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Mimic 2 I DATE DUE DATE DUE

 

 

 

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NW? 2334

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MAY 0 3 2010

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1m cams-peso“

 

(9'

01'

ROI

ABSTRACT
LONGITUDINAL PLASTIC WAVE PROPAGATION
IN ANNEALBD ALUMINUM BARS
by

Leonard Efron

In this investigation aluminum rods were subjected to
dynamic compressive impact loading of duration of the order
of 500 microseconds in order to study the propagation of
longitudinal plastic waves, Two independent series of tests
were conducted, In the first, an electro-magnetic transducer
was used, while in the second, etched foil resistance strain
gages yielded records of surface strain at the same gage
locations. Strain rates on the order of IOOiran/sec were
reached.

Test results indicated that any given level of velocity
or strain propagates along the bar with a constant velocity,
not affected by the strain rate within the small range of
strain rates encountered. However, the velocities of propa-
gation observed differed noticeably from those predicted by
von Karman rate-independent theory based on the static curve,
Good agreement was found between the propagation speeds ob-
served for different levels of velocity (averaged over all
tests) and predictions of von Karman theory based on a single
dynamic stress-strain curve differing from the static curve°

That the apparent applicability of a single dynamic curve

tn

'1'.)

it

”an
6‘56

and rate-independent theory to this kind of plastic wave pro»
pagation is consistent with rate-dependent theory for a ma»
terial with a very slight rate dependence, was demonstrated

by the results of computer solutions for rate-dependent theory.

The wave propagation speed versus strain level plots
from the transient strain records showed consistently lower
propagation speeds than those based on the velocity records.
It is believed that the strain gage response actually lags
behind the strain in the material, but considerably more
evidence is needed before final conclusions can be drawn
about the lag in the strain measurements and the reasons for
it,

The velocity recording technique for non-magnetic materials
is believed to give good results, but it may be possible to
modify it to make it more nearly a routine type of test,

In order to apply the strain-rate-dependence theory to
the experimental measurements made, it is necessary to have
boundary values at x = 0. To avoid the threemdimensional
difficulties associated with the stress at the actual impacted
end of the bar and test one-dimensional theory in a region
where it should be applicable, it was decided to take the
first gage station (six diameters from the impacted end of
the bar) as x = O, and use the recorded velocity there as a
boundary condition to predict the velocity versus time at
gage stations further along the bar,

A numerical computer solution was obtained using the

tr
:2
a]
cc
wi

U18

C0
C01
a l
ra‘

10:

rate-dependent theory with a power law for rate dependence,
The computer solution did predict a constant wave propaga~
tion speed for any given level of velocity, but the constant
values predicted did not agree well with the experimental
values from the velocity records. This lack of agreement
appears to be mainly the result of using a rather poor fit
to the static curve in the computations, since von Karman
rate-independent theory using the same fitted static curve
also gave poor agreement with the experiments. Since the
computer solutions with rate-dependent theory were consistent
with a single dynamic curve, and since the velocity measure-
ments correlate with a single dynamic curve, it appears that
a little ingenuity in curve-fitting could produce agreement
between the rate theory and the experiments.

For the case of linear strain rate dependence, previously
considered by Malvern (1950), a new computer solution for a
constant stress boundary condition indicated the formation of
a plateau of constant strain in agreement with von Karman
rate-independent theory, if the load is applied for a duration
long enough for the material at the impacted end to reach

equilibrium.

LONGITUDINAL PLASTIC WAVE PROPAGATION
' IN

ANNEALBD ALUMINUM BARS

by

Leonard Efron

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

in
Mechanics

Department of
Metallurgy, Mechanics and Materials Science

1964

 

n—h
*‘

Cw
fir.-

ACKNC-l; LEDGIKELNTS

I wish to express my sincere appreciation to Professor
Lawrence B. Malvern who suggested this problem. I shall
always be grateful for his invaluable guidance and counsel
throughout this research.

Thanks are also given to Professor William A. Bradley,
Professor George E. Mase and Professor David Moursand for
their services on my guidance committee. Appreciation is
also expressed to Professor Charles S. Duris for his sug-
gestions concerning numerical methods, to Dr. William W.
Lester for his suggestions and assistance in developing the
velocity transducer used in this study, and to Mr. William
T. Bean for his advice regarding resistance strain gage
technique. Note is also made of the assistance rendered
to me by my fellow graduate students in Metallurgy and
Applied Mechanics.

The project was supported by the National Science Founda-
tion under Grant No. 6-24898.

To my wife Joy, who is as perfect as her typing, I will
always be thankful for her understanding, encouragement and

assistance in the completion of this dissertation.

ii

LIST OF

CHAPTER
1.1
1.2

CHAPTER
2.1
2.2

2.3

2,4
CHAPTER
3.1
3.2
3,3
3,4
3.5
3.6
3.7

3.8

TABLE OF CONTENTS

FIGURES . . . . . . . . . . . . .
I INTRODUCTION . . . . . . . . . .
Purpose . . . . . . . . . . . .
Background . . . . . . . . . . . .
II FUNDAMENTALS . . . . . . . . .
Strain-Rate-Independent Theory . .
Strain-Rate-Dependent Theory . . .
Numerical Procedure . . . . . . .
(a) General Interior Point . . . .
(b) Impact End Point . . . . . . .
(i) Velocity Boundary Condition
(ii) Stress Boundary Condition
(iii) Strain Boundary Condition
(c) Plastic Wave Front . . . . . .
The Iteration Scheme . . . . . . .
III THE EXPERIMENT . . . . . . . .
General Description . . . . . . .
Specimens . . . . . . . . . . . .
Velocity Transducer . . . . . . .
Surface Strain Transducer . . . .
Static Test Procedure . . . . . .
Velocity Transducer Calibration .
Velocity-Gage Tests . . . . . . .

Strain Gage Tests . . . . . . . .

iii

Page

13
13
17
26
28
28
29
29
30
3O
31
36
36
39
39
43
48
54
59
62

 

|
III
‘lr

CH

BI

AP]

3.9 Elec

iv

tronics and Recording Equipment . . . .

CI-IA PTBR I v RESULTS 0 O O O O O O O O O O O O O O O O

4.1 Stat
4.2 Velo
4.3 Surf
4.4 Disc
4.5 Nume
(a)

ic Stress-Strain Curve . . . . . . . . .
city Test Results . . . . . . . . . . .
ace Strain Results . . . . . . . . . . .
ussion of Test Results . . . . . . . . .
rical Results . . . . . . . . . . . . .

Linear Overstress Rate Dependence Theory.

(b) Power Law Rate Theory with Input Data

from Velocity Transducer . . . . . . . .
(i) Convergence Difficulties . . . . . .
(ii) Discussion of Calculation Results

Based on Power Law Rate Theory with
Input Data from Velocity Transducer.

CHAPTER V SUMMARY AND CONCLUSIONS . . . . . . . . .

BIBLIOGRAPHY
APPENDIX .
A.l Lead

ing Wave Front .

(a) Linear Overstress Rate Dependence

(MALRATE) . . . . . . . . . . . . . . .

(b) Power Law Rate Dependence (POWRATE) . . .

A.2 Computational Procedure . . . . . . . . . .

A.3 Inte

A.4 Computation Flow Chart . . . .

A.5 Portran Programs

rpretation of Code Words . . . . . .

Page
64
66
66
68
79
81
84

84

89

89

93
101
105
110

110

110
113
116
120
123
125

(2.1)

(2.2)
(2.3)

(3.1)
(3.2)
(3.3)
(3.4)

(3.5a)
(3.5b)

(3.6)

(3.7)
(3.8)
(3.9)
(3.10)

(3.11)
(3.12)
(3.13)
(3.14)
(3.15)

(3.16)

LIST OF FIGURES

Schematic Representation of the Strain
Distribution Due to Constant Velocity
Impact in a Rate Independent Material . . .
The Characteristics in the X,T-Plane . . .

Finite Difference Grid Used in Numerical
SOIution O O O O O O O O O O O O O 0’ O O 0

Schematic of Loading Set Up . . . . . . .
Schematic of the Velocity Transducer . . .
Wire Arrangement at Gage Section . . . . .

Magnet Arrangement Providing Four Gage
Stations . . . . . . . . . . . . . . . . .

General View - Hyge and Test Set Up . . .

General View - Electronics and Recording
Equipment 0 O O 0 O O O O O O O O O O O 0

Details - Velocity Transducer and Alignment
Jig O O O O O O O O O O O O O O 0 O O O 0

Schematic of Strain Gage Bridge . . . . .
Range Extender - unloaded . . . . . . . .
Range Extender - Loaded . . . . . . . . .

Details of the Static Stress-Strain Test
SEt Up 0 O O O O O O O I I O O O O O O O 0

Schematic of Magnet Support Fixture . . . .
Field Mapping Set Up . . . . . . . . . . .
Location of Magnetic Field Centers . . . .
Polygonal Approximation of Wire Loop . . .

Schematic of Wave Form Produced by Eddy
currents O O O O O O O O O O O O O O O O 0

Details - Dynamic Bridge Elements . . . .

Page

16

24

27
37
41
42

44

45

46

47
49
51
51

53
55
57
58
58

61
63

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

(4.9)
(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

(4.16).

vi

Static Stress-Strain Properties
(Aluminum 1100-0) . . . . . . . . . . . .

Trace Records - Particle Velocity
TranSducer O O O O O O O O O O O O O O O O

' Typical Record of Particle Velocity vs Time

Wave Propagation Speed vs Particle Velocity
(Based on Averaged Data from Six Velocity
Records) 0 O O C O O O O O O I O C O O O 0

Data Of Fig. (4.4)] o o o o o o o o o o 0

[Based on von Karman Theory and

Static and Dynamic Stress-Strain Curves
(Dynamic Curve Based on von Karman Theory
and Velocity Tests) . . . . . . . . . . .

Particle Velocity vs Strain . . . . . . . .

Trace Records - Etched Foil Resistance
Strain Gages . . . . . . . . . . . . . . .

Wave Propagation Speed vs Strain Level . .

Strain Distribution (After Malvern)
(T = 10204 H Sec) 0 O O O O O O O O O O 0

Solution - Constant Stress Boundary Condition
and Linear Overstress Rate Dependence . . .

Solution - Constant Stress Boundary Condition
and Linear Overstress Rate Dependence . . .

Comparison of Fitted and Experimental
Static Stress-Strain Curves . . . . . . .

Computer Solution Compared with von Karman
Solution and Experiment . . . . . . . . .
Level Lines of Velocity for Solutions with
Velocity Test Data . . . . . . . . . . . .
Computed Stress vs Time After Yield

at x = O .

Page

67

7O

71

74

76

77

78

8O
83

86

87

88

95

96

97

98

 

Dc

{0

CHAPTER I
INTRODUCTION

1.1 Purpose

The mechanical behavior of engineering materials has
long been known to exhibit marked differences under condi-
tions of impact and high rates of loading as compared to
the results obtained during static testing. Theories taking
into account the effects of strain rate in stress-strain re-
lations were offered as early as 1909. The concept of a
rate-of-strain dependence in dynamic deformations of metals
was naturally extended to studies of stress wave propaga-
tion.

It is the purpose of this investigation to study the
propagation of longitudinal plastic waves in aluminum rods,
caused by dynamic compressive impact loading of duration of
the order of 500 microseconds. Cross section particle velo-
city and surface strains from two independent series of tests
are examined with special attention to the possible existence
of strain rate effects.

The data from the velocity transducers is compared with
predictions of a strain-rate-independent theory and also a
strain-rate-dependent theory. Consideration is given to the
possibility of using a single dynamic stress-strain curve
for the material to account for the wave propagation observed.

In order to apply the strain-rate-dependence theory

 

f 1'
ba
bc

th

tr;

fit

the

to the experimental measurements made, it is necessary to
have boundary values at x = 0. However, at the actual end

of the bar, the stress state is three-dimensional and a load~
time history obtained from the transmitter bar would not be
the proper boundary condition for the one~dimensiona1 wave

* . .
27 has found that the one-dimen51onal

propagation. Bell
wave is formed in a distance along the bar equal to about
one diameter. In order to avoid the three-dimensional dif-
ficulties and test the one-dimensional theory in a region
where it should be applicable, it was decided to take the
first gage station (6 diameters from the impacted end of the
bar) as x = O, and use the recorded velocity there as a
boundary condition to predict the velocity versus time at
the other three gage stations further along the bar.

All calculations are for a semi-infinite rod and the
transient experimental data are all obtained before any re-
flections arrive from the far end.

Associated with this study is a remexamination, using
the high speed CDC-3600 digital computer, of some previous
solutions of strain—rate-dependent longitudinal plastic wave
propagation. Iterative procedures were used to solve the
governing system of nonlinear equations. Difficulties in

computation were encountered. The criteria for convergence

 

*Superscript numerals indicate references as listed in the
Bibliography.

IL-

 

I
II
I
In

in

to:

on

the

ini

91a

(17

Ven.

290'

teer

of the iteration process was found to be a function of strain-

rate and the degree of material strain rate dependence.

1.2 Backggound

 

Thomas Young (1773-1829) included in his Course of

 

Lectures on Natural Philosophy and the Mechanical Arts,

 

London, 1807, a discussion of one-dimensional waves in an
elastic bar due to longitudinal impact with another bar

and concluded that C7'= %¥ where C7'is the stress at impact
due to an imposed boundary velocity v. The quantities E
and c are Young's Modulus and the elastic wave propagation
speed respectively, which are material constants. This
correct result perhaps marks the beginning of the history
of stress wave propagation in solids.

In 1821 Navier (1785-1836), then Professor of Mechanics
in Paris, presented a memoir giving the equations for vibra-
tory motion of an elastic medium composed of particles acting
on one another with forces directed along the lines joining
them, and proportional to the product of displacement and
initial distance between them. This paper for a particular
elastic solid was followed by a series of works by Cauchy
(1789-1857), Poisson (1781-1840), Green (1792-1841), St.-
Venant (1797-1886), Stokes (1819-1903), Lord Kelvin (1824-
1907), Lord Rayleigh (1842-1919) and others during the nine-
teenth century.

Their researches were carried on not only in an attempt

to discover the laws governing vibrating bodies, but to un-
derstand the nature of light, the transmission of which was
believed due to the vibrations of a perfectly elastic aether.
Thus, many of the early studies of stress wave propagation in
an elastic medium were prompted by an interest in electro«
magnetic phenomena. The twentieth century opened with our
understanding of the governing equations for longitudinal
waves (irrotational dilatation), transverse waves (equi~
voluminal distortion) in extended elastic bodies and Rayleigh
surface waves in the form known to us today.*

In an extended elastic medium obeying Hooke's Law,

longitudinal waves are propagated with a velocity

c = M it (1.1)

p

where A and M are Lame's constants and p is the mass den-
'sity, whereas one dimensional wave analysis applied to longi-

tudinal vibrations of rods yields
c = ___. (1.2)

where E is Young's Modulus.

 

*For a review of the early history of elastic wave propaga-
tion, see:

Whittaker, E. T., A History of the Theories of Aether and
Electricity, Vol.4I, Nelson, London, 1951 and Harper,
N. Y., 1960, Chapter V.

 

 

Love, A. E. H., The Mathematical Theoryof Elasticity,
Dover, N. Y., 1944, Introduction and Chapter XIII.

 

tc
te

in

HM

This latter result is only approximate since we assume
that plane sections of the rod remain plane and the stress is
uniform across the section.

As physicists gave up their quest for the elusive aether,
interest in stress wave propagation slackened. However,
technological advances began making use of metals and other
materials past their proportional and elastic limits and into
the plastic region. It also became apparent that many ma-
terials of interest exhibited mechanical properties under
conditions of dynamic loading which differed significantly
from the properties determined during static loading tests.

L. H. Donnell1 (1930) introduced the first scheme for
treating longitudinal wave propagation in a medium with a
stress-strain relation deviating from Hooke's Law. Stress
waves in a long bar were analyzed by a superposition tech-
nique in which the stress wave was treated as a succession
of incremental steps in stress. Each increment was assumed
to travel at a velocity determined by the slope of the ma-
terial static stress-strain curve at the stress level of the

increment. The wave velocities thus obtained are

c= I (’0' ‘5‘ (1.3)

which reduces to Equation (1.2) for a Hookean material.
World War II brought a surge of interest in elastic-
plastic wave propagation. Studies were made in the light of

developments in armor-piercing shells and armor plates. The

er

all
Shc

as

in;
(lot

the

plas

problem was treated independently by von Karmanza3 (1942)

in the United states, Taylor4 (1940) in England, and
Rakhmatulins (1945) in Russia. The von Karman-Taylor theory
assumes a single-valued strain-rate~independent stress«strain
curve which is concave towards the strain axis (thus pre~
cluding the possibility of shock waves being built up) and
assumes that radial inertia effects are negligible.

Whereas von Karman used Lagrangian co-ordinates, Taylor
treated the problem using an Eulerian co-ordinate system,
but later showed that by a suitable transformation, the two
solutions are identical.

Experiments were carried out shortly after the develop-
ment of this theory by Duwez and Clarkb. The results showed
some discrepancies from the predictions, which it was sug-
gested might be attributable to strain-rate effects in the
material.

The hypothesis of material strain-rate dependence had
already been offered. It had been suggested that stress
should be considered as a function of strain rate as well
as the level of strain as early as 1909.

Among the proposed functional relationships was a logaru
ithmic relationship suggested empirically by both P. Ludwik7
(1909) and H. Deutler8 (1932). L. Prandtlg (1928) reached

the same conclusion as the result of a physical theory of

plastic flow. The relation may be written

0(e'é)=0;(€)+k1né (1,4)

 

I i

01

wh

str
the

by

the.
The

rate

Thus

Dias

where 0’; (E) is the stress at a strain of 6‘ when the strain
rate, é", is unity. The factor k could be a function of
strain.

Another relation which has been proposed is a power law

of the form
0'<€.€)=0;<€)é‘n (1.5)

where n may be a function of strain.

In a more general form the relation can be written

0' =¢<€puépo (1.6)

where the subscript refers to nominal plastic strain and

10’11 developed a one-dimensional

strain rate. L. Malvern‘
theory for longitudinal stress-wave propagation as in a rod,
by rewriting Equation (1.6) as

O

Eo€p=g(0’,€) (1.7)

where EO (Young's Modulus) is introduced for convenience.
The elastic components of the deformation are considered

rate independent, and hence we obtain
0

5,66 =G’ A (1.8)
Thus, the constitutive equation which is the flow law when

plastic deformation occurs is given by

0 O

€=ce+ép
(1.9)
Eo€=0‘-tg(0’,€)

N

th

Wh

tn

Pl;
to

rat

On ,
311C]

0ft

Malvern gave a numerical solution for the case of a linear

strain-rate-dependence

0' =00 “it", (1.10)

where (76 represented the static stress-strain relation

0-,=f<€)
Thus

6'

p k [04(5)]

and

Eoé=&'+k[0'-f(€)] (1.11)

The relation f(€f) was chosen to represent approximately

a hardened aluminum alloy, and one result of the solution

is that small plastic strains propagate at a velocity greater
than that predicted by the strain-rate-independent theory,
whereas larger strains are progressively retarded.

The special case C76 = constant had previously been
treated numerically by Sokolovsky.1‘2'll
Most experimental work has shown the formation of a
plateau of uniform strain at the impact end of rods subjected
to constant velocity loadings, as predicted by the strain-
rate-independent theory. The numerical solution of Malvern

for a strain-rate-dependent theory, which was carried out

on desk calculators, did not indicate the formation of any

such region of uniform strain in the first 100 microseconds

of the impact.

