AN EXPERIMEHTAL DEERMINATEQN 0? THE
STRESS$TRAIN S‘WAS’N RATE RELATIORfi? OF
SE‘JWE. fiETAiS

Thesis he the W 6? Ph. D.
MKHIGAN STATE UNZ‘E’ERSW‘!
Ukéc Sh Linéhzzim

39M}

{HES-IS

This is to certify that the

thesis entitled

AN EXPERIMENTAL DETERMINATION
OF THE STRESS-STRAIN STRAIN RATE RELATIONS
OF SEVERAL METALS
presented by

ULRIC S . LIND HO LM

has been accepted towards fulfillment
of the requirements for

DOCTOR OF PHILOSOPHYdegree in APPLIED MECHANICS

 

DmNOVEMBER 15, 1960

 

0-169

 

 

 

   
     

  

LIBRARY

Michigan State
University

 

ABSTRAC T

In order to formulate an analytic solution to the problem of plastic
wave propagation in metals more in formation on the rate dependence of
the stress-strain relationship is necessary. In the present investigation
stress-straineurves for three metals, lead, aluminum, and copper,
were obtained over a range of strain rates from 10"4 in/in/sec to 103
in/in/sec. These curves were all obtained at a nearly constant strain
rate.

At the lower strain rates a universal testing machine was used.
For the dynamic tests an adaptation of the method of Kolsky was employed
where by short compression specimens were compressed between two
sections of a split Hopkinson pressure bar. Loading was achieved by
means of impact from a third striker bar which produced a long, flat-
topped loading pulse. By means of strain gage pick-ups mounted on the
pressure bars, a continuous record of the stress and strain in the speci-
men throughout the test could be obtained. Due to the nature of the loading
pulse the elastic protion of the dynamic stress-strain curves could not be
determined.

Over the entire range of strain rates covered the stress-strain
curves for all three metals formed a homologous set with an increase in
stress produced by an increase in the rate of straining for any given
strain. The dependence of stress on the strain rate was found to follow

the logarithmic law of Prandtl

0': oo+klog€ ,

when 60 and k are functions of strain 00(6 ) is the stress-strain curve at

unit strain rate. k( 6) was found to be an increasing function of strain,

indicating that the tangent modulus also increased with an increase in
strain rate. A measure of the rate sensitivity of a mate rial is given by
k/O'o. Values of k, 00, and k/O'0 are given at several values of strain

for lead, aluminum and copper.

AN EXPERIMENTAL DETERMINATION
OF THE STRESS-STRAIN STRAIN RATE RELATIONS
OF SEVERAL METALS

by
x99

A“

Ulric s.4 Lindholm

A THESIS

Submitted to the School of Advanced Graduate Studies
of Michigan State University of Agriculture and
Applied Science in partial fulfilment of
the requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Applied Mechanics
1960

Approved i 22%;“? g m

 

3.6/ch2

’I"-,‘;AI
93-4" I

ACKNOWLEDGEMENTS

I especially wish to express my gratitude to Dr. L. E. Malvern
who originally suggested the problem and gave valuable advise through-
out the project, and to Dr. C. A. Tatro for his continuous assistance in
solving the experimental problems encountered and reading of the manu-
script. I am also indebted to the other members of my committee,

Dr. G. Mase and Dr. C. Wells.

Sincere thanks are also due to Mr. J. Hoffman and the Division
of Engineering Research for their co-operation and support of this work.

I would also like to thank Mr. R. Jenkens and Mr. E. Thompson

for their assistance in building the apparatus.

1. INTRODUCTION AND PURPOSE

In general stress may be thought of as a function of strain, strain

rate, and temperature such that
a = F (e, e’, T).

For usual engineering applications the temperature is assumed to be
held within comparatively narrow bounds, and also the loads are taken to
be applied "statically", where the effect of strain rate may be small.
Thus, in these cases the stress is taken only as a function of the strain
as indicated by the static stress-strain curve. It has long been real-
ized, however, that the stress-strain curve is affected by varying the
temperature and/or the rate of straining. Thus, there has been a con-
siderable amount of work done and data obtained toward determining the
magnitude of these effects. Stress—strain curves have been obtained in
both the high and low temperature domains. Also the effect of strain
rate from the static range down to creep or stress relaxation tests has
come under considerable scrutiny. For very high strain rates, such as
those that would come into effect in impact or wave propagation problems,
however, there is considerably less definite information. This is due
principally to the difficulty of the experimental problem of measuring
stresses and strains when they are applied in very short periods of time.
It is only in recent years that an attack on the problem has been made
possible through the advent of electronic measuring ani recording devices.

The purposes of this investigation then are to:

(1) Experimentally determine stress-strain curves for several
metals at very high strain rates, these curves extending well into the

-1-

plastic range, with the tests to be conducted at constant (room) temper-
ature.

(Z) Correlate these stress-strain curves with curves at "slow"
rates obtainable on a standard universal testing machine.

(3) Attempt to deduce a functional relationship between stress,
strain, and strain rate at constant temperature.
2. HISTORICAL BACKGROUND

2.1. Interest In The Problem

 

The interest in stress-strainastrain rate effects has been from
several points of view. First, from the influence of the rate of loading
on the standard static tension or compression test, a factor which may
be of considerable importance for many materials. Secondly, the recog-
nition of the importance of impact loading and the development of impact
tests, such as the Charpy or Izod tests, showed that the maximum stress
and the energy absorbed by specimens subjected to impact were greater
than in the case of static loading. Also stresses greatly exceeding the
static yield stress could be sustained by materials without plastic deform-
ation if the loads applied were of extremely short duration. These facts
indicated that a different stress-strain relationship was needed to account
for these phenomena. A third and comparatively recent interest in dy-
namic stress-strain relations has grown out of the development of the
theory of plastic wave propagation. It is from this aspect of the problem
that the present investigation was initiated.

An analytic solution to the plastic wave propagation problem was
first obtained during World War II in this country by von Karman1 and in
England by Taylor2 working independently. A principal conclusion of
their analysis was that for a non-decreasing axial impact load each in-

-2-

crement of stress would be propagated at the velocity c such that

1 do
= A - ——
C JP de » (1)

where P is the density and gig- is the slope of the stress-strain curve
at the given stress level. This relation had also been proposed earlier
by Donnell3. It is seen that in the elastic region this reduced to the well
known elastic wave velocity co = JE/P . The von Karman - Taylor
theory assumed a unique stress-strain relation indpendent of the rate

of strain. This relation was taken as the static stress-strain curve as
dynamic relations were not known. Experimental verification of the
theory4 showed some deviations from the predicted behavior. These
deviations indicated that a solution incorporating a strain rate effect
would perhaps be more accurate.

Such a solution was obtained in 1951 by Malverns. Malvern
assumed that the stress could be taken as a function of the plastic strain
6p and the plastic strain rate ép , such that

o
0' = 4) (6p, (P)

If this is solved for 3p , we get

2p = g (a. e) (2)

and by including the elastic portion of the strain

§§= fiS—‘thgw, e) (3)

where E is the modulus of elasticity and g(<r,€) is an assumed or experi-
mentally determined function of the stress and strain. Malvern con-
jectured that the function g(<T,€) should be a function of <7 - f( 6) , where
0 = HE) represents a quasi-static or equilibrium stress-strain relation,
i. e. , one that is independent of time effects. Thus flow was assumed to
occur only when 0' é f(€land the plastic strain rate would be a function
only of the excess of instantaneous stress (I over the equilibrium stress
at the corresponding strain. It may be noted that this type of relation
will only be valid for cases of short time or impact loading, because a
true equilibrium or strain rate independent stress-strain relationship is
not feasable due to creep effects. Due to the lack of experimental evidence
and the need to make the mathematical problem tractable, the function
g(0, E) was simplified to be linearly proportional to G - f(€) . Thus,

Equations (2) and (3) become

0

6p = k[0' - f(€)] (2a)
and
g:- = ég-g-I- k[0'- f(€)] (3a.)

One conclusion from this strain rate dependent theory is that the
leading wave front of each increment of plastic stress will travel at the
elastic wave speed and not according to Equation (1) as predicted by the
rate independent theory. This conclusion was later verified experiment-
ally by Sternglass and Stuart6 who showed that for a material prestressed
into the plastic range, a small additional increment of impact stress

-4-

would be propagated at the elastic velocity, not at the velocity determined
from the slope of the stress-strain curve at the prestress level. Other
differences from the strain rate independent theory need further experi-
mental verification.

