AN EXMREMENTAL STUDY OF." ALUMINUM AT HiGH
3&th RATES AND EkE‘VATED TEMPERATURES

Thais Got flu beam 04 ML D.
MfCl‘K’lfiAN SfATE UNNERSiTY
Jerry L. Chiddisfar
1961

This is to certify that the

thesis entitled

AN BXFERIi-‘EET'ETAL STUDY OF ALITT-TII‘TITT’. AT HIGH
STRJXIFI RATES MED ELE‘JI’XTT‘TD 'I‘ET”PT?RATI‘3ES

presented by

JERRY L. CHI DDISTE

has been accepted towards fulfillment
of the requirements for

DOCTOR OF PI‘iILOSOPHY degree in APPLIED TJ‘EWATJICS

@6445

Major professor

Date gm AA]; /76/

0-169

 

 

LIBRARY

Michigan State
University

 

A BSTRACT
AN EXPERIMENTAL STUDY OF ALUMINUM AT HIGH STRAIN

RATES AND ELEVATED TEMPERATURES

by
Jerry L. Chiddister

In this investigation short cylindrical specimens of aluminum
were subjected to impact loads by means of a split Hopkinson pressure

bar apparatus in which a third bar was used as a striker bar to

produce a long flat—topped loading pulse. Tests were conducted at

strain rates from 300 to 2000 inches per inch per second and at six

0 .
temperatures from 30 to 550 C. True stress-true strain curves were

obtained at nearly constant strain rates. The curves extended to

approximately 5 percent strain for the lowest strain rates and to

over 25 percent strain for the highest strain rates. Reliable data

could not be obtained over the elastic portion of the stress-strain

curves due to the nature of the loading pulse. Continuous records

of stress and strain were obtained throughout the tests by means

of strain gages mounted on the pressure bars. The area surrounding

the specimen was heated by an electric resistance -type furnace.

Data was taken from the stress-strain curves to construct
semi—logarithmic and 10g~10g plots of stress versus strain rate.
Both coordinate systems produced quite linear distributions of the

data, which indicated that either the 10garithmic flow law,

0' =Oo+klog 6

or the power law,

.n
0:0' 6
0

could be used to represent the data. In these laws 0 o’ k and n

are functions of strain and temperature and (I o is the stress at

unit strain rate. Values of 0' 0 were determined for both laws by

extrapolating back to the abscissa of unit strain rate. Values of k/o o

and n, which characterize the strain-rate sensitivity of the material,
are given for a strain of 5 percent at all test temperatures. The data
was not sufficient for determining the variation of these parameters
with the level of strain.

When the results were correlated with the data of Alder and

Phillips, obtained from compression tests at lower strain rates, the

power law was found to give a better fit to the plotted points.

Stress-temperature curves were plotted at several strain

rates and for several levels of strain.

AN EXPERIMENTAL STUDY OF ALUMINUM AT HIGH
STRAIN RATES AND ELEVATED
TEMPERATURES

by

U
Jerry L." Chiddister

A THESIS

Sdbmitlte—‘d to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOC TOR OF PHILOSOPHY

Department of Applied Mechanics

1961

ACKNOW LEDGE MEN TS

I wish to express my gratitude to Dr. L. E. Malvern
who suggested the problem and gave valuable guidance and counsel
throughout the project. Special thanks are also due to Dr. C.A. Tatro
for assistance in solving the prcblems of instrumentation. Sincere
thanks are due to the other members of my guidance committee, Dr.
C. 0. Harris, Dr. G. E. Mase and Dr. C. P. Wells.

I would also like to thank Mr. Donald Childs and Mr. R.
Jenkins for assistance in constructing the apparatus and machining

the specimens.

TABLE OF CONTENTS

LIST OF TABLES .................... V
LIST OF FIGURES .................... vi
CHAPTER I INTRODUCTION . ............ l
1 1 General Discussion and Purpose ,,,,,, l
l. 2 Some Theories of Dynamic Plastic DeformationZ
l. 3 History of Experimental Techniques , , , , 8
1.3.1 Hopkinson Pressure Bar ,,,,, 9
1.3.2 Compression Impact ,,,,,,, 13
1.3. 3 Tensile Impact ,,,,,,,,,, 18
1.3.4 Plastic Wave Propagation ,,,,, 20
CHAPTER II EXPERIMENTAL METHOD ,,,,,,, 22
2- 1 General Description and Objectives . 22
2. 2 Pressure Bar System ............ 23
2. 3 Impact Loading Machine .......... 30
2. 4 Furnace and Temperature Control ..... 32
Z. 5 Strain Measuring and Recording Equipment 32
2. 6 Specimens ................. 35
Z. 7 Derivation of Stress and Strain in the
Specimen . ................ 35
CHAPTER III EXPERIMENTAL PROCEDURE ..... 44
3-1 Determination of the Elastic Wave Velocity. 44
3 2 Dynamic Calibration of Strain Gages . . . . 45
3 3 Specimen Size and Material Effects ..... 48
3 4 Test Procedure ............... 52
3.5 Reduction of Data .............. 54
CHAPTER IV RESULTS ................ 57
4.1 Oscilloscope Records ............. 57
4. Z Deformation in Specimens ......... 59
4. 3 Strain-Time Curves ............. 59
4. 4 Stress-Strain Curves ------------ 64
4. 5 Stress-Strain Rate Curves --------- 67
4 6 Stress- -Temperature Curves -------- 34
CHAPTER V SUMMARY AND CONCLUSIONS ..... 90
BIBLIOGRAPHy........... .......... 92
APPENDIX A CALCULATION OF WAVE .
REFLECTIONS DUE TO TEMPERATURE CHANGES 100
APPENDIX B EXPERIMENTAL SUPPORT FOR THE
METHOD USED IN CALCULATING SPECIMEN ‘ 111

S TRAIN .......................

iii

Verification of the Relation e :6
Comparison of Calculated and Meeasured
Specimen Strains . . - - --------

iv

Page
111

TABLE I

TABLE 2

LIST OF TABLES

Page
Values of k, (I o’ k/ 0 , n and 00'
in the Legarithmic andQPower Laws
at 5 Percent Strain . . . . . ....... 83

Values of n Obtained by Alder and Phillips . 83

16.
17.
18.

19.
20.

21.

22..
23.
24.

25.
26.

2.7.

28.

LIST OF FIGURES

Schematic Diagram of Kolsky Apparatus . .

Schematic Diagram of the Pressure Bar System
of Krafft, Sullivan and Tipper

General Arrangement of Habib Apparatus

Mechanical Arrangement of Manjoine- -Nadai

Apparatus .......
Experimental Test Setup
General View of Test Setup -
Close-up of Pressure Bars and Furnace
Dimensions of Striker and Pressure Bars
Pressure Bar Supports -
Strain- Time Records Showing Effect of

Dispersion ----- - - -
Schematic of Hyge Shock Tester
Barrel with Striker Bar -
Strain Gage Bridge ----------
Breakdown of the Strain-Time Records - -
Schematic of Specimen Showing Stresses and

DiSplacements - - -----

Striker and Incident Bar .....
Schematic Diagram of Calibration Apparatus - -

Comparison of Strains Determined from Impact
Velocity and the Incident Strain Gage Station -
Effects of Non—isotropy in Specimen Material
Typical oscilloscope records for 1/2 inch
diameter by 1/4 inch long aluminum specimens
Typical deformed specimens at each test
temperature
Strain- Time Curves at 30 OC
Strain- Time Curves at 550 OC ..........
Stress-Strain Curves at Various Test
Temperatures ...............
Stress-Strain Curves at Several Strain Rates
Stress Versus LOgO Strain Rate for 5 Percent

True Strain at 300 C
Stress Versus Log Strain Rate for 5 Percent

True Strain at 1500 C ..............
Stress Versus Log Strain Rate for 5 Percent

Ti'u'e Strain at 250 C ..............

Page
12

12
14

14
24
25
26
27
27

29
31
31
34
38

41
46
46

49
51

58
6O
61
62

65

66
68
68

69

Z9.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45..

46.

47.
48.

49.

50.

 

Stress Versus LOg Strain Rate for 5 Percent

True Strain at 350 C . . . . . . ........

Stress Versus Log Strain Rate for 5 Percent
True Strain at 4500 C . . .
Stress Versus Log Strain Rate for 5 Percent
True Strain at 5500 C ...... .
Log Stress VerSus Lag Strain Rate for

5 Percent True Strain at 300C . . . . . . . . .

Log Stress Versus Log Strain 0Rate for
5 Percent True Strain at 1500 C . . .
Log Stress Versus Log Strain Rate for
5 Percent True Strain at 2500 C
Log Stress Versus Log Strain CRate for
5 Percent True Strain at 350 C. .......
LOg Stress Versus Log Strain Rate for
5 Percent True Strain at 450 0.C
Log Stress Versus Log Strain Rate for
5 Percent True Strain at 550 O.C
Semi-logarithmic Stress Versus Strain Rate
Curves at 10. 54 Percent True Strain . .
Log-Log Stress Versus Strain Rate Curves

at 10. 54 Percent True Strain .........

Dependence of Strain Rate Effect Coefficient
k/O 0 on Temperature at 5 Percent True Strain
Dependence of Strain Rate Effect Exponent n

on Temperature at 5 Percent True Strain . . . .

Stress Versus Temperature at O. 05 Inches Per

Inch True Strain .................
Stress Versus Temperature at 0.1054 Inches Per

Inch True Strain .................

Stress Versus Temperature at a Strain Rate of

2000 Inches Per Inch Per Second .........
Bar with a Discontinuous Change in E ......

Modulus of Elasticity Versus Temperature for

18-8 Stainless Steel Pressure Bars .......

Close-up of High Temperature Gage Installation

Pulse Records at Center of Furnace in a Long. . .
Pressure Bar .................

Records Used for Measuring Temperature
Reflections in a Long Pressure Bar ....... ..
Strain Waves Reflected from a Heated Section
of a Stainless Steel Bar ..............

vi 1

Pa ge

70

71

72

73

73

74

74

75

76

78

79

85

85

86

105

51.

52.

53..
54.
55.
56.

Temperature Distribution in Pressure Bar . . . .

Measuring from Center of Furnace . .
Record of Incident, Reflected and Transmitted

Pulses . . .................
Record from Strain Gage Mounted on Specimen

Comparison of 61 and 6 . .
Close- -up of SpecimIen wiflh Gage Mounted on it
Specimen Strain Versus Time . . . . .

viii

Page
109

112
112
113
115
117

CHAPTER I
INTRODUCTION

1. 1 General Discussion and Purpose

 

The mechanical behavior of engineering materials under
conditions of dynamic loading is significantly different from that
under conditions of static loading. Among the first indications of
this was the fact that energy absorbed in the Charpy and Izod tests
depended on the rate of impact. It is also well known that mechanical
properties change markedly with temperature. The separate prob-
lems of determining these properties under dynamic loading conditions
and of determining them at elevated temperatures have been invest-
igated by a number of people. However, comparatively little has
been reported on the combined problem of determining mechanical
properties at impact rates of loading while the Specimen is at an
elevated temperature.

As might be expected, the problems involved in obtaining
experimental data, especially stress-strain curves of metals under
dynamic loading conditions were naturally directed toward increasing
the Speed of the quasi-static type testing machines. The range of
loading speeds attainable by this method is quite limited because at
higher Speeds it becomes difficult to separate the inertia forces of
the moving machine parts from the actual load on the. specimen. This
problem required new methods of measuring loads, and at the same
time new methods were required for accurately determining the strains
at these higher Speeds. If loading rates are increased further into ‘
the impact range, stress wave propagation effects in the Specimen
must also be taken into consideration if an accurate stress-strain

curve is to be obtained.

The problems involved in high temperature testing are not
quite as formidable as those arising in impact testing, but they offer
a considerable challenge when attempted in combination with impact
testing.

The purpose of the present experiment is to obtain stress-strain
curves of nearly pure aluminum at constant true strain rates of the
order 1000 per second and at several temperatures from room tem-
perature to near the melting point of the metal, the tests to be per-
formed on short cylindrical Specimens in compression impact. A
desired objective is to obtain accuracy sufficient to determine the

dependency of stress on the rate of strain for a particular value of

strain at each test temperature.

1. 2 Some Theories of Dynamic Plastic Deformation

A brief history of the development of the theory of plastic wave
pr0pagation will be given here, as it is a field which cannot rightfully
be separated from the field of dynamic testing of materials in the
anelastic range. Those who are interested in obtaining further in—
formation are referred to an extensive review of the literature in these
associated fields completed recently by Hopkins (1961). U

The theory of one dimensional elastic wave pr0pagation was
developed in the early part of the last century, but the theory of
plastic wave propagation is of more recent origin. It was Donnell (1930)
Who first devised a method of treating waves of propagation in a
material which does not obey Hooke's law. He treated the stress wave
in a long bar as a superposition of stress increments. Each increment

was assumed to travel at a velocity which depended on the Slope of

 

 

Surnames followed by dates in parenthesis refer to BibliOgraphy

the static stress—strain curve of the bar material at the stress level

of the increment. These velocities were given by the relation

’1 d
C23-a-E- (1.1)

where p is the mass density of the material, 0 is nominal stress and
e is nominal strain. This of course reduces to the well known relation
c0 = E/p for the case of elastic waves.

The problem of plastic wave pr Opagation was not developed
further until World War II when the details of the above method, in-
cluding solutions for the problem of longitudinal impact of a semi-infinite
bar, were worked out independently in this country by White and Griffis
(1948) and von Karman and Duwez (1950), in England by Taylor (1958)
and in Russia by Rakhmatulin (1945). The main assumptions of their
theory, sometimes known as the von Karman-Taylor theory, are
the following: There is one stress-strain curve for the bar material,
independent of strain rate; the stress-strain curve is concave towards
the strain axis, strain being a Single valued function of stress; and
radial inertia effects are negligible.

Soon after the deveIOpment of the theory, experiments were
carried out on long bars for the purpose of checking the principilo re-
sults (Duwez and Clark, 1947). The results of these experiments
agreed fairly well with the theory, but there were some discrepancies.
For instance the measured strains near the impact ends of the bars
varied from the predicted values. It was suggested that this variation
was due to strain rate sensitivity of the bar material.

It had been proposed earlier by several investigators that
stress at a constant temperature could be taken as a function of plastic

strain 6 p and plastic strain rate é p, such that

0 = E a g) 1.21
<1>( p p ( ,

The problem now remained for someone to incorporate a function of
this type into a solution of the plastic wave propagation problem. It
was Malvern (1950, 1951) who first did this. Malvern assumed that
for a particular level of plastic strain, the plastic strain rate é
could be taken as a function of G - 0‘0 where 0’ is the stress under
dynamic conditions and 0‘O is the stress given by the usual
quasi-static stress-strain curve. This function may be written in

the form
; :f(o-o) (1-3)
p 0

or in the form

0

\ca

. 0'

O=OO+F(6 -E4 (1.4)
The theory was worked out for this quite general flow and an example
was given in which a simplified version of the flow law was used. In
the simplified law é was assumed to be linearly proportional to

O - 00, that is

€p:k(O’-00; (1.5)

which may also be written in the form

0 =00+k (L6)

1
In Equations (1. 5) and (1.6) k and k1 are. coefficients which may de-
pend on the material, the level of strain and the temperature.