 

Th
ga
Bel
pre
inc
ind
pul
the
eXp
3P?!
Howe
and

9135

Drop.

the l

13

Experiments by Bell‘ on steel bars and Sternglass

and Stuart14 with copper strips involved the propagation of
incremental impact loads superimposed upon static loads in

excess of the elastic limit. The wave fronts were found to
propagate at the elastic wave velocity and not at the lower
speed predicted by von Karman theory.

15

Alter and Curtis subjected lead bars to impact loading

16 with a stepped increase in

using a Hopkinson Pressure Bar
diameter. Due to reflections within the bar, the result was
a plastic preloading to the lead followed by a second impact.
The wave front of the second disturbance was found to propa-
gate at the elastic wave velocity. More recent studies by

Bell and Stein17

of incremental loading waves in dynamically
pre-stressed aluminum, using a similar set-up in which the
increment exceeded the original elastic limit of the material,
indicated that only the initial portion of the subsequent
pulse travelled at the elastic wave speed. The remainder of
the pulse appeared to propagate at the plastic wave speeds
expected from rate-independent theory. All these results
appeared to contradict the rate-of—strain independent theory.
However, other studies of wave propagation in lead by Bodner

and Kolsky18

suggest that lead should be treated as a visco-
elastic material for small amplitude plastic waves.

Attempts to establish a physical basis of plastic wave
propagation in crystalline solids based on the laws governing

the generation and motion of dislocations have been made by

 

nc
tr
ra

nu

ma

the
st)
sh:
The
Mir.
men
Pen
Str
rat:
Strz

the

tod

a1Um

10

. 1 l 4
Campbell, Dorn, Hauser, Simmons, et a1, 9 2 They have shown

that Equation (1.9) is a good approximation for the material
constitutive equation. The experiments conducted by this
group using a Hopkinson split pressure bar device showed that
g(0’,€) was independent of stress and strain histories, but
not a simple function of CTFf(ET). In discussion of these
theories, Dorn has said that they suggest a greater strain
rate effect in pre-strained aluminum than in annealed alumi-

num, and this agrees with their experimental observations.23’24

25m27 has

Using a diffraction grating technique, Bell
made studies of constant velocity impact on annealed aluminum
bars and found agreement with the strainurate-independent
theory on the basis of a "dynamic" rather than the static

28 made studies of

stress-strain curve° Kolsky and Douch
short bars of pure copper, pure aluminum and aluminum alloy.
They found no appreciable strain rate dependence for the alum
minum alloy. For the copper and pure aluminum their measure~
ments indicated a rate-of-strain dependence, but a rate—inde~
pendent theory gave reasonable agreement if a single dynamic
stress-strain relation appropriate to the actual range of
rates of straining in the test was used. The copper at low
strain rates did, however, exhibit a strain rate effect of
the nature predicted by Malvern.

Lindholm,29

in a series of tests in which short (length
to diameter ratios from 0.2 to 2.0) specimens of high purity

aluminum were subjected to strain cycling at widely variant

11

strain rates, has shown that the prior strain rate history
of the specimen has a significant effect on plastic flow be-
havior when reloading is at a high strain rate. Dynamically
reloaded specimens indicated an annealing recovery effect
with a characteristic time on the order of seconds. These
findings do not agree with the concept of a single dynamic
stress-strain relation.

A strain-rate-dependent theory would result in the
higher increments of strain in a pulse being propagated at
slower speeds than predicted by the rate-independent theory
based on the static curve. Such apparent slowing has been
observed?0 but it is the contention of some researchers
that the apparent decrease in wave propagation speed for
large strains is due to the failure of the measuring devices
to faithfully follow the deformation.

Strain gages of both the wire and foil type have been
used to successfully monitor "static" strains into the plastic
range, but controversy still exists as to their ability to
respond accurately to large strains at high strain rates.

It has been suggested by Bell31

that the strain rate depen-
dence indicated from earlier wave propagation experiments
was due to a lag in the gage response. When compared to
measurements made with his diffraction grating technique,
he found that wire resistance strain gages gave errors which

were related to the maximum slope of strain-time curve.

Tests at a strain rate of 1000 in/in/sec and a maximum

12

amplitude of 2.5% indicated errors of 26% at a point onem
half inch away from the impact end of a one-inch diameter
specimen. At a distance equal to 3% diameters from the
impact end (with a much lower strain rate) the error was on
the order of 10%.

The problem at hand is to determine the three dependent
variables (stress, strain, velocity) in terms of the two
independent variables x and t. With the exception of the
techniques of dynamic photoelasticity with birefringent
materials or use of the Hopkinson Pressure Bar, we are re»
stricted experimentally to techniques for measuring strain
or particle velocity. The higher the strain rate and strain
magnitude, the higher the required frequency response of the
transducer.

For further background information concerning dynamic
stress-strain relations and anelastic stress waves, the
reader is referred to references 32 through 35 as listed in

the Bibliography.

 

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im
at
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are
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adi

CHAPTER II

FUNDAMENTALS

2.1 Strain-Rate-Independent Theory

The rate-independent theory of von Karman was derived
for a long, thin unstretched wire subjected to an impulsive
tension load at one end. The analogous treatment for an
impulsive compressive load, also using Lagrangian co-ordin-
ates, is described below. The governing equations for one-
dimensional longitudinal stress wave propagation in a bar
are obtained by assuming that plane sections remain plane
and that the stress is uniform across them. Lateral inertia
effects are assumed negligible. These assumptions make the
one-dimensional theory incorrect in the immediate vicinity
at a suddenly impacted end of the bar, since, as Bell27
has shown, a three-dimensional wave pattern exists there.
For this reason, in comparing experiments with one-dimen-
sional theory, we will take input data from the first gage
station three inches from the end.

Lagrangian Co-ordinates will be used. Let u(x,t) be
the displacement at time t of the cross section initially at

a distance x. Loading occurs at the section x = 0. Then

€= % (2.1)

v = -%¥ (2.2)

13

14

where he is the strain and v is the particle velocity at the
section under consideration. Compressive stress and strain
are reckoned positive, while displacement and particle veloa
city are considered positive when they are to the left
(negative x-direction). In the cases treated in Chapter IV,
the compressive wave moving to the right produces negative
displacement and velocity.

Differentiating the first equation with respect to time

and the second with respect to x, we obtain the equation of

continuity
b6 av
bt -. bx (2°3)

The equation of motion for an element of the bar gives

If strain is assumed to be a single valued function of

stress, we can rewrite (2.4) as

Ozu = fl bzu

. (2.5)
at2 de 3x2

which we recognize as the one dimensional wave equation for

 

waves propagating with the velocity c =‘\[—3%: p . One

obvious solution is
u = vlt + 61x . (2.6)

which corresponds to a constant velocity impact at x s 0

on a semi-infinite bar and from Equation (2.1) represents a

 

an
in

poi

at
32

int

15

constant strain 61.

d C7
(1 €
nitude 61 will propagate at a speed. c1 given by

 

Letting S = we see that a compression wave of mag-

2
2 x S
c = =1— (2.7)
1 t2

where S is evaluated at E = 61.

Thus, the complete solution requires the consideration
of three regions for the case of constant velocity impact
at x = 0.

(a) 0<x<c1t where €=€1
(b) c t<x<cot where the relation 9%- =‘\/B— holds

1

and co is the elastic wave propagation velocity

(c) x) cot where E = 0

In the field of the solution [region (b) above] , C7'and
6 are positive, decreasing toward the right, while u and v
are negative, increasing toward the right (i.e. decreasing
in magnitude), and v is decreasing with time at any one
point (increasing in magnitude).

Unless the constant velocity imposed at x = 0 has a
finite rise time from v = 0, we have a discontinuity in strain
at x = cot. Here, S = E0 (Yourg's Modulus) and we have A

2

Bo. . .
co = ZT'Wh1Ch 15 the result in the elastic case.

Fig. (2.1) shows the relation between 6 and B =%
in the three regions.

16

 

 

 

 

o b-——-—

a .26

, °o (3 4;

no (2.1) SCHEMATIC REPRESENTATION OF
THE STRAIN DISTRIBUTION our-z
To CONSTANT VELOCITY IMPACT

IN A RATE INDEPENDENT MATERIAL

17

242 Strain-Rate-Dependent Theory

In addition to Equations (2.3) and (2,4) a third equa-
tion is provided by the constitutive equation of the material.
We will assume the flow law in the general form given by

Equation (1.9)
so. g—f— = Pr??- ‘+ g(0’,'€) (2.8)

Thus, we now have a system of three quasi-linear first
order differential equations. Although the equations are
linear in the derivatives, the term g(0’,€) may be non-_
linear in G'and € ,

The system is rewritten

-_fi_ = 0 (2.9)

which we see is of the form

i i Summed i l,2,..,,N
Lk [11] = akiux + bkiut = Gk . (2,10)
N Eqs. k l,2,..,,N

where the subscripts x and t denote partial derivatives.
The system will be shown to be hyperbolic and hence

suitable for numerical solution by the method of character-

37’38 We seek a combination

istics.
L2 AkLk = 41:91: Summed k = l,2,..,,N

(2.11)

18

such that it represents interior differentiation in only

X
one direction, the direction given by the ratio ‘3? =.¥2_ ,
P

where the subscripts denote differentiation with respect to
the parameter p along the curve being sought. If such a

direction exists, we have

 

dx
Akaki 33
Xkbki 9.1
(113
Thus,
Ak(akidt - bkidx) = o (2.12)

which represents N equations for i = l,2,..,,N.
Returning to Equation (2.11) we multiply by dx to ob-
tain

de as AK aki dxnx1 t Ak'bkid’i 11t1

and then substitute from Equation (2.12) to obtain

i i
de Akakiux dx + Akakiut dt

A kakidui = AkadX

with the final result

Ak(akidu1 - dex) = 0 (2.13)

Equation (2.11) is now multiplied by dt and the pro-
cedure repeated to obtain an additional relation

Akwkidul-det) s o (2.14)

19

Equations (2,12), (2,13), (2.14) are N + 2 homogeneous
linear algebraic equations for the N multipliers )Ak, For
these equations to be satisfied by a non-trivial solution,
it is required that all the NxN determinants of the co-
efficients of Ak vanish,

The determinant obtained from Equation (2.12) will de-
fine the characteristic base curves in the base x,t-plane,
while the others furnish the interior differential equations,
holding along the characteristic base curves.

For the system of Equations (2.10) we have

q I. . I 1

 

 

 

 

 

 

IO’ 0 o o -1 E00
ui= 6 an: 00-1 bu: 010
v 1 0 ol 0 0-D
_- 4
3(0’,€).
Gk: o
o

 

 

From Equation (2.12)

A1(a11dt-b11dx) + 12(a21dt-b21dx) + A3(a31dt-b31dx) = 0

I
O

A1(‘113‘1t‘1’13d’0 t 42(323dt-b23dX) + II 3(a33dt—b33dx) = o

 

SUI

and

WhEI

wave

Substituting from above

20

(dx)}t1 41(0)).2 + (aw/I3 = o (2.15)
C-Eodx)Al + («m/12 + (0M3 = o (2.16)
(0) A1 + (-dt)/I2 + (pdx)/{3 = 0 (2,17)

and we require

dx
-Eodx

O

 

0

-dx

-dt

dt

0

l)dx

 

= dx(-pdx2 + Eodtz) = 0

Hence, the characteristic curves are

dx

dx

dx

0
+cdt (2.18)

-cdt

- [E
where c = 2; represents the speed of propagation of the

wave front,

Since there are as many distinct families of character-

istics (all of which are real) as the order of the system,

the system is completely hyperbolic, The method of charac-

teristics is therefore applicable for the solution.

Equations (2.13) and (2.14) are

(-gdx)/\1 + (-dv)/\2 +(c10')A3 = o (2.19)

("dJ‘i' Bode ~ gdt)A1 ‘I' (d€)A2 + (-pdV)A3 = O

(2.20)

 

an

 

dx

OI

sin

tion

 

Thus]

21

The determinant formed by Equations (2.19), (2.16)

and (2,17) is

-gdx -dv dCr

2 ._
-Eodx -dx 0 = dx(EodC7dt + pgdx - EOKDdxdv) _ o

 

 

0 -dt pdx

We now examine the conditions required for the bracketed
term to vanish along the characteristic curves. Along
dx = cdt we have

2
Bodo’dt + pgczdt - Eopcdtdv = 0

or
d0' - p cdv = -gdt

since

>_ 2
Eo -,£)c

Along dx = -cdt we similarly obtain
dCT'+l)cdv = ~gdt
Finally, we consider the determinant formed by Equa-

tions (2.15), (2.20) and (2.17)

 

 

dx 0 dt
-d0’+Bod€ -gdt- ..d€ -p dv- = dx(pd €dxspdvdt)
0 -dt pdx +dt2(d0’-Eod€+gdt) = 0

Thus, along dx = 0, we have

dG’ - Eod€ = -gdt

 

 

and

are

C)

pli
HOW
pro

to

22

The differential equations defining the characteristics
and the interior differential equations holding along them

are summarized below,

Characteristic Interior Diff, Equation

Diff, Equation Along the Characteristic
dx = cdt d0’ -‘ pcdv = -g(0’,€)dt (2.21)
dx = -cdt dO’ + pedv = -g(0’,€)dt (2.22)
dx = 0 do' - Bode = -g(0',€)dt (2.23)

The form of g(0‘, 6) will not in general permit an ex-
plicit integration of the interior differential equations.
However, the system can be treated by numerical integration
procedures. For this purpose, the following transformations

to non-dimensional variables is introduced.

3:30...
0

13:6

Vzi
co

T = kt

Xzix
co

_ 1
G - TEE-g 8(0'.€)

/E
where co = 2;. is the elastic wave propagation speed and

o -1 o u
k has units of sec and a magnitude chosen for convenience

 

depe

tion

tics
solu1
of tr
three

tics,

shock

by th

"IEre

and ,9
from e
of an .
salts 1

23

depending on the form of g(CT,€').
The characteristic curves and interior differential equa-

tions are now given by

dS - dV = -GdT along the curve dX = dT (2,25)
dS + dV = -GdT " " " dX = -dT (2.26)
dS - dE = -GdT " ” " dX = 0 (2.27)

From Fig. (2.2) we see that there are three characteris-
tics passing through each point in the X,T-plane. Thus, the
solution at any point P can be obtained if we have knowledge
of the dependent variables at points A, B and C by solving
three difference equations along the appropriate characteris-
tics.

The conditions across the leading edge of an elastic
shock wave traveling in the positive direction, represented

by the line x = cot in the x,t-plane are

A 0’ = -~p coAv (2.28)
A v = - COAE (2.29)
A 0": pcOZAG‘ = EOA€ (2.30)

where A7, A6 and Av are the jumps in stress, strain

and velocity as the wave passes. The first condition results
from equating impulse to change of momentum for the traversing
of an element of the bar by the shock wave, The second re-
sults from continuity of the displacement across the shock,

and the third follows from the first two and co =1/E9- .

 

Z4

 

 

 

 

 

 

 

 

__111 x

FIG (2.2) THE CHARACTERISTICS IN THE x,T-PLANE

 

Th1
cec
ju:

qu

alo

tio:

all
tha1

Stre

one

the

use I

91131

25

Thus, in a semi-infinite bar with an undisturbed region pre»
ceding the shock, we obtain 0': pc02€ = -pcov on x = cot
just after the shock passes, and the interior differential
equation along the characteristic can be integrated to obtain

(see Malvernll)

do...

   

6', (2.31)

along x = cot where C70 = ~[Dcovo is the stress at x = 0,
t = 0.
If there is no shock wave, but rather a gradual transi-

tion from an elastic to a plastic stress wave, we have

0' = 0’}, = ,OCOZE' = -pcov

all along the curve x = Co(t-ty) where t = ty is the time
that the loading at the boundary x = 0 reaches the yield
stress Cry.

For the case of point P along x = 0, we assume at least
one of the dependent variables to be prescribed and hence
the equations along X = 0 and dx = -dT will be sufficient
for solution,

In writing the finite difference equations, we must
use the average value of G along the element of the appro-

priate characteristic curve.

26

2.3 Numerical Procedure

 

Rewriting Equations (2.25), (2.26) and (2.27) as dif-
ference equations along the appropriate characteristics as

shown in Fig. (2.3) we obtain

II
I
D
o-a
a)
.0

I
I
H
C)
'0
N‘l'N-l-
G)
0

(SP "' Sa) " (VP - Va)
(Sp "' Sc) + (Vp - Vc) - (2.32)

-(SD - sb) + (up - Eb) = AT(Gp + Gb)

These equations are solvable by iteration techniques

in the form

(3,1 - s.) - (v,i - v.) = -%AT(Gpi'1 + Ga) (2.33)
(Spi - Sc) + (vpi - vc) = -%AT(Gpi'1 + cc) (2.34)
-(spi - Sb) + (upi - Eb) = [jrccpi‘l + Gb) (2.35)

We begin with an initial guess for the value of Gp
i i i . i _ i i i
and solve for SD , Ep , Vp after which Gp - Gp (Sp ,Ep )
may be evaluated,
i+l
p 9

the iteration continued until the new values of Sp, Ep, and

Vp differ by less than some pre-determined amount from the

The process is then repeated to obtain S etc, and

values in the preceding iteration, Three types of “typical
points" must be considered for a wave propagating in a semi-
infinite bar, or before reflections occur in a finite bar:

(a) general interior point, (b) impacted end (x = 0) and

Z7

 

 

 

 

 

4
V

C") P viz,

2‘1

A

B
(A?
, 345.

 

 

"" X

FIG (2.3) FINITE DIFFERENCE GRID USED IN
NUMERICAL SOLUTION

28

(c) plastic wave front.
(a) General Interior Point

The equations for a general interior point are

1 s, - va - %AT(Ga + Gpl'l)

U)
Ho
I
<
H

U)
’U
H
+
<:
'U
h

i - sC 4- VC - t-Arcec + Gpi‘l) <2-36>
‘S i + E i 2 ”Sb + Eb + [XT(Gb + Gpi-l)
the solution of which is

s1 %(D +D

; 1-1
a C) - .ATCP

p

Epi = in)a + DC) + Db + %Z)TGpi”1 (2.37)
i _ 1

vp - sC + vC - 2A‘TGC

where

._ .1.

pa — sa - va - ZZXTGa

Db = -Sb + Eb-+ [Ircb (2.38)
_ .2».

DC - sC + vc - ZZXTGC

(b) Impact End Point
Since we are not involved with any reflections from
the striker bar, we need only consider propagations along
dX = 0 and dX = -dT at the boundary X = 0. Thus, we will
have only two equations available. We consider individually

the solution for three possible boundary conditions.

 

pa:
In
011

thI

The

The

The (

Solutj

29

These boundary conditions are described here for the im~
pact end of the bar, according to the one-dimensional theory.
In the solution presented in Art, 4,5(b), we will use the velo~
city boundary condition applied at the first gage station
three inches from the impact end.