Other relations than the linear strain rate dependence of Malvern
have been proposed. Prandtl7 on theoretical grounds derived a relation-

ship of the form

3 = AeBU,
P

where A and B are constants. Again including the elastic strain this

give 5

0/

8€_l 0’ Bo
F‘Et+Ae ‘

01

Previously Ludwik8 and later Deutler9 added experimental support to
this form.
Some experimental evidence10 has also shown that a better fit

to the data may be obtained by taking

where A and n are again constants.
To compare these three proposed forms of strain rate dependence

let us write them in the form

o=o+k€ (4)

Here we are assuming that the plastic deformations are large and the
elastic components of the strain may be neglected. In Equation (4), 00
represents the equilibrium stress, i. e. , the stress at which E is taken
as zero. In Equations (5) and (6), 0'0 is the stress corresponding to unit

k,k

strain rate. ‘70, 1 and n may in general be functions of the strain

2
e . These three relations will be referred to as the linear, logarithmic,
and exponential strain rate laws respectively.

2. 2 History Of Experimental Techniques

 

Experimental verification of the above relations for strain rate
dependence becomes a difficult task at high strain rates. Inertia effects
in the loading mechanism and the recording of the transient stress and
strain in the specimen pose problems which are not easily solved. At
very high strain rates wave propagation produces an uneven stress and
strain distribution in the specimen which also must be taken into account
if the true stress-strain relationship is to be obtained. Although a
solution of these problems has been put within experimental reach
through the advent of modern electronic and optical measuring and re-
cording devices, a completely satisfactory testing method has yet to be
devised.

In order to give an idea of the problem, a brief resume’of some

-6-

the experimental approaches that have been made will be presented here.
The list is by no means complete but will introduce the reader to the
general methods of dynamic testing.

2. 2. l. Hopkinson Pressure Bar

One of the first methods of measuring transient stresses was de-
vised by B. Hopkinson11 in 1914 and has come to be known as the Hopkinson
pressure bar. A cylindrical steel bar was suspended ballistically with a
small "time piece" of the same material and diameter as the bar wrung
onto one end with a thin film of grease. An impact load applied to the
opposite end of the bar caused a compression pulse to travel down the bar.
This compression pulse would be transmitted into the time piece, but
its reflection from the free face of the time piece being of opposite sign
would, when it again reached the interface, cause a tensile stress across
the grease joint causing the time piece to fly off. Thus the momentum
captured in the time piece corresponded to a length of pulse twice the
length of the time piece. The maximum pressure in that segment of the
pulse could also be found from the momentum captured. Using time pieces
of different lengths, the pressure-time relations of a pulse were in-
vestigated. However, a continuous pressure-time curve was not deter-
mined.

Davieslz made a complete investigation of the Hopkinson pressure
bar and also devised an electrical means of recording the pressure pulse.
He used the free end of the pressure bar as one side of a parallel plate
capacitor, thus converting the small displacements of the end of the bar
produced by a pulse into an electrical output which could be recorded on
a cathode ray oscilloscope. Continuous displacement-time and, if the

-7-

bar remained elastic, pressure-time curves could be obtained by this
method. Davies also investigated the limiting values of pulse length
which could be propagated down a cylindrical bar without the pulse be-
coming distorted.

A further adaptation of the pressure bar technique was made by
Kolsky13 in order to determine the stress-strain relations in materials
at high rates of loading. A diagram of Kolsky's apparatus is shown in
Figure l. A stress pulse produced by the detonator was propagated down
the length of the bar. The pulse was assumed to remain elastic and travel
with the velocity co = «IE/p . Then assuming the pulse was propagated
without distortion and that the stress was uniform over a given cross
section of the bar, conditions which would be met if the pulse duration
was long compared with the time required for a pulse to travel a distance
equal to the radius of the bar, the stress 0' at any point would be given
by

O'=PCV:

where v is the particle velocity. At the free end of a bar the particle
velocity is twice as great as the particle velocity associated with the
pulse along the length of the bar. Thus if we let u be the displacement
of the end of the bar as measured by the parallel capacitor,

_ .du
0'— l/ch-d—t- (7)

Kolsky inserted thin specimens between the two sections of the
bar. When an incident stress pulse of magnitude “I struck the specimen,

-3-

 

 

Anvil Collar Inerttiah Cylindgical Paral el late
con ensefi' con en e11;
microp one microp one
3 m \
P Collar Specimen

 

 

 

 

F

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lFeed unit I Feed unifl
weep ’
Unit
lAmplifier l [Amplifier I
l . e
I
l CRO

 

 

Figure 1. General arrangement of Kolsky apparatus.

Lamps

Bark 9 ® .Specime

‘

 

 

 

 

l

( Anvil

1

== - -
. / ———————
Piston (S ;) <; ;?
Photocells

'I‘o amplifier
and oscillosc0pe

 

 

 

 

 

Figure 2. General arrangement of Habib apparatus.

-9-

a portion of the pulse was reflected with magnitude GR and part trans-
mitted with amplitude GT . Assuming the stress constant across the

thin specimens,

o-o =0" (8)

Now if 1.1.1 is the displacement of the face of the main bar in contact with

the specimen at time t, then

91th (0'I +cr R)dt (9)

and if uz is the displacement of the face of the extension bar in contact

with the specimen,

1:

_ l
112 — P7 0 GT dt . (10)

If the thickness of the specimen is 1, the strain 6 in the specimen is

6 = -—-1— (11)

Thus using Equation (8), (9), (10), and (11),

t
2

Since -—fot our dt = uT was the displacement measured with the specimen
included and -——f: oldt = uI the displacement with the specimen absent,
a strain-time curve could be calculated from

-10-

A stress-time curve was obtained by taking the gradient of the u curve,

T
see Equation (7). Thus the stress-strain curve was obtained for the con-
ditions of the experiment.

Due to the nature of the pulse produced by the detonator, the strain

rate in the specimen varied throughout the test requiring a correction in

the stress for lateral inertia effects. Kolsky determined this correction

tobe
2
Z d e
O'-O'€ 1‘ l/Z Vzr p2 Tit—Z : (13)

where 0‘6 was the stress required to produce strain 6 when no kinetic
energy was imparted to the specimen and 0’ was the measured stress.
V and r are Poisson's ratio and the radius of the bar.

A few difficulties with this method may be noted. Since the speci-
mens were very thin, O. 05 and 0. 10 cm. thick for lead and copper, friction
effects may have been important although a. thin grease film was used to
minimize this effect. Alsothe determination of the stress requires a
numerical differentiation of the displacement-time curves which may
introduce error. Further, two separate detonations were required to
determine a single stress-strain curve. An attempt to remedy this was
made by using a cylindrical capactior concentric with the main bar to measure
the radial expansion due to the incident pulse, but the results from this
proved to be unreliable due to distortions introduced by the bar.

-11-

2. Z. 2. Compression Impact

Habib14 has indirectly arrived at stress-strain relations under
impact conditions by measuring the energy absorbed by small copper
cylinders when they were impacted as shown in Figure 2 by a piston fired
from a compressed air gun. The energy absorbed by the specimen was
determined from the difference in the impact and rebound velocity of the
piston as measured by photo-electric cells. By plotting the energy ab—
sorbed E against the deformation x as determined from the initial and

final dimensions of the specimen, energy-deformation curves were ob-

tained which could be differentiated to derive the force F, since

x
Ezf Fdx

O

or

F dE/dx .

Each point on an energy—deformation curve was obtained from a different
specimen. The energy absorbed and the deformation in the specimen
could be varied by changing the mass and the velocity of the impacting
piston.

Habib's results showed that the energy required to produce a
given deformation always increased as the rate of straining increased.
Also, the increase in dynamic stress over the corresponding static stress
was a function of both the strain rate and the strain. However, as pointed
out by Lee and Wolfls, the assumption of uniform strain in the specimen
when deriving stress-strain relations based on energy absorption may
produce only an apparent rise in the stress-strain curve if the strain

-12-

distribution is actually non-uniform.

Turnbow and Ripperger16 have used a setup similar to Habib's
except that they measured the stress and strain in the specimen directly.
This was done by placing strain gages directly on the specimen to measure
strain, and by replacing the solid anvil by a Hopkinson type pressure bar
instrumented with strain gages to measure the force transmitted by the
specimen. The strain rates could be varied by changing the piston
velocity. With sufficient mass in the piston essentially constant strain
rates were achieved. With this method a continuous record of the strain
could only be obtained up to the limit of the strain gages on the specimen,
also proper alignment so as to insure planeness of impact becomes a
difficult problem. Data from the tests performed were not sufficiently
reliable to affirm the linear strain rate theory of Malvern.

Alder and Phillips10 carried out compression tests at constant
true strain rates between 1 and 100 in. /in. /sec. on a specially designed
testing machine. In this machine the specimen was compressed to 50%
reduction in height between two platens, the lower one of which was driven
by a rotating cam, whose speed could be varied. The upper platen was
attached to a load measuring device which consisted of two birefringent
glass blocks. The change in birefringence upon application of load was
recorded photographically on a rotating drum mounted on the driving
shaft. The results of tests performed on aluminum, copper, and steel
at various temperatures showed that in the range of strain rates used
the data could best be fitted to the exponential type relationship although
the logarithmic law also gave reasonable approximation.

2. 2. 3. Tension Impact
Perhaps the first investigators to record stress and strain directly

-13-

for dynamic loading conditions were Manjoine and Nadail'z’la’19 in 1940.

