1. Comments on the Constitutive Equation 125
A.2. Rate Independent Incremental Plasticity Theory ..... 127
A. 3. Rate Dependent Plasticity Theory
A.4. Dimensionless Expressions for the Functions
0(s,&) and iji(s,A) 137
IV


92
by the longitudinal velocity is much greater than the transverse
velocity specified at the boundary. Therefore, the transverse veloc
ity specified at the boundary (impact end of the tube) may be too low.
The longitudinal velocity is plotted in Figure 4.7 and the
longitudinal strain in Figure 4.8. Both of these quantities have a
region (at distances from the impact end greater than 2.0 diameters)
where the value is constant. This corresponds to the interval of time
between the passing of the fast wave and the arrival of the slow wave.
It takes 2.0 diameters for the two waves to become distinct because of
the finite rise time of the velocities at the boundary. That is, the
fast wave originating at the boundary at the time T = T^ must overtake
the slow wave originating at the boundary at the time T=0 before the
two waves become completely separated.
The effects of radial inertia are most evident in the longitudinal
strain. This can be seen in Figure 4.8 where the longitudinal strain
at the impact end is about 25 percent greater when radial inertia is
included than when it is not included. Qualitatively this is the result
that would be expected. Under compressive loading, the tube is allowed
to expand instantly when radial inertia effects are not included. How
ever, the presence of radial inertia initially resists expansion of the
tube (thereby decreasing the value of the longitudinal strain at any
given time) and then continues the radial expansion beyond the equi
librium point (causing the longitudinal strain to increase beyond the
equilibrium value). This behavior is evident in Figure 4.8. The
maximum value of the longitudinal strain when radial inertia effects
are included approaches the same value as when radial inertia effects
are not included at distances greater than 2.0 diameters from the


51
(3.2.12)
and the equations (B.4.7) along the nonvertical characteristics
(c = c ) for the slow waves can be written as
s
0 = D
6c
(1 -v2)
dV. + dT +
V
(1
2
V )
(-) t (s,A)dT
S o
']
(3.2.13)
The equations (3.2.5), (3.2.6), and (3.2.7), or (3.2.8), or
(3.2.9) are the equations along the characteristics for the fully
coupled waves written in differential form in terms of the dimension
less variables. The solution to this set of equations will be obtained
numerically by writing them in finite difference form, and then solving
the resulting set of algebraic equations simultaneously. When the
waves are uncoupled, the equations along the characteristics are given
by equations (3.2.10), (3.2.11), (3.2.12), and (3.2.13). These equa
tions will also be written in finite difference form and solved (when
applicable) in the same manner as described for fully coupled waves.
The procedure for obtaining these finite difference solutions is out
lined in the next sections.
3.3 Numerical Grid for Characteristic Solution
Since the slopes of the characteristic lines at any point in the
characteristic plane depend upon the state of stress at that point and
upon the history of the deformation at the corresponding location along
the axis of the tube, the equations for the characteristic lines cannot


20
and equations (2.18) become
0(s,A)J
e =2 +
n (2ax ~ ae)2
x,t
4s
Tv (2cjxCTe)( ax,t-lj + ^2 0(SA)J CTe,t
4s"
-6(2a cr )tc (2a aQ)
~-2 0 (2.20)
4s
2s
p (2a ac) (a 2aQ)
! v x G x 0
" 8, t
L
E
0 (s
4s
A)]ax,
m + 0(S,A) JCTg
4 s2
r6(2a0 -CTy)T(
+ L
0(s
,A)]
0X, t
2a0 ax
+ - j(s,A)
2s
(2.21)
Gx, t _
3t (2a a)
0x x 0
-2
4s
1 +
18t\
yx
0 ( S
0(S
,A)]
,A)]
CT +
X, t
'-3Tex(2£70-CJx)
4;2
0(S
'4)]V
T0X,t +
0X
\|r(s,A)
(2.22)
4s -1 2s
The equations (2.7) to (2.12) and equations (2.20) to (2.22) form
a set of nine simultaneous hyperbolic quasilinear partial differential
equations for the nine unknowns o x 0 bx x t x b
e A special case of this system of equations is the set of equations
6x
obtained by neglecting radial inertia effects. When radial inertia
effects are ignored, the variables a, s and v are not included
0 0 T
directly in the problem formulation. This case can be incorporated
into the more general formulation by multiplying a., s v and their
0 0 r
derivatives by the dummy variable "a," where "a" has the value of 1
when radial inertia effects are included and the value of 0 when radial
inertia effects are neglected. Also the equation of motion, the kine
matic equation and the constitutive equation for motion in the radial
direction must be multiplied by "a." Doing this, and defining the
quantities


119
for which the two wave speeds (for the fast wave and the slow wave)
were identical. These critical values of the tangent modulus were
found for several values of Poisson's ratio. When radial inertia
effects were not included, the values obtained when v = .30 was the
same as that given by Clifton (1966). This critical value of the
tangent modulus was always smaller when radial inertia effects were
included (with the hoop stress set equal to zero) than when radial
inertia effects were not included.
Next the solution for this wave propagation problem in the
characteristic plane was found. In order to do this, it was necessary
to use numerical computation methods. To make the numerical solution
more general, the variables used were defined in terms of dimensionless
quantities, and then the equations for the characteristics and the
equations along these characteristics were written in terms of these
dimensionless variables. A uniform numerical grid was established and
the equations along the characteristics were written in finite differ
ence form for each grid element. These finite difference equations
were then solved to find expressions for the stresses and the veloc
ities at each point in the numerical grid. The strains were calculated
at each grid point by using the values of the velocities. A computer
code was written to help solve this wave propagation problem for differ
ent sets of initial conditions and boundary conditions. All of the
numerical results were obtained by specifying the initial static pre
stress in the tube and then giving the values to the velocities at the
impact end.


Longitudinal Strain,
Time, T
Figure 4.22 Longitudinal Strain Versus Time for Data Set 3
114


69
along c = is
0 = A
4i
.c (AxP 1'xRRB) + 2 (SxP~SxRRB)
f c.
ill
f
(2S aS + 2S
xP GP xRRB
2aS )} 1
0RRB J ll + C J J
+ aA
i|_^ rP + ^ rRRB T 'll2^SxP 2aS6PTSxRRB 2aS0RRB^/ J Ll +
(3.4.12)
i "j r 2at i
along c = + cg is
0 = -
1 -v
2
2 + (TP TLBB)
^4ri<
+ 2 [_2(TP + lLBB')
1 v
][iff]
(3.4.13)
along c = c is
s
0 =
1 -v
2 (V6P V 0RBB) + (TP TRBB)
6c2i1(
s
1 V
I [I (3.4.14)
3.5 Solution to the Finite Difference Equations
The solution to the finite difference equations of Section 3.4 are
given here for any iteration. The solutions consist of expressions for
Vxp, vgp< vrp Sxp, Sgp, and in terms of known quantities, including
quantities calculated during a previous iteration. The solutions given
in this section are obtained using Cramer's rule as shown in Appendix D,
and the definitions of the variables used in Appendix D will not be
repeated here.


178
v + + Z (2 + a ~ )ls aZ j 1
XP LZq 7 V A4i /J xP 7 L
A 1 A .
2aA-iJsep+2aZ-^v
+ 2a
'4i'
5 A, rP
4i
V + i Z ^2 + a ~ ') I S + aZ
cf xRRB \_Z3 7 V A4i /J xRRB 7
1 S + aZ 1 + 2a -=il
yj xRRB 7 L A, J
4i
0RRB
2 aZ - V
5 A rRRB
4i
- z2 Vep + <1 + Z10,TP = Z2Ve.,BB+ (1 -zio>tlbb
Z2VSP + (1+Z10)TP= VeRBB+ (1-Z10)TRBB
By letting
Flf c
2f
3f
Z7 P + 2a A~
4i
F4f ^ 22
5 A
2i
4i
F = F + Q F
5f 3f ^3 4f
F = 7 (2 + a
6f z3 7l a4.
>
and
F2s = 1 + Z10
F3s = 1 Z10
(D. 2. 5)
(D. 2.6)
(D.2.7)
(D. 2. 8)
(D.2.9)


150
or as
where
MZ = d
(B.2.15)
M =
Op 0000 -1 00000
OOOapOOOOOOOO
OOOOOp 0000 -1 0
-1 000000A OaAOA
12 3
0 0 0 0 0 0 0 aA 0 aA 0 aA_
2 4 5
0 0 0 0 -1 0 0 A 0 aA_ 0 A
3 5 6
x,£ t,g 0000000000
0 0 ax, a^§ 0 0 0 0 0 0 0
0 0 0 0 x,g t,g 0 0 0 0 0 0
0 0 0 0 0 0 x,g t,g 0 0 0 0
0 0 0 0 0 0 0 0 ax,^ at,- 0 0
0000000000
XS *§
(B.2.16)
z = 1v v 4.t v v V. V. | a a *, aQ *, t .
x,x vx,t vr,x 'r,t v0,x v0,t ^x,x wx,t u0,x w0,t 0x,x T0x,1j
(B.2.17)
and
L aCTe r r N\ dVx dVr dV0 dax da0 dT0x
d = |o, o, -V H--V> _2^9x "Hr a^r 1 "dT dT a-dT^T
o o
(B.2.18)


60 CONTINUE
EXPO = SXO*PHIQ
ETP3 = -.5*A*EXP1
C PL A SO = PEL TAG SBO
ETXPO = 1.5*TAUP*PHIQ
E Xu = SXO + EXPO
ETu = A*lET PO NU*SXO)
ETX1 = Q A T A U C * ETXPO
CG 7 0 MX = 1MXMAX
T(MX ) = (MX 1 ) *OELT
TAU(MX) = TAUO
SX(MX ) = SXO
ST(MX ) = 0.0
VX(MX) = 0.0
VT(MX) = 0.0
VR(MX) = 0.0
SB(MX) = SB.)
CF(MX) = CFG
CS(VX) = CSu
A 1(MX) = All
AP(MX) = A2 0
A3(MX) = A30
A 4(MX) = A41
AS(MX) = A 50
A6(MX ) = A6 0
EX(MX) = HXO
cT(MX) = ETC
ETX(MX) = FJXO
EXP(MX) = EXPO
ETP(MX) = E T PO
ETXP(MX) = ETXPO
ITER(MX) 1
PHIO(MX) = PH IP
PSIO(MX) = 0.0
208


148
and assume that the six variables CT ct, T. v v and v. are known
x 8 0x x r 8
along this curve as functions of the parameter §. Then the derivatives
of these functions can be computed as
dv
"Hr = vx,x x§ + vx,t t$
dve
dg ve,x xg + ve,t
dv.
r
= v x,_ + v t, _
r,x § r,t §
d a
x
-jr* = ct x + CT t,-
dg x,x I x,t §
da.
"5T CTe,x x,g + CTe,t ti
dT
6x
dc~ = T6x,x X§ + T8x,t tfg
(B.2.13)
where the subscript following the comma denotes differentiation with
respect to that variable, i.e.,
v
x, x
9v
x
=
V
X,
dv
X
t ~ ~5F
dx
§ dg
The six equations (B.2.13) and the equations (2.24), (2.25), (2.26),
(2.34), (2.35), and (2.36) form a set of twelve simultaneous partial
differential equations which can be written in matrix form as


37
TABLE 1 (Continued)
Gamma
Delta
50
60
O
O
t>
00
; o
o
90
-90
0.37111
0.28868
0.19747
0.10026
0.00000
1
00
o
0
0.47710
0.37111
0.25386
0.12889
0;00000
l
<1
o
o
0.56858
0.44228
0.30253
0.15360
0.00000
-60
0.64279
0.50000
0.34202
0.17365
0.00000
o
o
m
!
0.69747
0.54253
0.37111
0.18842
0.00000
o
O
1
0.73095
0.56858
0.38893
0.19747
0.00000
I
CO
o
o
0.74223
0.57735
0.39493
0.20051
0.00000
o
o
Cj3
0.73095
0.56858
0.38893
0.19747
0.00000
-10
0.69747
0.54253
0.37111
0.18842
0.00000
0
0.64279
0.50000
0.34202
0.17365
0.00000
O
o
0.56858
0.44228
0.30253
0.15360
0.00000
20
0.47709
0.37111
0.25386
0.12889
0.00000
30
0.37111
0.28867
0.19747
0.10026
0.00000
O
O
0.25386
0.19747
0.13507
0.06858
0.00000
50
0.12889
0.10026
0.06858
0.03482
0.00000
O
O
CD
0.00000
0.00000
0.00000
0.00000
0.00000
o
O
-0.12889
-0.10026
-0.06858
-0.03482
0.00000
00
o
o
-0.25386
-0.19747
-0.13507
-0.06858
0.00000
CD
O
o
-0.37111
-0.28868
-0.19747
-0.10026
0.00000


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
R. C. Fluck
Associate Professor of Agricultural
Engineering
This dissertation was submitted to the Dean of the College of Engineering
and to the Graduate Council, and was accepted as partial fulfillment
of the requirements for the degree of Doctor of Philosophy.
August, 1973
Dean', College of Engineering
Dean, Graduate School


APPENDICES


11
This concludes a brief survey of the history of the development
of plastic wave propagation theory. No attempt was made to give
a complete historical background. For more information the reader is
directed to Hopkins (1961), Kolsky (1963), Olszak, Mroz, and Perzyna
(1963), and Cristescu (1967a, 1968).
The remainder of this dissertation will be devoted to solving
the problem of combined stress wave propagation in a thin-walled tube
when radial inertia effects are included. A general quasilinear
constitutive equation for multiple states of stress will be presented,
and it will be shown to be a generalization of the constitutive equa
tions of both Lipkin and Clifton (1970) and Cristescu (1972). But
first the wave propagation problem itself must be developed.


Shear Strain,
Figure 4.20 Shear Strain Versus Time for Data Set 2
111


03
Figure 2.1
Coordinate System for the Thin-Walied Tube


54
T
Figure 3.5
Regular Element in Numerical Grid


175
Now substituting equations (D.1.3) and (D.1.16) into equations (D.1.12)
to (D.1.15) and letting
B7f B5f + Q3B6f
B = B + Q B
7s 5s *3 6s
RHSA1 = RHSA aD B
3 6f
RHSBl = RHSB aD^B
3 6i
RHSCl = RHSC aD B_
3 6s
RHSD1 = RHSD aD B. ,
3 6s s
(D.1.17)
the four equations along the nonvertical characteristics can finally
be written in the form given below. The equation
along c = + c^ is
BlfV B2fV + D312TP + D4f2SxP aB7£S8P EHSA1
(D.1.18)
along c = c^ is
Blf'xp + B2f V + D3f2Tp + D4f2SXP aB7fS0P = RHSB1
(D.1.19)
along c = + c^ is
- B V B V + D_ T + D S aB = RHSCl
Is xP 2s 0P 3s2 P 4s2 xP 7s 0P
(D.1.20)
and along c = c^ is
BlsVxP + B2sV0P + D3s2TP + D4s2SxP aB7sS0P
RHSDl .
(D.1.21)


134
Expressions for the plastic work for uniaxial loading (s = ct ) will
x
be found separately in the three different iegions. First
= 0
if a £ a
(A. 2.22)
When ct < ct < ct the plastic work is
y z
l
-2e2
p p /. p r z ii
W = I a. .de. = 1 ct de = J X, ct ~ do
ij ij ^ x x ^ 1 X L Q+ EJ
X X
o
X
p
W
and
dw
F' (U?)
dF
dW1"
dc
*
f2 6
L
z
+
a*-
(A. 2.23)
When ct ^ ct the plastic work is
z
2ct
f r* f r* x 1
W = J O' d = f CT x0 ( -) dCT
^ x x J x 2 n+2 E >
/ =
*2
[-
L30
2 *3 3. 1 *2 2
72 3
and
dW
/ P
F (W) dCT
* ^2
*
' %
1
(A. 2.24)
Now, using equations (A.2.23) and (A.2.24) in equation (A.2.12) and
comparing the results with equation (A.1.1), the expressions for the
functions 0(s,A) and ijr(s,A) become


25
If the wave speeds in equation (2.48) are denoted by
(2.49)
and
b (b'" 4aA^)3
'}]
l
s
(2.50)
where is the fast wave speed and is the slow wave speed, then
the slopes of the characteristic lines are given by
(twice)
c = c
f
c
c
s
Equations (2.47) and (2.48) are the six equations for the character
istic lines for the set of six simultaneous, hyperbolic, quasilinear
partial differential equations of (2.38). When radial inertia effects
are not included (that is, when a=0), the equations (2.47) vanish
identically and the remaining four simultaneous equations of (2.38) have
the characteristics given by equations (2.48). For this case (a=0) and
when 0(s,A) is obtained from incremental, rate independent plasticity
theory with isotropic work-hardening, equations (2.48) reduce to those
given by Clifton (1966) as shown in Appendix B.
The equations along the characteristics can be obtained in two
different ways, both of which are discussed in Appendix B. The result
ing equations along the characteristic lines of equation (2.48) are


IB = Al (MX)
3n 1
302
A2B = A2(MX)
A 3 B A3(MX)
A 4 B A4(MX)
A 5 B = A5 (MX)
A6B = A6(MX)
SBB = SB(MX)
P S I 3 = PSIO(MX)
PMIB = PHIO(MX)
SMAXB = SMAX(MX)
LEL TAB = DELT A(MX )
1F(SD3.LT.l.UE-08)G0 TU 301
PSI3B = PSIB/SBC
GO TG 332
CONTINUE
PSIBB =0.0
CONTINUE
PSIBC = 375 *PSIBB
C3 = VR13 U3*STB
C4 = CELT2*D3
A 1 0 C :
A 2 B C =
A3BC =
A4BC
A 5 3 C =
A6BC =
T AUKR
c X k 3
STR3 =
VXRQ =
VTRB :
VRKB
I F ( MX
IFIMT
. 375*A1B
. 375 A2B
. 3 7 5 A 3 B
: 3 7 5 A 4 B
. 375 A53
. 3 75 A6D
= CLRB*TAUR
: CLRB*SXR +
: ULRB*STR +
: CLRB*VXR +
= CLRB*VTR +
: CLRB*VRR +
.GT.DGU TU
+ CLRR I *TAUB
CLREI*SXB
CLRBI*STB
CLRBIVXC
CLRBI* V T B
CLRBI*VRB
310
GT.JR ISE)GO TO 303
210


199
For the static case, A can be obtained from equations (A.2.13)
and (A.2.21) directly from the two uniaxial stress-strain curves.
For the generalization of the Lipkin and Clifton (1970) curve,
P / vn
A = B (s s^> (E. 2.5)
and for the generalization of the Cristescu (1972) curve,
1
-2es
z
- (s s ) + A s
p' y y
-sJ- 6)
If the only stresses present initially are ct and T then
X 0x
o2 o2
X + 31 ex
and
o / o2 o2
s = / S + 3t
AJ X
(E.2.7)
and A is calculated from either equation (E.2.5) or equation (E.2.6)
as
or
and
AP
A = X
,Po / o vn
A B (s s >
y
i
p2e2 ~i [ / oxz i
1 [ir (s y + \ s]-*2 [(fry SJ
P k
> (E.2.8)
o Po o
A = A + s
J
and the initial plastic strains are given by equation (E.2.4) as
_Po 3 AP /Sij
ij 2 sO V E


GAM1 = GAMO + UGAM
1 = 1 + 1
I F( I 1 1 ) 2,5,5
5 CONTINU
JJ = J 1
11=1-1
CO 6 1=1,11
GAM I I ) = GAM(I)* 1 BO./PI
6 CONTINUO
CO 7 J=lfJJ
CELTA!J) = DELTA(J)*1BG./PI
7 C U N T I N U _
WRIT 2!6 102) A,BET A NU
W NIT C(6,103)
WRITE!6,1C4) (GAM( I), 1 = 1, II )
CO d J=1,JJ
WRITE(* 105) DELTA!J),(CFC2!I,J ),1 = 1, I I )
3 CONTINUE
KM IT (6, 106)
HR 1T c{6, 104) !G A M( I ), 1 = 1, I I)
CU 9 J=1,JJ
WRITC(,105 ) DELTA!J),(CSC21I,J ) ,I = 1, I I )
9 CONTINUE
GO TC 1
19 CONTINUE
KRITE!6,107)
WRIF E(6, 1^4 ) (GAM! I ), 1 = 1, I I )
CU 11 J=1,JJ
KRITt(6,105) DELTA(J),(5X(I,J),1=1, II)
11 CONTINUE
WRITE(6, 108 )
WRITtlto,104)!GAM( I), 1 = 1, II )
CU 12 J=i,JJ
WRITE(6, 1C5 ) DEL TA(J),(ST( I J ) 1=1, II)
167


LIST OF TABLES
Table
1
2
3
Page
Normalized Longitudinal Stress
x
Normalized Hoop Stress
s
Normalized Shear Stress
s
36
38
40
vi


/
X
Figure 3.8 Numerical Representation of the Characteristic Lines in a Regular Grid Element


193
Case III: Mixed boundary conditions
When S and V. are known at a boundary point, the expressions
xP 9P
for the hoop stress and the shear stress are found from equations
(D.2.3) and (D.2.15) to be
s0p = ^-(RHSEEM D1Sxp) (D.6.12)
Tp = (RHSDE Z2V ) (D.6.13)
2s
and once SQ is known, the longitudinal velocity is seen from equation
UP
(D.2.13) to be given by
V = F- (D'6'14)
Case IV: Mixed boundary conditions
When V and T are known at a boundary point, using equation
xP P
(D.2.15), the tangential velocity is given by
V0P = Z¡(RHSDE ~ F2STP>
Using equation (D.6.5), equation (D.2.13) can be written as
(D.6.15)
F_S + aF S. = RHS12 .
2f xP 5f 0P
However, this equation and equation (D.2.3) are the same two equations
solved for Case II above. Therefore, the longitudinal stress and the
hoop stress are given by equations (D.6.9) and (D.6.10) when radial
inertia effects are included (a = l), and when radial inertia effects
are not included (a = 0) the hoop stress vanishes and the longitudinal
stress is given by equation (D.6.11).


5
effects and temperature dependence. He also presented extensive data
for one-dimensional loading and combined stress loading at several
strain rates and temperatures which were used in empirically determin
ing the constants used in his generalized constitutive equation.
While these more general constitutive equations were being developed,
it was shown by Malvern (1965), by Wood and Phillips (1967), and by
Efron and Malvern (1969) that the semi-linear equation of Malvern-
(1951a, 1951b) did indeed predict a region of constant or nearly con
stant strain near the impact end if the solution was obtained long
enough after impact. Suliciu, Malvern, and Cristescu (1972) have shown
that a region of constant strain is not possible for the semi-linear
constitutive equation but may be approached asymptotically. They have
also shown that a region of constant strain is possible when the quasi-
1inear constitutive equation is used. However, in the interpretation
of experimental results it has been difficult to differentiate between
a region of truly constant strain and a region in which the constant
strain is approached asymptotically.
The experiments of Sternglass and Stuart (1953) Alter and Curtis
(1956) and others were believed by many investigators to be proof of
the strain-rate dependence of some materials. This led to the exten
sive development of constitutive equations just discussed. However,
other investigators sought to explain the experimentally observed phe
nomena by including radial inertia in the formulation of the wave prop
agation problem. Plass and Ripperger (1960) introduced radial inertia
effects into the problem of longitudinal impact and used the constitu
tive equation of Malvern (1951a, 1951b). In order to find a character
istic solution, all of the variables were averaged at each cross section


162
The equation along c = c is
0 =
dv +
x
~2 dcrx + x dt 1A4
rv.
PC4
+a(A)
L_r
\
dt
and along c = cg is
-2
0 = + pcg dv0 + dtQv + 2pco i|f dt
9x
's 0X
(B.4.7)
J
For the uncoupled waves, the equations along the vertical character
istics (c=0) can be written directly from equation (B.2.26) as
acr.
6 ^ j
dt = an dv
r r
o
v
a( Tlr_ Idt = aA0 dcr + aA. dan
\r Y 0/ 2 x 40.
o J
(B.4.8)
B.5. Elastic Waves
A special case where the waves are uncoupled occurs when 0(s,A) =0.
In this case the waves are elastic and may be either loading at stress
levels within the yield surface or unloading. Since 0(s,A) =0,
equations (2.23) become
(B.5.1)


201
E.3. Calculation of A
In order to find the solution (stresses and velocities) at a grid
point P, a value of A must be calculated during each iteration. This
can be done by calculating the strains at point P from the velocities
(as shown in Section 3.6) and then calculating the value of A from the
strains at points P, R, B, and L (see Figures 3.5 and 3.6). The value
of A can also be found from the functions §(s,A) and \[r^(s,A) by eval
uating these functions using the values of s and A obtained during the
previous iteration. The second method will be used so the strains will
not have to be calculated during each iteration, and the finite differ
ence expression for the value of A at the point P will now be found.
From equation (2.14), the inelastic strain rate is
s
and then
2
J
From equation (2.16)
and using equation (3.2.1), the dimensionless form becomes
J^$(s,A) H+2i)ro(s,A)JdT+ s
(E.3.1)


154
Using equation (2.48) in this equation yields
4 2f 1 T -2 -2-
0 = a pc [-i[pc (A4A6_aA5)_A4
%
db
*[ --
VrvJ-ar-=i [<
pc
[P;2(i446-^)-iJ 1*. + a[,
pe2(A4A6-^)-Sj
do
x
"dT
pc (AgAg AgAg) A2j
dt
~~ V df
o
+ 2 [pc2(a2A5-34)j i6x
This is an ordinary differential equation along the characteristic lines
of equation (2.48). This equation can be written in differential form
by multiplying by the increment d§ along the characteristic line. Doing
this, and noting that the equation is satisfied if the term in the
bracket vanishes, this equation finally reduces to
0 = | dvx_i5 [aV5_vJdve+ [fvvvj^ex
[p^'hV^s-h] +xdt
pc
+ a[p2(A2A6-A3A5)~2] (/"O^ 2 [pc2(aV5-A3A4)]'t'exdt
o
(B.2.23)
which is identical to equation (B.2.11) above. Note that equation (2.48)
represents four characteristic lines and that equation (B.2.23) repre
sents different equations along each of these four lines since along
each line c has a different value. The process just described can be
carried out for values of n from 2 to 12. Doing this, two additional
forms of the equations along the characteristics are found to be


SUCAGUTINE PS I(PK,S*D,DY,YS.BETA,XKtHAT,DZ)
IFIK.GT.O )GO TO 20
P = 0.0
GO TC 25
2 j CONTINUE
C THE CATA USED HERE ARE FROM CRISTESCU (1972) CASE XII
071 = SwRT(UZ)
IF(D.GT.CY)GO TO 21
YS = ABS(YS)
F = YS
GO TC 23
21 CONTINUE
I F(C.GT.DZ)GO TO 2 2
F = YS + .5*BETAMD DYJ/0Z1
CO TO 23
22 CONTINUE
F = BETA*SORT(D )
23 CONTINUE
IF(S.GT.F.ANC.C.GE.CY)G0 TO 24
F = 0.0
GO TC 25
24 CONTINUE
XK = XK:*(1.0 EXP(-D/DHAT ) )
P = XK ( S F )
25 CONTINUE
RETURN
END
230


1314
FORMAT(/3X,
VX =
6cl5.5/( 10X,8E15.5) )
1015
FORMAT(/3X,
' VT =
',8E15.5/1 10X.8E15.5) )
1016
FORMAT(/ 3X,
VR =
*,8E15.5/1 10X,8E15.5> )
1017
FORMAT(/3X,
CF =
',8E15.5/( IDX,RE15.5) )
l 0 1 ri
FORMAT(/3X,
' cs =
'8E 15.5/( 1DX,815.5 ) )
1019
FORMAT(/3X,
SB =
,8E15.5 / ( inx be 155 ) )
1C 2 >j
FORMAT(/ 3X T
EX =
,8E15.5/t 11X,BE 15.6) )
1021
FORMAT(/3 X,
' ET =
,8E15.5/( 10X,E15.5 ) )
102 2
FORMAT(/3X,
FTX =
',8E15.5/ 10X,8E15.5 ) )
1023
FORMAT{/3X,
! EXP =
'v0E15.5/( 10X.8E15.5) )
102 4
FORMAT(/3X,
' ETP a
',8E i 5.5/( 10X.8E15.5 ) )
102 5
FORMAT(/3X,
ETXP =
'8E15.5/<10X.8E15.5M
10 2 6
FOkMAT(/3X,
' ITER =
,01 15/( 10 X 8115) )
102 7
FORMAT(/3 X,
'DELTA
= '(8E15.5/( 10X t 8E15.5 ) )
1 0 2 H
FORMAT(/JX,
CPLAS
= ,8tl5.5/( 10X,8E15.5) )
1020
FORMAT(/3X,
PHI =
*8E15.5/( 10X.8E15.5) )
13 30
FORMAT(10X,
' JBEIL
= 'tlltllXf'D/ = ',rfi.6,7X, 'DHAT = F8.6,5X XM
i
= ',FK.8,
5 X,'XN
= *F10.85Xt XLAM = ,F8.6/)
1031
FORMAT(1GX,
MI = '
, I 3,12X MJ = ,I 3,12X, I PUNCH = ,I 38X 'KAS2 =
J.
I2/ )
10 3 2
FORMAT(10 X,
X = ,
F 9.6 5 X, T = F9.6,5X,'DIFF = ,L13.5,5X,'DNOM
1
= 'ttl3.5.
5 X,ERRUR = ,E13.5//)
103 3
FORMAT( 1 )
10 34
FORMAT(615,
2F10.6)
10 3 5
FORMAT(7 E 11
. 6 )
10 36
FOKMAT(/3X,
'OGAM =
',6E15.5/( 10X,8E15.5) )
1037
FORMAT(/3X,
DEX =
,8E15.5/(10X.8E15.5) )
1038
FORMAT{/3X,
'STRES
= ',PE15.5/( 10X.8E15.6) )
IO 3 > FORMAT(//////50X '
1040 rORMAT(//////50X, '
STOP
ENL
WAVES ARE UNCOUPLED ******#**#)
WAVES ARE COUPLED ********#)
227


APPENDIX D
SOLUTION TO THE FINITE DIFFERENCE EQUATIONS
IN THE CHARACTERISTIC PLANE
In this appendix, the expressions for the stresses and velocities
at a grid point P will be determined. In doing this, many new terms
will be defined. In order to simplify the interpretation of the
computer code in Appendix E, these new terms will be defined in exactly
the same manner in which they are used in this computer code.
D.l. Equations for Fully Coupled Waves
When the waves are fully coupled, the finite difference equations
can be simplified by grouping known quantities together. Thus, letting
Q3 = 2AT(1 -v2) ^
D = V Q S
3 rB j 9b
A2Q 2(A2P + A2B^
A4Q 2(A4P + A4B)
A5Q 2(A5P + A5B'>
and
RHSE = >Tj^2V + (S 2aSrsJ -^l + A_S_+A_SftT,+ A_T
I
rB xB
0B s
oB~]
3
B
2Q xB 4Q 0B 5Q B
(D.1.1)
(D.l.2)
the equations (3.4.2) and (3.4.3) along the characteristics (c = 0) can
be written as
aVrP = a(D3"Q3S6P)
(D.l.3)
171


CHAPTER 5
SUMMARY
In this dissertation the propagation of inelastic waves of combined
stress was discussed. The equations were developed to describe the
propagation of both longitudinal and torsional waves along a thin-walled
tube. These equations were written in such a manner that radial inertia
effects could be either included or not included in the problem formula
tion simply by setting a single parameter equal to either 1 or 0.
A generalized quasilinear constitutive equation was presented for mul-
tiaxial stress states and two different forms of it were used to study
two different materials. Both materials studied were assumed to obey
the von Mises yield criterion and to be isotropically work-hardening.
Once the problem was formulated the plastic wave speeds were then
determined as functions of the state of stress. For the case when
radial inertia effects were not included, the results of Clifton (1966)
were reproduced. When radial inertia was included, quite different
results were obtained, even with the hoop stress set equal to zero.
Both plastic waves (the slow wave and the fast wave) were found to
propagate faster when radial inertia effects were included (even with
the hoop stress as zero) than the corresponding waves when radial
inertia was not included for the same state of stress. It was also
found that when loading occurred in a principal stress plane, there
was often a critical value of the tangent modulus (or work hardening)
118


23
where
A w, = B w, ^ b
A =
P
0
0
0
0
0
0 0 0 0 0
ap 0 0 0 0
0 p 0 0 0
0 0 A1 aA2 A3
0 0 aA aA aA
Z "x O
0 0 A aA_ A.
3 5 6
(2.38)
(2. 39)
0 0 0 -1 0 0 -¡
0 0 0 0 0 0
B =
0
-1
0 0 0 0 -1
0 0 0 0 0
0 0 0 0 0 0
0 0 -1 0 0 0
(2.40)
v
X
V
r
a
X
cr
9
L
Gx
(2.41)
and


LIST OF REFERENCES
Alter, B. E. K. and Curtis, C. W. (1956), "Effect of Strain Rate on the
Propagation of a Plastic Strain Pulse along a Lead Bar," Journal
of Applied Physics, Vol. 27, pp. 1079-1085.
Bell, J. F. (1951), "Propagation of Plastic Waves in Prestressed Bars,"
Technical Report No. 5, Johns Hopkins University, Navy Contract
N6-ONR-243-VIII.
Bell, J. F. (1960), "Propagation of Large Amplitude Waves in Annealed
Aluminum," Journal of Applied Physics, Vol. 31, pp. 277-282.
Bell, J. F. (1963), "Single, Temperature-Dependent Stress-Strain Law
for the Dynamic Plastic Deformation of Annealed Face-Centered
Cubic Metals," Journal of Applied Physics, Vol. 34, pp. 134-141.
Bell, J. F. and Stein, A. (1962), "The Incremental Loading Wave in the
Prestressed Plastic Field," Journal de Mecanique, Vol. 1,
pp. 395-412.
Bianchi, G. (1964), "Some Experimental and Theoretical Studies on the
Propagation of Longitudinal Plastic Waves in a Strain-Rate-
Dependent Material," Stress Waves in Anelastic Solids, I.U.T.A.M.
Symposium, Brown University (H. Kolsky and W. Prager, Eds.),
Berlin: Springer-Verlag, pp. 101-117.
Clark, D. S. and Wood, D. S. (1950), "The Tensile Impact Properties of
Some Metals and Alloys," Transactions of the American Society of
Metals, Vol. 42, pp. 45-74.
Clifton, R. J. (1966), "An Analysis of Combined Longitudinal and
Torsional Plastic Waves in a Thin-Walled Tube," Proceedings of
the Fifth U.S. National Congress of Applied Mechanics, ASME,
N.Y., pp. 465-480.
Clifton, R. J. (1968), "Elastic-Plastic Boundaries in Combined
Longitudinal and Torsional Plastic Wave Propagation, Journal of
Applied Mechanics, Vol. 35, pp. 782-786.
Convery, E. and Pugh, H. L. D. (1968), "Velocity of Torsional Waves in
Metals Stressed Statically into the Plastic Range, Journal of
Mechanical Engineering Science, Vol. 10, pp. 153-164.
Cristescu, N. (1959) "On the Propagation of Elastic-Plastic. Waves in
the Case of Combined Stress," Prikladnaia Matematika i Mekhanika,
Vol. 23, pp. 1124-1128.
Cristescu, N. (1964), "Some Problems in the Mechanics of Extensible
Strings," Stress Waves in Anelastic Solids, I.U.T.A.M. Ss^mposium,
Brown University (H. Kolsky and W. Prager, Eds.), Berlin:
Springer-Verlag, pp. 118-132.
231


APPENDIX E
COMPUTER PROGRAM FOR CHARACTERISTIC PLANE SOLUTION
E.l. General Description of the Program
This program is written to solve the finite difference equations of
Chapter 3. Two types of grid elements are used: the regular grid ele
ment shown in Figure 3.5 and the boundary grid element shown in Fig
ure 3.6. These two types of elements are fitted together to form the
complete grid network of Figure 3.1, and the solutionsto the finite
difference equations are obtained at each point in this grid network
using equations developed in Appendix D and rewritten in Chapter 3.
To obtain the solution at a grid point P, however, the values of each
variable must be known at the points L, B, and R for a regular element
and at the points B and R for a boundary element. For this reason,
some initial conditions must be specified.
The initial conditions are specified in the region
X S T
since the fastest elastic wave can propagate with a speed no greater
than
For the problem considered here, this amounts to specifying all of the
variables at each grid point along the line
X = T,
that is, at the points (0,0), (AX,AT) (2AX,2AT) (a^AX.a^AT).
The initial conditions used are discussed in the next section.
194


151
Now, since all of the components of M and d are known at a given point
(x,t), the solution to equation (B.2.15) can be found directly if the
determinant of M does not vanish. However, if the determinant of M
vanishes, the solution for Z cannot be found. This is represented by
| M j = 0 (B.2.19)
which is the equation for the characteristic lines along which discon
tinuities in the variables may propagate. Evaluation of this determi
nant results in the characteristic equation (2.45). When equation
(B.2.19) is satisfied, the system of equations (B.2.15) will be shown
below to reduce to a set of ordinary differential equations along the
characteristic lines, that is, along each characteristic line one ordi
nary differential equation involving the six variables v^, Vq, v^, , and t must be satisfied. If M is defined as the matrix M with
0' 0x n
t h
its n column replaced by d, then the solution to equation (B.2.15) can
be written in terms of Cramer's rule as
M
Z = (B.2.20)
11 I M I
t h
where Z is the n element of Z. However, by equation (B.2.19), the
n
denominal;or vanishes and for a solution to equation (B.2.15) to exist,
it is necessary that
| M 1=0 (B.2.21)
i ~n 1
Equations (B.2.21) are twelve equations, one for each column in M which
is replaced. Next, the calculations involved in equation (B.2.21) will
be carried out as an example for the case where n=l. This yields


33
and using equation (3.1.4) these become
ct = s cos
x
y j^cos 6
- sin 6
/
V o
n
j
~\
aCTg = s cos y |^cos 6 +
- = s sin y
0X
^3
1 ,1
sin §
4 J
(3.1.14)
y
Now substituting equations (3.1.12) and (3.1.14) into equation (2.23),
-if i i 2 2 1 2~^
A^ = j 1 + (g- 1) () (cos y) (2 cos 6 sin 6 cos 6 sin 6) J
V3
JS
_ 1 P 1 1 2 2 1
A = v + (?r 1) cos y(2 cos 6 sin 6 cos 6 sin 6)
2 E L 4 p T
V3
i 2 r n
(cos 6 sin 6-2 cos 6 sin 6) I
/3
*/3
*/3
- i n i 6 i 2 i ~i
A-j = (-r) ( sin y) (cos y) (2 cos 6 sin 6 cos 6 sin 6)
3 E L 4 a/3 J
A4 = I [X + X) (cos 6 si

\ = | [2d +v) + ~ (|-1)(| sin2Y)]
sin 6-2 cos 6 sin 6) cos2Y
\/3 J
1 2
sin y) (cos y) (cos 6 sin 6-2 cos 6 sin 6) I
vy3 v^3 V3 J
or
i r ii 2 2
At = |^1 + ^(^ 1) (cos v) (cos 6 2 y3
2
sin 6 cos 6 + 3 sin 6)
J
A =
i |^v + i(i 1)(cos2y)(-cos26 + 3 sin"6)J
3 1 ~]
)(-g--l)(sin y cos y) (cos 6 -*/3 sin 6) I
3 E L 2 3
1) cos v(cos6 + 2 cos 6 sin 6+3 sin 6)
]
>
(3.1.15)
A.
A,
= ~ p 1) (sin y cos y) (-cos 6 Sin 6) J
1
1
[^2(1 + v) + 3(1-D sin Yj


7 A = C 2 D 4 S 2 + C1*B7S
ZB = 02 C4F 2 + C1*B7F
DELI") = B 2F Z A B2S*ZB
VT I = ( RHS 9 *7 A RHS1CWB + RHS 11 ( D4S2* B 7F-D4F 2 *B 7S ) )/DE L 1 0
SXI = (B2FM 02RHS107S*RHS11)-B2S*(D2*RHS9+37F*kHSl1) )/UEL10
ST I = (B2F *(D4S2*RHS11-D1*RHS10)-R2S*(D4F2*RHS11-D1*RHS9))/0EL10
GU TC 390
585 IFIKASE.NE.2)GO TO 536
RHS4 = RHSB A*D3*H6F D1F*VXI B2F*VTI
RHS 5 = RFSD A*D3*36S B1S*VXI B2S*VTI
CEL7 = U3F2*U4S2 D4F2*D3S2
T AUI = (RHS4 *04S2 RHS5*D4F2)/DEL7
SXI = (RHS5*D3F2 RHS4 D 3S 2 ) /DF L 7
ST l ^ 0.0
GU TU 390
586 IE(KASE.NE.i)GO TO 587
RHS6 = RHSB AD3*86F 32F*VTI 04F2*SXI
R HS 7 = RHSD A*D3*B6S B2S*VTI 04S2*SXI
GE 19 = B1F*D3S2 BLSD3F2
VXI = IRHS6*D3S2 RHS7 *D3F2 ) 70EE9
TAU = (RHS7*B1F RHS6*B13 )/OEL9
ST1 = 0.0
GU T 390
587 CONTINUE
RHS9 ^ RHSB AC3*R6F B1F*VXI D3F2*TAUI
RHS10- RHSO AD3*6S B1S*VXI D3S2*TAUI
CECIL = B2F-*C4S2 R2S*04F2
VT I = ( RHS9 *04S 2 RHS 10D4F 2 )/PEL 11
SXI = (RHS19*B2F RHS9*B2S)/DEL 11
ST I = 0.0
GO TO 39 U
384 CONTINUE
IF(KASE.E0.2.OR.KAS EEQ4)GO TO 594
STi = A*(RHS E EM D1*SXI)/D2
219


122
One set of data was used with a multiaxial generalization of the
constitutive equation of Cristescu (1972). These data consisted of
applying a static pretorque to the tube and then impacting the tube
longitudinally. YVhen no strain-rate dependence was included, the
results were qualitatively the same as for the previous case.
However, when strain-rate dependence w'as included for the constant
velocity impact, the longitudinal strain was found to be reduced while
the stress trajectories were not significantly affected by strain-rate
dependence.
The major contributions of this dissertation were the development
of the equations for combined stress wave propagation with radial
inertia effects included, the introduction of a general quasilinear
constitutive equation for multiaxial stress states, and the writing of
a computer code capable of solving this wave propagation problem in
the characteristic plane.
The theory and results presented in this dissertation suggest the
need for further research in order to more fully understand plastic
stress wave propagation. First the computer code in this dissertation
could be made more general by including the boundary element for the
opposite end so that tubes of finite length could be studied. Also,
the initial assumptions can always be improved; in this work the most
restrictive assumption is that the material work-hardens isotropically.
This is valid only as long as reversed loading is not approached. But
for many cases, waves of reversed loading can occur, and obviously the
introduction of more realistic work-hardening assumptions may be


Change in Longitudinal Strain, As
112


138
*\
(5,A) = o
0 (s,A) = X
1
i
-2sSz n
~~d+ x
r.2s 1
2L3+2
eJ
(A.4.5)
and again using equation (3.2.1), these functions for this case can be
written in dimensionless form as
and
i|r (s,A) = \|r(s,A) = 0
o -
C1
(s,A)
Y(s,a)
f(s,A) = E0(s,A)
1
i2E | r i
(A.4.6)
where
-2es
3' =
F
(A.4.7)
(A. 4.8)
The expressions for 0(s,A) and ijf(s,A) are given in Section A. 3
for the Case XII studied by Cristescu (1972). In dimensionless form
these become
r
0 if s^F(A) or A *o (A. 4.9)
[s-F(A)] if s>F(A) and A >A
V C1
where


I'-[ELASTIC WAVE PROPAGATION
UNDER COMBINED STRESS STATES
By
CHARLES DANIEL MYERS
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1973

