Constitutive Behavior of Tantalum and Tantalum-Tungsten
Alloys
SHUH RONG CHEN and GEORGE T. GRAY Ill
The effects of strain rate, temperature, and tungsten alloying on the yield stress and the strain-
hardening behavior of tantalum were investigated. The yield and flow stresses of unalloyed Ta and
tantalum-tungsten alloys were found to exhibit very high rate sensitivities, while the hardening rates
in Ta and Ta-W alloys were found to be insensitive to strain rate and temperature at lower temper-
atures or at higher strain rates. This behavior is consistent with the observation that overcoming the
intrinsic Peierls stress is shown to be the rate-controlling mechanism in these materials at low tem-
peratures. The dependence of yield stress on temperature and strain rate was found to decrease, while
the strain-hardening rate increased with tungsten alloying content. The mechanical threshold stress
(MTS) model was adopted to model the stress-strain behavior of unalloyed Ta and the Ta-W alloys.
Parameters for the constitutive relations for Ta and the Ta-W alloys were derived for the MTS model,
the Johnson-Cook (JC), and the Zerilli-Armstrong (ZA) models. The results of this study substantiate
the applicability of these models for describing the high strain-rate deformation of Ta and Ta-W
alloys. The JC and ZA models, however, due to their use of a power strain-hardening law, were
found to yield constitutive relations for Ta and Ta-W alloys that are strongly dependent on the range
of strains for which the models were optimized.
I. INTRODUCTION
THE
microstructure/property relationships of tantalum
and tantalum-based alloys continue to attract scientific and
engineering interest due to their high density, melting point,
excellent formability, good heat conductivity, good fracture
toughness (even at low temperatures), corrosion resistance,
and weldability.V] Since 1950, numerous studies have
probed the microstructure-chemistry/property response of a
large number of tantalum and tantalum-based alloys, both
in single-crystal and polycrystalline form.[' 51 Tantalum,
like all bcc metals, exhibits deformation behavior that is
markedly influenced by impurities, alloying additions, crys-
tallographic texture, temperature, and strain rate. t~81 Tan-
talum and its alloys are increasingly being utilized in
defense-related applications where their mechanical prop-
erties under high strain-rate deformation are attractive. In
this article, a wide range of data on unalloyed tantalum and
Ta-W alloys subjected to high strain-rate compression at
various temperatures will be presented. Yield and flow
stresses are shown to be sensitive to the changes in tem-
perature and strain rate at low temperatures and/or high
strain rates. A large Peierls stress in bcc materials has been
proposed as the rate-controlling mechanism in this temper-
ature and strain-rate regime. [7.91 This large intrinsic lattice
resistance results in restricted movement of screw disloca-
tions; long straight screw segments are often observed in
this class of materials after deformation, t3,7,~~ This suppres-
sion of cross-slip of screw dislocations results in linear
glide and, therefore, a lower degree of dynamic recovery.
The strain-hardening rates in this class of materials at low
temperature or high strain-rate loading states are seen to be
temperature insensitive [2.71 The addition of alloying solutes
to tantalum raises its yield and flow stresses through solid
solution strengthening, tvl The overall work-hardening rates
are increased relative to unalloyed Ta due to dislocation-
solute interactions. The temperature and strain-rate depend-
ence of the yield and flow stresses, as well as the
strain-hardening rate, are changed upon solute additions in
commercially pure tantalum. While a large number of stud-
ies have probed the mechanical behavior of a broad spec-
trum of tantalum alloys, details of the underlying defor-
mation mechanisms remain poorly understood and, in some
cases, controversial.
The availability of modem high-speed computers makes
it possible to develop more sophisticated material consti-
tutive model descriptions capable of modeling complex
problems. [1~,12,~3j An accurate description of a materials re-
sponse over a wide range of loading environments, as well
as having predictive capabilities outside the measured
range, is in great demand. The material properties unique
to bcc metals and alloys bring many challenges for the de-
velopment of physically based constitutive models. The in-
fluence of impurities and the effect of tungsten alloying on
the constitutive behavior of Ta and Ta-W alloys will be
presented in this article. Several currently utilized consti-
tutive models, namely the mechanical threshold stress
(MTS) model, tl41 the Johnson-Cook (JC) model,r15] and the
Zerilli-Armstrong (ZA) model,V6] are examples of consti-
tutive models currently implemented in a range of finite
element codes, such as EPIC, t17j MESA, t~Sl and DYNA, t~91
were examined. The same data set is used to derive the
parameters for each model enabling direct comparisons be-
tween each model.