(i) Velocity Boundary Condition: V(0,T) = Vo(t)

The equations for this point are

i i .. 1-1
s +v _Dc-tAer

p p
-spi + Epi = D1) + Arcpi‘l (2.39)
vp = v0

The solution here is

i _, 1-1
sp - (I)C - v0) - é-ATGP

i _ __ 1-1
Ep - (D0 + Db v0) + tA’er (2.40)
vp = vo

(ii) Stress Boundary Condition: S(O,T) = So(t)
The equations for this boundary condition are

S + v i = D _ 11"].
p . me.

p
-sp + up1 = D1) + Arcpi‘”1 (2.41)
sp = so

solution of which yields

3O

sp = 50

Tip1 = Db + 30 + Amp“1 (2.42)
i i-l

vp = Dc - SO - %ATGP

(iii) Strain Boundary Condition: E(0,T) = E1(t)
This final boundary condition enables us to obtain the solu-
tion S(O,T) for all T by solving Equation (2.27) which is
now an ordinary differential equation for S in the indepen-

dent variable T, For the complete solution, we write the

equations
Spi + Vpi = Dc .. %ATGpi'1
-5p1 + up = ob + [tropi'l (2.43)
Ep = El
whose solution is
sp1 = (-1)b + E1) - [)TGpi'l
Ep = E1 (2.44)
Vpi = (Db + I)C - E1) + g—A'rcpi'l

(c) Plastic Wave Front
For an impact loading with a finite rise time, the
conditions at the leading edge have already been described

and transformed to

31

along

where ty is the time at which C7'= Cry at x = 0,

A shock wave propagating along X = T is treated by
transforming Equations (2.28), (2.29) and (2.30) to obtain
[35 = -[Xv , (2.45)
[5v = -[§E (2.46)
As = AB (2.47)
Rewriting Equation (2,25)

_dT = dS - dV 2ds

G = G
bt '
W8 0 am S d
_1 = 5
2T ETETES (2°48)
S(o.o)

along X = T which is the transformation of Equation (2.31),

2.4 The Iteration Scheme

A system of equations

f (x x ,,,., x ) 0 k = l 2 ,..,m
{ k 1' 2 n } ' ' (2.49)
is called normal if m = n. If the system is reduced to the

form

‘(xj = QDjCXI. x2..... Xn)} j = 1.2.....n (2.50)

we can use the method of iteration to construct a series of

solutions

32

by means of the formula

{le‘I’l = ¢j<x11’ le'...'xn1)} i : 1,2,...,r_1 (2.51)

where, under certain conditions, the solution may be made
as accurate as one pleases for sufficiently large r.

The iterative scheme for non-linear equations will al-
ways converge if the following two conditions are satisfied.39

1. Denoting the system solution as
..}.I .}
I. or.

it is required that ‘{xji} be "close" to the solution with

the degree of proximity determined by the functions {¢j} ,
2. The second condition, which is the only one re-

quired of linear systems, is associated with the Jacobian

matrix of the system

' MIDI .3911 5951

 

 

 

OX1 3X2 an
M132
J{Xj} = 9x1 . : (2.52)

0 o

b¢n o 3" a‘ _.' . ‘1 . 0‘. a¢n
EIXI I b):

L -.

 

 

 

33

For a linear system the elements %§za~ of the matrix are
x O
J

constants, whereas for non-linear equations they will, in
general, be functions of x1, x2,,,,, xn,

A necessary and sufficient condition (if the first con-
dition is satisfied) for iterative convergence is that all
the eigenvalues of the matrix J evaluated at {xi} = {(Xj}
have moduli less than unity. This is a difficult condition
to check. It can only be hoped that if the moduli of the
eigenvalues of J {xjg} are less than unity, the eigenvalues
of J {0(j} will be likewise for {xjo} sufficiently close
to {ozj} .

An alternative, less difficult, sufficient (but not
necessary) condition for convergence of the iteration process
is that the sum of the elements of every column or every row
of the Jacobian matrix J( a1, “2“... an) be less than

unity. Thus, at least one of the following two systems of

inequalities must be satisfied.

 

 

Z bx <1 (2.53)
5= 1 J
or
b - '
: l-Sigil < 1 (2.54)
i = 1 J
for

34

A second iterative scheme known as Seidel's method

carries out the calculations according to

X11+1 = ¢1CX11o X21: . - . » an)

- - i i
X21+l =¢2(X11+1, x2 ’ . . . , Xn )

(2.55)

xn1+1 = qanX11+1, X21+1, . . . ' Xn-11+1a an)

using in each line those values of xj1+1 which are available.

The conditions of convergence for Seidel's method are
different from the previous iteration method, and hence,
one scheme may converge while the other diverges for the
same set of initial estimates. It is however known that
when any of the inequalities of Equation (2.53) are satis»
fied, the speed of convergence in Seidel's method is more
rapid than for conventional iteration.39

In the numerical solution Seidel's method was utilized
and the grid mesh Size was determined by the choice of (ST.
As ZXT appears in the terms 52x3 it was chosen so that at
least one of the two sets of ineéualities (2.53) and (2.54)
were satisfied. For any given flow law the required incre-
ment Z)T will decrease with increasing strain rate.

An additional aid to improving convergence was to in-

corporate a technique knowm as "Aitken'sfs2 - process."

35

This permits an improvement in the solution after five itera-
tions better than that which would be obtained by the usual
next iteration, The method is particularly suited where
convergence is oscillatory or like that of a geometrical
progression, which appeared to be the case for the problem
under consideration,

The equations were programmed in Fortran for the CDC-3600
computer at the Michigan State University Computer Center.

The program is described in the Appendix,

CHAPTER III

THE EXPERIMENT

3.1 General Description

 

An adaption of a commercial Hyge shock tester was used
to apply a compressive impact load having a duration of
approximately 560 microseconds to aluminum bars of half-
inch diameter. Two independent series of tests were con-
ducted. In the first series, transient surface particle
velocity records were obtained at four stations along the
plastically deforming specimen by means of electromagnetic
transducers. In the Second series, etchednfoil strain
gages were used to monitor surface strains at the four sta»
tions.

Ideally, the strain and velocity measurements should
have been conducted simultaneously on each specimen. This
was impossible since the magnetic field would have induced
an error signal in the foil gages as they translated during
the passage of the wave,

The basic system is shown schematically in Fig. (3,1).
Both the transmitter and striker are made of 9/16 inch dia-
meter steel drill rod. The Hyge shock tester is capable of
accelerating its piston and a five pound mass to a maximum
velocity of 1200 in/sec in a distance of about 12 inches.
At this point the piston is decelerated to zero in an addiu

tional four inches, while the striker is free to travel in

36

37

n3 Em @2543 no 9.25:3 2.8 or.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

muxoz_ n
2. can 4 . 59235 535.24 .
52:0
“My Mn N.“
«av»; v _._ 32°
.7. ,_F N._ , __._
-E
c. ,. mi
ii \75
32. o 45%,...“ 23m...
334: zmzamam 55.52%:
ZOFFOO Kuhmmb XOOIm wG>IV

mun—:0 @2370 mun-23m

 

it

tr

sh
co

St

the
t6!
im
to

The

twi

the

mat!

late
V' H.
firaPr

{IOm

38

its Omring guides. When the striker makes contact with the
transmitter bar AB, a compression wave is propagated outward
from A to both the right and left. Since the striker is
shorter in length than the transmitter, the bars remain in
contact until the wave reflected from the left end of the
striker returns to the interface, A wave normally incident

to a free surface is reflected with a change in sign and hence
the outgoing compression wave returns to the interface as a
tension wave, Tension waves cannot be passed across the
interface and hence the striker moves to the left with respect
to the transmitter after the reflected tension wave arrives.
The length of the compression wave transmitted is therefore
twice the length of the striker bar.

We now turn our attention to the transmitter bar. When
the wave front arrives at B, the acoustic and geometric mis-
match will cause part of the pulse to be reflected and part
to be transmitted as a compression wave into the specimen,
‘The transmitter and specimen remain in contact until the
tension wave reflected from the nearest free surface, C,
returns to B, The specimen then begins moving to the right
and is caught in the cotton filled tube, Sufficient energy
is still trapped in the transmitter to cause it to also trans-
late to the right, but the bumper, which is formed of a
wrapping of plastic electrical insulating tape, prevents it
from making a second contact with the specimen,

Friction in the O-ring guides and the spring action of

 

the

int

110
of

Ion
Pet
was
turl

face

650(

piec

dens

ing,
from

the 1;

well

39

the cotton decelerator bring the specimen to rest without

introducing additional plastic strains.

3,2 Specimens

 

~ The specimens were all prepared from extruded Alcoa
1100 P aluminum (28 aluminum) bars with a nominal diameter
of 0.5005 inches. Each bar was on the order of 58 inches
long and had both faces turned to provide a flat surface
perpendicular to the longitudinal axis. A one-inch piece
was cut from each end of the bar before facing and this in

turn was turned to a length of 1.000 2 .005 inches and also

faced, Both bars and short specimens were then annealed at
650° F for one hour and furnace cooled,

Employing a chemical balance, several of the inch long
pieces were weighed, first in air and then in water. The
density of the aluminum was thereby found to be

p = 2.531(10"4 lb sec/in4.

Static stress-strain curves were obtained, after anneal-
ing, from seven of the one-inch specimens. Four of them came
from opposite ends of two specimens to serve as a check on
the uniformity of material properties along each specimen as

well as between specimens,

3.3 Velocity Transducer

It is well known that a current will flow in a conductor

 

I8

is

in

per

V911
Ear:

tral

4O

moved through a magnetic field (for example, see reference 40).
The relation between the voltage generated and the magnetic

field is

e=—f/§--v—xd-I. (3.1)

where [3 is the magnetic field vector,‘f'is the vector

representation of length measured along the conductor and'V

is the velocity of the conductor with respect to the field,
Figs. (3,2) and (3,3) show a permanent horseshoe magnet

in position about a bar so that the magnetic field is per-

pendicular to the longitudinal axis of the bar. Two Nyclad-

covered strands of #30 copper wire are attached to the bar

as indicated so that when the cross-section at which they

are located has a velocity imparted to it by the passing stress

wave, a current will flow in each. This method has previously

been used by Ripperger and Yeakley41 to detect particle

velocities in aluminum bars subjected to short elastic pulses.

Earlier efforts at developing a similar magnetic-inductive

transducer were reported by Ramberg and Irwin,42
If the field is constant over the entire cross section,

the effectivelength of the conductor is the diameter of the

bar, Thus,}9 ,

tudinal waves and Equation (3.1) reduces to the scalar

L, and'v'are mutually perpendicular for longi-

product

e=flLv

where e is the potential difference between points B and E

41

SINGLE STRAND
COPPER WIRE

LEADS TWISTED __
TOGETHER

  

‘QI

 

 

 

 

 

 

 

 

 

 

ALUMINUM BAR

 

 

FIG (3.2) SCHEMATIG or THE VELOCITY TRANSDUCER

42

+A

 
     

LEADS TWISTEDi
TOGETHE R

 

 

 

 

 

 

 

 

 

 

 

FIG (3.3) WIRE ARRANGEMENT AT GAGE SECTION

43

in each wire for L = 2R. Thus, the induced voltage is direct-
ly proportional to the particle velocity,

Points A, C, D and P are rigidly fixed and hence, the
four sections AB, DC, DE and BF can only rotate when the
cross section translates. With these sections moving in
horizontal planes, we find Eat-I: perpendicular to Vand hence
their dot product vanishes and no contribution is made to
the voltage generated in each loop, The sections AP and CD
are rigidly cemented to a non-magnetic jig and hence do not
move. Above points A and C the leads are twisted together
so that any vibrations or motions will cause no additional
signal to be generated, Two semi-circular loops of wire
are used to eliminate any possible effects of bending in the
bar.

The leads are connected as indicated in Figs. (3,2) and
(3,3) to the input of a Tektronix D-Unit differential pre-
amplifier, thus giving a two-fold increase in signal to

e = gfgdv (3.2)
where d is the diameter of the semi-circular loop of wire.

Pour such measuring stations were set up using two
sets of magnets from military-surplus magnetron tubes.

Fig, (3.4) shows the gage arrangements and Figs. (3,5) and

(3.6) are views of the physical set up.

3,4 Surface-Strain Transducer

Annealed constantan etched-foil resistance strain gages

44

 

mzoEfim mods
«Sou. 92552:. pzmzmozammal 5232 $8 or.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a a W E a

 

 

 

 

 

 

 

45

um

.,_.§‘.0v ..,

 

Fig. (3.5a) General View - Hyge and Test Set Up

46

shard!
. 1

.cVI

 

Fig. (3,5b) General View - Electronics and Recording Equipment

an
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48

manufactured by Micro—Measurement, Inc, were selected for

the dynamic tests, Type EP403-125CA-120 gages having a length
of 0,125 inch and width of 0,182 inch were chosen, Series

EP gages are rated as accurate to 7% strain under static
loading conditions, Dynamic strains on the order of 2-3%
were anticipated,

The gages were mounted in pairs, diametrically opposed
to cancel any effects due to bending, at four stations which
corresponded to the spacing used in the velocity transducer
tests, The gages at each station were connected in series
and then installed as one arm of a Wheatstone bridge,

Fig, (3,7) is a schematic of the strain gage bridge set up,

3,5 Static Test Procedure

 

The one-inch samples were subjected to compressive load-
ing in an Instron testing machine, Recording accuracy of the
Instron load measuring system was calibrated at better than
1% in the range of interest, Strain was measured to 5%
using etched-foil strain gages mounted on the specimens,
Three combinations of gages and cements were used as an added
check, Load was applied at a strain rate on the order of
4 x 10"5 in/in/sec, Two tests were carried out with continu-
ous loading while two others involved alternate loading and
unloading as an aid in the determination of the apparent

Young's modulus of the material,

49

 

    

ACTIVE
GAGES.

PASSIVE
RESISTANCE
TO

’oETECTOR S

TE MPERATUR PASSIVE .
COMPENSATOR RESISTANCE

V

STORAGE
BATTERIES

 

 

width—www—

 

 

FIG (3.7) SCHEMATIC 0F STRAIN GAGE BRIDGE

50

The test combinations are tabulated below,

 

 

 

 

 

 

Parent Micromeasurement

Test Specimen Gage Type Bonding Cement Loading
1 ll EP-03-l25AD-120 Eastman 910 continuous
2 11 EA-13-125AD-120 W,T, Bean RTC alternate
3 13 EP-03-125AD-120 '" n " continuous
4 l3 EA-l3-125AD-120 " " " alternate
5 5 A EA-13-125AD-120 Eastman 910 continuous
6 7 EA-13-125AD-120 " " "
7 8 EA-l3-125AD-120 " " i "

 

 

 

Strain was read with a Baldwin Type-N static strain
indicator, This instrument has a range of t 3%, In order to
use the N unit over a range from zero to minus five percent
strain, the device illustrated in Figs, (3,8) and (3,9) was
used,

The linearly tapered cantilever beam within the channel
is made of beryllium copper which, after machining, was heat
treated to a Rockwell hardness of C-43, This indicated a
proportional limit on the order of 75,000 psi and a yield
strength of well over 100,000 psi, Ultimate strength is
about 175,000 psi, Young's modulus is 15 x 106 pSi,

Due to the taper, the maximum bending stress in the

beam, when loaded by the screw, occurs at a point two inches

to the right of the support, With the beam loaded as shown

51

LOAD SCREw
OANTILEVER /

fin SCREWS \..,

-
3'

 

 

 

 

 

4'STEEI.
CHANNEL

FIG (3.8) RANGE EXTENDER- UNLOADED

 

 

 

 

 

 

 

'FIG (3.9) RANGE EXTENDER-LCADED

52

in Fig, (3,9), foil gages were mounted to both the upper

and lower surfaces of the beam at the critical cross section,
After appropriate curing of the bonding agent, the load was
removed and the gages successively coated with Gagekote 1,

2 and 5* to provide electrical moisture and mechanical
protection respectively, The beam was then rotated 180
degrees about its longitudinal axis, replaced in the fixture,
and reloaded,

The type N unit indicated a total of about 21,000 micro-
strain for the sum of the magnitudes of strains on the two
surfaces. Stability was excellent, Switching to four ex—
ternal arm operation, the fixture gages were used for one-
half of the bridge, The one-inch sample specimen with its
two 120 ohm diametrically opposed gages connected in series
and an appropriate temperature compensator completed the
circuit as illustrated in Fig, (3,10),

With no load on the Instron, the fixture was adjusted
to permit the N unit to be balanced at a reading of +32,000
microstrain, As the Instron loaded the specimen, the con-
.tinuous load curve traced by the machine was marked at pre—
-detérmined increments of strain and a load-versus-strain
record obtained until the N-unit indicated a reading of
-l8,000 microstrain, A stress-strain curve to 5% strain

was thereby obtained,

 

*Supplied by w. T, Bean, Detroit, Michigan

53

 

(a) Instron, specimen, temperature compensator,
range extender and N-Unit

  
  
  

120 .n. 240 n.
ACTIVE
GAGES LOWER CANTI LEVER
120 [1 SURFACE
240 .0. 240 .n.
TEMPERATURE UPPER CANTI LEVER
COMPENSATOR SURFACE

(b) Strain Gage Bridge Elements

Fig. (3.10) Details of the Static Stress-Strain
Test Set Up

54

In order to check the linearity of the Nuunit when
using the cantilever fixture, the active gage was replaced
by a second temperature compensator loaded elastically in
bending, A balanced reading of +3l,360 microstrain, i,e,,
a strain of 640 microstrain, was attained, The copper
cantilever was then removed and the N-unit converted to two
arm operation, balanced, and the reading recorded, The
load was removed from the active gage and the N-unit balanced
again, The change was found to be 640 microstrain which
agreed with the previously obtained result, Changes in con—

tact resistance were found to be negligible,

3,6 Velocity Transducer Calibration

Two pairs of magnetron magnets were rigidly mounted within
magnesium spacers to an aluminum plate, Fig. (3.11) shows
the general set up, The magnets were locked in place with
the pole pieces aligned along parallel planes one inch apart,

The field mapping was done at the M,S,U° Cyclotron
Laboratory, The system made use of a Rawson rotating-coil
flux meter which was mounted on the crOSSmfeed of a servo-
controlled milling-machine carriage, A volume with a grid
spacing of 0,1" was mapped within each pole gap,

The output of the flux meter is proportional to the
magnetic field intensity, This signal was passed through
a voltageuto-frequency converter and the frequency was then

counted, As a count was finished, a coupler (built by the

55

$5pr Booms"... 5.23;. no 2.25:3 3.3 or.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

56

Cyclotron Lab) read the digital output of the frequency
counter and the digits were then punched out by an: IBM
card punch,

Calibration of the system was performed by checking
frequency counts against a bridge which can be used directly
with the flux meter and a null galvonometer to obtain the
flux density to the nearest gauss for the fields being
mapped, The arrangement is described in Fig. (3,12),

The centers of the magnetic fields are shown in Pig.
(3,13), If the wired cross sections on the specimen are
located 0,1 inch forward of this point, a translation of
0.2 inch would result in a variation in the magnetic field
through which the wire passes of less than 1%.

However, the field was not uniform across any section.
Hence, the loop of wire was approximated by a circumscribed
dodecagon and we have

12 — — _.
60 = Z 931 X Aw 'v (3.3)
.___1

1

where £3: is taken as the value of [B at the midpoint of

[Xii. Referring to Fig. (3.14) we have

e, = 1-2- [<32 +36 +38 wows (3 4)
+(B3 +35 + 9 +Bll)cos 30° +34 +310“:-

The appropriate units are eO in volts, [3 in webers/meter2

(l weber/m2 = 104 gauSS), [XL (hence d) in meters.

 

 

 

 

 

 

 

RAWSON m m
ROTATING COIL , , A /\ _ _
FLUX METER l 1.