TO PEGGY

ACKNOWLEDGMENTS
I would like to thank Professor Martin A. Eisenberg, Chairman of
the Supervisory Committee, not only for his untiring efforts during
the development and preparation of the material contained in this
manuscript, but also for being a counselor, teacher, and friend during
both my undergraduate and graduate studies. I am also indebted to
Professors L. E. Malvern and E. K. Walsh for their helpful criticism
and encouragement during my doctoral studies. In addition, I would
like to express my appreciation to the other members of my Supervisory
Committee: Professors U. H. Kurzweg, C. A. Ross, and R. C. Fluck.
A special word of thanks is extended to Professor N. Cristescu
for his many helpful discussions during the development of this
dissertation.
I thank my wife, Peggy, for her encouragement, moral support, and
understanding during the course of my studies. I also thank Peggy for
typing and proofreading the drafts of this dissertation. I appreciate
the efforts of Mrs. Edna Larrick for the final typing of the manuscript
and Mrs. Helen Reed for the final preparation of figures.
I acknowledge financial support from the National Defense Education
Act, the National Science Foundation, and the University of Florida
which made my studies possible.
I also acknowledge the Northeast Florida Regional Computing Center
for the use of its IBM 370 Model 165 digital computer without which
the scope of this work would have been greatly curtailed.
iii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
LIST OF TABLES vi
LIST OF FIGURES vii
ABSTRACT ix
CHAPTER 1. INTRODUCTION 1
CHAPTER 2. THEORETICAL DEVELOPMENT 12
CHAPTER 3. DIMENSIONLESS EQUATIONS AND NUMERICAL PROCEDURES . 29
3.1. Wave Speeds as a Function of the State of Stress .... 29
3.2. Characteristic Solution in Terms of
Dimensionless Variables 46
3.3. Numerical Grid for Characteristic Solution 51
3.4. Finite Difference Equations ... 58
3.5. Solution to the Finite Difference Equations 69
3.6. Calculation of the Strains ..... 77
CHAPTER 4. RESULTS AND DISCUSSION 80
4.1. Introduction 80
4.2. Grid Size Effects 81
4.3. Effects of Radial Inertia 84
4.4. Effects of Strain-Rate Dependence 110
CHAPTER 5. SUMMARY 118
APPENDIX A. CONSTITUTIVE EQUATIONS 125
A. 1. Comments on the Constitutive Equation 125
A.2. Rate Independent Incremental Plasticity Theory ..... 127
A. 3. Rate Dependent Plasticity Theory
A.4. Dimensionless Expressions for the Functions
0(s,&) and iji(s,A) 137
IV

TABLE OF CONTENTS (Continued)
Page
APPENDIX B. CHARACTERISTICS AND EQUATIONS
ALONG THE CHARACTERISTICS 140
B. 1. Equations for the Characteristics 140
B.2. Equations along the Characteristics 141
B. 3. Reducing Equations to Simpler Case 156
B.4. Uncoupled Waves 159
B. 5. Elastic Waves 152
APPENDIX C. PROGRAMS FOR DETERMINING THE PLASTIC
WAVE SPEEDS 164
APPENDIX D. SOLUTION TO THE FINITE DIFFERENCE EQUATIONS
IN THE CHARACTERISTIC PLANE 171
D.l. Equations for Fully Coupled Waves 171
D.2. Equations for Uncoupled Waves 17&
D.3. Solution at a Regular Grid Point
for Fully Coupled Waves 130
D.4. Solution at a Regular Grid Point
for Uncoupled Waves 183
D.5. Solution at a Boundary Point (X=0)
for Fully Coupled Waves 185
D.6. Solution at a Boundary Point (X=0)
for Uncoupled Waves 191
APPENDIX E. COMPUTER PROGRAM FOR CHARACTERISTIC PLANE
SOLUTION 194
E.l. General Description of the Program I94
E.2. Initial Conditions 196
E. 3. Calculation of A 201
E.4. Input Data 202
E.5. Listing of the Program 204
LIST OF REFERENCES 231
BIOGRAPHICAL SKETCH 236
v

LIST OF TABLES
Table
1
2
3
Page
Normalized Longitudinal Stress
x
Normalized Hoop Stress
s
Normalized Shear Stress
s
36
38
40
vi

LIST OF FIGURES
Figure
2.1 Coordinate System for the Thin-Walled Tube
2.2 Stresses on an Element of the Tube ....
3.1 Yield Surface Representation in Spherical Coordinates .
3.2 Plastic Wave Speeds as Functions of [3 and y for
Poisson's Ratio of 0.30
3.3 Values of 8 at v= 0 for which c-c -c
Y f s 2
3.4 Numerical Grid in the Characteristic Plane
3.5 Regular Element in Numerical Grid
3.6 Boundary Element in Numerical Grid
3.7 Location of the Characteristic Lines Passing Through P .
3.8 Numerical Representation of the Characteristic Lines
in a Regular Element
3.9 Representation of the Characteristic Lines in
a Boundary Element
4.1 Grid Size Effects on the Longitudinal Strain
at X = 3.75
4.2 Grid Size Effects on the Longitudinal Velocity
at X = 3.75 .
4.3 Grid Size Effects on the Stress Trajectories
at X = 3 75
4.4 Longitudinal Strain Versus Time at X = 3.75
for Data Set 1 . .
4.5 Change in Shear Strain Versus Time at X = 3.75
for Data Set 1
Page
13
14
31
41
45
53
54
55
57
59
60
83
85
86
88
89
vii

LIST OF FIGURES (Continued)
Figure Page
4. 6 Transverse Velocity Versus Time for Data Set 1
Without Radial Inertia 91
4.7 Longitudinal Velocity Versus Time for Data Set 1
Without Radial Inertia 93
4. 3 Longitudinal Strain Versus Time for Data Set 1 94
4.9Maximum Radial Velocity Versus X for Data Set 1
With Radial Inertia 96
4.10 Change in Shear Strain Versus Time for Data Set 1
Without Radial Inertia 97
4.11 Longitudinal Strain Versus X for Data Set 1 98
4.12 Stress Trajectories for Data Set 1
Without Radial Inertia 99
4.13 Strain Trajectories for Data Set 1
Without Radial Inertia 100
4.14 Shear Stress Versus Longitudinal Stress for Data Set 1
With Radial Inertia 102
4.15 Stress Trajectories for Data Set 1 With Radial Inertia . 104
4.16 Hoop Stress Versus Longitudinal Stress for Data Set 1
With Radial Inertia 105
4.17 Stress Trajectories for Data Set 2
Without Radial Inertia 107
4.18 Stress Trajectories for Data Set 2 With Radial Inertia . 108
4.19 Hoop Stress Versus Longitudinal Stress for Data Set 2
With Radial Inertia 109
4.20 Shear Strain Versus Time for Data Set 2 Ill
4.21 Change in Longitudinal Strain Versus Time for Data Set 2 112
4.22 Longitudinal Strain Versus Time for Data Set 3 114
4.23 Stress Trajectory at X = 0 for Data Set 3 116
4.24 Stress Trajectory at X = 25 for Data Set 3 117
viii

Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy
INELASTIC WAVE PROPAGATION UNDER
COMBINED STRESS STATES
By
Charles Daniel Myers
August, 1973
Chairman: Dr. M. A. Eisenberg
Major Department: Engineering Science, Mechanics
and Aerospace Engineering
The purpose of this dissertation was to investigate the effects
of radial inertia and material strain-rate dependence on the propa
gation of inelastic waves of combined stress along a thin-walled tube.
A general quasilinear constitutive equation for multiaxial stress (and
strain) states was introduced. The equations of motion and the
strain-displacement equations, along with the constitutive equations,
were written to form a set of nine simultaneous hyperbolic, quasilinear,
partial differential equations. This set of equations was reduced to
a set of six equations which was then used to determine the expres
sions for the characteristic lines and the equations along the char
acteristic lines.
For combined torsional and longitudinal loading, two distinct
wave speeds were found. The values of these two wave speeds were
found as functions of the state of stress. Including radial inertia
effect in the formulation of the problem was shown to significantly
increase the wave speeds for a given stress state. Also certain
IX

critical combinations of Poisson's ratio and the "effective tangent
modulus" caused the two wave speeds to be equal when the shear stress
vanished.
The equations for the characteristics and the equations along the
characteristics were written in terms of dimensionless variables.
These equations were then written as first order finite difference
equations. A computer code was written in the Fortran IV language,
and several problems were solved using an IBM 370 model 165 digital
computer. In order to obtain these solutions two particular forms of
the constitutive equation were used; one form represented a strain-
rate independent material while the other form represented a strain-
rate dependent material.
The strain at the impact end was considerably larger when radial
inertia effects were included than when radial inertia effects were
not included in the problem. However, radial inertia effects were
found to have little influence on the solution more than two diameters
from the impact end. The strain at the impact end was lowered by
including strain-rate dependence of the material. For any particular
set of initial conditions and boundary conditions, the stress trajec
tories behaved in the same manner, at least qualitatively, whether or
not radial inertia effects or strain-rate dependence were included.
The details of the stress trajectories were more complicated when
radial inertia effects were included since the trajectories were
three-dimensional.
x

CHAPTER 1
INTRODUCTION
Stress wave propagation is the mechanism by which forces and
displacements are transmitted from one part of a structure to another.
Stress waves arise when a transient force is applied to a structure,
and they propagate through the structure reflecting (at least partially)
back into the structure whenever they encounter a boundary. After
several reflections the amplitude of the stress waves diminishes and
the structure reaches a state of equilibrium. In many engineering
problems the time required to reach equilibrium is very short, and for
practical purposes the structure can be assumed to reach equilibrium
instantly. Problems in which the forces are applied slowly or in which
the state of stress is required a long time after the forces are applied
are examples of instances when wave propagation effects may be neglected.
However, in many cases, the forces are applied rapidly (such as during
impact loading or explosive loading), and failure is most likely to
occur in the structure almost immediately after the application of these
forces. In these cases when it is necessary to determine the state of
stress during and immediately following the loading, wave propagation
effects may be significant and should be included in the analysis of
the problem. In order to understand the development of the theory of
stress wave propagation and the application of this theory to modern
engineering problems, it is instructive to review briefly the history
of wave propagation research.
1

The first serious attempt (at least in this century) to understand
nonlinear wave propagation in solids was made by Donnell (1930). In
this paper, Donnell used energy principles and impulse-momentum expres
sions to find the particle velocity and the elastic wave speed for
longitudinal waves. He also predicted that if a material with a bilin
ear stress-strain curve were impacted at the end by a stress above the
yield stress, two stress waves would propagate with distinct velocities
However, after the publication of this paper interest in wave propaga
tion subsided until the early 1940's.
A more general theory of longitudinal stress wave propagation was
developed independently by Taylor (1940) von Karman (1942) and
Rakhmatulin (1945) by assuming that the material exhibited a nonlinear
stress-strain curve above the yield point. This stress-strain curve
was assumed to be independent of the rate of straining. Using this
theory the velocity of propagation of the longitudinal waves was found
to be given by
'1 da
p de
c
where c is the wave speed, p is the density of the material, a is the
stress, and e is the strain. Thus represents the slope of the stress
de
strain curve or the tangent modulus. This theory also considered the
stress-strain curve of the material which was obtained for the static
case to be valid in the dynamic case. With this assumption, the stress
and strain followed a unique functional relationship as long as no
unloading occurred. Because of this, the tangent modulus could be
written as a function of the stress (or strain) only, so that the
velocity of propagation then became a function of the level of stress

3
(or strain). This immediately led to the conclusion that a given
level of stress (or strain) propagated at a specific speed, and the
stress wave changed shape as it propagated along a prismatic bar for
stresses in the nonlinear region of the stress-strain curve. For a
bilinear stress-strain curve, the results of Donnell (1930) were again
predicted. However, these theories did not account for the lateral
inertia effects in the bar or the dependence of the stress-strain curve
on the rate of strain, and so more complex theories and constitutive
equations were proposed to account for these phenomena.
By the late 1940's many investigators including Davis (1938),
Manjoine (1940) and Clark and Wood (1950) had experimentally observed
the effect of the rate of strain on the stress-strain curve for several
materials. In order to incorporate this strain-rate effect into the
constitutive equations used to study plastic wave propagation,
Sokolovsky (1948a, 1948b) and Malvern (1949, 1951a, 1951b) independ
ently introduced one-dimensional constitutive equations in which the
stress was a function of the plastic strain and the plastic strain rate.
By selecting a particular form of this constitutive equation, Malvern
(1951a, 1951b) was able to obtain a numerical solution which predicted
several experimentally observed phenomena. However, his numerical solu
tion did not apparently predict a region of constant strain near the
impact end such as had been observed by Duwez and Clark (1947) and
others. This new strain-rate dependent constitutive equation also
predicted that, if a bar were strained statically above the yield stress
and then impacted, the first increment of strain would propagate with
the elastic wave velocity and not the velocity given by the tangent

4
modulus, in the strain-rate independent theory. Since this prediction
was quite different from that of the strain-rate independent theory
several investigators tried to verify one or the other. Bell (1951)
published the results of his experiments with aluminum which showed
that, for a bar stressed above the yield point, the initial strain
pulse propagated with the elastic wave velocity. These results were
in accordance with the strain-rate dependent model of Malvern (1951a,
1951b) as were the experimental results of Sternglass and Stuart (1953)
which were obtained using copper, Alter and Curtis (1956) which were
obtained using lead, Bell and Stein (1962) which were obtained using
aluminum, and Bianchi (1964) which were obtained using copper.
Encouraged by these experimental results, many investigators
continued the development of more general constitutive equations to
describe material behavior. Perzyna (1963) generalized the semi-linear
constitutive equation of Malvern (1951a, 1951b) to multiaxial states of
stress. At about this same time Cristescu (1964) introduced full quasi-
linear constitutive relations for a one-dimensional problem. This
quasilinear equation was used immediately by Lubliner (1964) to show
that the strain-rate independent constitutive equation of Taylor (1940),
von Karman (1942), and Rakhmatulin (1945), and the strain-rate depend
ent constitutive equation of Malvern (1951a, 1951b) and Sokolovsky
(194Sa, 1948b) were both special cases of this more general constitu
tive equation. Later Cristescu (1967a) gave a generalization for multi
dimensional stress states of the quasilinear constitutive equation as
well as an extensive summary of the developments in dynamic plasticity
until that time. Lindholm (1967) developed a constitutive equation
for combined stress states of aluminum which included strain-rate

5
effects and temperature dependence. He also presented extensive data
for one-dimensional loading and combined stress loading at several
strain rates and temperatures which were used in empirically determin
ing the constants used in his generalized constitutive equation.
While these more general constitutive equations were being developed,
it was shown by Malvern (1965), by Wood and Phillips (1967), and by
Efron and Malvern (1969) that the semi-linear equation of Malvern-
(1951a, 1951b) did indeed predict a region of constant or nearly con
stant strain near the impact end if the solution was obtained long
enough after impact. Suliciu, Malvern, and Cristescu (1972) have shown
that a region of constant strain is not possible for the semi-linear
constitutive equation but may be approached asymptotically. They have
also shown that a region of constant strain is possible when the quasi-
1inear constitutive equation is used. However, in the interpretation
of experimental results it has been difficult to differentiate between
a region of truly constant strain and a region in which the constant
strain is approached asymptotically.
The experiments of Sternglass and Stuart (1953) Alter and Curtis
(1956) and others were believed by many investigators to be proof of
the strain-rate dependence of some materials. This led to the exten
sive development of constitutive equations just discussed. However,
other investigators sought to explain the experimentally observed phe
nomena by including radial inertia in the formulation of the wave prop
agation problem. Plass and Ripperger (1960) introduced radial inertia
effects into the problem of longitudinal impact and used the constitu
tive equation of Malvern (1951a, 1951b). In order to find a character
istic solution, all of the variables were averaged at each cross section

6
and these averaged variables were used. The results of this work were
given by Tapley and Plass (1961) but were somewhat inconclusive. More
work including radial inertia effects was published by Hunter and
Johnson (1964), and a year later DeVault (1965) showed that, at least
qualitatively, many observations formerly attributed to a material
strain-rate effect could be explained by including radial inertia effects
in the formulation of the problem of longitudinal impact of a bar.
Shea(1968) obtained good agreement between theory and experiment for the
propagation of longitudinal waves in a lead bar. He used the strain-
rate dependent constitutive equation of Malvern (1951b) and the
"correction" for radial inertia proposed by DeVault (1965). Mok (1972)
used the same averaging technique for the variables as Plass and
Ripperger (1960) for the problem of longitudinal impact of a bar with
radial inertia effects included. He used the strain-rate independent
constitutive equations and agreed in essence with DeVault (1965) that
radial inertia effects could explain, at least qualitatively, those
experimental results usually attributed to strain-rate sensitivity
of the material. Since radial inertia is always present in an experi
ment using longitudinal impact it seemed that the only way to conclu
sively determine strain-rate effects in a material would be to perform
the experiments using a torsional wave.
In an effort to determine the strain-rate dependence of various
materials, several investigators have recently conducted theoretical
and experimental studies concerning the propagation of torsional waves.
Convery and Pugh (1968) gave the results of their experiments in which
a tube was stressed statically above the yield stress in torsion and

7
then subjected to a suddenly applied incremental torsional load. The
strain caused by this incremental load was found to propagate with the
elastic shear wave velocity. This seemed to be proof that the strain-
rate dependent theory was correct, but Convery and Pugh (1968) cau
tioned against that conclusion. For Bell (1960, 1963) and Bell and
Stein (1962) had asserted that (based on experimental results with
annealed aluminum), while an increment of strain may propagate with the
elastic wave velocity, the larger amplitude strains propagate with the
wave velocity predicted by the strain-rate independent theory.
Nicholas and Garey (1969) tested aluminum samples in torsion at high
strain rates and found very little strain-rate dependence. However,
Yew and Richardson (1969) were able to measure some strain-rate depen
dence in copper.
Another problem which was encountered in wave propagation studies
f
was that of unloading. The two most common unloading cases were when
the applied load was reduced and when waves were reflected from a bound
ary. Unloading was examined for longitudinal plastic wave propagation
by Lee (1953) using the strain-rate independent constitutive equation
and by Cristescu (1965), Lubliner and Valathur (1969), and Cristescu
(1972) using the quasilinear constitutive equation. In all of these
investigations, regions of unloading and boundaries between regions of
unloading and loading in the characteristic plane were predicted but
the results have not been verified experimentally.
Many investigators in recent years have become interested in the
behavior of materials under combined stress and, more specifically, the

8
propagation of waves of combined stress. One of the first discussions
of combined stress wave propagation was given by Rakhmatulin (1958).
In this paper he developed the equations which must be solved for
elastic-plastic wave propagation under combined stress. Strain-rate
independent constitutive equations were used and only the problem for
the elastic case was solved. He found that the shear wave did not
affect the longitudinal wave in the elastic case. A similar discussion
of combined stress wave propagation was presented by Cristescu (1959).
Until now nothing has been said about the plasticity theory used.
ITie two plastic strain theories were the total strain theory proposed
by Hencky (1924) and the incremental strain theory proposed by Prandtl
(1924) and Reuss (1930). These two plasticity theories along with many
other developments in plasticity theory were presented in detail by
Hill (1950). The different plasticity theories were not presented
earlier because in many cases both theories gave the same results.
For instance, when a strain-rate independent constitutive equation was
used, the two plasticity theories led to identical results when one
dimensional (either longitudinal or torsional) stress wave propagation
was studied, when combined stresses were used if the loading was pro
portional, or even when unloading occurred in one-dimensional problems.
However, when strain-rate dependent material behavior of nonpropor
tional loading under combined stresses was considered, most investiga
tors used the incremental strain theory. Shammamy and Sidebottom
(1967) showed that the incremental strain theory more accurately pre
dicted the experimental results when various metal tubes were subjected
to nonproportional static loading in tension (compression) and torsion.

9
Interest in the propagation of waves of combined stress continued
and Clifton (1966) presented the results of his study of combined longi
tudinal and torsional plastic wave propagation in a thin-walled tube.
Strain-rate independent material behavior and incremental strain
theory were used while radial inertia effects were ignored. The thin-
walled tube allowed Clifton to eliminate any dependence on the radial
coordinate so that a solution could be obtained in the characteristic
plane. (Earlier, Plass and Ripperger (1960) had used a rod and averaged
the variables, over the cross section in order to eliminate the dependence
on the radial coordinate.) The results of this investigation were
based on a simple wave solution which resulted from applying a step
velocity impact at the end of the tube. Clifton (1966) found that when
the tube was stressed into the plastic range, an impact at the end of
the tube caused waves with two different speeds to propagate. These
waves were called the fast wave and the slow wave, and each wave was
found to carry both longitudinal and torsional stresses. Two special
cases were examined. The first case involved statically prestressing
the tube above the yield stress in torsion and then applying a longi
tudinal velocity at the end. In this case the fast wave caused almost
neutral loading, that is, as the fast wave passed a point on the tube,
the shear stress decreased and the longitudinal stress increased in such
a way that the stress state at that point remained close to the initial
loading surface. Then as the slow wave passed the same point, loading
occurred so that the stress path was normal to the initial loading sur
face. The second case was for a tube with a static longitudinal plastic
prestress impacted by a torsional velocity at the end. In this case the
fast wave caused unloading along the longitudinal stress axis followed

10
by an increase in shear stress at a constant value of longitudinal
stress and then the slow wave caused loading such that the stress path
was normal to the initial loading surface. Clifton (1966) also found
that for a given initial loading surface, the two wave speeds depended
upon the particular stress state on the initial loading surface, and
that for one particular initial loading surface the fast and slow wave
speeds were equal when the shear stress vanished.
This work of Clifton (1966) was a significant step forward in the
investigation of waves of combined stress. An extension of this work
was presented by Clifton (1968) in which the simple wave solution was
used along with unloading at the impact end. In this way certain unload
ing boundaries for combined stress states were determined. Two years
later Lipkin and Clifton (1970) published their experimental results
from combined stress wave propagation tests and compared these results
to the simple wave solution developed earlier. Agreement between the
simple wave theory and the experiments was fair.
Cristescu (1967b) formulated the problem of combined stress wave
propagation in a thin-walled tube using general quasilinear constitu
tive equations but again ignoring radial inertia effects. The equa
tions for the characteristic lines and the equations along these char-
axteristic lines were determined. No numerical results were given
but the two waves (fast wave and slow wave) were shown to be coupled
during loading. Again Cristescu (1971) showed that the coupling of the
waves of combined stress depended on the constitutive equations and
yield conditions used.

11
This concludes a brief survey of the history of the development
of plastic wave propagation theory. No attempt was made to give
a complete historical background. For more information the reader is
directed to Hopkins (1961), Kolsky (1963), Olszak, Mroz, and Perzyna
(1963), and Cristescu (1967a, 1968).
The remainder of this dissertation will be devoted to solving
the problem of combined stress wave propagation in a thin-walled tube
when radial inertia effects are included. A general quasilinear
constitutive equation for multiple states of stress will be presented,
and it will be shown to be a generalization of the constitutive equa
tions of both Lipkin and Clifton (1970) and Cristescu (1972). But
first the wave propagation problem itself must be developed.

CHAPTER 2
THEORETICAL DEVELOPMENT
The specific problem to be considered here is that of the propagation
of inelastic waves of combined stress along a semi-infinite thin-walled
tube, with the effects of radial inertia included. The material consti
tutive equation used is a generalization for multiple states of stress
of the quasilinear constitutive equation used by Cristescu (1972) for
a single stress component, and is a special case of the very general
quasilinear constitutive equation given by Cristescu (1967a). The coor
dinate system used is shown in Figure 2.1, and the stresses on an ele
ment of the tube are shown in Figure 2.2, where r^ is the mean radius of
the tube.
The problem is assumed to be axisymmetric so that there is no
dependence on 9. Since the tube considered is thin-walled, the stresses
a T and T are assumed to be negligibly small as are the strains
r r0 rx
and e The strain e is not included in the problem. Stability
r 0 rx r
of the tube wall and thermal effects are not included in the formulation
of the problem, and only small strains are used. The strain rate is
assumed separable into elastic, plastic, and visco-plastic parts. The
radial displacement is very small compared to the tube radius, and
plane sections of the tube remain plane. The material is assumed to be
isotropic and homogeneous, to obey the von Mises yield condition, and
to be isotropically work-hardening. All unloading is assumed to be
elastic.
12

03
Figure 2.1
Coordinate System for the Thin-Walied Tube

CT
r
Figure 2.2
Stresses on an Element of the Tube

15
The equations of motion in the cylindrical coordinates shown in
Figure 2.1 are given by
1 1
0* *f T 4- T 4. T p U
x,x rx,r r 0x,0 r rx r x,tt
T + u + T + ~(o o'.) = p u
rx,x r,r rr0,0 rr 0 H r,tt
T0x,x + Tr0,r + r CTe,e + r Tr0 ~ P U0,tt
which, under the assumptions given above, become
CT = P U 4-4-
x,x r x,tt
(2.1)
P u
r, tt
T0x,x p U0,tt
(2.2)
(2.3)
where the subscripts following the comma represent partial differentia
tion with respect to the variables x (the coordinate along the tube
axis) or t (time). The density of the material is p and u^, u^, and
u are the displacements of any point in the x, r, and 0 direction,
o
respectively.
For the cylindrical coordinates of Figure 2.1, the strain-
displacement equations are given by
e = u
r r,r
ee 3 r(u9,e + V
e = u
X x,x
, V1 1 ,
Sr9 = 2(7 Ur,0 + U0,r 7 V

16
e
rx
e
ex
= i(u +
u
)
2 x, r
I
>x
= x-(u. +
1
u
2 0,x
r
X
CD
)
and under the above restrictions, these reduce to the following three
equations
Defining
respectively,
0 =
X
u
x,x
(2.4)
e0 =
1
u
r r
o
(2.5)
0x =
1

2 0,x
(2.6)
the velocities v ,
X
v and v. as u u ,
r 0 x,t r,t
and U0,t
equations
(2.1) to
(2.6) become
o =
X X
p v ,
x, t
(2.7)
1!
CD
t>
-u
l
P Vr,t
(2.8)
T0X,X
P V0,t
(2.9)
£X, t
V
X, X
(2.10)
e0,t =
1
v
r r
o
(2.11)
60x,t
1
V
2 0, x

(2.12)
Under the assumptions used here, the variables no longer depend on r,
so that the problem becomes two-dimensional (the independent variables
are x and t) and can be solved by the method of characteristics.
The equations necessary for completion of the set of simultaneous
partial differential equations describing the behavior of the body are

17
the constitutive equations. Cristescu (1972) uses a full quasilinear
constitutive equation for a single longitudinal stress as
3e
at
1 da
E 'St
+ 0(a,e)
3a
"3t
:(a,e)
(2.13)
As a generalization of this equation to a constitutive equation
governing multiaxial states of stress and strain, the following equation
is used
e
ij
1 + v
E
a. .
ij
v
E
6 a +
ij kk 2
0(s,A)s + \ji(s
,A)J
ij
s
(2.14)
where the dot represents partial differentiation with respect to time,
s is the deviatoric stress, 6 is the Kronecker delta, v is Poisson's
ij iJ
ratio, E is Young's modulus, 0(s,A) and i|f(s,A) are material response
functions as yet unspecified, and s and A are defined as
s = /- s. .s. (2.15)
V 2 xj ij
A
/I
V 3
J
P *P
e e
ij iJ
dt +
E
(2.16)
P
and e is the inelastic portion of the strain rate which, using
equation (2.14), can be written as
['
0(s,A)s + ^(s
IS..
(2.17)
when the elastic, plastic, and visco-plastic portions of the strain
rate are assumed to be separable. The constitutive equation (2.14)
is a special case of the equation
e
ij
_pk!
f. a + g. .
ij hi ij

18
given by Cristescu (1967a). The form of equation (2.14) was chosen as
the general constitutive equation because it contains terms which may be
considered separately as elastic, plastic, and visco-plastic strain-
rate terms, because the inelastic strain-rate tensor is proportional
to the corresponding deviatoric stress tensor, and because it reduces
to the form of equation (2.13) when the only stress present is the longi
tudinal stress. This simplification to the form of equation (2.13) is
shown in Appendix A.
The functions 0(s,A) and i(r(s,A) are functions which depend on the
particular material being studied. The function 0(s,A) is a measure of
the rate insensitive inelastic work-hardening, and the function iji(s,A)
is a measure of the visco-plastic strain rate due to the strain-rate
sensitivity of the material. In the classical rate independent plastic-
- - *
ity theory, i|i(s,A) vanishes. When s < 0 or when s < u (the current
"yield stress"), 0(s,) is set equal to zero. The unloading conditions
when i|r(s,A) = 0 are stated in equation (A. 3.1).
Two separate materials are modeled in the numerical work done.
One is a 3003-H 14 aluminum alloy used in the experimental work of
Lipkin and Clifton (1970). This material is assumed to be insensitive
to strain rate and the functions 0(s,A) and i)/(s,A) are obtained using
the classical Prandtl-Reuss incremental plasticity theory with iso
tropic work-hardening and the stress-strain curve for uniaxial tension.
(See Appendix A.) The other material used is a commercially pure
aluminum dead annealed at 1100F. This material is assumed to be
strain-rate sensitive, and the functions 0(s,A) and jf(s,A) are
obtained from the data given by Cristescu (1972). (See Appendix A.)

19
Since the stresses cr f T and T are assumed to vanish,
r r8 rx
equation (2.14) as applied to the present problem reduces to
\
I I
e :,t = 1 CTx,t-| e,tV<2ffx-'V L0<5'i,5 + *<:-4)J
2 s
X,
ee,t = -iCTx(t + iCTe,t + ^(2ae-CTx)L
1 + V
3
+ T,
'9x,t E 9x,t 8x
2s
0 (s A) s + i|f(s.A)J
0(s,A)s + ty(s,A)l
(2.IS)
where the deviatoric stresses are
s = s = (2ct ctq)
x 11 3 x 9
Se = S22 = 3<29 ax1
Sr = S33 = 3(x+V
S0x S12 S21 T0x
s = s = 0 .
r 0 rx
Using these deviatoric stresses, the expression for s becomes
9s 9 3
S ~~ dt 9t L2 SijSij_
"¡e i rs ~¡ssr3 1
jj ~~ 2 |_2 SklSklJ ~St L2 SijSijJ
1 3 P s s 1- 3 3 r
s =
2s
_3_ 5
4s
S11S11 + S22S22 + S33S33 + S12S12 + S21S21
]
2 2 2 2
- q(o- + - ot _3 x o x 0 0
J
-L [<2ax V4x,t+ <% -x)4e,t+ 6TexTex,t]
(2.19)

20
and equations (2.18) become
0(s,A)J
e =2 +
n (2ax ~ ae)2
x,t
4s
Tv (2cjxCTe)( ax,t-lj + ^2 0(SA)J CTe,t
4s"
-6(2a cr )tc (2a aQ)
~-2 0 (2.20)
4s
2s
p (2a ac) (a 2aQ)
! v x G x 0
" 8, t
L
E
0 (s
4s
A)]ax,
m + 0(S,A) JCTg
4 s2
r6(2a0 -CTy)T(
+ L
0(s
,A)]
0X, t
2a0 ax
+ - j(s,A)
2s
(2.21)
Gx, t _
3t (2a a)
0x x 0
-2
4s
1 +
18t\
yx
0 ( S
0(S
,A)]
,A)]
CT +
X, t
'-3Tex(2£70-CJx)
4;2
0(S
'4)]V
T0X,t +
0X
\|r(s,A)
(2.22)
4s -1 2s
The equations (2.7) to (2.12) and equations (2.20) to (2.22) form
a set of nine simultaneous hyperbolic quasilinear partial differential
equations for the nine unknowns o x 0 bx x t x b
e A special case of this system of equations is the set of equations
6x
obtained by neglecting radial inertia effects. When radial inertia
effects are ignored, the variables a, s and v are not included
0 0 T
directly in the problem formulation. This case can be incorporated
into the more general formulation by multiplying a., s v and their
0 0 r
derivatives by the dummy variable "a," where "a" has the value of 1
when radial inertia effects are included and the value of 0 when radial
inertia effects are neglected. Also the equation of motion, the kine
matic equation and the constitutive equation for motion in the radial
direction must be multiplied by "a." Doing this, and defining the
quantities

21
1 (2ox aCTfi)2
AX = E + =2 0(S'A)
4s
A2 = ~ Le +
(2ax aCT0> (ax 2acG)
1
0(s,A) J
- 6Tex<2ox V
A3 = 0(s,)
4s
1 (CTX 2aCTfi)2
A4 = E + ^ 0(sA)
4s
r6Tex(CTx ~ 2aCT8)
4s2
0 ( S
,A)J
- 2(1 +v) 36T0x .x
A6 = + ~=2 0(SA)
4s
the nine simultaneous equations (2.7) to (2.12) and equations
(2.22) can be written as
cr = p v
X, X r X, t
T- CTe = ^r.t
O
6x,x ^V0,t
e = v
x,t x,x
ae
9, t
a
v
r r
o
0X, t
1
V
2 8, x
'x, t
2x aCTe
+ Vex.t +
ae
8, t
2ao CTX
V*,t + Ve.t + Vex.t + a T
' r\
ox, t
2 A3x,t + 2 A5CT8,t 2 A6'8x,t
3t
6x
+ i|Ks>A)
(2.23)
(2.20) to
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
(2.31)
(2.32)
2s

22
Eliminating the strain rates from the last six of these equations,
and defining
, 2ctx ~ acJe s
- t(s,A)
X 2s
2acfi a
ij/g = ^ (s ,A)
2s
3t
0x ,i,
Gx
?(s,A)
(2.33)
2s J
the system of nine equations reduces to the following system of six
equations for the unknown variables a aA, T. v vA, and v
x 0 Gx x 9 r
where
a =
X, X
4->
K
>
Q_
ii
CD
b
al I, O
a p v
r,
Gx,x
PV8,t
V =
X, X
V*,t
a
v =
r
o
a u
2 x,
ii
X
CD
>
Vx.t
s =
(cr2 .
X
aAT,
X
(2.
34)
+ avlr 0
(2.
35)
2\|f _
' 0x
(2.
36)
3tL>*
(2.37)
Since the equations (2.24) to (2.26) and (2.34) to (2.36) form
a system of hyperbolic equations, they can be solved by the method of
characteristics. To do this, first the equations for the characteristic
lines must be determined, and then the equations along these characteristic

23
where
A w, = B w, ^ b
A =
P
0
0
0
0
0
0 0 0 0 0
ap 0 0 0 0
0 p 0 0 0
0 0 A1 aA2 A3
0 0 aA aA aA
Z "x O
0 0 A aA_ A.
3 5 6
(2.38)
(2. 39)
0 0 0 -1 0 0 -¡
0 0 0 0 0 0
B =
0
-1
0 0 0 0 -1
0 0 0 0 0
0 0 0 0 0 0
0 0 -1 0 0 0
(2.40)
v
X
V
r
a
X
cr
9
L
Gx
(2.41)
and

24
b =
a cr.
0
-
x
av
r 9
- 2i|r
6x
(2.42)
and if the slope of the characteristic line (or the wave speed ) is
denoted by c where
H v
(2.43)
c =
dt
then the equation for the characteristic lines is given by
c A B = 0
(2.44)
and from the calculations shown in Appendix B, equation (2.44) yields
|^a2p c^J j^(p c2) [a] (pc2) {b} + J = 0 (2.45)
where
s + SciA^A^Aj sA^A^.
^ A1A4 aA2 + A4A6 ^5
aAoAc A0A
2 o o 4
>>
V
Setting the first factor in (2.45) equal to zero,
dx
ac = a = 0 (twice)
dt
and, setting the second factor in (2.45) equal to zero,
i
c =
"2ap
f- -2 A")
jb (b 4aA4)e
IT .
(2.46)
(2.47)
(2.48)

25
If the wave speeds in equation (2.48) are denoted by
(2.49)
and
b (b'" 4aA^)3
'}]
l
s
(2.50)
where is the fast wave speed and is the slow wave speed, then
the slopes of the characteristic lines are given by
(twice)
c = c
f
c
c
s
Equations (2.47) and (2.48) are the six equations for the character
istic lines for the set of six simultaneous, hyperbolic, quasilinear
partial differential equations of (2.38). When radial inertia effects
are not included (that is, when a=0), the equations (2.47) vanish
identically and the remaining four simultaneous equations of (2.38) have
the characteristics given by equations (2.48). For this case (a=0) and
when 0(s,A) is obtained from incremental, rate independent plasticity
theory with isotropic work-hardening, equations (2.48) reduce to those
given by Clifton (1966) as shown in Appendix B.
The equations along the characteristics can be obtained in two
different ways, both of which are discussed in Appendix B. The result
ing equations along the characteristic lines of equation (2.48) are

26
O =
(p c2) (A
A .X -aA2) -A. dv -pd aA A -A A dvQ-
46 5' dj x M l. 2 o 3 4j 9
3A A A A
Z D O <£
]dV
pe
(p c2) (A4A6-aA2) -A4J dav +
(p c2) (A A -aA2)
x L 4 6 5
d]
-(2o acr )
x 0
2s
']
i|>(s,A) dt
v (o 2actq)
o -] r-v (.0 -j
+ al (pe ) (A A.-A A )-A + t(s,A) dt
2 6 o 5 2J [_r _l
o 2s
-2
2(pe )
~A3A4J [^(s,)Jdt
aA2A5
2s
(2.51)
0 =
(P2)(A2A6-A3A5)-A2]dVx^
(pe ) (A1A5-A2A3)-A5 ¡dvf
J
4£(Ps2)(A1A5-23)-xjdTex+
pe pe
ex' -2Llfc|(Ve-i3M%
ti
d][
(2a aa )
x 0
2s
i|Ks>A)Jdt
-1j[(pS2)2(V6--
pe
+ 2
1 FV '-u j -i
a2) (pc )(A +A ) + l j ZL + t(s,A) dt
x D L_r0 2s -J
i r3t
.(PC )(AiA5-A2A3)-A5J
0X
^ 2
, A)J <
t(s,A) dt
(2.52)
2-3
0 = p c
n
aAA -AA I dv o c
2 5 3 4J x
(p c2) (A1A4-aA2) -A J dv
e
[(pc2)(A1A4-aX2)-Xjd
Vs-vJ
9x
, -2.
+ (p c )
aA A -A A. da
. 2 5 3 4J x
a -2,2
+ (p c )
-(2a aa.)
x 6
2s
,A)J'
f(s,) dt

27
_2 r -2 - - ry (d 2aa )
+ a(pc ) (pc ) (A1A¡5-93)-A I j ^ + - iHs,A) ldt
^ o 2s
s,A)J'
-2 r -2 -2-1 r1 sx i
+ 2(p Oj^Cp O (14-aA^)-A4J i)r(s,A)Jdt .
(2.53)
These three equations each represent four equations, one equation
in differential form along each of the four charaxeristic lines of
equation (2.48). When the waves are coupled, equations (2.51), (2.52),
and (2.53) are identical. That is, by multiplying equation (2.52) by
the quantity
(p-2)(ay5 y4>
(pc") (A A. A A ) A
1 3 O
and using equation (2.48), equation (2.51) is obtained; or by multiply
ing equation (2.52) by the quantity
pc|_(pc ) (A1A4 aA2) A4J
(p c")(xa5 23) a5
and using equation (2.48), equation (2.53) is found. When the numerator
and denominator of these multiplying quantities do not vanish, equations
(2.51), (2.52), and (2.53) are identical. However, when the waves
become uncoupled, a phenomenon discussed in Appendix B, A and A vanish.
o
In this case the multiplying factors used above become undefined and
the equations (2.51), (2.52), and (2.53) are not the same. When
A -A 0 the equations (2.51) and (2.53) reduce to equations (B.4.4)
3 5
and (B.4.6), respectively. Under these conditions, equation (2.52)
also reduces to the form of equation (B.4.4).

28
The equations along the characteristics of equation (2.47) may be
obtained (since along these characteristics there is no variation in x)
directly from equations (2.25) and (2.35) by multiplying each of these
equations by the increment of time, dt. These equations then yield
a8
- dt = a p dv (2.54)
r r
o
av
dt = aA^do + aA.don + aA^dT^ + ailr.dt (2.55)
r 2x 40 5 9xT0
o
The equations for the six characteristics and the equations along
these characteristics, along with the appropriate initial conditions
and boundary conditions, represent a complete mathematical formulation
of the problem, and the solution to these equations is the solution to
the problem posed here. The solution to these equations will be
obtained by using a finite difference numerical technique which will
be discussed in the next chapter.