Briefly their apparatus consisted of a tension specimen connected on
one end to a force measuring bar and the other end to a tup which could
be struck by a hammer attached to a heavy fly wheel. Displacement of
the specimen and deformation in the force measuring bar were both re-
corded photoelectrically and fed simultaneously to the horizontal and
verticle deflection plates of a cathode ray oscilloscope. In this manner
the stress-strain curves were recorded directly. Constant strain rate
tests, at low rates, were also carried out on another machine, so

that a range of strain rates from 10'6 to 103 in. /in. /sec. was covered.
Tests were conducted on copper, aluminum, and mild steel at room and
elevated temperatures. The results indicated that when stress was
plotted against the logarithm of the rate of strain a relatively linear
relationship with positive slope was obtained at the lower strain rates,
indicating a logarithmic law to be valid in this region. Above about
10-1 in. /in. /sec. , however, the slope of the curves gradually increased
so that they bent upward and no longer remained linear. This nonlinearity
was most pronounced for tests at elevated temperatures.

A serious problem encountered in the recording technique of
Manjoine and Nadai was that serious oscillations occured in the records
at the high strain rates due to elastic vibrations in the force measuring
system. Another disadvantage to the tension test is that the specimen
must of necessity be of appreciable length in order to grip it and to apply
the load. Thus at the very high strain rates the distribution of strain
over the length of the specimen and also the rate of straining is not uni-
form, due to wave propagation effects in the specimen.

-14-

This problem in dynamic tensile testing has been discussed by
Clark and Duwez20 who have also attempted to design a tensile test that
would eliminate wave propagation effectsZI. Their technique consisted
of subjecting thin walled cylinders to an internal pressure pulse. This
pulse was obtained by impacting a piston which compressed the fluid
within the cylinder. Stress-time but not strain-time was recorded for
these tests, thus limiting their value.

2. 2.4. Plastic Wave Propagation

Some investigators have derived dynamic stress~strain relations
directly from measurements of plastic waves propagated in long rods.
These techniques all utilize the von Karman-Taylor theory, particularly
the relationship expressed in Equation (1).

An example of this type of technique is given by the tangent modulus
method of Campbellzz. In this method a long bar was subjected to a
tensile impact by means of a freely falling weight, causing a stress pulse
to travel the length of the bar. The pulse was measured directly as a
function of time by two wire strain gages placed a given distance apart
on the bar. As seen from the expression for wave velocity, Equation (1),

the tangent modulus 5-:- is given by the relation

do _ 2
(daezen '-

where c is the velocity of propagation at the strain level 6n . Since

n
the stress in the bar could not be measured directly, it was found by
numerically integrating the curve of the tangent modulus plotted against

E . Thus

-15-

€
0' : [ (ii-1%) d5

0

With these values for (I and 6 the stress-strain curve could be plotted.
The experimental results reported by Campbell were at the time only
exploratory and were subject to considerable error.

23

Johnson, Wood, and Clark have also used the von Karman- Taylor

theory to determine strain in compression impact tests. However, due to
the uncertainty of the basic assumption that c = % ad?“ the results of
these tests using wave propagation must be taken with some reservation.
The method also does not lend itself well to obtaining data at a constant
strain rate.

3. EXPERIMENTAL METHOD

3 . 1. Desired Objectives

 

From the numerous methods of attack on the problem that have
been employed in the past there does not seem to be any one that is com-
pletely satisfactory. This is an indication of the difficulty of the problem.
Before describing the experimental methods employed in this investig-
ation to obtain stress-strain-strain rate relations it may be best to out-
line just what our objectives should be in designing an experiment that
will give us reliable and useful information. We may list them as follows:

1. Perform tests such that stress-strain curves at as nearly
constant strain rate as possible may be derived.

2. Obtain each complete curve from a single specimen.

3. Record instantaneous stress and strain continuously through-
out the test.

4. Obtain stress and strain records well into the plastic range.

-16-

5. Obtain dynamic curves up to 1000 in. /in. /sec.

6. Minimize wave propagation effects that would cause non-uni-
formity in the distribution of stress, strain, and strain rate in the
specimen.

In the present investigation an attempt was made to meet all of
the above objectives.

3.2. Dynamic Tests

 

With the above objectives in mind an adaptation of the method of
Kolsky was used in which short specimens are compressed between two
segments of a Hopkinson pressure bar. A schematic of the test setup is
shown in Figure 3. Two changes from Kolsky's original setup were made.
First, the impact is supplied by a third striker bar fired against the
incident bar. This produces a long, flat-topped pulse rather than the
shorter rounded pulses produced by an explosive charge. Second, strain
gages are mounted directly on the incident and transmitter pressure bars
to record the strain pulse both before and after it has passed through the
specimen. Thus only a. single firing is required to perform each test.
Also, strain-time and therefore stress-time, since the pressure bars
remain elastic, are recorded rather than displacement-time as determined
from the capacitor type pickup of Kolsky.
3.2.1. Impact Loader

The stress pulse was initiated in the pressure bar by impacting it
with a striker accelerated by means of a commercial Hyge Shock Tester
manufactured by Consolidated Electrodynamics Corporation, see Figure 4.
A schematic sketch of this equipment is shown in Figure 5a.

The acc eleration of the piston is achieved in this manner: A

given set pressure is applied to chamber B by means of compressed

-17..

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-19-

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Piston shaft Barrel Striker bar

Figure 5b. Barrel with striker bar.

-20_

nitrogen. This pressure is locked in and exerts a force over the entire

area of the back face of the piston. A load pressure is then applied to
chamber A. This pressure acts only over the reduced piston area deter-
mined by the orifice at C. As the load pressure is increased to about four
times the set pressure the forces acting on both faces of the piston be-
come equal. An additional increase in the load pressure breaks the seal
at C and the load pressure expands over the entire face of the piston
resulting in a large accelerating force.

The acceleration is regulated by the acceleration pin which con-
trols the rate at which the gas is allowed to expand over the piston. The
piston is stopped by means of the decel eration pin which fits into the
orifice at D. The chamber B is partially filled with oil to aid the
deceleration. The entire system is controlled from a separate control
panel which contains the pressure gages and control valves.

The striker bar itself rides in a "barrel" which is mounted on
the end of the piston shaft as shown in Figure 5b. The barrel is a
circular tube which is threaded onto the end of the piston shaft. The
striker bar rides freely within the barrel supported on two rubber "0”
rings. These were lightly greased so that there would be no binding
of the striker bar as it moved.

The spacing between the Hyge and the incident pressure bar was
such that as the piston decelerated the striker bar would just break
loose from the brass plug at the base of the barrel before it struck the
incident bar but would still remain supported in the ”0" rings. It was
essentially then a free bar upon impact. The velocity of the striker bar
could be varied by regulating the set pressure in the Hyge.

-21-

The striker bar was made from 3/4 inch diameter steel drill rod,
being of the same material and diameter as the pressure bars. For the
majority of the tests a striker bar 16 inches long was used, although
this could be varied. The length. of the striker bar determines the length
of the pulse produced in the pressure bar. This pulse length will be
equal to the time required for an elastic wave to travel twice the length
of the striker bar, due to the fact that the striker bar will rebound when
the original compression pulse has been reflected from the free end and
again returns to the impact face as a tensile force. In order to prevent
a non-axial impact, the impact face of the striker bar was slightly
rounded. This prevented impact from occurring at one edge of the
striker or incident bar.

3. 2. 2. Pressure Bars

The arrangement and dimensions of the two pressure bars are
shown in Figures 6 and 7a. Both bars were made from 3/4 inch diameter
steel drill rod. The lengths of the incident bar and transmitter bar were
34 7/8 inches and 24 inches respectively. The faces in contact with the
specimen were carefully turned on a lathe to a flat surface and then
finished with a fine emery paper. These surfaces were made as smooth
as possible in order to minimize friction effects between the bars and
the specimen.

The incident and transmitter bars were each supported at two
points by means of rubber "0" rings mounted in steel support blocks as
shown in Figure 7b. No distortion or reflection of the pulses was found
to be produced by this means of support. In order to insure uniform
compression of the specimens, the bars were carefully aligned so that

-22-

 

Figure 6. Close-up of Striker and Pressure Bars

-23-

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-24-

the contact faces with the specimen would be parallel. A lead pad was
placed after the transmitter bar to take up the shock from the bars after
the pulse had passed through the system.

The position of the strain gages on the pressure bars was deter-
mined so that a continuous record of a single pulse could be obtained
before reflections from an interface returned to the gage to mar the
record. This required a length of bar equal to or greater than the length
of the striker bar following each gage station. This was obtained by
placing the gages 18 inches and 3 inches from the specimen on the incident
and transmitter bars respectively. By placing the gages in this manner
the serious oscillations produced in the records of many investigators
who have used short force measuring bars was eliminated.