U 1.4 1.4. L4
_ ! F flfixé COIL
["‘1 l""1 [-1 n r
L- 7
‘ V TI l \ 7

I H

h“ 7 -

VOLTAGE 5:: I B N
7° CARD PUN
FREQC

'FREOUENCY
COUNTER

 

O
I d

c

BIIIG‘NC' NULL
- GALVO NONETER
BRIDGE I

FOR CALIBRATION

 

 

 

 

 

 

 

 

 

 

CROSS FEED
L * " .H MILLING
' .5 MACHINE
TABLE 1} CARRIAGE
1.. -
III-_—
L ’ ' I

 

FIG (3J2) FIELD MAPPING SET UP

58

 

 

 

 

 

 

 

 

 

 

 

 

 

2.5"

 

 

 

 

 

 

 

 

 

 

 

 

7T—
in

 

00' Jfi arqlfimiir‘,,,.

3.2 S" I 3.325" 3J5"

+ GEOMETRIC CENTER -
oMAGNETIC CENTER (APPROXJ

FIG (3J3) LOCATION OF MAGNETIC FIELD V
CENTERS

 

D. 0.5" "
'DDOI? DerII '

IO

 

 

7

AL APPROXIMATION
OF WIRE LOOP

FIG (3J4) POLYGON

59

Equation (3.4) can be solved for v in the form

v = keo (3.5)

The following table gives the constants obtained for

the four gage stations.

 

 

 

 

 

x (m/Eec/' Max. Variation [gayerage
Station, ,(in) volt)»r‘ in 0.2 inch (gauss)
+1 3.000 118.6 no.2 = -O.2% 3350
2 ' £56,250 131.0 -0.6 = -O.5% 3250
3 9.S75 6138.8 -o.4 = -o.3% 2850
4 12.725 . 120.8 -o.4 = -o,4% 3275'

 

 

 

 

 

 

 

Slight errors in either vertical or lateral placement
Of the wires would also result in variations in k on the

order of less than 1%.

3.7'IVelocity-Gage Tests

Circumferential scribe marks were made on the specimens
at the appropriate intervals with the first mark three inches
(Six diameters) from the impact end. The wires were then
bonded to the bar at the scribe marks using Armstrong C-4
epoxy cement with activator D. Curing was at room tempera-

ture for at least 36 hours. Heat from an incandescent lamp

60

was used to aid curing during the first 12 hours. .An alu-
minum jig served to maintain proper alignment during both
cementing and testing. Special spacers which prevented
lateral motion during cementing and moving were removed

just prior to testing. In order to minimize the possibility
Of vibrations Of the horizontal portions of the lead wires
[see Figs. (3,2) and (3,3)] mass was added at these sections
by applying a liberal coating of vaseline.

It was found that eddy currents were set up within the
specimen in the region between the pole pieces during the
passage of the stress wave, These currents modified the
magnetic field so that when an uninstrumented aluminum bar
was subjected to loading, a signal was detected in a free
hanging loop of wire placed near it. No such signal was
detected when a non-conducting polyethylene rod was substi-
tuted in place Of the aluminum. The wave form generated was
similar to that produced by a conventional magnetic pickup
and is shown in Fig. (3.15). The maximum amplitude of the
spikes was less than one-half millivolt and their presence
was never detected on any of the records from instrumented
specimens. '

A series of tests were run in which particle velocities
in the range 250 to 650 in/sec were obtained. The electronic
circuitry and recording equipment used also monitored the

strain gage tests and will be discussed later.

ARRIVAL OF
WAVE FRONT

 

61

START OF
UNLOADING

 

 

 

A

 

‘7"""

FIG (3J5) SCHEMATIC OF WAVE FORM
PRODUCED BY EDDY CURRENTS

62

3.8 Strain Gage Tests

TO increase the output of the strain gage bridge used
in the dynamic tests, the passive half of the bridge con-
sisted Of 352-ohm resistors in the form of foil strain gages.
The entire bridge was thus made relatively current insensi-
tive. With two 120-Ohm gages in series on the specimen and
a 240-ohm temperature compensator, a 50% increase in output
over a 240-ohm resistor bridge was attained. Two 12-volt
storage batteries in series powered the bridge and kept power
dissipation in the acceptable range for these gages.

The two 352-Ohm gages for each station were mounted
on opposite sides Of l/l6-inch-thick strips Of spring steel.
These were mounted as cantilever beams in the fixture illus-
trated in Fig. (3.16). The beam was loaded until the bridge
was nulled (possibly requiring a 1800 rotation of the beam
about its longitudinal axis). A sensitive center-zero gal-
vonometer was used for this purpose.

The Baldwin Type-N static strain indicator was used to
record the residual longitudinal strain by recording before
and after balanced readings at each gage station.

A digital voltmeter was connected across the batteries
to monitor voltage immediately prior to and after each test,
but was not in the circuit during the test itself to prevent
any possible noise in the system from this source.

A series of four tests were conducted in which the

maximum dynamic strains varied over the range from about

63

wucoaoam omomum umsoaun I mamauoa

AoH.mV .mms

 

64

0.5% to 1.0%. The first two were at the lower value, one
using Eastman 910 and the second W. T. Bean RTC epoxy as
bonding agents. No noticeable difference in response was
noted. Eastman 910 was then used for the tests at higher
strain levels because it simplified the specimen preparation.
All gages were coated to provide electrical insulation and

moisture and mechanical protection as in the static tests.

3.9 Electronics and Recording Equipment

 

Output from both the velocity and strain transducers was
fed to Tektronix D—unit plug-in differential amplifiers in
rack-mounted Tektronix 127 pre-amplifier power supplies.

The frequency response (t3db) of the D-unit is 350kc at

a gain of 100 and increases to 2mc at a gain of 2 when used
with the 127 power supply with the push-pull output cables
terminated in l70-ohms. Using single ended output reduced
the signal gain by half. -For all tests the D-unit was set
at 20 mv/cm, and the single ended output resulted in a gain
of 2%.

The signal from each D=unit was then input to a Tektronix
M-unit in a Tektronix Type 551 oscilloscope. The M-unit is
an electronic switching unit which enables four signals to
be displayed simultaneously on one beam of a scope. When
all four channels are used in the chopped mode, the switching
rate was found to be 960kc. This rate was found to be relia

ably constant and thereby provided a timing mark.

65

A Polaroid camera was used to make a permanent record
of each test. After the graticule markings were photographed,
the grid intensity was set to zero and the shutter locked
open in the bulb position with the scope set on single sweep~
external trigger. The trigger was provided by a barium
titanate element clamped to the steel transmitter bar. The
exact position was chosen so as to allow for any delay in the
scope sweep mechanism,

The entire system was calibrated prior to each test
using the internal square wave calibration signal in the
scope. Several photOs of square waves applied to each channel
revealed that "eye error" was less than the rated t3% cali-

bration accuracy Of the square wave.

CHAPTER IV
RESULTS

fig; Static Stress-Strain Curves

Five Of the one-inch sample specimens tested produced
load-Strain curves which varied by an amount on the order
Of the trace width of the Instron continuous pen recorder.
These are shown as a single curve, the lower curve in i I
Fig. (4.1). The two samples from parent specimen nO.ll
produced stress-strain curves which agreed with one anothér,
but appeared to indicate a condition Of work hardening when
compared to the curve from the other specimens. This
curve is shown as the upper curve in Fig. (4.1), but was
ignored in the least squares curve fitting in Obtaining
0;, = f( 6 ).

It should be noted that the slope of the upper cUrve
is very nearly equal to that Of the lower curve everywhere
except for the region in the neighborhood Of E = 0.005.

d0'

Since the slope 3?? will determine the speed of propagation
of any level of strain in strain-rate independent theory,
the speed predicted for most strain levels would still be
about the same for both curves, according to the rate-inde-
pendent theory. From the static tests involving alternate
loading and unloading, Young's Modulus was found to be

E0 = 9,4x106 psi and hence, the specimens had a predicted
elastic wave propagation speed Of 1.93x105 in/sec.

66

67

“CI I00: IDZEDJdv mmrrmmaomn. 22¢ka mmmmkm o_k<._.m
3:).3 2.415%

2.3 0....

 

co. no. No. .6. o
_ _ T .
«23.3% oz...
KI :62 53.33.

0

O.

 

N.

"8:! con ssaals

68
A best fit in the form
0’, = mm = A63 (4,1)

was Obtained for the lower stress-strain curve data for
strains up to two percent, as this is the range of the
strains in all the dynamic tests. The resulting power law

relation between stress and strain is
0’0: 39,4006‘0-366 (4.2)

The fitted curve of Equation (4.2) is compared in
Fig. (4.13) to the lower experimental static curve of
Big. (4.1). For strains in the range from yield to about
C = 0.001 in/in, the fitted curve exhibits a steeper slope
than the experimental curve, and therefore, the rate-inde-
pendent theory based On this fitted curve will predict higher
propagation speeds for these levels of strain than would be
predicted with the actual experimental curve.

A second fit was made using only the data between
6’: 0.005 and €’= 0.02. This gave the power law
0'0 = 29,40060'311.

4.2 Velocity Test Results

The translation of the bar during the passage of the stress
pulse resu1ted in strain in the horizontal elements AB, BC,
DE and,EF of Fig. (3.2) at each gage station. Translation of

two tenths of an inch results in a 10% strain in these

69

elements. The change in resistance of the copper wires
for the time interval of interest was small with respect
to the one Megohm input resistance of the Duunit,

Each specimen translated about one foot before being
brought to a rest by the braking action of the cotton filled
tube. The copper wires would invariably shear at the point
where they enter the aluminum support jig, The epoxy bond
appeared to hold up well except for occasional yielding at
the points B or E indicated in Figs, (3,2) and (3.3),

Fig. (4.2) shows three typical oscilloscope trace rem
cords obtained with the velocity transducer. Fig. (4,2a)
illustrates a test in which gage failure occurred after the
sweep was completed. The gain for gage station one in the
test was half of that for the other three channels, and each
station has a different calibration factor due to differences
in magnetic field strength. In Fig, (4,2b) we note slight
disturbances occurring simultaneously at stations one and
two and later at station three, The final trace record,

Fig. (4.2c), illustrates transducer failure occurring first

at the magnet forming stations one and two followed by failure
at stations three and four. Failure occurs with catastrophic
suddenness,and its onset is thus readily detectable,

The initial step, which propagates with no attenuation,
is the leading elastic wave. This is then followed by the
moreslowly rising plastic stress wave. The reduced data

in Fig, (4.3) for specimen No.7 shows that a nearly constant

7O

station
one

I"

.. » I fle'i'o I I Irr‘sI'Ist'I -I I‘rIALHI/IFF>HH H—H—A—o—Hr» 4O mv

20mv all others

 

 

PIG. (4.2) TRACE RECORDS - PARTICLE VELOCITY TRANSDUCER
(SWITCHING RATE 960 KC)

71

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8.33 m...» _ _

mom owe com oioh om.

    
   
 

 

 

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(OBS INI I AIIOO'IBA 3'1OILUVd

72

level of final velocity is reached, but in general, this
value showed a slight decrease with increasing x. Station 4
did not appear to have reached equilibrium in any of the
tests.

Negligible variation was noted in the speed of wave
propagation for a given level of velocity on any single
specimen. A larger variation was found between specimens.
This scatter did not appear to be associated with the maxi-
mum velocity (and hence, strain rate) of the specimen as is

shown in the table below.

 

 

 

Particle Max. Variation in Wave Prop. Vel. Number
Velocity of
m/sec Along Spec. Between Spec. Specimens

2 19.6% 18.5% 5*

3 12 % 37.5% 6

4 12.5% :5 % 6

5 :1.5% 35.7% o

o 1-2 % :2 % 4

7 32.5% :3 % 4

8 12.5% 34.7% 3

 

 

 

 

 

 

*The reading Obtained from one spECimen, in the region of the
"knee," was discarded. The slope was small and hence, the
errors in reading horizontal distances between traces was

arge.

73

NO data is presented for particle velocities representa-
tive of the elastic range of the material, since a variation
of i1 microsecond (the approximate limit of trace reada-
bility) in horizontal distance between station records
represents a variation of about 28% in wave propagation speed.
The data for all these tests was averaged and the rela-
tion between c and v thus obtained is shown in Fig. (4.4).
The elastic wave speed appears to be 196,500 in/sec giving

E0 = 9.77): 106

psi for the apparent dynamic Young's Modulus.
This is approximately four percent greater than the value,
E0 = 9.4x106 psi, determinedrfrom the static stress-strain
records.

The vertical bars indicate the scatter in the experimen-
tal records. There was virtually no scatter for values of
particle velocity up to about 50 in/sec. .An examination of
Fig. (4.3) shows that between particle velocities Of 50 in/sec
and 100 in/sec,, the velocity versus time records exhibit a
region in which we have an inflection point. This region of
rapidly decreasing, then increasing, slopes increases in
length at successive stations. Determination of wave propa-
gation speeds here, for any level of particle velocity, is
therefore subject to maximum error and this explains the
greater scatter in this region.

This curve was examined in light of the von Karman

theory by taking the derivative Of Equation (2.1)

74

away—00mm >2004w> x5 30¢... <h<o ouo<¢m><
20 omwdo. >.—._OO..m> m40_h¢<m m) ommam ZO_P<0¢A_O¢& m><3 afiev or.

.uum\z: >2004u>
OOo 00¢ 08 OON . OO. O

 

f

A a . 4 a . a a i a a a

5:3.» 44:61:25 H

 

O

0
III
(OBS/NI g0!)

 

03368 NOIAVOVBOBd 3AVM

in the form

D

n}
C>
<:
‘0

 

= = __._ 4,4
c(€,) .QQE. [SV' ( )
dE

Thus, Fig. (4.5) of g—g—I’versus 6 was Obtained, This curve

was integrated with respect to 6' and the resulting dynamic
stress-strain curve is plotted in Fig. (4.6) with the pre-'
viously Obtained static stress strain curve.
In Fig. (4.7) the 6 versus v curves are compared for
(1) von Karman theory based on the experimental
static curve
and
(2) von Karman theory applied to work backward from
the Observed c versus v. This could be interpreted
as based on von Karman theory using the single
dynamic stress-strain curve of Fig. (4.6).
Comparison of the two shows that disagreement between
them is slight. The single dynamic curve predicts propa»
gation speeds in agreement with the averaged velocity test
data, since it was in fact derived by working backward from
the averaged velocity test data. The deviations of the
measured propagation velocities in the individual tests from

these averaged values did not appear to have any systematic

76

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DP

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77

65“.... C804? oz<

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$2.3 2.35-335 c.2425 oz< 0.25 83 or.

‘ 2.5.3.0
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WIS/NI 30” ‘ALIOO‘IBA 310l18Vd

79

relation to the strain rate levels, as was shown in the

tabulated comparisons at the beginning of this section,

g;2 Surface Strain Results

Records from the four strain gage tests are shown in
Fig. (4.8). We note the initial elastic wave followed by
the more slowly rising plastic portion of the pulse, The
final negative step in two of the trace records indicates the
arrival of the unloading wave front. The records end before
the reflection from the far end has yet reached station 4.

At no gage station in any test did the strain appear to reach
equilibrium. This condition is readily noticeable at station 40

The residual strains in specimens 9 and 10, for which
static measurements were made with the N-unit immediately
prior to and after the dynamic tests, are tabulated on page 81
and compared with the final level of the transient record
photo.

Post-test micrometer measurements of bar diameter
indicated a plateau of residual strain extending from x = O
to beyond the fourth gage station in all tests. The slight
slope of the plateau indicated by the strain gage readings
from specimens nOs.9 and 10 was too small to be detected by
this means. The bar then tapered until a point was reached
which had no detectable residual strain. The lengths of
the total region of residual strain and of the plateau were

related to the magnitude of the impact loading,

80

 

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RESIDUAL MICROSTRAIN
AT STATION
Specimen
1 2 3 4
N-Unit 7110 6995 6830 ’ 6520
9
Photo 7170 -7080 6940 f 6760
‘ N-Unit 8965 8725 8560 8245
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Photo 8950 8780 8710 8490

 

 

 

 

 

 

 

 

 

345_ Discussion of Test Results

One feature of the test records on which some attention
should be focused is the "knee" in the curve marking the
transition from elastic to plastic wave propagation. All
strain levels in the elastic range propagate at a single
speed. Such a "knee" is not indicated in any of the results

reported by Be1126'27

using his diffraction grating technique
on annealed aluminum bars. A comparison of my static stress-
strain curve with that of Be1127'43 for "dead-annealed"
aluminum shows that his was relatively softer, and had a

much lower yield point,

82

Wave propagation speeds for various levels of strain
as found from the velocity and surface strain tests are
compared to those predicted by the rate-independent theory
from the static stress-strain relation in Fig, (4.9), The
curve for the velocity tests is obtained from Fig, (4,5)
and the relation c =1/%g: p , This is equivalent to
using the single dynamic curve of Fig. (4.6), and shows
that the single dynamic curve predicts that strains below
about €'= 0.0055 have higher propagation speeds than pre-
dicted with the static curve, while strains above 6': 0.0055
have lower propagation speeds,

It is readily seen that the foil gages indicated markedly
lower propagation speeds than the velocity transducer at all
levels of plastic strain. It should, however, be noted that
the rise time of the strain pulse was fairly constant for
all tests, and hence, the maximum strain and the strain rate

at any gage section varied proportionately, The average e
strain rates for the four foil gage tests are approximately

in the ratio 3,5:4:6:9, The apparent lag in the strain record
at any propagation speed for a constant rise time pulse ap-
pears to be inversely related to the strain rate, This does
not appear to agree with Bell's observations31 (see Art, 1.2)
that the lag in resistance strain gage records is propor-
tional to strain rate, Again noting that rise times of the
strain pulse in all tests are fairly constant and that records

of residual strain from the trace records and static N-Unit

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84

measurements show close agreement, we see that the gages
seem to have a time constant which causes them to lag be-

hind in response,

fiég Numerical Results
(a) Linear Overstress Rate Dependence Theory
It has been noted in Section 1,2 that the numerical
solution obtained by Malvern10 for a hardened aluminum alloy
subjected to constant velocity impact did not indicate the
formation of a constant strain plateau in the neighborhood

of x = 0, The material constants used were

0; = {(6) = 20,000 - 39-
E0 = 107 psi
0'), 9-104 psi

2.5 x 10'"4 lb secZ/in4

,o

The constitutive equation then becomes

Eoé=0.'+k(0'-20,000+11é9.)