Malvern's numerical calculations based on a linear dependence
on strain rate did not appear to agree well with the kind of experimental
data available at the time. They did, however, predict that a small
dynamic increment superposed on a plastic strain distribution would
travel at the elastic wave velocity and not at the appropriate plastic
wave velocity as predicted by the von Karman-Taylor theory. This

point was verified by Sternglass and Stuart (1953) who applied elastic

tensile pulses to copper bars which had been preloaded into the plastic
region. Verification was also given by Alter and Curtis (1956) in
experiments with lead bars in compression and by Riparbelli (1956)

in experiments with copper wires loaded by tensile impact.

Malvern suggested that his theory might give better agreement
with experimental results if stress is assumed to be a non-linear
function of strain rate. One such functional relationship, derived
on the basis of a mechanical model of a solid, has beenpproposed by

Prandtl (1928). His relation may be written in the form

where (70 is the stress corresponding to unit strain rate and k2 is

a coefficient which may depend on the material, the level of strain
and the temperature. This relatien was also prOposed on empirical
grounds by Ludwik (1909) and by Deutier {1932)}. Prandtl first

arrived at the more general hyperbolic sine law

g :Clsinh “10 (1.8)

which reduces to the 10gar ithmic law, equation ('1. 7) for large values

of stress. Here C1 and (1 also depend on the level of strain, the

, 1
temperature and the structure of the material. A hyperbolic sine
law was also suggested by Nadai {Nadai and McVetty, 1943) on the
basis of the results of c: esp tests performed at elevated temperatures.

Still another flow law which has received considerable support

from experiments is the power law

0:0 6 (1.9)

Where 00 is the stress at unit strain rate and n depends on the
material, the temperature and the level of strain. Alder and Phillips
(1954)- found that the power law provided a slightly better interpretation

of their data from compression tests on c0pper, al/uminum and steel

over a wide range of temperatures. Sokolov (1946 ~1950), who per-
formed compressiontests on tin, lead, aluminum, zinc, copper,
nickel, brass and several steels at room and elevated temperatures,
found the power law to give much better agreement with his plotted
data than the 10garithmic law. This was true except for high melting
point steels whose behavior at room temperature could be described
quite well with either relation. The power law has also been used
to express yield stress as a function of strain rate (Hollomon and
Jaffe, 1947) and to predict the critical stress at which creep initiates
(Vitovec, 1957).

Several investigators have included the absolute temperature
T in the plastic flow laws which they have proposed. Among these

proposed relationships are the following in simplified form:

 

 

a O
E = C2 Sinh '1: 11.10)
% (130
ézc3€ sinh—;f- (L111
' - c “04(5) 0) ‘1 12
e — 48 T (- l
: ' 1 X . '
0 (:5 +1.31" n e (113)
0 -(°8 k4‘r '114‘
_ 6 T) l' i”
. k5T
6 i
O —OO(—€.-l (1‘15)
0
Wh : k a k 1
ere C2, C3, C4, C5, C12. <13, (14, 5» k3 4 5 00 and 60 may

depend on the material and the level of strain. Equation (1. 10) was
derived by Eyring (1936) from his general equation deve10ped from the

theory of reaction rates. Equation (1.11) is a general equation of flow

developed by Freudenthal (1950), also from considerations of a rate
process. The relations (1.12) and (1.13) were derived from theoretical
studies of single crystals by Becker (1926) and Orowan (1934) and by
Seeger (1954a, 1954b) respectively. Zener and Hollomon (1944) pera-
formed experiments on Several steels over a wide range of strain
rates and at temperatures between ~1900C and + 200C and their re—
sults gave good agreement with equation (1.14). This relationship
indicates an equivalence between an increase in strain rate and a
decrease in temperature. Sokolov (1949) arrived at (1.15) when he
included the effects of temperature on his experiments. Equation
(1.15), in another form, was derived earlier on a theoretical basis

by Witman and Stepanoff (19 39). It should be pointed out that for a
particular temperature (1.10) and (1.11) reduce to the form of the
hyperbolic sine law, equation (1. 8):. Also note that, when T is constant
(1.14) and (1.15) reduce to the power law, equation (1. 9).

Attempts to formulate theoretical flow laws are based on
considerations of activation energy, the energy level a particle must
attain in order to move across a potential barrier. If the energy
distribution over the atoms can be expressed by the Boltzmann-aMaxwell-a
Gibbs distribution function, then the rate of strain, which is pro-n

portional to the rate of activation, can be expressed by the relation

 

(1.16)
where U is the activation energy and R is the universal gas constant
(Freudenthal, 1950). It is seen that Equations (1.10), (1.11) and (1.12)
are simply special cases of Equation (1.16) where some special form
0f 11(0) has been used. Several examples of this type of function are
considered in connection with yield stress, recovery in the annealing
process and creep by Cottrell {1953).

MacGregor and Fisher- (1946) suggested combining the effect

of strain rate and temperature so that the behavior of metals could
be determined by a series of tensile tests involving only one variable.
Their experiments on steel indicated that the true stress in a tension
specimen depends only on the true strain and a "velocity -modified"
temperature

Tm = T(l - k 1n 56 ) (1.17)

o
where 5'0 is taken as 100 ‘ 10- per second and k is a constant.

A relation which includes strain as well as stress, strain
rate and temperature was derived by Lubahn (1947). His pr0posed

relation for plastic flow is

D C
T (E-FTlni)
6 6

o (1.18)

where C, D, E, F, G and 60 are constants of the material. The
derivation is based on three empirically supported basic relations,
one of which is the power law.

Noting the variety of the proposed plastic flow laws, one may
doubt the usefulness of any of them. It can hardly be expected that
one flow law would predict the behavior of every metal under all
conceivable conditions. One reason for this is the change in structure
which occurs in many metals with a change in temperature. Never-
theless, a relationship which predicts the behavior of a particular
metal for a sufficiently wide range of strain rates and temperatures

should be of considerable value.

1. 3 History of Experimental Techniques

 

The experimental methods which have been used to study
materials at loading speeds in the impact range may be placed in five

categories: (a) Hopkinson pressure bar (b) compression impact

(c) tensile impact ((1) plastic wave prOpagation and (e) torsion impact.
The first four, with emphasis on the HOpkinson pressure bar and
compression impact, will be discussed briefly here. For those who
are interested in torsion testing, reference is given to Work and
Dolan (1953) and to Goldsmith (1960) and the literature cited in these
references.
_l_.__3_._l_ HOpkinson Pressure Bar

B. Hopkinson (1914) was probably the fir st person to devise
an effective method for measuring the forces involved in a high velocity
impact situation. He determined the maximum pressures produced
by the detonation of high explosives and by the impact of bullets. The
bullets were fired against the end of a cylindrical steel bar producing
an elastic pulse which proceeded along the bar and into another shorter
bar of the same diameter. This second section of the bar was in effect
a momentum trap since the compressive pulse due to the bullet was
reflected as a tensile pulse from the free end of this short section
and when the amplitude of this tensile wave, arriving again at the
intersection of the bars, became greater than the amplitude of the
compressive pulse continuing toward the free end, the short end
piece would separate from the longer section and continue mOVing
with the momentum of the trapped portion of the pulse. By taking
end pieces of different lengths and measuring the momentum trapped
in each, the area under the pressure-time curve over corresponding
intervals could be obtained, the time interval being determined from
the length of the end piece and the known elastic wave velocity. In
general the exact shape of the pressure-time curve could not be
obtained by this method since the starting points of these intervals
could not be located in time, but the important values of maximum
pressure and total duration of impact could always be obtained. The

momentum in the end pieces was measured by catching the pieces in

a box supported as a pendulum from the ceiling, the angle of swmg
through which the box traveled be mg the measured quantity The
longer inc1dent bar was also supported 1n pendulum fashion and the
smaller end pieces were held 1n place before impact by a magnetic
force

The Hopkinson pressure bar testing method was not improved
conS1derably until the deveIOpment of modern electronic equ1pment.
Davies (1948) made a cr1tica1 study of the method and dev1sed an
electrical means to obta1n a continuous record of the part1cle d1s—
placements due to an elast1c pulse. The plane surface at the free
end of the pressure bar served as one side of a parallel plate capac1tor
which was connected to a cathode ray OSCIllOSCOpe. This gave a
d1rect measurement of the displacement of the end of the bar. The
pressure-time curve could be obtamed from a photOgraphic record
of these displacements. 11' a pressure wave is assumed to travel
at the elastic wave veloc1ty c0 : \fETp— in an uninterrupted portion
of a bar, then the pressure or stress 0 may be calculated from

du .
U I pC-OC-l-t— (119,

where u is the part1c 1e displacement at a pomt 1n the bar as the

wave passes that pomt, and t 15 time. Now the displacements of the
free end of a bar W111 be [W1ce as lar ge as those due to the or1g1na1
pulse because the displacemcrts of the reflected pulse are superposed
upon those of the or 1g1na1 pulse Therefore, the stress 1n Dav1es
method is calculated from the relation

1 du1

Oz-ZQCO'aT- (12.0

where u is the measured particle displacement at the end of the bar

1
Using the Pochhammer v-Chree equations for d1sper~31on in an

Infinitely long cy11ndr1ca1 bar. (See Pochhammer, 1876 and Chree, 1889.

11

Davies showed that the stress pulses are propagated without dis»-
tortion only if the wavelengths of the elastic waves involved in the
propagation of the pulse are large when compared with the diameter
of the bar. He showed that this condition is also necessary for an
even distribution of longitudinal diSplacements and stress at a cross
section of the bar.

In Davies‘s study of the pressure bar technique an experiment
was performed which adds still more to the validity of the method.

It was shown that the stress in a pulse is practically uniformly dis-
tributed over a cross-section of the bar at a distance of about four
diameters from the impact end even when the force applied is con-
centrated near the center of the end. This experiment was performed
by replacing the original momentum trap piece with a distribution of
small steel balls. This dynamic case of SaintVenant's principle has
also been verified to some extent by Flynn and Frocht (1961) with
preliminary photoelastic studies of elastic waves.

Kolsky (1949) investigated the mechanical properties of several
materials using a pressure bar technique which incorporated the
capacitor measuring device of Davies. A diagram of Kolsky's apparatus
is shown in Figure 1. The Specimen, a thin disc, was mounted be-
tween two steel pressure bars and the force was applied by means of
a small explosive char ge. An attempt was made to use a cylindrical
capacitor to obtain the longitudinal diSplacements of the incident bar,
but this did not prove reliable due to distortions introduced by the bar.
Without the use of this cylindrical capacitor, it was necessary to
obtain the displacement -time records of two pulses in order to
calculate the strain in each specimen. One pulse was fired without
a Specimen in place. The stress-time record was obtained by taking
the gradient of the displacement—time curve for the specimen in place.

Some other disadvantcages of Kolsky's method should be pointed

12

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Anvil Collar Inertia Cylindrical Parallel Plate
. \hSwitch / Capacitor Capaci or
I C0113” Specimen

Detonator .._:__ 7 J.

Feed Unit Feed Unit
'f' . .

Sweep Ampli ier Amplifier
Unit l

 

l

 

 

 

 

CRO

 

 

 

Figure 1. Schematic Diagram of Kolsky Apparatus

SR -4 Strain Ga ge 5

Gun Striker Incident . Transmitter
Spec1men Bar Throw-off

/ Bar / Bar \ Rod
fir. l A/ ,_
, -l i 3 L 1 TR I j lf/_2"

~— 7” —-' 7"-*f l

6” ~ 14" ——-~3/ 14..___.._ 7H—uf

 

 

 

 

 

 

 

 

Figure 2. Schematic Diagram of the Pressure Bar System of
Krafft, Sullivan and Tipper

13

out. For example, the nature of the loading pulse was such that
the strain rate in the specimen varied throughout the test. Also, the
effect of friction on the thin specimens used, 0. 05 and 0.10 cm. thick
by l. 0 inch in diameter for lead and copper, is an unknown quantity
even though it was minimized by applying a thin coat of lubricant to
each side of the Specimen.

Krafft, Sullivan and Tipper (1954) used a pressure bar technique
which is quite similar to the one used in the present investigation. A
diagram of their bar arrangement is shown in Figure 2. Tests were
conducted on high purity iron and a mild steel at temperatures from
- 195°C to + 100°C. The temperature was maintained by surrounding
the Specimen with a container through which a heating or cooling fluid
was circulated. Stress -strain curves were plotted for several tem-
peratures. Quite serious oscillations occurred in the strain records
obtained from the SR-4 strain gages mounted on the pressure bars.
The reasons for such oscillations and furthers details of this method
will be given in the next chapter.
Lu Compression Impact

Habib (1948) obtained energy-deformation curves for copper
at average strain rates from 100 per second to 3000 per second by
compressing small c0pper cylinders (0. 5 inches long by 0. 3255 inches
in diameter) with a high velocity projectile as shown in Figure 3. In " '
this apparatus the hardened steel piston is accelerated in a gun
barrel with compressed air . The strain rate was varied by changing
the velocity of the piston, and the total strain of the specimen for a
particular strain rate was varied by changing the size of the piston.
Several hundred Specimens were tested. The energy absorbed by
each was calculated from the piston velocity which was measured
before and after impact by means of pulses produced on an oscilloscoPe

trace by photocells. The energy absorbed by the anvil behind the

14

Copper

La
mps Specimen

Barrel

 

m oe____

L mum:

O

9\
Piston 'hotocells \

To amplifier and oscillosc0pe

 

Anvil

 

 

 

/////////

 

 

 

Hardened face

 

 

 

 

 

plate
Figure 3. GeneralArrangement of Habib Apparatus
Hammers
Heavy Flywheel
//
© l_i A Specimen

 

 

 

 

 

 

 

 

Induction He atin g

1 II @ \‘ Coil
\ ' l \

Anvil

 

 

 

 

 

 

Mechanical Arrangement of ManjOine -Nadai
Apparatus

Figure 4.

specimen was assumed to be negligible. Stress—strain curves were
plotted for several strain rates, the stresses being determined from
the $10pe of the energy-deformation curves. The strain rate was
taken directly from the average piston velocity during contact with

the specimen. This velocity was assumed to be one -half the initial
velocity. Habib's results showed that the energy absorbed at a
particular level of deformation increased with an increase in strain
rate and that stress is a function of strain rate which changes with

the level of strain. It should be pointed out that plastic wave pro-
pagation effects in the specimens may have had considerable effect

on Habib's data. Habib's calculations based on the von Karman-Taylor
strain rate independent theory indicated that the plastic wave did not
have time to traverse the specimen in both directions during the time
of impact, when the lightest piston was used. This indicates a nonuuniform
strain distribution in the specimen which could produce an apparent
rise in stress. That is a rise in stress which is not due to an actual
material strain rate effect.