3. 2. 3. Strain gage bridges

At each gage station on the incident and transmitter bar two SR-4,
type A-8, wire resistance strain gages were placed at opposite ends of
a diameter of the bar. The gages were attached with Duco cement. Each
gage had a resistance of 120. 5 j . 3 ohms and a gage factor of l. 81 _+_ 2
per cent. At each station the gages were connected to the opposite arms
of a Wheatstone bridge circuit, see Figure 8. The dummy gages were
120 ohm precision resistors. The bridge voltage was supplied by a 6
volt wet cell with the exact voltage recorded before each test by means
of a voltmeter connected in parallel with the battery.

With the active gages placed in opposite arms of the bridge only
the direct pulse was measured while any bending components of the
stress were cancelled. It was assumed that for the long pulses used
the amplitude would be constant through any cross section of the bar,

-25-

 

 

 

 

 

I|I|I|r

 

 

 

 

Bat-Iry

 

 

Figure 8. Strain gage bridge with two active gages.

-26..

i. e. , the wave would remain plane, and that there would be no attenu-
ation of the pulse in the elastic pressure bars.
3. 2. 4. Strain Recording Equipment

The output from each bridge circuit was fed into a Tektronix type
53/54E plug-in preamplifier. These are high gain, ac-coupled preamp—
lifiers with a vertical sensitivity to'50 microvolts per centimeter and a
frequency response from 0. O6 cps. to 60 kc. They also permit attenu-
ation by means of outphasing of any undesired signal through differential
input connections.

The two plugwin preamplifiers were mounted in a Tektronix type
551 dual beam oscilloscope. The output from each gage station on the
pressure bars was thus recorded simultaneously on seperate channels
of the oscilloscope. The sweep speed of the oscilloscope could be varied
continuously from 1 microsec. /cm. to 5 sec. /cm. By this means a com-
plete test could be recorded on a single sweep of the dual oscilloscope beams.
This sweep was triggered by means of a piezoelectric crystal pickup
mounted before the first gage station on the incident bar.

A record of the oscilloscope trace was obtained by using a Dumont
type 302 oscilloscope record camera with Polaroid Pola Pan 400, type 44
film. Using an open shutter at a setting of f/l. 9 and an oscilloscope
sweep speed of 50 microsec. /cm. a clear picture of each trace was re-
corded.

3. 2. 5. Derivation of Stress and Strain in the Specimen

In order to determine the strain in the specimen as a function of
time we desire to know the displacements of the faces of the incident and
transmitter bars in contact with the specimen as a function of time. Let

-27-

these two displacements by u and 112 as in Figure 9. From the theory

1

of elastic wave propagation we have

v 1 du
where v is the particle velocity and co is the elastic wave velocity, a

constant. On solving this equation, the displacement u is given in terms

of the strain in the bar by

t
u = C0] edt . (15)
0

When a compressive pulse in the incident bar strikes the specimen, part
of the pulse will be reflected from the interface as a tension pulse while
part will be transmitted through the specimen to the transmitter bar as
a compression pulse. The relationship between the magnitudes of these
incident, reflected, and transmitted pulse will depend upon the physical
properties of the specimen. The effects of the numerous internal re-
flections in the short specimen will be absorbed in the reflected and
transmitted pulses, as the duration of these pulses is much greater than
the time required for a pulse to traverse the length of the specimen.

Thus, the displacement u of the face of the incident bar will be the re-

1
sult of both the compressive strain of the incident pulse 61 and the

tensile strain of the reflected pulse ER . Both these strains, however,

produce a displacement in the same direction, so that

t
‘11 = co] (61+6R) dt . (16)
0

-28...

 

 

 

 

 

 

 

 

 

Figure 9. Schematic of specimen showing stresses and displacements.

-29-

Similarly the displacement u2 produced by the compressive strain of the

transmitted pulse € T is

u = c 6 dt (17)

In these and the following equation 61, 6R, and 6T denote absolute

magnitudes of these quantities.

If we assume that the stress is constant across the specimen,

from continuity we have

GT - 0' = 0' (18)

Substituting this in Equation (16),

O

12
ul : C0 { (261- ET) dt . (19)
0

Taking 10 as the initial length of the specimen, the strain in the

specimen 65 is

 

Aco t
68 z I {0 (6I - 6.1,) (it (20)

-30-

where 61 and 6T are the strains measured as functions of time by the
strain gages on the incident and transmitter bars. Equation (20) is
equivalent to Equation (12) as derived by Kolsky if the relationship
E = pooZ is used.

The stress in the specimen is obtained directly from the record

of the transmitted pulse, since

=E€ . 21
a T ()

It may be noted that by direct measurement of the strains a differen-
tiation of the displacement-time curve to determine stress is eliminated.
However, this is replaced by the necessity of a numerical integration in
calculating strain in the specimen.

3. 3. Low strain Rate Tests

 

In order to obtain data over a wide range of strain rates and to
correlate the dynamic with "static" behavior, compression tests were
run at three different speeds on a standard testing machine.

3. 3. 1. Testing Machine

All tests at low strain rates were performed on a model FGT
Baldwin-Emery SR-4 Universal Testing Machine. The testing speed range
of this machine controlled by a variable speed motor is O. 025 to 9. 00
inches per minute. A general view of the machine is shown in Figure 10.

The specimens were compressed between two 3/4 inch diameter
steel columns. The faces of these two columns were ground flat and
finished with fine emery paper. The specimens were the same size as
those used in the dynamic tests so that the results could be compared.

3. 3. 2. Stress and Strain Recording
At strain rates of l. 67 x 10"3 and 3. 33 x 10.4 in. /in. /sec. con-

-31-

 

1ne wi

Baldwin-Emery Testing Mach

Figure 10.

Stress and Strain Recording Equipment

-32-

tinuous load-deflection curves were obtained by using Baldwin Automatic
Stress-Strain Recorder, Model MA-IB. On this recorder the strain axis

is driven by a microformer type extensometer. A Baldwin Model TS-M
extensometer was used. The extension arms of the microformer, see
Figure 11, rested on two clips attached to the upper and lower load columns.
Displacement was measured between two points each 1/8 inch from the
faces of the load columns. The strain in the steel columns between these
two points was negligable in comparison with the strain in the specimens
and was neglected.

The load axis on the automatic recorder is driven by the output
from the load cells in the testing machine itself.

In these tests a constant rate of straining was maintained by the use
of a Baldwin Strain Rate Pacer. In this instrument a needle is driven by
the output from the microformer at a rate corresponding to the strain in
the specimen. This needle is paced to the speed of a coaxially mounted
background wheel driven at a constant speed by means of a synchronous
motor. The speed of the background wheel could be varied by changing
gear ratios.

For strain rates on the order of 10"1 in. /in. /sec. the frequency
response of the Baldwin automatic recorder was insufficient so that a two
channel Brush pen oscillograph recorder was used instead. Stress-time
and strain-time were recorded simultaneously on the two channels. In
this setup the microformer extensometer was connected to a low voltage
differential transformer balance unit in a Brush universal amplifier, Model
RD 5612-00. The output from this amplifier was connected to one channel
of the oscillograph.

-33-

I . l -‘
H O ‘ | F t - ‘ H
‘ ‘, .3.“ ‘ln '- . .3‘3 lg; A” (M "
l I.‘I .pl , * C :4 It” {I i. -
u v 9' -, . ' , _
mg ”m" '

 

u
t n . - . ‘_¢‘..

 
 

-e I¢;' h)...'u.4‘-".-.‘..'-"-'~ ._ . .

Figure 11. Close-up of Testing Jig with Microformer

-34..

To measure load two semiconductor strain gages were mounted
on opposite sides of the bottom load column. These gages were Kulite,
Type DA-lOl with a gage factor of 118 i 5 and a resistance of 65 _'_|' 2 ohms.
Gages with this high gage factor were necessary because of the small strains
produced in the steel column by the loads required to compress the speci-
mens. As it was necessary to mount these gages on a flat surface, shallow
flats were ground on the round column. The two gages were connected to
opposite arms of a bridge balancing unit of a second Brush amplifier.
Precision resistors were used in the other arms of the bridge. The out-
put from this amplifier was fed to the second channel on the oscillograph.

The time axis of the oscillograph record was determined by the
chart feed speed. For the tests performed this speed was 125 cm. per
second.

3 . 4. Specimens

 

In the present investigation three different metals were tested;
lead, aluminum, and copper. These were selected as they were common
metals and are sufficiently softer than the steel pressure bars, a re-
quirement of this method if appreciable strains are to be produced in the
specimen. The crystal structure of all three metals is face centered
cubic.

The lead specimens were made from commercially pure lead
sheet stock. This lead was melted down and molded into blanks 1/2 inch
in diameter and 1/2 inch long. These blanks were then turned down and
faced on a high speed collet lathe to the proper thickness. The machined
specimens were annealed at 200°F for two hours.

The aluminum was from No. 1100 (ZS), 1/2 inch diameter aluminum

-35-

rod. The specimens were cut from the rod and faced on the collet lathe.
Annealing was at 600°F for two hours.