The assumed value of the constant R was

6 -1
sec

k = 10
which gives an increase in stress over the static value of
approximately ten per cent for a strain rate of 200 in/in/sec,

The boundary condition imposed at x = 0 was

v(0,t) = v0 = -600 ips

85

The strain distribution in the bar at t = 102,4 ’1 sec
is shown in Fig, (4,10). Attempts to carry out the solution
further into the x,t-plane using the CDC-3600 computer re-
sulted in an oscillatory behavior of the solution in the
neighborhood of x = 0, This may have been due to an accumu-
lation of errors in the finite difference method used,

As an alternative to this solution, the constant stress
boundary condition was investigated with the aid of the
CDC-3600 computer, The boundary condition Cr(0,t) = 18,650 psi
(which is the asympotically approached value of stress at
x = 0 according to von Karman theory for v( 0,t) = -600 in/sec)
gave results similar to Malvern in that no plateau seemed
to appear even after 300 [1 sec, However, lowering the impact
stress to 67(0,t) = 17,500 psi and then to Cr(0,t) = 15,000 psi
did produce the sought-for plateau, The results are illus-
trated in Figs, (4,11) and (4,12), The curves for the rate-
independent solution are for von Karman theory based on the
static curve,

It is seen that the velocity at x = 0 very quickly
asymptotically approaches a constant. The value is that
which would be predicted by von Karman rate-independent theory,
The significant feature of the solution is the appearance
of the constant strain plateau. Hence, contrary to what
has long been thought, the Malvern formulation for a rate—of-
,strain dependence does predict the formation of a region of

constant strain near the impact end of the bar for a nearly

86

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89

constant velocity imposed at x = 0, if the pulse is long
enough for the material near the impact face to reach equili»
brium, as has been previously pointed out by Mercado,44 By
applying the method of characteristics to a non-linear visco-
elastic model (with a linear plastic strain rate dependence)
representative of Fort Peck sand, Mercado has demonstrated
that a plateau of residual strain occurs close to x = 0 for

a constant stress boundary condition with appropriate values
of the dynamic constants,

It is noted from Figs, (4,11) and (4,12) that as the
stress (and hence velocity) boundary condition increases in
magnitude, the time required for a plateau to form increases
so that the time required for the boundary condition
(7': 18,500 psi would certainly be considerably greater than
100 microseconds,

As seen in Pig, (4,10), Malvern found a strain at x = 0
greater than that indicated from rate independent theory,

but this may have been due to the error accumulation,

(b) Power Law Rate Theory with Input Data from Velocity
Transducer

(i) Convergence Difficulties

 

The velocity record from station 1, specimen
No,7 [see Fig, (4,3)] obtained from the velocity transducer
was considered as a velocity boundary condition for a bar,

A strain-rate dependence of the form

0'=0’1én

90

was assumed, and the constant value of n = 0,017 used was
that obtained by Chiddister45 for strains on the order of
five per cent in the same material at room temperature,

Due to a problem of convergence of the numerical itera-
tive scheme, the static stressestrain curve was assumed to

be valid for 6 = 10'"2 in/in/sec and thus,
0'1 = (70(100)n

where CT; is the static stress-strain relation

0'0 = f(€) = 39,40060‘366. Thus, we finally arrive at

0.: 39,4006 0,366 (100é)0°°17

for use in the numerical solution,

For the computer solution, yield was chosen at
v = 1,3 m/sec = 51,18 i'n/sec because this point on the velo-
city records had been observed to propagate at the elastic
wave speed, This choice was made, since what was being
studied was the post yield behavior, and there was no obvious
point on the fitted static curve to choose for the yield
value, Conditions along the leading wave front are those
previously discussed in Art, (2,2) for a gradual transition
from an elastic to a plastic wave, except that the fitted
static curve implies a non-linear elastic behavior before
yield,

From Equations (2,38) for a general interior point and

Equations (2,40) for an impact end point with a velocity

91

boundary condition, we have

Bp

= C

1 +~%ZXTGP

- 1:.
sp .. c2 - 2ATGp

and hence, from Equation (2,51)

9191

 

2X1:
¢2=fi=
with
b 1-02.9.8
08 " 2 E
b¢'2=-.A_I_P_§
as 2 513

E = Cl + %ATG(E,S)

s = 02 - —§—ATG(1~:,S)

¢>

 

 

22
38

and the inequalities of Equations (2,53) and (2,54) become

a6. +

 

 

 

 

 

 

 

 

 

 

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[2.8.2
DE ”
.821 1 5192
be» ' 013
39:. ,, 2922
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)=/91<1

 

 

92

0 - Q
From 0': 0.16m we obtain G = 10 8 [C "'S—B'] where
B

C = if. and Q = n-1, For G(E,S) in this form we obtain
3:-EG ’ —b-—g=QG
b E as s

In order to satisfy the sufficiency conditions for conver-

gence of the iteration process, we desire

’83=QATG --:-<1

 

 

B4 = Q AT Glél<l

or

B1 =32 = $433 +£45<1

Thus, we note that as the strain rate, é', increases
and the degree of rate dependence, n, decreases (i.e. in-
creasing Q) we may require a decrease in the mesh grid as
determined by our choice of (ST, For specimen No,7 the
choice AT = 0.26 (At = 0,26 ,1 sec) allowed a complete
solution, but a value twice as large was found to be too
large for convergence everywhere in the x,T-plane for the
input data described,

A check on the propagation of errors in the solution
due to round off in the computer was obtained by obtaining
a solution with AT = 0,13 (At = 0,13# sec), The results
for stress, strain and velocity agreed with those of the

previous solution with [5T = 0,26 to five or six digits at

93

all points in the X,T~plane,

(ii) Discussion of Calculation Results Based on
Power Law Rate Theory with Input Data from

Velocity Transducer

Pig, (4,15) shows level lines of velocity in
the x,t-plane, Since the level lines are straight (except
for some slight curvature for v = 300 in/sec and
v = 350 in/sec near x = 0) the propagation speed for any
given level of particle velocity is predicted to be constant
as the pulse travels down the bar and thus independent of
the variations in strain rate encountered, The dashed lines
in Fig, (4,15) are the level lines predicted by the von
Karman rate-independent theory based on the fitted power-
law static curve of Equation (4,2), Comparison with the
solid curves shows that the rate-dependent solution predicts
higher propagation speeds for velocity increments up to a
little above 300 in/sec, The level lines of stress and
strain are not shown separately in the figure since they
so nearly coincide with velocity level lines as to be indis»
tinguishable in plotting, What this means is that for the
range of strain rates actually encountered in this solution
and for the very slight rate dependence implied by the rate
law with n = 0,017, the level lines are the same as would
be predicted by using a single dynamic stress~strain curve
instead of the static curve in the von Karman theory, except

for very slight differences observable where the level lines

94

plotted show a slight curvature, It was pointed out in

Art, (4,4) that the average strain rates at the first and
last gage stations were approximately in the ratio 3,5:9,

It is not surprising, therefore, that the predictions of the
rate-dependent theory can be correlated with a single dynamic
curve, since there is so little variation in the strain rates
in the solution, all of them being of the order of magnitude
of 10,000 times the rate of é = 10"2 in/in/sec assumed for
the "static" curve,

Additional information about the solution is contained
in Figs, (4,16) and (4,14), Fig, (4,16) shows calculated
stress versus time at x = 0 where the velocity input was
taken from the first velocity gage station for specimen No,7,

The solid curves in Pig, (4,14) show the input velocity at

x 0 and the calculated velocity versus time at x = 3,07in,
x = 3,58 in, x = 6,14 in, and x = 9,21 in, The dashed curve
is the experimental record from the second gage station at

x = 3,25111 on specimen No,7, Comparison of this with an
interpolation between the two calculated curves for

x = 3,0751: and x = 3,58iJI indicates that the calculated
propagation velocities are greater than the measured velo-
cities, The greater part of the discrepancy is believed to
be due to the use of the fitted power law for the static
curve, As is seen in Fig, (4,13), the slopes of the fitted

curve deviate considerably from those of the actual static

curve, The curve in Pig, (4,14) indicated by the small

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TIME AFTER YIELD ”LSED)
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350 IN/SEC
300 IN/SEC

—-é RATE DEPENDENT SOLUTION
—--— VON KARMAN SOLUTION

X (INCHES)

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circles is the predicted curve at the second gage station
according to the von Karman rate-independent theory based

on the fitted power-law static curve, The predictions are
for levels of strain from 6': 0.0005 to 6:: 0.0065 in
increments of 0,0005 in/in, The rate-independent theory
based on the fitted static curve thus also predicts a higher
propagation velocity than was measured on specimen No,7.

The small circles fall in between the dashed experimental
curve and the interpolated solid curve (not shown) between
x = 3,07i11 and x = 3,585Jh but they fall closer to the
interpolated curve than to the experimental curve, so that
neither theoretical prediction agrees very well with the
experimental curve, Part of this discrepancy may be due to
variation of the experimental behavior of specimen No.7

from the averaged velocity behavior of all the specimens,
which was seen in.Art. (4,2) to correlate well with a single
dynamic stress-strain curve prediction and fairly well with
a static curve prediction, The greatest part of the dis-
crepancy is, however, believed to be due to the use of the
fitted power law instead of the actual static curve,

The computer solution may also be rather sensitive to
the way the conditions representing yield were introduced
into the solution. Picking a slightly higher value to
repreSent yield seems likely to move the computed curves

nearer to the experimental curve°

 

100

It was, however, considered not worthwhile to repeat
the computer solution with a better fit on the static curve
or to adjust the assumed yield value, since it was already
clear from the results of Art, (4.2) that the averaged velo-
city data from the velocity experiments can be represented

by a single dynamic curve,

 

CHAPTER V

SUMMARY.AND CONCLUSIONS

Two independent series of dynamic plastic compression
impact tests were performed on half-inch diameter bars of
commercially pure aluminum, In the first series, an
electro-magnetic transducer was used to obtain measurements
of particle velocity at four stations along the bar, while
in the second series, etched foil resistance strain gages
yielded records of surface strain at the same gage locations.
Strain rates on the order of 100 in/in/sec were reached.

Test results indicated that any given level of velocity
or strain propagates along the bar with a constant velocity,
not affected by the strain rate within the small range of
strain rates encountered, However, the velocities of propa-
gation observed differed noticeably from those predicted by
von Karman rate-independent theory based on the static curve,
Good agreement was found between the propagation speeds ob-
served for different levels of velocity (averaged over all
tests) and predictions of von Karman theory based on a single
dynamic stress-strain curve differing from the static curve°

That the apparent applicability of a single dynamic
curve and rate-independent theory to this kind of plastic
wave propagation is consistent with rate-dependent theory
for a material with a very slight rate dependence, was

demonstrated by the results of computer solutions for rate=

101

102

dependent theory inArto (4.5). The nature of the process

is such that the range of strain rates encountered for most

of the observations is covered by a 3:1 ratio of strain rates,
and almost all of the plastic deformation occurs at rates

in the range from 3,000 to 10,000 times the "static curve"
strain rate.

The wave propagation speed versus strain level plots
from the transient strain records showed consistently lower
propagation speeds than those based on the velocity records.
It is believed that the strain gage response actually lags
behind the strain in the material, as previously observed
by Bell,31 but our records are not consistent with a lag
proportional to strain rate as reported by Bell, since, in
fact, the higher strain rate tests came nearer to agreeing
with the velocity measurements and the rateaindependent
theory than did the lower strain rate testso Considerably
more evidence is needed before final conclusions can be
drawn about the lag in the strain measurements and the reasons
for it,

Using the velocity records obtained from the first gage
station (six diameters from the impact end of the bar) as
an input boundary condition to predict values at stations
farther along the bar, a numerical computer solution was
obtained using the rate-dependent theory with a power law
for rate dependence and the power n m 0,017 found in dynamic

stress-strain tests on short specimens of the same material

103

performed by Chiddister,45 This computer solution did pre»

dict a constant wave propagation speed for any given level

of velocity, but the constant values predicted did not agree
well with the experimental values from the velocity records.
This lack of agreement appears to be mainly the result of
using a rather poor fit to the static curve in the computa~
tions, since von Karman rate-independent theory using the
same fitted static curve also gave poor agreement with the
experiments. Since the computer solutions with rate-dependent
theory were consistent with a single dynamic curve, and since
the velocity measurements correlate with a single dynamic
curve, it appears that a little ingenuity in curve-fitting
could produce agreement between the rate theory and the
experiments. This did not seem to be worth the effort.

Such agreement between rate theory and the experiments
would not of course prove that the rate theory was correct
and von Karman theory based on a single dynamic curve was
incorrect, since the two would predict virtually the same
thing for a material with little rate sensitivity. The
rate-independent theory is easier to apply and therefore
preferable for a situation like this. Any real test of
the rate-dependent theory must come in a situation with a
greater range of strain rates in the test and for a material
with more strainurate sensitivity than annealed aluminum°

Further study in this area should include experimental

and theoretical wave propagation studies and dynamic tests

104

on materials not in the soft annealed condition, which may
exhibit more strain rate dependence as suggested by Dorn et a1.

Ferrous materials are also known to be rate sensitive,
but in wave propagation tests of the kind described in this
report, anomalies occur because of the yield delay time,6
It might be possible to study tensile pulse propagation in
bars pre-loaded statically into the workuhardening range and
impacted while the static loading was continuing, Further-
more, the magnetic velocity transducer could not be used
on ferrous specimens, Further study is also in order on
transient strain recording in dynamic plasticity to develop
a simple strain recording technique not subject to the lag
exhibited by the strain gage records.

The velocity recording technique might be improved by
using stronger and more uniform magnetic fieldso These would
yield an increase in output signal level as well as allow
the use of wider gaps between pole pieces. The wider gap
would enable construction of a wire support system capable
of a greater translation during the passage of the waveo

The velocity recording technique for nonumagnetic mam
terials is believed to give good results, but if possible,
it would be desirable to modify it to make it more nearly

a routine type of testo

10.

105

BIBLIOGRAPHY

Donnell, L. H., "Longitudinal Wave Transmission and
Impact,“ Trans. ASMB, Vol. 52, 1930, pp. 153-167.

von Karman, T., "On the Propagation of Plastic Defor-
mation in Solids," NDRC Report No. A-29
(OSRD No. 365), 1942.

von Karman, T. and Duwez, P., "The Propagation of
Plastic Deformations in Solids," J. Appl. Physics,
Vol. 21, 1950, pp. 987-994.

Taylor, G. I., "The Plastic Wave in a Wire Extended

by an Impact Load, " The Scientific Pa ers of
Sir Geoffrey Taylor, Cambridge Un1v. grass, 1958

pp.157-379.

 

Rakhmatulin, K. A., "Propagation of a Wave of Un-
loading," Prik1.Mat. i Mek., Vol. 9, 1945,
pp. 91- 100 (in Russian).

Translation.All - T2/15, Brown Univ., 1948.

Duwez, P. and Clark, D. 8., "An Experimental Study
of the Propagation of Plastic Deformation Under
Conditions of Longitudinal Impact," Proc. Am.
Soc. Test. Mat., Vol. 47,1949, pp. 502-532.

Ludwik, P., "Uber den EinfluSs der Deformation-
geschwindigkeit bei Bleibenden Deformationen mit
Besonderer Berucksichtigung der Nachwirkungser-
scheinungen, " Physikalische Zeitschrift, Vol.10,
1909, pp. 411-417.

Deutler, H., "Experimentalle Untersuchen uber die
Abhangigkeit der Zugspannungen von der Verfor-
mungsgeschwindigkeit, " Physikalische Zeitschrift,
Vol. 33, 1932, pp. 247- 259

Prandt1,L., "Bin Gedankenmodell Zur Kinetischen
Theorie der Pesten Korper, " Zeitschrift fur
.Angewandtke Mathematik und Mechanik,Vol.8, 1928,
pp. 85-106.

Malvern, L. 3., "The Propagation of Longitudinal
Waves of Plastic Deformation in a Bar of Material
Exhibiting a Strain-Rate Effect, " J. Appl. Mech.,
Vol. 18, 1951, pp. 203-208.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

106

Malvern, L. 3., "Plastic Wave Propagation in a Bar
of Material Exhibiting a Strain Rate Effect,"
Quart. Appl. Math., Vol. 8, 1950, pp. 405=411.

Sokolovsky, V. V., "The Propagation of Elastic-
Viscous-Plastic Waves in Bars," Prikl. Mat. 1
Mek., Vol. 12, pp. 261n280, 1948 (Russian).
Translation A11~T6, Brown University, 1949.

Bell, J. P., "Propagation of Plastic Waves in Prew
Stressed Bars," Dept. of Mechanical Engr., The
Johns Hopkins University, Baltimore, Tech. Rep.
No. 5, Navy Contract N6-ONR-243, 1951.

Sternglass, E. J., and Stuart, D. A., "An Experimental
Study of Propagation of Transient Longitudinal
Deformations in Elastoplastic Media," J. Appl.
Mech., Vol. 20, 1953, pp. 427-434.

Alter, B. E. K. and Curtis, C. W., "Effects of Strain
Rate on the Propagation of a Plastic Strain Pulse
Along a Lead Bar," J. Appl. Phys., Vol. 27, 1956,
pp. 1079-1085.

Davies, R. M., "A Critical Study of the Hopkinson
Pressure Bar," Phil. Trans. Roy. Soc. Lond.,
Ser. A, Vol. 240, 1948, pp. 375~457.

Bell, J. P. and Stein, A., "The Incremental Loading
Wave in the Pre-Stressed Plastic Field," J. de
Mecanique, Vol. 1, No. 4, Dec. 1962.

Bodner, S. R. and Kolsky, H., "Stress Wave Propagau
tion in Lead," in Proc. Third U. S. Nat. Congr.
Appl. Mech., New York, 1958, ASME, pp. 495a501.

Simmons, J. A., Hauser, P., and Dorn, J. E.,
"Mathematical Theories of Plastic Deformation
Under Impulsive Loading," Univ. of California
MRL Publication, Series No. 133, Issue No. 1,
December, 1959.

Campbell, J. D., Simmons, J. A. and Dorn, J. E.,
"On the Dynamic Behaviour of a Prank-Read Source,"
Univ. of California MRL publication, Series No. 133,
Issue No. 2, May, 1959 (also J. Appl. Mech., Vol° 28,
Sept. 1961, p. 447).

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31

107

Hauser, P. E., Simmons, J. A., and Dorn, J. E.,
"Strain Rate Effects in Plastic Wave Propagation,"
Univ. of California MRL publication, Series No. 133,
Issue No. 3, June, 1960.

Hauser, F. E. and Winter, C. A., "An Experimental
Method for Determining Stress-Strain Relations at
High Strain-Rates," Univ. of California MRL publi-
cation, Series No. 133, Issue No. 4, December, 1960.

Hauser, P. E., Simmons, J. A., and Dorn, J. E.,
“Strain Rate Effects in Plastic Wave Propagation,"
Response of Metals to High Velocity Deformation,
Interscience, New York, 1961, pp. 93-114.

Rajnak, S. and Hauser, P., "Plastic Wave Propagation
in Rods," Symposium on Dynamic Behavior of Materials,
ASTM Special Technical Publication No. 336, 1963,
pp. 167-179.

 

Bell, J. P., "Diffraction Grating Strain Gauge,"
Proc. SESA, Vol. 1, 1959, pp. 51-64.

Bell, J. P., "Propagation of Large Amplitude Waves
in Annealed Aluminum," J. Appl. Phys., Vol. 31,
No. 1, 1960, pp. 277-282. >

Bell, J. P., "Study of Initial Conditions in Constant
Velocity Impact," J. Appl. Phys., Vol. 31, No. 2,
1960, pp. 2188-2195.

Kolsky, H. and Douch, L. S., "Experimental Studies
in Plastic Wave Propagation," J. Mech. Phys. Solids,
Vol. 10, July/Sept. 1962, pp. 195-223.

Lindholm, U. 8., "Some Experiments with the Split
Hopkinson Pressure Bar," Technical Report No. 1,
Contract No. DA-23-072-0RD-1674, SwRI Project
No. 02-1102, Southwest Research Institute, San
Antonio, Texas, March, 1964.

Ripperger, E. A. and Karnes, C. H., "Plastic Impact
on Short Cylindrical Specimens," 8%mposium on
gynamic Behavior of Materials, AS Spec1a echnical
ublication No. 336, 1963, pp. 180-194.

Bell, J. P., Discussion, "Experimental Studies of
Plastic Wave Propagation in Bars," Plasticity,
Proceedings of the Second Symposium on Naval
Structural Mechanics, Brown University, 1960, p. 485.

 

 

32.

33.

34

35.

36.

37.

38.

39.

40.

41

42

43

108

Hopkins, H. 6., "Dynamic Anelastic Deformations of
Metals," Appl. Mech. Rev., 1961, Vol. 14, No. 6,
pp. 417-431.

Kolsky, H., "Stress Waves in Solids," J. Sound and
Vibr., 1964, Vol. 1, pp. 88-110.