Lee and Wolf (1951} made a detailed analysis of the wave pron
pagation in Habib‘s impact tests and concluded that the strain dis-
tribution effects were negligible for most of Habib's tests but an
error of up to 10 percent was possible when the lightest piston was
used. It was pointed out that a non-uniform strain distribution causes
an increase in energy absorption for a fixed reduction in length. They
concluded in general that if several traverses of the plastic wave up
and down the specimen occur during straining, then essentially
uniform strain conditions will occur and the effects of specimen inertia
may be neglected when considering the use of an average strain rate.

Several investigators at the University of Texas have used
an apparatus similar to Habib's for studying metallic cylinders under

impact conditions. The basic apparatus consisted of an air gun

16

which fired a steel piston directly against the specimens and a
Hopkinson type pressure bar to measure the force transmitted by the
specimen. The pressure bar was instrumented with resistance strain
gages. Ripperger (1960b) used the apparatus to study the effects of
strain rate and specimen configuration on the yield stress of copper
and tellurium lead. He used specimens from O. 0625 to l. 0 inches in
diameter and from O. 5 to 2. 0 inches long with piston velocities from
35 to 110 feet per second. The dynamic yield stresses measured
varied considerably hinder supposedly identical conditions, but the
results were sufficiently accurate to conclude that specimen con-
figuration has an important influence on the measured stresses. He
also concluded that lead is relatively insensitive to changes in strain
rate, a conclusion which was not supported by Kolsky (1949).

Karnes (1960) used the apparatus at the University of Texas
for experiments in which he mounted resistance strain gages directly
on the Specimens. Copper Specimens of 0. 5 inch diameter and lengths
of 0.5 , l. 0 and 2.. 0 inches were used. The gages were mounted
longitudinally at both ends of the Specimen in order to determine the
plastic wave prOpagation effects. These gages were inherently limited
to measuring strains of up to about 1. 5 percent. Karnes' experimental
results, as given for the three specimen sizes in Figures 46, 48 and 49
of his report, indicate that at a piston velocity of 65 feet per second
the wave propagation effects become. negligible, at least in the case
of the 0.51 inch long Specimen, after the elastic wave has traveled
twice the length of the specimen. These results shown the reduction
of wave propagation effects with a reduction in specimen length. His
Figure 48 shows that the strain rate at either end of the 0. 5 inch long
Specimen is approximately the same after two traverses of the wave
even though the level of strain is higher at the impact end. This re-

sult supports the use of an average specimen strain in tests of very

 

A\.

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17

short specimens at high strain rates. It might be of interest to pomt
out here that the impact velocities and specimen lengths used in the
present investigation are less than those used by Karnes.

Tapley (1960) made a theoretical analysis of the plastic wave
propagation in the 0.5 and 1. 0 inch long c0pper Specime nts used by
Karnes. The analysis was based on an impact velocity of 65 feet per
second in order to compare results. He used the Malvern strain rate
theory and a modification if it which included the effects of radial motion
The results gave good agreement With experiment for the O. 5 inch long
Specimen but only fair agreement for the longer spec imen, An important
result is that the theory predicts an essentially uniform longitudinal
stress distribution for the 0.5 inch specimen after the elastic wave
travels tw1ce the length of the Specimen.

The UniveISity of Texas apparatus was also used by Guyton
\‘1960, for experiments on the effects of radial friction on the Spec1men
faces, variations in the mate rial, and specimen configuration on the
results of impact tests. His experiments were carried out with c0pper
SpeCimenS. He concluded that radial friction definitely affects the
dynamic yield stress for length "-tC~-I'adlus ratios as small as one -half
but is unnoticeable for ratios as high as four. The experiments con-
sistently indicated that dynamic yield Stress increases for an increase
in speCimen radius and that the effects of cold working in the machining
of Specimens were negligible

Additional information on recent plastic, wave prOpagation
research at the UniverSity of Texas may be found in reports by Plass
(1960; and Ripper ger (1960a... In the former report Plass developed a
theory of plastic wave propagation including the effects of strain rate,
radial inertia and shear.

Alder and Phillips {1954} used a compression machine in which

the load platform at one end of the specimen was driven by a logarithmic

 

. LI.-

'4:

p..-

.a—

 

18

shaped cam. The load was measured by the birefringence created in two
glass blocks, the change in birefringence being recorded photographically
on a rotating drum. Tests in which copper, aluminum and steel Specimens
were compressed to 50 percent in height at strain rates from 1.34 per
second to 39. 3 per second were performed. These tests were also
conducted over a wide range of temperatures. This was accomplished
by enclosing the Specimen in a guard ring box which was heated in an
electric furnace or cooled by immersion in a low boiling point liquid
and then transferred to the compression machine just before the test.
The specimen remained in the box during the test, being compressed
by two platens which were free to move longitudinally in the box while
being in direct contact with both the Specimen and the load platforms of
the machine. The results of thesefttests, as stated earlier, agreed
quite well with both the logarithmic: and the power law, the latter giving
a slightly better fit to the plotted data. Strain rate sensitivity was found
to increase with temperature for all three metals.

Sokolov (1946 4950}: used a modified Charpy impact machine
for the high strain rate compression tests discussed in section 1.2. A
striker tip was mounted on the pendulum of the machine and an anvil was
added to take the force transmitted by the specimen. His impact tests
were performed at average strain rates up to 120 per Second. The
Specimens were 20 millimeters long and 10 millimeters in diameter and
the stress-strain curves were obtained from energy-deformation curves
as in Habib‘s experiment.
if: Tension Impact

Tensile impact tests have been performed by many investigators,
most of the tests being performed to determine the effects of loading rate
on such prOpertieS as yield strength, energy absorption, reduction in

area and elongation. Few investigators have actually determined

19

force-elongation or stress—strain curves during high impact tensile
loadings. Manjoine and Nadai (1940) (see also Nadai and Manjoine,
1941 and Manjoine, 1944) were probably the first to effectively measure
the load and extension during these short time tests. They deve10ped
a high speed rotary impact machine whose mechanical parts were much
like those of a machine designed earlier by‘Mann (1936). A diagram of the
Manjoine and Nadai machine is shown in Figure 4. The impact was
applied by two hammers mounted on a heavy flywheel which was rotated
by a variable speed direct current motor. At a predetermined Speed
the hammers were allowed, by means of a trigger mechanism, to
engage an anvil fastened to the bottom of the Specimen. The load-ex «-
tension curve was obtained directly from an oscilloscope by an ingenious
photoelectric cell arrangement. The apparatus also included an induction
heating coil around the Specimen which could produce temperatures up
to 120000.

By using another machine for the slower strain rates, ex—
tensive data was collected on copper, aluminum and steel at strain
rates from 10-6 per second to 103 per second and at a series of temper—
atures. When ultimate Stress was plotted versus the logarithm of strain
rate the curves for copper and aluminum showed a marked nonwlinearity
in the region of high strain rates, the stress increasing in this region.
This effect was more pronounced as the temperature was increased.
These results in the region of high strain rates must be taken with
caution since they may have been affected by plastic wave propagation
in the Specimens. This problem was discussed by Clark and Duwez
(1948). It should also be mentioned that large oscillations, which seemed
to be due to the inertia of the Specimen supports, appeared in the re»
cords at extremely high strain rates.

More recently, Austin and Steidel (1959) developed an ex-

 

midi

  

’5»
. i.

.qr‘

ti.

20

plosive impact tensile tester capable of producing strain rates up to

17, 000 per second. The load was measured with SR-4 strain gages
near the stationary end of the Specimen and the strain and deformation
were recorded by means of a high Speed movie camera. Stress-strain
curves for an aluminum alloy and cold rolled steel at room temperature
were obtained. Here again, the plastic wave prOpagation effects in

the Specimens could introduce error in the load measurements.

Clark and Duwez (1950) devised a method for obtaining im-
pact rates of strain in tension which virtually eliminated plastic wave
propagation effects in the specimen. In this method circumferential
strain is induced in a thin—wall hollow cylindrical specimen by an
internal fluid pressure. A nearly constant strain rate is produced by
compressing the liquid (mercury) with a piston moving at constant
velocity. Tests were made— on steel specimens at strain rates up to
200 persecond, but unfortunately the data has limited usefulness since
only stress-time records were obtained.

1.3.4 Plastic Wave Propagation

 

Stress-time curves have also been obtained indirectly from
the strain-time records of plastic waves in longitudinal bars. If the
propagation velocities at successively increasing levels of strain are
determined for a plastic: wave then the tangent moduli 32 of the stress-

5
strain curve can be calculated from

1 d
c21/_ ——0 (1.2)
p e

the relationship proposed in the von Karman—Taylor theory. If the
tangent moduli are plotted against strain and the resulting curve is
integrated, a dynamic stress-strain curve is obtained.

Campbell (1952) used this method to determine dynamic

Stress-strain curves for hard and soft copper. He performed experiments

 

 

(.ll‘

in

‘Mi

--.E

‘I

[a
t...

21

on long bars impacted in tension by a freely falling weight. The
strain-time records were determined at two points near the impact
ends by means of resistance strain gages. The propagation velocities
at successive levels of strain were then obtained by plotting these
strain—time curves on the same time axis and measuring the time re-
quired for each level of strain to be propagated from one gage station
to the other. The dynamic stress strain curves from these experiments
showed stress values much lower than those of the static curves. How-
ever, Campbell suggested that the calculated stresses were in error

by as much as 25 percent due to the lack of refinement in experimental
technique.

Johnson, Wood and Clark (1953) and Bell (1959, 1960) also
determined dynamic stress—strain curves by the tangent modulus
method. Their experiments were performed on soft aluminum under
compression impact. Bell measured strains with a diffraction grating
method in which. light is reflected by a fine grating ruled directly on
the surface of the specimen. The method utilizes the fact that the
angle of diffraction is dependent upon the number of lines per inch of
the grating. Bell‘s experiments have also given support to the von
Karman-Taylor strain rate independent theory.

The plastic wave propagation method of determining dynamic
stress-strain curves discussed above can be used only under conditions
Where the variations in strain rate during the test are known not to
influence the wave propagation, because the use of c :\/-Fl)- 3% is
based on the existence of a single dynamic stress—strain relation

applying throughout the test.

 

.uu

 

.\«
.A.‘

 

CHAPTER II
EXPERIMENTAL METHOD

2. 1 General Description and Objectives

 

The basic apparatus of the present investigation was deve10ped
and used by Lindholm (1960) for high strain rate experiments on copper,
lead and aluminum at room temperature. A few minor changes have
been made in the basic apparatus and a furnace has been added for
heating the specimens Lindholm's objectives in developing the apparatus
were to: (a) obtain stress-strain curves at as nearly constant strain
rate as possible, (b) obtain each complete curve from a Single specimen,
recording stress and strain continuously during the test, (c) obtain
stress and strain records well into the plastic range, (d) obtain
dynamic curves up to 1000 in/in. /sec. , and (e) minimize wave prop~
agation effects in the Specimen. His apparatus, which was an adaptation
of the Split Hopkinson pressure bar method used by Kolsky, met all of
these objectives. In Lindholm's apparatus, the loading pulse was obtained
by longitudinal impact with a steel striker bar which had been accelerated
by means of a commercial impact testing machine The strain pulse
was monitored by means of strain gages mounted on the pressure bars
as in the Krafft, Sullivan and Tipper experiment discussed in Section
1. 3.1.

For the present Study changes in the apparatus were made
to: (a) obtain dynamic stress-strain curves at a series of temperatures
from room temperature to near the melting point of aluminum, (b)
develop accuracy sufficient to determine cha‘hges in stress with changes
in strain rate at each test temperature, and (c) accurately determine
Specimen temperature at the time of the test. An electric furnace was

placed around the pressure bars, covering the section containing the

22

 

23

specimen. A thermocouple placed near the specimen was used to monitor
the temperature at the time of the test. The accuracy of the apparatus
was improved by using foil strain gages with a higher output than those
used by Lindholm. A schematic drawing of the apparatus is shown in
Figure 5 and a general view of the test setup is shown in Figure 6.

2. 2 Pressure Bar System.

 

The arrangement and dimensions of the pressure bar system
are shown in Figures 7 and 8. The striker bar and the pressure bars
were made of 3/4 inch diameter centerless ground, type 303, stainless
steel rod, stainless steel being used to reduce oxidation at the higher
temperatures. This 18-8 stainless steel also has good impact prOperties
at all temperatures. The pressure bars were supported in rubber "0"
rings, which were in turn supported in steel pillow blocks constructed
for this purpose (see Figure 9). The “0” rings allowed sufficiently
free lateral expansion of the bars, so that no reflections of the strain
waves from the supports were observed in the strain records.

The length of the bars and the locations of the gage stations
were calculated to permit the recording of complete strain pulses
without interference from reflected pulses returning from the bar ends.
The problem of reflected pulses due to temperature gradients in the
bars was also considered; this is discussed in Appendix A.

It was also necessary to consider the effect of temperature
on the performance of the strain gages. The gages were located so
that the operating temperatures were well within the safe operating
temperature listed by the manufacturer.

The ends of the bars which were to be in contact with the
specimen were carefully turned in a lathe and finished with a fine
emery paper in order to reduce the effects of radial friction during

compression of the Specimen. The bars were aligned longitudinally

24

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Figure 6. General View of Test Setup

26

 

Figure 7. Close—Up of Pressure Bars and Furnace

27

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28

by measuring from the ”I” beam Shock tester supports and then more
carefully by placing a light behind the specimen opening with the
pressure bars in contact. A lead pad was placed at the far end of the
transmitter bar to absorb the energy of the pulse.

The 16-inch long striker bar produced a 32-inch long flat-topped
pulse in the incident pressure bar, since the striker bar remained in
contact with the pressure bar until the compression pulse was reflected
from the free end of the striker bar and returned to the interface as
a tensile pulse. The impact end of the striker bar was rounded slightly
to assure axial impact and to reduce the effects of diSpersion on the
strain record of the pulse by lengthening the rise time of the pulse.
Figure 10 shows strain records obtained at two stations on a Solid
steel pressure bar. Each photograph records a single pulse; the upper
trace is from the first gage station at a point 22 inches from the impact
end, while the lower trace shows the same pulse when it reaches the
second station 58 inches from the impact end. Figure 10a is for the
rounded striker bar, the arrangement actually used in the study, while
Figure 10b is for a nearly flat-ended striker bar. The oscillations at
the leading edge of the record from the Second station are dispersion
effects of the high-frequency components in the rise portion of the main
pulse. The flat-ended striker bar produced a pulse with a shorter rise
time so that the shorter wave length components are more significant
and therefore, according to the Pochhammer-Chree theory (see
Davies, 1948), the dispersive effects are greater. This explains the
greater oscillation in Figure 10b.

The problem of diSpersion in elastic bars has been studied
by many investigators. One of the more recent studies was made by
Curtis (1960). Suggestions for eliminating strain record disturbances

caused by dispersion and other factors were given by Krafft (1955) who

29

<— Time

-5 Millivolts

 

I l 50 Microseconds

(a) Rounded Striker Bar

: 5 Millivolts

 

l [.50 Microseconds
4——— Time

(b) Nearly Flat-ended Striker Bar

Figure 10. Strain-Time Records Showing Effect of Dispersion
The impact velocity is approximately 25 feet
per second.