II. EXPERIMENTAL PROCEDURE
SHUH RONG CHEN and GEORGE T. GRAY III, Staff Members, are
with the Los Alamos National Laboratory, Los Alamos, NM 87545.
Manuscript submitted March 24, 1995.
The materials used in this investigation were commer-
cially pure (triple electron-beam) annealed unalloyed Ta
plate, Ta bar stock, Ta-2.5 wt pet W (Ta-2.5W), Ta-5 wt
U S GOVERNMENT WORK
2994--VOLUME 27A, OCTOBER 1996 METALLURGICAL AND MATERIALS TRANSACTIONS A
NOT PROTECTED BY U.S COPYRIGHT
Table I. Alloy Compositions (in Weight Parts per Million A.
JC Model t~51
Unless Otherwise Noted)
C O N H W Nb Ta
Ta-A
TM
9 44 18 < 1 < 150 123 bal
Ta bar 10 65 20 <5 <25 80 bal
Ta 12 <50 <10 <5 60 250 bal
Ta-2.5W 18 98 <10 <5 2.4 wt pct 330 bal
Ta-5W 15 <50 <10 <5 5.2 wt pct 65 bal
Ta-10W 11 63 <10 <5 9.6 wt pct 385 bal
pct W (Ta-5W), and Ta-10 wt pct W (Ta-10W) supplied
by Cabot Corporation, Boyertown, PA, with compositions,
as listed in Table I. The unalloyed Ta, Ta-2.5W, and Ta-
10W materials were prepared by melting 254-mm diameter
or greater ingots. The Ta-5W, in contrast, was cast in a
smaller research-scale electron beam melter, triple melted
similar to the other ingots. All the ingots were forged into
billets, the billets were annealed and cut prior to cross roll-
ing. The plates were straight rolled in the final finishing
passes. Each material was supplied in 6.35-mm-thick plate
form. The as-tested microstructures of the 6.35-mm-thick
unalloyed Ta, Ta-2.5W, Ta-5W, and Ta-10W plates exhib-
ited equiaxed grains with average grain sizes of 42, 45, 48,
and 42/zm, respectively.
The mechanical responses of the tantalum materials were
measured in compression using solid-cylindrical samples
6.35 mm in diameter by 6.35 mm in length, lubricated with
molybdenum disulfide grease. Compression samples were
machined from the plates in both the through-thickness and
in-plane longitudinal orientations. An initial description of
the influence of tungsten alloying on the mechanical prop-
erties and texture of Ta was published previously.tZ01 Quasi-
static compression tests were conducted at strain rates of
10 -3 and 10 -1 s -1, at 77 and 298 K, respectively. Dynamic
tests at strain rates of 1000 to 8000 s -1 were conducted
from 77 to 1273 K in a vacuum utilizing a Split-Hopkinson
pressure bar. t2q The inherent oscillations in the dynamic
stress-strain curves and the lack of stress equilibrium in the
specimens at low strains make the determination of yield
inaccurate at high strain rates.
The shear modulus was calculated for bcc Ta using the
formula
(Cll -- C12 "q- C44 )
/z = 3 [1]
where C,j are the elastic constants, t22~ For simplicity, an em-
pirical equation tz3~ was used to fit the data to incorporate
the temperature dependence of/x in the form of
D
/x = /z o - [2]
exp (~) - 1
where /x0, D, and To are fitting constants. The same tem-
perature-dependent shear modulus was used for the Ta-W
alloys.
IlL DESCRIPTION OF MODELS
The constitutive equations used in this study have the
following forms.
o- = (A + B. ep")(1 + C. In k*)(1 - T *m) [3]
where k* is a nondimensional strain-rate value, ep is the
plastic strain, and 7"* is
(T - Troom)/(Tmol, -
Troom). The value
of Tis in degrees kelvin, and A, B, n, C, and m are constants
for this model.
B. ZA Model for bce Materials t16j
0-= Co+ C,.exp(-C3"T+ Ca'T'lnk)
+ C5" ep" [4]
where Co, C1, C3, C4, C5, and n are constants for this model.