Lee, E. H., "The Theory of Wave Propagation in
Anelastic Materials," in N. Davids, ed., Int.
Symposium on Stress Wave Propa ation in Solids,
Interscience, 1960, pp. 199-22 .

Craggs, J. W., "Plastic Waves," in Sneddon-Hill, eds.,
Progress in Solid Mechanics, Vol. II, Interscience,
1961, PP. 141-197. -

 

Kolsky, H., Stress Waves in Solids, Oxford, 1953
(Dover, 19635, CH. VIT.

Courant, R. and Hilbert, D., Methods of Mathematical
Physics, Vol. II, Interscience, New York, 1962.

Courant, R. and Friedrichs, K. 0., Sugersonic Flow
and Shock Waves, Interscience, 1 , C . I .
Zaguskin, V. L., Handbook of Numerical Methods for the

Solution of Algebraic andfTranscendental Equations,
Pergamon, New York, 1961.

 

 

Pugh, E. M. and Pugh, E. W., Principles of Electricity
and Magnetism, Addison-Wesley, Reading, Mass., 1960,
pp. 216-226.

 

Ripperger, E. A. and Yeakley, L. M., "Measurement
of Particle Velocities.Associated with Waves
Propagating in Bars," Experimental Mechanics,
Vol. 3, No. 2, February, 1963, pp. 47-56.

Ramberg, W. and Irwin, L. K., "A Pulse Method for
Determining Dynamic Stress Strain Relations,"
Proc. Ninth International Congress of Applied
Mechanics, Brussels, 1957, Vol. 8, pp. 480-489.

Bell, J. P., "An Experimental Study of the Interrelation
between the Theory of Dislocations in Polycrystalline
Media and Finite Amplitude Wave Propagation in
Solids," Tech. Report No. 7, Air Force Contract
AF49(638)-423, Johns Hopkins University, March, 1961.

109

44. Mercado, E. J., "Plane Wave Propagation in Nonlinear
Viscoelastic Materials," Thesis, Rensselaer Poly-
technic Institute, 1963.

45. Chiddister, Jerry L., "An Experimental Study of
Aluminum at High Strain Rates and Elevated Tempera-
tures," Thesis, Division of Engineering Research,
Michigan State Univ., 1961.

APPENDIX
THE COMPUTER PROGRAM

In the following pages an asterisk (*) will denote
multiplication and a slash(/) will denote division so that
the equations in which they appear will be similar to the

corresponding Fortran statements in the computer programs.

A.1 Leading Wave Front
For a transition from an elastic to plastic impact,

the conditions along the leading wave front are given by
0"=0’y

6:6},

where the subscript y refers to the values at yield.
Assuming rate independent theory applicable until

yield occurs, we have

6;
-_-v-_- 1.9.9.:
v y .1; £7 d6, d6?

When the impact is initially plastic, we must consider
a shock wave propagating along X = T (x = cot).
(a)_Linear Overstress Rate Dependence (MALRATE)
From Equation (1.11) I

1506‘ = +k[0'- f(€)]

110

 

 

111

g(0’o€) = k[a"/f(€)]
k = 106 sec"1

(i.e. 10'6 sec = unit dimensionless time)

((6) 20,000 - 19—

 

€
0"- 20,000 + 10
G = _g_. = . I.
k‘EO E0
= s _ 20,000 , 10
E0 EOE
= - B +15
8 a
where
10 2 x 104

A=~§;, B-T

From Equation (2.48)

 

l1
awe

S S
‘=' ds = sds
. ——-x f 2
S(0,0) 5 ' B +Ԥ s - 35 + A
which yields

1‘ .. ._ __10.000 10.000
0’- 10,000 070,0) - 10,000

. 1n 0’(0,0) - 10,000 ,, 10.000 )]

10,000 (7' - 10, 000

= 2[P - Q + 1n (P/Q)]

 

112

 

 

where
P = 10.000
0’- 10,000
Q _ 10,000

‘ 0’(0,0) - 10,000

At any mesh point along X = T we have T = (J-l) * T.

Hence,

TAX = J [P - Q + 1n<P/Q)]

2
_1=—_
AT

Thus, 7%; = f(P) is solved by the "Method of False

Position" and »0'= 0’(P) = 0’(cot,t) = 0’(T,T) is obtained

along X = T."

 

TAX

 

P1

 

 

 

 

 

113

 

. . . _ TAX
P 1+1 = P 1 _ P 1 _ F2
2 2 ( 2 P1) 92 - F1
i+1 i i+1 . .
When P2 - P2 <6 we assume that P2 15 suff1

ciently close to P3 giving us the value of stress at any
given time (TAX *‘ZSTO along the leading wave shock front.
(b) ~Power lam: Rate Dependence (POWRATE)
0’<€,é) = 0'1<€,1>é“ '

Assuming that the static-stress strain.curve is
O
-1

applicable at €= 10.2 sec we have
0;, = 0'1(0.01)n
01 == 0’,<o.01>'n
and
076.61) -"-’ 00(100 ftp)" for ~é§0.01 sec"l
O’(€ , 6") = (7'o 080.01 sec-1
Hence 4

O
. n
p (76 .
And the constitutive equation becomes
0 . '
Bo€=0'+10‘213 [i,q
0
Therefore,

g(0'.€) = 10'2130 ‘%) Q

 

 

114

Choosing k = 106 see”1 so that unit T is 10'6 second,

we obtain

._._ -8_Q_’_Q
61°19.)

For a static stress-strain relation expressible in

the form
B
0'0=A€ ,B<1
we have Q
- E S
0:108 £5
AE
=10”8 (Ci)Q
EB
where
E
=_9.
C A

For a plastic shock wave along X = T, we substitute

into Equation (2.48) to obtain

S
T=-2x108I _._—(£—
8 (csl‘B)Q

O
8 .
= .2449.— S -
99-9 - 1{———__j—[c(s)1'3 Q m}
where

_ SO
- [ccs )1'B] Q ' 8° = S(O'O)
O

115

For B<l, Q>l, C>>1 and O<S ((1 we note that

dT 2
__=_—<0
dS G

Hence, the graph of T/[ST versus 5 appears as shown

 

 

 

 

 

 

below.
'r/AT
F2
TAX
Fl
S
and

 

'1 ' F2-TAX
$1+=S+(81-S)
2 1 2 1112-91

The process is repeated as has been described previously

for the case of linear rate dependence.

116

A.2 Computational Procedure

 

 

 

 

V ’
, w J
2’» \_ \go:::::::::°;:°:2

Point P is denoted by the array identification (2,N).

Points A, B and C are identified by (1,N), (1,N-1) and

(2,N-1) respectively. The variable N is given by the

"column" number, K.

117

The solution works outwards from the origin and
proceeds along a "row," J = constant. For K = N = 1, we
make use of the equation for a gradual elastic-plastic
wave transition interface (see last equation of Art. 2.2)
or use the schemes of Sec. A.1 for shock waves. For
K = N = J the boundary conditions will determine the
appropriate relations [Equations (2.40), (2.42) or (2.44)]
to be solved.

If we have not reached the last row, we will proceed
to the next row and the values just obtained in the former
"computation row 2“ become the elements of the next "compu-
tation row 1."

In the vicinity of the origin, we chose the values of
variables at B or C as initial estimates of the variables
at P. Elsewhere, initial estimates in the field for K<:5
make use of the first and second0 derivatives of the variable
along the column K = constant. For K2>5 we make use of the
derivatives along J = constantr

The iterations appeared to converge as an oscillatory
geometric progression. Hence, the following Aitkin -, 6 2
process was used.36

Assume the solution is {a} and iterated estimate {3k} .
For oscillatory convergence, a - ak is assumed to decrease

approximately as the sequence of numbers

prk cos(k¢ + 9)

118

Such a sequence is the sum of two geometrical progressions

with complex conjugate ratios

r (cos ¢ + i sin¢)
r (cos ¢ - i 510$)
ak+1 - ak decreases approximately as the sequence

k cos (k¢+ 91)

where p1 and 61 are in general different from p and 0 ,

q1

q2

but r and ¢ are the same for the sequence {a - ak} .

The improved value, 2H1, I is given by .

2

Aak_!(Aak ' r Aak-l)
2 2

Azak-l " r A ak-Z

 

ak+1 = 3k '

Here

Aak-z Aak-l

Aa _ a
2 k 1 ; A‘k 1.. Aak-Z A31: - (A9-
_. Aak 3 i A91: 2 Aak- 3 Aak- 1 ' (Aakiz 2)2

 

 

 

Aak-Z Aak-l

 

where

Aai = ai+1 ' 31

2 _
A ai " Aa14-1 " Aai

 

119

 

Thus,

._ (a4 — a3) [(a5 - a4) - r2(a4 - 213)]

as = a4 " 1 j.
[(a5 - a4) - (a4 - a3)] - :2 [(a4 - a3) - (a3 - 32)]

where

2 _ (a3 " 32) (a5 ‘1' a4) " (a4 " 3.3)2

 

r
(a2 - a1) (a4 - a3) - (a3 - a2)2
If necessary, this process is repeated after four more
Seidel iterations. Each time the Aitkin - 6 2 process is
used, the magnitude of the correction term denominator in
the equations for as is checked to make sure that it is
greater than zero. If the denominator vanishes, we omit

the Aitkin - 6 2 process and perform another Seidel itera-

tion.

120

Interpretation of Code Words

Words Common to Both Programs

LOOK

JAZZ

M1

M2
M3
M4

LL

IMPACT

MARK

 

II II II II
-b O.) N

Grid Row Number

Grid Column Number

V(0,T) = VOL = constant
V(0,T) is a known variable
S(O,T) is a known variable
E(O,T) is a known variable
S(O,T) is tabulated

V(0,T) is tabulated

S(O,T) is 0:) - then zero
V(0,T) has exponential rise

Maximum permissible iterations

First Row to be printed after initial
impact point (1,1)

First Column in first row to be printed out
Increment along column for print out
Increment along row for print out
Last row after which

stress = 0 for LOOK = 3, JAZZ = 2
or

strain = constant for LOOK = 4

Plastic Impact - Calculations Required
Along X = T

Elastic-Plastic transition - Conditions
Constant along X = T

Value changes from 1 to 2 when 0": f(€)
at an interior or impact boundary point

 

121

 

 

T AT = 106 At
BO Young's Modulus (psi)
RHO Mass Density (lb/secz/in4)
YIELD Yield Stress (psi)
VMAX Vmaan/seC)
(For case of exponential
TFIX tfix(llsec) velocity rise at X = O,
v|x=o = a% of vmax when
CENT .Ola t = tfix)
VOL v(0,t) = constant (in/sec)
STRS 0’(0,t) = constant (psi)
ALPHA Constants in v(0,t) = vmax[1 - e'€£?+ch)]
BETA (calculated in computer)
Program Malrate
A
G = S-B + A.
B E
' s - c 104
C Along X = T, P = , C =.__.
C E
. o
D (SP, Increment in P along X = T
P 1
Q 1 T = 2[P - Q + Ln(F*P)] , F = q‘1
P .
4
Program Powrate
A
Jo = A68
B
E
C = .2
C A
D (SP, Increment in P along X = T

 

TIM

 

122

P=SalongX=T
7:0'lén,0=;11-
NOTUSED

Normal Run

Program will restart after last row
with new mesh

Program is a restart (J )l 1)

Factor by which AT is multiplied in
going from MAD = 2 to MAD = 3

 

123

5:4 Computation Flow Chart

The computation flow chart on the following page gives
a schematic representation of the process by which the actual
computer program carries out the solution in the characteristic
X,T-plane. Using the flow chart and the "comment" cards
within the actual program, it is hoped that the reader will
be able to follow the Fortran coding.

The routine for changing the mesh size (i.e. increasing
lXT) during computation is presently only found in PROGRAM
POWRATE but could easily be incorporated in PROGRAM MALRATE.

124
can

[ nu la Mme. L

Pratt Comm
Due-lac Bounty Coalition I

 

 

 

had In an
Hm Initial Conflue-

 

 

 

 

 

 

 

 

 

 

mum".
....... 6 ml

 

 

 

 

 

“‘1" lo a - m
1 km Pom
' calculation
. 1. W
" an.“ sauna '”
I must
no I. km.
'33::- .. g)... Yon '° ms Point "' mm
no.

 

 

 

 

 

If m
00
Correct Och-Hm

COMPUTATION FLOW CHART

 

 

 

125

£12 Fortran Programs

The following Fortran programs (with illustrative data
cards) have been programmed for the CDC-3600 and use an
iteration convergence criteria of agreement to ten signifi-
cant figures for two successive iterations. The following
items should be taken into account before attempting to use
either program.

1. The programs are presented with the warning that
all of the possible boundary conditions have not been used
to date, and hence, errors may exist.

2. Computer.must provide a minimum of ten digit
storage capability plus sufficient storage for program and
variables.

3. Beware of time requirements.

 

 

 

 

Approximate
Impact Machine Time
PROGRAM L B.C. on CDC-3600
MALRATB 401 0’(O,t) = constant 4%: minutes
velocity data from
POWRATE 501 Gage Station One 20 minutes
Spec. No.7

 

 

 

 

 

 

4. Proper choice of [ST‘is not necessarily known
a'priori. Given an initial set of input data for which the

choice of [ST gives convergence only into a small (or not

 

126

at all) region of the characteristic X,T-plane, interpola-

tion of already punched data may be used with another choice

of (ST. The illustrative examples given below were the schemes

actually used with the velocity input data from the velocity

transducer.

TO HALVBéLT

201

202

203

READ 4, (V(2,N), N=J,L,2)
00 202 N=J,L,2

v(2,N) = -V(2,N)/CO

LLL = L-l

no 203 N=2,LLL,2

V(2,N) = 0.5*CV(2,N-1) + V(2,N+1)>
CONTINUE

TO QUARTER [ST

201

202

203

READ 4,(V(2,N). N=J.L.4)

no 202 N=J,L,4

V(2,N) = —V(2,N)/CO

LLL = L-3

DO 203 N=2,LLL,4

TEMP = V(2,N+3) - V(2,N-l)
V(2,N) = V(2,N-1) + 0.25*TEMP
V(2,N+1)

V(2,N-l) + O.50*TEMP

V(2,N+2) V(2,N-1) + 0.75*TEMP

CONTINUE

 

 

 

127

PROGRAM MALRATE

1 FORMAT (lXoBHRQUoBXoéHCOLUMNo9Xc6HSTRESSo13X96HSTRAIN9

111X88HVEL0CITYo9XoIOHG‘FUNCTIONoBXolOHITERATIONS/)
FORMAT c1x.13.5x.13.4x.4(E16.10.3x1.3x.13>

FORMAT (14(14.1X))

FORMAT (S(Elé.10))
FORMAT(IXQZHJ=91393X92HK=91303X97HNUMBER89I3)

meUN

1I394X94HMAX=O13/1X92HK39I394X95HJAZZ‘O1294X93M72=0I39

FonflAT ClXoZHch1394X05HLOOK39IZO4X03HM1=91304X03HM3=9

24X03HM4=9I394X02HL=QI393X93HLL30I393X07HIMPACT301303X0

34HMAD=QI3/) ’
6 FORMAT (1X93HEO=9E1601004X04HRH0395160IOO4X93HC0‘9
151601094X92HT=9516010/1
7 FORMAT (1x02HA=CE16O1094XOZHB=OE1601004x02HC39EI60109

14X92HD=QEI601004X92HP=9516010/0/1X96HYIELDSQE160IOQ4XO

25HVMAX=QEIGo1094X95HTFIX=9E160IOO4X05HCENT=9E16010/9
3/1X92H030E1601094X92HF=951601004X94HTIM=95160IO/e/)
8 FORMAT (IXO6HALPHA=QE1601094XO5HBETA=QE16010‘4X9
116HYIELD REACHED ATOIXQEIéOlOOZXQ
225HMICROSECONDS AFTER IMPACT/0/)
9 FORNAT ‘/1XO25HG FUNCTION = olE‘O7 AT J=OI493X02HK=9
114/)
DIMENSION 5(29401)0E(294°1)0V(20401’OG(29401)OSP(5)9
ISPP(5)9EP(5)QEPP(5)
READ 3gJ9K.LgLOOK.JAZZ.MAX.M1.M2.M3.M4,LL.IMPACT
READ 4.T.EO.RHO.A.B.C.D.P.YIELD.VMAX.TFIX.CENT.G.F
CO=SORTF(EO/RHO)
FRINT SOJOLOOKQMIOMSOMAXQKOJAZZOM29M4OLQLLOIMPACT
PRINT 69EOoRHOQC09T
FRINT.79A039CoDsp9YIELD9VMAX9TFIX9CENTcooF
NUMBER=O
MARK=1
E13000
528000
E33000
E48000
$13000
$23000
53:000
54:000
DO 98 N=194OI
S(ION)=OOO
S(ZeN)=OoO
E(ION)=OOO
E‘ZON)=OOO
V(ION)3OOO
V(ZvN)=OoO
G(19N)=OoO
98 G‘ZQN):OOO
DO 99 Nzlos

99

100

101

200
201

202

210

220
225

300
301

302

310

311

128

EP(N)‘OOO ,

EPP(N)‘O¢O

SPINI‘OOO

SPR‘NI'OOO

VELOCITY o STRESS OR STRAIN BOUNDARY CONDITIONS AT X30
GO TO (IOOQZOOQ3OOO4OOIvLOOK

IMPACT VELOCITY IS CONSTANT

READ 40VOL

V(IOI)8VOL/CO

E(Iol)8~V(lcl)

5(19113-V1191)

DO 101 N=IQL

V‘Z‘N)8V(I‘l)

GO TO 500

IMPACT VELOCITY IS A KNOWN VARIABLE
GO TO (ZOIOEIOIOJAZZ

IMPACT VELOCITY IS A TABULATED VARIABLE IN/SEC
READ 40(V(20N)9N=10L)

DO 202 N=IOL

V(ZoN):V(20N)/CO

GO TO 225

INPACT VELOCITY IS GIVEN AS AN EXPONENTIAL FUNCTION
TENP=IoO/(IoO-CENT)'
ALPHASLOGF(TEMP)/TFIX

TENPIEO*VMAX
BETA:LOGF(TEMP/(TEMP+CO*YIELD)I
TEHPBBETA/ALPHA

PRINT BOALPHAOBETAOTEMP

DO 220 N=IQL

WER=2*(N-l)
TEMPZEXPF(ALPHA*T*WER+BETA)
V(ZnN)=VMAX*(1.0-1.0YTEMP)/CO
V(Iol)8V(291)

S(Iol)8~V(IoI)

E(Iol)8-V(Iol)

S(ZvII3SIIQI)

E(291)3E(191)
GC201)SS(201)-B+A/E(291)

GO TO 500

IMPACT STRESS IS A KNOWN VARIABLE
GO TO (3OIQSIOIQJAZZ

IMPACT STRESS IS TABULATED PSI
READ 49(5120N)0N819L)

DO 302 N=IoL

S(ZvN):S(29N)/EO

GO TO 325

IMPACT STRESS IS A CONSTANT -~ THEN ZERO
READ 4OSTRS

DO 311 N=19LL

S(ZQN)=STRS/EO

312
313
325

400

412

413
425

500

1000

1001
1002

1010

129

LLzLL+1

IF(Lb’LI 31293120325

DO 313 N=LLOL

S(ZcN13000

S(1o1)=S(291)

E(101)=S(101)

V110113‘SIICI)

WILL BE RECOMPUTED BELOW IF INITIAL IMPACT IS PLASTIC
E(291)*E(1911

V(201)8V(101)
GIZQ118$(2911‘B+A/E(201)

