Page 1 of 8
European Journal of Applied Sciences – Vol. 10, No. 5
Publication Date: October 25, 2022
DOI:10.14738/aivp.105.13102. Partom, Y. (2022). Shear Modulus and Yield Stress Change with Pressure and Temperature. European Journal of Applied Sciences,
10(5). 114-121.
Services for Science and Education – United Kingdom
Shear Modulus and Yield Stress Change with Pressure and
Temperature
Yehuda Partom
Retired from RAFAEL, P.O. Box 2250, Haifa, Israel
ABSTRACT
It is well known that the shear modulus (G) and the yield stress (Y) of metals
increase with pressure (P) and decrease with temperature (T). Steinberg [1], in his
popular compendium of dynamic material properties, assumes for
Y/Y0(P,T)=G/G0(P,T) linear relations based on tests at ambient conditions. But
recent tests of high-pressure dynamic loading of certain metals yielded results that
generally deviate from Steinberg’s equations. Here we use a different approach to
estimate G/G0(P,T). As a first approximation we let G/G0 follow from the assumption
of constant Poisson ratio (n). This leads to G/G0=K/K0, where K is the isentropic bulk
modulus. With this assumption we compute the longitudinal sound speed of
tantalum along its principal Hugoniot curve, and compare the result to recent
measurements. There is a slight disagreement which we correct by assuming
(second approximation) that Poisson’s ratio decreases slightly with pressure, and
increases slightly with temperature. As K is always available in a hydrocode run
from the equation of state, so are therefore also G/G0 and Y/Y0.
INTRODUCTION
Derivatives of the shear modulus (G) with respect to pressure (P) and temperature (T) at
ambient conditions have been determined for many materials. Steinberg [1] extrapolates these
derivatives to high pressure and temperature by:
(1)
Steinberg, as well as others before him, also assumed that:
(2)
where Y is the yield stress. In [2] and [3] they determine changes of yield stress and shear
modulus of copper and tantalum at high pressure dynamic loading, and the results generally
deviate from the predictions of Steinberg [1].
( )
1
3
0
0 0
0 0 0 RT
G 1 AP B T T
G
1G 1G A B
GP GT
-
æ ö r = + ç ÷ - -
è ø r
æö æö ¶ ¶ = = ç÷ ç÷ - èø èø ¶ ¶
( ) ( )
0 0
Y G P,T P,T Y G=
Page 2 of 8
115
Partom, Y. (2022). Shear Modulus and Yield Stress Change with Pressure and Temperature. European Journal of Applied Sciences, 10(5). 114-121.
URL: http://dx.doi.org/10.14738/aivp.105.13102
One way to determine G(P,T) is by assuming that Poisson ratio (n) does not change, or changes
only slightly, with pressure and temperature. Assuming (as a first approximation) that
n(P,T)=n0=constant, we get:
(3)
where K is the isentropic bulk modulus and cb is the sound speed. Eq. (3) leads to:
(4)
In any dynamic situation simulated by a hydrocode, r,cb and therefore also K are known for any
(x,t). Therefore, K/K0 and using Eq. (4) also G/G0 can be determined directly without need to
evaluate their separate dependence on P and T.
Generally, it is known that Poisson’s ratio decreases slightly with pressure and increases
slightly with temperature. For our second approximation we therefore assume that:
(5)
Where the coefficients nP and nT are quite small. Eq. (5) is a reasonable approximation only as
long as the state (P,T) is away from melting. When (P,T) is close to melting, we may use the
relation:
(6)
where nm(P) is determined from:
(7)
and where Tm(P) is the melting curve. Here we use Eq. (5) as we compare to test results only
up to 50GPa, which is far away from melting.
One way to determine nP and nT for a certain metal is through planar impact tests, in which the
longitudinal sound speed cL is measured [4,5]. CL is given by:
(8)
where the bulk sound speed cb is known from the equation of state, and where cs is the shear
wave speed. Measuring cL yields cs and therefore also G.
3
2
2
b
G 12 Const.