Copper specimens were similarly machined from 1/2 inch diameter
commercially pure soft copper bar stock and were annealed at 650°F for
two hours.

The thickness of all the specimens used in the dynamic and low
strain rate tests was 0. 250 inches which was determined so that con-
sistent results could be obtained as will be described later. Using 1/2
inch diameter specimens with the 3/4 inch diameter pressure bars assured
that uniform displacement of the faces of the specimen would be main-
tained throughout the test. Originally 1/2 inch diameter specimens were
used with 1/2 inch diameter pressure bars, but as the strain increased,
the material squeezed out from between the bars formed a ring which it
was felt would affect further compression of the specimen.

4. EXPERIMENTAL PROCEDURE

4.1. Dynamic Tests

 

4. 1.1. Determination of Elastic Wave Velocity

In order to calculate strain in the specimens the elastic wave
velocity co in the steel pressure bars must be determined. This was
done by measuring the time required for a single pulse to travel twice
the length of the incident bar. The output from the strain gage station
on the incident bar was fed through a Tektronix Type 122 Low Level
Preamplifier to a Hewlett Packard Electronic Counter. The preamplifier
was necessary in order to supply enough voltage to trigger the counter.
With the incident bar supported as a free-free bar a strain pulse with
a short rise time (on the order of 15 microseconds) was produced upon

-36-

impact from the striker bar. A time count on the counter was started

by the initial strain pulse traveling down the bar and was stopped by

the pulse when it again reached the gage station after having been reflected
from the far end of the bar, back to the impact surface, and again to the
gage station, a distance of twice the length of the bar. The sharp rise
time of the pulse provided an accurate trigger for the counter so that
consistant results within one microsecond were obtained. By this method
co was determined to be 198,000 in. /sec.

4. l. 2. Dynamic Calibration of Strain Gages

 

For the elastic impact between the striker and pressure bar we
may derive an expression between the impact velocity and the maximum
strain in the compression pulse. Consider the pressure bar to be station-
ary and to be struck by the striker bar traveling at velocity vs. Both
bars have the same cross sectional area A, density P , and elastic wave
velocity co. At the moment of impact a wave of compression travels away
from the interface into both bars at the velocity co. Behind the wave
front there will be a region of constant strain which will be maintained
until unloading occurs due to reflection from the opposite end of the
shorter bar.

At time t the compression will have travelled a distance cot into
each bar as in Figure 12. If F is the contact force between the two bars
and we apply the impulse momentum equation to each bar we obtain for

the pressure bar

Ft ~.-- (pAcotlvp,

where vp is the particle velocity acquired by the mass (PAcot) due to

-37-

 

 

(a) I I

 

 

 

 

 

 

a.) |

 

set—4H —4

Striker bar Pressure bar
Figure 12. Striker and presaire bar (a) at moment of impact, and
(b) after time t.
Striker bar ®fLamps Trigger Pressure bar
(3

\ £=/ R

v“

8 Phototubes

 

 

 

 

 

 

 

Strain gage station

 

I Bridge I

 

 

IAmplifier I

. r
_—I CRO

Electronic I Camera I

 

 

 

 

 

 

 

 

 

 

 

 

Counter

 

Figure 13. Setup for calibration of strain gages.

-33-

the impulse Ft. Similarly during the same time the striker bar receives

impulse
- Ft = (pAcOt)(Vp-VS)

By equating the impulses we get
v

s (22)
v 3 —
P 2

For a region of uniform strain cot in the pressure bar, the displacement

will be given by vpt. Therefore the strain is

€:——=— . (23)

To dynamically calibrate the strain gages in this experiment, a
series of tests were run at varying impact velocities. The output from
a strain gage bridge as recorded on the oscilloscope was then calibrated
against the maximum strain as computed from Equation (23).

In order to measure the impact velocity of the striker bar a
photocell setup was employed as shown in Figure 13. Two IP42 photo
tubes with separate light sources were placed behind two narrow slits
one inch apart immediately before the impact face of the pressure bar.
The output from the first tube after being amplified was connected to
the start input on an electronic counter. The amplified output from the
second tube was connected to the stop input of the counter. As the
striker bar successively cut the two light beams a time count by the
counter was started and stopped, thus recording the time interval for
the striker to travel one inch. As the striker bar at this point was

-39-

riding free in the barrel, any acceleration or dec eleration in this short
distance was neglected. The impact velocity vs was then taken as one
inch divided by the time recorded on the counter. By varying the set
pressure on the Hyge, striker velocities between 17 and 37 ft. /sec. were
achieved.

Strains calculated from the equation

Vs

= — 23
6 2c ( a)
0

were then compared with results using the strain gage bridge equation

2
= A 4
€ VOF v , (2 )

 

where V0 is the applied bridge voltage, F is the gage factor for a single
gage, and Av is the change in voltage produced by the strain 6 as deter-
mined from photographic records of the oscilloscope trace. It was found
that if the manufacturers gage factor of 1. 81 was used the difference be-
tween Equations (23a) and (24) was less than 2 per cent over the range
of impact velocities used, see Figure 14. Thus Equation (24) was used
to calculate strain in the ensuing tests.
4. 1. 3. Check on the Uniformity of Stress in the Specimen

In order to check on the assumption of Equation (18) that the stress
across the specimen is uniform, it was necessary to obtain a clear record
of all three pulses, incident, reflected, and transmitted. The incident
and reflected pulses could both be recorded on the single gage station on
the incident bar if a striker bar somewhat shorter than the distance be-
tween the gage station and the specimen was used. The incident pulse
would then have completely passed the gage station before its reflection

-40-

(microin. /in. )

MAXIMUM STRAIN

1400

1200

1000

800

600

400

200

Figure 14.

 

 

 

/
/‘/
fl
/{;
/‘
8
v
— S -------- 0
8—2c
e:_ZA_V ________ A
V F
o
l l l l I
10 20 30 40 50

IMPACT VELOCITY (Ft. /sec.)

Comparison of strain determined from impact velocity and
the strain gage bridge equation.

-41-

returned from the interface with the specimen.

In tests on lead specimens of thicknesses up to 0.250 inches it
was found that during the flat portion of the incident pulse the assumption
of Equation (18) was fairly good, there being only a slight decrease in
stress, generally less than 3 per cent. During the sharp rise time,
however, there tended to be considerable error. This was probably
due to a greater attenuation of the higher frequency components of the
pulse, both in the specimen and in the elastic pressure bars. For this
reason and due to the lack of accuracy in measurement during the sharply
rising portion of the pulse, values obtained at low strains are somewhat
in doubt. Thus with the type of pulse used in this investigation the
elastic behavior of the specimens cannot be accurately determined.
However, above about 2 per cent strain there will be little error in using
Equation (20) to determine strain.

4. l. 4. Evaluation of the Effect of Friction

When using very short compression specimens the effect of fric-
tion is always an unknown quantity. In order to determine whether there
was a minimum thickness that could be used that would be independent of
friction or size effects a series of tests were run on lead specimens
ranging in thickness from 0. 108 to 0. 300 inches. Any effect of the thick-
ness on the dynamic stress-strain curves could then be noted.

Specimens of five different thicknesses, 0.108, 0. 200, 0. 250,
0.275, and 0. 300 inches were tested. The results of one series of tests
at a single impact velocity are given in Figure 15. When-testing the
specimens a thin coat of lubricant was applied to each face. The best
lubricant that was found for this purpose was molybdenum disulfide,

-42-

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.2 05mg

 

x .E\ de 22mg
mm. mm. mm. ON. 0H. NH. mo. V0.
4 _ J _ _ _ _ _
<4
:-
o
‘
O ‘0. O
4 ...... .58 .o u H
O o “ I
< IIIIII :mNN O H H o
O o
o ...... .53 .o u H
o D
O llllll :CCN .0 H H (ON 5
O
D llllll :wcd .0 H H
BEA . s I
do
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<04
.04
L
D od- 04
‘ 4
o
.04
2 0‘
1 ea
0 D .l
b

 

SSHHLS

(13d 0001)

-43-

commercially called Molykote. In static compression tests on very thin
specimens this lubricant was found to be much superior to other types
that were tried.

From Figure 15 it is seen that at the two lower thicknesses a con-
siderable increase in the stress at the lower strains occurs, although
above about 5 per cent strain these curves coincide well with those of
the thicker specimens. Since this series of curves was obtained with
the same impact velocity and therefore incident loading pulse, there is
some variation in the strain rate, the thinner specimens having a higher
strain rate due to the fact that they have a shorter gage length. In order
to ascertain that the high initial increase in stress in the thin specimens
may not have been due to a strain rate effect only, the same series of
tests was run at different impact velocities. These tests showed that
the large initial jump in stress was characteristic of the thin specimens
and not of the strain rate. If very thin lead specimens, say 0. 030 inches,
were used, the initial jump in stress, as shown by the transmitted pulse,
appeared as a large oscillation on the stressmtime record. On the records
for the thicker specimens this oscillation was still noticeable, see
Figure 16, but is considerably damped out. This phenomena was much
more pronounced in lead than in the other metals tested.

