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European Journal of Applied Sciences – Vol. 9, No. 3
Publication Date: June 25, 2021
DOI:10.14738/aivp.93.10500. Partom, Y. (2021). Non Shock Dynamic Compression of Tantalum. European Journal of Applied Sciences, 9(3). 623-633.
Services for Science and Education – United Kingdom
Non Shock Dynamic Compression of Tantalum
Yehuda Partom
Retired from RAFAEL, P.O. Box 2250, Haifa, Israel
ABSTRACT
We start by simulating the response of tantalum to dynamic non-shock loading
using a standard hydrocode, and the results we get are not consistent with test data.
Standard hydrocodes use what we call the flowstress approach to describe the
dynamic viscoplastic response of materials. We therefore transform our hydrocode
to use what we call the overstress approach to dynamic viscoplastic response. It
turns out that using the overstress approach makes the hydrocode more successful
in reproducing non-shock dynamic compression test data.
INTRODUCTION
In recent years we see more reports on gradual (non-shock) dynamic loading of various metals
to high pressure [1,2]. The advantage of gradual dynamic loading over shock loading is: 1) it’s
possible to control the loading rate independently of pressure. This is important as strength
increases considerably at very high strain rates [3]; 2) When a material undergoes an isentropic
loading path (not a shock), it heats up at a slower rate than from a shock (there is no shock
heating). It’s therefore possible to determine the influence of pressure on strength separately
from the influence of temperature.
In Fig. 1, taken from [1], we see the response of tantalum targets to non-shock loading at
uniaxial strain (plane strain loading), where pressure is under 20GPa.
We see from Fig. 1 that details of the initial response depend on the initial state of the tested
material. Tantalum samples in tests designated (a) went through an annealing process, and
contain a relatively small number of defects and dislocations. The elastic wave in these samples
has an initial spike which then connects smoothly to the plastic part of the signal. On the other
hand, tantalum samples in tests designated (b) went through a cold-rolled process, and
therefore contain a large number of dislocations. There is no elastic spike in the response
signals of these tests. Instead, the rise to the elastic wave is gradual, and so is the initial rise
from the elastic wave to the plastic wave. Trying to reproduce the results of Fig. 1 using a
hydrocode that contains a regular strength model yields inaccurate results.
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European Journal of Applied Sciences (EJAS) Vol. 9, Issue 3, June-2021
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Figure 1. Copied from [1]. Particle velocity histories at different depths into the target. (a)
annealed tantalum. (b) cold-rolled tantalum.
In Fig. 2 we show an example of such a simulation: 1) we use a Mie-Gruneisen EOS [4] with
constant rG and with Steinberg’s parameters [5]; 2) we use an initial strength of Y0=0.77GPa
and an initial shear modulus of G0=69GPa; 3) we change strength and shear modulus with
pressure and temperature according to [6]:
(1)
Where K=bulk modulus, r=density and c=sound speed; 4) there are no strain hardening or
strain rate hardening; 5) the boundary particle velocity rises at a constant rate during 0.25μs
and stays there. Also, as mentioned above, we use the regular strength model included in
standard hydrocodes. We refer to such models as flowstress approach strength models [7].
2
2
0 0 0 00
YGK c
YG K c
r === r
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Partom, Y. (2021). Non Shock Dynamic Compression of Tantalum. European Journal of Applied Sciences, 9(3). 623-633.
URL: http://dx.doi.org/10.14738/aivp.93.10500
Figure 2. Particle velocity histories every 1mm, from constant velocity rate loading of the
boundary, using the standard flowstress approach with radial return according to Wilkins [8].
We see from Fig. 2 that as expected, a constant level elastic precursor wave is formed. The
elastic precursor wave then gradually joins the plastic wave in a way that has no resemblance
to the curves of Fig. 1. In what follows later we use what we call the overstress approach to
dynamic plasticity [7] to reproduce the same tests as above. As strain rate hardening is an
integral part of the overstress approach, we first apply in what follows next the same amount
of strain rate hardening to simulations with the flowstress approach. We do this mainly for
comparison purposes. With the overstress approach the effective deformation rate is given by:
(2)
But with the flowstress approach, the equivalent stress is the flow stress. We therefore get:
(3)
We repeat the simulations of Fig. 2, but with strain rate hardening according to Eq. (3). We use
A=0.1/μs and α=2.4. We show the results in Fig. 3.