CHAPTER 3
DIMENSIONLESS EQUATIONS AND NUMERICAL PROCEDURES
3.1 Wave Speeds as a Function of
the State of Stress
In this chapter the numerical schemes used to find the solution to
the wave propagation problem of Chapter 2 will be presented. In this
first section the dependence of the wave speeds on the stress state
will be shown. The stresses a T ,, and T have already been assumed
r r9 rx
negligibly small so that the scalar representation of the stress state
is given by equation (A.1.2) as
s =
2
a
x
aa C- +
x b
(3.1.1)
Next, the new variables o', o', and T' will be defined so that the
X 0 0X
surface s = constant can be represented in terms of these variables as
a sphere, and the stress state on this surface in terms of these new
variables can be described in terms of spherical coordinates. Now
defining,
o
X
+
ace)
/ a/3 ,
On = (aff.
CTx)
y
(3.1.2)
equation (3.1.1) can be written as
s
i
/ 2 / 2 / 2~I 8
,CTx + CTe + Texj
(3.1.3)
29

30
and defining the angles y and 5' as shown in Figure 3.1 these new
variables defined in equation (3.1.2) can be written as
q s cos v cos 6
x '
Cg = s cos y sin 6 \ (3.1.4)
V = Sin Y >
The angle y is the complementary angle to the one normally used in
spherical coordinates. It is used here to facilitate comparison of
results obtained later on to already published results.
From equations (3.1.2) and (3.1.4),
CT0 s cos y sin 6 ^ ^ g 2^x+aCT0^
<7^ S COS y cos 6 a/3
CTx+aCTe
vo, N y3 (aa. a )
r^e-V 9 x
(3.1.5)
so that the a and a axes are located by
x 0
c -axis : o = 0 and tan 5 =
x 0
ar-axis : ct = 0 and tan 6 =
0 x
¡3
V3
and 6 = -60
and 6 = +60 .
(3.1.6)
In order for the equations (3.1.4) to reduce automatically to the
simpler case when radial inertia is not considered, the angle 6 is
defined as
6 = a6' + (a-1)60 (3.1.7)
so that when radial inertia effects are included, a=l, and 6 = 6/, and
when radial inertia effects are not included, a=0, and 6 --60 which
from equation (3.1.6) automatically causes crQ to vanish as it should.
9

/
7 ex
X
Figure 3.1
Yield Surface Representation in Spherical Coordinates
co
k-*

32
Using the uniaxial stress-strain curve in the form of equation
(A.2.13), the universal stress-strain curve can be written as
| + B n
e y
(3.1.8)
and letting E^(s) be the tangent modulus of this curve, yields
dA
E^Cs) ds
1 /- xn-l
E + Bn
(3.1.9)
and from equation (A.2.18) this becomes
77 + 0(s)
E
Et(s)
or
0 (s)
Et(5)
1
E
(3.1.10)
Now 3 = ¡3(s) is defined so that
where
E (s) = 0(s) E
t
0 £ 3(s) 1
(3.1.11)
and when 3 = 1, the material is elastic, and when 3 = 0, the material
is perfectly plastic. Using equation (3.1.11) in equation (3.1.10),
0(s)can be written as
0(s) = \ ( 1) (3.1.12)
E 3(s)
Inverting equations (3.1.2), the stresses are given by
a = o' al ^
X X J3 6
ae CTx + CTe
1 /
0X yg. 0X
(3.1.13)

33
and using equation (3.1.4) these become
ct = s cos
x
y j^cos 6
- sin 6
/
V o
n
j
~\
aCTg = s cos y |^cos 6 +
- = s sin y
0X
^3
1 ,1
sin §
4 J
(3.1.14)
y
Now substituting equations (3.1.12) and (3.1.14) into equation (2.23),
-if i i 2 2 1 2~^
A^ = j 1 + (g- 1) () (cos y) (2 cos 6 sin 6 cos 6 sin 6) J
V3
JS
_ 1 P 1 1 2 2 1
A = v + (?r 1) cos y(2 cos 6 sin 6 cos 6 sin 6)
2 E L 4 p T
V3
i 2 r n
(cos 6 sin 6-2 cos 6 sin 6) I
/3
*/3
*/3
- i n i 6 i 2 i ~i
A-j = (-r) ( sin y) (cos y) (2 cos 6 sin 6 cos 6 sin 6)
3 E L 4 a/3 J
A4 = I [X + X) (cos 6 si

\ = | [2d +v) + ~ (|-1)(| sin2Y)]
sin 6-2 cos 6 sin 6) cos2Y
\/3 J
1 2
sin y) (cos y) (cos 6 sin 6-2 cos 6 sin 6) I
vy3 v^3 V3 J
or
i r ii 2 2
At = |^1 + ^(^ 1) (cos v) (cos 6 2 y3
2
sin 6 cos 6 + 3 sin 6)
J
A =
i |^v + i(i 1)(cos2y)(-cos26 + 3 sin"6)J
3 1 ~]
)(-g--l)(sin y cos y) (cos 6 -*/3 sin 6) I
3 E L 2 3
1) cos v(cos6 + 2 cos 6 sin 6+3 sin 6)
]
>
(3.1.15)
A.
A,
= ~ p 1) (sin y cos y) (-cos 6 Sin 6) J
1
1
[^2(1 + v) + 3(1-D sin Yj

34
The elastic wave speeds are defined from equations (B.5.4)
; = /r ^
C1 =
p (1 v )
j
"2 Jo J
and the wave speeds from equation (2.48) can be written in
less form as
c'2 = ^ ^ {- [B ^~ 4V J )/
(G/p)
,2-yy [5 44;]
or
/2 1+vf Jr /*7T2 2 r 3 -
: = -33- I E b V (E
E a
b) 4(EA4)(E a)
J
By defining the dimensionless functions from equation (3.1.
A. = E A.,
1 1
i = 1,2,...,6
and
3 o 2 2
a' E a = A, A.A + 2aA AA,_ A'A. aA A aA A
146 235 34 15 26
b' = E)2£ = AA aA2 + A4A6 aA2 ,
then the wave speeds in dimensionless form become
/2 1 + v
7" _b' J b'2 ~ 4A4a' ] = (r
" \2
c \
as
(3.1.16)
dimension-
(3.1.17)
15) as
(3.1.18)
(3.1.19)
(3.1.20)
A computer program was written to solve this equation for the two
positive wave speeds as functions of the angles y and 6 for specified

35
values of v and 3. This program is listed in Appendix C. This pro
gram also calculated the values of the normalized stresses a /s,
x
a./s, and T /s as functions of v and 6. These results are given in
0 8x
Tables 1, 2, and 3. The wave speeds are shown in Figure 3.2 for
the case when 6 = -60, which corresponds to oa = 0. Also plotted in
Figure 3.2 are the results given by Clifton (1966). It is obvious
that the results are not the same and that including radial inertia
effects in the formulation of the problem can have significant effects
on the wave speeds and that, for any given state of stress, the waves
are always faster when radial inertia effects are included. The
results plotted in Figure 3.2 do not correspond to the case when
a = 0. That is, although crQ = 0 when 6 =-60 aQ does not necessarily
b 0
vanish for this case. When a = 0, the results obtained were identical
to those of Clifton (1866) as they should be, since a=0 corresponds
to the absence of radial inertia effects.
An interesting phenomenon can be observed by remembering that the
physical presence of radial inertia is due to the Poisson effect.
That is, the longitudinal (fast) wave speed would be expected to be
the same when a = 0 (no radial inertia effects) as when v = 0 (the cause
of the radial inertia effects vanishes). However, in the fomulation
of this problem it is tacitly assumed that Poisson's ratio for the
inelastic portion of the material behavior is i, or that the material
behavior in the inelastic range is incompressible. When the material
is elastic (3 = 1), this Poisson effect can be studied directly.
Comparing equations (3.1.15) with (B.5.1) when 3=1, the wave speeds
are given by equation (B.5.4) as

36
TABLE 1
/CTx\
NORMALIZED LONGITUDINAL STRESS (
\ >
s
Delta
Gamma
0o
o
O
rH
20
30
o
O
1
O
o
1
0.57735
0.56858
0.54253
0.50000
0.44228
1
GO
O
o
0.74223
0.73095
0.69747
0.64279
0.56858
o
O
I
0.88455
0.87111
0.83121
0.76605
0.67761
o
O
O
1.00000
0.98481
0.93969
0.86603
0.76604
-50
1.08506
1.06858
1.01963
0.93969
0.83121
O
O
1
1.13716
1.11988
1.06858
0.98481
0.87111
-30
1.15470
1.13716
1.08506
1.00000
0.88455
1
co
o
o
1.13716
1.11988
1.06858
0.98481
0.87111
-10
1.08506
1.06858
1.01963
0.93969
0.83121
0
1.00000
0.98481
0.93969
0.86602
0.76604
O
o
0.88455
0.87111
0.83121
0.76604
0.67761
20
0.74223
0.73095
0.69746
0.64279
0.56858
co
o
0
0.57735
0.56858
0.54253
0.50000
0.44228
o
o
0.39493
0.38893
0.37111
0.34202
0.30253
50
0.20051
0.19746
0.18842
0.17365
0.15360
60
0.00000
0.00000
0.00000
0.00000
0.00000
O
O
-0.20051
-0.19747
-0.18842
-0.17365
-0.15360
o
O
00
-0.39493
-0.38893
-0.37111
-0.34202
-0.30254
90
-0.57735
-0.56858
-0.54253
-0.50000
-0.44228

37
TABLE 1 (Continued)
Gamma
Delta
50
60
O
O
t>
00
; o
o
90
-90
0.37111
0.28868
0.19747
0.10026
0.00000
1
00
o
0
0.47710
0.37111
0.25386
0.12889
0;00000
l
<1
o
o
0.56858
0.44228
0.30253
0.15360
0.00000
-60
0.64279
0.50000
0.34202
0.17365
0.00000
o
o
m
!
0.69747
0.54253
0.37111
0.18842
0.00000
o
O
1
0.73095
0.56858
0.38893
0.19747
0.00000
I
CO
o
o
0.74223
0.57735
0.39493
0.20051
0.00000
o
o
Cj3
0.73095
0.56858
0.38893
0.19747
0.00000
-10
0.69747
0.54253
0.37111
0.18842
0.00000
0
0.64279
0.50000
0.34202
0.17365
0.00000
O
o
0.56858
0.44228
0.30253
0.15360
0.00000
20
0.47709
0.37111
0.25386
0.12889
0.00000
30
0.37111
0.28867
0.19747
0.10026
0.00000
O
O
0.25386
0.19747
0.13507
0.06858
0.00000
50
0.12889
0.10026
0.06858
0.03482
0.00000
O
O
CD
0.00000
0.00000
0.00000
0.00000
0.00000
o
O
-0.12889
-0.10026
-0.06858
-0.03482
0.00000
00
o
o
-0.25386
-0.19747
-0.13507
-0.06858
0.00000
CD
O
o
-0.37111
-0.28868
-0.19747
-0.10026
0.00000

TABLE 2 NORMALIZED HOOP STRESS
Gamma
Delta
0
H-1
O
o
O
O
CO
o
o
O
O
o
! o
CT>
-0.57735
-0.56858
-0.54253
-0.50000
-0.44227
o
O
CO
1
-0.39493
-0.38893
-0.37111
-0.34202
-0.30253
-70
-0.20051
-0.19746
-0.18842
-0.17365
-0.15360
-60
0.00000
0.00000
0.00000
0.00000
0.00000
o
O
in
i
0.20051
0.19747
0.18842
0.17365
0.15360
o
O
1
0.39493
0.38893
0.37111
0.34202
0.30254
1
CO
o
o
0.57735
0.56858
0.54253
0.50000
0.44228
1
to
o
o
0.74223
0.73095
0.69747
0.64279
0.56858
-10
0.88455
0.87111
0.83121
0.76605
0.67761
0
1.00000
0.98481
0.93969
0.86603
0.76604
h-1
o
o
1.08506
1.06858
1.01963
0.93969
0.83121
20
1.13716
1.11988
1.06858
0.98481
0.87111
CO
o
o
1.15470
1.13716
1.08506
1.00000
0.88455
o
o
1.13716
1.11988
1.06858
0.98481
0.87111
50
1.08506
1.06858
1.01963
0.93969
0.83121
60
1.00000
0.98481
0.93969
0.86602
0.76604
70
0.88455
0.87111
0.83121
0.76604
0.67761
00
o
o
0.74223
0.73095
0.69746
0.64279
0.56858
CO
o
o
0.57735
0.56858
0.54253
0.50000
0.44228

39
TABLE 2 (Continued)
Delta
Gamma
o
O
60
70
00
o
o
90
1
CD
O
o
-0.37111
-0.28867
-0.19747
-0.10026
0.00000
1
00
o
o
-0.25386
-0.19746
-0.13507
-0.06858
0.00000
I
<1
o
o
-0.12889
-0.10026
-0.06858
-0.03482
0.00000
-60
0.00000
0.00000
0.00000
0.00000
0.00000
1
cn
O
o
0.12889
0.10026
0.06858
0.03482
0.00000
o
O
1
0.25386
0.19747
0.13507
0.06858
0.00000
o
O
CO
1
0.37111
0.28868
0.19747
0.10026
0.00000
-20
0.47710
0.37111
0.25386
0.12889
0.00000
o
O
H
1
0.56858
0.44228
0.30253
0.15360
0.00000
0
0.64279
0.50000
0.34202
0.17365
0.00000
O
o
0.69747
0.54253
0.37111
0.18842
0.00000
to
o
o
0.73095
0.56858
0.38893
0.19747
0.00000
CO
o
o
0.74223
0.57735
0.39493
0.20051
0.00000
o
o
0.73095
0.56858
0.38893
0.19747
0.00000
50
0.69747
0.54253
0.37111
0.18842
0.00000
o
o
CD
0.64279
0.50000
0.34202
0.17365
0.00000
o
O
0.56858
0.44228
0.30253
0.15360
0.00000
00
o
o
0.47709
0.37111
0.25386
0.12889
0.00000
90
0.37111
0.28867
0.19747
0.10026
0.00000

40
TABLE 3 NORMALIZED
SHEAR STRESS
(v)
S'
Gamma
-F Tx
Value of
s
Gamma
T
.. 9x
Value of
s
0
o
o
CJl
o
o
0.44228
O
O
tH
0.10026
60
0.50000
20
0.19747
70
0.54253
GO
o
0
0.28868
o
O
CO
0.56858
0
o
0.37111
90
0.57735
c =
p (1 av )
c
s
2p(l + v) C2
It is now obvious that the fast wave speed is the same when a=0 and
v^O as when a = 1 and v = 0. The slow wave speed (and consequently c^)
is the same when a=l as when a=0, although it does depend on v.
Because of this dependence of c^ on v, the dimensionless fast wave
speed of equation (3.1.20) will have values when a = 0 and v^O differ
ent from those when a =1 and v = 0.
Also in Figure 3.2 it can be seen that when a=0 and y = 0 the
fast and slow wave speeds are the same for 3= .385 and v= .30. There
is usually some value of |3 for which the two wave speeds are equal
at y = 0 for each combination of values of v and 6. The condition for
which this is true can be obtained from equation (3.1.20) and is

Plastic Wave Speeds,
Figure 3.2 Plastic Wave Speeds as Functions of ¡3 and y for Poisson's Ratio of 0.30

42
b'- 4a'A = 0 .
4
When y = > equations (3.1.18) and (3.1.19) are
11 2 r- 2
A = 1 + (-pr 1) (cos 6-2 a/3 sin 6 cos 6 + 3 sin 6)
1 4 P
A^ = v + ~(tt 1) (cos"6 3 sin26)
A
a3 = 0
1 i ' 2
A = 1 + (-g- 1) (cos 6 + 2 a/3 sin 6 cos 6+3 sin 6)
T l)
4V3
A5 =
Ag = 2(1+v)
J
and
a = A A .A aA A
14 6 2 6
A1A4 + A4A6
aA,
and using the same manipulations as in Appendix B, Section 4,
equation (3.1.21) becomes
0 = WV + ^2
where A^, A^, A^, and Ag are given by equation (3.1.22). Now
f ^ = f^(6) = i |^cos^6 2 f/3 sin 6 cos 6 + 3 sin26^
(6)
i j^cos26 + 2 ^3 sin 6 Cos 6+3 sin26^
f = f (6) = -i r
3 3V 4 L
z = z(P ) = -g-
c c 6
cos26 3 sin6
]
(3.1.21)
(3.1.22)
(3.1.23)
(3.1.24)
defining
/ (3.1.25)

43
the critical value of 3 is found from
0 = 1 + vj [2(1+'') -(1+Vi>]+ a L-(v+zcVJ
'= Zc [flf2 + 0I1] + Zc [f2(1+',)-1}
J 2C1 + v) -1+ av2J .
(3.1
The expressions for the critical value of 3 will now be found
for the two separate cases of a=0 and a=l. First, when a=0,
equation (3.1.24) becomes
A6 = \
2(1 + v) = 1 + z f
c 1
1 + 2v
3. =
\
(3.1
c z +1
c
and in this case § = -60 from equation (3.1.7) and
f^ = i j^cos(-60) 2 */3 sin (-60) cos (-60) +3 sin2(-60)J
h =1
and
P = i
Hc (1 + 2v) + 1
1
3_ -
c 2(1 + v)
.26)
. 27)
(3.1.28)

44
When a = 1,
2 2 ~ 2 2
(cosJ5 + 2 */3 sin 6 cos $ + 3 sin 6) (3 sin 6 cos
and equation (3.1.26) becomes
-[2(1 + v) 1 -i-v2]
z
c [(2) (1 + v) -1] f2 f1 + 2vf3
- (1+v)2
(3.1.29)
z
2
(1 + v) sin 6 cos 6 + 3v sin 6
A short computer program was written to calculate the critical
values of 3 (using equation (3.1.28) when a = 0 and equations (3.1.29)
and (3.1.27) when a = l) for various values of v and 6 when y = 0.
This program is shown in Appendix C and the results are plotted in
Figure 3.3. The only values of 3 which are physically possible are
between 0 and 1 and therefore only values of 3 in this range are
plotted in Figure 3.3. For all other values of 6, there is no phys
ically possible critical value of 3; that is, there is no value of
3 such that the fast and slow wave speeds are equal at y =0.
For the case when cQ = 0 (6 = -60), for any value of v the critical
W
value of 3 is smaller when radial inertia effects are included.

Critical Value of
Figure 3.3 Values of |3 at Y = 0 for which c = c = c
' f s 2

46
3.2 Characteristic Solution in Terms
of Dimensionless Variables
In order to make the numerical solution in the characteristic
plane more general, the equations for the characteristics and the
equations along the characteristics given in Chapter 2 will be written
in terms of dimensionless variables. The dimensionless variables
used are
S =
x
cr
x
TT
v
X
V
X
c =
) (3.2.1)
(s,A) i|/(s,A)
Z1
§(s,A) = E0(s,A)
I (s,A)
Y(s,A) =
where c^ given by equation (3.1.16) is the elastic longitudinal wave
speed when radial inertia effects are included. In terms of these
variables, the functions defined by equations (3.1.18) can be written
from equations (2.23) as

47
(2S aS )
A1 = EA1 = 1 + S(s,A)
A = EA
2 2
-[
4s
(2S aS.)(S 2aS)
x G x 9
v + f(s
4s^
6t(2S aS )
A3 EA3 = $(s,A)
4s
(S ^ 2aS )
A4 = EA4 = - X 2 ?(s,A) + 1
4s
6T(S 2aS )
fl5 = s = 2 ~ 4 4s
- 9t
Ag = EAg = 2(1 + v) + $(s,A).
s
,&)]
y
(3.2.2)
Using these and equations (3.1.19), the fast and slow wave speeds can
be written from equations (2.49) and (2.50) as
Cf =
! 2a P
c =
s
C1 -2a p
<*'2 '*>*}
or
^ V
(b'2 4,,'a/1
- [7^ {' <*'2 '*>*} ]
\
(3.2.3)
J
and the wave speeds in dimensionless form are given by equations
(3.2.3) and (2.47) and (2.48) as

48
c =
c = c V (3.2.4)
S (
c = 0 (twice)
J
where the wave speeds were written in dimensionless form by dividing
the wave speeds by c^. This was done because c is the fastest wave
speed possible in the problem considered here, and thus all the
dimensionless wave speeds have values in the range
-l £ c <: l .
Mien radial inertia effects are not included, the maximum value of the
fast wave speed is
Cf
max
so that, for a = 0, the maximum value of the dimensionless wave speed is
cfl
max
V E/o
p (1 v )
Next, the equations along the characteristics for fully coupled
waves will be written in dimensionless form. Along the vertical char
acteristics (c = G), the equations can be written from equations (2.33),
(2.54) and (2.55) as
-aS0 dT
I
L
2(1
v2)
]
dV
r
or
- a [2(l-v2)S ] dT = adV
u r
(3. 2.5)

49
and
r- /2aSg S -j
a (2V dT) = a! AdS + A dS. + A dT + ( \¡J (s,A)dT .
r L 2 x 4 0 5 \s/ o J
(3.2.6)
The equations along the nonvertical characteristics (c = c_^, c^)
can be written in three different forms from equations (2.51), (2.52)
and (2.53). In dimensionless form these become, respectively,
2 i r* -1
0 =
c
1 V
x 2LaA2VA3\JdV9
1 V
+ [aA2A5"A3A4] dT+ ^2
c 1 v
5> -fllldSx
]
^ 1 -v
-| r-2S as
l_hJ vs'A,dT
/ 2 \ i p S 2aS
' ^)(A2A6 A3-V AJ [2Vr +
1 v
2
o(S,A)J
dT
+ 6
*)[
3A A A A
2 5 3
J[i o dT
(3.2.7)
and
0 =
I l(t5-2)(Vs-a3a5>-a2]dvx-1 [(r^KvAMj.
1 v 1 V
(A A A A ) A IdS
- aJ
, (AA AA
LA, 27 2 6 3 5
1 v
5> AJ J
-2S aSg
^c(s,A) |dT
,A)J'
(^) [(:
2' Lf (Aia6 A2) M-5; (a1 + a6) + i
1 v 1 V
]

50
r
S 2aS.
I 2V + -i J
L r s
(s, A) IdT
+ 6
- 1 v
27 (A1A5 A2A3)
a5][} I."'1*]"
(3.2.8)
and
0 = -
2 2
(1 -v )
aA A -A A, dV
2 5 3 4J x 2
1 v
(-4) -AJdve
1 V
[(r4)-AJdT+ (r^s) [A2A5-A3AJdSx
1 V 1 V
2 % 2 p i -2S aS. -j
777) l^vvJL-t-- o 2 \ p / 2 v -t r S 2 aS A 1
7C) [i-" )(AlA5-A2A3>-A5][2Vr +^LT-? to'84]
i v 1 V
+ a
+ 6 (:
dT
-4) [(tAKv^-vI ¡} t0]dT
1 V 1 V
(3.2.9)
When the equations are uncoupled (A = A. =0), the equations along the
3 5
characteristics are given by equations (B.4.7) and (B.4.8). In dimen
sionless form, the equations (B.4.8) along the vertical characteristics
(c = 0) become
- a[2(a v2)Sd] dT = adV (3.2.10)
0 r
and
/2aSp S v
a[2VrdT] = a[A2dSx +A4dSQ+ { 1 -J *o(s,A)dT] .
(3.2.11)
The equations (B.4.7) along the nonvertical characteristics (c = c^)
for the fast waves are given by

51
(3.2.12)
and the equations (B.4.7) along the nonvertical characteristics
(c = c ) for the slow waves can be written as
s
0 = D
6c
(1 -v2)
dV. + dT +
V
(1
2
V )
(-) t (s,A)dT
S o
']
(3.2.13)
The equations (3.2.5), (3.2.6), and (3.2.7), or (3.2.8), or
(3.2.9) are the equations along the characteristics for the fully
coupled waves written in differential form in terms of the dimension
less variables. The solution to this set of equations will be obtained
numerically by writing them in finite difference form, and then solving
the resulting set of algebraic equations simultaneously. When the
waves are uncoupled, the equations along the characteristics are given
by equations (3.2.10), (3.2.11), (3.2.12), and (3.2.13). These equa
tions will also be written in finite difference form and solved (when
applicable) in the same manner as described for fully coupled waves.
The procedure for obtaining these finite difference solutions is out
lined in the next sections.
3.3 Numerical Grid for Characteristic Solution
Since the slopes of the characteristic lines at any point in the
characteristic plane depend upon the state of stress at that point and
upon the history of the deformation at the corresponding location along
the axis of the tube, the equations for the characteristic lines cannot

52
be determined before the solution (in terms of stresses) is known.
Because of this, the slope of the characteristic lines and the solution
to the problem must be determined at each point simultaneously. This
is done by using the iterative numerical technique described below.
The numerical grid shown in Figure 3.4 will be used. There are
two types of elements in this grid: boundary elements and regular
elements. All of the regular elements are alike, and all the boundary
elements are like the right-hand side of a regular element. A detailed
picture of a single regular element is shown in Figure 3.5, and a
single boundary element is shown in Figure 3.6. The grid is defined
in terms of the dimensionless variables given in equations (3.2.1) and
(3.2.3). It is diamond shaped with the straight outer lines corre
sponding to the characteristic lines for elastic longitudinal waves
with radial inertia effects included. These outer characteristic lines
have slopes of either c=+1 or c= -1, which can be seen from equations
(3.2.1) and (B.5.4). The vertical straight line corresponds to the
two vertical characteristic lines, and the straight inner nonvertical
lines correspond to the characteristic lines (through the point P) for
the elastic shear waves. For both types of elements, the problem reduces
to that of determining the values of the stresses and velocities at the
point P, when their values at the points B, R, and L are known.
This grid with all the elements constant in size simplifies the
writing of the finite difference equations. The diamond shape allows
the vertical characteristic lines to automatically connect point P (at
which the solution is desired) with point B (at which the solution is
known) and makes the finite difference equations along the vertical

53
T
Figure 3.4 Numerical Grid in the Characteristic Plane

54
T
Figure 3.5
Regular Element in Numerical Grid

55
T
Figure 3.6 Boundary Element in Numerical Grid

56
characteristic lines very easy to obtain. The boundary lines for each
element are c = l This is the smallest value of c which insures that
all characteristic lines passing through the point P will intersect
the line L~B between the points L and B if the lines have a positive
slope at P or will intersect the line R-B between the points R and B
if they have a negative slope at P. This is true since all of the
waves considered here will propagate with a speed less than or equal
to the speed of an elastic longitudinal wave with radial inertia effects
included. A larger value of c could be used, but the element size
would increase (for a given distance along the T axis), and the solution
would be inherently less accurate.
The straight lines representing the elastic shear wave character
istic lines are added to the grid elements as a convenience. The
results of Section 3.1 show that the fast wave speed always occurs in
the range
C2 Cf C1
and the slow wave speed always occurs in the range
Therefore, these characteristic lines c = c divide each element so
that a characteristic line through P lies in one of the upper triangles
(P-L-LB or P R RB) if it is for a fast wave and in one of the lower
triangles (P B LB or P B RB) if it is for a slow wave. This is
shown for the regular grid elements in Figure 3.7. These characteristic
lines for the fast and slow waves will not, in general, be straight.

T
Figure 3.7 Location of the Characteristic Lines Passing Through P
m
-i

58
3.4 Finite Difference Equations
General Discussion
While the actual characteristic lines for the fast and slow waves
are seldom straight, they can be represented as straight lines within
each grid element without introducing significant errors if the grid
elements are small. From the discussion in Section 3.3, it is known
that the slope of the. characteristic lines at any point cannot be
determined before the solution at that point is known. Because of
this the solution at the point P (Figures 3.5, 3.6, and 3.7) must be
obtained by an iterative technique. Within any grid element, the slope
of each characteristic line will be constant during each iteration
although the slope of each characteristic line will change from one
iteration to the next as the solution at P is approached. These
straight lines are used to represent the characteristic lines for
th
c = c^ and c = c^ during each iteration and are shown for the i iter
ation as c = c-f. and c = cs. in Figure 3.8 for a regular grid element
and in Figure 3.9 for a boundary grid element. The points LLB, LBB,
RBB, and RRB are the intersections of the lines shown in Figure 3.8.
/ /
Each element has its own coordinate system X T which is also shown
in Figures 3.8 and 3.9, and the finite difference equations are written
in terms of this local coordinate system so that the finite difference
equations for each element are the same.
First order finite difference equations will be written along each
characteristic line. The coefficients of the dependent variables in
these equations will in general be functions of the stresses and §(s,A).
Thus, in order to linearize the equations, the coefficients for each

/
X
Figure 3.8 Numerical Representation of the Characteristic Lines in a Regular Grid Element

60
/
f n rn
1 > a
Figure 3.9 Numerical Representation of the Characteristic Lines
in a Boundary Grid Element

61
iteration will be calculated using the solution obtained in the
previous iteration. In this way the coefficients are always known
quantities.
One other scheme will be used with the coefficients in the finite
difference equations in order to reduce the time required for compu
tation. Normally each coefficient used is the average value of that
coefficient at the end points of the interval over which the finite
difference equations are written. As an example, consider the charac
teristic line from point LLB to point P, and let one term in the finite
difference equation along this characteristic line be
U(S
xP
- S ).
xLLB
As a rule the value of the coefficient is calculated as
P
+ U.
LLB
) .
If this method is used, the coefficients of each variable in the equa
tions along the characteristics of positive slope will be different
from the coefficients of the corresponding variables in the equations
along the characteristics of negative slope. For instance, one term
in the equation along c = +C£^ can be represented as
VSXP SxLLB>
and the corresponding term in the equation along c= -Cf as
UR(SxP SxRRB)
L
R
+ w
+ URRB)
where

62
and Up is the value of the coefficient at point P calculated from the
solution from the previous iteration.
When the coefficients are calculated in this manner, the number of
equations which must be solved simultaneously cannot be conveniently
reduced below five. However, if the coefficients are calculated in
such a way that the coefficient of any variable in the equation along
c = +cf^ is equal to the coefficient of that same variable in the
equation along c = -c-f. (so that U = U etc.), then by adding these
two equations and subtracting one from the other, two different equa
tions can be obtained, each with fewer variables than the two original
equations. If this procedure is applied to the equations along c = cSi>
then the set of five simultaneous equations can be reduced to at most
a set of three simultaneous equations and a set of two simultaneous
equations. This is shown in Appendix D. Since this set (or sets) of
equations must be solved during each iteration, the savings in compu
tation time is significant.
One way to make the coefficients of similar terms equal is to
calculate the coefficients from values of the variables obtained at
point P during the previous iteration such as
U = U = U.
L R P
The coefficients will be calculated in a somewhat more accurate manner
by using a weighted average of the value of each coefficient between
point P and point B, that is
UL = UE = 1DP + (1 al)UB

63
This gives the value of the coefficients at a point nearer the center
of each grid element. For this work, the value of is chosen arbi
trarily as .625, so that the point at which the coefficients are calcu
lated is at approximately the same location along the T-axis
(Figures 3.8 and 3.9) as the centers of the four characteristic lines
C = Cfi, cSi<
The values of all quantities at the points LB, LLB, and LBB will
be obtained by linear interpolation between the points L and B.
Similarly, the values of all quantities at the points RB, RRB, and
RBB will be obtained by linear interpolation between the points R and
B. From Figure 3.8, the times T T and T can be written as
-L £ *3
2c
2c,
2c.
T, =
1 1 + c
AT
T =
2 1 + c.
AT
= r
3 1 + c
AT
and the interpolation constants for the points LB and RB are
2c,
CLRB
AT 1 + c.
Cl,RBI = 1 -
T2 ^
AT ~ 1 + c.
Using subscripts to denote the grid point, the values of any quantity F
at the points LB and RB are
FT = CLRB-Ft + CLRBI F
LB L B
F = CLRB*F_ + CLRBIF
RB R B

64
The
interpolation constants
CONI =
C0N2 =
for
the points LLB, LBB, RBB,
T2 2(Cf c2)
T2 = (1 + Cf)(1 C2>
Cs(1 + V
and RRB are
C0N3 = 1 CONI
C0N4 = 1 C0N2
so that the value of F at
F
LLB
F
LBB
F
RRB
F
RBB
each of these points is
= CONIF, + C0N3F
L LB
= C0N2F + CON4F
LB B
= CONI*I + C0N3F_
R RB
= C0N2F + C0N4F .
RB B
For Fully Coupled Waves
When the equations are fully coupled, the equations along the
nonvertical characteristics (c = c^, c ) are given by either
equation (3.2.7) (3.2.8), or (3.2.9). Equation (3.2.7) will be used,
and the values of A., A, A., A_, and A of equation (3.2.2) will
1 2 o 4 5 o
th
be calculated for the i iteration as described for U earlier in this
section and defined as A, , A., A., A.. , and A. respectively.
li 2i 3i 4i 5i 6i
th
The coefficients will then be defined for the i iteration as

65
If
2f
Is
V 2y(A4iA6i
1 V
2
1 V
2
(A A -
2/ 2i 6i
1 v
2
(A,. A,,. -
2/ 4i 6i
aA .) A .
5i 4i
A A ) A .
3i 5i 2i
aA ) A
5i 4i
R
2s
1 v
2/(A2iA6i
A A ) A .
3i 5i 2i
R^ = aA .A A.A.
fs 2i 5i 3i 4i
(3.4.1)
Thus, the finite difference equations when the waves are fully coupled
can be written directly from equations (3.2.5), (3.2.6) and (3.2.7),
and using the last subscript to represent the point in the numerical
grids of Figures 3.8 and 3.9 the equation
along c = 0 is
2a(1 v2) [ SP 2 9B] 2AT = a(Vrp V^)
(3.4.2)
along c = 0 is
-V + V
2a a SXB,
A4P + N + A5B^,
2 /(S9P S0B + V 2 ) P 'b}
r2aS S 2aSn S i
+ 1 SP g toP 22_22 *J (T,]
§{-
S /
P
B
(3.4.3)

66
along c
0 :
along c
0
+ c is
If
(XxP VxLLB)
c _R
f fs
(V V )
2 0P 0LLB;
(1 v^)R
R_ (T_ T ) +
fs P LLB
If
(SxP SxLLB)
+ R-.-p'I'-, ^(2S aS+ 2S aS ) 1 ~j
If Y1 L2 xP @P xLLB 0LLB J Ll + cj
+ aR J V + V + ilr -¡^-(S 2aS+S 2aS )
2fL rP rLLB *1 12 xP eP xLLB 0LLB
}]
It7
2AT
+ c
-1 6efRfs*! fl ,
+ 2 I2 'p + 'LLB'
f-1
v
2AT "j
_1 + c J
(3.4.4)
c is
R-. CR
-1-f. (V V ) + --- (V V )
C xP xRRB 2 0P 0RRB
f 1 v
(1 v2)R
+ Rfs(Tp lRRB) +
If
2
(SxP SxRRB^
:*i B
+ R, .iL I (2S aS. + 2S aS )
lfyl L2 xP 0P xRRB 0RRB
][r¥y
+ aH2f[VrP + 'rERB + ltf }]
f 2AT 1 6cfRfs*l ri 1 r_2iTl
Ll + v2 L2 P RRB J Ll + c^J
(3.4.5)

67
along c = + c is
s
~ (AxP \xLBB) ~ ~2 0P 8LBb'^
s 1 v
(1 v2)R,
+ Rfs(TP TLBB > +
(s s )
xP xLBB
+ R. t, ^(2S ~ aS + 2S aS ) ,
lsYl [_2 xP 0P xLBB 9LBB J |_1 + c
>JM
+ aR2s [VrP + VrLBB + *l{s(S*P 2aSGP + SxLBB 2W) J
[rrT"J
6c2R \|r
s f s 1
s 1 v
2 L2(tp + tlbb}J Ll + c_J
][l
2AT "1
(3.4.6)
along c = c is
s
Ri c R*
0 = -ii(V -V ) + -S-f (V V )
c xP xRBB 2 GP 0RBb'
s 1 v
(1 v^)R
+ Rfs(Tp trbb) +
Is
(SxP SxRBB)
!is+i B
+ R, V. i (2S aS + 2S aS )
n xP 0P xRBB 0RBB
+ aR V +V + \I,t:(S 2aS + S 2aS )
2s L rP rRBB Yll2 xP 0P xRBB 8RBB
/I,
J
r bat i 6%RfS*i ri.. T ,ir 2at i
Ll + cJ+ 2 b P+ RBbJLi+cJ
S 1 v s
(3.4.7)
where the values of i|r / and s^/ obtained at point P from the previous
iteration are vised and
;1 1
*oP' N *oB
'I'-, = + (1 at)
V B
(3.4.8)
with a^ defined earlier in this section.

68
For Uncoupled Waves
When the waves become uncoupled as described in Section B.4, the
equations along the characteristic lines have a simpler form given by
equations (3.2.10) to (3.2.13). Using the averaging technique already
described in this section for the coefficients, the equations along
the characteristic lines for uncoupled waves simplify. The equation
along c = 0 is
(3.4.9)
along c = 0 is
+
(2aSeB-sxBH
xB toB
(3. 4.10)
along c = + c^ is
+ ilr i (2S -aS + 2S aS^
Tll2 xP 0P xLLB 0LLB
+ aA V +V + ilf 1
2iL rP lLLB yll
>} ^ ] *4i
dLLB^Jj Ll + c^J
(3.4.11)

69
along c = is
0 = A
4i
.c (AxP 1'xRRB) + 2 (SxP~SxRRB)
f c.
ill
f
(2S aS + 2S
xP GP xRRB
2aS )} 1
0RRB J ll + C J J
+ aA
i|_^ rP + ^ rRRB T 'll2^SxP 2aS6PTSxRRB 2aS0RRB^/ J Ll +
(3.4.12)
i "j r 2at i
along c = + cg is
0 = -
1 -v
2
2 + (TP TLBB)
^4ri<
+ 2 [_2(TP + lLBB')
1 v
][iff]
(3.4.13)
along c = c is
s
0 =
1 -v
2 (V6P V 0RBB) + (TP TRBB)
6c2i1(
s
1 V
I [I (3.4.14)
3.5 Solution to the Finite Difference Equations
The solution to the finite difference equations of Section 3.4 are
given here for any iteration. The solutions consist of expressions for
Vxp, vgp< vrp Sxp, Sgp, and in terms of known quantities, including
quantities calculated during a previous iteration. The solutions given
in this section are obtained using Cramer's rule as shown in Appendix D,
and the definitions of the variables used in Appendix D will not be
repeated here.

70
At a Regular Grid Point for Fully Coupled Waves
The solution to the finite difference equations along the charac
teristic lines at a regular grid point in the case of fully coupled
waves is given here. The longitudinal and transverse velocities from
equations (D.3.5) and (D.3.6) are
VxP = ^ (D2s RHSBA D2f RHSDC)
(3.5.1)
vcr, = 7 RHSDC D RHSBA) .
0P A^ If Is
(3. 5. 2)
When radial inertia effects are included, the stresses at point P
are given by equations (D.3.13), (D.3.14), and (D.3.15). These stresses
are
TP A,
SF + RI1SG(DlD7f D2D4f)
* EHSH ]
(3.5.3)
SXP = T2 [SF(A5QD7s D2D3s> + RHSG + KHSH(D3sD7f D3f
*.>]
(3.5.4)
i r.
0P A0 LR v~l~3s 5Q"4s' "~'"5Q~4f ~l"3f'
RHSF(D D,
AD ) + RHSG(A D D D )
RHSH(D3fD4s D4f
D3s>]
(3.5.5)
and the radial velocity of equation (D.1.3) is
aVrP = a(D3 3S9P>-
(3.5.6)
When radial inertia effects are not included, the hoop stress,
Sgp, and the radial velocity, V^, automatically vanish, and the shear

71
stress and the longitudinal stress are given by equations (D.3.18) and
(D.3.19), respectively, as
TP = (D4s RHSF D4f RHSG) (3.5.7)
SxP = f At a Regular Grid Point for Uncoupled Waves
When the waves are uncoupled, the solution to the finite differ
ence equations has a much simpler form. In this case, the shear stress,
the transverse velocity, and the longitudinal velocity of equations
(D.4.1), (D.4.2), and (D.4.3), respectively, are
1
T -
P 2F
(RHSCE + RHSDE)
(3. 5. 9)
2s
V
6P 2Z
V
2
1
xP 2F
(RHSDE RHSCE)
(RHSBEM RHSAEM) .
(3.5.10)
(3.5.11)
If
When radial inertia effects are included, the longitudinal and
hoop stresses from equations (D.4.8) and (D.4.9) are
1
S = 7 (D RHS3 F RHSEEM)
xP A 2 5f 2
5
(3.5.12)
!ep r 5
(3.5.13)
and the radial velocity is again given by equation (3.5.6).
When radial inertia effects are not considered, both the hoop stress
and radial velocity vanish, and the longitudinal stress of equation (D.4.10)
is
S
xP
RHS3
F
2f 2
(3.5.14)

72
At a Boundary Point (X=0) for Fully Coupled Waves
In a boundary element, there are only four characteristic lines
(c = 0, c = 0, c = -c c = -c ) and consequently only four equations along
S X '
these characteristic lines. Since the equations along the character
istic lines are written in terms of six unknown variables at point P,
the solution at a boundary point can be obtained only if two of these
variables are prescribed at each boundary point. The hoop stress and
the radial velocity do not enter the formulation of the problem when
radial inertia effects are omitted, and therefore these variables are
not specified at the boundary. Thus, the four remaining variables, two
of which may be specified at any boundary point, are the longitudinal
stress, the longitudinal velocity, the shear stress, and the transverse
velocity. From a purely physical standpoint, it is also reasonable to
specify the longitudinal and transverse variables at the boundary since
these are the quantities which are normally associated with the impact
at the end of the tube and which can be measured more readily than
radial velocity and hoop stress. Only two of the four variables S ,
V T and V can be specified at any one boundary point. Furthermore,
X 0
at a given boundary point V and S cannot both be specified since they
are not independent. Also, both T and V cannot be given at the same
o
boundary point. Therefore, four combinations of variables to be speci
fied on the boundary will be considered: for Case I, and T will be
given at the boundary, for Case II, V and VQ will be given, for Case III,
S and V will be given, and for Case IV, V and T will be given. The
X 0 X
solution to the finite difference equations at a boundary point for each
of these four cases when the waves are fully coupled is given below.

73
Case I: Traction boundary conditions
When S^_p amd are known, then from equations (D.5.1)
and (D.5.6) the solution to the finite difference equations
Sep = (RHSH DlSxp A5QTp)
vxp = r4 (b2s RHS1 B2fRHS2)
vep = r (D.5.5),
at P is
(3.5.15)
(3.5.16)
(3.5.17)
and is given by (3.5.6).
Case II: Kinematic boundary conditions
When V^p and Vgp are given, the solution at P is given by
equations (D.5.11), (D.5.12), and (D.5.13) when radial inertia effects
are included as
= [^RHS4(D/1_ + B_ D.,) RHS5 (D^nD^+B^^D, )
P A,
4s2 2 7s V
4f2 2 7f 1J
A RHSH(D4s2B7f D4f2B?s)
]
(3.5.18)
i r
I R
'xP A L v3s22 1 *5Q7s' '3f2"2 ~7f"5Q'
S = I RHS4(D0_0D0 + A_DJ + RHS5(D0J,D0 + B^^A^)
A6
RHSH(D3s2B?f D3f2B'
*>]
(3.5.19)
^ = \ RHS4(D0 A^D,, ) RHS5 (DOJCOD., A^D^0)
0P A_ L
D
3s2 1 5Q 4s2
3f2 1 5Q 4f2
RI,SH (3. 5.20)
and V is given by equation (3.5.6). When radial inertia effects are
rP
not included, V and SD are zero and the solution given by equations
rP oP
(D.5.16) and (D.5.17) is

74
TP = h, (D4s2RHS4 D4f2RHS5) (35-21)
SxP r7 (D3f2RHS5 D3s2RHS4) (3-5'22)
Case III: Mixed boundary conditions
When S and V. are known, the solution when radial inertia
xP 0P
effects are included is given by equations (D.5.22), (D.5.23), and
(D.5.24) as
V
xP
h [RHS6 RIIS7
o
+ RHSS(D3s2B7f D3f2B7s)]
(3.5.23)
-.r,
Tp = ^Blf(D2RHS7 + B?sRHS8) Blg(D2RHS6 + B7fRHS8)J
8
GP
Blf
]
(3.5.24)
(3.5.25)
and V again is found from equation (3.5.6). When radial inertia
effects are not included, vrp=^Qp~ anc^ from equations (D.5.27)
and (D.5.28), the solution at P becomes
VxP = ^ T = (B7 RHS7 B RHS6) (3.5.27)
P If Is
Case IV: Mixed boundary conditions
When V and T are known at the boundary, the solution at P is
xP P J
found from equations (D.5.33), (D.5.34), and (D.5.35) when radial
inertia effects are included to be

75
V6P = iT~ LRHS9 RHS10
+ KHS11(D4s2B7s D4f2B7s)]
SxP = 4^ [B2fS11> B2s(D2RHS9 + B^RHSll)]
S6P = 4^ LB2£ (3. 5.28)
(3.5.29)
(3. 5. 30)
where again is given by equation (3.5.6). When radial inertia
effects are not included, and vanish, and from equations (D.5.38)
and (D.5.39), the solution at P is found to be
V S = (B RHS10 B rhS9) (3.5.32)
xP A.,., 2f 2s
At a Boundary Point (X=0) for Uncoupled Wraves
When the waves are uncoupled, the solutions to the finite differ
ence equations are obtained at the boundary points for the same four
cases outlined above. When radial inertia terms are included in the
formulation of the problem, the expression for is given by
equation (3.5.6), and in all cases when radial inertia terms are not
included both V and vanish. In all four cases the solutions can
rP 0P
be found in Appendix D.
Case I: Traction boundary conditions
When Tp and S^p are known at a boundary point, then from equations
(D.6.1), (D.6.2), and (D.6.3) at that point

76
Case II:
V0P = h,
S.0 = ~ (RHSEEM D S ) (3.5.34)
P D 1 xP
Cl
VXP = Flf <3'5'35>
Kinematic boundary conditions
When V^p and Vgp are prescribed at a boundary point, then the
solution at that boundary point is given by equations (D.6.4), (D.6.9),
(D.6.10) ,
and- (D.6.11). When radial inertia is included the solution is
S = (D RHS12 F RHSEEM) (3.5.36)
XP A12 3 5
= - (F RHSEEM D RHS12) (3.5.37)
0P A12 2f 1
and when no radial inertia effects are included the solution becomes
The shear
stress in both cases is
Tp = Y~ (RHSDE Z2 V0p) (3.5.39)
Case III:
Mixed boundary conditions
When S^p and Vgp are known at a boundary point, then from
equations
(D.6.12), (D.6.13), and (D.6.14), the solution at that point
is
Tp = (RHSDE Z2V p) (3.5.40)
2s
S0P = (RHSEEM DlSxP) (3.5.41)
2
VxP = Flf
77
Case IV: Mixed boundary conditions
When V^p and are given at a boundary point, then the solution
at that point is given by equations (D.6.9), (D.6.10), (D.6.11) and
(D.6.15), i.e.,
v9p = 4 <3'5'43)
and S^p and Sgp are given by equations (3.5.36), (3.5.37), and
(3.5.38).
3.6 Calculation of the Strains
At any grid point P, the solution is obtained by an iterative
technique. Once this is done, the values of S S., T V V and
Xu X P
Vg are known at P as well as at points L, B, and R (see Figures 3.5
and 3.6). The strains at point P can be computed very easily from
equations (2.27), (2.28), and (2.29). These equations can be written
in dimensionless form using equation (3.2.1) as
9e Sv
x x
T ~ X
(3.6.1)
Se
i Bve
9x
dT 2 5x
(3. 6. 2)
de6
~W = 2Vr
(3.6. 3)
For a regular grid element, these equations can be written in
finite difference form as

78
e e v v
xP xB XR xL
2AT 2AX
e. s v v
0xP 0xB 1 0R 0L
2AT ~ 2 2AX '
e ~ v + v
0P 0B rP rB
2AT ( 2
where the final subscript on each variable denotes the point in the
grid element where that variable is evaluated. Now, since the outer
grid lines defined in Section 3.3 have slopes of c = ~ = 1, AX and
AT are equal so that the expressions for the strains at point P are
£^=e + V V
xP xB xR xL
£0xP G9xB + 2 (V9R ~ ^'0L)
CSP = 66B + 2 For a boundary grid element, equations (3. 6,1),
(3.6.3) can be written in finite difference form as
e -e v (V + V )
xP xB xR 2 xP xB
2AT
AX
e0xP £0xB 1
2AT 2
^V0R 2 (V0P + V0B)
AX
-]
(3.6.4)
(3.6.5)
(3.6.6)
(3.6.2), and
g9P ~ e6B
2AT
and again since AX = AT, the strains at the boundary point P are given
by

79
e = e + 2V V V
xP xB xR xP xB
:9xP eexB + V0R 2 E0P = e0B + 2 iT
where equations (3.6.6) and (3.6.9) are the same expression.
(3.6.7)
(3. 6. 8)
(3.6.9)