GO TO 500

STRAIN HISTORY IS GIVEN

READ 40(E(29N)9N=19LL1
IF(LL-L) 412.425.425

STRAIN IS NOW CONSTANT
LLL=LL+1

DO 413 NSLLLOL

E(20N18EIZQLL)

S(IOI’3€(ZOI)

EIICII=E(2OI)

V(19118-E(201)

WILL BE RECOMPUTED BELOW IF INITIAL IMPACT IS PLASTIC
5(20118E(291)

V(201)8-E(291)
GIZ.1):S(201)‘B+A/E(201)

GO TO 500

PRINT OUT OF IMPACT CONDITIONS
PRINT 1

STRESS=S(101)*EO
VELOC=V(1¢1)*CO
G(191)8$(101)-B+A/E(101)

PRINT ZQJQKQSTRESSOEI191)0VELOC0G(111)
GO TO FIRST POINT AFTER IMPACT
J8J+1

GO TO (I00001045’9IMPACT

CALC. ALONG X:CO*T FOR PLASTIC INITIAL IMPACT
TT=2.0/T

FISTT*(P-G+LOGF(F*P))

TAX=J“I

PIBP

P3P+D

F2=TT*(P-Q+LOGF(F*PII
DISFZ'TAX

IF(DI) 10109101001020

F1=F2

GO TO 1002

1020 IF(DI-01E-09) 10409104001030
FALSE POSITION ITERATION
1030 R=F2-F1

 

1040

V1045

1050

1055

1060

1100
1110

1112

1114

1120

1130

130

PIP-(PflP11*DI/R

F2:TT*(P“Q+LOGF§F*P))

DI'FR‘TAX

NUMBERaNUMBER+I

GO TO 1020

S(ZoI):C+C/P

E12011=S(20I)

V(201)‘-S(201)

GIZ.1)=S(201)‘B+A/SIZOI)

F1=F2

081001*D

DO WE PRINT THIS ROW

IF(J-M1) 10600105001060

DO WE PRINT THIS COLUMN

IF(K-H2) 10600105501060

PRINT THIS POINT

M2=M2+M3

STRESSSE0*S(Z0K1

VELOC3C0*V(20K1

PRINT 20J0K0$TRESS0EIZ0K10VEL0C0GIZ0K)0NUMBER
MOVE TO THE NEXT POINT IN COLUMN

K=K+1

N080

NUMBER80

GO TO 1100

ARE WE IN THE FIRST FIVE ROWS

IFIK-S) 11100111001120

ARE WE IN THE LAST TWO ROWS IN THIS COLUMN
IFIJ-K-Z) 11140111201112

INITIAL ESTIMATES FOR THE FIRST FIVE ROWS
DELSxSPIK1+SPP(K)

DELESEPIK1+EPPIKI
DELG=DEL5-A*DELE/(E(10K)*E(10K)1
28GII0K1+DELG

GO TO 1130

INITIAL ESTIMATES FOR THE LAST TWO ROWS IN THIS COLUMN
Z=GCZOK‘1)

GO TO 1130

ESTIMATE FOR GENERAL INTERIOR POINT
DELS=200*$(29K‘1)‘300*S(20K‘2)+S(20K“3)
DELE=200*E(20K‘1)‘3.0*E(20K‘2)+E(20K~31
DELGSDELS~A*DELE/IE(Z0K‘1)*E(20K'I)1
28G(20K-I)+DELG

GO TO 1130

CONSTANTS WHICH ARE FUNCTIONS OF ADJACENT STATE POINTS
DC=SIZ0K~1)+V(20K-l)-05*T*G(20K*1)
OBS-$110K‘I)+E(I0K~1)+T*G(10K—I)
DA=SII0K)“V(I0K1-05*T*G(10K)

INTERIOR OR END POINT

IFIJ-K) 20470115001140

 

 

131

INTERIOR POINT CALCULATION
1140 C2805*10A+DC1
CItC2+DB
TEMP80"T*Z
E48C1+TEMP
S48C2‘TEMP
Z=S4~B+A/E4
GO TO 1240
END POINT -- CHECK BOUNDARY CONDITION
1150 GO TO (1160011600118001190)0LOOK
VELOCITY BOUNDARY CONDITION
1160 C4=DC-V(20K1
C3=C4+DB
TEMP=.5*T*Z
E4=C3+TEMP
S4=C4-TEMP
Z=S4-B+A/E4
GO TO 1260
STRESS BOUNDARY CONDITION
1180 X=S(20K)
C5:DB+X
E4=CS+T*Z
C68X-B
Z=C6+AIE4
GO TO 1280
STRAIN BOUNDARY CONDITION
1190 X=E(20K)
C7=X~DB
CBc-C7+DC
SA=C7-T*Z
C9=-B+A/X
ZSS4+C9
GO TO 1220
END POINT *- STRAIN B. C. -- SEIDEL 0R AITKEN CORRECTION
1220 IFINO-4) 12240122401221
AITKEN CORRECTION METHOD
1221 DELSI=SZ-51
DEL52=S3-52
DELS3=S4-S3
DELS4=SS~S4
RSSQ:(DEL$2*DEL$4-DELS3*DELS31IIDELSI*DELSS-DELSZ*DEL52)
DELSS280ELSS’DEL52
DELSSBzDELS4~DEL53
DENOMSBDELSS3~RSSOiDEL$52
DENOMINATOR CHECK
IF(ABSF(DENOH$1) 12240122401223
DENOMINATOR DOES NOT VANISH
1223 SS=S4~DEL53*(DELSA-RSSO*DELS3)/DENOHS
ZBSS+C9
N0=1

 

 

132

NUMBERINUNBER+1
GO TO 1220
C SEIDEL ITERATION
1224 $1252
$2353
S3854
$4855
1225 $5=C7‘T*Z
2355+C9
NUMBERtNUMBER+1
DIFF=ABSF155‘S4)
C CONVERSION CHECK TO TEN DIGITS
IF(55-0001) 12260122801228
1226 IF(55-00001) 12270122901229
1227 IF(55-0000011 12310123001230
1228 IFIDIFF~01E-101 12350123501237
1229 IFIDIFF-oIE-II) 12350123501237
1230 IF(DIFF-0lE-12) 12350123501237
1231 IFIDIFF-olE-13) 12350123501237
1235 V(Z0K)8C8+05*T*Z
S(20K1355
GI20KI=Z
C SEARCH FOR FIRST POINT WHERE G‘FUNCTION VANISHES
GO TO (20300203210MARK
2030 IFIZ) 20310203102032
2031 PRINT 90J0K

MARK=2
2032 CONTINUE
C ARE WE IN THE FIRST FIVE ROWS
IF(K-5) 12360123601300
C NEW PARTIAL DERIVATIVES ALONG THE ROW

1236 TEMP=S(20K)-S(10K)
SPP(K)8TEMP-SP(K)
TEMP=E(20K1-E(10K)
EPP(K)8TEMP-EP(K1
EPIK1=TEMP
GO TO 1300

C COUNTER FOR AITKEN CORRECTION
1237 NO=N0+1
C ARE THE ITERATIONS BOUNDED
IFINUMBER-MAX) 12200122002047
C INTERIOR POINT -— SEIDEL OR AITKEN DELTA SQUARE ITERATION
1240 IF(NO-4) 12440124401241
C AITKEN CORRECTION METHOD

1241 DELEI=E2~E1
DELE2=E3-E2
DELE3=E4-E3
DELE4=E5~E4
DEL31852~SI
DEL52=S3-52

 

 

1242

1243

1244

1245

1246
1247
1248
1249
1250
1251
1255

133

DELS3SS4-53

DELS4=55-S4
RESQ=(DELE2*DELE4-DELE3*DELE3)/(DELEI*DELE3‘DELE2*DELE21
RSSO:(DEL52*DELS4*DELS3*DEL53)/(DELS1*DELSS-DEL$2*DELSZ)
DELEE2=DELE3-DELE2
DELEE3=DELE4~DELE3
DELSSZcDELSB-DELSZ
DELSSB:DELS4*DELS3
DENOMESDELEE3-RESO*DELEE2
DENOMSaDELSSB~RSSO*DELS$2
DENOMINATOR CHECK IN AITKEN METHOD
IFIABSFIDENOM511 12440124401242
IF(ABSF(DENOME)1 12440124401243
DENOMINATORS DO NOT VANISH
E5=E4*DELE3*(DELE4-RESO*DELE3)/DENOME
SS=S4-DELSB*(DELS4-RSSO*DELSS)/DENONS
Z=SS-B+A/E5

NO=1

NUMBERBNUMBER+1

GO TO 1240

SEIDEL ITERATION

E1=E2

E2=E3

E3=E4

E4=E5

51:52

52:53

53:54

54:55

TERN=05*T*Z

E5=C1+TERM

TEMPI=~B+A/E5

Z=S4+TEMP1

TERM=05*T*Z

$58C2~TERM

Z=55+TEMPI

NUMBERaNUMBER+1

DIFF:ABSF(E5-E4)

CONVERGENCE CHECK TO TEN DIGITS
IF(E5-0001) 12460124801248
IF(E5-00001) 12470124901249
IF(E5-000001) 12510125001250
IF(DIFF~01E-10) 12550125501257
IF(DIFF-olE-11) 12550125501257
IF(DIFF-01E-12) 12550125501257
IF(DIFF-01E-13) 12550125501257
S(20K1355

E(20K)=E5

V(20K)=05*(~DA+DC)

G(20K)=Z

 

 

2000
2001

2002

1256

1257

1260

1261

1262

1263

134

SEARCH FOR FIRST POINT ATTAINING STATIC S‘E VALUE
GO TO (20000200210MARK

IFCZ) 20010200102002

PRINT 90J0K

MARK=2

CONTINUE

ARE WE IN THE FIRST FIVE ROWS

1F(K*5) 12560125601045

NEW PARTIAL DERIVATIVES ALONG THE ROW
TEMP=S(20K)-S(10K)

SPPIK13TEMP-SPIK)

SP(K)=TEMP

TEMP=E120K)-E(I0K)

EPPIK13TEMP~EPIK1

EP(K)3TEMP

GO TO 1045

COUNTER FOR AITKEN CORRECTION

NO=NO+1

ARE THE ITERATIONS BOUNDED
IFINUMBER*MAX) 12400124002047

END POINT -- VELOCITY Bo Co -* SEIDEL OR AITKEN CORRECTION
IF(N0-4) 12640126401261

AITKEN CORRECTION METHOD

DELE1=E2-E1

DELE2=E3-E2

DELE38E4-E3

DELE4=E5~E4

DEL51852*51

DEL52=S3-52

DELS3=S4~S3

DELS4355—S4
RESO:(DELE2*OELE4-DELE3*DELE3I/IDELEI*DELES‘DELE2*DELE21
RSSG=(DELS2*DELS4-DELS3*DELS31/(DELSI*DELS3-DEL52*DELS21
DELEE2=DELE3*DELE2

DELEE3IDELE4-DELE3

DELSSZBDELS3‘DELSZ

DELSS38DELS4“DELS3
DENOMEaDELEE3-RESO*DELEE2
DENOMS=DELSS3-RSSQiDELSS2

DENOMINATOR CHECK

IFIABSFIDENOM511 12640126401262
IFIABSF(DENDME)) 12640126401263
DENOMINATORS DO NOT VANISH
E5=E4“DELE3*(DELE4-RESO*DELE3)/DENOME
S5854-DELS3*(DELS4-RSSQ*DELS3)IDENOMS
Z=SS-B+A/E5

N081

NUMBERSNUMBER+1

GO TO 1260

SEIDEL ITERATION

 

 

135

1264 E1=E2
E2tE3
E3=E4
E4885
51:52
$2=53
53854
84:55
1265 TERM=05*T*Z
E5=C3+TERM
TEMPI=~B+A/E5
Z=S4+TENPI
TERM=05*T*Z
$58C4-TERM
Z=SS+TEMP1
NUMBERaNUMBER+1
DIFF=ABSFIE5-E4)
C CONVERSION CHECK T0 TEN DIGITS
IF<E5~0001) 12660126801268
1266 IF(E5-00001) 12670126901269
1267 IFIE5-000001) 12710127001270
1268 IF(DIFF-01E-10) 12750127501277
1269 IF(DIFF-01E-11) 12750127501277
1270 IF<DIFF~01E-12) 12750127501277
1271 IF(DIFF-olE-13) 12750127501277
1275 S(20K)=55
E(20K18E5
G(20K)8Z
C SEARCH FOR FIRST POINT ATTAINING STATIC S-E VALUE
GO TO (20100201210MARK
2010 IF(Z) 20110201102012
2011 PRINT 90J0K

MARK=2
2012 CONTINUE
C ARE WE IN THE FIRST FIVE ROWS
IF(K-5) 12760127601300
C NEW PARTIAL DERIVATIVES ALONG THE ROW

1276 TEMP=S(20K)~S(10K1
SPP(K)3TEMP-$P(K)
SPIK)=TEMP
TEMP=E(20K)-E(10K)
EPP(K)=TEMP~EP(K)
EP(K)=TEMP

GO TO 1300
C COUNTER FOR AITKEN CORRECTION
1277 NO=NO+1
C ARE THE ITERATIONS BOUNDED
IF(NUMBER~MAX) 12600126002047
C END POINT -- VELOCITY 80 C0 -- SEIDEL OR AITKEN CORRECTION

1280 IF(NO-4) 12840128401281

 

 

136

C AITKEN CORRECTION METHOD

. 1281 DELEIBEZ“EI
DELE2=E33E2
DELE3=E4‘E3
DELE48E5~E4
RESO:(DELE2*DELE4'DELE3*DELE3)/(DELE1*DELES-DELE2*DELE2)
DELEE2=DELE3‘DELE2
DELEE3=DELE4-DELE3
DENOME=DELEE3-RE50*DELEE2

C DENOMINATOR CHECK
IF(ABSF(DENOME1) 12840128401283
C DENOMINATOR DOES NOT VANISH
1283 E5= E4-DELE3*(DELE4JRESO*DELE3)/DENOME
Z=X-B+A/E5
NO=1
NUMBERSNUMBER+1
GO TO 1280
C SEIDEL ITERATION
1284 E1=E2
E2tE3
E3=E4
E48E5
1285 E5=C5+T*Z
Z=C6+A/E5
NUMBERsNUMBER+1
DIFF:ABSF(E5-E4)
C CONVERSION CHECK TO TEN DIGITS

1F(E5-0001) 12860128801288
1286 IF(E5~00001) 12870128901289
1287 IF(E5‘000001) 12910129001290
1288 IF(DIFF*0IE-IO) 12950129501297
1289 IF(DIFF-01E-11) 12950129501297
1290 IFIDIFF‘olE-IZ) 12950129501297
1291 1F(DIFF-01E-13) 12950129501297
1295 V(20K18DC-X-05*T*Z
E(20K13E5
G(20K)=Z
C SEARCH FOR FIRST POINT ATTAINING STATIC S—E VALUE
GO TO (20200202210MARK
2020 IF(Z) 20210202102022
2021 PRINT 90J K

MARK=2
2022 CONTINUE
C ARE WE IN THE FIRST FIVE ROWS
IF(K-5) 12960129601300
C NEW PARTIAL DERIVATIVES ALONG THE ROW

1296 TEMP=SI20K)-S(10K)
SPPIK18TEMP-SPIK)
SPIKI=TEMP
TEMP=E(20K1-E(10K1

 

1297

1300

1310

1320

1330

1340

1350

2047

2046

2045

137

EPPIKI=TEMP*EPIK1

EPIK)=TEMP

GO TO 1300

COUNTER FOR AITKEN CORRECTION
NO=N0+1

ARE THE ITERATIONS BOUNDED
IFINUMBER-MAXI 12800128002047

ARE WE PRINTING THIS ROW

IF(J-MI) 13300131001330

DO WE PRINT THIS COLUMN

IFIK-M21 13300132001330

PRINT THIS POINT

STRESS=E0*S(20K)

VELOC‘CO*V(20K)
PRINT20J0K0STRESS0E120K)0VELOC0GI20K10NUMBER
NEXT ROW AND COLUMN TO BE PRINTED
M1=M1+M4

M2=1

HAVE WE CALCULATED THE LAST ROW
IFIJ“L) 13400204702047

STORAGE TRANSFER BEFORE MOVING ON TO NEXT COLUMN
DO 1350 N310J

S(I0N)=S(20N)

E(ION)=E(20N1

V(10N)=V(20N)

G110N1=G120N1

GO TO FIRST POINT IN NEXT COLUMN

J=J+1

K=1

NUMBERzo

CHECK B. C. -- IS LEADING.UAVE FRONT CALC. NEEDED

GO TO (10010104510IMPACT

HAVE WE REACHED LAST ROW

IF (J-L) 20460204502046
ERROR STOP -- OUTPUTS STORAGE
PRINT 410JOKONUMBER

PROGRAM OVER

STOP

END

END

 

138

 

oo moooooooomb. ~o+mnmmnnnmmn~. oo
moooooooooo.smo wooooooooou. so mooooooomcu. oo
wooooooooom. moamooooooooo.. moamoooooooomm. mo

«000 ~0¢0 0~00 0300 H000 «~00

m0 m0000000m>~0
w00000000000 MO w000000000~0
w000000000—0 N01w000000000~0
w000000000H0 00 NOOOOOOOOOM0
OMOO N000 0000 ~0¢0 ~000 «000

zauhuozou >I<OZDDD mmmka kz<kmzou II wk<aJ<z £<awoaa In <k<o MJaz<m

 

139

PROGRAM POWRATE

1 FORMAT I1X03HR0W03X06HCOLUNN09X06HSTRESS013X06HSTRAIN0
111X08HVEL0CITY09X010HG FUNCTION08X0IOHITERATIONS/I

2 FORMAT I1X0I305X0I304X04IE1601003X)03X0I3)

3 FORMAT (14(1401X11

31 FORMAT (10(1X01411
32 FORMAT (6(1X0E16010)1
33 FORMAT (4(E1601011

4 FORMAT (5(E16010)’

41 FORMAT (1X02HJ30I303X02HK=0I303X07HNUMBER=0I31

5 FORMAT 11X02HJ80I304X05HL00K801204X03HM1=01304X03HM3=0
II304X04HMAX=0I3/1X02HK=01304X05HJAZZB0I204X03M72=0130
24X03HM430I304X02HL=0I303X03HLL=0I303X07H1NPACT=01303X0
34HMAD=013/) ‘

6 FORMAT IIX03HE0=0E1601004X04HRH030E1601004X03HC0=0
1E1601004X02HT=0E16010/)

7 FORMAT I1X02HA80EI601004X02HB=0E1601004X02HC80E160100
14X02HD30E1601004X02HP80E16010/0/1X06HYIELDB0E1601004X0
25HVMAX=0E1601004X05HTF1X=0E1601004X05HCENT80E16010/0
3/1X02HQ30E1601004X02HF30E1601004X04HTIM=0E16010/0/)

8 FORMAT I1X06HALPHA=0E160IO04X05HBETAB0EI601004X0
116HYIELD REACHED AT01X0E1601002X0
225HMICROSECONDS AFTER IMPACT/0/1’

9 FORMAT l/1X025HG FUNCTION a .1E~07 AT J801403X02HK=0I4/)

91 FORMAT (/0/)
DIMENSION 5(2060110E120601)0V(20601)06120601105PI601)0
ISPPI60110EPI6OI)0EPPI6011
MARK=1
E13000
E23000
E33000
E43000
$13000
$23000
S3=000
‘548000
D0 98 N310401
S(10N18000
3(20N13000
E(I0N13000
E(20N)‘000
V(I0N18000
V(20N13000
G110N1=000