.Ai
M n§

..\v

\.v.

30

also observed that slight deviations from parallelism between the impact
surfaces of the bars have little effect upon the oscillations.

2.3 Impact Loading Machine

 

The striker bar was accelerated by means of a commercial
Hyge Shock Tester, type HY-3422, manufactured by Consolidated
Electrodynamics Corporation. The tester is shown in Figure 6 and a
schematic sketch is shown in Figure 11. The piston diameter is 3 inches
and the full stroke is 16.75 inches.

The acceleration of the piston is achieved in the following
manner: A given set pressure is applied to chamber B by means of
compressed nitrogen. This pressure acts over the full area of the
piston, pushing it against a ring seal at C. A load pressure is then
applied to chamber A, this pressure being restricted to a smaller
area of the piston by the ring seal thus causing a smaller force to
act on this side of the piston. When the load pressure reaches a
magnitude of 4. 2 times that of the set pressure, the forces acting on
the piston become equal. If additional load pressure is applied, the
Seal at C is broken and the load pressure expands over the entire
face of the piston causing a large acceleration.

The acceleration is regulated by a conical pin which regulates
the amount of gas supplied to the piston face during the initial part of
the acceleration. The deceleration is aided by a hydraulic fluid which
occupies approximately one -half of the volume of chamber B. The
deceleration is accomplished by a pin which gradually closes the area
of the escape orifice D through which the hydraulic fluid is forced. The
system is controlled from a separate control panel which contains pressure
gages and control valves.

A ”barrel" was designed and threaded onto the end of the

piston shaft to support the striker bar. A diagram of this barrel is

31

To loagressure control To pressure control

 

 

 

 

Pis on

 

c
‘31

 

 

 

 

 

 

 

 

 

 

 

 

 

 

c EL
A mm
T /
ji_ 7 I F1 8
I / l 1 r
Acceleration Deceleration 7
pin pin Piston Shaft

Figure 11. Schematic of Hyge Shock Tester

 

 

 

 

 

 

 

 

Brass plug "0" ring supports
l 1 \ 1
H E
Fr A ‘
I / l
Piston shaft Barrel Striker bar

Figure 12. Barrel with Striker Bar

32

shown in Figure 12. The striker was supported within the barrel
by rubber "0" rings which were lightly lubricated with petroleum
jelly to allow free sliding of the bar at the end of the piston stroke.
The spacing between the shock tester and the incident bar was such
that the striker bar was in free motion within the "0" rings just
prior to impact.

2.4 Furnace and Temperature Control

 

A Type M-1012 electric resistance combustion tube furnace
manufactured by Hevi-Duty Electric Company was used to heat the
area around the Specimen. The furnace had an internal diameter of
1.75 inches and the heating element was 12 inches long. A hinged
cover was provided for free access to the Specimen (see Figure 7).
The furnace was capable of producing temperatures to 19500F for
intermittent operation and to 18500F for continuous operation. The
temperature within the furnace was monitored by means of a Chromel-
Alumel thermocouple which was welded directly to the transmitter
bar at a distance of 3/8 inches from the specimen. A variable trans-
former was connected to the furnace to maintain Specimen temperature
at the desired level. The furnace was capable of reaching the maximum
temperature used in the experiment in approximately 15 minutes and
essentially steady state conditions were attained in the pressure bars
within 30 minutes.
3i Strain Measuring and Recording Equipment

The pressure bars were instrumented with SR-4 Type
FAP-25-35 foil strain gages with a gage length of 1/4 inch. The
distortion of the pulse introduced by the 1/4 inch gage length was
considered negligible Since the gage length was small in comparison
with the rise portion of the pulse. The resistance of these gages as

given by the manufacturer was 350 ohms _+_ 2. 5 ohms and the gage

’ C3 ' (.1, 1’) .

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(Ir)

ll!

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33

factor was 2.16 i 1 percent. Two gages were mounted at each gage
station. They were placed diametrically opposite one another and
longitudinally along the bar. This effectively eliminated the measurement
of bending strains when the two gages at a station were placed in
opposite arms of a Wheatstone bridge as Shown in Figure 13. The
dummy gages in the other two arms of each bridge were 350 ohm
precision resistors. The bridge voltage was supplied by two 12 volt
wet cell batteries connected in series, and voltage was recorded
by means of a voltmeter which could be read to the nearest 0.905 volt.

Two Tektronix Type 53/54 E plug-in preamplifiers were
used to amplify the signals from the bridges. These are high gain
differential preamplifiers with a vertical sensitivity to 50 microvolts
per centimeter and a frequency response from 0. 06 to 60, 000 cycles
per second. The differential feature permits attenuation of any un-
desired signal by means of outphasing.

The preamplifiers were mounted in a Tektronix Type 551
dual beam oscilloscope which permitted the Signals from the two
gage stations to be displayed at the same time with one horizontal
sweep of the beams. The amplifier of this sc0pe has a vertical rise
time of O. 012 microseconds. The sweep speed could be varied con-
tinuously from 1 microsecond per centimeter to 5 seconds per centimeter
The scope was triggered by means of a piezoelectric crystal which
Was mounted on the incident bar about 4 inches ahead of the first
gage station.

The strain pulses were permanently recorded with a
Dumont Type 302 oscilloscope record camera using Polaroid Pola

Pan 400, type 44, film. An open shutter with an aperture setting

of f/2.8 was used.

34

 

 

Volt -
meter

 

 

 

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Batteries

resistor

 

CRO

 

 

 

 

 

   

3509
resistor

  
    

Active
gage

  

 

Figure 13. Strain Gage Bridge

35

2.6 Specimens
The specimens were all machined from the same 1/2 inch

 

diameter bar of extruded aluminum. This was number 1100 F aluminum

(ZS aluminum) with the following composition as certified by the

 

distributor:
Element Percent by Weight
Aluminum 99. 00 max.
Copper 0. 20 max.
Silicon and Iron 1. 00 max.
Manganese 0. 05 max.
Zinc 0. 10 max.

The specimens were carefully turned to 0. 495 inches : 0. 002 inches
in diameter and 0. 250 inches : 0. 005 inches in length on a high speed
collet lathe. Special care was taken to assure parallelism between
the end faces of the specimens, nonparallelism being limited to less
than 0. 001 inch in all cases and usually less than half of this value.
Annealing was at 4000C for one hour in an electric furnace.

The diameter of the specimens was sufficiently smaller than
the 3/4 inch diameter pressure bars to insure full contact of the
Specimen faces with the parallel faces of the pressure bars during
compression.

2. 7 Derivation of Stress and Strain in the Specimen

 

The average specimen strain is determined from the dis-
placement -time records of the pressure bar faces in contact with the
Specimen. These displacement-time records are in turn obtained from
the strain-time records of the incident and transmitted pulses as ob-
tained from the two gage stations. From the theory of elastic wave

propagation in a longitudinal bar we have

1 Chi

V
E- C a? (“3
0

36

where v is the particle velocity and CO is the elastic wave velocity
which is constant for a particular temperature. Solving this equation

for u, the particle diSplacement, we have
u = C 5 dt (2. 2)

which shows that the particle displacements in a bar which is originally
at rest can be obtained by integrating the strain-time history curve at
the point under consideration.

In order to determine the strain-time histories at the faces
of the bars from the strain-time records taken from the remote gage
stations we must consider the reflections of the incident and trans-
mitted pulses from the Specimen interfaces and from the temperature
gradients in the pressure bars on either side of the Specimen. It
Should also be pointed out that the magnitude of the strain in a pulse
changes as the pulbe proceeds into a section of bar having a different
temperature, and consequently, a different Young‘s modulus. This
means that the amplitude of the incident pulse as measured at the
first gage station must be multiplied by a correction factor in order
to obtain the strains occurring at the end face of the bar within the
furnace. If a strain wave is propagated into a region where the
temperature is above room temperature, we can determine the

transmitted strain 5 t from the relation

__. z (2.3)

Where 6 , is the incident strain and E1 is Young‘s modulus at the
1

higher temperature T Similarly, the reflected strain e can be
r

1.
determined from

37

 

 

61 Vi: +V'E'3'

These relations are derived on the basis of a single discontinuous

(2.4)

jump in temperature but experimental evidence indicates that they
are also valid for a continuous temperature distribution in which
the total change occurs in a section of bar which is long in comparison
with the bar diameter. The derivation of Equations (2. 3) and (2. 4)
and experimental evidence of their validity is presented in Appendix A.
In the present investigation the transmitted strain pulse is a maximum
20 percent higher than the incident pulse, and the maximum reflected
strain pulse is 6 percent of the incident pulse.

In order to reduce the amount of writing in the following
derivation of the displacements at the faces of the Specimen let us
denote the interfaces of the Specimen and the pressure bars as a and
[3, a being the interface with the incident bar. Let us also denote the
temperature distributions in the bars as x and g , )\ being the distribution
in the incident bar as shown in Figure 14. In this figure, compressive
strain pulses are plotted above the axes and the arrows indicate the
direction of propagation but not necessarily the direction of particle
velocity. The pulses propagated past the gage stations are listed
below their respective stations. If we correct the pulse records
obtained from the gage stations by a temperature factor which can
be calculated from Equation {2. 3) we will have the strains these
pulses cause at a and 6. Let these strains at a and B be identified
by the subscripts a and B. For instance, let 5 10 be the strain at a
later time at a. due to the incident gage station pulse 5 I. Only first-order
temperature reflections are considered, as the reflection of a reflection
is assumed to be a negligible quantity.

The incident pulse 5 I is partially reflected at K, at Q +{3

8
3

mpuooom oEMHudmmnum 93 mo GBvadoum .«A oufiwmh

AsoEmovmm mo opmm a adv

w Scum vouoofioh NH. w «TH. w

 

<
l

a

.o + a Son 300 on w
H U a .- MB W
.‘ll

 

Asoemoomm wo opmm dumv A 50.3

NH. w wouoofioh Mm w +3H

>

Til

A 503 wouuofio." Hm w

b
T

W

ommsm usogocm
mo upon Homepages—duh. 7H.

/._

All

.w

 

n C.

_ F

cofldun omna nah nouumfimanH /

 

cofisfluummr oududnomgofi

A
\‘
\\
I. l?\
\. a
\

a + a Sch voted?— .ow {mm

 

 

nllllnflnllvl w
1
TH.
w Eoum wouoofioh w lNMW
(

a + d 85.3 wouoofloh w

A Scam @oaooflou w M
w
bill... .
w IIIIIVI

omfifinm an OBUGH

 

L

nomad: .ommm awn «Q0303

’II‘.\
a i11|||l .r

39

and at I; . These three reflected pulses are denoted by e e

I
R, R1

and e reSpectively. At the transmitter bar gage station we

R2
have 5 T1’

on this is e

the transmitted part of the incident pulse and superposed
h'
T2' the part of 5 R1 w 1ch was reflected from x and

transmitted back through the specimen. We also have 5 T3 which is the

double reflection of e + 5 from g and from 6 + 0.. The total

T1 T2
reflected pulse as seen at the incident gage station is the Sum of

€'+€ +6 ands

R R1 R2 R3’ 6R3
R2 from x and from o. + 6. The pulses e
a

the reflections of E R1a+ E RZu T16 + E T26

respectively, act at the pressure bar faces but do not appear in the

being a double reflection of 5 R1 +

' and e B” which are

6
from )\ and e from 2;
strain-time records taken from the gage stations Since they are
traveling away from their reSpective stations.

If e 'o is determined and is added to the incident and reflected
strain pulses acting at 0. which can be obtained from the incident gage
station record and if e B is determined and is added to the superposed
transmitted pulses acting at 6, we will be able to calculate the strain-
time histories of points 0. and 6 and the diSplacements can then be
determined from Equation (2. 2}. If the temperature distribution iS
assumed to be a step function with several discontinuous jumps in
temperature (x and I; were each divided into 5 increments in the
present case), then the reflection pulses Ga! and e Hi3 can be Obtained
by superposing the reflections frome ach jump in temperature. The
time axis for each of these increments of reflection is shifted by an
amount equal to the time of travel between the temperature jumps.
This time is easily determined Since the elastic wave velocity is
known at every point along the bar. The origin of the total reflected
pulse is determined from the distance between the Specimen and

the fir st temperature jump-

40

If we let 6 R0. be the sum of the reflected strains occuring

at a then we see from Figure 14 that

6 = 6 (2. 5)

-~ 6 -'~ 6
R0. R10 RZa R30
where the strains are taken as absolute values.

If 6 TB is the sum of the transmitted strains at the interface

6 then
___, .. _ 2.6
5m €T16 Eras €T36 ( )
Now let us denote the longitudinal diSplacements of the
pressure bars at o. and 6 as u and u;3 respectively as shown in
a
Figure 15, then we have
t
: + - ’ dt 2 7
U0. Co. (61a 6 Rd 6 o. ( )
0

Where c is the elastic wave velocity at (1. Here e la and 5 R0 are
0.

positive since a compressive wave traveling in the positive direction,
that is away from the impact end, and a tensile wave traveling in the

n8gat1ve direction cause a p051t1ve displacement. The strain 5 a

has a minus Sign before it Since a tensile wave traveling in the p081t1ve

direction causes a negative displacement. Using this same reasoning

we have

(2.8)

I

: —- ‘ _ ')dt
u C (6TF3 6‘3.
0

Where c!3 is equal to c since the temperature is the same on both

(1
Sides of the Specimen.

If the stresses corresponding to the previously considered

Strains are denoted by like subscripts, and if we assume that the

total stress on either side of the specimen is the same then We

41

 

 

 

 

 

 

 

 

 

 

 

 

‘1 \ I“ L —>- ‘3
o
01 ————————o~r O‘
0‘ *— T1[3
0 l +—————-1
R a ——> o T25
ORZQ ___.> 0T36
_—._N
€ Ha l ' ‘ 06 ”[3
+———'-
J
u
—> ua i—‘y B
Flgure 15. Schematic of Specimen Showing Stresses

and Displacements

42

have
- + + - = — -
a la 0 Rla 0 R2a U R3a 06h 0T16 0T26 UT36 as
or by substituting from Equations (2. 5) and (2.6),

(3.10}

-o - ._ +
In Ra Oe'a 0T5 06"9

Dividing Equation (2.10) by Ea, Young‘s modulus at o. and 6, we have

0

- - 6 ' = + " 2.11‘
6 Id 6 Ru 0. 6 T6 6 s ( ’
Solving this for e and substituting into Equation (2. 7) we have
a
t
= 2 — - 2 ' - .. dt 2. 12
u(1 Ca [ ( 61a 6 T6 6 a 6 F3) ( )

If the length of the Specimen is taken as LO then the nominal strain
in the Specimen is

—9————9- (2.13)

 

2c t
o.
: .. - ‘ dt 2.
63 Lo (6IOL 6TF3 6a) ( 14)
o
where 5 and g are obtained from the strain pulses e and 6
I0. T6 I

recorded at the incident and transmitter bar gage stations, but
corrected in amplitude as in Equation (2. 3) because of the transmission
through the thermal gradient and e ‘a is a reflection pulse which can

be determined from e la and 5 T6. Equation (2.14) is used to

calculate all Specimen strains in the present study. This equation
Shows that it is not necessary to obtain a record of the reflected

pulses at the incident gage station in order to determine the specimen
strain. Time in Equation (2.14) is measured from the instant the

Incident strain pulse reaches the Specimen. This is measured on the

PhOtOgraphic records beginning with the initial rises of the incident

43

and transmitted pulses.
The stress in the specimen is obtained from the strain

at the face of the transmitter bar in contact with the Specimen,

that is 6 T6 + e ". Since this bar remains elastic we have
0 =E(6 +6”) ”-15
s F3 TB 5 ’
where e " is a pulse which can be determined from c I and e T°

Note that if the temperature is uniform over the pressure
bars, which is the case for the room temperature tests, then 5'
a

and e ”L3 do not exist and Equations (2.13) and (2.14) become

 

2c t
o
= ~~ dt 2.
68 L (61 ET) ( 16)
o
and
= 2. 7
08 E 6T ( l)

where 5 and e T are the actual strains occuring at the gage stations.
The applicability of Equation (2. 16) to the pressure bar
method used in this investigation was verified experimentally. A
description and the results of this work is given in Appendix B. A
comparison of e - 5 R against 6 T for room temperature is also

given in this appendix.