The athermal stress term
Co
can be replaced with a Hall-
Petch relation
o" o + k 9 d -~/2,
where d is the grain size, to
address a grain size dependence on the yield stress.t~61 In
the ZA model, it is presumed that the work-hardening rate
is independent of temperature and strain rate. Both of the
preceding models use a power-law stress-strain relationship
that exhibits continual work hardening without approaching
a saturation in the flow stress at large strains.
For the JC and ZA models, computer programs were de-
velopedt24J to optimize the fitting constants to the stress-
strain data over a wide range of temperatures and strain
rates. A range of corresponding constants is given to cal-
culate the stress at a certain strain followed by a comparison
of this value to the experimental values. This process is
repeated for every curve of interest until the best agreement
to the entire set of stress-strain data is achieved. A param-
eter indicating the degree of fit is defined as
k
]~
I~.co,cul.to~ (e,) - ~oxp ..... ~1 (e,)l
6 = t = 1 O'exp ..... tal (•,)
k [5]
Four points representing the characteristic hardening be-
havior for each stress-strain curve were taken to compare
the calculated stresses at the corresponding strain values.
For the modeling results presented in this article, fits with
deviation parameters of better than 4 pct were achieved.
C. MTS Model
The framework and detailed description of the MTS
model are given elsewhere.t14.25.261 A summary of the MTS
model is presented here to facilitate comparisons with the
JC and ZA models and discussion of the results. Plastic
deformation is known to be controlled by the thermally
activated interactions of dislocations with obstacles. In the
MTS model, the current structure of a material is repre-
sented by an internal state variable, the mechanical thresh-
old (~r), tz71 which is defined as the flow stress at 0 K. The
mechanical threshold is separated into athermal and thermal
components:
6 = 6-~ + Z gr, [6]
where the athermal component b-, characterizes the rate-
independent interactions of dislocations with long-range
barriers, such as grain boundaries, dispersoids, or second
phases. The thermal component b', characterizes the rate-
dependent interactions of dislocations with short-range ob-
METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 27A, OCTOBER 1996--2995
stacles
(i.e.,
forest dislocations, interstitial, solutes, Peierls
barrier,
etc.)
that can be overcome with the assistance of
thermal activation. The summation of the contributions
from different obstacles does not need to be linearY~ The
flow stress of a constant structure at a given deformation
condition can be expressed in terms of the mechanical
threshold as
o" = o" a + 2~ o', _ o" a + 2~ S(k,T) &'
[7]
/z /~ p~ /z /~0
where the athermal component is a function of temperature
only through the shear modulus, and the factor S specifies
the ratio between applied stress and mechanical threshold
stress. This factor is smaller than one for thermally acti-
vated controlled glide because the contribution of the ther-
mal activation energy reduces the stress required to force a
dislocation past an obstacle. In the thermally activated glide
regime, the interaction kinetics for short-range obstacles are
described by an Arrhenius expression of the form
= k0 exp (~)-AG [8]
The free energy (AG) is a function of stress and a phenom-
enological relation was chosen [27]
[
AG = gol.6b 3 1 \fr,/tZo / d
[9]
go in units of/zb 3 is the normalized activation energy for
the dislocations to overcome the obstacles. It is also an
indication of the sensitivity of overcoming this obstacle to
changes in temperature and strain rate. The terms p and q
are parameters with the ranges 0 < p < 1 and 1 < q < 2.
They detail the glide resistance profile in the higher and
lower activation energy regions, respectively.tin Upon re-
arrangement, we have the following relation between ap-
plied stress (~r,) and mechanical threshold stress (b-t) at a
constant structure...
[
(g~
e__o.][
1- In.