98 GI20NI=OOO
D0 99 N310401
EPINIPOOO
EPPINI=OOO
SPIN13000

99 SPP1N1'000

 

999

100

101

200

201

202

210

220
225

300

301

302

140

READ 30J0K0L0LOOK0JAZZ0MAX0MI0M20M30M40LL0IMPACT0MAD
READ 40T0EO0RH00A0B0C0D0p0YIELD0VMAX0TFIX0CENT000F0TIM
CCEO/A

COBSORTFIEO/RHOI

FRINT 50J0LOOK0M10M30MAX0K0JAZZ0M20M40L0LL0IMPACT0MAD
PRINT 60EO0RHO0CO0T
FRINT'70A0B0C0D0p0YIELD0VMAX0TFIX0CENT000F0T1M
NUMBER=0

VELOCITY 0 STRESS OR STRAIN BOUNDARY CONDITIONS AT X30
GO TO (10002000300040010LOOK

IMPACT VELOCITY IS A CONSTANT

READ 40VOL

V(10118-VOL/CO

E110113‘VII01)

$110118‘V110I)

DO 101 N=10L

V(20N13V1101)

GO TO 500

IMPACT VELOCITY IS A KNOWN VARIABLE

GO TO (201021010JAZZ

IMPACT VELOCITY IS A TABULATED VARIABLE IN/SEC
READ 401V120N10N=JOL011

DO 202 N=J0L01

V(20N18VI20N1/CO

GO TO 225

IMPACT VELOCITY IS GIVEN AS AN EXPONENTIAL FUNCTION
TEMPRIOO/(100*CENT1

ALPHASLOGFITEMPI/TFIX

TEMP=EO*VMAX

BETA=LOGF(TEMP/(TEMP+CO*YIELD)I

PRINT 80ALPHA0BETA0TEMP

DO 220 N=JOL

WER82*(N-1)

TEMPBEXPF(ALPHA*T*WER+BETA)
V(20N):VMAX*(100-100/TEMP1/CO

V(10113VI201)
E(101)=(*SORTF(RHO/IA*BI)*B+100)*V(1011*005*C0)**

1(200/(B+100))

S(l0I)IIA*(E(IOI)**B)1/EO
S(201):S(101)

E(20113E1101)
G1201)IolE-O7*(C*S(20I1/(E(2011**B))**G
GO TO 500

IMPACT STRESS IS A KNOWN VARIABLE
GO TO (301031010JAZZ

IMPACT STRESS IS TABULATED PSI
READ 4015(20N10N8JOL1

DO 302 N=J0L

S(20N185120N1/EO

00‘T0 829

 

310

311

312
313
325

400

412

413
425

500
503
504

1000

1001
1002

141

READ 40STRS

DO 311 NIJOLL

S(20N18STRS/EO

LLaLL+1

IF(LL-L) 31203120325

DO 313 N8LLOL

$120N13000

511011851201)

E(I011‘SII01)

V1101)8*S(101)

WILL BE RECOMPUTED BELOW IF INITIAL IMPACT IS PLASTIC
E(201)'E(101)

V(201)8V(10I)
G120])801E‘07*(C*S(2011/(E12011**811**0
GO TO 500

STRAIN HISTORY IS GIVEN

READ 401E120N10N=J0LL1

IF(LL-L) 41204250425

STRAIN IS NOW CONSTANT

LLLaLL+1

DO 413 N=LLL0L

E(20N)=E(20LL)

S(I0IJIEI201)

E110118E02011

V110118-E1201)

WILL BE RECOMPUTED BELOW IF IMPACT IS INITIALLY PLASTIC
$120113E1101) '
V120113V1101)
G(2011801E~07*(C*S(201)/IE(201)**8))**O
GO TO 500

PRINT OUT OF IMPACT CONDITIONS

PRINT 1
61101)nolE-07*(C*S(101)/IE(101)**8)1**0
STRESS:S(101)*EO I

VELOCIVII01)*CO

PRINT 20J0K0STRESS0EII01)0VELOC0GI1011
GO TO FIRST POINT AFTER IMPACT

J=J+1

GO TO (100001045101MPACT

CALC- ALONG X8C0*T FOR PLASTIC INITIAL IMPACT
TT=(20O/T)*01E+O9/(0*(100-81-100)
DOT=S(101)/(C*S(1011**(100-B))**Q
TEMP2P/(C*P**(10O“B))**G
F1=TT*(TEMP-DOT)

TszJ-I

P2=P

P=P+D

TEMP8P/(C*P**(10O‘B)’**Q
F2=TT*(TEMP-DOTI

DIBFZ-TAX

 

1010

1020
1030

1031

1040
1041

1045
1050

'1055

1060

1100
1110
1111

1113
1120

, 1025

1129

142

IFIDI) 10100101001020
F1=F2
GO TO 1002
IF(DI‘0IE“09)
P2=P1
P1=P
R=F2-F1
P2=P1+IP2-P1)*DI/R
TEMP=P2/IC*P2**(100‘B))**O
F1=TT*(TEMP-DOT)

DIGTAX~F1

NUMBERaNUMBER+1

IF(DI-01E‘O9) 10400104001031

P=P2

512011=P

E120118P

V1201)3*P

6(201)=0IE~07*(C*P/(P**81)**Q

F1=F2

D8099*D

DO WE PRINT THIS ROW

IF (J-MI) 10600105001060

DO WE PRINT THIS COLUMN

IF(K-M2) 10600105501060

PRINT THIS POINT

M2=M2+M3

STRESSIEO*S(20K)

VELOC=CO*V(20K)

PRINT 20J0K0STRESS0EI20K)0VEL0C0G120K)0NUMBER
MOVE TO THE NEXT POINT IN COLUMN

K=K+1

NO=O

NUMBERSO

GO TO 1100

IFIJ-K~31 11100112001125

IF(J-K-1) 11110111301113
E4=E(l0K-11-EP(K-1)

S4=SIIOK-1)-SP(K-I)

GO TO 1129

E4=ECI0K1+EP(K-I)-EPP(K‘1)
$435110K)+SP(K~I)-SPP(K‘II

GO TO 1129

E4=E110K1+EPIK1

5485(10K)+SP(K)

GO TO 1129

E4=E(10K)+EP(K)+EPP(K)
S4=SLIOK)+SP(K)+SPP(K)

GO TO 1129

IMPACT STRESS IS A CONSTANT -- THEN ZERO
ZzolE-07*(C*54/(E4**B))**Q

10410104101030

 

1130

1140

1150

1160

1180

1190

1191
1192

1193
1235

2030
2031

2032

143

CONSTANTS WHICH ARE FUNCTIONS OF ADJACENT STATE POINTS

-DC=S(20K¢11+V120K-1)“.5*T*G(20K*11

DBa-SIIOK‘1)+E110K“I1+T*G(10K‘11
DA=SII0K)‘V(10K1‘05*T*G110K1
INTERIOR OR END POINT

IFIJ-K) 20470115001140

INTERIOR POINT CALCULATION
C2=05*(DA+DC)

C1=C2+DB

TEMP=05*T*Z

E4=C1+TEMP

S4=C2-TEMP
Z301E‘07*(C*54/(E4**B))**0

GO TO 1245

END POINT “- CHECK BOUNDARY CONDITION
GO TO (116001160011800119010LOOK
VELOCITY BOUNDARY CONDITION
C4=DC‘V(20K1

C38C4+DB

TEMP:= 05*T*Z

E48C3+TEMP

S4=C4-TEMP
ZS.1E‘07*1C*54/1E4**BII**Q

GO TO 1265

STRESS BOUNDARY CONDITION
X=SIZOK1

C5=DB+X

E4=C5+T*Z
Z=0IE-O7*(C*x/(E4**BI1**Q

GO TO 1285

STRAIN BOUNDARY CONDITION
X=E120K1 ’

C7:X*DB

C88-C7+DC

SUN=C/(X**B)

1F1J~L1 11920119101192
55:11050E+08*ABSF(E120K)-E(10K4}))/T)**(1.0/Q))/SUN
GO TO 1193
S5=((025E+08*ABSF(E(20K+1)‘EII0K’1)1/T)**(100/O))/SUN
Z=01E-07*155*SUN)**Q
V(20K1=C8*05*T*Z.

S(20K1855

G120KI=Z

SEARCH FOR FIRST POINT ATTAINING STATIC S~E VALUE
GO TO (20300203210MARK
IF‘Z-OIE-07) 20310203102032
PRINT 90JOK

MARK=2

CONTINUE

ARE WE IN THE FIRST FIVE ROWS

 

1236

1240

1241

1242

1243

1244

1245

144

IFIK-5) 12360123601300 .

NEW PARTIAL DERIVATIVES ALONG THE ROW
TEMP=SI20K1*$(10K1

SPPIKIITEMP~SPIK1

TEMP=E(20K)»E(10K1

EPPtKIITEMP-EPIK)

EP(K)=TEMP‘

GO T0-1300

INTERIOR POINT ~~ SEIDEL OR AITKEN DELTA SQUARE ITERATION
IF(NO-4) 12440124401241

AITKEN CORRECTION METHOD

DELE1=E2~E1

DELE2=E3-E2

DELE3=E4-E3

DELE4=E5~E4

DELSIBSZ-SI

DEL52=S3-52

DEL53=S4-S3

DELS4=SS~S4
RESO:(DELEZGDELE4-DELE3*DELE3)/(DELE1*DELE3-DELE2*DELE21
RSSQ:(DEL52*DELS4-DEL53*DEL531/(DELSI*DELSS-DEL52*DEL521
DELEEZIDELE3-DELE2

DELEE3=DELE4~DELE3

DELSS2IDELS3-DELSZ

DELSSB=DEL54~DEL83
DENOME=0ELEE3-RESO*DELEE2
DENOMS=DELSS3-RSSO*DELSSZ
DENOMINATOR CHECK IN AITKEN METHOD
IFIABSF(DENOMS)) 12440124401242
IFIABSFIDENOMEJI 12440124401243
DENOMINATORS DO NOT VANISH
E5=E4~DELE3*(DELE4-RESO*DELE3I/DENOME
SS=S4-DEL53*(DELS4-RSSO*DELS3)IDENOMS
Z:.1E-07*(C*SS/(E5**B)1**Q

NO=1

NUMBERSNUMBER+1

GO TO 1240

SEIDEL ITERATION

E1362

£2053

E3254

E4=E5

$1852

$2853

$3854

S4=S5

TERM=05*T*Z

E5=C1+TERM

TEMP1=C/(E5**B)
Z=01E-07*(S4*TEMP1)**O

 

1246
1247
1248
1249
1250
1251
1255

2000
2001

2002

1256

1257

1260

1261

145

TERM305*T*Z

S5302‘TERM

2801E‘07*IS5*TEMP11.*O
NUMBERINUMBER+1

DIFFSABSFIE5‘E4)

CONVERGENCE CHECK TO TEN DIGITS
IFIE5“0001) 12460124801248
IFIE5‘00001) 12470124901249
IFCE5‘000001) 12510125001250
IFCDIFF~01E‘10) 12550125501257
IFIDIFF-olE-III 12550125501257
IFIDIFF-01E-12) 12550125501257
IFIDIFF‘olE‘13) 12550125501257
S(20K1855

E(20KIIE5

V(20K1805*(-DA+DC)

G(20K)3Z

SEARCH FOR FIRST POINT ATTAINING STATIC S‘E VALUE
GO TO (20000200210MARK

IFIZ'oIE‘07) 20010200102002

PRINT 90JOK

MARK=2

CONTINUE

ARE WE IN THE FIRST FIVE ROWS

IFIK-5) 12560125601045

NEW PARTIAL DERIVATIVES ALONG THE ROW
TEMP=S(20K)-S(10KI

SPPIKISTEMP-SPIK)

SPIKIPTEMP

TEMP=EI20KI-EIIOK)

EPPIK18TEMP-EPIKI

EPIKI=TEMP

GO TO 1045

COUNTER FOR AITKEN CORRECTION

NO=N0+1

ARE THE ITERATIONS BOUNDED
IFINUMBER“MAXI 12400124002047

END POINT -- VELOCITY Bo C0 ‘* SEIDEL OR AITKEN CORRECTION
IFINO‘4) 12640126401261

AITKEN CORRECTION METHOD

DELE13E2‘E1

DELE2=E3-E2

DELE3=E4-E3

DELE43E5*E4

DEL51352-SI

DEL52353-52

DELS3354-S3

DELS4355-S4
RESOBIDELEZ‘DELE4*DELE3*DELE3I/(DELEI*DELE3-DELE2*DELE2I
R5508(DELS2*DELS4-DELS3*DEL53I/IDELSI*DELS3-DELS2*DELS2I

 

146

DELEEZnDELEa-DELEZ
DELEES-DELE4-DELE3
DELSSZIDELSB-DELSZ
DELSSS=DELS4‘DEL53
DENOME=DELEE3~RESQ*DELEEZ
DENOMS:DELSSS-RssaioELSS2
C DENOMINATOR CHECK
IF(ABSF(DENOMS)1 12640126401262
1262 1F(ABSF(DENOME)) 12640126401263
C - DENOMINATORS 00 NOT VANISH
1263 E5=E4-DELE3*(DELE4—RESO*DELE31IDENOME
SS=S4~DEL$3*(DELS4-RSSO*DEL53)/DENOMS
Z=.1E-07*(C*55/(E5**81)**O
NO=1
NUMBEReNUMBER+1
GO TO 1260
C SEIDEL ITERATION
1264 E1=E2
E28E3
E33E4
E4=E5
$1852
$2833
$3854
$4855
1265 TERM=05*T*Z
E58C3+TERM
TEMP1=C/(Efi**81
2801E~07*(S4*TEMP11**0
$5=C4-TERM
ZaolE-07*(55*TEMP1)**O
NUMBERsNUMBER+1
DIFF=ABSF(E5-E4)
C CONVERSION CHECK T0 TEN DIGITS
IF(E5~0001) 12660126801268
1266 IF(E5-00001) 12670126901269
1267 IFIE5-000001) 12710127001270
1268 IFIDIFF‘olE-IOI 12750127501277
1269 IF(01FF~01E~11) 12750127501277
1270 IFIDIFF-01E*12) 12750127501277
1271 IFIDIFF-olE-IS) 12750127501277
1275 S(20K1355
E(20K18E5
G120K18Z
C SEARCH FOR FIRST POINT ATTAINING STATIC S-E VALUE
GO TO (20100201210MARK
2010 IFIZ-01E-071 20110201102012
2011 PRINT 90J0K
MARK82
2012 CONTINUE

 

147

C ARE WE IN THE FIRST FIVE ROWS
IF(K‘5) 12760127601300
C NEW PARTIAL DERIVATIVES ALONG THE ROW

I276 TEMPBSI20KI‘SII0K)
SPPIK18TEMP-SPIK)
SPIKISTEMP
TEMPBEI20KI‘EII0K)
EPPIK):TEMP-EP(K)
EPIK1=TEMP
GO TO 1300
C COUNTER FOR AITKEN CORRECTION
1277 NO=NO+1
C ARE THE ITERATIONS BOUNDED
IF(NUMBER-MAX) 12600126002047
C END POINT ‘- STRESS B. C. ‘- SEIDEL OR AITKEN CORRECTION
1280 IFINO-4) 12840128401281
C AITKEN CORRECTION METHOD
1281 DELE1=E2*E1
DELE2=E3-E2
DELE3=E4-E3
DELE4=E5-E4
RESO=(DELE2*DELE4-DELE3*DELE31/(DELE1*DELE3’DELE2*DELE2)
DELEEZSDELE3“DELE2
DELEE3BDELE4—DELE3
DENOME=DELEE3-RESQ*DELEE2

C DENOMINATOR CHECK
IFIABSF(DENOME)) 12840128401283
C DENOMINATOR DOES NOT VANISH

1283 E5=E4~DELE3*(DELEa-RE$O*DELE3)/DENOME
ZzolE-07*(C*X/(E5**B))**Q
NO=1
NUMBER:NUMBER+1
GO TO 1280
C SEIDEL ITERATION
1284 E1852
E2=EB
E3=54
E4=E5
1285 E5=C5+T*Z
=.lE-O7*<C*XI(E5**B))**Q
NUMBERaNUMBER+1
DIFF=A8$FIES-E4)
C CONVERSION CHECK TO TEN DIGITS
IF(Es-0001) 12860128801288
1286 IFtES-O-OOI) 12870128901289
1287 1F(E5-0.000I) 12910129001290
1288 IF(DIFF-olE-10) 12950129501297
1289 IFtDIFF-clE-ll) 12950129501297
1290 IFtDIFF-olE-IZ) 12950129501297
1291 1F(DIFF-.lE-13) 12950129501297

 

 

1295'

2020
2021

2022

1296

1297

1300

1310

1320

1330

1340

1350

2097

148

V(20K18DG-X-05*T*Z

E(20K13E5

6(20K1lz

SEARCH FOR FIRST POINT ATTAINING STATIC S‘E VALUE
GO TO (20200202210MARK

IFCZ-olE-O7) 20210202102022

PRINT 90JOK

MARK=2

CONTINUE

ARE WE IN THE FIRST FIVE ROWS

IF(K-5) 12960129601300

NEH PARTIAL DERIVATIVES ALONG THE ROW
TEMPSS(20K)-S(10K)

SPPCK13TEHP-SP(K)

SPCKI=TEMP

TEMP=E(20K)-E(10K)

EPP(K)8TEMP-EP(K)

EP(K)=TEMP

GO TO 1300

COUNTER FOR AITKEN CORRECTION

N0=N0+1

ARE THE ITERATIONS BOUNDED
IFINUHBER-MAX) 12800128002047

ARE WE PRINTING THIS RON

IFCJ'M1) 13300131001330

DO WE PRINT THIS COLUMN

IF(K-M21 13300132001330

PRINT THIS POINT

STRESSBEO*S(20KI

VELOCICO*V(20K)
PRINT20J0K0STRESS0E120K)0VEL0C0G(20K)0NUMBER
M18M1+M4

M281

HAVE HE CALCULATED THE LAST ROI
IF(J-L) 13400204702047

STORAGE TRANSFER BEFORE MOVING ON TO NEXT COLUMN
DO 1350 N810J

S(10N)SS(20N)

E(l0N)sE(20N)

V(I0N)8V(20N)

G(10N)'G(20N1

GO TO FIRST POINT 1N NEXT COLUMN
JIJ+1

K31

NUMBER80

CHECK B. C. -- IS LEADING WAVE FRONT CALC. NEEDED
GO TO (100101005101MPACT

HAVE UE REACHED THE LAST ROW

IF(J‘L1 20460204502006

ERROR STOP -- OUTPUTS STORAGE

m“-

C

2046

2045
2044

2043

2042

149

PRINT 410JOKONUM8ER
PROGRAM OVER
IF(MAD~21 20420204402042
MITsTIM

DO 2043 N'IOLOM1T
NN=1+(N-1)/MIT
SP(NN)3TIM*SP(N)
SPP(NN)3TIM*SPP(N)
EPINN)‘EP(N)*TIM
EPP(NN)'EPP(N)*TIM
S(I0NN)=S(20N)
E(10NN13E120N)
V(I0NN)'V(20N)
G(10NNI.G(20N)
S(20NN)‘S(20N)
E(20NN)=E(20N)
V120NN)=V(20N)
G‘20NN1=G(20N)
PRINT 91

GO TO 999

CONTINUE

STOP

END

END

 

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