CHAPTER 4
RESULTS AND DISCUSSION
4.1 Introduction
In Chapter 2 the problem of inelastic wave propagation was
formulated and the equations for this problem were found. In Chapter 3
these equations were written in finite difference form and from them
expressions for the stresses and the velocities at the points in the
numerical grid (Figure 3.4) were determined. Next a computer code
(shown in Appendix E) was written to facilitate the calculation of
the stresses, velocities, and strains at the grid points in the charac
teristic plane. Now, in this chapter the results obtained by using
this computer code will be discussed for several different combinations
of initial conditions and boundary conditions.
The computer code is written so that the boundary conditions are
specified by reading in values of two variables at each grid point
along the boundary (X=0). By specifying the boundary conditions in
this manner, any variable given as one of the boundary conditions can
have any functional shape. All of the data presented in this chapter
were obtained using the kinematic boundary conditions (Case II), that
is, by assigning values to the longitudinal velocity (V ) and the trans
verse velocity (VQ) at the impact end of the tube. Furthermore, the
same functional form was chosen for the two velocities in each case.
This form consists of assuming that each of the velocities at the
80

81
boundary increases linearly up to its final value (denoted by Vx^
or Vgf) during a period of time called the rise time (T ) and then
remains constant. That is
'f
if 0 < T < T.
R
R
Ve (X = 0, =/
if T > T.
R
r
if 0 < T < T.
R
V (X = 0) = /
if T > T.
R '
Now that the computer code is set up, it would be advantageous to
compare the results from it to data which have already been published.
This is done in the following section by using the data of Lipkin and
Clifton (1970), and some interesting effects of the size of the numer
ical grid are noted. Then, finally, the effects of radial inertia and
strain-rate dependence on the propagation of inelastic stress waves are
discussed.
4.2 Effects of Numerical Grid Size
Lipkin and Clifton (1970) published the results of three different
experiments where a thin-walled tube was given an initial static shear
stress and then impacted longitudinally. In this section the initial
conditions and boundary conditions from one of these experiments will
be used and the results obtained from the computer code will be compared

82
with the experimental and theoretical results of Lipkin and Clifton
(1970). The data which will be used are
o
T = initial static shear stress = 3480 psi
o
a = initial static longitudinal stress = 0
x
vx = final longitudinal boundary velocity = 500 ips
v6f
final transverse boundary velocity = 23 ips
t
R
rise time = 9.6 p sec
which can be written in terms of the dimensionless quantities for input
to the computer code as
6x
= .0003480
= .002404
= 0
V.
= .0001106
T = t
R 2r R
= 4.00 .
o
No radial inertia effects or rate dependence will be considered in
this section.
The results from three different computer runs will now be made.
Each computer run used these initial conditions and boundary conditions
but had different grid sizes. The three grid sizes used were
AX = AT = .25, AX = AT = .125, and AX=AT= .05. The longitudinal strain
versus time obtained by using the computer code in Appendix E is shown
in Figure 4.1 along with the experimental results and the simple wave
solution of Lipkin and Clifton (1970). From this it can be seen that

Longitudinal Strain,
Figure 4.1 Grid Size Effects on the Longitudinal Strain at X = 3.75
oo
CO

84
for the small grid size the strain follows closely the strain obtained
by Lipkin and Clifton (1970) for a simple wave with an instantaneously
applied velocity at the boundary. For the larger grid sizes the strain-
versus-time curve is smoother and follows more closely the experimental
results of Lipkin and Clifton (1970). Apparently, the larger grid
sizes tend to smooth out the data and eliminate the distinction between
the fast and slow wave speeds. For instance, in Figure 4.1, the' simple
wave solution of Lipkin and Clifton (1970) exhibits a region where the
longitudinal strain has the constant value of 0.00085. The strain
remains at this constant value from just after the fast wave passes
until the arrival of the slow wave.
From these computer runs other quantities of interest can also be
plotted and the same grid size effect can be observed. This is shown
in Figure 4.2 for the longitudinal velocity versus time. The grid size
has a much smaller effect on the stress trajectory than on the time
history curves. The stress trajectory is shown in Figure 4.3.
Because the details of the solution depend on the size of the
numerical grid, all subsequent computer runs will be made using a small
grid. This small grid size necessitates a large amount of computer
time to obtain a solution more than 1.0 diameter from the impact end,
and most of the results given below are obtained near the end of the
tube.
4.3 Effects of Radial Inertia
In order to determine the effects of radial inertia, four separate
computer runs were made using the computer code in Section E.5. The
generalization of the uniaxial stress-strain curve of Lipkin and Clifton

Longitudinal Velocity, V
Figure 4.2 Grid Size Effects on the Longitudinal Velocity at X = 3.75
CO
CJi

Shear Stress, T x10
Figure 4.3 Grid Size Effects on the Stress Trajectories at X = 3.75
cc
Gi

87
(1970) was used. This constitutive equation (shown in Appendix A) was
for strain-rate independent material behavior.
The first two computer runs (one including and one not including
radial inertia effects) were made using the initial conditions and the
boundary conditions which Lipkin and Clifton (1970) used in one of
their experiments. These input data used were
f T =4.00
R
AX = AT = .050
s = 0
Data Set 1 /
T = .0003480
V = .002404
Xf
V. = .0001106
l Gf
These data represent a tube with an applied static pretorque
(above the yield stress) impacted longitudinally at one end. The time
history curves of the longitudinal strain and the change in shear
strain are shown in Figures 4.4 and 4.5, respectively, for the section
of the tube 3.75 diameters from the impact end. The simple wave solu
tion and the experimental results of Lipkin and Clifton (1970) are
also shown in these figures. It can be seen in Figure 4.4 that the
longitudinal strain obtained in this work follows the experimental
results more closely than does the simple wave solution. Most of the
improvement over the simple wave solution is the result of using
a finite rise time (T = 4.0) for the impact velocity. The fast wave
R
has passed the point X=3.75 at the time when the longitudinal strain

Longitudinal Strain,
Figure 4.4 Longitudinal Strain Versus Time at X = 3.75 for Data Set 1
oo
00

Change in Shear Strain, Ay
Figure 4.5 Change in Shear Strain Versus Time at X = 3.75 for Data Set 1
oo
CD

90
has reached the value of 0.00085. For the simple wave solution this
time is approximately T-6 where for the finite rise time (T =4) this
time is approximately T=10. The difference in time when the fast wave
has passed can thus be accounted for by the finite rise time.
As the slow wave passes a point on the tube, the longitudinal com
pressive strain begins to increase to values larger than 0.00085. The
higher levels of strain (e ^ -.008) occur later (in the results given
here) than in the simple wave solution. Again this can be accounted for
by the finite rise time.
The inclusion of a finite rise time in the theoretical solution
gives results which resemble the experimental data more closely than
the simple wave solution. It can also be seen that including radial
inertia effects in the formulation of the problem gives longitudinal
strains which are somewhat closer to the experimental data than the
corresponding strains when radial inertia effects are ignored.
The change in shear strain versus time curve in Figure 4.5
exhibits the same rise time effect as the longitudinal strain. The
results obtained here are much closer to the experimental data than
the results for the simple wave solution. The final value of the
shear strain appears to be low. Since this shear strain is calculated
from the values of the transverse velocity, it may be that the final
value of the transverse velocity should be larger.
This can be seen more easily by examining the transverse velocity
at several distance from the impact end as shown in Figure 4.6.
A transverse velocity is induced when the tube is impacted with a
longitudinal velocity, if the tube is statically preloaded in torsion.
From Figure 4.6 it can be seen that the transverse velocity induced

Transverse Velocity, VQx10
Figure 4.6 Transverse Velocity Versus Time for Data Set 1 Without Radial Inertia
CD

92
by the longitudinal velocity is much greater than the transverse
velocity specified at the boundary. Therefore, the transverse veloc
ity specified at the boundary (impact end of the tube) may be too low.
The longitudinal velocity is plotted in Figure 4.7 and the
longitudinal strain in Figure 4.8. Both of these quantities have a
region (at distances from the impact end greater than 2.0 diameters)
where the value is constant. This corresponds to the interval of time
between the passing of the fast wave and the arrival of the slow wave.
It takes 2.0 diameters for the two waves to become distinct because of
the finite rise time of the velocities at the boundary. That is, the
fast wave originating at the boundary at the time T = T^ must overtake
the slow wave originating at the boundary at the time T=0 before the
two waves become completely separated.
The effects of radial inertia are most evident in the longitudinal
strain. This can be seen in Figure 4.8 where the longitudinal strain
at the impact end is about 25 percent greater when radial inertia is
included than when it is not included. Qualitatively this is the result
that would be expected. Under compressive loading, the tube is allowed
to expand instantly when radial inertia effects are not included. How
ever, the presence of radial inertia initially resists expansion of the
tube (thereby decreasing the value of the longitudinal strain at any
given time) and then continues the radial expansion beyond the equi
librium point (causing the longitudinal strain to increase beyond the
equilibrium value). This behavior is evident in Figure 4.8. The
maximum value of the longitudinal strain when radial inertia effects
are included approaches the same value as when radial inertia effects
are not included at distances greater than 2.0 diameters from the

Longitudinal Velocity, V x10
Figure 4.7 Longitudinal Velocity Versus Time for Data Set 1 Without Radial Inertia
CD
CO

Longitudinal Strain,
Figure 4.8 Longitudinal Strain Versus Time for Data Set 1
<£>

95
impact end. Experimental results similar to these results were
presented by Bell (1960) for an aluminum bar under uniaxial compres
sive loading.
The distance from the impact end when radial inertia effects
become insignificant can be more easily determined from the radial
velocity. Since the radial inertia is proportional to the time rate
of change of the radial velocity, the maximum value of the radial
velocity at each tube location should give a reasonably accurate
comparison of the radial inertia at each of these locations. The
maximum value of the radial velocity is shown in Figure 4.9. From
this it can be seen that radial inertia effects are about four times
greater at the impact end than at 2.0 diameters from the impact end,
and that the effects of radial inertia decrease asymptotically toward
zero as the distance from the impact end increases.
The change in shear strain is shown at various locations in
Figure 4.10, and the longitudinal strain profile is shown at differ
ent times in Figure 4.11. In this last figure, the strain plateau has
developed near the impact for times greater than T= 20 when radial
inertia effects are not included. However, when radial inertia effects
are included, a strain plateau near the impact end is just beginning
to develop at the time T=20.
The stress trajectories and the strain trajectories are shown in
Figures 4.12 and 4.13 for the case when radial inertia effects are not
present. These trajectories are shown at several distances from the
impact end. The stress trajectory at distances greater than 1.0 diam
eter from the impact end behaves in essentially the same manner as the
ones given for the simple wave solution by Lipkin and Clifton (1970)

Maximum Radial Velocity, V x10
3 6
Distance from Impact End, X
T
Figure 4.9 Maximum Radial Velocity Versus X for Data Set 1 With Radial Inertia
CD
05

Change in Shear Strain, y
Figure 4.10 Change in Shear Strain Versus Time for Data Set 1 Without Radial Inertia
CD
<1

Longitudinal Strain,
Figure 4.11 Longitudinal Strain Versus X for Data Set 1

Shear Stress, T x10
6 _
4
Figure 4.12
Stress Trajectories for Data Set 1 Without Radial Inertia
to
o

Change in Shear Strain
100

101
and by Clifton (1966). That is, for a tube under a static torsion
load, a longitudinal impact creates a period of neutral loading as
the fast wave passes followed by loading normal to the yield surface
as the slow wave passes. During the neutral loading the shear stress
decreases while the longitudinal compressive stress increases. The
stress trajectory at the boundary (impact end) does not follow this
simple wave solution and changes character in the first diameter from
the end. This behavior is probably due to the particular selection of
the boundary conditions, coupled with the fact that near the impact end
both fast and slow waves pass the same point along the tube at almost
identical times. Therefore the tube is attempting to undergo both
neutral loading and loading simultaneously. The sharp corner on the
stress trajectory at X = 0 corresponds to the time T = T =4.0 which is
R
the peak of the input velocity ramp.
When radial inertia effects are included, there are two additional
variables: the hoop stress (Sg) and the radial velocity (V^). For
this case the stress trajectory will not necessarily be planar.
In spite of this, the stress trajectory (T versus S ) projected onto
the plane SQ=0 is shown in Figure 4.14. It appears at first that
0
the stress trajectory at X=1.0 represents loading followed by unload
ing and then by reloading. Recalling the definition of the general
ized state of stress from equation (A..1.2)when radial inertia is included
as
2 2 2 2
s = S + S S S. + 3T
x 6 x 9
the quantity S can be defined by
2 2 2
s = s + s s S-
N x 8 x 0
(4.3.1)

Shear Stress, t x10
Figure 4.14 Shear Stress Versus Longitudinal Stress for Data Set 1 With Radial Inertia
102

103
where now S represents the component of the stress vector (s) in the
N
Sr-S^ plane. By plotting a new stress traj ectory of T versus S we
x b N
can effectively represent this three-dimensional stress trajectory as
a two-dimensional one; this is shown in Figure 4.15. Again from this
figure it can be seen that at distances greater than 1.0 diameter from
the impact end the tube exhibits neutral loading followed by loading.
This is the same general behavior as the case when radial inertia is
not included, except that now the stress trajectory is not planar.
This is easier to visualize if these same stress trajectories are pro
jected onto the plane T = 0; the stress trajectory at X=1.0 is shown
in Figure 4.16. Since in any experimental wave propagation problem of
this type radial inertia effects are present, interpretation of the
experimental data without considering the hoop stress could lead to the
false conclusion (for a set of initial conditions and boundary condi
tions similar to these) that unloading occurred. Therefore, extreme
care must always be exercised when interpreting experimental results.
The other two computer runs made used artificial data where the
initial conditions and the boundary conditions were described by the
input data
" tr = -40
AX = AT .025
S = -.0006040
J X
Data Set 2 <
o
T =0
V =0
Xf
V0f = 00100 .

Shear Stress, T x10
Figure 4.15 Stress Trajectories for Data Set 1 With Radial Inertia
104

Hoop Stress, SQxlO
105
Figure 4.16 Hoop Stress Versus Longitudinal Stress
for Data Set 1 With Radial Inertia

106
These data represent the impacting of the end of a tube with a torsional
(transverse) velocity pulse when the tube is preloaded in compression
above the yield stress. First the stress trajectories obtained by not
including radial inertia effects will be examined. The stress trajec
tories for this case are shown in Figure 4.17 at various distances from
the impact end. Here the stress trajectory develops (at distances from
the end of the tube greater than 1.0 diameter) into the type given qual
itatively by Clifton (1966), that is, the tube undergoes unloading
along the S^-axis followed by an increase in T at a constant value of
S^_ until the yield surface is reached followed by loading normal to the
yield surface. Again the stress trajectory does not exhibit this
behavior near the impact end, but rather approaches this behavior as
the distance from the impact end of the tube increases. As explained
above, the stress trajectory near the end of the tube does not behave
in the manner given by Clifton (1966) using the simple wave solution
because the fast and slow waves are not distinct near the end of the
tube and because the input boundary conditions force the solution near
the impact end.
The stress trajectories (T versus S ) are shown in Figure 4.18
N
for the case when radial inertia effects are present. The stress
trajectories are plotted using the combined hoop stress and longitu
dinal stress of equation (4.3.1) since this gives an effective three-
dimensional representation of the stress trajectories. The stress
trajectories exhibit a "ringing" effect after loading has taken place.
This "ringing" is caused by the hoop stress as can be seen from
Figure 4.19.

Shear Stress, T x10
4.0 _
Final Stress State
Figure 4.17 Stress Trajectories for Data Set 2 Without Radial Inertia
107

Shear Stress, T x10
108

Hoop Stress, SQxlO
109
Figure 4.19 Hoop Stress Versus Longitudinal Stress for Data Set 2
With Radial Inertia

110
The shear strain and the change in longitudinal strain are plotted
against time in Figures 4.20 and 4.21, respectively. Both of these
strains exhibit the period of constant strain between the passing of
the fast and slow waves. When radial inertia effects are not included
the shear strain at the boundary has a smaller value than at other
points along the tube. Physically this is difficult to realize, and
it may have been caused here by the manner in which the boundarycondi
tions are given since the strains are computed directly from the veloc
ities and the velocities are specified at the boundary. The change in
longitudinal strain (Figure 4.21) is interesting; initially the tube is
compressed so that the longitudinal strain is negative. As the fast
wave passes a point on the tube, the change in longitudinal strain is
positive, that is, the compressive strain decreases. This decrease in
compressive strain corresponds to the initial decrease in compressive
stress shown in Figure 4.17. Then as the slow wave passes a point on
the tube the change in longitudinal strain is negative. This repre
sents a further compressive loading of the tube and corresponds to
the region of loading normal to the yield surface shown in Figure 4.17 .
4.4 Effects of Strain-Rate Dependence
In order to investigate the effects of strain-rate dependence,
four computer runs were made using the computer code in Appendix E.
These four data sets were obtained using the generalization of the
stress-strain curve of Cristescu (1972). Two computer runs were made
without strain-rate dependence (one with and one without radial inertia
effects) and two with strain-rate dependence (one with and one without

Shear Strain,
Figure 4.20 Shear Strain Versus Time for Data Set 2
111

Change in Longitudinal Strain, As
112

113
inertia effects). For all four cases the initial conditions and the
boundary conditions were specified by
. 40
T
R
AX = AT = .025
o
0
x
o
.0000900
T
.000250
x
f
These input data represent a tube with a static pretorque which
is impacted longitudinally at the end. The longitudinal strain versus
time curves obtained by using these data are shown in Figure 4.22.
When no strain-rate dependence is included, the results are the same
qualitatively as those obtained earlier for the Lipkin and Clifton
(1970) data. However, when strain-rate dependence is included (and
the tube is subjected to constant velocity impact), the maximum longi
tudinal strain is reduced.
This lower value of total longitudinal strain may be better under
stood by considering a longitudinal impact of an unstressed tube of
a strain-rate dependent material. The stress-strain curve for such
a material is steeper at higher strain rates; and therefore, if the
maximum applied stress is the same for two different loading conditions
the total strain will be larger for the condition when the rate of load
ing (or straining) is smaller. For this combined stress case, the

Longitudinal Strain,
Time, T
Figure 4.22 Longitudinal Strain Versus Time for Data Set 3
114

115
highest rate of loading is at the impact end, and using the analogy
for longitudinal loading, the maximum value of the strain would be
expected to be lower here than at other positions along the tube where
the loading rate is lower.
The stress trajectories are shown for the four cases in
Figures 4.23 and 4.24. The behavior of these trajectories is similar
to the behavior already discussed. It can be seen that strain-rate
dependence does not significantly alter the stress trajectories except
at the impact end where the loading appears to be a little more severe
for a strain-rate dependent material.

Shear Stress, T x10
Figure 4.23 Stress Trajectory at X = 0 for Data Set 3
116

Shear Stress, 7 x10
1. 2
rr
Classical
With Radial Inertia
With Rate Dependence
With Radial Inertia and Rate Dependence
Figure 4.24 Stress Trajectory at X = .25 for Data Set 3
117

CHAPTER 5
SUMMARY
In this dissertation the propagation of inelastic waves of combined
stress was discussed. The equations were developed to describe the
propagation of both longitudinal and torsional waves along a thin-walled
tube. These equations were written in such a manner that radial inertia
effects could be either included or not included in the problem formula
tion simply by setting a single parameter equal to either 1 or 0.
A generalized quasilinear constitutive equation was presented for mul-
tiaxial stress states and two different forms of it were used to study
two different materials. Both materials studied were assumed to obey
the von Mises yield criterion and to be isotropically work-hardening.
Once the problem was formulated the plastic wave speeds were then
determined as functions of the state of stress. For the case when
radial inertia effects were not included, the results of Clifton (1966)
were reproduced. When radial inertia was included, quite different
results were obtained, even with the hoop stress set equal to zero.
Both plastic waves (the slow wave and the fast wave) were found to
propagate faster when radial inertia effects were included (even with
the hoop stress as zero) than the corresponding waves when radial
inertia was not included for the same state of stress. It was also
found that when loading occurred in a principal stress plane, there
was often a critical value of the tangent modulus (or work hardening)
118

119
for which the two wave speeds (for the fast wave and the slow wave)
were identical. These critical values of the tangent modulus were
found for several values of Poisson's ratio. When radial inertia
effects were not included, the values obtained when v = .30 was the
same as that given by Clifton (1966). This critical value of the
tangent modulus was always smaller when radial inertia effects were
included (with the hoop stress set equal to zero) than when radial
inertia effects were not included.
Next the solution for this wave propagation problem in the
characteristic plane was found. In order to do this, it was necessary
to use numerical computation methods. To make the numerical solution
more general, the variables used were defined in terms of dimensionless
quantities, and then the equations for the characteristics and the
equations along these characteristics were written in terms of these
dimensionless variables. A uniform numerical grid was established and
the equations along the characteristics were written in finite differ
ence form for each grid element. These finite difference equations
were then solved to find expressions for the stresses and the veloc
ities at each point in the numerical grid. The strains were calculated
at each grid point by using the values of the velocities. A computer
code was written to help solve this wave propagation problem for differ
ent sets of initial conditions and boundary conditions. All of the
numerical results were obtained by specifying the initial static pre
stress in the tube and then giving the values to the velocities at the
impact end.

120
The first numerical results were obtained for data given by
Lipkin and Clifton (1970) for one of their experiments. A rise time
of about 10 microseconds was used for the velocity impact. Several
grid sizes were used and the larger grid sizes were found to smooth
out the time histories of the variables. Because of this the strains
obtained using the larger grid sizes were very similar to the experi
mental results while the strains using smaller grid sizes were similar
to the simple wave solution. Since the larger grid sizes smoothed out
the numerical data, a small grid size was used for the remainder of the
computer runs.
Numerical data were first obtained using the generalization of
the Lipkin and Clifton (1970) strain-rate independent constitutive
equation. Two separate sets of initial conditions and boundary condi
tions were used. One was a tube prestressed statically above the yield
stress in torsion and impacted at the end by a predominantly longitu
dinal velocity pulse. The other was a tube prestressed statically
above the yield stress in compression and then impacted at the end by
a torsional velocity pulse. The rise time for the velocity pulse in
the first case was 10 microseconds and in the second case was 1 micro
second. Two computer runs were made with each data set to generate the
results both with and without radial inertia effects.
For the first case when the static prestress was torsional, the
stress trajectories were found. When radial inertia was not included
the stress trajectory at distances greater than 1.0 diameter from the
impact end behaved in exactly the manner predicted by Clifton (1966)
from his simple wave solution. That is, the stress path at any point

121
along the tube exhibited neutral loading as the fast wave passed,
followed by loading normal to the yield surface as the slow wave
passed. However, at distances closer than 1.0 diameter from the
impact end the stress trajectory at a point along the tube followed
a path in stress space which represented a combination of neutral
loading and loading. This behavior was caused by the finite rise time;
since the velocity pulse was not a step function, it took about 1.0
diameter for the two waves to separate. In other words, the fast
wave starting at T = T overtook the slow wave starting at T = 0 in
R
about 1.0 diameter.
When radial inertia effects were included, the stress trajectories
became more complicated as they were no longer planar. For this case
the longitudinal stress was plotted against the shear stress and it
appeared that at distances greater than 1.0 diameter from the impact
end unloading occurred. However, when the stress trajectories were
plotted in the three-dimensional stress space the behavior was found
to exhibit neutral loading followed by loading.
The strains and velocities were shown at several tube locations
and constant state regions were observed at points along the tube
after the fast wave had passed and before the slow wave had arrived.
Also, radial inertia was shown to have little effect more than two
diameters from the impact end.
For the second case when the impact velocity was torsional, the
same types of results were obtained. Here the stress trajectory
obtained was like the one predicted by Clifton (1966) with his simple
wave solution.

122
One set of data was used with a multiaxial generalization of the
constitutive equation of Cristescu (1972). These data consisted of
applying a static pretorque to the tube and then impacting the tube
longitudinally. YVhen no strain-rate dependence was included, the
results were qualitatively the same as for the previous case.
However, when strain-rate dependence w'as included for the constant
velocity impact, the longitudinal strain was found to be reduced while
the stress trajectories were not significantly affected by strain-rate
dependence.
The major contributions of this dissertation were the development
of the equations for combined stress wave propagation with radial
inertia effects included, the introduction of a general quasilinear
constitutive equation for multiaxial stress states, and the writing of
a computer code capable of solving this wave propagation problem in
the characteristic plane.
The theory and results presented in this dissertation suggest the
need for further research in order to more fully understand plastic
stress wave propagation. First the computer code in this dissertation
could be made more general by including the boundary element for the
opposite end so that tubes of finite length could be studied. Also,
the initial assumptions can always be improved; in this work the most
restrictive assumption is that the material work-hardens isotropically.
This is valid only as long as reversed loading is not approached. But
for many cases, waves of reversed loading can occur, and obviously the
introduction of more realistic work-hardening assumptions may be

123
expected to affect significantly the theoretical results. The final
test of any theory (along with its assumptions) is how well it pre
dicts the real behavior of materials. This can only be determined by
comparison of the theoretical results with experimental results.
But experimentation in the field of dynamic plasticity under combined
stresses has only recently begun. Much work remains to be done before
enough experimental data have been gathered so that investigators can
determine a unified theory of inelastic wave propagation under combined
stress. At least a beginning has been made.

APPENDICES

APPENDIX A
CONSTITUTIVE EQUATIONS
A.l. Comments on the Constitutive Equation
The constitutive equation used in the general formulation is given
by equation (2.14) as
3
1 + v
e, = - a. .
- 6
.a,, +
ij E ij E ij kk 2
j~0(s,A)s + \|r(s,A)J (A.1.1)
where s is the scalar representation of the stress state given by
¡3
/ s s
*J 2 ij ij
and s is the deviatoric stress given by
ij
s. a. -6 .a .
xj ij 3 ij kk
Under the assumption that a = T = T = 0, s reduces to
r rx r0
/ 3 r 2 2 2
s = J 2 LS11 + =22 + s33 + 2S
i!
and with
S11 = S,
3(2cx aCTe}
s22 se
3(2acre CJx)
33
= s
3(ax + raB>
S12 S21 S0x T0x
this becomes
2 2 2
cT -aa oG + aac + 3T
x x 0 6 0x
(A.1.2)
125

126
If the only stress present is a then
x
S = CT
and from equation (A.1.1) the plastic portions of the strain rates are
P *P *P
e~ = = = 0 .
'6x 9r
rx
P P 1-P
e. = e = e
9 r 2 x
If, in addition, each plastic strain component is monotonically increas
ing (or monotonically decreasing), then the expression for A in equa
tion (2.16) becomes
CT
._ .'P.2 P 2 P.2 x
A = /- I / CO + (eQ) + (er) dt +
P , X
e dt +
x E
A -
P e
e + e
x x
A = e
and equation (A.1.1) reduces to
1 3 f
e = ct + ¡
x E x 2 L
0(ct e )ct + A(ct
X, X X X
, X J
3 CTx
c = ci+0(CT,e)cr+t(CT,e)
x Ex xxx xx
(A.1.3)
which is the same as equation (2.13) used by Cristescu (1972).
Again, when the only stress present is ct the function 0(cr e )
X XX
can be set equal to zero and
'Kct e ) = i g(o e )
xx E xx
to obtain the semi-linear constitutive equation of Malvern (1951a,1951b) as
1 1 ,
e = ct +-g(CT,e)-
x E x E xx

127
If 0(s,A) is assumed to vanish in equation (A.1.1), and if
t (s ,A) = yf (y l) ,
\''3 K
then the constitutive equation (A.1.1) reduces to the generalized semi-
linear constitutive equation of Perzyna (1963).
The constitutive equation (A.1.1) can also be reduced to the equa
tion obtained from the rate independent equations of incremental plastic
ity as v/ill be shown in the next section.
A.2. Rate Independent Incremental Plasticity Theory
The constitutive equation derived in this section is obtained when
the material is loading in the manner outlined by Malvern (1969).
If the material is strain-rate independent, if it exhibits isotropic
work-hardening, and if its yield function, f, depends only on the state
of stress, then
f(C. .) = F (A.2.1)
ij
where F is the work-hardening parameter. If F, and, consequently, the
size of the yield surface, is assumed to depend only on the total
P
plastic work, w then
F = F(vF) = a* (A. 2.2)
where vF = f a.. de^. (A.2.3)
J ij iJ
*
and ct is the magnitude of the stress vector.
During loading the material is assumed to obey the Prandtl-Reuss
flow law
ds.
ij
dX s..
ij
(A.2.4)
where d\ is a scalar function, and the von Mises yield condition
f = a
rr
y 2 s s
ij ij
s
(A.2.5)

128
so that'the assumed flow law is derivable from the plastic-potential
equation
o 3f(o\ .)
deP = dX y -1J-
(A. 2.6)
The form of the von Mises yield condition, equation (A.2.5), is chosen
so that under uniaxial tension, f = a .
x
From equations (A.2.3) and (A.2.6) an expression for d\ can be
obtained as
dW*3 = a. ,deP = a. ,d\
and
1J 1J ij oaTj
dX
dW1"
df
ij So..
ij
de
p
P iJ
ij df
^mn da
mn
Now, from equations (A.2.1) and (A.2.2)
so that
and
da, = ^ dW1" = dvF = f' (vP) dvF
dCTki kl 9W
F
dr
dvF
1 Sf
33 CTki
de
F'GV13) ~wkl
df df
p v
1J f'(W)P a
mn oa
mn
(A.2.7)

129
From eqation (A.2.5),
Bf
3a 3a
pq pq
Bf 3
r# t
L2 1J ljJ
Ba 2f pq
pq
(A. 2. 8)
Now from equation (A.2.8) it can be seen that equations (A.2.4) and
(A.2.6) are identical when
d\ = ~ d\ .
The first invariant of the deviatoric stress is
1 ,
s = a -o6 cr, = a a,, = 0
pp pp 3 pp kk pp kk
so that
Bf
a -K = a
pq ^ pq
Bf
[-J
pq 3a
= f .
(A. 2. 9)
pq
Using equations (A.2.8) and (A.2.9), the expression for the plastic
strain increment, equation (A.2.7), becomes
[h si J [ %i] daki
or
de .
ij
f' (w1
9s s
deP .
ij
ij kl
4f3 f' (/)
9s s
P
ij kl
ij
4f3 f' (/)
[f]
da
kl
kl
(A.2.10)
where
3a,
3eP.
P ij
ij 3t kl 3t
kl

130
Next, a expression for s, cr will be found as
kl kl
s,, = (a o 6, t a ) cr. ,
kl kl kl 3 kl nn kl
z 01
SklCTkl 3 f dt '
Using equations (A.2.11) in (A.2.10),
strain rate becomes
(A.2.11)
the expression for the plastic
P
e
ij
2f2 F' O'?)
s .
ij
(A. 2.12)
The function F/(W ) will be obtained by assuming that the material
obeys a "universal stress-strain law." The functions s and A are chosen
in such a form that if the only stress present is then s is equal to
a and A is equal to e as shown in Section A.l, and therefore the
x x
"universal stress-strain curve" is the same as the stress-strain curve
for simple tension with s replacing ct and A replacing e .
X X
Two forms of the uniaxial stress-strain curve will be considered.
The first form is the type used by Clifton (1966) and by Lipkin and
Clifton (1970). For this case the static and dynamic curves are both
given by
where
(A. 2.13)
r o if
a < ct
x y
(A. 2.14)
and a is the tensile stress,
x
yield stress, E is the elastic
e
(ct ct ) if cr
" x y x
is the tensile
x
tensile modulus,
> CT
strain, cr is the
y
and B and n are
constants used to adjust the curve to represent various materials.

131
The slope of the curve defined in equation (A.2.13) is continuous for
n>l, and this equation becomes bilinear when n = l. The term may be
thought of as a "plastic modulus. From equation (A.2.13) the strains
are separable into the elastic and plastic portions
ct
e x
- T
P T^/ V11
e = B\ ct ct /
x X y
(A.2.15)
so that under uniaxial stress, the plastic work is given by
= J a. .deP. = f a deP = i s dP
ij xj J x x
ct* p
r*
= j a I Bn(a ct )n 1 da "1 = I sTb^s ct )
i XL X y xj o L.
v
n-1 dij
where o is the maximum stress on the "universal stress-strain curve"
which is reached during loading, and P .
W is zero. Thus
_* *
CT O ~
P p .n1 p n1
w = Bn s(s-o ds = ¡ Bn s(s a ) ds
ti y 0 y
OCT
y
and integrating this by parts
*
'"I-
W*3 =Hs(s-cr)n|_ B (s-CT__)nds
y CT
/ = a ,] B [j (J dy)ntlJ .
/ = bTct*(CT* CT..)n A- (CT*-CT_.)n+1l
n+1
J

132
r a* a -i
/ = B(a* a )n I a* -Zl
y L n +1 J
W = (a*-a )(na*+a )
n+1 y y
and
dW13
(vf)
dF
dvf
da da
dW
i- = Bn(a*) (a* a )n 1
f'(/) y
(A.2.16)
Using equations (A.2.5) and (A.2.16) in (A.2.12), the expression for
the plastic strain rate for the static stress-strain curve of equation
(A.2.13) becomes
oD / \n-l 3a*
.P 3Bn ~5t
e. =
ij
2a
ij
(A. 2.17)
Comparing equation (A.2.17) with equation (A.1.1), the functions
0(s,A) and i|r(s,A) for the rate independent plasticity theory using
equation (A.2.13) are
t(s,A) = 0
0 (s,A) = Bn(s ay)n
(A.2.18)
where s = a The constitutive equation (A.1.1) during loading for
this case is
1+v v c 3 / \n1 ij
e. = a.. 6 . a, + Bn( s a ) s -
ij E ij E ij kk 2 y -
(A.2.19)
The second form of the uniaxial stress-strain curve is the one
used by Cristescu (1972) as the relaxation boundary for dynamic load
ing. Although Cristescu (1972) uses a quasistatic curve which is

133
slightly different from the relaxation boundary, the relaxation boundary
will be used here for both the dynamic loading and the static preloading.
This relaxation boundary is defined as
r
Ee
if e s e
x y
1 _i
o = f (e ) = / a + g e 2 (e e ) if e s: s
x \ y 2 z xv V
x
x z
3 e2
V. x
if e e
z X
or as
r
X
-<
1
E CTx
l
2e2
(o a ) + e
,+ x y y
3^
V-
. 1 ,2
( O' )
p+ X
if O' =£ CT
x y
if a ^ a ^ o
y x z
if 0^0
Z X
(A. 2.20)
where o and a are stresses on the uniaxial stress-strain curve corre-
y z
sponding to the strains and respectively. The slope of this
curve is discontinuous at the point e = e This equation can be
x y
written in the form
e
x
1
E
o
x
+ X.
(a.
o ) + e
y y
i
E
(A.2.21)
where
Xr
if
o
£
a or
X
y
if
CT
<
q
A
Q
y
x :
if
o
<
a
X
z
if
o
a
x z

134
Expressions for the plastic work for uniaxial loading (s = ct ) will
x
be found separately in the three different iegions. First
= 0
if a £ a
(A. 2.22)
When ct < ct < ct the plastic work is
y z
l
-2e2
p p /. p r z ii
W = I a. .de. = 1 ct de = J X, ct ~ do
ij ij ^ x x ^ 1 X L Q+ EJ
X X
o
X
p
W
and
dw
F' (U?)
dF
dW1"
dc
*
f2 6
L
z
+
a*-
(A. 2.23)
When ct ^ ct the plastic work is
z
2ct
f r* f r* x 1
W = J O' d = f CT x0 ( -) dCT
^ x x J x 2 n+2 E >
/ =
*2
[-
L30
2 *3 3. 1 *2 2
72
3
and
dW
/ P
F (W) dCT
* ^2
*
' %
1
(A. 2.24)
Now, using equations (A.2.23) and (A.2.24) in equation (A.2.12) and
comparing the results with equation (A.1.1), the expressions for the
functions 0(s,A) and ijr(s,A) become

135
tys.A) = O
^0
if S <
0(s ,A) =<
2£2
z
i if cr < s < a /
E y z
(A.2.25)
2s 1 ..
- if a ^ s
~+2 E z
For this case the constitutive equation (A.1.1) during loading becomes
1+v
e. = o
ij E
x
Ks f 6ijk+ tiivr e} + 0
3 Sij
2 1+2
(A.2.26)
The expressions (A.2.18) and (A.2.25) are valid only when the
material is loading. During unloading, the constitutive equation
(A.1.1) is still valid, except that 0 (s ,A) = i|f (s ,A) = 0. This results
from assuming that the unloading is elastic.
A. 3. Rate Dependent Plasticity Theory
The constitutive equation (A.1.1) is shown in Section A.l to
reduce to the constitutive equation (2.13) when the only stress present
is a Therefore, the data from Cristescu (1972) Case XII, can be used
directly, and the expressions for 0(s,A) and \|r(s,A) can be written as
r
K(A)
>]
t(s,A) = <
s f (A) if s>f(A) and A^ A
if s s f (A) or A < A
and
V.
(A.3.1)
(A.3.2)

136
where
if A ^ A
(A) = l
8+ A
CJ + V A 2 (A-A ) if A SA SA
y 2 z y y :
V
, 1
3+Ae
if A ^ A
z
A =
z
3+A
/ 2 q+2
a -a/ a A P
y y y
K(A) = Kj_l exp (-A)J
a = m +
n\ZA~
X =
0 if s ^ f(A) + o' or < 0
1 if s > f(A) + o7' and > 0
ot
+
o + +
A = h + X
(A. 3.3)
(A.3.4)
(A. 3.5)
(A.3.6)
(A. 3.7)
(A. 3.8)
and A is the yield strain of the material in tension, y y
stress in tension, and ¡3' K A, m, n, h, and \ are constants of the
material. Because of the particular selection of the form of s and A,
the values of these constants, as determined experimentally by
Cristescu (1972) for uniaxially loaded 1100F aluminum, can be used in
the treatment of multiaxial states of stress for this same material.