We see from Fig 3 that what strain rate hardening does is: 1) it increases the elastic precursor
wave level, and 2) it does not reproduce a waveform that is similar to the test results (Fig. 1).
In what follows later we perform similar simulations using the overstress approach.
p eq qs
eff
0
Y
d A
Y
a
æ ö s - = ç ÷ è ø
1
p
0 eff
eq qs
qs
Y d Y Y1
Y A
é ùa æ ö = s = + ê ú ç ÷ ê ú è ø ë û
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European Journal of Applied Sciences (EJAS) Vol. 9, Issue 3, June-2021
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Figure 3. Particle velocity histories from constant velocity rate loading on the boundary. The
computation uses the flowstress approach with a strain rate hardening similar to the one used
later with the overstress approach
INSERTING THE OVERSTRESS APPROACH INTO OUR HYDROCODE
We introduce the overstress approach into a standard and old code that we use (PISCES2D) for
the case of uniaxial symmetry. We perform the needed changes through the user subroutine
EXYLD that evaluates the flowstress. We enter this subroutine for every computational cell and
for each time step. Entering the subroutine, we know the values of the following variables: 1)
components of the total deformation rate dij, from which we compute the components of its
deviator δij in the usual way; 2) components of the stress deviator at the beginning of the time
step sijb (b for beginning), from which we compute the equivalent stress by (seq)2=1.5sijsij; 3)
the current shear modulus according to G/G0=K/K0 [6], from which we compute the yield stress
by Yqs=Y0G/G0. We then compute the effective plastic deformation rate by:
(4)
the coefficient λ by:
(5)
and the rate of change of the components of plastic deformation rate and elastic deformation
rate by:
(6)
We get a system of three time-dependent ODEs for the deviatoric stress components, for each
computational cell and for each time step. These equations are:
(7)
p eq qs
eff
0
Y
d A
Y
a
æ ö s - = ç ÷ è ø
p
3 eff
2
eq
d l = s
p p
ij ij ij
e p
ij ij ij
d s = d = l
d = d -d
e
ij ij s 2G (ij xx, yy,xy) ! = d =
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Partom, Y. (2021). Non Shock Dynamic Compression of Tantalum. European Journal of Applied Sciences, 9(3). 623-633.
URL: http://dx.doi.org/10.14738/aivp.93.10500
We integrate this system of ODEs across the time step with a standard ODE solver (we use the
4th order Runge-Kutta solver). At the end of such a time step we get from the integration the
components of the deviatoric stress sije (e for end), from which we compute the flow stress
Ye=seqe by Eq. (3). We then return the flow stress Ye to the code, which computes the
components of the new deviatoric stress (for each cell) with the radial return approach
originally built into the code. At this stage the code also takes into account rigid rotation of the
axes, using the usual rotation correction equations.
In Fig. 4 we show results of the same problem as in Fig. 3, computed with the overstress
approach, as described above.
Figure 4. Particle velocity histories every 1mm into the target, from a constant acceleration of
the boundary, using the overstress approach to dynamic viscoplasticity
We see from Fig. 4 that: 1) the elastic precursor wave and the transition to the plastic wave are
similar to the lower part (b) of Fig. 1; 2) there is a small decay of the elastic wave, which we also
discern in part (b) of Fig. 1.
We conclude that the overstress approach to dynamic viscoplasticity may perform better than
the flowstress approach in predicting test results.
