• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Theoretical developoment
 Dimensionless equations and numerical...
 Results and discussion
 Summary
 Appendix
 Constitutive equations
 Characteristics and equations along...
 Programs for determining the plastic...
 Solution to the finite difference...
 Comuter program for characteristic...
 Reference
 Biographical sketch
 Copyright














Title: Inelastic wave propagation under combined stress states
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Permanent Link: http://ufdc.ufl.edu/UF00086029/00001
 Material Information
Title: Inelastic wave propagation under combined stress states
Physical Description: x, 236 leaves. : illus. ; 28 cm.
Language: English
Creator: Myers, Charles Daniel, 1945-
Publication Date: 1973
 Subjects
Subject: Wave-motion, Theory of   ( lcsh )
Stress waves   ( lcsh )
Strains and stresses   ( lcsh )
Engineering Sciences thesis Ph. D
Dissertations, Academic -- Engineering Sciences -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 231-235.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00086029
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000577578
oclc - 13990362
notis - ADA5273

Table of Contents
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Tables
        Page vi
    List of Figures
        Page vii
        Page viii
    Abstract
        Page ix
        Page x
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
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        Page 11
    Theoretical developoment
        Page 12
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    Dimensionless equations and numerical procedures
        Page 29
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    Results and discussion
        Page 80
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    Summary
        Page 118
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    Appendix
        Page 124
    Constitutive equations
        Page 125
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    Characteristics and equations along the characteristics
        Page 140
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    Programs for determining the plastic wave speeds
        Page 164
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    Solution to the finite difference equations in the characteristic plane
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    Comuter program for characteristic plane solution
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    Reference
        Page 231
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    Biographical sketch
        Page 236
        Page 237
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    Copyright
        Copyright
Full Text















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




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