137
A. 4. Dimensionless Expressions for the Functions
0(s,A) and ijr(s,A)
Using the definitions for the dimensionless variables in equation
(3.2.1), the expressions for 0(s,A) and i|f(s,A) given in Sections A. 2
and A.3 can be converted into dimensionless form. First, when the
uniaxial stress-strain curve in equation (A.2.13) is used, the functions
0(s,A) and i|r(s,A) are given by equations (A.2.18) as
(A.4.1)
and using equation (3.2.1) the dimensionless form of these variables is
r
(A.4.2)
and
5(s,A) = E0(s,A)
EBn(s ct )n
y
n / \n-l
- E Bn(s s )
y
$(s,A) = Bn(s Sy)n 1
(A. 4.3)
where
n
(A.4.4)
B = BE
When the uniaxial stress-strain curve of equation (A.2.21) is
used, the functions 0(s,A) and i|r(s,A) are given in equation (A.2.25) as

138
*\
(5,A) = o
0 (s,A) = X
1
i
-2sSz n
~~d+ x
r.2s 1
2L3+2
eJ
(A.4.5)
and again using equation (3.2.1), these functions for this case can be
written in dimensionless form as
and
i|r (s,A) = \|r(s,A) = 0
o -
C1
(s,A)
Y(s,a)
f(s,A) = E0(s,A)
1
i2E | r i
(A.4.6)
where
-2es
3' =
F
(A.4.7)
(A. 4.8)
The expressions for 0(s,A) and ijf(s,A) are given in Section A. 3
for the Case XII studied by Cristescu (1972). In dimensionless form
these become
r
0 if s^F(A) or A *o (A. 4.9)
[s-F(A)] if s>F(A) and A >A
V C1
where

139
and
F(A) = f(A)
E
r K(A)
o
XK
XKo [l exp (-4-)]}
r K
o o
J
$(s,A) = E0 (s A) = E0(A)
EX
/ a \3/2-| 2/3
3/2-1 2/3
() }
- 1
a/E
$(s,A) = X
-r / v 3/2-, 2/3
4-vi+t(Â¥) 1
- 1
xa
(A. 4.10)
(A.4.11)
where
a
xa =
(A.4.12)

APPENDIX B
CHARACTERISTICS AND EQUATIONS ALONG THE CHARACTERISTICS
B.l. Equations for the Characteristics
The equations necessary for determining the characteristics are
given by equations (2.38) to (2.44). Thus, the characteristics c are
given by the equation
or
acp
Co
0
0
1
0
0
0
acp
0
0
0
0
0
0
cp
0
0
1
1
0
0
O 1
>1
H-1
acA2
CA3
0
0
0
acA2
acA.
4
acA_
5
0
0
1
O
>
CO
acA,_ cA^,
5 6
cp
0
1
0
0
0
cp
0
0
1
1
0
CA1
acA2
CA3
= 0
0
0
acA2
acA
4
acA
0
0
1
CA3
acA
o
CA6
0
(B.1.1)
140

141
or
a(cp )'
or
a(cp )
or
,
-8
O
O
1
0 Cp 0
1
0 cA^ acA2 cA2
1 0 acA,
2 Zl3
+ acp
0 acA acA acA
-i nr D
0 0 acA acA
4 5
1 cA acA cA
0 1 acA
cA
5
6
5 6
cA. acA cA
0 cAw acA
12 3
1 2
ac.4 acA, acA
2 4 5
-
- 2
a (cp )
0 acA acA,
2 4
- acp
cA acA,. cA
1 cA acA
3 5 6 J
3 5
3-d~- -
-2-
- -2 -2- "1
2 2-4
cl A A,A + 2aA^
¡ir
- A A -
aA A aAA
an c
L 1 4 6 2
3 5
3 4
1 5 2 6J
M 1
2- lT- -2*1
_ r
1
c ¡ A A aA 1-
L 4 6 5J
acp|_
- ao\J
= 0
cp 0 1
0 acA^ acA^_
1 acA cA
o o
V4 all\
and using the definitions (2.46), this becomes
2 3-6 2 2-4 2 -2-
apc(a)-apc(b)+apcA4 =0
|^a2oc2J j (p c2) 2 (a) (pc2) (b) + aJ = 0
(B.1.2)
which is the equation for the characteristics given in equation (2.45)
B.2. Equations along the Characteristics
The equations (2.48) along the nonvertical characteristics can be
obtained in two different ways. One form of the equations along the
characteristics will be found below by each method as an example; then
the other forms of these equations will be given. Using the first

142
methodthe system of equations is written in the form of equation (2.38)
and the equations along the characteristics are given by
1T A dw = ir b dt
(B.2.1)
where 1 is the left eigenvector and is determined from
1T (cA B) =0
(B.2.2)
which becomes
L1! *2 x3 x4 x5 X6J
Cp
0
0
1
0
0
0
acp
0
0
0
0
0
0
cp
0
0
1
1
0
0
CA1
acA
2
CA3
0
0
0
acA
2
acA^
4
acA
5
0
0
1
CA3
acA_
5
CA6
This is the matrix form of the following six simultaneous algebraic
equations for the six components of the left eigenvector
pl1 + l4 = 0 ^
acpl.
= 0
P13 + 16 =
^1 + CA114 + acA215 + CA316 =
acAl + acA 1 + acA_l_ = 0
2 4 4 5 5 6
1 + cA 1 + acA 1 + cA 1 = 0
3 3 4 5 5 6 6
(B.2.3)
However, from equation (2.38) the equation for the characteristics is
cA B
0

143
so it is apparent that the simultaneous equations (B.2.3) obtained from
equation (B.2.2) are not independent. Therefore, one of the unknowns
can be chosen arbitrarily, and the others can be solved in terms of it.
From equation (B.2.3) ,
12 = 0 (B.2.4)
and arbitrarily selecting
14 = 1 (B.2.5)
equation (B.2.3)^ requires that
1x= ~ ~ (B.2.6)
P c
Using equations (B.2.5) and (B.2.3) the last two of equations
(B.2.3) reduce to
which yield
acA,l + acA 1 = -acA
4 5 5 6 2
acA_l +
5 5
ft'S
cp
*6 = CA3
LA6 52]
A3A5 A2 l_ o c*
5 Z~TZ 1-I -2
A4LA6-J-^S
pc
(B.2.7)
and
X6 =
aA A A A.
D Z O 4
, aA2
4 e -21 5
pc
and equation (B.2.3) becomes

(B.2.8)
X3 =
pc
aA2A5 ~ A3A4
-A4 LA6 J ^5
pc
(B.2.9)

144
The fourth equation of equations (B.2.3) has not been used. However,
when equations (B.2.5), (B.2.7), and (B.2.8) are substituted into it,
the resulting equation is the characteristic equation (B.1.2) and no
new information is obtained. From these results, the left eigenvector
can be written as
1 =
1
pc
1
pc
0
aA A A A
Ad o 4
- -2 4
A A aA
4 o 5 -Z '
Pc
^3^5 ^2^6 + -2
V6 ^5
A4
-2
pc
aA A A A
Ad - -2 A4
A4A6 ^5
(B.2.10)
A dw
(2.
39) and
(2
41),
p
0
0
0
0
o ~
dv
X
0
ap
0
0
0
0
dv
r
0
0
P
0
0
0
dvQ
0
0
0
A1
^2
>1
CO
da
X
0
0
0
aA2
^4
^5
dCTe
0
0
0
A3
aA
D
>i
02
1
K1
CD
ir-
1

A dw
pdv
ap dv
pdv
Aldx+aA2da8+VT0x
| aA da + aA^da.+ aAdT.
j 2 x 4 9 5 0x
LA3dax+aA5dOe + VT0x
and using this and equation (2.42), the equations along the character
istics given by equation (B.2.1) become
[hVshVs]
p dv
ap dv
pdv
Aid,J*+aA2doe + A3dTe*
aA2dax+ aA4da9 + aA5dT0x
A3dax+aA5% + VTex
L
HVshVeJ
0
acr.
a-
2i|f
0x
or,

146
1 1
- dv
x -
c
'"aA2A5 A3A4
~i r
-2 -
c A A aA A /o c
4 6 5 4
2J dve + LAidcrx+ aA2dC7e
3' ej
+ AdT
K
AA_ -
A A +
3 5
2 6 -2
DC
A 4 -
-2 ^4
aA -
4 6
5 -2
pc -
f-
aA~A
- ia.
2 5
3 4
A
_
-2 4
A A -
aA^
L 4 6
5 -2_
p c

a" *

- 2
AA -
A A + -
3 5
2 6 -2
pc
[
aA d 2 x 4 0 5 0
J
A3dax+aA5% + VT0.x
J-
lit dt
Tx
- -2 4
A4A6 ^5 ~^2
Pc J
r
al ) ldt
o
0X
aA2A5 A3A4
- -2 A4
A,A aA
46 5-2.
pc
dt
r
and multiplying through by the term jp c A A aA
-2 A4
4 6 5 -2,
pc
A 1 >>
4]}.
this form
of the equations along the nonvertical characteristics becomes
-2
0 = -_-[(P c2) (A4A6 a2) A J dvx e|_ [aA^ 3 J dv
0
[<
+ I D
4 6 5
A,
dax+aA2do0 + A3dTe
J
2 i r -
+ I AA_ AA + 2 j aAdcf_ + aA^dcr^ + aA,_dT,
3 5 2 6
pc
2 x 4 0 5 0x
J[-]
[p 2J [ai2I5 -V j [Vx + aA5doe + A6dT0
j
[
+ I (pc2)(A4A6-aA2) -A4
'][
p c2 I I aA2A5 A3A4
1] +xdt {-
J [2,eJ
A3A5-A2A6 +
pc o

147
and regrouping the terms in this equation yields
r
-2. -
= rL(PC-)(A4A6-aA-) A
c
dv^ -pc
LaV5-vJdv
r -2 t t T t2 7 t 7 t2t t r r r2:
(pc)'
+ |_(pC) (A1A4A6-aA1A+aA2A3A5-aA2A6+aA2A3A5-;A4)-(14-a2)jdaj
+ [(Pi2)(W6-^5+W5-Â¥4V^5'V4V -(VVVV]adCTe
[(p;
+ ¡ (pc ) (A3A4A6-aA3A5+aA3A5-aA2A5A6+aA2A5A6-A3A4A6) (AgA4 aA2A5J d 0X
[<
+ | (p c2) (A4Ag-aA2) -aJ 4x dt a j^(p c) (A3A5-A2Ag)+A2 || l|r I dt
f (pc2)[aA2A5-A3A4][2^0J dt .
Using the characteristic equation (2.48), the above equation along these
nonvertical characteristics can be written in differential form as
0- r[-j
dvx-po|_aA2A5-3jdv0
-h [(pH2) (4 -a2)-jdax+ [a25-34JciTex
PC
[(pc2)(4A6-aX2)-4] [tlrjdt a[(Pc2)(35-26)+2] [--^Jdt
o
J[2*eJdt
-2
+ (p c )
aA2A5_A3A'
(B.2.11)
The second method for obtaining the equations along the nonvertical
characteristics may be shown as follows. Consider a curve in the x t
(characteristic) plane described parametrically by
x = x(§)
t = t(§)
(B.2.12)

148
and assume that the six variables CT ct, T. v v and v. are known
x 8 0x x r 8
along this curve as functions of the parameter §. Then the derivatives
of these functions can be computed as
dv
"Hr = vx,x x§ + vx,t t$
dve
dg ve,x xg + ve,t
dv.
r
= v x,_ + v t, _
r,x § r,t §
d a
x
-jr* = ct x + CT t,-
dg x,x I x,t §
da.
"5T CTe,x x,g + CTe,t ti
dT
6x
dc~ = T6x,x X§ + T8x,t tfg
(B.2.13)
where the subscript following the comma denotes differentiation with
respect to that variable, i.e.,
v
x, x
9v
x
=
V
X,
dv
X
t ~ ~5F
dx
§ dg
The six equations (B.2.13) and the equations (2.24), (2.25), (2.26),
(2.34), (2.35), and (2.36) form a set of twelve simultaneous partial
differential equations which can be written in matrix form as

Op OOOO-IOOOO
OOOapOOOOOOO
OOOOOp 0000 -1
-1 000000 A O aA 0
X Cj
0 0 0 0 0 0 0 aA2 0 aA4 0
0 0 0 0 -1 0 0 A 0 aA^ 0
d 5
x,g t,^ 0 0 0 0 0 0 0 0 0
0 0ax,^at,^0 0 0 0 0 0 0
0 0 0 0 x,_ t,_ 0 0 0 0 0
§ ?
0 0 0 0 0 0 x,g t,g 0 0 0
0 0 0 0 0 0 0 0 ax,^ at,g. 0
0000000000 x,,
0
V
X,X
0
0
Vx,t
a cr0
0
r
o
0
V
r, x
0
>1
CO
Vr ,t
-Ilf
T X
a5
V0, x
a(r--+e)
o
6
V0,t
-2^0x
0
a
X X
dv
X
db
0
x, t
dv
r
a d§
0
CTe,x
dv0
d£
0
CTe,t
da
X
d§
0
T0X,X
da0
a di
t§
T0x,t
dT0x
dT
(B.2.14)
149

150
or as
where
MZ = d
(B.2.15)
M =
Op 0000 -1 00000
OOOapOOOOOOOO
OOOOOp 0000 -1 0
-1 000000A OaAOA
12 3
0 0 0 0 0 0 0 aA 0 aA 0 aA_
2 4 5
0 0 0 0 -1 0 0 A 0 aA_ 0 A
3 5 6
x,£ t,g 0000000000
0 0 ax, a^§ 0 0 0 0 0 0 0
0 0 0 0 x,g t,g 0 0 0 0 0 0
0 0 0 0 0 0 x,g t,g 0 0 0 0
0 0 0 0 0 0 0 0 ax,^ at,- 0 0
0000000000
XS *§
(B.2.16)
z = 1v v 4.t v v V. V. | a a *, aQ *, t .
x,x vx,t vr,x 'r,t v0,x v0,t ^x,x wx,t u0,x w0,t 0x,x T0x,1j
(B.2.17)
and
L aCTe r r N\ dVx dVr dV0 dax da0 dT0x
d = |o, o, -V H--V> _2^9x "Hr a^r 1 "dT dT a-dT^T
o o
(B.2.18)

151
Now, since all of the components of M and d are known at a given point
(x,t), the solution to equation (B.2.15) can be found directly if the
determinant of M does not vanish. However, if the determinant of M
vanishes, the solution for Z cannot be found. This is represented by
| M j = 0 (B.2.19)
which is the equation for the characteristic lines along which discon
tinuities in the variables may propagate. Evaluation of this determi
nant results in the characteristic equation (2.45). When equation
(B.2.19) is satisfied, the system of equations (B.2.15) will be shown
below to reduce to a set of ordinary differential equations along the
characteristic lines, that is, along each characteristic line one ordi
nary differential equation involving the six variables v^, Vq, v^, , and t must be satisfied. If M is defined as the matrix M with
0' 0x n
t h
its n column replaced by d, then the solution to equation (B.2.15) can
be written in terms of Cramer's rule as
M
Z = (B.2.20)
11 I M I
t h
where Z is the n element of Z. However, by equation (B.2.19), the
n
denominal;or vanishes and for a solution to equation (B.2.15) to exist,
it is necessary that
| M 1=0 (B.2.21)
i ~n 1
Equations (B.2.21) are twelve equations, one for each column in M which
is replaced. Next, the calculations involved in equation (B.2.21) will
be carried out as an example for the case where n=l. This yields

152
r o p
ac
1 0
M, -
alr-*e
o
-2^9x 0
dv
df
dv
I
"dT
dv
s-
e
d9
da
y.
d§
da
0
dT
0X
d?
0 0 0
0 ap 0
0 0 0
0 0 0
0 0 0
0 0-1
0 0 0
ax,g at,- 0
0 0 x, g.
0 0 0
0 0 0
0 0 0
0
0
p
0
0
0
0
0
0
0
0
-1
0
0
0
0
0
0
0
0
x§
0
0
0 0 0 0
0 0 0 0
0 0 0 -1
A 0 aX2 0
a 0 aA. 0
2 4
A 0 a A 0
o o
0 0 0 0
0 0 0 0
0 0 0 0
t, 0 0 0
0 ax,^ at,^ 0
0 0 0 x, ^
A,
0
0
0
0
0
0
or, after some manipulation

153
^ 3 5 r ______ o __ i) __ r)
_M^ ) = -a p x,_ p- |^A^A,A+2aAAA^-AA-aAH -aA.
46 235 34 15 26
]
4 2 2
3
dVQ
P ^ > p
Xr
d£
4 2
4
da
X
a p t,.
X£
d§
4 2 2
a p t,g
4
x, _
s
[-*:
4 2 2
3
dv
X
a p
XF
a
d§
4 ^ 3
2
da
X
p i
X§
d§
aA2A5_A3A4
J
r.
de I AlA4A6+2ail2A3A5_A3A4_aA2A6 &A
3
)(A2A6-A3A5)-2^0x(A2A5a-A3A4)
]
4 3 2 dT 0x
1 pt§ x§ dT L^Wa
J
5V^]+ a4pt,§ x,§ [^(r^eX] = -
Along the characteristic lines of equation (2.47), x,_ = 0 and this
equation is identically satisfied. Along the characteristic lines of
equation (2.48), t,F is never equal to zero since this would correspond
5
to an infinite wave speed. Therefore, dividing by t,P and noting that
dx/d£ dx
t, ~~ dt/d§ ~ dt
S
= c
(B.2.22)
the above equation becomes
dv
0 = aV2
iV^]
dv
d
]+ if hA-vJ
pc(aA2A5-A3A4) j + I aAA_-AA,
da
x
"dT
[p c2 (^46+2a235-2X4-a26-a142) (A^-^]
4/ig-drt3^ -4j c -r a \ ~ y g jj3 c ^2A6_A3A5^'
+ 2iojp£2(a52s5-y4)]t,5} .

154
Using equation (2.48) in this equation yields
4 2f 1 T -2 -2-
0 = a pc [-i[pc (A4A6_aA5)_A4
%
db
*[ --
VrvJ-ar-=i [<
pc
[P;2(i446-^)-iJ 1*. + a[,
pe2(A4A6-^)-Sj
do
x
"dT
pc (AgAg AgAg) A2j
dt
~~ V df
o
+ 2 [pc2(a2A5-34)j i6x
This is an ordinary differential equation along the characteristic lines
of equation (2.48). This equation can be written in differential form
by multiplying by the increment d§ along the characteristic line. Doing
this, and noting that the equation is satisfied if the term in the
bracket vanishes, this equation finally reduces to
0 = | dvx_i5 [aV5_vJdve+ [fvvvj^ex
[p^'hV^s-h] +xdt
pc
+ a[p2(A2A6-A3A5)~2] (/"O^ 2 [pc2(aV5-A3A4)]'t'exdt
o
(B.2.23)
which is identical to equation (B.2.11) above. Note that equation (2.48)
represents four characteristic lines and that equation (B.2.23) repre
sents different equations along each of these four lines since along
each line c has a different value. The process just described can be
carried out for values of n from 2 to 12. Doing this, two additional
forms of the equations along the characteristics are found to be

155
- | [(pE2)(26-A35)-2]dvj!4 [<1VV3>-5]dve
> A,A !
-2
P c
i r ~2 -
l''V'lf3' 5j u 9x + ~=2 L(p C ) A2A6" VV _A2J dCTx
(pe2) (1 -A AJ-.4 I d
[(PE2)(A2A6-A3A5)-A2]?xdt+ 2 [(DE2)(A1A5-A2A3)-A5]^0xdt
~2 j^(p c')2(AiA6-2) (pe2) (A. T
pe
i+V+1] (/^e)dt (B-2-24)
and,
2-3
O = pc~ [j^2A5-A3A4J dvx pe ^(pc2)(A1
Jdve
[^(p C2) (A1A4-a2) - J dTfa + -2
5-vJ
r
(PC >LaA2A3-A3A4J^x
J
-2 2
+ (p c )
,aA2A5
t dt + 2(p c)
[(pc2)(A1A4-aA2)-A4]1J/exdt
+ ap^^p^XA^-^-j (^-*e)dt
(B.2.25)
Equations (B.2.23), (B.2.24), and (B.2.25) are all forms of the same
equation when the waves are fully coupled. Uncoupled waves will be
discussed in Section B.4.
The equations along the vertical characteristics (2.47) can be
written directly from equations (2.25) and (2.35), since along these
characteristics only t varies (x is constant) and partial differential
equations with all partial differentiation with respect to t become
ordinary differential equations. Thus, along the vertical character
istics of equation (2.47), equations (2.25) and (2.35) may be written as
a ov
and
'a dv
0 r
= St
aVr dCTx dCTfi dTPx
= aA + aA - + aA
r 2 dt 4 dt
o
s dt + a*e

156
of in differential form these become
e
dt = ap dv
r
r
o
(B.2.26)
B.3. Reducing Equations to Simpler Case
When radial inertia effects are not included and when the functions
0(s,A) and \|f(s,) are obtained from the rate independent incremental
plasticity theory, then the equation (B.1.2) for the characteristics
and the equations (B.2.23) and (B.2.25) along the characteristics
reduce to those given by Clifton (1966) for the von Mises yield condi
tion as will be shown below.
Under these conditions, equation (A.1.2) becomes
-2
(B.3.1)
s
and Clifton's expression for k can be written as
Now Clifton defined the function H(k) for the von Mises yield condition
as
H (k)
1
3

157
and this function H(s) can be written in terms of 0(s,A) by using
equations (A.2.16) and (A.2.18) to get
H(s)
H(s) =
3Bn(s-a )
y
n-1
-2
s
30(s)
or
1 -2
0 (s ,A) = 0(s) = S H(s) .
(B.3.2)
Using equation (B.3.2) in equation (2.23), and remembering that
a = 0,
where
and
A5 2 CTxTGx H
V 5 + 3Te* H
(B.3.3)
2(1+y) = 1
E G
H = H (s) .

158
Using equation (B.3.3) in equations (2.46)
a = A.
rv1 ax
3Tt
Sx 2 2 2 \
(22 2\1
i
6T
at
+
S!
a
+
- H + a T H -
E x 0x /
(Ve* H )J
a = A.
r~
LEG
a2 3tB
x 0x
+ 3GH + ~E
= ]
- 1 x 1 2
b A4 |^ + -g-H + + 3T H
]
G 0x
and the equation (2.45) for the characteristics becomes after dividing
by A^ (which is always greater than zero)
(P^2)2 ih + 3^H+ ^H) (PS2)(i+^H+ ^+3T0xH) + 1 = O (B-3-4)
which is the characteristic equation given by Clifton (1966).
The equations along the characteristics can be reduced to the two
forms given by Clifton (1966). When a = 0, and ijj = |Q=fQ =0,
x 0 0x
equation (B.2.23) becomes
0=- \ [(pc2)X4X6-4]dvx+pc[3jdve-|]34]dTex+ 4(pc2)(/Ic)-jda,
4 6 4J x
or dividing by A
0 = [(p52)6-i][- i dvx + A_ dCTJ + a3 [P;cive-dT6x]
c pc
and using equation (B.3.3), this equation along the characteristic
becomes
[(I
+ 3T H
P c
J + [V0XH][P
ex-/-~r2j[dc7x~DcdvT H npdve-dT
6 9x
J (B.3.
5)
0

159
Also, under the conditions given here, equation (B.2.25) becomes
(p o (A.AJ-A, dv. +
2-3 r~-
i -r
0 = P c |_~A3A4
JdvxpcL
[(p52XA1A4)-A4JdTex-pi2 [vjda,
-2
and dividing by (p c ), this becomes
o=[h-^][d'
pc
P c dvft J + A3 |^p c dvx doj
6x ^ 0
and again using equation (B.3.3) this is
2
0 = [^2 ^iTH}][pSdv8-dTexJ+[v9xH][p3dvx-dCTJ- Equations (B.3.5) and (B.3.6) are the equations along the character
istics given by Clifton (1966) for the von Mises yield condition.
B.4. Uncoupled Waves
When the stress waves are fully coupled, the equations along the
characteristics given by equations (B.2.23), (B.2.24), and (B.2.25) are
equivalent. Each of these equations represents one equation along each
of the four nonvertical characteristics. The waves are fully coupled
when none of the coefficients of the variables in these equations vanish.
However, it can be seen that some of the coefficients vanish in these
equations when A =A =0. This condition can occur when loading is
o o
along the TQ axis (a =cto = 0), or when loading is perpendicular to the
OX X W
T- axis in the plane T. =0, or when loading occurs at stress levels
0x 0X
within the yield surface (0(s,A) =0), or when unloading occurs (0(s,A) = O)

160
Under these conditions (A = A =0), the equation for the nonvertical
o o
characteristics of equation (2.45) becomes
(pc)2(14A6 aA2A6) (pc2) (^ + a2) + 4 = 0
and as in equation (2.48)
-2 2 WA -
C =
2p(X146 aX2X6)
-2 ^A1A4 + A4A6 ~ ^2^ ^ i-A4^A6 Al^ + aA2^
2pA6(AiA4 aA2)
and using equations (2.49) and (2.50),
-2 (A1A4 + A4A6 aA2) + (A4A6 ~ A1A4 + aA2)
2PA6(A1A4 aA2}
-2
Cf
A.
p (V4 ~ ^
(B.4.1)
-2 (A1A4 + A4A6 aA2) ~ (A4A6 A144 + aA2)
2PA6(A1A4 ^
-2 1
c =
PAC
(B.4.2)
and the four nonvertical characteristics are
c = cf, cs.
The equations along the characteristics will be obtained from
equation (B.2.23). For A =A =0, this equation becomes
o O
o = [(P;2)(6)-i][a4][-
dv + ~ da + i dt
x -2 x Tx J
c pc
[;2)V1][¡2]Lr-*Jdt
+ a
(B.4.3)

161
However, along the characteristic lines for the slow wave speeds
(c = cg) equation (B.4.3) is identically satisfied by using equa
tion (B.4.2) and no information can be obtained from equation (B.4.3)
along the characteristic lines c = c Along the characteristic lines
for the fast waves (c= c^) given by equation (B.4.1), the equations
(B.4.3) become
0 =
(pcp6-lJ [ J [t idvx+ -A- dax+ dt]
Pc.
+ aj^(p c)Ag-l
(B.4.4)
In this case when the waves are uncoupled, equation (B.2.23) yields
no information about the variables vA and T. and by inspection, neither
9 9x
does equation (B.2.24). Therefore, we will next investigate the equa
tion along the characteristics of equation (B.2.25). This equation for
A_ =A_= 0 reduces to
O
0 = [(pc2)^^ aA2) T
<¡[
-2
^Cdve+ dTex + 2p hxdt
*]
(B.4.5)
However, along the characteristic lines for the fast wave speeds
(c = c^.) of equation (B.4.1), equation (B.4.5) is identically satisfied
Along the characteristic lines for the slow wave speeds (c = c ) given
by equation (B.4.2), equation (B.4.5) becomes
0 =
^2' '"'ij r
f sdve + dV + 2fcs't'exdt
]
(B.4.6)
Thus, in the case of uncoupled waves the nonvertical characteristic
lines are given by equations (B.4.1) and (B.4.2) and the equations along
these lines are given by (B.4.4) and (B.4.6) which may be simplified.

162
The equation along c = c is
0 =
dv +
x
~2 dcrx + x dt 1A4
rv.
PC4
+a(A)
L_r
\
dt
and along c = cg is
-2
0 = + pcg dv0 + dtQv + 2pco i|f dt
9x
's 0X
(B.4.7)
J
For the uncoupled waves, the equations along the vertical character
istics (c=0) can be written directly from equation (B.2.26) as
acr.
6 ^ j
dt = an dv
r r
o
v
a( Tlr_ Idt = aA0 dcr + aA. dan
\r Y 0/ 2 x 40.
o J
(B.4.8)
B.5. Elastic Waves
A special case where the waves are uncoupled occurs when 0(s,A) =0.
In this case the waves are elastic and may be either loading at stress
levels within the yield surface or unloading. Since 0(s,A) =0,
equations (2.23) become
(B.5.1)

163
and the equations (B.4.1) and (B.4.2) for the characteristic lines
become
-2
Cf
(B.5.2)
and
-2 G
c =
S p
(B. 5.3)
which are
(B.5.4)
J
Also, for the elastic case we have iji = ij/ = i(r = 0 and the
x 0 0x
equations (B.4.7) and (B.4.8) along the characteristics further reduce.
The equation along c = c^ is
0 = I + dv + i da j i
I x -2 xl E
c_ P c
,v \
(-L)dt
E Vr /
o
(B.5.5)
along c
cs is 0 = + pcs dv0 + dT0x
(B.5.6)
along c = 0 is
aa
dt = ap dv
r r
(B.5.7)
and along c = 0 is
av
r ,, av a
dt = da + da
r E x E 8
where c^ and C£ are given by equation (B.5.4).
(B.5.8)

APPENDIX C
PROGRAMS FOR DETERMINING THE PLASTIC WAVE SPEEDS
In this appendix the listings are given for the two computer
programs used in the first section of Chapter 3. The first program
listed is the one which calculates the plastic wave speeds as functions
of v, 3, y, and 6 for the case when radial inertia is included or for
the case when radial inertia effects are not included. It is also used
to calculate the values of the normalized stresses as functions of y
and 6. The second program listed is the one which calculates the crit
ical value of 8 (as a function of v) at y = 0 for which c =c =c .
I s z
164

c
c
c
c
c
c
c
c
c
c
c
PROGRAM T CALCULATE; THE PLASTIC WAVE SPEEDS
THE EQUATIONS USED ARE (3.1.7), (3.1.18), (3.1.19), AND (3.1.20)
A = ALPHA = 0 FOR CLIFTON'S CASE
A = ALPHA = 1 FOR RADIAL INERTIA EFFECTS
0 E T A = SLOPE OF THE STRESS-STRA IN CURVE IN TENSION DIVIDED BY
YOUNGS MODULUS
NU = POISSONS RATIO
DIMENSION DELTA(20), GAM(IC), A1(10,2C), A2(10,20), A3(1C20)
DIMENSION A4(10,73), A5( 10,20), A6( 10,20), ABAR(10,20)
DIMENSION B B A R(ID 2 0) RAD(10,2?), CFC2(13.20) CSC2(10,20)
DIMENSION SX( 10 23), ST110.20), TAU(ll)
REAL NU
INTEGER A
1 REAC(StlOl) BETA,DEL GAM0,DGAM,DDFL,NU, A
I F ( NU. GT.C.5)GO TO 10
E = SORT(3.0 )
PI = 3.141593
1 = (l./BETA) 1.0
1 = 1
2 GAM(I) = GAMS
C = COS(GAM( I ) )
S = S I N ( G AM ( I ) )
C2 = C * 2
S 2 = S * 2
CS ^ C*S
TAU(I) = S/t
J = 1
CELL = DEL
3 DELTA(J) = ADELL+ (A l)*(PI/3.0)
165

CD = CS(CELTAlJ))
SD = S IN ( DEL T A(J ) )
CD2 = CD *2
SD2 = S D 2
CSC = CD* SD
S X( I ,J) = C *(C D SD/b)
ST( I,J ) = C *(CD + SD/b)
A1 ( I J ) = 1. + .25*Z*C2*(CD2 2.*E*CS0 + 3.C-*SD2)
A2(I,J) = -NU .25*Z*C2*(3.*S02 CD2 )
A 3 ( I J ) = .5G*E7*CS*(CC E*SD)
A 4 ( I J ) = 1. .2WC2MCD R 2.£*CSD + 3.0*SD2)
A5(I,J)=.50*E*Z*CS(CD+E*SD)
A6 ( I J ) = 2.0 ( 1 * iMU) + 3 0 Z S 2
ABAR(I,J)=A1(I,J)*A4(I,J)*A6(I,J)-A3(I,J)*A3U,J)*A4 I(I,J)*A3( I,J)* A 5( I,J )-Al( I,J )*A5(I,J)*A5(1,J)-A2(I ,J)*A2( I ,J)*A6(I
2, J ) )
RBAR(I,J >-A4l I,J)*(A6(I,J)+Al(I,J))-A*{A2(I J)*A2(I ,J)+A5(I,J)*A5(
1 I J ) )
WAD2 = BBAR( I,J ) *BBARlI,J ) 4.* ABAR( I,J ) *A4(I,J )
IE(RAB2.GT.0.'-')G0 TU 13
RAD 3 = -RAD2
RAD4 = 110.*SQRT(RAD3)
I F(ORAR( I ,J) .LT.RAD4JG TO 13
RAD { I J ) = .i.O
GO TC 14
13 CONTINUE
RAD( I J ) = SQRT(RAD?)
14 CONTINUE
CFC2 ( I J ) =SORT( ( ( 1. 4-UU ) /ABAR( I J ) ) ( BBAR ( I J ) 4-RAD { I J) ) )
CSC2(I J)= S Q R T( ((l.+NU)/ABAR(I,J) )*(BBAR( I,J)-RAD(I J)) )
DELL= DELL+ DDEL
J = J + 1
IFIJ-20) 3,4,4
4 CONTINUE
166

GAM1 = GAMO + UGAM
1 = 1 + 1
I F( I 1 1 ) 2,5,5
5 CONTINU
JJ = J 1
11=1-1
CO 6 1=1,11
GAM I I ) = GAM(I)* 1 BO./PI
6 CONTINUO
CO 7 J=lfJJ
CELTA!J) = DELTA(J)*1BG./PI
7 C U N T I N U _
WRIT 2!6 102) A,BET A NU
W NIT C(6,103)
WRITE!6,1C4) (GAM( I), 1 = 1, II )
CO d J=1,JJ
WRITE(* 105) DELTA!J),(CFC2!I,J ),1 = 1, I I )
3 CONTINUE
KM IT (6, 106)
HR 1T c{6, 104) !G A M( I ), 1 = 1, I I)
CU 9 J=1,JJ
WRITC(,105 ) DELTA!J),(CSC21I,J ) ,I = 1, I I )
9 CONTINUE
GO TC 1
19 CONTINUE
KRITE!6,107)
WRIF E(6, 1^4 ) (GAM! I ), 1 = 1, I I )
CU 11 J=1,JJ
KRITt(6,105) DELTA(J),(5X(I,J),1=1, II)
11 CONTINUE
WRITE(6, 108 )
WRITtlto,104)!GAM( I), 1 = 1, II )
CU 12 J=i,JJ
WRITE(6, 1C5 ) DEL TA(J),(ST( I J ) 1=1, II)
167

12 CONTINUE
WRITE(6109)
WRITE(6,110)(GAM( I), 1 = 1, I I)
WRITEI6, 111 ) (TAUl I ), 1 = 1, II )
101 F0RMAT(6F1G.4,15)
10? FORMAT('I',5X,'ALPHA = I 1 /6X 'BETA = ,F6.4/6X,NU = ',Fb.4)
103 FORMAT( / / / 5 O X ,FAST PLASTIC WAVE SPEEDS, CF/C2'/)
104 FORMAT(2X,'DELTA*,3X,'GAMMA = 10(F4.1,7X)/)
105 FORMAT(F8.24Xf10FI 1.5)
106 F0KMATI///5JX,SLOW PLASTIC WAVE SPEEDS, CS/C2'/)
107 FORMAT (' 1',50XNORMAL IZED AXIAL STRESS'/)
108 FORMAT (///5-jX,'NORMALIZED HOOP STRESS'/)
109 FORMAT(///50X,NORMALIZED SHEAR STRESS'/)
llu FORMAT(10X, 'GAMMA =,10 IF4.1,7X )/)
111 FORMAT(12X,10 FI 1.5)
STOP
END
168

o o n c~. o
THIS PROGRAM CALCULATES THE CRITICAL VALUES OF BETA FuR WHICH
CF/C2 = CS/C2 = l.C AT GAMMA = 0.0
THIS PROGRAM USES EQUATIONS (3.1.7), (3.1.27), (3.1.28), AND
(3.1.29)
DIMENSION DELTA(20) BETA(2u)
REAL NU
INTEGER A, AA(20)
1 REALMS.101) NU,CUTOFF
I F ( CUT Or F G E 1.0 ) GO TO 4
A = I
E = SORT(3.0)
PI = 3.141593
J = 1
CEL = -PI/2.
2 CFLTA(J) = A*DEL (A l)*(PI/3.0)
C = CCS( EL T A(J ) )
S = SINIDFLTA(J))
S 2 = S 2
CS = CS
CENOM = (l.+NU)*E*CS + 3.*NU*S?
IF(A.EQ.I)GO TO 5
Z = 1. + 2.Nil
CO TC 6
5 CONTINUE
C = ABS(DENOM)
IF(C.GT.1.0 E-4)CO TO 8
BETA(J) = 0.0
GO TO 7
8 CONTINUE
Z = -((1. NU ) *2 )/DENOM
6 CONTINUE
BETA!J) = l./( 1 .*Z )
169

7 CONTINUE
A A ( i ) = A
J J + I
CEL = DEL + PI/1H.0
IF(J.LT.2D)GO TU 2
A = u
IF(J.ED.20)GO TO 2
WRIT E(6,102)
CO 3 J=1,20
CELT A(J) = DELTA!J)*180./PI
WRITE(6103) DELTAJ ) ,AA(J), NU.BETAIJ)
3 CONTINUE
CC TC I
4 CONTINUE
Id FOkMAT( 2 F 10.3 )
102 FORMAT(' IFOR EACH VALUE F DELTA, THE APPROPRIATE VALUE OF Bn TA IS
1 THE ONE BETWEEN 0 AND +1'//' THE VALUc UF BETA GIVEN IS TH~ ON
AT WHICH CF/C2 = CS/C2 = 1.0 AT GAMMA = C//' A = ALPHA = 0 FO
3CLIFT0NS CASE'/' A = ALPHA 1 FOR RADIAL INERTIA EFFECTS////1
4, DELTA',7X,ALPHA*15X, 'NU13X, CRITICAL VALUE OF BETA /)
103 FORMAT IF 20.2,I 13,F2 > 3.F30.6)
STOP
END
o
uj a£ in

APPENDIX D
SOLUTION TO THE FINITE DIFFERENCE EQUATIONS
IN THE CHARACTERISTIC PLANE
In this appendix, the expressions for the stresses and velocities
at a grid point P will be determined. In doing this, many new terms
will be defined. In order to simplify the interpretation of the
computer code in Appendix E, these new terms will be defined in exactly
the same manner in which they are used in this computer code.
D.l. Equations for Fully Coupled Waves
When the waves are fully coupled, the finite difference equations
can be simplified by grouping known quantities together. Thus, letting
Q3 = 2AT(1 -v2) ^
D = V Q S
3 rB j 9b
A2Q 2(A2P + A2B^
A4Q 2(A4P + A4B)
A5Q 2(A5P + A5B'>
and
RHSE = >Tj^2V + (S 2aSrsJ -^l + A_S_+A_SftT,+ A_T
I
rB xB
0B s
oB~]
3
B
2Q xB 4Q 0B 5Q B
(D.1.1)
(D.l.2)
the equations (3.4.2) and (3.4.3) along the characteristics (c = 0) can
be written as
aVrP = a(D3"Q3S6P)
(D.l.3)
171

172
and
i|r / 2\Jf /
ToP YoP
aRHSE = -2aATV n+ a(Ao^ AT)S +a(A, + a AT)S
rP 2Q Sp/ xP 4Q s /
GP
+ aA T .
5Q P
Now defining the quantities
(D.1.4)
Q4 = 2AT Q3
op'
D-, = A AT
1 2Q sp/
lit /
D = A + 2a -5
2 4Q s /
A
AT + Q,
(D.1.5)
and
D4 = 2AT D3
RHSH = RHSE + D,
4 J
the two equations along the vertical characteristics (c = 0) can be
reduced to a single equation by substituting equation (D.1.3) into
equation (D.1.4). This becomes
aDlSxP + aD2S0P + aA5QTP = aRHSH (D.1.6)
The equations along the nonvertical characteristic lines are
given by equations (3.4.4), (3.4.5), (3.4.6), and (3.4.7). These can
be written in a much simpler form by defining the following quantities
as A
Z1 =
Z2 ^
Z3 =
1 v
c
s
2
Z4
1 v
AT
1 -
Z5 1 + c,
7 AT
O 1 + C
(D.1.7)
J

173
and
Z7 = VZ5
Z8 VZ6
Z =
6*VZ7
Z10 = 6 Z4'Z8
Z11 = V(2,Rif + aR2f} }
z-.o = zc* (2 R + aR0 )
12 8 Is 2s y
and then
If
If
B2f = VRfs
B3f = Rfs*(1-V
R,
If
- Z*
4f Z3 11
B5f = V(Rlf+2aB2f> '
B61 = 2'VR2f
y
and
Is
Is c
B2s Z2*Rfs
B3s = Efs-(1-Z10>
lS
34s = ~Z~ ~ Z12
4
"N
B5s = Vis+2aR2s) (
B- = 2*ZC*ROc
6s 6 2s
y
and finally
D3f2 = BfS,(1 + Z9)
D3s2 = BfS-<1 + Z10>
Rlf
4f2 ~Z~+ Z11
R-
D = + Z
4s2 Z 12
4
The equations along the four nonvertical characteristic lines
(c = c^., cg) can now be written as given below. The equation
along c = + c^ is
BlfVxP B2fV0P + D3f2TP + D4f2SxP aB5fS0P + aB6fVrP
_BlfVxLLB "" B2fV0LLB + B3fTLLB+ B4fSxLLB
+ aB5fS0LLB aB6fVrLLB
(D.1.8)
(D.1.9)
(D.1.10)
(D.1.11)
(D.1.12)

174
along c = c is
BlfVxP + B2fV9P+ D3f2TP + D4f2SxP aB5fS0P + aB6fVrP
BlfVxRRB+ B2fVGRRB+ B3fTRRB+ B4fSxRRB
+ aB5fS0RRB aB6fVrRRB
(D.1.13)
along c = + cs is
" BlsVxp B2sV6P + D3s2TP + D4s2SxP aB5sS0P + aB6sVrP
"B1sVxLBB ~ B2sV6LBB+ B3sTLBB^ B4sSxLBB
+ aB5sS9LBB aB6s^rLBB
(D. 1.14)
and along c=-c is
s
B V + B V +D T + D S aB_ S. + aB V
Is xP 2s 0P 3s2 P 4s2 xP 5s 0P 6s rP
BlsVxRBB+ B2sV0RBB+ B3sTRBB+ B4sSxRBB
+ aB5sS0RBB_aB6sVrRBB '
(D.1.15)
Next, let
RHSA BlfVxLLB B2fV0LLB+ B3fTLLB+ B4fSxLLB+ aB5fS0LLB
aB6fVrLLB
RHSB BlfvxRRB+ B2fVeRRB+ B3fTRRB+ B4fSxRRB+ aB5fS0RRB
aB6fVrRRB
RHSC b1svxLBB B2sVeLBB+ B3sTLBB+ B4sSxLBB+ aB5sS0LBB
aB6sVrLBB
RHSD B2sV0RBB+ B3sTRBB+ B4sSxRBB+ B5sS0RBB
HB6s^ rRBB
J
(D.1.16)

175
Now substituting equations (D.1.3) and (D.1.16) into equations (D.1.12)
to (D.1.15) and letting
B7f B5f + Q3B6f
B = B + Q B
7s 5s *3 6s
RHSA1 = RHSA aD B
3 6f
RHSBl = RHSB aD^B
3 6i
RHSCl = RHSC aD B_
3 6s
RHSD1 = RHSD aD B. ,
3 6s s
(D.1.17)
the four equations along the nonvertical characteristics can finally
be written in the form given below. The equation
along c = + c^ is
BlfV B2fV + D312TP + D4f2SxP aB7£S8P EHSA1
(D.1.18)
along c = c^ is
Blf'xp + B2f V + D3f2Tp + D4f2SXP aB7fS0P = RHSB1
(D.1.19)
along c = + c^ is
- B V B V + D_ T + D S aB = RHSCl
Is xP 2s 0P 3s2 P 4s2 xP 7s 0P
(D.1.20)
and along c = c^ is
BlsVxP + B2sV0P + D3s2TP + D4s2SxP aB7sS0P
RHSDl .
(D.1.21)

176
The four equations (D.1.18), (D.1.19), (D.1.20), and (D.1.21) along
with equation (D.1.6) now become a set of five simultaneous algebraic
equations which must be solved during each iteration for the unknowns
VxP V0P TP SxP and Sep point P of a regular grid element. The
reason for calculating the coefficients by the method outlined in
Chapter 3 is now apparent. If the coefficients had been calculated in
the normal manner, these five equations would have to be solved simul
taneously for each iteration at a regular grid point when the waves are
fully coupled. However, by calculating these coefficients by the method
used here, these equations reduce to two separate sets of simultaneous
equations, one set with two equations and one set with three equations
(this will be shown in Section D.3), and the computation time required
to solve these two sets of equations is significantly less than the
time to solve one set of five equations. Once the solution is found
at a regular grid point, the radial velocity is obtained from equation
(D.1.3). This will be done in Section D.3.
At a boundary point the solution is obtained by specifying two of
the variables (as described in Chapter 3) and then solving equations
(D.1.19), (D.1.21), and (D.1.6) simultaneously. Again equation (D.1.3)
is used to get V This will be done in Section D.5.
D.2. Equations for Uncoupled Waves
When the waves are uncoupled, the finite difference equations
along the characteristic lines are given in equations (3.4.9) to
(3.4.14). These expressions will now be simplified by grouping known

177
terms together. Using equation (D.1.1), equation (3.4.9) along the
vertical characteristic (c=0), becomes
aVrP = a(D3-W which is the same as equation (D.1.3). Next, using equations
(D.1.1) and (D.1.5), defining
\
S 2aS. -i
L2 '
S 2aS
RHSEE = AT| 2V + ill
rB s_ toB.
B
J
A2QSxB + A4QS0B
and
(D. 2.2)
RHSEEM= RHSEE + 2T D
J J
and using equation (D.2.1) in equation (3.4.10), the other equation along
the vertical characteristic (c = 0) is
aD S + aD S = aRHSEEM (D.2.3)
1 xP 2 0P .
The finite difference equations along the nonvertical character
istic lines (c = c^, c^) will be simplified by using equations
(D.1.7) and (D.1.8) in equations (3.4.11) to (3.4.14) and noting that
A. is not zero so that the four equations reduce to
4i
v*p+[
1 + 2Z + aZ n
Z3 7 7 A4i-'
xP
aZv[1 + 2aS¡i]
A
S. + 2aZ_ -- V =
GP 5 A . rP
4i
V +
c xLLB
f z (2 + a S + aZ j 1 + 2a -Hi 1
LZ3 7 \ A4. /J xLLB 7L A4iJ
0LLB
- 2aZ i V
5 A . rLLB
4i
(D.2.4)

178
v + + Z (2 + a ~ )ls aZ j 1
XP LZq 7 V A4i /J xP 7 L
A 1 A .
2aA-iJsep+2aZ-^v
+ 2a
'4i'
5 A, rP
4i
V + i Z ^2 + a ~ ') I S + aZ
cf xRRB \_Z3 7 V A4i /J xRRB 7
1 S + aZ 1 + 2a -=il
yj xRRB 7 L A, J
4i
0RRB
2 aZ - V
5 A rRRB
4i
- z2 Vep + <1 + Z10,TP = Z2Ve.,BB+ (1 -zio>tlbb
Z2VSP + (1+Z10)TP= VeRBB+ (1-Z10)TRBB
By letting
Flf c
2f
3f
Z7 P + 2a A~
4i
F4f ^ 22
5 A
2i
4i
F = F + Q F
5f 3f ^3 4f
F = 7 (2 + a
6f z3 7l a4.
>
and
F2s = 1 + Z10
F3s = 1 Z10
(D. 2. 5)
(D. 2.6)
(D.2.7)
(D. 2. 8)
(D.2.9)

179
and
RHSAE Fif^xLLB+ F6fSxLLB aF3fS0LLB + aF4fVrLLB
RHSBE = F V + F _S aF S + aF V
If xRRB 6f xRRB 3f 0RRB 4f rRRB
RHSCE = ZV + F T
2 0LBB 3s LBB
RHSDE = Z V. + F T
2 0RBB 3s RBB
and
RHSAEM = RHSAE + aD F
O I
RHSBEM = RHSBE + aD F
o ^xl
> (D.2.10)
y
(D.2.11)
and by substituting equation (D.2.1) into equations (D. 2.4) and (D. 2.5),
the equations (D.2.4), (D.2.5), (D.2.6) and (D.2.7) along the nonvertical
characteristics become as follows. The equation
along c = + c
is
-FlfVxP+F2fSxP+aF5fS0P=RHSAEM
(D.2.12)
along c = cp
is
FlfVxP+F2fSxP+aF5fS0P=RHSBEM
(D.2.13)
along c = + c
s
is
- Z2V0P + F2sTP = RHSCE
(D.2.14)
and along c =
- c is
s
Z2V0P + F2sTP=RHSDE *
(D.2.15)
Equations (D.2.3) and (D.2.12) to (D.2.15) form a set of five
simultaneous algebraic equations for the unknowns V^, Vgp, Sxp, sgp
and T Tlieir solution will be obtained at a regular grid point and
a boundary grid point in Sections D.4 and D.6, respectively. In all
cases the radial velocity, V will be found from equation (D.2.1)
after the hoop stress, Sa at the grid point is known.
0P

180
D.3. Solution at a Regular Grid Point
for Fully Coupled Waves
For this case the solution will be obtained by solving equations
(D.1.6) and (D.1.18) to (D.1.21) simultaneously. These equations can
be reduced to a set of two simultaneous equations and a set of three
simultaneous equations as follows. Sutracting equation (D.1.18) from
equation (D.1.19) and subtracting equation (D.1.20) from equation.
(D.1.21) results in two equations involving only the unknowns V and
Vgp. They are
2BlfVxP + 2B2fV6P = RHSB1 RHSA1 (D.3.1)
2EL V + 2B = RHSD1 RHSCl (D.3.2)
Is xP 2s 0P
and if
II
tH
Q
2Blf
D2f "
2B2f
Dls =
2Bls
2s =
2B2s
RHSBA
= RHSBl
- RHSA1
= RHSB
- RHSA
RHSDC
= RHSD1
- RHSCl
= RHSD
- RHSC
(D. 3.3)
then these equations can be written in matrix form as


r .
Dlf
D2f
VxP
RHSBA
^Dls
D2s _
_V9P_
RHSDC
Solving this, the expressions for the longitudinal
velocities at a regular grid point are given by
(D.3.4)
and transverse
V = (D RHSBA D RHSDC) (D.3.5)
xP 2s 2f
V.n = -i- (D.. RHSDC D RHSBA)
6P A^ If Is
(D.3.6)