It seems doubtful that the variation in stress at low strains was
due to friction, a factor which should become more important at the
higher strains. Another measure of the friction effect is obtained, how—
ever, by the amount of barreling in the compressed specimens. Only
slight barreling was noticed at the higher strains indicating that friction
effects were small.

-44-

5. 0 millivolts
(a) Lead
0. 5 millivolts

5. 0 millivolts
(b) Aluminum

2 . 0 millivolts

 

50 microseconds

5. 0 millivolts
(c) Copper
2. 0 millivolts

 

50 microseconds

<—— Time

Figure 16. Typical oscilloscope records for 1/4 inch long lead,
aluminum, and copper specimens. Incident and
transmitted pulses are shown on the upper and
lower channels respectively.

-45-

Above 0. 250 inches the variation of the thickness in the lead speci-
mens does not seem to affect the stress~strain curves. For this reason
a specimen thickness of 0. 250 inches was used in all of the dynamic and
low strain rate tests.

4.1. 5. Testing Procedure

In performing the dynamic tests the following procedure was
followed:

1. The oscilloscope with plug-in preamplifiers was calibrated
for amplitude by means of the square-wave calibrator output incorporated
in the oscilloscope. The sweep rate of the oscilloscope had been cal-
ibrated with one microsecond timing markers previous to these tests by
a factory representative. These calibrations were performed against
a graticule-scale placed over the face of the cathode ray tube.

2. A specimen with a thin coat of Molykote on each surface was
placed between the two faces of the incident and transmitter pressure
bars. The specimen was centered by means of a spacing jig. With the
pressure bars firmly pressed against the specimen the surface tension
of the Molykote was sufficient to hold it in place.

3. The oscilloscope was set on single sweep lockout so that the
entire test was recorded on one sweep. This required a sweep speed of
50 microsec. /cm. The sweep was set to trigger on the output from the
crystal mounted about 6 inches before the first gage station.

4. With a fixed set pressure locked in the Hyge the load pressure
was gradually increased.

5. Just before the Hyge fired the shutter on the oscilloscope
camera was opened and was held open until after the firing and the test

was completed.
-46-

6. Upon releasing the load pressure in the Hyge, the piston was
again seated and ready for another test.

In general it was necessary to check the set pressure on the Hyge
by means of the pressure gages on the control panel before each test in
order to assure that a given impact velocity would be reproduceable.
Even then variations were apt to occur.

For lead and aluminum four specimens each were run at four dif-
ferent set pressures; 15, 30, 45, and 60 pounds. For copper two speci—
mens each were run at 15 and 60 pounds set pressure.

4. l. 6. Reduction of Data

Measurements from the photographic records of the oscilloscope
traces were made using a Pye two—dimensional measuring microscope
accurate to 0. 01 mm. Amplitudes of the incident and transmitted pulses
were measured vs. time. The time axis for the two pulses was made to
coincide by using the fixed distance between the two gage stations and
the wave velocity in the bars.

Nominal strain was calculated from Equation (20) by plotting
(eI - e T ) vs. t and then numerically integrating this curve. True or

logarithmic strain 6n was then determined from

en = 1n(l+€)

Taking into account the change in diameter between the specimen and
the pressure bar and assuming there is no volume change in the specimen
during compression, the true stress (Tn in the specimen is
2
D
z: -— l + c
0'n d2 E 6T i 1

-47-

where D and d are the initial diameters of the pressure bars and the
specimen respectively, E is the modulus of elasticity of the pressure
bars taken as 30 x 106 psi. , and € is the nominal strain in the specimen.
All stress-strain curves plotted are for true stress and true strain.

4. 2. Low Strain Rate Tests

 

Before performing the compression tests on the Baldwin testing
machine the dial load indicator was calibrated with a Morehouse Proving
Ring and found to be accurate within one per cent. The load scales on
both the automatic stress-strain recorder and the Brush oscillograph
were then calibrated from the dial indicator. The deflection scales on
both these recorders were calibrated against the displacement between
the two faces of the load columns as measured with an Ames dial gage
accurate to 0. 0001 inch.

Specimens used were the same as those used in the dynamic tests.
Molykote was used to lubricate the surfaces in each test.

‘3 and 3.33 x 10’4 in. /in./

At the two lower strain rates of l. 67 x. 10
sec. the strain rate was controlled throughout the test by means of the
strain rate pacer and the manual load controls on the testing machine. Al-
though by this method a constant nominal rather than true strainrate was
obtained, in plotting the results this difference was neglected as it was not
found to be significant up to the maximum strains attained. With a specimen
centered between the load columns, the oscillograph was turned on and the
testing machine thrown in gear. The test was completed in about one second.
A small gap was left between the specimen and the upper load column before
starting the test so that the machine could come up to full speed before
starting to compress the specimen. The load and deflection channels

on the oscillograph were calibrated just prior to each test.

-43-

Two specimens of each metal were run at each strain rate. Load
and deflection data from all the records were converted into true stress-
strain curves.

5. RESULTS

5.1. Oscilloscope Records

 

Typical records of oscilloscope traces for lead, aluminum, and
copper are given in Figure 16. In these records the time scale reads
from right to left. The upper trace on each record is the output from
the gage station on the incident bar, while the lower trace is from the
gage station on the transmitter bar. Along with the incident pulse, the
main portion of the reflected pulse may also be seen on the upper trace,
although its leading edge runs into the tail of the incident pulse. The
shape of the reflected pulse corresponds to 61 - 6T , the function which
is integrated to give the strain in the specimen, see Equation (20).

As was noted previously for the case of lead, a small initial rise
occurs at the beginning of the transmitted pulse for each metal. This
initial rise occurs during the period when there is a rapid acceleration
in the strain due to the sharp rise time on the loading pulse. The initial
apparent increase in stress thus may be attributed to inertia effects in
the specimen, as seen from Equation (13).

In lead the initial rise is followed by a series of small oscil-
lations, the period of which is from 8 to 10 microseconds. This cor-
responds to the period of an elastic wave in the lead traveling twice the
length of the specimen. Thus, these oscillations would appear to be
associated with internal reflections in the specimen. As they are very

small in amplitude and rapidly damped out, it is difficult to make any

-49-

definite conclusions. It is also possible that they arise due to the change
in diameter between the specimen and the transmitter bar, although this
is doubtful as they were also present when 1/2 inch diameter specimens
were used with 1/2 inch diameter pressure bars.

From the oscilloscope records it may also be noted that the length
of the transmitted pulses are greater than the incident pulses, especially
in the softer metals. This increase in length may be attributed to internal
damping in the specimen. There is actually a continued increase in stress
in the transmitted pulse for a short interval after the stress in the incident
pulse has begun to decrease. This is somewhat obscured in the records
shown because the return of the reflected pulse cuts off the tail of the
incident pulse. With the 16 inch striker bar the stress should start to
decrease approximately 160 microseconds after the initiation of the pulse.
The initial decrease in stress in the transmitted pulses, however, occurs
somewhat after this 160 microseconds depending upon the material.

5. 2. Deformation in the Specimens

 

A photograph of several deformed dynamic specimens is shown
in Figure 17. The appearance of the dynamic and the static specimens
was similar after testing. Up to the maximum strains attained in this
investigation there was little evidence of barreling.

Two different types of lead specimens are shown because of their
marked difference in appearance. One type (specimens Pb250-l5, 17.,
and 28) shows definite lines of flow radiating from the center of the speci-
men and a very coarse radial surface indicating that plastic flow did not
take place uniformly throughout the specimen. The other lead specimens
(Pb250-10, 20, and 27) have a more uniform appearance similar to the

-50-

 

1mens

Test Spec

Dynamic

ion in

17. Deformat'

Figure

-51-

aluminum (A1250-24) and the copper (Cu250-l4). This difference in the
two types of lead specimens did not appear to be a function of the strain
rate or the maximum strain, but rather due to some variation or non-
uniformity in structure between the different specimens. As each lead
specimen was molded individually variations were liable to occur. The
specimens showing heavy flow lines generally required a lower stress
to sustain the same strain than the more uniform Specimens. This re-
sulted in some scatter in the data for lead.

5. 3. Strain-Time Curves

 

Strain-time curves for the lead, aluminum, and copper specimens
for the dynamic tests are given in Figures 18, 19, and 20 respectively.
Each curve represents an individual test at the indicated Hyge set pressure.
It is seen that accurate control over the strain rate with set pressure was
not obtained. However, this was not important as only a range in rates
was desired.

After the first 20 microseconds, which corresponds to the rise
time on the loading pulse, the strain-time curves become nearly linear
for lead and aluminum. For copper, due to its greater hardness, there
is some deviation from linearity, but for the purposes of this experiment
an average strain rate was taken. It was felt that the deviations were
not of sufficient magnitude to affect the conclusions.