CHAPTER III
EXPERIMENTAL PROCEDURE

3.1 Determination of the Elastic Wave Velocity

In order to determine the stress and strain in the Specimen
it was necessary to determine the elastic wave velocity in the pressure
bars. The elastic wave velocity at room temperature (260C) was
determined by measuring the time required for a pulse to travel twice
the length of the incident bar. The time was measured with a Model
226 B Universal Counter-Timer manufactured by Computer Measurements
Company. This electronic counter was triggered with the signal from
the strain pulse in the bar. The Signal from the strain gage bridge
was fed into a Tektronix Type 122 Low Level Preamplifier to provide
the necessary voltage to trigger the counter. The counter was started
when the initial compressive pulse reached the gage station and was
stopped when it returned to the station again as a compressive pulse
after being reflected from both ends of the bar. The counter was
operated with the start and stop triggers connected in common to
insure a time measurement from the same amplitude level of the
pulse. A time of 407 microseconds was consistently recorded for the
80 inch wave travel which gives a wave velocity of 196,600 inches

per second.

It was also necessary to know Young's modulus of the bar
material. This was obtained from the relation cO : M where
p was determined by weighing a 10 foot Section of the stainless steel
bar. The value for E turned out to be 28.56 x 106 pounds per square
inch which compares favorably with 28 x 106 pounds per square inch
given by the Metals Handbook published by the American Society for
hAetals.

44

45

The electronic counter was also used to check the sweep
speed of the dual beam oscilloscope. There was less than one per-
cent error in the speed.

3. 2 Dynamic Calibration of Strain Gages

The strain gages were dynamically calibrated by comparing
the measured strain with the strain calculated from an equation ex-
pressing the strain as a function of impact velocity. The impact
velocity equation may be derived by considering the impulse and
momentum of the collision between the striker bar and the incident
bar. If at the moment of impact the incident bar is stationary
and the striker bar is moving with velocity vs, then a compression
wave will proceed from the interface into both bars at a velocity Co
as shown in Figure 16. The strain behind the wave fronts remains
constant and uniformly distributed over the bars until the pulse is

reflected from the free end of the shorter striker bar.

The wave frort in the incident bar will have traveled a
distance cot in time t and the mass of the bar behind the wave front
will be moving with the velocity vi which is the particle velocity at
every point behind the wave front. Equating the impulse with the

momentum in the bar we have

: i ’ ti '
Ft (pACO [Vi (3 l)

where F is the force acting at the interface, and A is the cross-sectional
area of the bar. Similarly, we have for the striker bar

2 ' t' _ . .
Ft (pAco ) (vi vs) (3 2)

Where vS is the velocity of the striker bar. By equating the impulses

We get

5 (3.3)

46

 

(a)

 

 

 

 

 

6'

(b) ‘F

 

 

 

 

 

 

kc t c t
o + o " ’(
Striker bar Incident bar

Figure 16. Striker and Incident Bar (a) at moment of impact
(b) after time t.

Striker bar

 

 

 

 

 

 

 

 

\ Sw1tches Trigger Inci7nt bar
/ LR
V 1.1 \
5

Strain gage station

 

 

Bridge
F

Amplifier

CRO

 

 

 

 

 

 

 

 

 

Electronic

Counter

 

1
Camera

 

 

 

 

 

 

 

Figure 1?. Schematic Diagram of Calibration Apparatus

47

since the bars are of the same diameter and material. The strain
is uniformly distributed behind the wave front, therefore, the strain
in the incident bar is simply the particle displacement at the interface

divided by the distance traveled by the wave front or

 

vit vi
6 = c t : -C— (3. 4)
O 0

If EQuation (3. 3) is substituted into Equation (3. 4) we have strain

as a function of impact velocity, that is

V

S
E _ T: (3.5)

o
The amplitudes of the pulses measured by the strain gages

were calculated from the strain gage bridge equation

2Av (3.6)

Where Av is the bridge output produced by the strain 5 , V0 is the
voltage supplied to the bridge and F iS the gage factor of the strain
gages.

Several tests were performed in which the striker bar velocity
was determined by means of an electronic counter. The counter was
used to measure the time required for the striker bar to travel a
fixed distance. It was triggered to start and to stop by two mechanical
switches which were placed near the end of the striker bar travel
as Shown in Figure 17. The switches were constructed of Spring
brass and their performance was checked during several tests by
comparing their signals with the timing marks from the counter on

an oscillosc0pe photograph -

A comparison of the strains calculated from Equations

48

(3. 5) and (3. 6) is given in Figure 18. The straight line is from
Equation (3. 5) and the points represent the strains calculated from
the photographic records by means of Equation (3. 6). The abscissas
of these points are the measured impact velocities. The velocities
were measured over distances of 1.5, 2 and 2. 5 inches. The re—
sults Show that the differences in the strains from the two equations
were less when the measuring distance was decreased and that the
measured strains in nearly every case were less than those cal-
culated from impact velocity. This was probably due to a decreasing
striker bar velocity near the end of the stroke caused by the friction
drag of the “0" ring supports. If the striker bar was Slowing down,
the longer measuring distance over which an average velocity was
obtained would result in a measured velocity higher than the actual
velocity. Therefore it is believed that the velocities measured over
the l. 5 inch distance were approaching the actual velocity of the bar.
With the l. 5 inch gage length, the difference between the strains from
the two equations was less than 2 percent for impact velocities from
25 to 45 feet per second. The Strain Gage Equation (3.6) was used
to calculate pressure bar strain in the following tests.
i._3 Specimen Size and Material Effects

Some Specimen Size and material effects were observed
When an attempt was made to increase strain rate by reducing the
size of the specimen. The highest strain rate obtainable with the
apparatus used in this investigation was dependent upon the yield

Point of the pressure bar material and the length of the Specimen.

The yield point of the bar material limited the elastic strain obtainable

in the bars and therefore limited the specimen strain rate. This

can be seen by differentiating both Sides of Equation (2.13) with re:-

Spect to time to obtain

..-v

‘1; -

Ix.

V-

I-BAR STRAIN (micro in. /in.)

1600

1400

1200

1000

800

600

400

200

49

 

 

 

I
A
I
l
.. A
o . . . 1.5" gage length
' A 2. 0" gage length
I . . . 2.5" gage length
A V
_ s
— 6 — 2c
0
A
1 1 l 1
100 200 300 400 300 —6I)0
STRIKER BAR VELOCITY (in. /sec.)
FiSure 18. Comparison of Strains Determined from Impact

Velocity and the Incident Strain Gage Station

50

 

-€') (3.7)

Here 2c /L is a constant for a particular temperature and e
a o

d
is greater than 5 T6 or 5' . Thus an upper limit is placed on e s
a
by the permissible magnitude of e I . The diameter of the pressure
(1

bars was measured at several points, including points within the furnace,
after each temperature run and no evidence of yielding was found.
It is also seen from Equation (2.13) that a decrease in Specimen length
also increased the strain rate. Some Specimens were made with a
25 percent reduction in length with intentions of making use of this
fact. It was also necessary to reduce the diameter by 25 percent
since tests performed by Lindholm (1960) showed that a reduction
in the length to radius ratio from the standard ratio of one, used
in his experiment and in the present investigation, introduced
noticeable radial friction effects. Guyton (1960) also found that
friction effects were Significant for length to radius ratios as small
as one -ha1f. The smaller specimen is shown in Figure 19a along with
a standard size Specimen.

This attempt to extend the range of strain rates did not
give results which could be correlated with the data from the standard
size specimens. Five of the smaller specimens were tested at various
strain rates at room temperature and the stress-strain curve from
each one showed stresses which were approximately 5 percent higher
than those obtained from standard specimens. The smaller specimens
were machined from the same 1/2 inch diameter bar of aluminum
that the standard specimens were machined from and were annealed
under the same conditions. A possible explanation for the difference

In stresses is that the bar material varied across the cross-section

Figure 19.

Effects of Non-Isotropy in Specimen Material
The solid arrows in part (c) indicate locations
at mark placed on the bar before the specimens
were cut off and the dotted arrows indicate the
alignment of the specimens with the pressure

bar surfaces.

 

liez'et

52

of the bar. Some support is given for this explanation by the fact

that the smaller specimens showed quite a bit of radial distortion
which was not as pronounced with the standard specimens (see Figure
19b).

The radial distortion in the Specimens was at first be-
lieved to be due to nonparallel faces on the ends of the pressure
bars which were in contact with the Specimen, but evidence was
obtained which indicates this was not the reason for it. Three
small Specimens were cut in succession from a bar which had been
turned to 0.185 inches in diameter and marked with a small long-
itudinal groove by means of the tool in the lathe. When these
Specimens were compressed with the pressure bar apparatus, each
Specimen was rotated through an angle of 45 degrees with respect
to the previously tested specimen. The results of these tests as
shown in Figure 19c indicate that the distortion is oriented with the
bar material and not with the end faces of the pressure bars. This
is shown more clearly when the three Specimens are placed directly
on tOp of one another with the marks in line.

3. 4 Test Procedure

 

The furnace, the strain gage bridge voltage, the amplifiers
and the oscilloscope were turned on at least one -half hour before a
test was performed in order to obtain nearly steady state operating
conditions. In performing the tests the following procedure was
used:

1. The oscilloscope sweep speed was set at 50 micro-seconds
per centimeter and the trigger was placed on single sweep lockout
in order to obtain a complete re cord of the test with one sweep of the
oscillosc0pe trace.

2. The vertical gain adjustments of the amplifiers were set

53

in accordance with the impact tester set pressure used for the test.
3. The length and diameter of the Specimen were measured
to the nearest 0. 0002 inch with a micrometer and the measurements
were recorded.
4. The parallel faces of the Specimen were covered with
a light coat of lubricant and the specimen was centered between the
two pressure bars. The Specimen was held in place by a small
longitudinal force applied to the incident bar by means of rubber
bands.
5. A set pressure was applied to the forward chamber of
the Hyge unit and locked in.
6. The specimen temperature was checked and adjusted.
The specimen was held at the test temperature at least 5 minutes
to obtain a uniform temperature distribution.
7. The load pressure was increased to near the firing
pressure.
8. The camera shutter was Opened and held open until
after the Hyge unit fired.
9. The load pressure was released and the Hyge piston
was retracted to be ready for the next test.
10. The strain gage bridge voltage was read and recorded.
11. The Specimen was removed and the end faces of the
pressure bars were wiped clean for the next test.
Specimens were tested at 30°, 150°, 250°, 350°, 450°
and 550°C. At each test temperature, a series of tests was per-
fOl'med in which the Hyge set pressure was varied to obtain an evenly
distributed range of strain rates from approximately 300 to approximately
2000 inches per inch per second. The number of Specimens tested

at each temperature depended on the results obtained at that temper-

 

u-v
-

5“.

 

 

‘1’
(I)

in.

r.

 

54

erature. All tests at a particular temperature were completed and
the stress-strain curves were plotted before starting tests at the
next test temperature.

In order to maintain control over specimen temperature
during the room temperature tests, the furnace was used to heat
the Specimens slightly. The Specimen temperature was held at 300C
while the actual room temperature varied from 250 to 280C. Temper-
ature reflections in the bars were neglected for these small temperature
increases.

Molykote (commercial molybdenum disulfide) was used for
Specimen lubrication at test temperatures of 300, 1500, 3500 and
450°C. It was necessary to use powdered graphite at 2500 C because
at this temperature the Molykote formed a hard coating on the faces
of the specimen and the pressure bars which was difficult to remove
between tests. A powdered glass with a low softening point was used
as a lubricant at 5500C. The glass powder was mixed with alcohol
to form a paste which was easily applied to the faces of the Specimen.
The glass was composed of 80 percent Pb0 and 20 percent B203 by
weight. It was prepared by melting the constituents together and

then powdering the resultant pieces of glass with a mortar and pestle.

3. 5 Riduction of Data

 

Measurements were taken from the photOgraphic records
using a Pye two—dimensional traveling microscope accurate to
0. 01 millimeter. Slight corrections were made for photographic and
oscillosc0pe screen distortions. Amplitudes of the incident and
transmitted pulses were measured at a series of points along the
time axis. The spacing of the points was determined according to
the need for accuracy at the particular section of the curve. The
time axes for the two pulses were made to coincide by aligning the

initial rise portions of the curves.

55

The strain pulses e ' and e " were determined by the
a

B
method described in Appendix A. The pulse 5 ”‘3 was obtained by

calculating the reflection of the transmitted pulse 5 from the

T13

temperature gradient in the transmitter bar as Shown in Figure 14.
In order to obtain 5 ' it was necessary to find the reflected pulse
a

e R . The pulse 5 R was approximated by the sum of the pulses
a (1

6R0, 6 '0 and e ”B which was obtained by adding 6 ”B to both sides
of Equation (2. ll), transposing e I and changing signs. Let c g
0 a

be the assumed value for e R then

H = + l _ ll : _ 2 ll .
6RCL 6RC1 60 6‘3 610 6TB 6‘3 (3 8)
The strain e ' was calculated from 5 Re and then substituted back
a
into Equation (2. 11). This Showed that the error in 5 ' as determined
a

by this procedure was insignificant for the range of temperatures
used in this investigation.
Nominal Specimen strain was calculated from Equation

(2.14) by plotting e _, - - e ' versus time and then numerically
in

E T6 (1
integrating this curve with a planimeter. True or logarithmic strain

6 n Was determined from

1
en-ln m (3.9)

Taking into account the change in diameter between the specimen and
the pressure bars and assuming there is no volume change in the
Specimen during compression, the true stress 0 n in the Specimen
is

2

D
O “ T E

n d [3(6 +€")(1-‘€s() (3.10)

T13 5

Where D and d are the initial diameters of the pressure bars and the

Specimen reSpectively. All stress-strain curves plotted are for

 

56

true stress and true strain.