181
where
A1 = DlfD2s DlsD2f
(D.3.7)
Two equations involving only stresses can be obtained by adding
equations (D.1.18) and (D.1.19) and by adding equations (D.1.20) and
(D.1.21). Doing this and letting
D3f
2D3f2
D4f =
2D4f2
D7f =
2B7f
3s -
2D3s2
4s =
2D,
4s2
7s =
2B,
7s
RHSF = RHSA1 + RHSBl
RHSG = RHSC1 + RHSD1
these equations become
D3fTF + D4fSxP + aD7fS0P = RHSF
(D. 3.8)
(D. 3. 9)
D T + D S + aD = RHSG
3s P 4s xP 7s 0P
(D.3.10)
The third equation required to solve for the stresses is equation (D.1.6)
which is
aD S + aD S + aA T = aRHSH .
1 xP 2 0P 5Q P
When radial inertia effects are included, that is, when a=1,
these three equations can be written in the matrix form
D3f
D4f
D7f
T
P
-rhsf
D3s
4s
D7s
SxP
=
RHSG
_A5Q
D1
D2_
_SeF_
RHSH
(D.3.11)

182
and with
42 = 3f + D7f
the stresses at a regular gird point are given by
TP = ^[RHSF(D2D46-D1D7S) + RHSG(DlD7f-D2D4f)
+ raSH(D4fDjs-D4sD7f)]
'xP h,
LrhSf + BiSE<3sD7f-V7S)]
6P A2 L
RHSF(JL D A D ) + RHSG(A D D, D )
1 3s 5Q 4s 5Q 4f 1 3f
+ RHSH(D3ffD4s-D4fD3s)
]
and the radial velocity, V^p, is given by equation (D.1.3).
When radial inertia effects are not included, that is, when
equation (D.1.6) vanishes, and the two remaining equations (D.3.
(D.3.10) for the stresses, can be written in matrix form as
D3f
I
Q
T
P
RHSF
D3s
I
w
Q
RHSG
and letting
a = D D D D
3 3f 4s 3s 4f
(D.3.12)
(D.3.13)
(D.3.14)
(D.3.15)
a = 0,
9) and
(D.3.16)
(D.3.17)

183
the stresses at a regular grid point are given by
> = 4¡(D4sBHSF-D4fRHSG)
SxP = i¡ (D.3.18)
(D.3.19)
and the hoop stress, S ,-and the radial velocity, V are both zero.
H 0P J rP'
D.4. Solution at a Regular Grid Point
for Uncoupled Waves
For this case equations (D.2.3) and (D.2.12) to (D.2.15) must be
solved simultaneously. Once this is done V can be calculated from
equation (D.2.1). An expression for the shear stress is found by
adding equations (D.2.14) and (D.2.15) and dividing by to get
1 P
RHSCE + RHSDE
2F
(D.4.1)
2s
The tangential velocity is obtained by subtracting equation
(D.2.14) from equation (D.2.15) and dividing by 2Z^ to get
ep
RHSDE RHSCE
2Z
(D. 4.2)
By subtracting equation (D.2.12) from equation (D.2.13) and
dividing this by the expression for the longitudinal velocity
is found to be
V
RHSBEM RHSAEM
xP
2F
(D. 4. 3)
If

184
Adding equations (D.2.12) and (D.2.13) and letting
F = 2F
2f 2 2f
F
5f 2
2F
5f
RHS3 = RHSAEM + RHSBEM J
results in the equation
F2f2SxP + aF5f2SGP RHS3
(D. 4.4)
(D. 4.5)
When radial inertia effects are included (a = l), equations
(D.2.3) and (D.4.5) can be written as
F2f2
F5f2
"SxP~
RHS3
_ D1
D2
_ Sqp _
RHSEEM
(D.4.6)
and by defining
D F
2 2f2
- DlF5f2
(D.4.7)
the expressions for the hoop stress and the longitudinal stress can
be written as
S = j-(D RHS3 F RHSEEM) (D.4.8)
xP A 2 5f2
5
S0p = r'(F2f2RHSEEM D!RHS3)- (D.4.9)
5
When radial inertia effects are not included (a = 0), equation
(D.2.3) vanishes, the hoop stress and the radial velocity are zero,
and the expression for S is found directly from equation (D.4.5) to be
xP
S
xP
RHS3
F
2f2
(D.4.10)

185
D.5. Solution at a Boundary Point (X=0)
for Fully Coupled Waves
At a boundary point, the solution when the waves are fully coupled
is obtained by specifying two of the variables at the point P and then
using equations (D.1.6), (D.1.19), and (D.1.21). Once again the
radial velocity is calculated using equation (D.1.3) after SQri has been
0P
determined for all cases considered. The solution will now be devel
oped for each of the four sets of boundary conditions discussed in
Chapter 3.
Case I: Traction boundary conditions
When S and T are specified, then the hoop stress can be deter-
xP P
mined directly from equation (D.1.6) and is
sep = 5-dfflSH - a5(jV .
(D.5.1)
Once this is known, two new quantities can be defined as
RHS1 SB1 D3f2Tp D4f2Sxp + aB7fSep
RHS2 = RHSDl D T D. S + aB S_
3s2 P 4s2 xP 7s 0P
(D. 5.2)
so that equations (D.1.19) and (D.1.21) can be written in matrix form as
Blf
B2f
VxP
RHS1
_ Bls
B2s
1
<
CD
'P
l
RHS2
(D.5.2)
With
*4 = (D.5.4)

186
the velocities at point P are given by
Vxp = (B2sRHS1 B2fRHS2)
4
V6P = A- 4
Case II: Kinematic boundary conditions
When V^p and V are known at a boundary point, by letting
and
RHS4 = RHSB1 -
RHS5 = RHSD1 EL V Bn V.
Is xP 2s 0P
the three equations for the stresses from equations (D.1.6), (D.
and (D.1.21) can be written as
D3f2TP + D4f2S*P B7fS6P = RHS4
+ D S aB S. = RHS5 \
3s2 P 4s2 xP 7s 0P /
aA T + aD S n + aD = aRHSH
5Q P 1 xP 2 0P J
When radial inertia effects are included (a=1) these equations
be written in matrix form as
D3f2
D4f2
B7f
r t n
p
" RHS4
D3s2
D4s2
"B7s
SXP
=
RHS5
1
>
Oi
D1
D2
_ s6p _
RHSH
_ _
so that with
A6 D3f2(D2D4s2 + D1B7s) D4f 2 (D2D3s2 + A5QB7s)
B7f(DlD3s2 A5QD4s2)
(D.5.5)
(D.5.6)
(D.5.7)
1.19),
(D. 5.8)
can
(D. 5. 9)
(D.5.10)

187
the expressions for the stresses become
Tp = i [rHS4(D2D4s2+ DiB7s) RHS5(D2D4f2+ W
6
+ RHSH(D4s2B7f-D4f2B,
*>]
(D.5.11)
xP _
b
- RI,S4 - RHSH ]
(D.5.12)
Sep = rr[RHS4 RHS5(DlD3f2 A5QD4f2^
4 RHSH(D3f2D4s2 04f2D3s2)j
(D.5.13)
When radial inertia effects are not included (a=0) the radial
velocity and hoop stress vanish as does the third equation of (D.5.8).
The remaining two equations of (D.5.8) become
D3f2
D4f2
1
o,
t-
L
RHS4
D3s2
4s2
1
CO
$
L
RHS5
(D.5.14)
and letting
A7 ~ D3f2D4s2 3s2D4f2
(D.5.15)
the two remaining stresses are given by
TP = ; (D.5.16)
SxP = A7(D3f2RHS5"D3s2RHS4)
(D.5.17)

188
Case III: Mixed boundary conditions
When S^p and VQp are known at a boundary point
RHS6 = RHSB1 B2fVep D4f2Sxp
RHS7 = RHSD1 B V D S
2s 0P 4s2 xP
RHS8 = RHSH D S
1 xP
(D.5.18)
so that equations (D.1.19), (D.1.21), and (D.1.6) can be written as
VxP + D3f2TP aB7fS0P = EHS6
B-> V + D T aB S_ = RHS7
Is xP 3s2 P 7s 6P
Vp + aD2sep = aRHSB y
CD. 5.19)
When radial inertia effects are included (a=l) these equations become
[hi
D3f2
_B7f
V
xP
RHS6
:
Bls
D3s2
"B7s
T
P
RHS7
L
A5Q
3 j
r
w
CD
"0
L
RHS8
(D.5.20)
and if
A8 Blf (D2D3s2 + A5QB7s) Bls('D2D3f2 + A5QB7f}
(D.5.21)
the expressions for the unknown variables become
V.. = j^RHS6(DD0_0 + A^BJ RHS7 (DnD0J,0 + A^B,,^)
xP A
8
2 3s2 5Q 7s
2 3f2 5Q 7i'
+ KHS8(D3s2B7f D312B,
*>]
(D.5.22)

189
] r n
Tp = [Blf(D2RHS7 + B7sRHS8) Bls(D2RHS6 + B7fRHS8) J
8 ~
S6P " [Blf o
(D.5.23)
(D.5.24)
When radial inertia effects are not included (a = 0), the last of
equations (D.5.19) vanishes, Vrp and Sgp are zero, and the first two
equations of (D.5.19) become
B_ ^
D
V
RHS6 j
If
3f 2
xP
B-,
Do o
T
RHS7
Is
3s2
P
1
-J
so that with
BlfD3s2 BlsD3f2
the solution is
V
xP
T
P
= ¡r
9
= t^-(B, RHS7 B, RHS6)
If Is
Case IV: Mixed boundary conditions
When V^p and Tp are specified at a boundary point,
RHS9 = BHSB1 BlfVxp D3f2Tp >
RHS10 RHSD1 BlsVxp D3s2Tp l
(D.5.25)
(D. 5.26)
(D. 5.27)
(D. 5. 28)
(D. 5. 29)
RHS11 = RHSH AT
o y p

190
so that equations (D.1.19), (D.1.21), and (D.1.6) can be written as
B2fV6P +
D4f2SxP
aB7fS6P =
RHS9
B2sV6P +
D4s2SxP
aB7sS0P =
RHS10
(D. 5. 30)
aDlSxP +
aD2S9P =
aRHSll .
When radial inertia effects are included (a=l) these can be
written as
B2f
D
4f 2
"B7f "
vep
RHS9
B2s
4s2
B7s
SxP
=
RHS10
0
D1
2 _
_S0P_
RHS11
(D.5.31)
and when
S10 =B2f B2s(D2D4f2 + DlB7f>
(D.5.32)
the solution is given by
1
V
9P A. LBHS9(D2D4S2 + D1B7s> 11,5101D2D4f2 + DlB7f>
10
+ RHSll(B7fD4s2 B7S
4f2)J
(D. 5. 33)
'xP ~~ A |_B2f(D2RHS1+ b7sRHS11) B2s(d2RHS9 + B?fRHSll) I (D.3.34)
10
= ~ [B2f (D4s0RHS11 -D1RHS10) B2s(D4f2RHSll -D1RHS9)J (D. 5. 35)
8P
10
When radial inertia effects are not included (a=0), the last of
equations (D.5.30) vanishes as do the hoop stress and the radial
velocity, and the first two equations of (D.5.30) become

191
B2f
4f2
" V
RHS9
_ B2s
D4s2
s
xP
RHS10
so that with
11 B2fD4s2
B2sD4f2
the solution is given by
V
S
0P
xP
4s2
RHS9
2f
RHS10
D4f2RHS10)
B RHS9)
2s
(D. 5.36)
(D. 5.37)
(D. 5. 38)
(D. 5.39)
D.6. Solution at a Boundary Point (X=0)
for Uncoupled Waves
When the waves are uncoupled, the solution at a boundary grid point
is obtained by specifying two of the variables at the point and then
solving simultaneously the three equations (D.2.3), (D.2.13), and
(D.2.15). This will be done below for the four cases discussed in
Chapter 3. In all cases the radial velocity is calculated from equa
tion (D.2.1) after the hoop stress is found.
Case I: Traction boundary conditions
When S^p and are known, equations (D.2.3) and (D.2.15) yield
the expressions
S6P = ^-(RHSEEM ~ Vxp) (D.6.1)
V0p = ^-(RHSDE F2sV (D.6.2)
2
and once is known, the longitudinal velocity is found from equa
tion (D.2.13) to be
V = (RHSBEM 2F S ^ aF S0,_) .
xP F 2 xP 5f 9P
(D.6.3)

192
Case II: Kinematic boundary conditions
When V and V. are known, equation (D.2.15) becomes
xP SP
Tp = A-CRHSDE Z2V0p) (D.6.4)
If
RHS12 = RHSBEM .V _
If xP
then equation (D.2.13) is
F2fSxP + aF5fSGP = RHS12
(D.6.5)
(D.6.6)
and when radial inertia effects are included (a=l), equations
(D.6.6) and (D.2.3) can be written as
F
F_
S
RHS12
2f
5f
xP
D.
D
S.
RHSEEM
1
2
0P
(D.6.7)
By letting
J12
D F
2 2f
DlF5f
(D. 6. 8)
the expressions for the longitudinal and the hoop stresses can be
written as
S = -i-(D RHS12 F RHSEEM) (D.6.9)
xP A12 2 5f
S = ^(F-RHSEEM D..RHS12) (D.6.10)
6P 2f 1
When radial inertia effects are not included (a=0), equation
(D.2.3) vanishes along with S0p and V^p, and the expression for
is found from equation (D.6.6) to be
RHS12
(D.6.11)

193
Case III: Mixed boundary conditions
When S and V. are known at a boundary point, the expressions
xP 9P
for the hoop stress and the shear stress are found from equations
(D.2.3) and (D.2.15) to be
s0p = ^-(RHSEEM D1Sxp) (D.6.12)
Tp = (RHSDE Z2V ) (D.6.13)
2s
and once SQ is known, the longitudinal velocity is seen from equation
UP
(D.2.13) to be given by
V = F- (D'6'14)
Case IV: Mixed boundary conditions
When V and T are known at a boundary point, using equation
xP P
(D.2.15), the tangential velocity is given by
V0P = Z¡(RHSDE ~ F2STP>
Using equation (D.6.5), equation (D.2.13) can be written as
(D.6.15)
F_S + aF S. = RHS12 .
2f xP 5f 0P
However, this equation and equation (D.2.3) are the same two equations
solved for Case II above. Therefore, the longitudinal stress and the
hoop stress are given by equations (D.6.9) and (D.6.10) when radial
inertia effects are included (a = l), and when radial inertia effects
are not included (a = 0) the hoop stress vanishes and the longitudinal
stress is given by equation (D.6.11).

APPENDIX E
COMPUTER PROGRAM FOR CHARACTERISTIC PLANE SOLUTION
E.l. General Description of the Program
This program is written to solve the finite difference equations of
Chapter 3. Two types of grid elements are used: the regular grid ele
ment shown in Figure 3.5 and the boundary grid element shown in Fig
ure 3.6. These two types of elements are fitted together to form the
complete grid network of Figure 3.1, and the solutionsto the finite
difference equations are obtained at each point in this grid network
using equations developed in Appendix D and rewritten in Chapter 3.
To obtain the solution at a grid point P, however, the values of each
variable must be known at the points L, B, and R for a regular element
and at the points B and R for a boundary element. For this reason,
some initial conditions must be specified.
The initial conditions are specified in the region
X S T
since the fastest elastic wave can propagate with a speed no greater
than
For the problem considered here, this amounts to specifying all of the
variables at each grid point along the line
X = T,
that is, at the points (0,0), (AX,AT) (2AX,2AT) (a^AX.a^AT).
The initial conditions used are discussed in the next section.
194

195
Once the initial conditions are known, the solution can be found
at each grid point along the line
T = X + 2AT
beginning with the point (0,2AT). Once this solution is known, the
solution can be found at the point (AX,3AT) and then at (2AX,4AT),
(3AX, 5AT) ,...,( (a^-l)AX, (aVl)AT) The solution is then found along
the line
T = X + 4AT
in the same manner, and then along succeeding lines until the solution
is found at the point (0,2a^AT).
At each grid point (except along the line X = T) the solution
must be obtained by an iterative technique using the equations from
Chapter 3 (or Appendix D). Since some of the coefficients in these
equations are determined from the values of some of the functions cal
culated at P for the previous iteration, these coefficients must be
specified for the first iteration. This is done in every case by assum
ing that the waves are elastic, that is, by assuming on the first iter
ation (i=1) that
$(s,A) = (s A) = 0
and
Ali = 1
A.. = 1
4i
A2i -
A = 0
3i
A = 0
5i
A = 2(1+ v) .
6i
Using these values, the solution can be found at P. The values
variables obtained at P in this manner are then used to compute
of the
the
values of the coefficients for the second iteration to obtain a second

196
solution at P. This is done until two successive solutions at P differ
by some value which is specified in the input. Once this convergence
is reached, this iterative technique is begun at the next grid point.
E.2. Initial Conditions
Initially the tube is assumed to be at rest, loaded by a state of
constant stress (and, therefore, constant strain). Designating the
initial state by a superscript o, the initial conditions are given in
terms of the dimensionless variables as
V = V = V = 0
x 0 r
and since the body is at rest
i|f = 0
o
and from equation (2.8),
s.o
It is assumed that initially the tube is in a state of static
prestress (T = T, S = S) and that this stress state is reached with-
x x
out any unloading so that the strains can be uniquely determined.
Furthermore, to simplify matters, it will be assumed that this initial
state of stress (and strain) is reached by proportional loading so that
if the initial stress state is given by a?, (or s. for the deviatoric
ij iJ
stress), then the stress state at any time during the static preloading
is given by
a. = C(\)o. .
iJ iJ
s = C(X)s.
ij ij
\
J
(E.2.1)

197
where \ 'is a parameter along the static loading path and ij ij iJ
o
and s. are stresses which have not been made dimensionless. The incre-
ij
ment of plastic strain during this static loading is given by equation
(A.2.4) as
de. = s d\
ij ij
or
deP. = C(\)s?.d\
ij ij
and the plastic strain along the loading path becomes
P or1
e. = s. C(\)d\
ij iJ ^
where \ is a dummy variable, or
p = _H.
'ij C(\)
C(\)d\
and letting
U =
C(\)
C(\)d\
this plastic strain becomes
Now,
P
e. = cus .
13 ij
P P 2
e e = (ju s..s..
ij ij id ij
and from equation (2.15)
P P 2 2-2
e. e. = 77 ul) s
ij ij 3
or solving for (ju
P P
e. e. .
u = I
2 -2
s
(E. 2.2)
(E. 2.3)

198
Equation (2.16) defines the plastic part of the strain as
or
and from equation
ing becomes
s
(E.2.2), the plastic strain during the static preload-
P
e .
ij
s .
ij
(E. 2.4)

199
For the static case, A can be obtained from equations (A.2.13)
and (A.2.21) directly from the two uniaxial stress-strain curves.
For the generalization of the Lipkin and Clifton (1970) curve,
P / vn
A = B (s s^> (E. 2.5)
and for the generalization of the Cristescu (1972) curve,
1
-2es
z
- (s s ) + A s
p' y y
-sJ- 6)
If the only stresses present initially are ct and T then
X 0x
o2 o2
X + 31 ex
and
o / o2 o2
s = / S + 3t
AJ X
(E.2.7)
and A is calculated from either equation (E.2.5) or equation (E.2.6)
as
or
and
AP
A = X
,Po / o vn
A B (s s >
y
i
p2e2 ~i [ / oxz i
1 [ir (s y + \ s]-*2 [(fry SJ
P k
> (E.2.8)
o Po o
A = A + s
J
and the initial plastic strains are given by equation (E.2.4) as
_Po 3 AP /Sij
ij 2 sO V E

200
so that
.Po
3
AP
(2
0-
o
<3
S
X
2
o
s
\3
o
s
X
Po
3
AP
/
(
1 s)
1
AP
0 "
2
o
s
V
3 x/
2
o
s
Po
:0x ~
3
2
APO
o
o
T
A
(E.2. 9)
J
and the total prestrains are then given by
o o Po
e = S + e
xxx
o o Po
e9 = vSx + Ee
o ... o Po
e0x <1 + V)T + e0x
>1
J
(E.2.10)
For the two stress-strain curves considered here, the expressions
for $(s,A) are given in equations (A.4.3) and (A.4.7). Therefore, the
initial values of §(s,A) are given by §(s) for these two cases as
x. o. / o vn-l
s(s) = Bn(s s )
and
(E.2.11)
i
(~2 i r- O |
-I**,
respectively.

201
E.3. Calculation of A
In order to find the solution (stresses and velocities) at a grid
point P, a value of A must be calculated during each iteration. This
can be done by calculating the strains at point P from the velocities
(as shown in Section 3.6) and then calculating the value of A from the
strains at points P, R, B, and L (see Figures 3.5 and 3.6). The value
of A can also be found from the functions §(s,A) and \[r^(s,A) by eval
uating these functions using the values of s and A obtained during the
previous iteration. The second method will be used so the strains will
not have to be calculated during each iteration, and the finite differ
ence expression for the value of A at the point P will now be found.
From equation (2.14), the inelastic strain rate is
s
and then
2
J
From equation (2.16)
and using equation (3.2.1), the dimensionless form becomes
J^$(s,A) H+2i)ro(s,A)JdT+ s
(E.3.1)

202
or
dA ,, ds ds
dT §(s,A) dT + 2ij.'o(s,A) + dT
and omitting the functional dependence, this is
dA _. ds
dT = <+1) di+ 2V
(E.3.2)
Along the vertical characteristics from point B to point P, this
equation can be written in finite difference form as
-| rs s
AP AB
2AT
-il
($p + 1) + ($B +
1} J [-W5]* I [2*op+2U
or,
, A r, ?p + -b
ap = ,sb + L1 +
0[Sp-SB] + [*oP+toB] W '
(E. 3. 3)
This expression is used to calculate the value of A at point P during
each iteration.
If A had been calculated from the plastic strain rates, it would
P *p
have been necessary to include expressions for e and e .
r e
E.4. Input Data
In each computer run, the grid size (AX and AT) must be specified
as well as Poisson's ratio (v), the rise time for the pulse at the
boundary (T = (JRISE 1)AT), the maximum number of grid points in the
X direction (MXMAX), and whether or not radial inertia effects and/or
strain-rate dependence is included. Other data which are used are
obtained directly from Lipkin and Clifton (1970) and Case XII of
Cristescu (1972).
When using the data of Lipkin and Clifton (1970), the generalized
stress-strain curv becomes

203
A = s+ E¡(s-s)n
y
(E.4.1)
in terms of the dimensionless variables, where
n = 1.923 ^
v= 33^si = 3 39 x1Q-4
y 10 psi
- n -10 7 1 923 4
B = BE =(6.1313 xlO ) (10 ) = 1.77236 x10 )
(E.4.2)
When using the data of Cristescu (1972), the generalized stress-
strain curve is found from equation (A.2.21) to be
i
-2ez nr
7 s \2 1
(s s ) + A s + xo
L q' y y J 2 L
P
& -SJ
(E.4.3)
where
and
s
y
a
ys
E
1068.24 psi
6
10.2 X 10 psi
1.047294 x 10
When the data of Cristescu (1972) are used, several dimensionless
quantities can be defined for Case XII. This case is for dynamic load
ing only, and the equations for §(s,A) and ^(SjA) are given in
Section A.4. Other dimensionless quantities for this case are

204
O
S = #
y e
1500 psi
10.2 X 10 psi
-4
1.47059 x 10
-4
A = e = 1.47059 x 10
y y
A = e_
.002575
P =
5.6 X 10 psi
10.2 x 106 psi
, 0054902
A = 6 = .0004
m 3.25 x 10 psi
xm = =
E 10.2 X 10 psi
.0031863
(E.4.4)
xn
4
n 9 x 10 psi
J7 0
10.2 X 10 psi
= .0088235
h = .000004
X = .00005
r k 3 -1.
xko = 4^ = (---2.5- -1-n-)-(A. ec. 2-= .0012
c 2.08 x 10 in/sec
where the dimensionless coefficient for the coefficient in the rate
dependent term was obtained for a .5-inch diameter aluminum tube.
E.5. Listing of the Program
In this section is the listing of the computer program which was
used to obtain the solution in the characteristic plane to the wave
propagation problem presented in this paper.

o o r> o n n r- o o r¡ o r. r< n o o
DIMENSION SX ( 601 ) ST( 601 ) ,TAU( 6.' 1 ) VX( 61 ) VTI601 ) VR (601 ) ,trX(60l )
1 F;T ( CU ) FTX ( 6'U ) 0KLTA ( 601 ) HPLAS( 6'Jl ) ,CF ( oOl ) ,CS ( 6M ) t SB (6-n > ,
2E X P(6? 1 ) ,FTP(6 3 1 ) ,uTXP(601 ), PSI' (601),PHIU(601) ,1 TeK(601) ,Al (601 ) f
3 A2 ( 60 1 ) f A 3 I 6C l ) A4( 60 1 ) A5( 60 1 ) A6 ( 601 ) X ( 601 ) T ( 60 1) SMAX(6rl )
DIMENSION DELLHB(120),DELGAM(6'>1),DEX(601)
OI MANSION STRES(601)
DIMENSION SiP( 5? ),STRAIN( 50),VXP( 50) VTP( 50
PEAL N,riU
INTEGER A,UNCOUP
C
C UNCOUP = 1 WHEN THE TRANSVERSE ANO THE LONGITUDINAL WAVES.ARE UNCOUPLED
UNCOUP = O WHEN THE TRANSVERSE AND THE LONGITUDINAL WAVES ARE CUUPLED
A O POR NO RADIAL INERTIA EFFECTS
A = 1 FOR RADIAL INERTIA EFFECTS
JRATE = 0 FOP RATE INUEPENOENT CASE
JRATE = 1 FOR RATE DEPENDENT CASE
JDLL = C FOR CLIFTONS DATA
JBELL = 1 FOR BELLS DATA
IN NCNDIMENS 10NALIZINC THE PARAMETERS THE VALUES OF YOUNGS
MODULUS WHICH WERE USED WERE -
>z 1", 200, roo PSI FOR BELLS DATA
C = lO,v0O,''O0 PSI FUR CLIFTONS DATA
kL Afi (5,1C02) UNCOUP
Rf A0(5 1 Or1) A,JR AT 6,MXM AX,NU,H 1,H2,M3,U4,DEL X,PELT,N XKO
R t A D ( 5,1 0 C 2 ) JR ISE,XVFIN,TVFIN ,SXO,TAU j *SMALLBBARSY
REAL' ( 5,10f 2 ) KASE,SYS
PE All (5,10 02) JBELL,DZ,BETA, CHAT,XM,XN,H,XLAM
KEAD(5,lCn3) INCRX, I NCRT, I 0,MI,MJ,I PUNCH,I PUNI,IPUN2,INCRXP,INCRTP
W R I T E ( 6 ,1005) A,JRATE,DELX,CELT,MXMAX,SMALL
205

WRITE!6, 1030 ) JGELLtDZ fOHAT ,XM,XNXLAM
WRI T E ( 6 100 6 ) NU.SY,BDAR,N,XKC t RETA
WRI r E ( 6 100 7 ) JRISEfXVFINt TVFIN .SX-J.TAUO.H
WRITE(6,1006) H1.H2.H3.F4,I.SYS
WRI r::(6, 1' 3 1 ) MI.MJ, I PUNCH KASE
IF(UNCUUP.GT.O) WRITE(6,1C39)
IF! JNCGUP.LE.O) WRITE!6,1040)
I F( I PUNCH.EQ.O)GO TO 15
WRITZ ( 7, 1034) A, MXMAX,I PUNI,I PUN 2,INCRXPINCRTP,DELX.OELT
15 CONTINUE
C THIS BEGINS THE CALCULATION OF CONSTANTS USED
M M A X = 2 *MX MAX
NMAX = MMAX 1
1M A X = NMAX 2
Q = 1. Nil * 2
C2 = SCR T(.3*(1 .-NU ) )
uA = 1. + NU
CELT2 = 2.* D E L T
I 2 A = 2 A
Cl = 2 C A
C2 = C/2.
C3 = Q*D ElT 2
C4 = 03*CELT 2
FV1 1. + C 2
CV2 = 1. C2
CL R B = 2 *C2/l)V 1
CLRPI = 1. CLRB
C Y = ABS(SY)
CO 30 L= 1 MX MAX
3J XIL) = (L 1 ) *CELX
C THIS BEGINS THE CALCULATION OF INITIAL CONDITIONS
LINE =
LINER = I NCRTP 1
LINEO = I NCR T 1
206

S BO = SORT (SXO* 2 + 3.*TAUC**2)
CALL PH II PHIP,PHIQtOJBELL, SBOt DCLTAO SYS,BBAR,N,H,XLAM,BETA,DY,XM
1fXNtDZ,0)
IF(S BO.GT .1. E-0 8 ) GO TO 60
FI = 0.J
F 2 = 0.0
GO TO 65
60 CUNUNUL
Fl = sxo/suo
F 2 = TAUO/SBC
66 CONTINUE
G1 = PHIP*F1**2
G2 PHIP *F 2 * 2
C 3 = PHIPF1*F2
A10 = I. + GL
A 2 0 = -(NO 5C*G1 )
A 30 = 3 G3
A 40 = I. + 2 6 G1
A50 = -1.5 G 3
A6'j = Q l + 9 G 2
A 2 0 2 = A 2 C * 2
A 5 0 2 = A 5 ~ 2
IHUNCOOP.GT .0) GO TO 67
AO = A40(A10*Ao''-A30**2)+A*(2.*A20*A30*A5C-Ain*A502-A6,,*A2 0?)
P. = A 4 3 ( A 1 0 + A 60 ) A A 5 C 2 + A2G2)
COE F = 02/AO
PCCr = BC * 2 4. A 4 G A 0
RAO = SORT!ROOT)
CFO = SORT{CEF *(BO + RAO))
CSJ = SORT! COEFM BO RAD))
GO TO 6B
67 CONTINUE
CFO S0RT(0A4?/(A10*A40 A*A202)>
CSO = SORT!0/A60 )
207

60 CONTINUE
EXPO = SXO*PHIQ
ETP3 = -.5*A*EXP1
C PL A SO = PEL TAG SBO
ETXPO = 1.5*TAUP*PHIQ
E Xu = SXO + EXPO
ETu = A*lET PO NU*SXO)
ETX1 = Q A T A U C * ETXPO
CG 7 0 MX = 1MXMAX
T(MX ) = (MX 1 ) *OELT
TAU(MX) = TAUO
SX(MX ) = SXO
ST(MX ) = 0.0
VX(MX) = 0.0
VT(MX) = 0.0
VR(MX) = 0.0
SB(MX) = SB.)
CF(MX) = CFG
CS(VX) = CSu
A 1(MX) = All
AP(MX) = A2 0
A3(MX) = A30
A 4(MX) = A41
AS(MX) = A 50
A6(MX ) = A6 0
EX(MX) = HXO
cT(MX) = ETC
ETX(MX) = FJXO
EXP(MX) = EXPO
ETP(MX) = E T PO
ETXP(MX) = ETXPO
ITER(MX) 1
PHIO(MX) = PH IP
PSIO(MX) = 0.0
208

CEL r A(MX) = CELTAO
CPLAS(MX) = DPL AS')
SMAX(MX) = snc
CEL GAM(MX) = 0.0
C EX(MX) = 0.0
STKES(MX) = SORT(SX0**2)
70 CONTINUE
CELLHR(l) = CELTAO
C THIS ENOS THE CALCULATION OF INITIAL CONDITIONS
I W R I T E = 1
GO TU A01
9^ CONTINUE
KUM = 1
100 CONTINUE
MX = 1
JERROR 0
CO TO 300
200 CONTINUE
MX = MX + 1
'30 3 CONTINUE
MT = MX + KLJUNT
T(MX) = M T D E L T
SXk = SX(MXfl)
STR = ST(MX + 1 )
V X R = VX(MX 1 )
VTR = VT(MX+1)
VRR = VRIMX+l)
TAUR = TAU(MX + 1 )
T AUG = TAU(MX)
SXC = S X(MX)
STB = ST I MX)
VXD = VX{MX)
VT B = VT(MX)
VRB = VR(MX)
to
o

IB = Al (MX)
3n 1
302
A2B = A2(MX)
A 3 B A3(MX)
A 4 B A4(MX)
A 5 B = A5 (MX)
A6B = A6(MX)
SBB = SB(MX)
P S I 3 = PSIO(MX)
PMIB = PHIO(MX)
SMAXB = SMAX(MX)
LEL TAB = DELT A(MX )
1F(SD3.LT.l.UE-08)G0 TU 301
PSI3B = PSIB/SBC
GO TG 332
CONTINUE
PSIBB =0.0
CONTINUE
PSIBC = 375 *PSIBB
C3 = VR13 U3*STB
C4 = CELT2*D3
A 1 0 C :
A 2 B C =
A3BC =
A4BC
A 5 3 C =
A6BC =
T AUKR
c X k 3
STR3 =
VXRQ =
VTRB :
VRKB
I F ( MX
IFIMT
. 375*A1B
. 375 A2B
. 3 7 5 A 3 B
: 3 7 5 A 4 B
. 375 A53
. 3 75 A6D
= CLRB*TAUR
: CLRB*SXR +
: ULRB*STR +
: CLRB*VXR +
= CLRB*VTR +
: CLRB*VRR +
.GT.DGU TU
+ CLRR I *TAUB
CLREI*SXB
CLRBI*STB
CLRBIVXC
CLRBI* V T B
CLRBI*VRB
310
GT.JR ISE)GO TO 303
210

30 3
3 0 4
305
3 1C
IFCKASE.EQ.1) READ!5,10 C 4 ) SXI.TAUI
IFIKASE.EQ.2)
IF(K AS L .EQ.3)
IF(KASE.EQ.4)
GO TO 334
CONFINUE
IF(KASE.EQ.I)
IF(KASE.EQ.1)
IF(KASE.EQ.2)
IF(KASE.EQ.2)
IF(KASE.EQ.3)
IF(KASE.EQ.3)
IF(KASE.EQ.4)
IFIKASE.EQ.4)
R F AD( S, 100 4)
READ! 5 1004)
READ(5, 1004)
SXI = XVFIN
TAUI = TVFIN
VXI = XVFIN
VII = TVF1N
SXI = XVFIN
V r I = TVFIN
VXI = XVFIN
TAUI = TVFIN
CGNT I NUE
IF(MT.GT .2)GO TO 30 5
DEL TAP DELTAB
GO TO 32^
VXI,VTI
SXI,VTI
VXI,TAUI
DELTAP = 2.* DEL T AB DELLHB(MT-3)
CG 10 320
CONTINUE
sxl = sxirx-i)
STL = ST(MX-1)
VXL = VX(MX 1)
VTL = VT(MX-l)
VRL = VR(MX-l)
TAUL = TAU(MX-l)
CELTAP = DELTAIMX+1) + DELTA(MX-l)
TAULB = CLRB*TA'JL + CLRB I *T AUB
SXLB = CLRB*SXL + CLRBI*SXB
STLO = CLRB*STL + CLRBI*STB
VXLB = CLRB*VXL + CLRCI*VXR
VTL B = CLRB*VTL CLRBI *VTB
VRL B = CLRO*VRL + CLRBI*VRB
DELTAB

320 CONTINUE
I = 1
A1P = 1.0
A2 P = -NU
A 3 P = 0.0
A 4 P = 1.0
A5P = 0.0
A6P = Q1
PHIP = 0.0
CO TC 355
34t CONTINUO
1 = 1 + 1
I F { I.LE.MDGO TG 344
IF(JERROR.GT.0)GO TO 341
JERRCR = JERROR + 1
WRIT E(6 1033)
341 WRIT E(6110 3 2) X(MX ) ,T(MX ),DIFF,DENOM,ERROR
GU TC 430
344 CONTINUE
CALL PHI(PHIP,PH IQ,JRATE,JQuLL, 50PC,DEL TAP, SY,BtfAR,N, H XLAM, 3ETA, D
1Y,XM,XN,DZ,10)
I E ( S BPC. GT. SMAX3.UR I .GE .MJ )GU TO 351
PHIP = 0.0
351 CONTINUE
IE(SBPC.GT.1.CE-08)GO TO 353
FI = 0.0
F 2 = 0
F 3 = 0.0
GO TC 354
353 CONTINUE
FI = (2.*SX I A S T I J/SCPC
F 2 = (SXI I 2A*ST I ) /SBPC
F 3 = TAU1/S3PC
354 CONTINUE
212

Y 1 = 2 5 PH I P
Y2 = Yl* F1
Y 3 = PH IP *F 3
YA = 1.5 Y3
A 1P = 1.0 + Y 2 F1
A 2 P = { NU + Y 2 F 2 )
A3P = Y A F1
AA P = 1.0 + Y1*F2**2
A5 P = -Y A *F 2
A6P = Q1 + 9.*Y3*F3
355 CONTINUE
!F( PHIFJ.LT. L.0E-O5.OR.PHIP.LT.1.0E-05JG TO 356
1 F( I.CT.2 ) GO TO 360
356 CONTINUE
All
-
Al P
A 2 I
A2P
A3 I
=
A3 P
AA I
A'+P
A 5 I
A 5 P
A 6 I
-
A6P
A2C
-
A2P
m A (.
=
AtP
A 5 C
=
A5P
CO
TO
365
CONTINUE
All
=
.625*A1P

A IRC
A2 I
=
.625*A2P
+
A2BC
A 3 I
-
.625*A3P
+
A3 OC
AA I
=
625*AAP
+
AA RC
A 5 I
,625*A5P
-f
A5BC
A 6 I
=
.625*A6P
+
A6BC
A2C
=
. 5 ( A 2 P
+
A2B )
AA 0
-
. 5 ( A A P
+
AAB )
A5G
=
. 5 ( A 5 P
+
A5B )
213

365 CONTINUE
366
367
RHSc = DLLTM2. *VRB-PSIR0*( 12A*5TB-SXB) ) + A2U SXB + A4Q* S TB + A50 TAU6
RHSH = RHSE 04
A 2 I 2 = A 2 I * 2
A512 = A 5 I *2
IF(UNCOUP.GT.0) GO TO 366
A3 = A4I(A1I*A6I-A3I**2)+A*(2.*A2I*A3I*A5I-A1I*A5I2-A6*A213)
BR = A4IMA11 + A61 } A*(A2I2 + A 51 2)
C02F = Q2/A3
ROOT = 03**2 4.*A4l*AB
RAC = SURTI ROOT)
CFI = SURTI COFFM BB + R A D ) )
CSI = SURTICUEFMBB RAO))
GO TC 367
CONTINUE
CFI = SURTIQ*A4I/I AlIA4I A A 2I2) )
CSI = SURTIQ/A6I)
CUNTINUc
CV 3 = 1. + CFI
CV4 = I. + CSI
CUN I = 2 IC FI C2 ) /IDV2*0V3)
C0N2 = 0V1*CSI/I C2*0V4)
CON 3 = L. CON 1
CON4 = 1. CON 2
TAURRO = CON I T AUR + C0N3*TAUR0
SXRRB = C ON I SX R + CN3*SXR0
STKRB = CON I*ST R + C0N3STRB
VXkRC = C ON1 V X R + CN3*VXR0
VTRRB = CON 1*V T R + C0N3*VTRB
VRRRB = CONI*VRR + C0N3*VRKB
TAURCB = C0N2*TAURB + C0N4*TAUB
SXRBB = C0N2*SXRB + C0N4*SXB
STREG = C0N2*STR8 + C0N4*STB
VXRBB = C CN2 *V X R B + C0N4*VXB
214

VTRBB = CON2 *VTRB CON4*VTB
VRKD3 = CON2*VRRB + CON4*VR8
Z1 = CFI/O
l = CSI/Q
Z 3 = Z 1 C F I
Z 4 = Z 2 C S I
Z 5 = CELT/DV3
Z = DELT/DV4
RFS = A*A2I*A5I A 3 I *A4 I
RFSA = A BS(RFS)
IF(RFSA.LT. 1 .OF-^6.OR.UNCOUP.GT.O) GO TO 360
CVS = A4 I *A6 I A* A 5 I 2
PV6 = A2 I A6 I A 3 I A 5 I
P1F = Z3nV5 A4I
RIS = Z4 *DV 1 A4 I
P2F = Z 3 CV6 A 2 I
R2S = Z4DV6 A? I
BIS = R1S/CSI
81F = RIF/C FI
B2S = Z 2 *RFS
B2F = Z1RFS
RAS = 2. Z6 R2S
6F = 2.*Z5*R2F
368 CONTINUE
IF(I.GT.1)GO TO 370
PS IP = 0.0
RATIO = 0 .0
R A T I 0 P = 0 0
GO TC 330
370 CONTINUO
CALL PS I(PS IP,JRATE,SBPCDEL TAPDY.SY.BLTAXKO,DHAT,DZ)
SBUIF = SBPC SBB
OELTAP = DELTAB + SBUIF + DELT2MPSIP + PSIB)
lFISGPC.LT.SMAXB.ANU.I.LT.MJ)GO TO 371
215

C fc L T A P = CELTAP + .5MPHIP + PHIO)*SBDIF
371 CUM I NUE
1F(SUPC.GT.1.0E-08)GO TU 372
RATICP 0.0
GU TO 373
372 CONTINUE
FAT ICP = PSIP/SBPC
3 73 CONTI.NUt
I F( I.GT.2 )G0 TO 375
RATIC = RAT I OP
GO TO 300
375 CONTINUE
RATIC = .625 *RATIOP + PSIBC
380 CONTINUE
77 = Z5*RATI0
ZR = Z6 K AT IU
79 6.*73*77
7 lu = 6.* Z4 Z 8
Cl = A20 OELT *RATIOP
L2 = AMJ + A*DELT2*RATI0P + 04
1F(RFSA.LT. 1 .0E-06.UR.UNCOUP.GT.O) GO TO 382
711
=
17* ( 2.* R1F
+ A R 2 F )
712
=
Z8*(2.*R1S
* A *R2 S )
CV7
=
UF/Z3
CV8
=
RIS/Z4
B3S
=
RF S *(1 -
Z 10 )
P3F
=
RF S*( 1 ~
79)
04 S
=
DVH Z 12
B 4 F
=
CV7 Zll
35 S
=
Z8*(RIS +
I 2 A R 2 S )
05 F
=
Z 7 ( R 1 F +
I2A*R2F )
0 7 S
=
05S + 03* B6S
B7F
=
B6F + G 3* B6F
D3S2
= R F S* ( 1. +
Z10)
216

H3F2 = R F S* ( 1. + 7.9)
G4S2 = 0V8 + 712
C4F2 = [)V 7 + 711
RHSB = 1 F V X RR B + B2F*VTRRG + B 3F* TAURK 4 B4F SXRRB +
1 A*(S5F* STRRB B6F *VRRRB )
RHS D = B 1SVXRBB 4 B2S* VTRBB 4 B 3S T AURBB 4 B4S*SXRBB +
1 A *(B5S*STRBB B6S VRR BB )
GO TC 333
382 CONTINUE
ARATIO = A2I/A4I
7 13 = 1 7Z3
714 = 7 7 ( 2 4 AARA TIU )
F2S = 1. 4 7 10
F3S = 1. 710
F1F =1./CFI
F2F = 713 4 714
F6F = 713 714
F3F = -7 7 *( I2A*ARATIU 4 1.)
F4F = -2. *75*ARATIO
F 5 F = F 3 F 4 Q 3 F 4 F
PHSBE = F1F*VXRRB 4 F6F SXRRB 4 A*{-F3F*STRRB 4 F4F *VRRRB)
RHSUE = 72*VTRBR 4 F 3S* T AURBB
KHSBEM = RHSBE 4 A*F4F*D3
PHSEt = (2.*VR8 PS I BB* l I 2A*STB SXB))*DELT 4 A2QSXB 4
RHSEeM = RHSEE 4 D4
383 CONTINUE
IF(MX.GT.1)CO TC 385
IF(RFSA.LT. 1.0E-06.OR.UNCOUP.GT.Q) GO TO 384
IFUASE.NE. 1 ) GU TU 500
RHSBM = RHSB C3F2*TAUI D4F2SXI
R FIS CM = RHSD C3S2*TAUI C4S2*SXI
R FIS EM = RHSH A5Q*TAUI D1*SXI
STI A*RHSEM/D2
RHS1 = RHSBM 4 A*(B7F*STI- B6F*D3)
A4Q*S TB
217