/x /x 0
For single-phase materials with cubic crystal structures, the
thermal component (o- 3 consists of the linear summation of
a term describing the thermal portion of the yield stress (o-,
= ~ry - o'o) and a term describing the evolution of the
dislocation structure ~, as a function of temperature, strain
rate, and strain. Eq. [7] can be written as
or
_ o'~ + 5 +
o'~= o'~
+ S,(~,T)&,
IX I~ I z I~ tz IXo
+ S~ (k,T) O, [11]
/Zo
The second term on the right-hand side of the equation
describes the rate-dependent portion of the yield stress,
mainly due to intrinsic barriers, such as the strong Peierls
stress in bcc materials at low temperatures or at high strain
rates. It is further assumed that this term does not evolve
after yielding. The term &~ in Eq. [11] evolves with strain
due to dislocation accumulation (work hardening) and an-
nihilation (recovery). This structure evolution, 0 =
d b'/de,
is written as
0 = Oo- Or (T,k,&)
[12]
where 0 o is the hardening due to dislocation accumulation
and
Or
is the dynamic recovery rate. The physical under-
standing of the work-hardening behavior of polycrystals is
still inadequate to unify this complex process and represent
it entirely by physically based parameters. Follansbee and
KocksU4~ have selected the following form to fit their ex-
perimental hardening data for Cu, and the same form was
found to provide a robust fit to hardening behavior in Ni E26j
and Ti-6A1-4V: f25j
I tanh
0=00 1-
[13]
where a approaching zero represents a linear variation of
strain-hardening rate with stress (Voce law). The saturation
threshold stress &~ is a function of temperature and strain
rate. Kocks t281 has proposed a description for 3% that has
the same form as that proposed by Haasent291 for the begin-
ning of dynamic recovery, which in tum, was based on
calculations by Schoeck and SeegerO01 of the stress de-
pendence of the activation energy for cross slip in fcc met-
als. The relation is written
goes t zb3 O'es
In-- -- --In-- [14]
/~sO kT b-~ o
where e~0, go~, and &~0 are constants.
IV. RESULTS AND ANALYSIS
A. Compression Test Results
The compressive stress-strain responses of unalloyed Ta
and Ta-W alloys at 25 ~ and 500 ~ as a function of Ta-
alloying content and strain rate are shown in Figure 1. The
lower yield and flow-stress levels in the tantalum and tan-
talum alloys are seen to increase with increasing strain rate.
As observed in many bcc materials at low temperature or
at high strain rate, these materials show very high strain-
rate sensitivities in their yield and flow stresses. The addi-
tion of tungsten to tantalum results in an increase in the
yield stress and flow stress in Ta-W alloys in a similar way
to increasing the strain rate for unalloyed tantalum. The
room temperature stress-strain response of unalloyed tan-
talum at a strain rate of 2500 s -~ is very close to that ex-
hibited by Ta-10W deformed quasi-statically at a strain rate
of 10 3 s-1 at 25 ~ (Figure 1). This observation is consis-
tent with the well-established solute strengthening effects
of tungsten alloying additions on Ta. [2,3,311
The strain-rate and alloying effects of W on the flow-
stress response of tantalum are quantified further in Figure
2, where the stress level at high strain rates, designated (hr),
and low strain rates, designated (Lr), at two different strain
values are plotted
vs
the tungsten alloying content in weight
2996~VOLUME 27A, OCTOBER 1996 METALLURGICAL AND MATERIALS TRANSACTIONS A
1500 .... , .... i .... , .... , .... , .... I ....
-
25~ 2500/s
25~ 0.001Is
....... 500~ 0.001Is ,,, 1" " Ta-10W
1200 .,-
,g.-
/ ~, ,~ Ta-5W
j,[~ ,, ,.,r r"
r -- Ta-2.5W
0.. LI 1 ~ -- "~ ""
I~ 600 ~ ......................
Y .......
............. ................
10"3--> 10"25 "1
0 I I I I I I ' ' ' I " ' ' ' I ' ' ' ' I ' t ' ' I ' ' ' ' I ' ' ' '
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
E
Fig. 1--Compressive stress-strain response of Ta, Ta-2.5W, Ta-5W, and
Ta-10W at strata rates of 2500 and 10 -3 s i at 25 ~ and at strain rates
of 10 -3 s t at 500 ~
2000
1500
13.
~" 1000
(E
(/)
500
I I I I
r ~(hr) at e=0.10
-- o- - o(Lr) at e=0.10
---~--- c(hr)-c(Lr) at e=0.10
= ~(hr) at e=0.18
-- D- - ~(Lr) at ~-0.18
----D--- (~(hr)-G(Lr) at e=0.18
I I
T = 25~
hr ~- 2500
8 "1
Lr=
10"3S "1
r.~ ~- s~ ..,,~ ~"
.,.- Pg_ s .... 43
.=.~ ............. [] .......