The range of strain rates attainable by this pulse type loading
is unfortunately limited. Significant extensions beyond the range of
600 to 3000 in. lin. lsec. obtained in this investigation are not feasable.

In order to lower the strain rate to 100 in. lin. lsec. , the duration of the
test and therefore the incident pulse must be on the order of l millisecond

-52-

 

 

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120

90

60

30

(microsec.)

TIME

Strain-time curves for lead.

Figure 18.

-53-

 

 

 

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Strain-time curves for aluminum.

Figure 19.

-54-

(in. /in. )

STRAIN

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-55-

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Figure 20. Strain-gtime curves for c0pper.

if a maximum strain of 10% is to be acheived. This would require a
striker bar of 100 inches in length with a corresponding increase in length
of the pressure bars, making the apparatus unwieldy. The amplitude of
the incident pulse must also be reduced to a level at which the strains
would become more difficult to measure. On the other hand, in order

to increase the strain rate the amplitude of the incident pulse must be in-
creased and the duration of the test shortened. The amplitude of the in-
cident pulse, however, is limited by the condition that the pressure bars
must remain elastic. Also by having large strains occuring in a short
time, most of the strain would be achieved during the rise time of the
loading pulse when the strain rate is not constant. The narrow range of
uniform strain rates attainable by this method is thus a limitation on the
usefulness of the method.

5 . 4. StressuStrain Curves

 

The dynamic stress-strain curves for lead, aluminum, and copper
are given in Figures 21 - 29. The curves are grouped according to the
Hyge set pressure used for each test and therefore according to strain
rate, see Figures 18 - 20.

The results show very good consistency, especially for aluminum
and copper where the specimens each came from a single bar of the mater-
ial and were more liable to be uniform in structure. Due to the manner of
preparation, the lead specimens were not as uniform, therefore giving more
scatter to the data. For example, in Figure 24 the specimen having the
lower stress—strain curve (Ph250~28) was one that exhibited uneven flow,
see Figure 17. The other specimen (Pb250-27) was of the type having a
uniform appearance.

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-65-

Over the range of strain rates. obtained in these dynamic tests there
does not appear to be any significant variation in the stress-strain curves
for any of the three metals tested. This may be due to the relatively small
variation that should be expected over this limited range in strain rates, as
will be seen when the dynamic curves are correlated with those obtained at
the lower strain rates. It is also possible, however, that there is an upper
or saturation limit on the rate of straining above which the stress required
to maintain a given strain remains contant. Such an upper limit has been
postulated by Burgersz‘4 based on the theoretical work of Prandtl7. Burgers
states that there is a maximum value of stress which a molecular system
can sustain. Above this maximum value sliding would be possible at any
strain velocity, even if there was no thermal agitation present. However,
due to the presence of heat motions, the force necessary for sliding at a
finite velocity remains below this maximum. In order to confirm this
maximum stress hypothesis, tests at much higher strain rates would have
to be performed.

In Figure 30, 31, and 32 an average dynamic stress-strain curve
is shown along with average curves obtained at the lower strain rates on
the Baldwin testing machine. It is seen that for each metal a homologous
set of curves is obtained with an increase in strain rate producing an in-
crease in the stress required to maintain a given strain.

For lead, all of the low strain rate specimens showed deformation
of the type having marked flow lines. For this reason in comparing the
dynamic with low strain rate behavior only the dynamic specimens of this
same type were used.

5. 5. Stress-Strain Rate Curves

 

In order to evaluate the validity of the proposed irain rate laws

as given in Equation (4), (5), and (6), the stress required to produce

-66-

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a given strain was plotted as a function of strain rate for several levels
of strain. These curves are given in Figures 33, 34, and 35. For each
of the three metals a linear relationship was obtained between stress and
the logarithm of the strain rate. In the case of aluminum there is an
apparent leveling off at the low strain rates. A similar leveling off might
be expected to occur for the other metals if lower strain rates into the
creep range were attained. Over the range of strain rates tested, how-
ever, the logarithmic law of Equation (5) appears to be valid for lead,
aluminum, and copper in the annealed state.

For comparative purposes the results of some other investigators
are included in Figures 34 and 35 for aluminum and copper. For aluminum
the results of Alder and Phillips10 show very good agreement with the
data of the present investigation. Their results were obtained on com-
pression specimens made from as—extruded aluminum rod and were
annealed for one hour at 400°C in vacuum. Results are given for 10 and
20 per cent strain. There is less strain rate effect evident in the results
from Alder and Phillips for copper than is indicated by the slope of the
data from these tests. Better agreement for copper is obtained with the
results from Nadai and Manjoine18 for ultimate stress vs. log strain
rate. Although the effect of strain rate seems to be the same, as
evidenced by the comparable slopes in Figure 35, the results of Nadai
and Manjoine are not strictly comparable. Their data is for the maximum
true stress attained in a tension test as a function of the average strain
rate at which the test was run. Since the strain at which the ultimate
stress is reached may vary with strain rate, the stress values are not
at a constant strain as in the present tests. The specimens of Alder

-70-

 

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-73-

and Philips were coldwdrawn phosphorous-~deoxidized copper, annealed
for 2 hours at 6000C in vacuum. Those of Nadai and Manjoine were
commercially pure cooper, annealed to 500°C in 5 hours in regenerated
gas. The stress values in. all cases are true stress.

From Figures 33, 34, and 35 it may be seen that (To, the stress
at unit strain rate, and k, the constant slope of the stress-log strain

rate curves, are functions of strain. Thus Equation (5) may be writter.

CV
‘J‘
v

ozob(€)+k(€)log E. (.

In this equation ‘TO( 6) is the stress—strain curve at unit strain rate.
The significance of k( 6) may best be visualized by looking at the tangent

modulusgg. This is given by

de
95. = doom + dk(e) 1 .
d6 d6 (16 0g 6

Thus, at a given strain, the change in the tangent modulus with strain
rate is dependent upon the gradient of k( e) at the strain. If k( 5) is a
constant there is no change in the tangent modulus. For k( 6) an in-
creasing function, an increase in strain rate will produce an increase

in the tangent modulus. On the other hand, if k( e) is a decreasing
function, an increase in the strain rate will produce a decrease in the
tangent modulus. Up to the highest level of strain achieved in these tests,
k( 6) is a monotonically increasing function. Hence an increase in strain
rate produced an increase in the tangent modulus for all values of strain.

The increase in the tangent modulus can be associated (see

-74-

reference 25) with recovery processes that take place within the de-
formed structure. Due to the fact that the energy content of the deformed
crystal lattice is greater than in the undeformed state, it will be in an
unstable condition. The excess energy of the deformed state is gradually
dissipated by the movement of unstable particles into positions of equil-
ibrium. The rate of this movement or recovery will be dependent upon
both time and temperature such that the extent of recovery will increase
with both an increase in time and an increase in temperature.

Another factor influencing the shape of the stress—strain curve
is the change from isothermal to adiabatic conditions as the rate of de~
formation is increased. At high rates of straining the heat produced by
the process of inelastic deformation does not have time to be dissipated,
thus causing a local increase in temperature. Nadai and Manjoine18
have measured with thermocouples the increase in temperature of pure
iron tensile impact specimens for which the test duration was less than
0. 002 seconds. At the point of fracture the temperature increase was
found to be about 50°C. Away from the fracture surface the temperature
increase was found to be less than half this value. Such an increase in
temperature will increase the rate of recovery and should thus tend to
flatten out the stress-strain curves at high strain rates. No evidence of
such a flattening out was indicated in the present tests, thus the time factor
was dominant over any increase in temperature that may have occurred
during the tests.

Values for (To and k at several values of e are given in Tables I,

-75-

2, and 3 for lead, aluminum and copper. By interpolating between these

values and using Equation (25), stress-strain curves may be constructed

for any strain rate at which the logarithmic law is assumed to be valid.
Equation (25) may be put in non-dimensional form by dividing through

by 60(5) . Thus,

k6 .
39-: 1+E-(--)-log€. (26)
O 0

Values for k/O'o are also given in Tables 1, 2, and 3. In Equation (26),
the coefficient of log ‘6' is a measure of the rate sensitivity of the mater--
ial . The larger this coefficient, the greater will be the change in stress
for a given change in the rate of straining at any value of strain. From
the values of k/o-o it is seen that lead is more sensitive to rate effects
than the aluminum or copper. It is interesting to note that copper is less
rate sensitive than aluminum at strains below 12 per cent but is more
sensitive at strains above this value. It is also interesting that for
aluminum this coefficient is almost constant.