The assumption of no volume change in the specimen was

checked by measuring specimens before and after compression at

several test temperatures. The error in all cases was less than

one percent.

 

 

CHAPTER IV
RESULTS

4.1 Oscilloscope Records

 

Typical records of oscilloscope traces for the Six test
temperatures are given in Figure 20. The time scale reads from
right to left in these photographs. The upper trace on each record
is the output from the gage station on the incident bar and the lower
trace is from the transmitter bar station. The reflected pulses at the
incident gage station are not Shown. They were adjusted out of the
range of the upper beam in order to utilize higher gain settings on
the preamplifier.

It is seen from Equation (2.15), which gives specimen
stress in terms of the transmitter bar strain, that the shape of
the stress-time curve for each Specimen is represented approximately
by the shape of the transmitted pulse record, since gé is small
in comparison with 5 TB. A direct comparison of the shapes of the
stress-time curves and their amplitudes at the different temperatures
cannot be made on a common basis from the photographs in Figure
20 since the gain settings and the amplitudes of the incident pulses
are not the same, although Figures 20a and 20e Show the effect of
temperature on the curve quite well. It is seen that aluminum has
a more clearly defined yield stress at the higher temperatures.

The oscillations which occur in the leading portions of
the transmitted pulses are apparently due to dispersion of the pulse
components as discussed in Chapter II, since the period of these
oscillations is approximately the same as those observed for the
incident pulse in the long one wpiece pressure bar. The amplitude

of these oscillations increases with temperature. This may be due

57

58

  

I | 50 microseconds

(a) 30°C

5 mv.

_ 2mv.

 

' 1 ‘30 microseconds

(c) 250° c

5 mv.

: 2mv.

 

I I 50 microseconds

(e) 450°C

  

 

I I 50 microseconds

(b) 150°C

 

l 1 50 microseconds

(f) 550°C

Figure 20. Typical Oscilloscope Records for 1/2. Inch Diameter
by 1/ 4 Inch Long Aluminum Specimens

Incident and transmitted pulses are shown on the
upper and lower traces respectively.

EEC

h u c “X.
.f‘ «A;
Ila Fit
nvhv 6 t

p
fl.

tr-

Lu

0.

. flak»

A‘h
vVIA

«IL

\

3) Am
F A
l...‘ . 41rd

59

to the shape of the transmitted pulses at the higher temperatures

( more high frequency components) or it may be due to effects in

the heated pressure bars. Attempts to reduce the amplitude of these
oscillations at the higher temperatures by further rounding of the
end of the striker bar failed.

The oscilloscope records also Show that the transmitted
pulses are longer than the incident pulses. This is due to the fact
that the incident pulse continues to exert a force on the Specimen
after its amplitude begins to decrease. This may be seen from a
graph of the incident, reflected and transmitted pulses where the
transmitted pulse is shown to be the result of taking the difference
of the magnitudes of the incident and reflected pulses. Such a graph
is given in Appendix B.

4. Z Deformation in the Specimens

 

A photograph of specimens tested at each temperature
is shown in Figure 21. The dark surface and ridges at the edges of
the 550°C specimens are due to the glass lubricant which bonded
to them. The specimens tested at the higher temperatures were
slightly barreled but there was almost no evidence of barreling at
room temperature. The small amount of barreling at the higher
temperatures was believed to have little effect on the amount of
longitudinal strain in the Specimens. The circumferential surface
did not fold under on any of the Specimens used in this experiment.
The barreling at the higher temperatures was probably due to a re-
duction in the effectiveness of the lubricants at these temperatures.
No irregularities appeared on the parallel surfaces of the Specimens.

4- 3 Strain-Time Curves

 

Strain-time curves for four typical specimens at 300C

and for four others at 5500C are given in Figures 22 and 23 respectively.

60

 

Figure 21 Typical Deformed Specimens at each Test Temperature
The strains are from 20 to 25 percent true strain.

An-u\nuuv Z—<~.~-~.u(. u'-.JM~.H.

TRUE STRAIN (in/ in)

.32

‘. 28

.24

.20

.16

.12

.08

.04

 

61

1080

6 I:

 

l 1 1 l J
30 60 90 120 150

TIME (microsec. )

0
Figure 22. Strain-Time Curves at 30 C

An~u\n-av Z~ <~.~-~-u1n ”L qu~.N.

TRUE STRAIN (in/ in)

.32

.28

. 24.

.20

.16

.12

.089

.04

 

 

62
0
g e 1080
é = 1010
é = 610
é = 400
l L l 1 g
30 60 90 120 150

TIME (microsee)

0
Figure 23. Strain-Time Curves at 550 C

63

The pressure bar method used in this experiment is seen to produce
nearly constant strain rates for aluminum at strains above approximately
0.5 percent. The linearity of the strain-time curves is due to the
shape of the transmitted pulse Since the specimen strain is cal-
culated from the area under the curve obtained by subtracting the
transmitted pulse from the incident pulse. Fortunately, the linearity
is quite good at all test temperatures even though the Shape of the
transmitted pulse changed with temperature. The curves for the
intermediate temperature tests are not given because they are
similar to the ones shown.

Lindholm (1960) found that the strain-time curves produced
for lead and copper by this method were not linear although they could
be assumed linear to a fair approximation.

The very low strains during the first 10 microseconds are
due to the smaller area under the rise portion of the incident pulse.

The split Hopkinson pressure bar method used produces
a limited range of strain rates. Significant extensions beyond the
range of 300 to 2000 inches per inch per second obtained in this in-
vestigation are not feasible. AS discussed earlier, the yield point
of the bar material places an upper limit on the strain rate and friction
effects prevents the use of a shorter Specimen to increase strain
rates. In order to obtain lower strain rates at the level of strains
used in this experiment, it would be necessary to increase the length
of the striker bar and to reduce the impact velocity. If the length
of the striker bar is increased it is also necessary to increase the
length of the pressure bars in order to avoid reflections of the longer
pulse in the strain records, and the apparatus would soon become
unwieldy. If the impact velocity is reduced, the force applied to the

Specimen soon becomes so small that it will produce only elastic

64

strains. The lower velocities would also produce lower strains in
the pressure bars which would require a refinement in measuring
technique. Lower Specimen strain rates could also be obtained by
increasing the length of the Specimen but this increases the possibility
of errors due to wave propagation effects in the Specimen.

4. 4 Stress-Strain Curves

 

Typical stress-strain curves for Specimens at all Six
test temperatures are given in Figure 24. Curves at four strain rates
are shown at each temperature. It is seen that increased temperatures
have a very pronounced effect on the magnitude of stress and that
strain rate has a much greater effect on stress at high temperatures
than at room temperature.

Due to the nature of the loading pulse, it was difficult to
obtain accurate data for low levels of strain especially at the higher
-; strain rates. The oscillations which appeared on the transmitted
pulse at the higher temperatures limited the accuracy at the lower
strains, but the accuracy obtained by taking an average curve through
the oscillations was quite adequate at strains of 3 percent and above
for the highest strain rates and was accurate for strains as low as 0.5
percent at the lower strain rates. The curves were extended to the
origin for the sake of appearance.

With the inaccuracy at the lower strains, it was impossible
to determine the effects of strain rate and temperature on Young's
modulus. .However, some work has been done on this problem. The
variation of E with temperature as determined statically and dynamically
was given for aluminum by Vosteen (1958) (see also Garofalo, 1960).

In Figure 25, a room temperature stress-strain curve
from the present experiment is shown along with curves at lower

strain rates obtained by Lindholm (1960) using the same type of

65

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67

aluminum and the same Specimen size as used in the present in-
vestigation. These lower strain rate tests were performed on a
Model FGT Baldwin-Emery SR-4 Universal Testing Machine. The
machine had a constant loading speed which could be changed by
means of a variable Speed motor. It should be mentioned that stresses
obtained by Lindholm in the impact range of strain rates, using an
apparatus Similar to the one used here, were all approximately 5
percent lower than those obtained in the present study. Part of this
difference is believed to be due to a slight difference in the Specimen
material. Tests performed in this investigation showed that stresses
at room temperature varied by as much as 8 percent between different
bars of supposedly the same material.

A factor which could influence the Shape of the stress-strain
curve is the change from isothermal to adiabatic conditions as the
strain rate is increased. An approximate calculation based on an
adiabatic condition was made for a Specimen in the present experiment.
For a strain of 25 percent the Specimen temperature increases about
120C which would cause the stress to decrease by about 250 pounds
Per square inch according to temperature curves determined in this

study. This stress change would have little effect on the Shape of

the stress ~Strain curve 5.

4.5 Stress-Strain Rate Curves

The stress-strain rate curves at each temperature were

Plotted on both semi—logarithmic and IOg-log coordinate systems in

order to evaluate the validity of the logarithmic and the power flow

laws as given by Equations (1. 7) and (1. 9) respectively. The semi-

10garithmic curves are shown in Figures 26-30 and the 10g-log curves

are shown in Figures 31—34. Each point in these figures re

n where the stress at 0. 05 inches

presents

a Separate test with a new speCime

TRUE STRESS (1000 psi)

TRUE STRESS (1000 p81)

68

13.5—

13.4—

13.3

13.2”

13.1“

 

13 Or

 

l 1 1 1 1 1 1 1 1 r

300 500 700 1000 2000
TRUE STRAIN RATE (in/in/sec)

Figure 26. Stress Versus Log Strain Rate at 5 Percent True

 

 

 

Strain
11.6 1" 150° C
%
11.5 —
11.4 h
11.3 '-
11.2 r
11.1 —
11 O 1 1 1 1 1 1 1 1 1 1
300 500 700 1000 2000
TRUE STRAIN RATE (in/1n/sec)
Flgure 27. Stress Versus Log Strain Rate at 5 Percent True

Strain

TRUE STRESS (1000 psi)

11.2 —

11.0

10.8 _

 

Figure 28.

 

69

250° c

1 1 1 1 1 1 L 1 1 1

300 500 700 1000 2000
TRUE STRAIN RATE (in/in/sec)

Stress Versus Log Strain Rate at 5 Percent True
Strain

TRUE STRESS (1000 psi)

Figure 29.

 

7O

 

350°C
0
o
o
O o
20
o
/O/
o
o
o
O O
l 1 L 1 1 1 1 1 1 1
300 500 700 1000 2000

TRUE STRAIN RATE (in/in/sec)

Stress Versus LOg Strain Rate at 5 Percent True
Strain

 

71

 

 

1 450°C

8.4 -

8.2 —
'63
o.
o
o
S, 8.0 7
U)
m
Li]
01
53'

_ o
m 7.8
D
131
E...
O
7 _ O
.6 OO
O
O
7.4 _ /
/O
l l J l 1 l 1 1 l 1
300 500 700 1000 2000
TRUE STRAIN RATE (in/in/Sec)
Figure 30. Stress Versus LOg Strain Rate at 5 Percent True

Strain

TRUE STRESS (1000 psi)

 

300

72

550° c

 

1 l 1 l l l

500 700 1000
TRUE STRAIN RATE (in/in/sec)

2000

Figure 31. Stress Versus Log Strain Rate at 5 Percent True

A - 3

,1:

.13

,1}.

.12

OGIO TRUE STRESS

4.11

Figure

4. 07

4. 05

TR U E S TRESS

f)

u.

o” 4.04

LO

Figure

73

 

 

1 1111131 1 L

 

300 500 700 1000 2000

TRUE STRAIN RATE (in/in/sec)

32. Log Stress Versus Log Strain Rate at 5 Percent True
Strain

150° c

 

 

11111111 11

300 500 700 1000 2000
TRUE STRAIN RATE (in/in/sec)

33. LOg Stress Versus Log Strain Rate at 5 Percent True
Strain

LOGIC TRUE STRESS

LOG10 TRUE STRESS

4. 04

4. 03

4. 02

4.01

Figure

3.99

3.98

3.97

3.96

3.95

Figure 35. Log Stress Versus L0g Stra

 

 

74

 

 

O
/
O
/
O O
1 1 1 1 1 1 1 1 1 1
300 500 700 1000 2000

TRUE STRAIN RATE (in/in/sec)

34. Log Stress Versus LOg Strain Rate at 5 Percent True

 

Strain
f 350°C
_ o
O o
o
/O
- o
o o/
1/ 1 1 1 l 1 1 1 1 1
300 500 700 1000 2000

TRUE STRAIN RATE (in/in/sec)
in Rate at 5 Percent True-

Strain

75

 

 

 

_ 450°C

3.93 ‘

3.92 -

3.91 T
8
H 3.90
Ci
{—1
(I)

O
‘3
m 3.89
{—1
O
O
87 °
14 3 88 _ OO
O
3 87 ‘- O
/
/O
3 86 p
1 1 1 1 1 1 m 1 1 1
300 500 700 1000 2.000

TRUE STRAIN RATE (in/in/sec)

Figure 36. Log Stress Versus LOg Strain Rate at 5 Percent True
Strain

TRUE STRESS

OGIO

.87

.86

.85

.84

.83

.82

.81

.80

.79

.78

.77

 

76

 

 

550° C o
O
O O
O
O

o

1 O 1_ 1 L 1 1 1 1 n L

300 500 700 1000 2000

TRUE STRAIN RATE (in/in/sec)

Figure 37. L0g Stress Versus LOg Strain Rate at 5 Percent True

Str ain

77

per inch strain was taken from the plotted stress-strain curve of the
specimen. It is seen in Figure 24 that this level of strain makes

use of the widest range of strain rates Since the lower strain rate
curves only extend to approximately 5 percent strain. The difference
in the lengths of the stress-strain curves is due to the fact that the
duration of the loading pulse iS the same for all tests.

Most of the scatter in the data is probably due to variations
in the material and the lubrication of the Specimen faces, though some
of the scatter is of course due to the limitations of the strain measuring
equipment and to errors in the measurements taken. The accuracy
improved considerably with an increase in temperature, but some
improvement was to be expected here since the same percentage
error at the lower stresses gives a smaller deviation. There are a
few points at the higher temperatures which lie quite far from the
curves. This is believed to be due to material and lubrication effects.
The Specimens were selected in a random fashion in order that the
influence of any gradual change in material along the bar would not
be systematic.

There is very little difference between the semilogarithmic
and the 10g-log curves. This is probably due to the comparatively
small range of strain rates obtained. However, when stress-strain
rate curves are plotted on these two coordinate systems for a true
strain 0.1054 inches per inch and compared with the results of Alder
and Phillips (1954) at the same level of strain, the 10g-log plot is seen
to be more linear. These semi10garithmic and log-log curves are
shown in Figures 38 and 39 respectively. Alder and Phillips‘ data
was obtained from compression tests at lower strain rates in which
the specimens were machined from extruded aluminum bars and

o .
annealed at 400 C for one hour in vacuum.

Tu.Tv.;..— V!...~. «J.

ENE NH-W.

all.