SPIKE FORMATION
In Fig. 1 (a) we see that spikes have formed at the starts of the elastic waves, but there are no
spikes at the starts of the elastic waves of Fig. 1(b). We suspect that spike formation has to do
with two parameters: 1) the rate of loading. We know that there is no spike at the start of the
elastic wave for shock loading. So probably a spike may form at a not so high loading rate. In
Fig. 1, tests (a) and (b) are of similar loading rates. It follows that loading rate is not the only
reason for spike formation; 2) the rate of plastic flow. For the annealed material of Fig. 1(a), the
initial rate of plastic flow is low, and a spike is formed. But for the cold-rolled material of Fig.
1(b) there are many dislocations, the rate of plastic flow is therefore high, and a spike is not
formed. In our computations we control the rate of plastic flow by changing the parameter A.
Increasing A increases the rate of plastic flow. As a result, the spike size decreases, and for high
values of A it disappears altogether. The opposite happens for decreasing A. In Figs. 5 and 6 we
show the influence of loading rate and the parameter A on spike formation.
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European Journal of Applied Sciences (EJAS) Vol. 9, Issue 3, June-2021
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Figure 5. Influence of loading rate (or loading time) on spike formation. For a short loading
time the spike disappears
Figure 6. influence of the coefficient A. For large dislocation density the coefficient A is high,
which causes a high plastic deformation rate at the same shear stress
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Partom, Y. (2021). Non Shock Dynamic Compression of Tantalum. European Journal of Applied Sciences, 9(3). 623-633.
URL: http://dx.doi.org/10.14738/aivp.93.10500
From Fig. 5 we see that for a short loading time there is no spike. For longer loading time a spike
appears and it increases with loading time (or with decreasing loading rate). From Fig. 6 we see
that by increasing the parameter A (increasing the plastic deformation rate at the same stress
level), we decrease the spike size until it disappears.
We conclude that by using the overstress approach to dynamic plasticity, it is possible to
reproduce the response of viscoplastic materials to impact and high-rate loading, including the
formation of a spike at the start of the elastic precursor.
SUMMARY
In recent years we see in the literature an increasing number of reports on gradual (non-shock)
dynamic loading of metals to high pressure. The loading instrument is magnetic, and it enables
to reach pressure levels of hundreds of GPa, and to perform measurements as in shock tests. In
[1,2] we find results of such dynamic gradual loading tests.
High pressure dynamic tests are usually reproduced by hydrocode simulations (forward
computation), and this is the common approach as long as material strength is relatively small.
But for tests where material strength is the main issue, it is many times the practice to use
backward analysis, and deduce material strength parameters directly from test data. Such a
backward analysis may be quite complex, as in [1].
Here we perform hydrocode simulations to analyze some of the tests reported in [1]. First, we
show that by using the flowstress approach to dynamic viscoplasticity, we’re not able to
reproduce the test results. We then convert our hydrocode to work with the overstress
approach to dynamic viscoplasticity. Using the converted code, we then show that in this way
we get particle velocity histories that are similar to the test results. We get elastic waves with
and without an initial spike, depending on the parameter A (Eq. 2), which determines the level
of plastic deformation rate as function of shear stress.
References
[1]. J.R. Assay et al, Yield strength of tantalum under ramp compression to 18 GPa, J. Appl. Phys. 106, 073515 (2014).
[2]. J.L. Brown et al., Flow strength of tantalum under ramp compression to 250 GPa, J. Appl, Phys. 115, 043530 (2014).
[3]. Y. Partom, Modeling stress upturn at high strain rates for ductile materials, DYMAT 2018, Arcachon France (2018).
[4]. J.K. Roberts and A.R. Miller, Heat and thermodynamics vol. 4, Interscience publishers (1954).
[5]. D.J. Steinberg, Equation of state and strength parameters of selected materials, LLNL, UCRL-MA-106439 (1996).
[6]. Y. Partom, Change of shear modulus and yield stress with pressure and temperature, SCCM (2015).
[7]. Y. Partom, Overstress and flowstress approaches to dynamic plasticity, DYMAT 2015, Lugano Switzerland (2015).
[8]. M.I. Wilkins, Calculation of elastic-plastic flow, Calif. Univ. Livermore Radiation Lab. (1963).