RHS2 = RHSDM + A*(B7S*STI R6S*D3)
CEL 4 = B 1 F B 2 S B1SB2F
VXI = (RHS1*82S RHS2*C2F)/DEL4
VTI = (RHS2*D1F RHS 1*BIS)/DEL4
GO TC 390
580 I F(A.EO.O )GU TO 585
IFIKASE.NE.2)GO TO 581
ZA = D4S2*D2 + C1*B7S
ZB = D3S 2 *02 + A5q* B7S
ZC = C35 2 *01 A50* D4S2
Cr L6 = D3F2*ZA D4F2*Z3 B7F*ZC
RHS4 = RHSB AD3*B6F B1F*VXI B2F*VTI
RHS5 = RHSD A D 3* B6S B1S*VXI B2S*VT1
TAUI = ( RHS4*Z A-RFS5* ( D4F 2 02 + 1* 3 7F ) -RHSH* {D4S2 3 7F-D4F2 *B7S ) ) /DL:L6
SXI=l-RHS4*ZB+RFS5*(03F2*D2+37F*A5)-RHSH*(03S2*B7F-D3F2*B7S))/DEL
16
ST1 = (RHS4*ZC-RHS5*(O3F2 *D1-4F2 *A5U)+RHSH*(D3F2U4S2-D4F2*D3S2) )/
1CEL6
GO TC 39 J
581 IFKASE.NE.3 ) GO TO 582
RHS6 = RHSB A*D3*B6F B?F*VTI D4F2*SXI
RHS7 = RHSD A*D3B6S B2S*VTI D4S2*SXI
RHS 8 = R H SH D1*SX1
ZA = C2*D3S2 + A 5Q B 7 S
ZB = D 2 D 3 F 2 + A 5 Q H 7 F
CtLB = 81F* Z A B 1 S Z B
VXI = {R H S6 Z A R H S 7 Z B + RHS8 *{D3S2*B7F 3F2*B7S) )/DEL8
TAUI = ( B1F*(RHS7*D2 + RHS8*B7S)-B1S*(RHS6*02 + RGS8*B7F) J/DEL8
ST I = (B1F*(D3S2*RHS8-A5Q*RHS7)-BlS*(D3F2*RHS8-A5Q*KHS6) )/EL8
GO TC 390
582 CONTINUE
RHS9 = RHSB A*D3*B6F B1F*VXI D3F2*TAUI
RHS10= RHSD A*D3*B6S B1S*VXI D3S2TAUI
RHS11= RHSH A 5Q*T AUI
to
oo

7 A = C 2 D 4 S 2 + C1*B7S
ZB = 02 C4F 2 + C1*B7F
DELI") = B 2F Z A B2S*ZB
VT I = ( RHS 9 *7 A RHS1CWB + RHS 11 ( D4S2* B 7F-D4F 2 *B 7S ) )/DE L 1 0
SXI = (B2FM 02RHS107S*RHS11)-B2S*(D2*RHS9+37F*kHSl1) )/UEL10
ST I = (B2F *(D4S2*RHS11-D1*RHS10)-R2S*(D4F2*RHS11-D1*RHS9))/0EL10
GU TC 390
585 IFIKASE.NE.2)GO TO 536
RHS4 = RHSB A*D3*H6F D1F*VXI B2F*VTI
RHS 5 = RFSD A*D3*36S B1S*VXI B2S*VTI
CEL7 = U3F2*U4S2 D4F2*D3S2
T AUI = (RHS4 *04S2 RHS5*D4F2)/DEL7
SXI = (RHS5*D3F2 RHS4 D 3S 2 ) /DF L 7
ST l ^ 0.0
GU TU 390
586 IE(KASE.NE.i)GO TO 587
RHS6 = RHSB AD3*86F 32F*VTI 04F2*SXI
R HS 7 = RHSD A*D3*B6S B2S*VTI 04S2*SXI
GE 19 = B1F*D3S2 BLSD3F2
VXI = IRHS6*D3S2 RHS7 *D3F2 ) 70EE9
TAU = (RHS7*B1F RHS6*B13 )/OEL9
ST1 = 0.0
GU T 390
587 CONTINUE
RHS9 ^ RHSB AC3*R6F B1F*VXI D3F2*TAUI
RHS10- RHSO AD3*6S B1S*VXI D3S2*TAUI
CECIL = B2F-*C4S2 R2S*04F2
VT I = ( RHS9 *04S 2 RHS 10D4F 2 )/PEL 11
SXI = (RHS19*B2F RHS9*B2S)/DEL 11
ST I = 0.0
GO TO 39 U
384 CONTINUE
IF(KASE.E0.2.OR.KAS EEQ4)GO TO 594
STi = A*(RHS E EM D1*SXI)/D2
219

VTL8B = C0N2 *VT LB + CCN4*VTB
VKLBB = C0N2 *VRLB + C0N4*VRB
IF(RFSA.LT.1.0E-06.UR.UNC0UP.GT.0) GO TO 386
CIS = 2.*B1S
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VXI = (RHSBEM F 2 F SX I AF6FSTI )/FlF
IF(KASE.LQ.3)G0 TO 692
VTI = (RHSDE F 2 S T AUI )/Z2

02 S
=
2.* B2 S
06 S
=
2. B6S
DIF
=
2.*B1F
C2F
-
2 B2 F
C6F
=
2. B6F
C3S
2.*D3S2
C 3 F
=
2.D3F2
04 S
=
2. 04 S 2
C4F
=
2 C4 F 2
DBS
=
-2.BBS
DBF
RHSA
=
-2 BBF
= -BIF*VXLLB B2F*VTLLB + B3F TAULLB
1 A*(B5F*STLLB B6F*VRLL3)
RHSC = -B1S*VXLBB B2S VTL BB B3S*TAULBB
L A*(B5S*STLBB B6S*VRLB6)
OFL1 = O 1F*D2S D15*D2F
RHSBA = RHSB RHSA
RHi.CC = RHS 0 RHSC
VXI = (D 2 S R H S B A 02F*RHSDC)/DEL 1
VTI = (01 F*RFSDC D 1S*RHSBA)/DEL 1
C7F = DBF Q3*C6F
C7S = DBS Q3C6S
RHSF = RHSA + RFSB A D 3 D 6 F
RHSC = RHSC + RFSD A*D3*D6S
GO TO 3B 8
64F*SXLLB +
+ 34 S* SXLBB +
386 CONTINUE
RHS C E = -Z2*VTLSB + F3S*TAULBB
RHSAE = -F1F*VXLLB + F6F*SXLLB + A*(-F3F* STL LB + F4F*VRLLB)
FHSAEM = RHSAE + A*C3*F4F
TAUI = (RHSCE + RHSDE )/(2.*F2S)
VTI = (RHSDE RHSCF )/(2.*Z2)
VXI = (KHSBEM RHS A EM ) /(2.*FIF)
F2F2 = 2.*F2F
FBF2 = 2.F5F
221

RHSi = RHSAEM + RHSBEM
380 CONTINUE
IF(A.E0.0 ) GO TO 337
IFIRFSA.LT.1.QE-G6.OR.UNCOUP.GT.O) GO T 389
ZA = D2*4S DI D7 S
ZB = ASQ C7S C2 l) 3S
ZC = 0 1 C 3S A5Q*D4S
CEL2 = U3F#ZA + D4F*ZB + l)7F*ZC
TAUI = (RHSF*ZA*RHSG* SXI = (RHSF*ZB+RHSG*(02*03F-A5Q*D7F)+RHSH*(D3S*07F-D3F* 07S) )/0LL2
ST I = (RHSF*ZC + RHSG*(A5Q*D4F-D1 *D3F)+RHSH*<3F*D4S-3S*4F) )/DE L2
GO TO 390
389 CONTINUE
CELS = D 2 F 2 F 2 D1*F5F2
SXI ( R H S3 02 RHSEEMF5F2)/DCL5
STI = (RHSEEM*F2F2 RHS3*01)/DFL5
GO TO 39 J
387 CONTINUE
IF(RFSA.LT.1 .OE-96.OR.UNCOUP.GT.O) GO TU 381
DEL 3 = 0 3 F* D4S D3S*D4F
STI = 0.0
TAUI = (RHSF*C4S RHSG*D4F )/DEL 3
SXI = (RHSGD3F RHSF*D3S)/DEL 3
GO TO 390
361 CONTINUE
SXI = RHS3/F2F2
STI J.O
394 CONTINUE
SBP(l) = SGRT(SXI**2 + 3.*TAUI**2 + A*STI*(STI SXI))
STRAIN!I ) = DELTAP
VXP{ I ) = VXI
V T P ( I ) = VT I
SBPC = SBP(I)
IF(I.EG.1)GO TO 340
222

riFFl = ADS(SOP{ I ) SB P( I 1 ) )
DIFF2 = ABS(STRAIN! I ) STRAIN! I-I) )
CIFF3 = ADS ( VXP ( I ) VXP(I-D)
CIFF4 = A BS(VT P( I ) VTP(I-l))
CIFF = U1*DIFF1 + H2 C 1 FF2 + H3*DIFF3 * H4*DIFF4
DENOM = A BS ( VX P ( I ) ) + ABS(VTP(I))+ ABSISBPUJ) +
FRRCR = CIFF/DENOM
IF(ERROR.GT.SMALL)GO TO 340
430 CONTINUE
IFIMX.GT.1)00 TO 391
£X(MX) = £X(MX) + 2*VXR VXI VXB
ET X(MX) = ETX(MX) + VTR .5*(V TI + VT3)
GC TO 392
391 CONTINUE
FX(MX) = EX(MX) + VXR VXL
ETX(MX) = ETX(MX) + .5*(VTR V TL)
392 CONTINUE
VR{MX ) = A*(D3 03*STI )
ET(MX) = ET(MX) + DELT2*(VR(MX) + VRB)
TAU(MX) = TAUI
VX(MX ) = VXI
VT f M X ) = VT I
SX(MX) = SXI
ST(MX) = ST I
Al(MX) = A1P
A 2(M X) = A2P
A 3 ( M X ) = A 3 P
A4 ( MX ) = A4P
A^tMX) = A5P
A6(MX ) = A6P
CF(MX) = CFI
CS(MX) = CSI
S 3(MX) = SBPC
EXP(MX ) = EX(MX) (SXI NU*STI)
AB S(STRAIN(I))
to
to

FTP(MX) = ET(MX) {STI A*NU*SXI)
LTXP(MX) = E T X ( M X ) QA* TAUI
ITER(MX) = I
PS 10(MX) = PS IP
PHI. (MX) = PH IP
UELTA(MX) = CELTAP
PL AS(MX ) = CELTAP SBPC
CELGAM(MX) = 2.*(E T X(MX) ETXO)
CtX(MX) = EX(MX ) cXC
STKES(MX) = SQRT(SXI**2 + A*STI*(STI
IF(SBPC.LT.SMAXO)GO TO 393
SMAX(MX) = SBPC
393 CONTI CUE
IF(MX.GT.1)GO TO 399
CELLFB(MTU) = CELTAP
399 CONTINUE
IF(KCUNT.EQ. IMAX ) GO TU 400
MTT = MX + MT
IF(MTOT.LT.NMAX )G0 TO 2C0
LINE = LINF + 1
IWRITE = 2
GU TO 4)1
397 CONTINUE
KUUNT = KOUNT + 2
IF(KCUNT.LT.NMAX)GO TO 100
400 CONTINUE
LINE = LIME + 1
IWR I TE = 3
491 CONTINUE
LINEO = LINEO + 1
IFLINEO.LT.INCRTJGO TO 402
LINEO = O
W R I T t ( 6 1QG0) LINE
WRITE(6,1010) (X(L).L-ltMX, INCRX)
SXI ) )
224

402
WRITEl6, 1009) (T(L)*L=1,MX, INCRX)
WklTE(6,lCll)
WRITE(6, 1012)
WRITE(6, 1038 )
WRITE(6,1013)
W R I r E ( 6 10 1 4 )
W R I T F ( 6 ,1015)
WRIT6(6, 1016)
WRIT E{6 i 1Cl7)
W K1 F c(6 1^18 )
WRITEI6, 1019)
WRITE(6, 1020 )
WRITE6,1021 )
WRITE(6, 102 2 )
WRITE(6, in?3 )
WR I TE (6, K'24 )
WRITc(6,102 9)
WRITE16, 1^26)
W R 1 T E(o 1027)
WRITE-(6, 102 8 )
WRITE(6,1029)
WRIT F(6, 1036)
WRITlId,1037)
(SX(L ) ,1=1,MX,INCRX)
(ST(L ) ,L= 1,MX,INCRX)
(STRES(L),L=1,MX, I NCR X)
(TAU(L ) ,L = ltMX, INCRX )
(VX(L ) L= 1 .MX,INCRX)
(VT(L ) L = 1 MX.INCRX)
(V R(L ) L = 1 M Xi INCRX)
{CF(L ) L = 1,MX,INCRX)
(CSC L ) ,L l ,MX, INCRX)
C SB(L ) i L = 1 *MX, INCRX)
(EX(L),L=1,MX, INCRX )
( E T(L ) L=i,MX, INCRX)
( ETXtL )L=1MX, INCRX)
(EX P(L ) ,L = 1,MX,INCRX)
(E T P{L ),L = 1,MX, INCRX)
(ETXP(L ) t1 = 1,MX,INCRX)
( ITER(L),L = 1,MX,INCRX)
(CFL T A (l_ ) L = 1 MX, INCRX )
(CPLAS(L ) ,L = i,MX, INCRX)
(PH 10(L),L = l,MX, INCRX)
(LELCAMCL),L=1,MX,INCRX)
l f: F X ( L ),L = 1,MX, INCRX)
CONTINUE
IF( 1 PUNCH.E.0)C0 T0 404
LIN E P = LINIP + I
IFCLINEP.LT. INCRTPJGO TO 4U4
LINEP =
WRITt t 7, 1^3 5) (X(L ) ,L = 1,MX, INCRXP )
WRITEC7,1C 3 5 ) (T(L),L=1,MX,INCRXP)
WRITE(7, 1036) (SB(L ) L= 1,MX,I NCRXP)
WRITEC 7, 103 5) (CELTA(L),L = 1,MX, INCRXP)
IF( I PUNI.EQ.C)GO TO 40 3
WRITEC7, 1C35) {SXIL ) ,L= 1,MX,INCRXP)
225

WRITE 7* 1C35) (EX(L ),L = 1,MX,INCRXP)
WRITE<7, 103 5 ) (VX(L),L=1, MX,INCRXP)
W R I T E ( 7 1335) (CEX(L ), L = 1,MXINCRXP )
IF ( A.EC1.C )G0 T 403
WRI TCl7, 103 5) (ST(L ) L= 1 ,MX,INCRXP)
WPITb(7, 1035) (ET(L ) ,L=lfMX,INCRXP)
WRI TE(7, 1035 ) (VR(L ) ,L=1 ,MX,INCRXP)
403 CUNTI l\IUL
I F ( IPUN2.EQ.OGO T0 404
WRITE!7,1035) (TAU(L ),L = 1MXvINCRXP)
WRI TE(7, 10 35 ) (VT(L ) ,L=l,MX INCRXP)
WRITE(7, 1035 ) (ETX(L),L=1,MX,INCRXP)
WRITEI 7, 103 5 ) (CELGAML),L = 1,MX,INCRXP)
404 CONTINUE
IF! IWRITE 2) 90, 397,405
4C5 CONTINUE
LODO FORMAT(' 1 ,2 X,'LINE =',I5///)
100 1 FORMAT!3 15,5F5.2,4F10.2)
1002 FORMAT!I5.7F10.3,15)
ICC 3 FORMAT(1015)
1004 FORMAT(2 E207)
1005 FORMAT( 1 ,9X,A = , I 1, 15X, 'JR ATE = ,I 1,11X,ELX = *,F9.6,4X,
I'OlLT = ,F9.6,4X, MXMA X = ,I 4,BX, SMALL = ',F8.6/)
1V.-6 FORMAT ( 19X, NU = F 5.3 1CX SY = F12.9,3X 1 BOAR F 9. 1,4 X N
1= ,F7.4,9X, 'XK:: = F 8.6,6 X BETA = ',F8.6/)
1007 FORMAT!1 O X,J RIS F
i,*SXO = ,F9.6,5X
iC C 8 FORMAT(1TX, 'HI =
1= F4.2, 11X, n
10 0 9 FORMAT(/3 X, 'TIME
1010 FORMAT(/ IX,1X =
1011 FORMAT!/3X,SX =
1012 FORMAT(/3X,ST =
1013 FUKMATI/3X,'TAU =
= I 4,BX, 'XVFIN = ',F 9.6,3 X, 'TVFIN = *,F8.6,3X
TAUO = ,F8.6,5X, H = ',F8.6/)
,F4.2,11X,H2 = ,F4.2, 11X, *H3 = ,F4.2,11X,H4
12, 13X, SYS = ,F 1 2.9/)
',4X,71F9.56X)F9.5/(8X8(6X,F9.5)))
',4X,7!F9.5,6X),F9.5/!8X,0(6X,F9.5)))
,8 E15.5/( 10X.8E15.5) )
' t 8E155/(10X,6E15.5))
.8E15.5/10X.8E15.5))
226

1314
FORMAT(/3X,
VX =
6cl5.5/( 10X,8E15.5) )
1015
FORMAT(/3X,
' VT =
',8E15.5/1 10X.8E15.5) )
1016
FORMAT(/ 3X,
VR =
*,8E15.5/1 10X,8E15.5> )
1017
FORMAT(/3X,
CF =
',8E15.5/( IDX,RE15.5) )
l 0 1 ri
FORMAT(/3X,
' cs =
'8E 15.5/( 1DX,815.5 ) )
1019
FORMAT(/3X,
SB =
,8E15.5 / ( inx be 155 ) )
1C 2 >j
FORMAT(/ 3X T
EX =
,8E15.5/t 11X,BE 15.6) )
1021
FORMAT(/3 X,
' ET =
,8E15.5/( 10X,E15.5 ) )
102 2
FORMAT(/3X,
FTX =
',8E15.5/ 10X,8E15.5 ) )
1023
FORMAT{/3X,
! EXP =
'v0E15.5/( 10X.8E15.5) )
102 4
FORMAT(/3X,
' ETP a
',8E i 5.5/( 10X.8E15.5 ) )
102 5
FORMAT(/3X,
ETXP =
'8E15.5/<10X.8E15.5M
10 2 6
FOkMAT(/3X,
' ITER =
,01 15/( 10 X 8115) )
102 7
FORMAT(/3 X,
'DELTA
= '(8E15.5/( 10X t 8E15.5 ) )
1 0 2 H
FORMAT(/JX,
CPLAS
= ,8tl5.5/( 10X,8E15.5) )
1020
FORMAT(/3X,
PHI =
*8E15.5/( 10X.8E15.5) )
13 30
FORMAT(10X,
' JBEIL
= 'tlltllXf'D/ = ',rfi.6,7X, 'DHAT = F8.6,5X XM
i
= ',FK.8,
5 X,'XN
= *F10.85Xt XLAM = ,F8.6/)
1031
FORMAT(1GX,
MI = '
, I 3,12X MJ = ,I 3,12X, I PUNCH = ,I 38X 'KAS2 =
J.
I2/ )
10 3 2
FORMAT(10 X,
X = ,
F 9.6 5 X, T = F9.6,5X,'DIFF = ,L13.5,5X,'DNOM
1
= 'ttl3.5.
5 X,ERRUR = ,E13.5//)
103 3
FORMAT( 1 )
10 34
FORMAT(615,
2F10.6)
10 3 5
FORMAT(7 E 11
. 6 )
10 36
FOKMAT(/3X,
'OGAM =
',6E15.5/( 10X,8E15.5) )
1037
FORMAT(/3X,
DEX =
,8E15.5/(10X.8E15.5) )
1038
FORMAT{/3X,
'STRES
= ',PE15.5/( 10X.8E15.6) )
IO 3 > FORMAT(//////50X '
1040 rORMAT(//////50X, '
STOP
ENL
WAVES ARE UNCOUPLED ******#**#)
WAVES ARE COUPLED ********#)
227

SUlii (JUT I NE PHI ( P PQ K, J 81 S U YS B t Y H XLAM, OE TA Y XM XN DZ I 0 )
IF(K.GT.O) GO TO 4
YS = ABS(YS)
IF(S.GT.YS)GO TO 1
SZ = BETA*SuRT( CZ)
IF( IO.EQ.O) C = S
P = .. j
PQ O.J
no tc o
1 CON T I NU -
IF(JB.GT.OIGO TC 2
/ = S YS
ZY = Z *(Y-1 )
C = 8 *ZY
P = C Y
IF( Ij.GT.OIGO TO 10
CP c*z
c = dp s
PQ = OP/S
G TC n
2 CONTINUE
I F ( m.GT .0 ) GO T 3
SZ = BETA*SQRT(CZ )
C = (S/GLTA)**2
IF(C.LT.CZ) C = (2.0*SQRT(DZ)/B E T A ) *(S YS) DY
PQ -- /S 1.0
3 CONTINUE
IF(S.LT.YS) P = 0.0
IF(S.Gfc.YS.AND.S.LE.SZ) P = 2.O SORT(DZ)/BETA 1.0
IF(S.GT.SZ) P = 2 O S/(B E T A * 2 ) 1.0
GO TC lu
4 CONTINUE
C THE DATA USED HERE ARE FROM CRISTESCU (1972) CASE XII
CZ1 = SORT(DZ)
228

SYH = YS + H
DC = (SYH/BETA)*2
I F(CC.GT.DZ) GO TO 5
DC = 2.*H*DZ1/BETA + DY
5 CONTINUE
I F ( DGTCY)GO TG 6
F = YS
GO TO 8
b CONTINUE
IF(D.GT.DZ)GO TO 7
F = YS + EET A*(C 0Y)/(2.*DZI)
GO TO 8
7 CONTINUE
F = 01T A S O R T ( )
3 CONTINUE
CSTAR = H + XLAM*(D DC)
FM = F * OS TAR
IFIS.GT.FMJGO TC 9
P = 0.0
00 TC 10
0 CONTINUE
XA = XM + XN*SvjRT(D)
P = (3.*(C DY DSTAR + (XA/3.)**1.5)**(2./3.))/XA 1.0
10 CONTINUE
RETURN
END
229

SUCAGUTINE PS I(PK,S*D,DY,YS.BETA,XKtHAT,DZ)
IFIK.GT.O )GO TO 20
P = 0.0
GO TC 25
2 j CONTINUE
C THE CATA USED HERE ARE FROM CRISTESCU (1972) CASE XII
071 = SwRT(UZ)
IF(D.GT.CY)GO TO 21
YS = ABS(YS)
F = YS
GO TC 23
21 CONTINUE
I F(C.GT.DZ)GO TO 2 2
F = YS + .5*BETAMD DYJ/0Z1
CO TO 23
22 CONTINUE
F = BETA*SORT(D )
23 CONTINUE
IF(S.GT.F.ANC.C.GE.CY)G0 TO 24
F = 0.0
GO TC 25
24 CONTINUE
XK = XK:*(1.0 EXP(-D/DHAT ) )
P = XK ( S F )
25 CONTINUE
RETURN
END
230

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231

232
Cristescu, N. (1965), "Loading/Unloading Criteria for Rate Sensitive
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233
Hunter, S. C, and Johnson, I. A. (1964), "The Propagation of Small
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234
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235
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BIOGRAPHICAL SKETCH
Charles Daniel Myers was born July 27, 1945, in Brevard County,
Florida. He was graduated from Terry Parker High School in Jacksonville,
Florida, in June, 1963, and received his Bachelor of Science degree in
Engineering Science from the University of Florida in June, 1968.
He was employed in the Structural Analysis Section of the Martin Marietta
Corporation in Orlando, Florida. While on a leave of absence from the
Martin Marietta Corporation, he received his Master of Engineering
Degree in Engineering Mechanics from the University of Florida in
December, 1970. He returned to the Martin Marietta Corporation in
September, 1972. He is a member of Tau Beta Pi (having served as presi
dent of the Florida Alpha Chapter during the year 1967-1968), Sigma
Tau, Omicron Delta Kappa, Phi Kappa Phi, and the Society of Engineer
ing Science. He was recently selected for Outstanding Young Men in
America 1973.
Charles Daniel Myers is married to the former Peggy Lee Hufham
of Jacksonville, Florida. They have one son, Daniel Lee.
236

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
MS A. Eisenberg, Chairman
^Associate Professor of Engineering Science,
Mechanics and Aerospace Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
L. E. Malvern
Professor of Engineering Science, Mechanics
and Aerospace Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
E. K. Walsh
Associate Professor of Engineering Science,
Mechanics and Aerospace Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
T4 Pfnv^w^cr
U. H. Kurzweg
Associate Professor of Engineering Science,
Mechanics and Aerospace Engineering

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
R. C. Fluck
Associate Professor of Agricultural
Engineering
This dissertation was submitted to the Dean of the College of Engineering
and to the Graduate Council, and was accepted as partial fulfillment
of the requirements for the degree of Doctor of Philosophy.
August, 1973
Dean', College of Engineering
Dean, Graduate School

Internet Distribution Consent Agreement
In reference to the following dissertation:
AUTHOR: Myers, Charles
TITLE: Inelastic wave propagation under combined stress states, (record
number: 577578
PUBLICATION DATE: 1973
I, Charles D. Myers as copyright holder for the
aforementioned dissertation, hereby grant specific and limited archive and distribution rights to
the Board of Trustees of the University of Florida and its agents. I authorize the University of
Florida to digitize and distribute the dissertation described above for nonprofit, educational
purposes via the Internet or successive technologies.
This is a non-exclusive grant of permissions for specific off-line and on-line uses for an
indefinite term. Off-line uses shall be limited to those specifically allowed by "Fair Use" as
prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as
to the maintenance and preservation of a digital archive copy. Digitization allows the University
of Florida to generate image- and text-based versions as appropriate and to provide and enhance
access using search software.
This grant of permissions prohibits use of the digitized versions for commercial use or profit.
Signature of Copyright Holder
Charles D. Myers
Printed or Typed Name of Copyright Holder/Licensee
Personal Information Blurred
Printed or Typed Phone Number and Email Address of Copyright Holder/Licensee
5~- Z7 &&
Date of Signature
Please print, sign and return to:
Cathleen Martyniak
UF Dissertation Project
Preservation Department



190
so that equations (D.1.19), (D.1.21), and (D.1.6) can be written as
B2fV6P +
D4f2SxP
aB7fS6P =
RHS9
B2sV6P +
D4s2SxP
aB7sS0P =
RHS10
(D. 5. 30)
aDlSxP +
aD2S9P =
aRHSll .
When radial inertia effects are included (a=l) these can be
written as
B2f
D
4f 2
"B7f "
vep
RHS9
B2s
4s2
B7s
SxP
=
RHS10
0
D1
2 _
_S0P_
RHS11
(D.5.31)
and when
S10 =B2f B2s(D2D4f2 + DlB7f>
(D.5.32)
the solution is given by
1
V
9P A. LBHS9(D2D4S2 + D1B7s> 11,5101D2D4f2 + DlB7f>
10
+ RHSll(B7fD4s2 B7S
4f2)J
(D. 5. 33)
'xP ~~ A |_B2f(D2RHS1+ b7sRHS11) B2s(d2RHS9 + B?fRHSll) I (D.3.34)
10
= ~ [B2f (D4s0RHS11 -D1RHS10) B2s(D4f2RHSll -D1RHS9)J (D. 5. 35)
8P
10
When radial inertia effects are not included (a=0), the last of
equations (D.5.30) vanishes as do the hoop stress and the radial
velocity, and the first two equations of (D.5.30) become


6
and these averaged variables were used. The results of this work were
given by Tapley and Plass (1961) but were somewhat inconclusive. More
work including radial inertia effects was published by Hunter and
Johnson (1964), and a year later DeVault (1965) showed that, at least
qualitatively, many observations formerly attributed to a material
strain-rate effect could be explained by including radial inertia effects
in the formulation of the problem of longitudinal impact of a bar.
Shea(1968) obtained good agreement between theory and experiment for the
propagation of longitudinal waves in a lead bar. He used the strain-
rate dependent constitutive equation of Malvern (1951b) and the
"correction" for radial inertia proposed by DeVault (1965). Mok (1972)
used the same averaging technique for the variables as Plass and
Ripperger (1960) for the problem of longitudinal impact of a bar with
radial inertia effects included. He used the strain-rate independent
constitutive equations and agreed in essence with DeVault (1965) that
radial inertia effects could explain, at least qualitatively, those
experimental results usually attributed to strain-rate sensitivity
of the material. Since radial inertia is always present in an experi
ment using longitudinal impact it seemed that the only way to conclu
sively determine strain-rate effects in a material would be to perform
the experiments using a torsional wave.
In an effort to determine the strain-rate dependence of various
materials, several investigators have recently conducted theoretical
and experimental studies concerning the propagation of torsional waves.
Convery and Pugh (1968) gave the results of their experiments in which
a tube was stressed statically above the yield stress in torsion and


184
Adding equations (D.2.12) and (D.2.13) and letting
F = 2F
2f 2 2f
F
5f 2
2F
5f
RHS3 = RHSAEM + RHSBEM J
results in the equation
F2f2SxP + aF5f2SGP RHS3
(D. 4.4)
(D. 4.5)
When radial inertia effects are included (a = l), equations
(D.2.3) and (D.4.5) can be written as
F2f2
F5f2
"SxP~
RHS3
_ D1
D2
_ Sqp _
RHSEEM
(D.4.6)
and by defining
D F
2 2f2
- DlF5f2
(D.4.7)
the expressions for the hoop stress and the longitudinal stress can
be written as
S = j-(D RHS3 F RHSEEM) (D.4.8)
xP A 2 5f2
5
S0p = r'(F2f2RHSEEM D!RHS3)- (D.4.9)
5
When radial inertia effects are not included (a = 0), equation
(D.2.3) vanishes, the hoop stress and the radial velocity are zero,
and the expression for S is found directly from equation (D.4.5) to be
xP
S
xP
RHS3
F
2f2
(D.4.10)


19
Since the stresses cr f T and T are assumed to vanish,
r r8 rx
equation (2.14) as applied to the present problem reduces to
\
I I
e :,t = 1 CTx,t-| e,tV<2ffx-'V L0<5'i,5 + *<:-4)J
2 s
X,
ee,t = -iCTx(t + iCTe,t + ^(2ae-CTx)L
1 + V
3
+ T,
'9x,t E 9x,t 8x
2s
0 (s A) s + i|f(s.A)J
0(s,A)s + ty(s,A)l
(2.IS)
where the deviatoric stresses are
s = s = (2ct ctq)
x 11 3 x 9
Se = S22 = 3<29 ax1
Sr = S33 = 3(x+V
S0x S12 S21 T0x
s = s = 0 .
r 0 rx
Using these deviatoric stresses, the expression for s becomes
9s 9 3
S ~~ dt 9t L2 SijSij_
"¡e i rs ~¡ssr3 1
jj ~~ 2 |_2 SklSklJ ~St L2 SijSijJ
1 3 P s s 1- 3 3 r
s =
2s
_3_ 5
4s
S11S11 + S22S22 + S33S33 + S12S12 + S21S21
]
2 2 2 2
- q(o- + - ot _3 x o x 0 0
J
-L [<2ax V4x,t+ <% -x)4e,t+ 6TexTex,t]
(2.19)


SUlii (JUT I NE PHI ( P PQ K, J 81 S U YS B t Y H XLAM, OE TA Y XM XN DZ I 0 )
IF(K.GT.O) GO TO 4
YS = ABS(YS)
IF(S.GT.YS)GO TO 1
SZ = BETA*SuRT( CZ)
IF( IO.EQ.O) C = S
P = .. j
PQ O.J
no tc o
1 CON T I NU -
IF(JB.GT.OIGO TC 2
/ = S YS
ZY = Z *(Y-1 )
C = 8 *ZY
P = C Y
IF( Ij.GT.OIGO TO 10
CP c*z
c = dp s
PQ = OP/S
G TC n
2 CONTINUE
I F ( m.GT .0 ) GO T 3
SZ = BETA*SQRT(CZ )
C = (S/GLTA)**2
IF(C.LT.CZ) C = (2.0*SQRT(DZ)/B E T A ) *(S YS) DY
PQ -- /S 1.0
3 CONTINUE
IF(S.LT.YS) P = 0.0
IF(S.Gfc.YS.AND.S.LE.SZ) P = 2.O SORT(DZ)/BETA 1.0
IF(S.GT.SZ) P = 2 O S/(B E T A * 2 ) 1.0
GO TC lu
4 CONTINUE
C THE DATA USED HERE ARE FROM CRISTESCU (1972) CASE XII
CZ1 = SORT(DZ)
228


139
and
F(A) = f(A)
E
r K(A)
o
XK
XKo [l exp (-4-)]}
r K
o o
J
$(s,A) = E0 (s A) = E0(A)
EX
/ a \3/2-| 2/3
3/2-1 2/3
() }
- 1
a/E
$(s,A) = X
-r / v 3/2-, 2/3
4-vi+t(Â¥) 1
- 1
xa
(A. 4.10)
(A.4.11)
where
a
xa =
(A.4.12)


Shear Stress, T x10
4.0 _
Final Stress State
Figure 4.17 Stress Trajectories for Data Set 2 Without Radial Inertia
107


FTP(MX) = ET(MX) {STI A*NU*SXI)
LTXP(MX) = E T X ( M X ) QA* TAUI
ITER(MX) = I
PS 10(MX) = PS IP
PHI. (MX) = PH IP
UELTA(MX) = CELTAP
PL AS(MX ) = CELTAP SBPC
CELGAM(MX) = 2.*(E T X(MX) ETXO)
CtX(MX) = EX(MX ) cXC
STKES(MX) = SQRT(SXI**2 + A*STI*(STI
IF(SBPC.LT.SMAXO)GO TO 393
SMAX(MX) = SBPC
393 CONTI CUE
IF(MX.GT.1)GO TO 399
CELLFB(MTU) = CELTAP
399 CONTINUE
IF(KCUNT.EQ. IMAX ) GO TU 400
MTT = MX + MT
IF(MTOT.LT.NMAX )G0 TO 2C0
LINE = LINF + 1
IWRITE = 2
GU TO 4)1
397 CONTINUE
KUUNT = KOUNT + 2
IF(KCUNT.LT.NMAX)GO TO 100
400 CONTINUE
LINE = LIME + 1
IWR I TE = 3
491 CONTINUE
LINEO = LINEO + 1
IFLINEO.LT.INCRTJGO TO 402
LINEO = O
W R I T t ( 6 1QG0) LINE
WRITE(6,1010) (X(L).L-ltMX, INCRX)
SXI ) )
224


T
Figure 3.7 Location of the Characteristic Lines Passing Through P
m
-i


152
r o p
ac
1 0
M, -
alr-*e
o
-2^9x 0
dv
df
dv
I
"dT
dv
s-
e
d9
da
y.
d§
da
0
dT
0X
d?
0 0 0
0 ap 0
0 0 0
0 0 0
0 0 0
0 0-1
0 0 0
ax,g at,- 0
0 0 x, g.
0 0 0
0 0 0
0 0 0
0
0
p
0
0
0
0
0
0
0
0
-1
0
0
0
0
0
0
0
0
x§
0
0
0 0 0 0
0 0 0 0
0 0 0 -1
A 0 aX2 0
a 0 aA. 0
2 4
A 0 a A 0
o o
0 0 0 0
0 0 0 0
0 0 0 0
t, 0 0 0
0 ax,^ at,^ 0
0 0 0 x, ^
A,
0
0
0
0
0
0
or, after some manipulation


WRITE!6, 1030 ) JGELLtDZ fOHAT ,XM,XNXLAM
WRI T E ( 6 100 6 ) NU.SY,BDAR,N,XKC t RETA
WRI r E ( 6 100 7 ) JRISEfXVFINt TVFIN .SX-J.TAUO.H
WRITE(6,1006) H1.H2.H3.F4,I.SYS
WRI r::(6, 1' 3 1 ) MI.MJ, I PUNCH KASE
IF(UNCUUP.GT.O) WRITE(6,1C39)
IF! JNCGUP.LE.O) WRITE!6,1040)
I F( I PUNCH.EQ.O)GO TO 15
WRITZ ( 7, 1034) A, MXMAX,I PUNI,I PUN 2,INCRXPINCRTP,DELX.OELT
15 CONTINUE
C THIS BEGINS THE CALCULATION OF CONSTANTS USED
M M A X = 2 *MX MAX
NMAX = MMAX 1
1M A X = NMAX 2
Q = 1. Nil * 2
C2 = SCR T(.3*(1 .-NU ) )
uA = 1. + NU
CELT2 = 2.* D E L T
I 2 A = 2 A
Cl = 2 C A
C2 = C/2.
C3 = Q*D ElT 2
C4 = 03*CELT 2
FV1 1. + C 2
CV2 = 1. C2
CL R B = 2 *C2/l)V 1
CLRPI = 1. CLRB
C Y = ABS(SY)
CO 30 L= 1 MX MAX
3J XIL) = (L 1 ) *CELX
C THIS BEGINS THE CALCULATION OF INITIAL CONDITIONS
LINE =
LINER = I NCRTP 1
LINEO = I NCR T 1
206


RHSi = RHSAEM + RHSBEM
380 CONTINUE
IF(A.E0.0 ) GO TO 337
IFIRFSA.LT.1.QE-G6.OR.UNCOUP.GT.O) GO T 389
ZA = D2*4S DI D7 S
ZB = ASQ C7S C2 l) 3S
ZC = 0 1 C 3S A5Q*D4S
CEL2 = U3F#ZA + D4F*ZB + l)7F*ZC
TAUI = (RHSF*ZA*RHSG* SXI = (RHSF*ZB+RHSG*(02*03F-A5Q*D7F)+RHSH*(D3S*07F-D3F* 07S) )/0LL2
ST I = (RHSF*ZC + RHSG*(A5Q*D4F-D1 *D3F)+RHSH*<3F*D4S-3S*4F) )/DE L2
GO TO 390
389 CONTINUE
CELS = D 2 F 2 F 2 D1*F5F2
SXI ( R H S3 02 RHSEEMF5F2)/DCL5
STI = (RHSEEM*F2F2 RHS3*01)/DFL5
GO TO 39 J
387 CONTINUE
IF(RFSA.LT.1 .OE-96.OR.UNCOUP.GT.O) GO TU 381
DEL 3 = 0 3 F* D4S D3S*D4F
STI = 0.0
TAUI = (RHSF*C4S RHSG*D4F )/DEL 3
SXI = (RHSGD3F RHSF*D3S)/DEL 3
GO TO 390
361 CONTINUE
SXI = RHS3/F2F2
STI J.O
394 CONTINUE
SBP(l) = SGRT(SXI**2 + 3.*TAUI**2 + A*STI*(STI SXI))
STRAIN!I ) = DELTAP
VXP{ I ) = VXI
V T P ( I ) = VT I
SBPC = SBP(I)
IF(I.EG.1)GO TO 340
222


172
and
i|r / 2\Jf /
ToP YoP
aRHSE = -2aATV n+ a(Ao^ AT)S +a(A, + a AT)S
rP 2Q Sp/ xP 4Q s /
GP
+ aA T .
5Q P
Now defining the quantities
(D.1.4)
Q4 = 2AT Q3
op'
D-, = A AT
1 2Q sp/
lit /
D = A + 2a -5
2 4Q s /
A
AT + Q,
(D.1.5)
and
D4 = 2AT D3
RHSH = RHSE + D,
4 J
the two equations along the vertical characteristics (c = 0) can be
reduced to a single equation by substituting equation (D.1.3) into
equation (D.1.4). This becomes
aDlSxP + aD2S0P + aA5QTP = aRHSH (D.1.6)
The equations along the nonvertical characteristic lines are
given by equations (3.4.4), (3.4.5), (3.4.6), and (3.4.7). These can
be written in a much simpler form by defining the following quantities
as A
Z1 =
Z2 ^
Z3 =
1 v
c
s
2
Z4
1 v
AT
1 -
Z5 1 + c,
7 AT
O 1 + C
(D.1.7)
J


129
From eqation (A.2.5),
Bf
3a 3a
pq pq
Bf 3
r# t
L2 1J ljJ
Ba 2f pq
pq
(A. 2. 8)
Now from equation (A.2.8) it can be seen that equations (A.2.4) and
(A.2.6) are identical when
d\ = ~ d\ .
The first invariant of the deviatoric stress is
1 ,
s = a -o6 cr, = a a,, = 0
pp pp 3 pp kk pp kk
so that
Bf
a -K = a
pq ^ pq
Bf
[-J
pq 3a
= f .
(A. 2. 9)
pq
Using equations (A.2.8) and (A.2.9), the expression for the plastic
strain increment, equation (A.2.7), becomes
[h si J [ %i] daki
or
de .
ij
f' (w1
9s s
deP .
ij
ij kl
4f3 f' (/)
9s s
P
ij kl
ij
4f3 f' (/)
[f]
da
kl
kl
(A.2.10)
where
3a,
3eP.
P ij
ij 3t kl 3t
kl


APPENDIX A
CONSTITUTIVE EQUATIONS
A.l. Comments on the Constitutive Equation
The constitutive equation used in the general formulation is given
by equation (2.14) as
3
1 + v
e, = - a. .
- 6
.a,, +
ij E ij E ij kk 2
j~0(s,A)s + \|r(s,A)J (A.1.1)
where s is the scalar representation of the stress state given by
¡3
/ s s
*J 2 ij ij
and s is the deviatoric stress given by
ij
s. a. -6 .a .
xj ij 3 ij kk
Under the assumption that a = T = T = 0, s reduces to
r rx r0
/ 3 r 2 2 2
s = J 2 LS11 + =22 + s33 + 2S
i!
and with
S11 = S,
3(2cx aCTe}
s22 se
3(2acre CJx)
33
= s
3(ax + raB>
S12 S21 S0x T0x
this becomes
2 2 2
cT -aa oG + aac + 3T
x x 0 6 0x
(A.1.2)
125


VTRBB = CON2 *VTRB CON4*VTB
VRKD3 = CON2*VRRB + CON4*VR8
Z1 = CFI/O
l = CSI/Q
Z 3 = Z 1 C F I
Z 4 = Z 2 C S I
Z 5 = CELT/DV3
Z = DELT/DV4
RFS = A*A2I*A5I A 3 I *A4 I
RFSA = A BS(RFS)
IF(RFSA.LT. 1 .OF-^6.OR.UNCOUP.GT.O) GO TO 360
CVS = A4 I *A6 I A* A 5 I 2
PV6 = A2 I A6 I A 3 I A 5 I
P1F = Z3nV5 A4I
RIS = Z4 *DV 1 A4 I
P2F = Z 3 CV6 A 2 I
R2S = Z4DV6 A? I
BIS = R1S/CSI
81F = RIF/C FI
B2S = Z 2 *RFS
B2F = Z1RFS
RAS = 2. Z6 R2S
6F = 2.*Z5*R2F
368 CONTINUE
IF(I.GT.1)GO TO 370
PS IP = 0.0
RATIO = 0 .0
R A T I 0 P = 0 0
GO TC 330
370 CONTINUO
CALL PS I(PS IP,JRATE,SBPCDEL TAPDY.SY.BLTAXKO,DHAT,DZ)
SBUIF = SBPC SBB
OELTAP = DELTAB + SBUIF + DELT2MPSIP + PSIB)
lFISGPC.LT.SMAXB.ANU.I.LT.MJ)GO TO 371
215


204
O
S = #
y e
1500 psi
10.2 X 10 psi
-4
1.47059 x 10
-4
A = e = 1.47059 x 10
y y
A = e_
.002575
P =
5.6 X 10 psi
10.2 x 106 psi
, 0054902
A = 6 = .0004
m 3.25 x 10 psi
xm = =
E 10.2 X 10 psi
.0031863
(E.4.4)
xn
4
n 9 x 10 psi
J7 0
10.2 X 10 psi
= .0088235
h = .000004
X = .00005
r k 3 -1.
xko = 4^ = (---2.5- -1-n-)-(A. ec. 2-= .0012
c 2.08 x 10 in/sec
where the dimensionless coefficient for the coefficient in the rate
dependent term was obtained for a .5-inch diameter aluminum tube.
E.5. Listing of the Program
In this section is the listing of the computer program which was
used to obtain the solution in the characteristic plane to the wave
propagation problem presented in this paper.


103
where now S represents the component of the stress vector (s) in the
N
Sr-S^ plane. By plotting a new stress traj ectory of T versus S we
x b N
can effectively represent this three-dimensional stress trajectory as
a two-dimensional one; this is shown in Figure 4.15. Again from this
figure it can be seen that at distances greater than 1.0 diameter from
the impact end the tube exhibits neutral loading followed by loading.
This is the same general behavior as the case when radial inertia is
not included, except that now the stress trajectory is not planar.
This is easier to visualize if these same stress trajectories are pro
jected onto the plane T = 0; the stress trajectory at X=1.0 is shown
in Figure 4.16. Since in any experimental wave propagation problem of
this type radial inertia effects are present, interpretation of the
experimental data without considering the hoop stress could lead to the
false conclusion (for a set of initial conditions and boundary condi
tions similar to these) that unloading occurred. Therefore, extreme
care must always be exercised when interpreting experimental results.
The other two computer runs made used artificial data where the
initial conditions and the boundary conditions were described by the
input data
" tr = -40
AX = AT .025
S = -.0006040
J X
Data Set 2 <
o
T =0
V =0
Xf
V0f = 00100 .