TRUE STRESS (117730 {351)

18

16

14

12

10

 

 

 

,. 78
/
Alder and Phillips (1954) / / . ,
3c 4'.“
L r— ‘ / . O.
9039/
/
0 /
: ’/
O . /
I . /
/
P" . /
/ / 000
/ 000 O
/ ° °
/
/ A
/ 4‘
o o /“ u:
L o o 6 00 /
0 /
o o I /
/
/ Am
/ / AAMAA
/ A
1— / A A
A /‘
A /
A / l/
/ / In"
I
I '~
J
/
r 0 /
A A00 C)/ tho
A [101:1EJ
A / on
A D
A / 0° /
/
A A /
1— / /
/ I /
/
/ ' . 0°C /
/ 69 /
/ ' /
/ I /
./ I /
/ O
o . 30 c
/
a /o 0 o .. . .lSOOOC
a / . .2500C
D /
L-D// A...3500C
/ . .4500C
D .550 C
_ ~— -'— Nadai and Manjoine (1941)
1 11111111 1 1111mm 111L11111 L111
70 100 1000

TRUE STRAIN RATE (in/in/sec)

Figure 38. Stress Versus Log Strain Rate at 10.54 Percent

True Strain

4.1

4.0

"1...”an Y~.-.u1u

Hmfl ‘- m, ..~1

o

N Vflva.

3.6

3.5

 

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10

LADC}
u)

 

 

 

r 79
Alder and Phillips (1954) , *5 .... 3.3.;
2 L- I A m
. 8 ° °
006°C 0° 00
1— ‘A
l o O o o O 5‘ “‘.‘
O O
A
A A
AAA“s A
0" A A
A ‘ ‘
A '-
I-'EH'I'
J
9 b A Dog
A DoDD
A a o
A D
A Do
8 F- A A
I
I
I
7 _.
I
I
I
e 1- '
o
o 300c;
5 _ o 0 150°C
D D A 250°C
A 3502C
I 450 C:
D 0 550°C
4 1‘ a
D 1 11111111 4 1114111L 141111111 11
10 100 1000
TRUE STRAIN RATE (in/in/sec)
Figure 39. Log Stress Versus Log Strain Rate at 10.54 Percent

True Strain

 

 

80

Figure 39 shows that the SlOpes of Alder and Phillips'
data compare well with the present data on a log—log plot, but the
magnitudes of stress in their data are lower at nearly every temper-
ature. There are several possible explanations for the differences
in stress. First, there may be errors in the measurements from
either or both experiments. Second, the stresses in the present
experiment may be higher due to specimen lubrication problems.
Third, the materials used in the two experiments did not have the
same chemical composition. Fourth, it may be that the curves are
actually Shaped this way. It is likely that the difference in the materials
and lubrication effects are to blame for most of the discrepancy. The
material used by Alder and Phillips was probably more pure than
that used here. It was 99. 21 percent aluminum compared to as low
as 99.00 percent in the material used here.

The effect of alloying elements on the stresses in aluminum
was studied experimentally by Dorn, Pietrokowsky and Tietz (1950).
The results of quasi-static tests up to 30 percent true strain at room
temperature Show for instance, that increasing the copper content
of nearly pure aluminum from a level of 0.065 percent to 0.120 percent
increases the stresses of the stress-strain curve by approximately
8 percent. Nearly all of Alder and Phillips' specimens contained 0.10
percent copper while the Specimens used in the present experiment
had a possible maximum of 0. 20 percent. The difference between
Alder and Phillips' results and the present results at room temperature
is approximately 4 percent.

Dorn, Pietrokowsky and Tietz also showed that grain Size
has a considerable effect on the stress, but this effect is believed to
cause a negligible difference in the results Since the annealing of the

Specimens in the present experiment was carried out under nearly

81

the same conditions as those in Alder and Phillips' experiment.

The interface friction that occurs in the plastic compression
of cylindrical Specimens was studied by van Rooyen and Backofen (1960).
They used primarily Number 1100F aluminum, and the lubricants tested
included molybdenum disulfide, the lubricant used for four temperature
runs in the present study. The interface Shear stress that occurred
over the face of their 2 inch diameter by 0. 5 inch long specimen using
molybdenum disulfide averaged about 10 percent of the normal stress
when the specimen was compressed to 10 percent reduction. If this
value for interface shear stress is used, an approximate calculation
for 10 percent reduction in the present experiment shows that about
4 percent of the energy of compression would be used to overcome
interface friction.

The results of tensile impact tests performed on aluminum
by Nadai and Manjoine (1941) are shown in the form of dotted lines
in Figure 38. Although these curves are not strictly comparable since
they represent ultimate stresses for which no correction is made for
reduction in area, the slopes of the curves are quite Similar to those
of the compression test data shown. Since the strain at which ultimate
stress is reached may vary with the strain rate, the stress values are
not at a constant strain as in the present tests. It is also difficult to
determine actual strain rate in the relatively long specimens used in
tension. Nadai and Manjoine used an average rate for which it is
necessary to assume a uniform strain distribution in the Specimen.

Since there is little difference in the linearity between the
Semi-logarithmic and the IOg-log plots of the data from the present
study, both the logarithmic and the power law will be used to represent
the data. These laws as given previously by Equations (1.7) and (1.9)

are given here in the slightly different forms

0:00(6,T)+k(6.T110gé (4~1)

82

and

.n ,T
0 :00‘(6.T)e (E ) (4.2)

where 0' o' O 0', k and n are shown as functions of strain and temper-
ature and not a function of the material Since the data is for aluminum
only. In these equations 0 o and O 0' are stresses at unit strain rate,
k represents the SIOpe of a semi-10garithmic plot of stress versus
strain rate at a particular level of strain and temperature, and n

is the slope of a 10g-10g plot of O /O‘ 0 versus strain rate at particular
values of strain and temperature. If Equation (4.1) is divided by O 0
we have

0' k .
0' 0 0g 6 ( 3)
O 0

where k/O 0 represents the SIOpe of a plot of 0/ 0 0 versus the logarithm
of strain rate. This shows that k/o o is a measure of the rate sensitivity
of the material. The exponent n is also a rate sensitivity parameter
as may be seen more readily if Equation (4.2) is written in the form
10g g—, z: n log 6 (4. 4)
o

No significant change in the Slope of the stress-strain rate
curves was observed between strain levels of 0.05 inches per inch
and 0.1054 inches per inch, and the stress-strain curves obtained at
low strain rates did not extend far enough to obtain reliable stress-
strain rate data for strains higher than about 10 percent . Therefore ,
the dependence of k and n on level of strain was not determined. The
change in k and n with temperature was determined at 5 percent strain
from the curves shown in Figures 26-37. The values of k and n at
the various test temperatures are given in Table l. The values of
0 o and 0 ‘ which were obtained by extrapolating back to the unit

0
strain rate abscissa are also given. The values of O o are lower than

 

 

83

TABLE 1. Values of k, a o’ k/O’ o n and 0' 0' in the Logarithmic

and Power Laws at 5 Percent Strain

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Temp. ,
_ - f: .1 -
0C k(ps., 00(p51) 00 n 00 (p51)
30 413 12,600 0.034 0.017 12,140
150 498 9.910 0. 050 0.022 10,020
250 795 8, 290 0.096 0.028 8,550
350 1110 5,990 0.185 0.040 6,520
450 1310 4, 070 0. 322 0. 073 4,820
550 2100 130 0.141 2, 510
TABLE 2. Values of n Obtained by Alder and Phillips (strains
are nominal).
Values of n for a strain of;
Temp. .
0C 10% 20% 30% 40% 50%
18 0.013 0.018 0.018 0.018 0.020
150 0.022 0.022 0.021 0.024 0.026
250 0.026 0.031 0.035 0.041 0.041
350 0.055 0.061 0.073 0.084 0.088
450 0.100 0.098 0.100 0.116 0.130
550 0.130 0.130 0.141 0.156 0.155

 

 

 

 

 

 

 

 

 

 

84

those of 00' and the magnitude of the difference increases with
temperature. The value of 0’ 0 at 5500C is no doubt too small to
be an actual stress value at this strain rate and temperature. There-
fore, this value was not used to calculate a value for the rate sensitivity
coefficient k/O’ o' This Shows that the power law gives a better
overall fit to the data.

Lindholm found that k for aluminum at room temperature
varied from 367 psi. at 0.03 inches per inch strain to 661 psi. at
O. 24 inches per inch strain and k/O‘ o varied only Slightly over this
range of strains. These were much smaller variations than either
the lead or c0pper showed. Alder and Phillips also found that the
variation of the strain rate effect exponent n with level of strain was
comparatively small in aluminum at room temperature but the variation
with strain increased markedly with temperature. This is seen in
Table 2 where the values of n as determined by Alder and Phillips are
given for comparison with the present data.

The values of k/o o and n from Table 1 are plotted versus
temperature in Figures 40 and 41. These curves cannot be represented
with accuracy by either a 10garithmic or a power function.

4.6 Stress-Temperature Curves

 

Stress versus temperature plots at true strains of 0.05 and
0.1054 inches per inch and at several strain rates are given in
Figures 42 and 43. These figures Show that the data may be re-
presented to a fair approximation by straight-line plots, although
the points for 150°C consistently fall below the lines while those for
2500 are consistently above it. It is possible that the actual Specimen
stresses are those given by the data, although there are several other
possible explanations for the apparent deviations of the given stress

values. For instance, the effectiveness of the molybdenum disulfide

k/o

0.4

 

85

 

O
0.3..
0.2
.1" O
0.1 - o
-0

L 14, 1 _J. 1
100 200 300 400 500

TEMPERATURE (°C)

Figure 40. Dependence of Strain Rate Effect Coefficient

 

k/o' on Temperature at 5 Percent
True Strain

 

 

.ISOF
.100"
O
.050L
/0
‘o—f 40/0
; 1 L l 1
100 200 300 ‘ 400 500

TEMPERATURE (°C)

Figure 41. Dependence of Strain Rate Effect Exponent n on

Temperature at 5 Percent Strain

TRUE STRESS (1000 psi)

86

20 r

1’ Sherby, Anderson and Dorn

A1. with 0.120% Cu. , 6' = 0. 002, . . . .
_ A1. with 0.520% Cu., é = 0.002 . . . . o
Trozera, Sherby and Dorn
. -3

1— High purity A1., 6 =l.94x10 . . . . 1:1

15 )—

10

 

 

 

100 200 300 400 500

TEMP. (°C)

Figure 42. Stress Versus Temperature at 0.05 Inches Per Inch
True Strain

87

20"

IS

TRUE STRESS __(1000 ps.)

 

Alder and Phillips (1954)

 

0 L 1 L 1 1
100 200 300 400 500

TEMPERATURE (°C)

Figure 43. Stress Versus Temperature at 0.1054Inches Per Inch
True Strain

88

lubricant no doubt changed with temperature. The powdered graphite
may have caused the apparently high measured stresses that occurred
at 2500C. Also, the viscosity of the glass lubricant used at 5500C
may have caused an apparent increase in stress with'an increase in
strain rate. This would also influence the values of k and n given in
the previous section.

Sherby, Anderson and Dorn (1951) have shown that alloying
elements and the amounts of them present in aluminum also have
a considerable effect on the shape of the stress-temperature curve.
Two curves from the data of Sherby, Anderson and Dorn which are
for aluminum with different amounts of copper are shown in Figure
42. A third curve for high purity aluminum which was taken from
Trozera, Sherby and Born (1957) is also shown in Figure 42. Trozera,
Sherby and Dorn obtained stress-strain curves from tensile tests on
high purity aluminum at temperatures from - 195 to 5450C- and at
strain rates from 9. 63 x10”7 to 0.167 inches per inch per second.

It is also possible that the apparent deviations in the stress
values were caused by a change in the measuring equipment between
temperature runs. There was approximately one week between
temperatures runs.

The data from Alder and Phillips as shown in Figure 43
are represented quite well by straight lines. The points for these
curves were obtained from straight lines drawn through stress-strain
rate points plotted from their tabulated data.

Stress versus temperature curves for several levels of

strain at a strain rate of 2000 per second are given in Figure 44.

TRUE STRESS (1000 psi)

20

15

 

89

 

 

)_.
1 EL 4 1 1
100 200 300 400 500
TEMPERATURE (°C)
Figure 44. Stress Versus Temperature at a Strain Rate of 2000

Inches Per Inch Per Second

CHAPTER V
SUMMARY AND CONCLUSIONS

An experiment was performed in which Short cylindrical
Specimens (1/2 inch in diameter by 1/4 inch long) of commercially
pure aluminum were subjected to impact loads at room temperature
and at elevated temperatures by means of a Split HOpkinson pressure
bar apparatus in which resistance strain gages were mounted on the
pressure bars to obtain records of the impact pulses. An electric
resistance type furnace surrounded the area of the Specimen. Nearly
constant strain rates, over the entire range of plastic strains reached
in the experiment, were obtained for a range of rates from 300 to
2000 inches per inch per second. True strains (f approximately
5 percent were obtained at the lower strain rates and strains of over
25 percent were attained at the highest strain rates. Six test temper-
atures from 30 to 5500C were used. True stress-strain curves were
plotted for each of the 108 specimens and points were taken from these
curves to plot true stress-true strain rate and true stress-temperature
curves at strain levels of 0. 05 and 0.1054 inches per inch. The
accuracy of the stress-strain curves for strains below 0. 5 percent
was not sufficient for determining Young's modulus.

The stress-strain rate curves could be represented quite

well with either the 10garithmic law

ozoo+klogé 15.11
or the power law

0 = o ' é (5. 2)

where 0 , 0 ‘, k and n are functions of strain and temperature.
0 o

The ratio k/o and the exponent n are the slopes of the linear
o

90

91

semi-logarithmic and the 10g-=log plots of 0/0' 0 and 0 lo 0' versus
strain rate, and are therefore measures of the rate sensitivity of
the material. The rate Sensitivity was found to increase with
temperature but stress-strain rate curves were not obtained over
a sufficiently large range of strains to determine the dependency of
rate sensitivity on the level of strain. Values of k, k/O o and n
were tabulated for all test temperatures at a level of 5 percent strain.
When the results were correlated with the data of Alder
and Phillips obtained from compression tests at lower strain rates,
the power law was found to give a better fit to the plotted points.
Further studies are suggested along several lines: (a)
increase the range of ccnstant strain rates to cover both lower
and higher rates, (b) perform impact tests at very low temperatures
as well as elevated temperatures over a wide range of strain rates,
(c) perform tests with a high purity aluminum in order to minimize
the stress increasing effects of alloys, (d) find lubricants which
have similar properties at various test temperatures and (e) obtain
accurate stress-strain curves over the range of strains which in-
cludes the elastic region and extends well into the plastic region

for all temperatures and strain rates.

1.

10.

92

BIBLIOGRAPHY

Alder, J. F. and Phillips, V.A. , ”The Effect of Strain Rate
and Temperature on the Resistance of Aluminum, COpper

and Steel to Compression," J. Inst. of Metals, vol. 83,
1954—55, p. 80.