0" ..... ~ ..... -o
u ........... t~ .................
kJ
0 , I , I , I , I , I , I ,
-2 0 2 4 6 8 10 12
Tungsten content (wt%)
Fig. 2--Room temperature flow stresses at strains of 0.1 and 0. ] 8 at strain
rates of 2500 and 10 -3 s ~ as a function of tungsten alloying content in
tantalum.
percent. The flow-stress levels at low strain rate are seen to
increase linearly with alloying content (dashed lines with
open symbols in Figure 2). At two different strain values
(0.1 and 0.18), the linearity between the flow stresses and
the tungsten content ]s preserved with the same functional
relationship suggesting that tungsten alloying does not alter
the strain-hardening behavior of these materials deformed
at low strain rates at room temperature. The flow stresses
obtained at the same strains under dynamic loading (solid
lines with solid symbols in Figure 2) increase nonlinearly
with respect to the alloying content compared to that ex-
hibited under quasi-static conditions. The difference in the
1500
1200
,.., 900
O.
I I I
Unalloyed-Ta
.o "'~176 ...... 2
1.-196~ 10"3/s
2.-196~ 1800/s
3. 25~ 1300/s
4. 200~ 2800/s
5. 400~ 2600/s
1 6.25~ 10"1/s
7. 600~ 2200/s
8. 800~ 3900/s
9. 1000~ 3000L,
3
600 ~ 4
7
..~. ....
8
300
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
E
Fig. 3~ompressive stress-strain curves of the unalloyed tantalum under
dynamic and quasi-static deformation at various temperatures.
stress levels at high strain rate, for strains of 0.1 and 0.18,
increases with increasing alloying content. This divergence
reveals the increase in strain-hardening rate commensurate
with increasing tungsten content in tantalum. The flow-
stress differences between high and low strain-rate com-
pression, given in Figure 2 as the finely dotted lines,
increase with increasing tungsten content, except for the
unalloyed tantalum which exhibits a higher difference than
from alloying with 2.5 pet tungsten. This discrepancy can
be explained in terms of the high sensitivity of the flow
stress in bcc materials to impurities at low temperature or
high strain rate. Alloying with an ample quantity of tung-
sten tends to overshadow the impurity effects observed in
less pure tantalum.t20] The flow-stress decrease after yield-
ing, which was interpreted as an influence of impurities,
during quasi-static deformation at 25 ~ disappears when
tantalum is alloyed with 2.5 or 5 pet tungsten or when the
materials are deformed at elevated temperatures, as shown
in Figure 1. The flatness of the slopes of the curves de-
scribing the stress differences further indicates that the ma-
jor influence of tungsten alloying on the mechanical
properties of tantalum is to: (1) substantially raise the yield
and flow stresses of tantalum with tungsten alloying, and
(2) either weakly, as in the case of the 2.5 or 5 wt pet
alloys, or moderately, as in the case of the 10 wt pet alloy,
affect the strain-hardening response of tantalum.
The effect of temperature and strain rate on mechanical
properties of unalloyed and alloyed tantalum was investi-
gated through a series of compression tests at different tem-
peratures and strain rates. The results for the unalloyed
tantalum are plotted in Figure 3. Below 200 ~ this material
exhibits a high temperature and strain-rate sensitivity. Increas-
ing the test temperature by the same increment (200 ~ but
over two different temperature ranges shows that the dif-
ference in stress response diminishes. The constitutive re-
sponse shows less temperature sensitivity at higher
temperatures, as demonstrated by the curves numbered 7,
8, and 9
vs
curves 3, 4, and 5 in Figure 3. This behavior
METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 27A, OCTOBER 1996---2997
1500
1200
900
t~
n
600
300
t
''''1''''1''''1''''1'''' I''''
T = 200~
F ....... T = 6oo~
l ~ = 2500 s "1
Ta-10W
J- Ta-5W
1 ~ ~..-
......
_ Ta-2.5W
/~ ~ ..... ~..--Ta-5W '
.......
......
V~.- --o _...o...o...~
o._. ....
ooo
oO~
0 , , , I , . . , I , , , . I . , , . I . . , , I . . . .