42
b'- 4a'A = 0 .
4
When y = > equations (3.1.18) and (3.1.19) are
11 2 r- 2
A = 1 + (-pr 1) (cos 6-2 a/3 sin 6 cos 6 + 3 sin 6)
1 4 P
A^ = v + ~(tt 1) (cos"6 3 sin26)
A
a3 = 0
1 i ' 2
A = 1 + (-g- 1) (cos 6 + 2 a/3 sin 6 cos 6+3 sin 6)
T l)
4V3
A5 =
Ag = 2(1+v)
J
and
a = A A .A aA A
14 6 2 6
A1A4 + A4A6
aA,
and using the same manipulations as in Appendix B, Section 4,
equation (3.1.21) becomes
0 = WV + ^2
where A^, A^, A^, and Ag are given by equation (3.1.22). Now
f ^ = f^(6) = i |^cos^6 2 f/3 sin 6 cos 6 + 3 sin26^
(6)
i j^cos26 + 2 ^3 sin 6 Cos 6+3 sin26^
f = f (6) = -i r
3 3V 4 L
z = z(P ) = -g-
c c 6
cos26 3 sin6
]
(3.1.21)
(3.1.22)
(3.1.23)
(3.1.24)
defining
/ (3.1.25)


7
then subjected to a suddenly applied incremental torsional load. The
strain caused by this incremental load was found to propagate with the
elastic shear wave velocity. This seemed to be proof that the strain-
rate dependent theory was correct, but Convery and Pugh (1968) cau
tioned against that conclusion. For Bell (1960, 1963) and Bell and
Stein (1962) had asserted that (based on experimental results with
annealed aluminum), while an increment of strain may propagate with the
elastic wave velocity, the larger amplitude strains propagate with the
wave velocity predicted by the strain-rate independent theory.
Nicholas and Garey (1969) tested aluminum samples in torsion at high
strain rates and found very little strain-rate dependence. However,
Yew and Richardson (1969) were able to measure some strain-rate depen
dence in copper.
Another problem which was encountered in wave propagation studies
f
was that of unloading. The two most common unloading cases were when
the applied load was reduced and when waves were reflected from a bound
ary. Unloading was examined for longitudinal plastic wave propagation
by Lee (1953) using the strain-rate independent constitutive equation
and by Cristescu (1965), Lubliner and Valathur (1969), and Cristescu
(1972) using the quasilinear constitutive equation. In all of these
investigations, regions of unloading and boundaries between regions of
unloading and loading in the characteristic plane were predicted but
the results have not been verified experimentally.
Many investigators in recent years have become interested in the
behavior of materials under combined stress and, more specifically, the


63
This gives the value of the coefficients at a point nearer the center
of each grid element. For this work, the value of is chosen arbi
trarily as .625, so that the point at which the coefficients are calcu
lated is at approximately the same location along the T-axis
(Figures 3.8 and 3.9) as the centers of the four characteristic lines
C = Cfi, cSi<
The values of all quantities at the points LB, LLB, and LBB will
be obtained by linear interpolation between the points L and B.
Similarly, the values of all quantities at the points RB, RRB, and
RBB will be obtained by linear interpolation between the points R and
B. From Figure 3.8, the times T T and T can be written as
-L £ *3
2c
2c,
2c.
T, =
1 1 + c
AT
T =
2 1 + c.
AT
= r
3 1 + c
AT
and the interpolation constants for the points LB and RB are
2c,
CLRB
AT 1 + c.
Cl,RBI = 1 -
T2 ^
AT ~ 1 + c.
Using subscripts to denote the grid point, the values of any quantity F
at the points LB and RB are
FT = CLRB-Ft + CLRBI F
LB L B
F = CLRB*F_ + CLRBIF
RB R B


Op OOOO-IOOOO
OOOapOOOOOOO
OOOOOp 0000 -1
-1 000000 A O aA 0
X Cj
0 0 0 0 0 0 0 aA2 0 aA4 0
0 0 0 0 -1 0 0 A 0 aA^ 0
d 5
x,g t,^ 0 0 0 0 0 0 0 0 0
0 0ax,^at,^0 0 0 0 0 0 0
0 0 0 0 x,_ t,_ 0 0 0 0 0
§ ?
0 0 0 0 0 0 x,g t,g 0 0 0
0 0 0 0 0 0 0 0 ax,^ at,g. 0
0000000000 x,,
0
V
X,X
0
0
Vx,t
a cr0
0
r
o
0
V
r, x
0
>1
CO
Vr ,t
-Ilf
T X
a5
V0, x
a(r--+e)
o
6
V0,t
-2^0x
0
a
X X
dv
X
db
0
x, t
dv
r
a d§
0
CTe,x
dv0
d£
0
CTe,t
da
X
d§
0
T0X,X
da0
a di
t§
T0x,t
dT0x
dT
(B.2.14)
149


10
by an increase in shear stress at a constant value of longitudinal
stress and then the slow wave caused loading such that the stress path
was normal to the initial loading surface. Clifton (1966) also found
that for a given initial loading surface, the two wave speeds depended
upon the particular stress state on the initial loading surface, and
that for one particular initial loading surface the fast and slow wave
speeds were equal when the shear stress vanished.
This work of Clifton (1966) was a significant step forward in the
investigation of waves of combined stress. An extension of this work
was presented by Clifton (1968) in which the simple wave solution was
used along with unloading at the impact end. In this way certain unload
ing boundaries for combined stress states were determined. Two years
later Lipkin and Clifton (1970) published their experimental results
from combined stress wave propagation tests and compared these results
to the simple wave solution developed earlier. Agreement between the
simple wave theory and the experiments was fair.
Cristescu (1967b) formulated the problem of combined stress wave
propagation in a thin-walled tube using general quasilinear constitu
tive equations but again ignoring radial inertia effects. The equa
tions for the characteristic lines and the equations along these char-
axteristic lines were determined. No numerical results were given
but the two waves (fast wave and slow wave) were shown to be coupled
during loading. Again Cristescu (1971) showed that the coupling of the
waves of combined stress depended on the constitutive equations and
yield conditions used.


BIOGRAPHICAL SKETCH
Charles Daniel Myers was born July 27, 1945, in Brevard County,
Florida. He was graduated from Terry Parker High School in Jacksonville,
Florida, in June, 1963, and received his Bachelor of Science degree in
Engineering Science from the University of Florida in June, 1968.
He was employed in the Structural Analysis Section of the Martin Marietta
Corporation in Orlando, Florida. While on a leave of absence from the
Martin Marietta Corporation, he received his Master of Engineering
Degree in Engineering Mechanics from the University of Florida in
December, 1970. He returned to the Martin Marietta Corporation in
September, 1972. He is a member of Tau Beta Pi (having served as presi
dent of the Florida Alpha Chapter during the year 1967-1968), Sigma
Tau, Omicron Delta Kappa, Phi Kappa Phi, and the Society of Engineer
ing Science. He was recently selected for Outstanding Young Men in
America 1973.
Charles Daniel Myers is married to the former Peggy Lee Hufham
of Jacksonville, Florida. They have one son, Daniel Lee.
236


142
methodthe system of equations is written in the form of equation (2.38)
and the equations along the characteristics are given by
1T A dw = ir b dt
(B.2.1)
where 1 is the left eigenvector and is determined from
1T (cA B) =0
(B.2.2)
which becomes
L1! *2 x3 x4 x5 X6J
Cp
0
0
1
0
0
0
acp
0
0
0
0
0
0
cp
0
0
1
1
0
0
CA1
acA
2
CA3
0
0
0
acA
2
acA^
4
acA
5
0
0
1
CA3
acA_
5
CA6
This is the matrix form of the following six simultaneous algebraic
equations for the six components of the left eigenvector
pl1 + l4 = 0 ^
acpl.
= 0
P13 + 16 =
^1 + CA114 + acA215 + CA316 =
acAl + acA 1 + acA_l_ = 0
2 4 4 5 5 6
1 + cA 1 + acA 1 + cA 1 = 0
3 3 4 5 5 6 6
(B.2.3)
However, from equation (2.38) the equation for the characteristics is
cA B
0


75
V6P = iT~ LRHS9 RHS10
+ KHS11(D4s2B7s D4f2B7s)]
SxP = 4^ [B2fS11> B2s(D2RHS9 + B^RHSll)]
S6P = 4^ LB2£ (3. 5.28)
(3.5.29)
(3. 5. 30)
where again is given by equation (3.5.6). When radial inertia
effects are not included, and vanish, and from equations (D.5.38)
and (D.5.39), the solution at P is found to be
V S = (B RHS10 B rhS9) (3.5.32)
xP A.,., 2f 2s
At a Boundary Point (X=0) for Uncoupled Wraves
When the waves are uncoupled, the solutions to the finite differ
ence equations are obtained at the boundary points for the same four
cases outlined above. When radial inertia terms are included in the
formulation of the problem, the expression for is given by
equation (3.5.6), and in all cases when radial inertia terms are not
included both V and vanish. In all four cases the solutions can
rP 0P
be found in Appendix D.
Case I: Traction boundary conditions
When Tp and S^p are known at a boundary point, then from equations
(D.6.1), (D.6.2), and (D.6.3) at that point


79
e = e + 2V V V
xP xB xR xP xB
:9xP eexB + V0R 2 E0P = e0B + 2 iT
where equations (3.6.6) and (3.6.9) are the same expression.
(3.6.7)
(3. 6. 8)
(3.6.9)


APPENDIX B
CHARACTERISTICS AND EQUATIONS ALONG THE CHARACTERISTICS
B.l. Equations for the Characteristics
The equations necessary for determining the characteristics are
given by equations (2.38) to (2.44). Thus, the characteristics c are
given by the equation
or
acp
Co
0
0
1
0
0
0
acp
0
0
0
0
0
0
cp
0
0
1
1
0
0
O 1
>1
H-1
acA2
CA3
0
0
0
acA2
acA.
4
acA_
5
0
0
1
O
>
CO
acA,_ cA^,
5 6
cp
0
1
0
0
0
cp
0
0
1
1
0
CA1
acA2
CA3
= 0
0
0
acA2
acA
4
acA
0
0
1
CA3
acA
o
CA6
0
(B.1.1)
140


81
boundary increases linearly up to its final value (denoted by Vx^
or Vgf) during a period of time called the rise time (T ) and then
remains constant. That is
'f
if 0 < T < T.
R
R
Ve (X = 0, =/
if T > T.
R
r
if 0 < T < T.
R
V (X = 0) = /
if T > T.
R '
Now that the computer code is set up, it would be advantageous to
compare the results from it to data which have already been published.
This is done in the following section by using the data of Lipkin and
Clifton (1970), and some interesting effects of the size of the numer
ical grid are noted. Then, finally, the effects of radial inertia and
strain-rate dependence on the propagation of inelastic stress waves are
discussed.
4.2 Effects of Numerical Grid Size
Lipkin and Clifton (1970) published the results of three different
experiments where a thin-walled tube was given an initial static shear
stress and then impacted longitudinally. In this section the initial
conditions and boundary conditions from one of these experiments will
be used and the results obtained from the computer code will be compared


Plastic Wave Speeds,
Figure 3.2 Plastic Wave Speeds as Functions of ¡3 and y for Poisson's Ratio of 0.30


/
7 ex
X
Figure 3.1
Yield Surface Representation in Spherical Coordinates
co
k-*


Hoop Stress, SQxlO
109
Figure 4.19 Hoop Stress Versus Longitudinal Stress for Data Set 2
With Radial Inertia


127
If 0(s,A) is assumed to vanish in equation (A.1.1), and if
t (s ,A) = yf (y l) ,
\''3 K
then the constitutive equation (A.1.1) reduces to the generalized semi-
linear constitutive equation of Perzyna (1963).
The constitutive equation (A.1.1) can also be reduced to the equa
tion obtained from the rate independent equations of incremental plastic
ity as v/ill be shown in the next section.
A.2. Rate Independent Incremental Plasticity Theory
The constitutive equation derived in this section is obtained when
the material is loading in the manner outlined by Malvern (1969).
If the material is strain-rate independent, if it exhibits isotropic
work-hardening, and if its yield function, f, depends only on the state
of stress, then
f(C. .) = F (A.2.1)
ij
where F is the work-hardening parameter. If F, and, consequently, the
size of the yield surface, is assumed to depend only on the total
P
plastic work, w then
F = F(vF) = a* (A. 2.2)
where vF = f a.. de^. (A.2.3)
J ij iJ
*
and ct is the magnitude of the stress vector.
During loading the material is assumed to obey the Prandtl-Reuss
flow law
ds.
ij
dX s..
ij
(A.2.4)
where d\ is a scalar function, and the von Mises yield condition
f = a
rr
y 2 s s
ij ij
s
(A.2.5)


144
The fourth equation of equations (B.2.3) has not been used. However,
when equations (B.2.5), (B.2.7), and (B.2.8) are substituted into it,
the resulting equation is the characteristic equation (B.1.2) and no
new information is obtained. From these results, the left eigenvector
can be written as
1 =
1
pc
1
pc
0
aA A A A
Ad o 4
- -2 4
A A aA
4 o 5 -Z '
Pc
^3^5 ^2^6 + -2
V6 ^5
A4
-2
pc
aA A A A
Ad - -2 A4
A4A6 ^5
(B.2.10)
A dw
(2.
39) and
(2
41),
p
0
0
0
0
o ~
dv
X
0
ap
0
0
0
0
dv
r
0
0
P
0
0
0
dvQ
0
0
0
A1
^2
>1
CO
da
X
0
0
0
aA2
^4
^5
dCTe
0
0
0
A3
aA
D
>i
02
1
K1
CD
ir-
1


TO PEGGY


Longitudinal Strain,
Figure 4.1 Grid Size Effects on the Longitudinal Strain at X = 3.75
oo
CO


TABLE OF CONTENTS (Continued)
Page
APPENDIX B. CHARACTERISTICS AND EQUATIONS
ALONG THE CHARACTERISTICS 140
B. 1. Equations for the Characteristics 140
B.2. Equations along the Characteristics 141
B. 3. Reducing Equations to Simpler Case 156
B.4. Uncoupled Waves 159
B. 5. Elastic Waves 152
APPENDIX C. PROGRAMS FOR DETERMINING THE PLASTIC
WAVE SPEEDS 164
APPENDIX D. SOLUTION TO THE FINITE DIFFERENCE EQUATIONS
IN THE CHARACTERISTIC PLANE 171
D.l. Equations for Fully Coupled Waves 171
D.2. Equations for Uncoupled Waves 17&
D.3. Solution at a Regular Grid Point
for Fully Coupled Waves 130
D.4. Solution at a Regular Grid Point
for Uncoupled Waves 183
D.5. Solution at a Boundary Point (X=0)
for Fully Coupled Waves 185
D.6. Solution at a Boundary Point (X=0)
for Uncoupled Waves 191
APPENDIX E. COMPUTER PROGRAM FOR CHARACTERISTIC PLANE
SOLUTION 194
E.l. General Description of the Program I94
E.2. Initial Conditions 196
E. 3. Calculation of A 201
E.4. Input Data 202
E.5. Listing of the Program 204
LIST OF REFERENCES 231
BIOGRAPHICAL SKETCH 236
v


128
so that'the assumed flow law is derivable from the plastic-potential
equation
o 3f(o\ .)
deP = dX y -1J-
(A. 2.6)
The form of the von Mises yield condition, equation (A.2.5), is chosen
so that under uniaxial tension, f = a .
x
From equations (A.2.3) and (A.2.6) an expression for d\ can be
obtained as
dW*3 = a. ,deP = a. ,d\
and
1J 1J ij oaTj
dX
dW1"
df
ij So..
ij
de
p
P iJ
ij df
^mn da
mn
Now, from equations (A.2.1) and (A.2.2)
so that
and
da, = ^ dW1" = dvF = f' (vP) dvF
dCTki kl 9W
F
dr
dvF
1 Sf
33 CTki
de
F'GV13) ~wkl
df df
p v
1J f'(W)P a
mn oa
mn
(A.2.7)


87
(1970) was used. This constitutive equation (shown in Appendix A) was
for strain-rate independent material behavior.
The first two computer runs (one including and one not including
radial inertia effects) were made using the initial conditions and the
boundary conditions which Lipkin and Clifton (1970) used in one of
their experiments. These input data used were
f T =4.00
R
AX = AT = .050
s = 0
Data Set 1 /
T = .0003480
V = .002404
Xf
V. = .0001106
l Gf
These data represent a tube with an applied static pretorque
(above the yield stress) impacted longitudinally at one end. The time
history curves of the longitudinal strain and the change in shear
strain are shown in Figures 4.4 and 4.5, respectively, for the section
of the tube 3.75 diameters from the impact end. The simple wave solu
tion and the experimental results of Lipkin and Clifton (1970) are
also shown in these figures. It can be seen in Figure 4.4 that the
longitudinal strain obtained in this work follows the experimental
results more closely than does the simple wave solution. Most of the
improvement over the simple wave solution is the result of using
a finite rise time (T = 4.0) for the impact velocity. The fast wave
R
has passed the point X=3.75 at the time when the longitudinal strain


Shear Stress, T x10
108


106
These data represent the impacting of the end of a tube with a torsional
(transverse) velocity pulse when the tube is preloaded in compression
above the yield stress. First the stress trajectories obtained by not
including radial inertia effects will be examined. The stress trajec
tories for this case are shown in Figure 4.17 at various distances from
the impact end. Here the stress trajectory develops (at distances from
the end of the tube greater than 1.0 diameter) into the type given qual
itatively by Clifton (1966), that is, the tube undergoes unloading
along the S^-axis followed by an increase in T at a constant value of
S^_ until the yield surface is reached followed by loading normal to the
yield surface. Again the stress trajectory does not exhibit this
behavior near the impact end, but rather approaches this behavior as
the distance from the impact end of the tube increases. As explained
above, the stress trajectory near the end of the tube does not behave
in the manner given by Clifton (1966) using the simple wave solution
because the fast and slow waves are not distinct near the end of the
tube and because the input boundary conditions force the solution near
the impact end.
The stress trajectories (T versus S ) are shown in Figure 4.18
N
for the case when radial inertia effects are present. The stress
trajectories are plotted using the combined hoop stress and longitu
dinal stress of equation (4.3.1) since this gives an effective three-
dimensional representation of the stress trajectories. The stress
trajectories exhibit a "ringing" effect after loading has taken place.
This "ringing" is caused by the hoop stress as can be seen from
Figure 4.19.


113
inertia effects). For all four cases the initial conditions and the
boundary conditions were specified by
. 40
T
R
AX = AT = .025
o
0
x
o
.0000900
T
.000250
x
f
These input data represent a tube with a static pretorque which
is impacted longitudinally at the end. The longitudinal strain versus
time curves obtained by using these data are shown in Figure 4.22.
When no strain-rate dependence is included, the results are the same
qualitatively as those obtained earlier for the Lipkin and Clifton
(1970) data. However, when strain-rate dependence is included (and
the tube is subjected to constant velocity impact), the maximum longi
tudinal strain is reduced.
This lower value of total longitudinal strain may be better under
stood by considering a longitudinal impact of an unstressed tube of
a strain-rate dependent material. The stress-strain curve for such
a material is steeper at higher strain rates; and therefore, if the
maximum applied stress is the same for two different loading conditions
the total strain will be larger for the condition when the rate of load
ing (or straining) is smaller. For this combined stress case, the


44
When a = 1,
2 2 ~ 2 2
(cosJ5 + 2 */3 sin 6 cos $ + 3 sin 6) (3 sin 6 cos
and equation (3.1.26) becomes
-[2(1 + v) 1 -i-v2]
z
c [(2) (1 + v) -1] f2 f1 + 2vf3
- (1+v)2
(3.1.29)
z
2
(1 + v) sin 6 cos 6 + 3v sin 6
A short computer program was written to calculate the critical
values of 3 (using equation (3.1.28) when a = 0 and equations (3.1.29)
and (3.1.27) when a = l) for various values of v and 6 when y = 0.
This program is shown in Appendix C and the results are plotted in
Figure 3.3. The only values of 3 which are physically possible are
between 0 and 1 and therefore only values of 3 in this range are
plotted in Figure 3.3. For all other values of 6, there is no phys
ically possible critical value of 3; that is, there is no value of
3 such that the fast and slow wave speeds are equal at y =0.
For the case when cQ = 0 (6 = -60), for any value of v the critical
W
value of 3 is smaller when radial inertia effects are included.


137
A. 4. Dimensionless Expressions for the Functions
0(s,A) and ijr(s,A)
Using the definitions for the dimensionless variables in equation
(3.2.1), the expressions for 0(s,A) and i|f(s,A) given in Sections A. 2
and A.3 can be converted into dimensionless form. First, when the
uniaxial stress-strain curve in equation (A.2.13) is used, the functions
0(s,A) and i|r(s,A) are given by equations (A.2.18) as
(A.4.1)
and using equation (3.2.1) the dimensionless form of these variables is
r
(A.4.2)
and
5(s,A) = E0(s,A)
EBn(s ct )n
y
n / \n-l
- E Bn(s s )
y
$(s,A) = Bn(s Sy)n 1
(A. 4.3)
where
n
(A.4.4)
B = BE
When the uniaxial stress-strain curve of equation (A.2.21) is
used, the functions 0(s,A) and i|r(s,A) are given in equation (A.2.25) as


77
Case IV: Mixed boundary conditions
When V^p and are given at a boundary point, then the solution
at that point is given by equations (D.6.9), (D.6.10), (D.6.11) and
(D.6.15), i.e.,
v9p = 4 <3'5'43)
and S^p and Sgp are given by equations (3.5.36), (3.5.37), and
(3.5.38).
3.6 Calculation of the Strains
At any grid point P, the solution is obtained by an iterative
technique. Once this is done, the values of S S., T V V and
Xu X P
Vg are known at P as well as at points L, B, and R (see Figures 3.5
and 3.6). The strains at point P can be computed very easily from
equations (2.27), (2.28), and (2.29). These equations can be written
in dimensionless form using equation (3.2.1) as
9e Sv
x x
T ~ X
(3.6.1)
Se
i Bve
9x
dT 2 5x
(3. 6. 2)
de6
~W = 2Vr
(3.6. 3)
For a regular grid element, these equations can be written in
finite difference form as


232
Cristescu, N. (1965), "Loading/Unloading Criteria for Rate Sensitive
Materials," Archiwum Mechaniki Stosowanej, Vol. 17, pp. 291-305.
Cristescu, N. (1967a), Dynamic Plasticity, New York: John Wiley and
Sons, Inc.
Cristescu, N. (1967b), "Dynamic Plasticity under Combined Stress,"
Mechanical Behavior of Materials under Dynamic Loads (U.S.
Lindholm, Ed.), New York: Springer-Verlag, pp. 329-342.
Cristescu, N. (1968), "Dynamic Plasticity," Applied Mechanics Reviews,
Vol. 21, pp. 659-668.
Cristescu, N. (1971), "On the Coupling of Plastic W'aves as Related to
the Yield Condition," Romanian Journal of Technical Science,
Applied Mathematics, Vol. 16, pp. 797-809.
Cristescu, N. (1972), "A Procedure for Determining the Constitutive
Equations for Materials Exhibiting Both Time-Dependent and Time-
Independent Plasticity," International Journal of Solids and
Structures, Vol. 8, pp. 511-531.
Davis, E. A. (1938), "The Effect of Speed of Stretching and the Rate of
Loading on the Yielding of Mild Steel," Transactions of ASME,
Vol. 60, pp. A137-A140.
DeVault, G. P. (1965), "The Effect of Lateral Inertia on the Propa
gation of Plastic Strain in a Cylindrical Rod," Journal of the
Mechanics and Physics of Solids, Vol. 13, pp. 55-68.
Donnell, L. H. (1930), "Longitudinal Wave Transmission and Impact,"
Transactions of ASME, Vol. 52, pp. 153-167.
Duwez, P. E. and Clark, D. S. (1947), "An Experimental Study of the
Propagation of Plastic Deformation under Conditions of Longitu
dinal Impact," Proceedings of ASTM, Vol. 47, pp. 502-532.
Efron, L. and Malvern, L. E. (1969), "Electromagnetic Velocity-
Transducer Studies of Plastic Waves in Aluminum Bars,"
Experimental Mechanics, Vol. 9, pp. 255-262.
Hencky, H. (1924), "Zur Theorie Plastischer Deformationen und der
Hierdurch im Material Hervorgerufenen Nebenspannungen,"
Proceedings of the First International Congress for Applied
Mechanics, Delft, pp. 312-317.
Hill, R. (1950), The Mathematical Theory of Plasticity, England:
Oxford.
Hopkins, H. G. (1961), "Dynamic Anelastic Deformation of Metals,"
Applied Mechanics Reviews, Vol. 14, pp. 417-431.


Shear Stress, 7 x10
1. 2
rr
Classical
With Radial Inertia
With Rate Dependence
With Radial Inertia and Rate Dependence
Figure 4.24 Stress Trajectory at X = .25 for Data Set 3
117


27
_2 r -2 - - ry (d 2aa )
+ a(pc ) (pc ) (A1A¡5-93)-A I j ^ + - iHs,A) ldt
^ o 2s
s,A)J'
-2 r -2 -2-1 r1 sx i
+ 2(p Oj^Cp O (14-aA^)-A4J i)r(s,A)Jdt .
(2.53)
These three equations each represent four equations, one equation
in differential form along each of the four charaxeristic lines of
equation (2.48). When the waves are coupled, equations (2.51), (2.52),
and (2.53) are identical. That is, by multiplying equation (2.52) by
the quantity
(p-2)(ay5 y4>
(pc") (A A. A A ) A
1 3 O
and using equation (2.48), equation (2.51) is obtained; or by multiply
ing equation (2.52) by the quantity
pc|_(pc ) (A1A4 aA2) A4J
(p c")(xa5 23) a5
and using equation (2.48), equation (2.53) is found. When the numerator
and denominator of these multiplying quantities do not vanish, equations
(2.51), (2.52), and (2.53) are identical. However, when the waves
become uncoupled, a phenomenon discussed in Appendix B, A and A vanish.
o
In this case the multiplying factors used above become undefined and
the equations (2.51), (2.52), and (2.53) are not the same. When
A -A 0 the equations (2.51) and (2.53) reduce to equations (B.4.4)
3 5
and (B.4.6), respectively. Under these conditions, equation (2.52)
also reduces to the form of equation (B.4.4).


132
r a* a -i
/ = B(a* a )n I a* -Zl
y L n +1 J
W = (a*-a )(na*+a )
n+1 y y
and
dW13
(vf)
dF
dvf
da da
dW
i- = Bn(a*) (a* a )n 1
f'(/) y
(A.2.16)
Using equations (A.2.5) and (A.2.16) in (A.2.12), the expression for
the plastic strain rate for the static stress-strain curve of equation
(A.2.13) becomes
oD / \n-l 3a*
.P 3Bn ~5t
e. =
ij
2a
ij
(A. 2.17)
Comparing equation (A.2.17) with equation (A.1.1), the functions
0(s,A) and i|r(s,A) for the rate independent plasticity theory using
equation (A.2.13) are
t(s,A) = 0
0 (s,A) = Bn(s ay)n
(A.2.18)
where s = a The constitutive equation (A.1.1) during loading for
this case is
1+v v c 3 / \n1 ij
e. = a.. 6 . a, + Bn( s a ) s -
ij E ij E ij kk 2 y -
(A.2.19)
The second form of the uniaxial stress-strain curve is the one
used by Cristescu (1972) as the relaxation boundary for dynamic load
ing. Although Cristescu (1972) uses a quasistatic curve which is


ACKNOWLEDGMENTS
I would like to thank Professor Martin A. Eisenberg, Chairman of
the Supervisory Committee, not only for his untiring efforts during
the development and preparation of the material contained in this
manuscript, but also for being a counselor, teacher, and friend during
both my undergraduate and graduate studies. I am also indebted to
Professors L. E. Malvern and E. K. Walsh for their helpful criticism
and encouragement during my doctoral studies. In addition, I would
like to express my appreciation to the other members of my Supervisory
Committee: Professors U. H. Kurzweg, C. A. Ross, and R. C. Fluck.
A special word of thanks is extended to Professor N. Cristescu
for his many helpful discussions during the development of this
dissertation.
I thank my wife, Peggy, for her encouragement, moral support, and
understanding during the course of my studies. I also thank Peggy for
typing and proofreading the drafts of this dissertation. I appreciate
the efforts of Mrs. Edna Larrick for the final typing of the manuscript
and Mrs. Helen Reed for the final preparation of figures.
I acknowledge financial support from the National Defense Education
Act, the National Science Foundation, and the University of Florida
which made my studies possible.
I also acknowledge the Northeast Florida Regional Computing Center
for the use of its IBM 370 Model 165 digital computer without which
the scope of this work would have been greatly curtailed.
iii


WRITE 7* 1C35) (EX(L ),L = 1,MX,INCRXP)
WRITE<7, 103 5 ) (VX(L),L=1, MX,INCRXP)
W R I T E ( 7 1335) (CEX(L ), L = 1,MXINCRXP )
IF ( A.EC1.C )G0 T 403
WRI TCl7, 103 5) (ST(L ) L= 1 ,MX,INCRXP)
WPITb(7, 1035) (ET(L ) ,L=lfMX,INCRXP)
WRI TE(7, 1035 ) (VR(L ) ,L=1 ,MX,INCRXP)
403 CUNTI l\IUL
I F ( IPUN2.EQ.OGO T0 404
WRITE!7,1035) (TAU(L ),L = 1MXvINCRXP)
WRI TE(7, 10 35 ) (VT(L ) ,L=l,MX INCRXP)
WRITE(7, 1035 ) (ETX(L),L=1,MX,INCRXP)
WRITEI 7, 103 5 ) (CELGAML),L = 1,MX,INCRXP)
404 CONTINUE
IF! IWRITE 2) 90, 397,405
4C5 CONTINUE
LODO FORMAT(' 1 ,2 X,'LINE =',I5///)
100 1 FORMAT!3 15,5F5.2,4F10.2)
1002 FORMAT!I5.7F10.3,15)
ICC 3 FORMAT(1015)
1004 FORMAT(2 E207)
1005 FORMAT( 1 ,9X,A = , I 1, 15X, 'JR ATE = ,I 1,11X,ELX = *,F9.6,4X,
I'OlLT = ,F9.6,4X, MXMA X = ,I 4,BX, SMALL = ',F8.6/)
1V.-6 FORMAT ( 19X, NU = F 5.3 1CX SY = F12.9,3X 1 BOAR F 9. 1,4 X N
1= ,F7.4,9X, 'XK:: = F 8.6,6 X BETA = ',F8.6/)
1007 FORMAT!1 O X,J RIS F
i,*SXO = ,F9.6,5X
iC C 8 FORMAT(1TX, 'HI =
1= F4.2, 11X, n
10 0 9 FORMAT(/3 X, 'TIME
1010 FORMAT(/ IX,1X =
1011 FORMAT!/3X,SX =
1012 FORMAT(/3X,ST =
1013 FUKMATI/3X,'TAU =
= I 4,BX, 'XVFIN = ',F 9.6,3 X, 'TVFIN = *,F8.6,3X
TAUO = ,F8.6,5X, H = ',F8.6/)
,F4.2,11X,H2 = ,F4.2, 11X, *H3 = ,F4.2,11X,H4
12, 13X, SYS = ,F 1 2.9/)
',4X,71F9.56X)F9.5/(8X8(6X,F9.5)))
',4X,7!F9.5,6X),F9.5/!8X,0(6X,F9.5)))
,8 E15.5/( 10X.8E15.5) )
' t 8E155/(10X,6E15.5))
.8E15.5/10X.8E15.5))
226


Shear Stress, t x10
Figure 4.14 Shear Stress Versus Longitudinal Stress for Data Set 1 With Radial Inertia
102


176
The four equations (D.1.18), (D.1.19), (D.1.20), and (D.1.21) along
with equation (D.1.6) now become a set of five simultaneous algebraic
equations which must be solved during each iteration for the unknowns
VxP V0P TP SxP and Sep point P of a regular grid element. The
reason for calculating the coefficients by the method outlined in
Chapter 3 is now apparent. If the coefficients had been calculated in
the normal manner, these five equations would have to be solved simul
taneously for each iteration at a regular grid point when the waves are
fully coupled. However, by calculating these coefficients by the method
used here, these equations reduce to two separate sets of simultaneous
equations, one set with two equations and one set with three equations
(this will be shown in Section D.3), and the computation time required
to solve these two sets of equations is significantly less than the
time to solve one set of five equations. Once the solution is found
at a regular grid point, the radial velocity is obtained from equation
(D.1.3). This will be done in Section D.3.
At a boundary point the solution is obtained by specifying two of
the variables (as described in Chapter 3) and then solving equations
(D.1.19), (D.1.21), and (D.1.6) simultaneously. Again equation (D.1.3)
is used to get V This will be done in Section D.5.
D.2. Equations for Uncoupled Waves
When the waves are uncoupled, the finite difference equations
along the characteristic lines are given in equations (3.4.9) to
(3.4.14). These expressions will now be simplified by grouping known


126
If the only stress present is a then
x
S = CT
and from equation (A.1.1) the plastic portions of the strain rates are
P *P *P
e~ = = = 0 .
'6x 9r
rx
P P 1-P
e. = e = e
9 r 2 x
If, in addition, each plastic strain component is monotonically increas
ing (or monotonically decreasing), then the expression for A in equa
tion (2.16) becomes
CT
._ .'P.2 P 2 P.2 x
A = /- I / CO + (eQ) + (er) dt +
P , X
e dt +
x E
A -
P e
e + e
x x
A = e
and equation (A.1.1) reduces to
1 3 f
e = ct + ¡
x E x 2 L
0(ct e )ct + A(ct
X, X X X
, X J
3 CTx
c = ci+0(CT,e)cr+t(CT,e)
x Ex xxx xx
(A.1.3)
which is the same as equation (2.13) used by Cristescu (1972).
Again, when the only stress present is ct the function 0(cr e )
X XX
can be set equal to zero and
'Kct e ) = i g(o e )
xx E xx
to obtain the semi-linear constitutive equation of Malvern (1951a,1951b) as
1 1 ,
e = ct +-g(CT,e)-
x E x E xx


52
be determined before the solution (in terms of stresses) is known.
Because of this, the slope of the characteristic lines and the solution
to the problem must be determined at each point simultaneously. This
is done by using the iterative numerical technique described below.
The numerical grid shown in Figure 3.4 will be used. There are
two types of elements in this grid: boundary elements and regular
elements. All of the regular elements are alike, and all the boundary
elements are like the right-hand side of a regular element. A detailed
picture of a single regular element is shown in Figure 3.5, and a
single boundary element is shown in Figure 3.6. The grid is defined
in terms of the dimensionless variables given in equations (3.2.1) and
(3.2.3). It is diamond shaped with the straight outer lines corre
sponding to the characteristic lines for elastic longitudinal waves
with radial inertia effects included. These outer characteristic lines
have slopes of either c=+1 or c= -1, which can be seen from equations
(3.2.1) and (B.5.4). The vertical straight line corresponds to the
two vertical characteristic lines, and the straight inner nonvertical
lines correspond to the characteristic lines (through the point P) for
the elastic shear waves. For both types of elements, the problem reduces
to that of determining the values of the stresses and velocities at the
point P, when their values at the points B, R, and L are known.
This grid with all the elements constant in size simplifies the
writing of the finite difference equations. The diamond shape allows
the vertical characteristic lines to automatically connect point P (at
which the solution is desired) with point B (at which the solution is
known) and makes the finite difference equations along the vertical


Shear Stress, T x10
6 _
4
Figure 4.12
Stress Trajectories for Data Set 1 Without Radial Inertia
to
o


26
O =
(p c2) (A
A .X -aA2) -A. dv -pd aA A -A A dvQ-
46 5' dj x M l. 2 o 3 4j 9
3A A A A
Z D O <£
]dV
pe
(p c2) (A4A6-aA2) -A4J dav +
(p c2) (A A -aA2)
x L 4 6 5
d]
-(2o acr )
x 0
2s
']
i|>(s,A) dt
v (o 2actq)
o -] r-v (.0 -j
+ al (pe ) (A A.-A A )-A + t(s,A) dt
2 6 o 5 2J [_r _l
o 2s
-2
2(pe )
~A3A4J [^(s,)Jdt
aA2A5
2s
(2.51)
0 =
(P2)(A2A6-A3A5)-A2]dVx^
(pe ) (A1A5-A2A3)-A5 ¡dvf
J
4£(Ps2)(A1A5-23)-xjdTex+
pe pe
ex' -2Llfc|(Ve-i3M%
ti
d][
(2a aa )
x 0
2s
i|Ks>A)Jdt
-1j[(pS2)2(V6--
pe
+ 2
1 FV '-u j -i
a2) (pc )(A +A ) + l j ZL + t(s,A) dt
x D L_r0 2s -J
i r3t
.(PC )(AiA5-A2A3)-A5J
0X
^ 2
, A)J <
t(s,A) dt
(2.52)
2-3
0 = p c
n
aAA -AA I dv o c
2 5 3 4J x
(p c2) (A1A4-aA2) -A J dv
e
[(pc2)(A1A4-aX2)-Xjd
Vs-vJ
9x
, -2.
+ (p c )
aA A -A A. da
. 2 5 3 4J x
a -2,2
+ (p c )
-(2a aa.)
x 6
2s
,A)J'
f(s,) dt


CEL r A(MX) = CELTAO
CPLAS(MX) = DPL AS')
SMAX(MX) = snc
CEL GAM(MX) = 0.0
C EX(MX) = 0.0
STKES(MX) = SORT(SX0**2)
70 CONTINUE
CELLHR(l) = CELTAO
C THIS ENOS THE CALCULATION OF INITIAL CONDITIONS
I W R I T E = 1
GO TU A01
9^ CONTINUE
KUM = 1
100 CONTINUE
MX = 1
JERROR 0
CO TO 300
200 CONTINUE
MX = MX + 1
'30 3 CONTINUE
MT = MX + KLJUNT
T(MX) = M T D E L T
SXk = SX(MXfl)
STR = ST(MX + 1 )
V X R = VX(MX 1 )
VTR = VT(MX+1)
VRR = VRIMX+l)
TAUR = TAU(MX + 1 )
T AUG = TAU(MX)
SXC = S X(MX)
STB = ST I MX)
VXD = VX{MX)
VT B = VT(MX)
VRB = VR(MX)
to
o


121
along the tube exhibited neutral loading as the fast wave passed,
followed by loading normal to the yield surface as the slow wave
passed. However, at distances closer than 1.0 diameter from the
impact end the stress trajectory at a point along the tube followed
a path in stress space which represented a combination of neutral
loading and loading. This behavior was caused by the finite rise time;
since the velocity pulse was not a step function, it took about 1.0
diameter for the two waves to separate. In other words, the fast
wave starting at T = T overtook the slow wave starting at T = 0 in
R
about 1.0 diameter.
When radial inertia effects were included, the stress trajectories
became more complicated as they were no longer planar. For this case
the longitudinal stress was plotted against the shear stress and it
appeared that at distances greater than 1.0 diameter from the impact
end unloading occurred. However, when the stress trajectories were
plotted in the three-dimensional stress space the behavior was found
to exhibit neutral loading followed by loading.
The strains and velocities were shown at several tube locations
and constant state regions were observed at points along the tube
after the fast wave had passed and before the slow wave had arrived.
Also, radial inertia was shown to have little effect more than two
diameters from the impact end.
For the second case when the impact velocity was torsional, the
same types of results were obtained. Here the stress trajectory
obtained was like the one predicted by Clifton (1966) with his simple
wave solution.


Critical Value of
Figure 3.3 Values of |3 at Y = 0 for which c = c = c
' f s 2


160
Under these conditions (A = A =0), the equation for the nonvertical
o o
characteristics of equation (2.45) becomes
(pc)2(14A6 aA2A6) (pc2) (^ + a2) + 4 = 0
and as in equation (2.48)
-2 2 WA -
C =
2p(X146 aX2X6)
-2 ^A1A4 + A4A6 ~ ^2^ ^ i-A4^A6 Al^ + aA2^
2pA6(AiA4 aA2)
and using equations (2.49) and (2.50),
-2 (A1A4 + A4A6 aA2) + (A4A6 ~ A1A4 + aA2)
2PA6(A1A4 aA2}
-2
Cf
A.
p (V4 ~ ^
(B.4.1)
-2 (A1A4 + A4A6 aA2) ~ (A4A6 A144 + aA2)
2PA6(A1A4 ^
-2 1
c =
PAC
(B.4.2)
and the four nonvertical characteristics are
c = cf, cs.
The equations along the characteristics will be obtained from
equation (B.2.23). For A =A =0, this equation becomes
o O
o = [(P;2)(6)-i][a4][-
dv + ~ da + i dt
x -2 x Tx J
c pc
[;2)V1][¡2]Lr-*Jdt
+ a
(B.4.3)


riFFl = ADS(SOP{ I ) SB P( I 1 ) )
DIFF2 = ABS(STRAIN! I ) STRAIN! I-I) )
CIFF3 = ADS ( VXP ( I ) VXP(I-D)
CIFF4 = A BS(VT P( I ) VTP(I-l))
CIFF = U1*DIFF1 + H2 C 1 FF2 + H3*DIFF3 * H4*DIFF4
DENOM = A BS ( VX P ( I ) ) + ABS(VTP(I))+ ABSISBPUJ) +
FRRCR = CIFF/DENOM
IF(ERROR.GT.SMALL)GO TO 340
430 CONTINUE
IFIMX.GT.1)00 TO 391
£X(MX) = £X(MX) + 2*VXR VXI VXB
ET X(MX) = ETX(MX) + VTR .5*(V TI + VT3)
GC TO 392
391 CONTINUE
FX(MX) = EX(MX) + VXR VXL
ETX(MX) = ETX(MX) + .5*(VTR V TL)
392 CONTINUE
VR{MX ) = A*(D3 03*STI )
ET(MX) = ET(MX) + DELT2*(VR(MX) + VRB)
TAU(MX) = TAUI
VX(MX ) = VXI
VT f M X ) = VT I
SX(MX) = SXI
ST(MX) = ST I
Al(MX) = A1P
A 2(M X) = A2P
A 3 ( M X ) = A 3 P
A4 ( MX ) = A4P
A^tMX) = A5P
A6(MX ) = A6P
CF(MX) = CFI
CS(MX) = CSI
S 3(MX) = SBPC
EXP(MX ) = EX(MX) (SXI NU*STI)
AB S(STRAIN(I))
to
to


141
or
a(cp )'
or
a(cp )
or
,
-8
O
O
1
0 Cp 0
1
0 cA^ acA2 cA2
1 0 acA,
2 Zl3
+ acp
0 acA acA acA
-i nr D
0 0 acA acA
4 5
1 cA acA cA
0 1 acA
cA
5
6
5 6
cA. acA cA
0 cAw acA
12 3
1 2
ac.4 acA, acA
2 4 5
-
- 2
a (cp )
0 acA acA,
2 4
- acp
cA acA,. cA
1 cA acA
3 5 6 J
3 5
3-d~- -
-2-
- -2 -2- "1
2 2-4
cl A A,A + 2aA^
¡ir
- A A -
aA A aAA
an c
L 1 4 6 2
3 5
3 4
1 5 2 6J
M 1
2- lT- -2*1
_ r
1
c ¡ A A aA 1-
L 4 6 5J
acp|_
- ao\J
= 0
cp 0 1
0 acA^ acA^_
1 acA cA
o o
V4 all\
and using the definitions (2.46), this becomes
2 3-6 2 2-4 2 -2-
apc(a)-apc(b)+apcA4 =0
|^a2oc2J j (p c2) 2 (a) (pc2) (b) + aJ = 0
(B.1.2)
which is the equation for the characteristics given in equation (2.45)
B.2. Equations along the Characteristics
The equations (2.48) along the nonvertical characteristics can be
obtained in two different ways. One form of the equations along the
characteristics will be found below by each method as an example; then
the other forms of these equations will be given. Using the first