Alter, B.E.K. , and Curtis, C. W. , ”Effect of Strain Rate on
the Propagation of a Plastic Strain Pulse Along a Lead
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APPENDIX A
CALCULATION OF WAVE REFLECTIONS
DUE TO TEMPERATURE CHANGES

In this appendix the equations giving the amplitude of the
transmitted and reflected strain waves when a strain pulse encounters
a discontinuous change in Young‘s modulus will be derived and
experimental evidence of the validity of these equations for continuous
changes in Young's modulus will be presented. These Equations
were given in Chapter II as Equations (2. 3) and (2. 4).

Consider the fictitious case of a long metallic bar of un-
iform cross-section in which the temperatures on opposite sides of
a particular cross-section are different, say T1 and T2. Since the
modulus of elasticity is a function of temperature, we have a bar
like the one shown in Figure 45 where Young's modulus suddenly
changes from E1 to E2 where the subscripts correspond to the
temperatures in the two sections. If a stress wave approaches the
abrupt change from the left as the arrow indicates, part of the wave
will be reflected and part of it will be transmitted through the

cross-section. From a summation of forces at the cross-section of

temperature change we have

(A. 1)

where the subscripts I, R and T correspond to the incident, reflected

and transmitted pulses respectively. Now if e I’ e R and e T re-

present the strains corresponding to the stresses of Equation (A. 1)

then using Hooke‘s law we have

E(€ +5 )ZEzeT (A.Z)

lOO

101

 

 

 

Figure 45. Bar with a Discontinuous Change inE.

30"

26)-

E (106 psi)

24L

0
12.00%

1 l L 1 1 J I l
100 200 360 400 500 600 700

TEMPERATURE (°c)
Figure 46. Modulus of Elasticity Versus Temperature for 18 -8
Stainless Steel Pressure Bars.

 

20

102

If the bar material remains continuous at the cross-section we must

also have

uI+uR =uT (A.3)

where 111, 11R and uT represent the displacements at the cross-section
which are due to the incident, reflected and transmitted stress waves
respectively. If we take the derivative of both sides of Equation (A. 3)

with respect to time and substitute from the elastic wave propagation

relation
1 du .
E - E a? ”'4’

we have
- = .5
(11(61 6R) CZET (A )

where c1 and c2 represent the wave velocities in the first and second
sections of the bar respectively and the negative sign represents a
negative velocity. It is easily shown by consideration of the temperature
coefficient of expansion of steel that the change in density p in the

bar is insignificant when compared to the change in E so that c

l
2 may be taken proportional

to VEZ . This is from the elastic wave velocity equation c = V E/ p.

Using these proportionalities in Equation (A. 5) we have

VEl (eI~ 6R): VEZ 6T (A.6)

From Equations (A. 2) and (A. 6) we can solve for the ratio of trans—

may be taken proportional to VEI and c

mitted pulse to incident pulse and the ratio of reflected pulse to

incident pulse, that is

 

—— = (A. 7)

103

and

6R_VEZ 'VE-l

Equations similar to these were derived by Rayleigh (1945) for the

 

(A.8)

determination of transmitted and reflected sinusoidal waves at a point
of sudden change in material.

Two separate experiments were performed in order to
determine the usefulness of these equations in the present investigation.
The first involved the mounting of two high temperature Nichro‘me
V foil strain gages at the center of the furnace on a long pressure
bar. A photograph of the gage installation is given in Figure 47.
These gages were used to determine the amplitude of a strain pulse
after it passed through the temperature distribution on the incident
side of the furnace. This amplitude was compared with the amplitude
of the incident pulse obtained from a standard incident gage station
approximately at room temperature. The high temperature gages
had a one -inch gage length and a resistance of 138. 4 _+_ 0.2 ohms at
room temperature. The room-temperature gage factor was 2. 3 :04.
These gages were mounted with Tatnall H ceramic cement (for use
at temperatures up to lSOOOF), and the short lead wires were welded
to copper wires which were insulated in ceramic tubing.

Typical oscilloscope records of the strain pulse obtained
from the high temperature gage station are shown in Figure 48.
Figure 48a is at room temperature and Figure 48b is at lZOOOF.

It is seen that the reflection of the pulse from the transmitter bar
side of the temperature distribution is superposed onto the rear
portion of the pulse, increasing the amplitude. The strain calculations

were based on measurements taken at a point immediately following

104

 

 

Figure 47. Close -up of High Temperature Gage Installation

_2mv.

__ 5mv.

 

50 microseconds I I 50 microseconds

(a) Room Temperature (b) 1200°F

Figure 48. Pulse Records at Center of Furnace in a Long Pressure Bar

    

1 mv. _ 1 mv.
5 mv. 5 mv.
I I 50 microseconds I l 50 microseconds
(a) Room Temperature (b) 1200°F

Figure 49. Records used for Measuring Temperatures Reflections in
a Long Pressure Bar

106

the rise portion of the pulse. In order to calculate the strain in the
bar from these measurements it was necessary to use a linear variation
of gage factor with temperature from the data of Brosius and Hartley
(1959) who tested Nichrome V foil gages up to 11000 F. The linear
variation was assumed to extend to lZOOOF for the present experiment.
The variation in gage resistance with temperature was also included
in the calculations.

The calculation of transmitted strain amplitude from Equation
(A. 7) included values of E taken from the E versus temperature curve
given in Figure 46. The slope of the curve in this figure was taken
from Garofalo (1960). Pulse records were obtained at 300, 600,
900 and 12000F and the difference between calculated and experimental
values was less than one percent for the temperatures used below
1200017, and the difference at lZOOOF was less than 3 percent.

The second experiment was performed in order to verify
Equation (A. 8) and to check a method devised to calculate reflected
waves. In this second experiment, the reflected portion of an in-
cident pulse was measured directly from photographs which were
obtained from a gage station mounted 28 inches from the center of
the furnace and 12 inches from the impact end of the long bar. These
photographs are shown in Figure 49. The pulse reflected from the
temperature gradients appears immediately after the incident pulse
on the upper channel in Figure 49b. The strain gages used at this
first gage station were SR-4 Type A-8 wire gages with a nominal
resistance of 120. 5 ohms and a gage factor of 1.81 i 2 percent. It
should be mentioned here that the small peak which appears just after
the rise of the upper channel pulses seems to be a characteristic of the
A-8 gages. This peak also appeared in Lindholm's (1960) data, which

was obtained with A-8 gages. The temperature at the center of the

107

furnace for Figure 49b was lZOOOF. The reflection pulse was
determined from the difference of the upper channel traces shown in
Figure 49.

The calculated reflection pulse was obtained by assuming
step functions for the temperature distributions on both sides of the
furnace and using Equation (A. 8) to calculate the increments of re—
flection at the discontinuous jumps in temperature. The increments
were superposed onto one another to obtain the reflected strain pulse
shown in Figure 50 as the solid line curve. The incident pulse was
assumed to have a trapezoidal shape to make the graphical construction
simpler. Time in Figure 50 is measured from the instant the incident
pulse reached the gage station and the increments of reflection are
shifted to the right by an amount equal to their time of travel from
the gage station to their corresponding points of reflection and return.
The discontinuous jumps in temperature were spaced according to
equal time intervals of wave travel. This meant that the points of
reflection had to be spaced closer tOgether at the hot section of the
bar since the wave velocity was lower there. This method of deter-
mining temperature gradient reflections was used for all elevated
temperature tests in the present investigation.

The temperature distribution which was used for the above
calculations is shown as the dotted step function in Figure 51. Actual
temperature distributions were measured by means of ten thermocouples
which were spaced at two inch intervals measuring from the center
of the furnace. The regularly used thermocouple located 3/8 inch
from the center of the furnace was used to determine the first point
on the graph.

The pulses transmitted through the solid bar in the furnace,

as shown on the lower traces of the photographs in Figure 49, were

108

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obtained from a standard transmitter gage station using foil gages.
The transmitted pulse in Figure 49b appears to be unchanged in am-
plitude and shape after passing through the hot section of the bar.
Calculations from Equation (A. 7) predict a decrease in amplitude of

0. 5 percent. A decrease in amplitude of approximately this magnitude

was detected by taking measurements.

APPENDIX B
EXPERIMENTAL SUPPORT FOR THE METHOD USED IN
CALCULATING SPECIMEN STRAIN

B.1 Verification of the Relation e - e = 6

I R T

The relation 5 I - 6 = 5 is obtained from Equation (2.11)

R T
when the pressure bars are at room temperature. Equation (2.11)
was a necessary part of the derivation of Equation (2.14) used for
calculating specimen strains Therefore, an experimental verification

of the relation 6 - 6 will provide support for the intergration

I R: 6 T
method of calculating specimen strains. The accuracy of the assumption
that the stress is equal on both sides of the specimen, which is the
basis of the above relation, is also important in determining stress
in the Specimen since the stress measurements are calculated from
the transmitter bar strain at the interface with the specimen.

The relation was checked by comparing actual strain
records of the incident, reflected and transmitted pulses which were
all recorded at the same time for a specimen at room temperature.
The oscilloscope record of the three pulses is shown in Figure 52.
The pulse reflected from the specimen is the inverted pulse, which
follows immediately after the incident pulse on the upper channel.
These traces were obtained from the regular transmitter gage station
and a station on the incident bar mounted 18 inches from the specimen.
A plot of all three pulses on the same time axis is shown in Figure 54.
The error between the reflected pulse as measured from the photo-
graph and the pulse obtained by taking the difference of the incident
and transmitted pulses is approximately 2 percent. Some of this
difference may be due to oscillosc0pe and photographic distortions.

It is seen that the transmitted pulse is a little longer than

111

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— 10 mv.

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| | 50 microseconds

é : 700 in./in. /sec.

Figure 52. Record of Incident, Reflected and Transmitted Pulses

‘— Time

100 mv.

 

' 1 100 microseconds

E = 710 in. /in./sec.

Figure 53. Record from Strain Gage Mounted on Specimen

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the incident pulse. This is a result of taking the difference between
the incident and reflected pulses. The amplitude of the reflected
pulse naturally starts to decrease at the same time that the amplitude
of the incident pulse begins to decrease and the difference between
the two curves, that is the transmitted pulse, remains approximatély
the same for an additional 10 microseconds.

The experimental results given in the next section provide
additional support for the use of 51- s R: e T in the derivation of

specimen strain.

E. 2 Comparison of Calculated and Measured Specimen Strains

 

 

An experimental verification of the integration method used
to calculate specimen strains was carried out by mounting strain
gages directly onto several standard specimens. One SR-4 Type
FAP-12-12 120 ohm foil gage with a gage length of 1/8 inch was
mounted longitudinally on each specimen. A close up of the specimen
and gage installation is shown in Figure 55. The gage was connected
to one arm of a Wheatstone bridge in which the other three arms were
120 ohm precision resistors. The signal from the bridge was am-
plified and fed into a Tektronix Type 532 single -beam oscilloscope,
and a record of the strain was obtained with an oscilloscope camera
as in the regular tests.

The dual beam oscilloscope was also used in this experiment
so that records of the strains at the incident and transmitted gage
stations could be obtained simultaneously with the specimen strain
record. The scopes were triggered with the same signal produced
by a piezoelectric element mounted ahead of the incident gage
station. The records for three specimens at room temperature and
at true strain rates of 485, 710 and 1010 per second were obtained.

The photOgraphically recorded specimen strain-time curve obtained

115

 

 

 

 

Close-up of Specimen with Gage Mounted on it.

Figure 55.

 

116

from the gage mounted on the specimen at a strain rate of 710 per
second is shown in Figure 53. The specimen true strain-time record
for this strain rate as calculated from the incident and transmitted
pulse records is given in Figure 56 along with a true strain record
calculated from the nominal strain of the photOgraphic record. This
figure shows that the strain measured by the gage on the specimen is
much less than the calculated strain. In the order of increasing
strain rates the measured specimen strains were low by 8.1, 16. 6
and 23.9 percent. Therefore, a direct comparison of calculated
specimen strains versus measured strains could not be made. But

a comparison of total Specimen strains could be made since total
strain in the Specimens could be determined from the three photo-
graphic records from the pressure bars, and micrometer measurements
of the Specimens before and after each test could be made for a
direct calculation of total specimen strain.

The total Specimen strain was obtained from the pressure
bar photOgraphic records in an indirect manner. It is seen in the
photographic record of specimen strain obtained from the gage mounted
on the Specimen that an additional increment of strain occurs after
the main pulse has passed. This increment of strain is due to the
reflected strain pulse in the incident pressure bar, which returns to
the specimen after another reflection from the striker end of the
incident bar, which is now a free end. This additional increment of
Specimen strain was calculated as a percentage increase over the
strain occurring during the passing of the incident pulse, and this
percentage increase was then applied to the specimen strain determined
by the integration method to obtain the total specimen strain.

This indirect method of calculating the total strain from the

integration method data by using a correction factor based on the

117

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specimen gage data was used because the pressure bar records do

not show the additional increment of strain caused by the pulse re-
flected from the striker end of the incident bar. The correction
factor will be valid if the percentage error of the strain gage mounted
on the Specimen is constant over the total range of Specimen strain.
The‘error appears to be constant at least for the strains due to the
incident pulse, as is seen in Figure 56. The total strains determined
in this manner for the three Specimens were compared with strains
calculated from micrometermeasurements before and after impact;
the error at a strain rate of 485 per second is 0.8 percent, at a

rate of 710 per second it is 1. 4 percent and at 1010 per second it is

8. 9 percent. It is believed that the two specimens tested at the lower
strain rates provide a valid check on the integration method. The
larger error with the third specimen is believed to be due to a faulty
measurement from the gage mounted on the Specimen. It was
difficult to establish a zero level of strain on the photOgraphic re-
cord for this Specimen because the initial part of the trace was
slightly obscured.

The second pulse, not shown by the pressure bar records,
does not affect the data presented in the results of the main in-
vestigation reported in Chapter 5, since we are there concerned
only with the transient behavior during the first pulse. The correction
factor was used only for the purpose of checking the integration
method Specimen strains against the micrometer measurements of
length in order to establish the validity of the integration measurements.

Strain gage errors at high strain rates for resistance
gages mounted on aluminum have also been encountered by other
investigators. Bell (see discussion at the close of the paper by

Ripperger, 1960a) found that wire resistance strainigages mounted

119

on aluminum gave measurements which were 26 percent low at a strain
rate of 1000 per second when compared with strains measured by

a diffraction grating technique. The level of strain in his experiments
was 2. 5 percent. This is in fair agreement with the 8.1, 16.6 and
23.9 percent errors at corresponding rates of 485, 710 and 1010 inches
per inch per second found in the present investigation. Cunningham
and Goldsmith (1958) found an error of approximately 8 percent

when the measured change in momentum of a 1/2 inch diameter steel
ball impacting on the end of a 1/2 inch square aluminum bar was
compared with the momentum calculated from a record of the strain
pulse obtained by means of wire resistance strain gages mounted on
the bar. The impact velocity of the ball was approximately 170 feet
per second. This error did not occur when the gages were mounted

on a steel bar.

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