In general practice it may be more useful to take the reference
curve 0'O(€) as the normal static stress-strain curve, 68(6) . Then the
increase in stress for an increase in strain rate would be proportional
to the difference (log a - log E‘s), where Es is the strain rate of the
static test. If, for example, we take k/cro as O. 05 and ES as 10-4in. /
in. /sec. , then the stress at a strain rate of main. lin. /sec. would be
1. 30 0's

The logarithmic law as first derived by Prandtl was an approxim-

ation for large values of stress of the more general law

SE, = C sinh oco (Z7)

-76-

TABLE 1. Values of 0’0, k, and lg/GO for lead.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. psi
6 «0 (ps1) k (——_T) k/a01sec1
sec
.03 2000 116 .0580
.06 2530 181 .0715
.12 3240 247 .0762
.18 3550 277 .0780
.24 3890 308 .0792
.30 4180 330 .0789
TABLE 2. Values of 0'0, k, and O‘O/k for aluminum.
s «0 (psi) k (—P—S‘_—1-1 k/cr01sec)
sec
:03 9400 367 .0390
.06 12200 444 .0364
.12 14600 547 .0375
.18 16100 615 .0382
.24 17300 661 .0382
TABLE 3. Values of 0' , k, and 0' /k for copper.
O O
6 , 0'o (psi) k (__ps_1:) k/‘Yo (sec)
J _- 4 sec 1
.03 13000 364 .0280
.06 19200 558 .0291
.12 29100 1081 .0371
.18 36600 1615 .0441

 

 

 

 

-77-

where C and cc are dependent on the temperature and the structure.
Prandtl's original derivation was based on a molecular model of the
dislocation process of the sliding of one row of molecules on another.
The hyperbolic sine relation of Equation (27) has also been derived by
Eyring‘?‘6 from a general equation for the rate of any process where
matter rearranges by overcoming a potential barrier, and has since
been shown by Nadaiz'7 to give a good fit to the data of creep tests.
The hyperbolic sine or the logarithmic law may thus be valid for many
materials over a wide range of strain rates.
6. SUMMARY AND CONCLUSIONS

Dynamic stress-strain curves were obtained for lead, aluminum,
and copper using a modification of the method of Kolsky whereby short
compression specimens are compressed between two sections of a
Hopkinson pressure bar. By using long flat topped loading pulses,
uniform strain rates were obtained for a range of strains between 3 and
30 per cent. The elastic portion of the stress-strain curve was not
accurately determined. A limitation of the method is that it is not poss-
ible to attain as wide a range in strain rates as would be desirable.

Stress-strain curves at constant strain rate were also obtained
over a range of rates on a standard testing machine. These results were
then correlated with the dynamic tests.

The stress required toproduct a given strain was found to follow

t he logarithmic law of Prandtl,

= k '
0' 0'0+ loge

where 0'0 and k are functions of strain. The coefficient k determines

-73-

the rate of change of the tangent modulus due to a change in strain rate.
k/o- o is a measure of the rate sensitivity of the material.

The tangent modulus increased with an increase in the rate of
straining for all three metals tested.

Theoretical basis for the logarithmic form of the rate dependence
can be made either from a mechanical model of the dislocation process
or from thermodynamical considerations of a rate process.

Further investigations are indicated along several lines: (1) 1n-
creasing the range of constant strain rates obtained in the dynamic tests.
First, in order to fill in the gap between rates attainable on a standard
testing machine and the rates achieved in the present tests, and second,
to check whether the stess—log strain rate curve levels off and reaches
a maximum value of stress that may be sustained by the material. (2)
Obtain constant strainrate tests in the elastic region in order to deter-
mind the dynamic modulus of elasticity and yield point. (3) Compare
the effect of dynamic strain hardening with static strain hardening. (4)

Include the third variable, temperature, in dynamic tests.

-79-

7 . BIBLIOGRAPHY

l. T. von Karman and P. E. Duwez, "The Propagation of Plastic Deform-
ation in Solids," J. Appl. Phys. , vol. 21, 1950, pp. 987-994.

2. G. 1. Taylor, "The Plastic Wave in a Wire Extended by Impact Load, "
The Scientific Papers of Sir Geoffrey Taylor, Cambridge Univ.
Press, 1958, p. 467.

 

3. L. H. Donnell, "Longitudinal Wave Transmission and Impact," Trans.
ASME, vol. 52, 1930, pp. 153-167.

4. P. E. Duwez and D. S. Clark, ”An Experimental Study of the Pro-
pagation of Plastic Deformation under Conditions of Longitudinal
Impact," Proc. ASTM, vol. 47, 1947, p. 502.

5. L. E. Malvern, "The Propagation of Longitudinal Waves of Plastic
Deformation in a Bar Exhibiting a Strain Rate Effect, " J. Appl.
Mech., vol. 18, no. 2, 1951, p. 203.

6. E. J. Sternglass and D. A. Stuart, "An Experimental Study of the
Propagation of Transient Longitudinal Deformations in Elasto—
plastic Media," Trans. ASME, vol. 75, 1953, p. 427.

7. L. Prandtl, "Ein Gedankenmodell zur Kinetischen Theorie der Festen
Korper, " Zeitschrift fur Angewandte Mathematik und Mechanik,
vol. 8, 1928, p. 85.

8. P. Ludwik, "Uber den Einfluss der Deformationgeschwindigheit bei
Bleibenden Deformationen mit Besonderer Beruecksichtigung der
Nachwirkungserscheinungen," Physikalische Zeitschrift, vol. 10,
1909, p. 411.

9. H. Deutler, "Experimentelle Untersuchen uber die Abhangigkeit der
Zugspannungen von Verformungsgeschwindigkeit, " Physikalische
Zeitschrift, vol. 33, 1932, p. 247.

10. J. F. Alder and V. A. Phillips, "The Effect of Strain Rate and Tem-
perature on the Resistance of Aluminum, Copper, and Steel to
Compression," J. Inst. of Metals, vol. 83, 1954—55, p. 80.

11. B. Hopkinson, "A Method of Measuring the Pressure Produced in
the Detonation of High Explosives or by the Impact of Bullets, "
Phil. Trans. Roy. Soc. Lond., Ser. A, vol. 213, 1914, p. 437.

12. R. M. Davies, "A Critical Study of the Hopkinson Pressure Bar,"
Phil. Trans. Roy. Soc. Long., Ser. A, vol. 240, 1948, p. 375.

13. H. Kolsky, "An Investigation of the Mechanical Properties of
Materials at Very High Rates of Loading, " Proc. Phys. Soc. Lond. ,
vol. 62B, 1949, p. 676.

-30-

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

E. T. Habib, ”A Method of Making High-Speed Compression Tests
on Small Copper Cylinders," J. Appl. Mech. , vol. 15, no. 3,
1948, p. 248.

E. H. Lee and H. Wolf, "Plastic-Wave Propagation Effects in High-
Speed Testing,” Trans. ASME, vol. 73, 1951, p. 379.

J. W. Turnbow and E. A. Ripperger, ”Strain Rate Effects on Stress—
Strain Characteristics of Aluminum and Copper, " Proc. Fourth
Midwest Conf. on Solid Mechanics, Austin Texas Univ. Press,
1959, p. 415.

M. Manjoine and A. Nadai, "High Speed Tension Tests at Elevated
Temperatures," Proc. ASTM, vol. 40, 1940, p.822.

A. Nadai and M. Manjoine, "High Speed Tension Tests at Elevated
Temperatures — Parts 11 and III," Trans. ASME, vol. 63, 1941,
p. A77.

M. Manjoine, ”Influence of Rate of Strain and Temperature on Yield
Stresses of Mild Steel," J. Appl. Mech. , vol. 11, no. 4, 1944,
p. A211.

D. S. Clark and P. E. Duwez, "Discussion of the Forces Acting in
Tension Impace Tests of Materials," J. Appl. Mech. , vol. 15,
1948, p. 243.

D. S. Clark and P. E. Duwez, "The Influence of Strain Rate on Some
Tensile Properties of Steel," Proc. ASTM, vol, 50, no. 3, 1950,
p. 560.

W. R. Campbell, "The Determination of Dynamic Stress-Strain Curves
from Strain Waves in Long Bars," Proc. SESA, vol. 10, no. 1, 1952,
p. 113.

J. E. Johnson, D. S. Wood, and D. S. Clark, "Dynamic Stress-Strain
Relations for Annealed ZS Aluminum under Compression Impact, "

J. Appl. Mech., vol. 20, 1953, p. 523.

J. M. Burgers, "Mechanical Considerations - Model Systems - Pheno-
menological Theories of Relaxation and of Viscosity, " First Report
on Viscosity and Plasticity, Amsterdam Acadamy of Sciences,
Amsterdam, 1935, p. 5.

 

A. M. Freudenthal, The Inelastic Behavior of Engineering Materials
and Structures, John Wiley and Sons, Inc. , New Yofic, 1950.

 

 

H. Eyring, "Viscosity, Plasticity, and Diffusion as Examples of
Absolute Reaction Rates," J. Chemical Physics, vol. 4, 1936, p. 283.

A. Nadai, ”The Influence of Time upon Creep. The Hyperbolic Sine
Creep Law, " Stephen Timoshenko Sixtieth Anniversary Volume,
Mac-Millan, New York, 1938, p. 155.

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