K 1
K c
- n = = + n
= r
0
00 0
GG G K
KK G K = =
( ) n = n0P T 0 + n P TT + n -
( ) ( )( )
2
n = n0P T 0 m 0 + n P TT PTT + n - + n -
( ) ( )
( )
P Tm 0
m 2
m 0
0.5 P T P T
P
TP T
-n -n - é ù ë û n = é ù - ë û
22 2 4
Lb S 3
2
S
cc c
c G
= +
= r
Page 3 of 8
116
European Journal of Applied Sciences (EJAS) Vol. 10, Issue 5, October-2022
Services for Science and Education – United Kingdom
In what follows we check (calibrate) our approach against data for cL along the principal
Hugoniot curve of tantalum measured in [5]. We do this only up to 50 GPa, because at that stress
there is a break in the cL curve.
G(P) FOR TANTALUM ALONG THE PRINCIPAL HUGONIOT
In Fig. 1 we show cL(P) data from [5] up to 50GPa and our curve-fit to it. (At 50GPa there is a
kink in the data curve which the authors of [5] attribute to a phase change).
Figure 1. Data of longitudinal sound speed of tantalum up to 50GPa along the Hugoniot curve.
Taken from [5].
We see from Fig. 1 that on this scale there’s some scatter in the data. The curve fit is a parabola,
starting at cL(0)=4.14km/s.
For computing the Hugoniot curve we use the usual Uu straight line. For tantalum we use the
parameters from [1]:
(9)
where U is the shock velocity and u is the particle velocity. The Hugoniot curve in the PV plane
is then:
(10)
where r is density and V is specific volume. We use the usual Gruneisen equation of state
referred to the Hugoniot with constant rG, where G is the Gruneisen parameter. The equation
of state is:
0
0
U c Su
c 3.4km s S 1.2
= +
= =
( )
2
h 00 2
0
0
V Pc 1
1 S V
V 1 16.69gr cc
e = r e = - - e
= r r =
Page 4 of 8
117
Partom, Y. (2022). Shear Modulus and Yield Stress Change with Pressure and Temperature. European Journal of Applied Sciences, 10(5). 114-121.
URL: http://dx.doi.org/10.14738/aivp.105.13102
(11)
where E is the specific internal energy, and the index h refers to the Hugoniot curve. It is easy
to verify that:
(12)
and we do not detail here the expressions for the partial derivatives of Eq. (12) along the
Hugoniot curve. We first calculate cL(P) along the Hugoniot curve assuming that v=v0=const.
We show the result of this calculation in Fig. 2, and compare it there to the data curve-fit shown
in Fig. 1.
Figure 2. Longitudinal sound speed of tantalum up to 50GPa.
Points: curve fit to data from [5].
Full curve: calculated with constant Poisson ratio.
We see from Fig. 2 that even for a constant Poisson ratio the agreement is quite good, especially
in view of the scatter in the data.
In Fig. 3 we show the results for G/G0=K/K0 obtained from the calculation of Fig. 2. We compare
this result to Steinberg’s curve from Eq. (10), with the coefficients given in [1], which are:
A=0.0145/GPa, B=1.3e-4/K.
( ) ( ) ( )
( )
0
h h
0
1
h h0 2
V E P,V E V P P V
E PV V
= + é ù - ë û G
= -
2 2
b
P EV c V
E P
+ ¶ ¶ = ¶ ¶
Page 5 of 8
118
European Journal of Applied Sciences (EJAS) Vol. 10, Issue 5, October-2022
Services for Science and Education – United Kingdom
Figure 3. Full line: G/G0=K/K0 along the Hugoniot curve from the calculation of Fig. 2.
Broken line: Steinberg’s curve.
We see from Fig. 3 that Steinberg’s line is below our line by about 25% in G/G0.
In Figs. 4 and 5 we show the influence of the pressure dependent coefficient nP.
Figure 4. Longitudinal sound speed of tantalum along the Hugoniot curve up to 50GPa. Curve
for nP=-1.e-4/GPa,
Compared to the curve with constant n.