0 0.05 0.1 0.15 0.2 0.25 0.3
E
Fig. 4~Compressivr stress-strata curves for Ta and Ta-W alloys
deformed dynamically at 200 ~ and 600 ~
2000
1500
'E
I1.
ff 1000
d
09
500
I
I I
I
: ~(LT)
at
e=0.05
-- o-
- c(hT)
at
e=0.05
----o--- a(LT)-G(hT) at e=0.05
= ~(LT) at e=0.20
-- ~ - t~(hT)
at
e=0.20
---~--- t~(LT)-a(hT)
at
e=0.20
I I
= 2500 s "1
LT = 200~
hT = 600~
O -- ~ ~ ..... .----~
......... I'1 ........... ....--O
0 ,
I , I , I , I , I I
-2 0 2 4 6 8 10 12
Tungsten content (wt%)
Fig. 5--Dynamic flow stresses at strains of 0.05 and 0.2 at temperatures
of 200 ~ and 600 ~ as a function of tungsten alloying content in the
Ta-W alloys studied.
is primarily due to dislocations overcoming the intrinsic
short-range obstacle(s), such as strong Peierls stress. The
intrinsic barrier(s) that exhibits very high temperature and
strain rate sensitivities becomes transparent to dislocation
motion at higher temperatures through thermal activation
process. The dependence of the flow stress on temperature,
therefore, dramatically decreases. However, the strain-hard-
ening behavior is essentially unchanged, even when de-
formed at 1000 ~ at a high rate, as compared to room
temperature and high strain rate.
The effect of temperature on the stress-strain response of
alloyed tantalum is illustrated in Figure 4, which is a plot
consisting of unalloyed Ta, Ta-2.SW, Ta-SW, and Ta-10W
deformed at a high strain rate at two different temperatures,
namely, 200 ~ and 600 ~ Within this range, all the ma-
terials show a relatively high sensitivity to the temperature
change. The strain-hardening rate decreased when the test
temperature increased for Ta-5W and Ta-10W. Less of an
effect was observed for the hardening behavior of unalloyed
Ta or Ta-2.5W. Figure 5 correlates the stress levels taken
from Figure 4 at two different strains of 0.05 and 0.2, re-
spectively, and plots them as a function of tungsten content.
Overall, the result is similar to that, as shown in Figure 2,
in which the strain rate was changed instead of the tem-
perature. This result further supports that the deformation
processes in Ta and Ta alloys can be explained in terms of
thermal activation theory, which requires that temperature
and strain rate are coupled in a form of
kT.
log (P~/~),
which is rearranged from Eq. [8]. Increasing (decreasing)
the strain rate logarithmically has the same effect as de-
creasing (increasing) the temperature linearly. The stress
increase observed at a low temperature, mainly due to solid
solution hardening of W, remains at high temperatures. The
change in strain-hardening rate varies from almost nothing
for unalloyed Ta to a moderate degree for Ta-10W, as in-
dicated by the separation of these two dotted lines in Figure
5, which represents the stress increment at strains of 0.05
and 0.2, respectively. If the strain-hardening rate remains
unchanged, these two points will be coincident, as is shown
for the unalloyed Ta. As deformation continues, the diver-
gence of the stress-strain curves results from changes in the
strain-hardening rate. This will be reflected in the separa-
tion of the two dotted lines and symbols, which is precisely
what is seen in Figure 5.
B. Constitutive Modeling
1. JC and ZA models
The basic approach to fitting the JC and ZA models is
to select a wide range of data to represent the temperature
and strain-rate sensitivity, as well as the hardening behavior
of a given material. A range of corresponding parameters
(.4, B, n, C, and m in the JC model; Co, C~, C3, C4, C5, and
n in the ZA model) varied by specified increments are then
tested in a computer optimization routine developed for the
personal computer to compare the calculated and experi-
mental stress levels at certain strains for all the data. Min-
imum deviation of the calculated stress from the experiment
is used as a guideline to judge the fitting. This process is
repeated and the range of the parameters is decreased as
well as the increment for each parameter. In general, to
calculate a few hundred thousand sets of parameters to fit
ten stress-strain curves takes about 5 minutes. Less than five
adjustments to the parameter range and increment were
found to give a best fit to the data. Because no single pa-
rameter is dominant or redundant, this optimization for the
whole spectrum of data of interest does yield a best set of
parameters without excessive prejudgement. A comparison
of the results derived for tantalum, from the tantalum data
of Hoge and Mukherjee, Egj by Zerilli and Armstrong [321 and
by the current method was recently presented, p4~ It was con-
cluded that the current parameter optimization method is as
satisfactory as or better than the method used originally.
The fit to the ZA model for unalloyed Ta is shown in
Figure 6. The experimental data at strains less than 0.2 were
treated as an isothermal condition for all strain rates. The
2998--VOLUME 27A, OCTOBER 1996 METALLURGICAL AND MATERIALS TRANSACTIONS A