CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
THE EXPANDING SHELL TEST:
NUMERICAL SIMULATION OF THE EXPERIMENT
Francois Buy, Fabrice Llorca
CEA Valduc, DRMN, BP14, 21120 Is sur Tille, FRANCE
Abstract. The behavior of materials at very high strain rates is a broad field of investigation. One of
the major problem is to impose a well known velocity and stress loading. The expanding shell,
because of its ability to impose a sudden loading with no ulterior stress, has been retained to analyze
the mechanical behavior of several metals through the free flight of a deforming structure. We
perform several simulations on various metals in order to evaluate the strain rates and velocity in the
sample. A second step consists in interpreting experimental data obtained on these metals.
TABLE 1. Characteristics of the tests
INTRODUCTION
Test
identification
One of the major problem linked to the
interpretation of high velocity mechanical testing is
the purity of the loading conditions. The expanding
spherical shell device developed at the CEA [1],
because of the high symmetry of the sample is a
very interesting facility in order to investigate the
mechanical behavior. However, the interpretation of
the results may be difficult because of some
parasitical phenomena such as the spalling of the
sample or the difficulty to check some hypothesis
necessary to the interpretation of the test. Thanks to
the simulation of the experiment, we will focus in
this paper on some important points of the test.
Material
Explosive
diameter
Sphere
thickness
3 mm
2 mm
3 mm
GUI
TA2
Copper
40 mm
Tantalum
40 mm
TI3
TA6V4
36 mm
Constitutive equations and equation of state
We used the Zerilli-Armstrong constitutive
equations [3] to represent the mechanical behavior
of the three tested materials. According to the
crystalline structure, equation (1) or (2) is used.
fee
(1)
bcc and hep
(2)
NUMERICAL CONDITIONS
The coefficients of the model have been
optimized from compression tests in the range of
temperature 77 -700 K and 10~3 to 103 s"1 [4].
Geometry
The experimental device is presented in a
companion paper [2]. 11000 elements compose the
assemblage in the numerical simulation. The tests
are conducted with Hesione, a CEA hydrodynamic
code used in its Lagrangian version.
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The first stage impose the initial kinetic energy
of the shell. We have tried to impose an initial strain
rate of 104 s-1 i.e. an initial velocity of the shell near
250 m.s"1.
TABLE 2. ZA coefficients for the three materials
Co
Ci
C2
Q
(MPa)
(MPa)
(MPa)
(10' K ' )
(10 5 K ' )
Cu
69.58
-
936.6
1.76
5.77
Ta
14.7
882
-
3.64
22.5
273 0.468
TA6V
0
1869
1.77
7.3
848
C4
C5
n
(MPa)
(-)
TABLE 3. Maximum pressure level in the shell
0.525
The equations of state are implemented in the
Hesione code and give results close from
Steinberg's [5].
We can distinguish three stages in the test:
the transmission of the shock wave to the sample
the reflection of the shock wave through the
device (fig 2),
the free flight of the shell (fig 3).
Ips
3p
5ps
TA2
TI3
pressure level
6.5 GPa
7.8 GPa
3.4 GPa
The following figures (4 to 7) exhibit the
expected behavior of the three materials.
(fig 1),
-2&&
GUI
The sample has to cope with the level of the back
stress coming from the free surface (t=9 us). We
found that for the dimensions listed in table 1,
spalling effects are avoided in the shell. However in
all the simulations, the driver is submitted to a
tension level it cannot bear and thus collapses.
RESULTS
initial state
Test
7|is
FIGURE 1. Transmission of the shock wave through the
explosive and the driver.
20
40 60
Time (us)
80
100
FIGURE 4. Expanding velocity curves for the three tests.
1600
FIGURE 2. Reflection of the shock wave in the shell and
damage of the driver.
1400
G—OCu
1200
V—VTA6V4
1000
800
400
200
0
FIGURE 3. Deformation of the shell during the free flight stage
from20usto 100 jus.
0.2 0.4 0.6
Plastic strain (-)
FIGURE 5. Flow stress evolution at the pole.
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0.8
100000
eff
rr
= —p —
r
and
7*A
Two problems may stem out from the
interpretation of the experimental signal. The first
one is that the measured velocity is the speed at the
free surface and may differ from the average speed
through the thickness. The second one is that the
oscillations measured by the experimental signal of
the velocity r lead to great variations on the
acceleration r and thus of the effective stress (%.
We have to fit the experimental data with an
interpolation function in order to estimate the
effective stress.
10000
1000
100 4
0.2
0.4
0.6
0.8
Plastic strain (-)
FIGURE 6. Strain rate evolution at the pole.
200
In order to evaluate the error introduced with
both hypothesis, we have performed the same
analytical treatment on the simulation data as the
one performed with the experimental data.
The easier way is to fit the data with a linear
function. The treatment of third test (Ti3) is
presented below (fig. 8).
Particle velocity (m.s )
300
0.2
0.4
0.6
Acceleration (10 m.s )
-11.0
0.8
Plastic strain (-)
FIGURE 7. Temperature evolution at the pole.
These values are in good agreement with the
experimental data. The interpretation of the
mechanical tests is discussed in a companion paper
PI.
0 20 40 60 80 100
Position (m)
0
10 20 30 40
Flow stress (MPa)
4- -* mid thickness
O—©free surface
—— model results
INTERPRETATION OF THE RESULTS
0 5 1015202530
Time (u,s)
We only focus in this part on the plastic behavior
of the sample, i.e. the third stage of the experiment.
During this stage, the initial kinetic energy is
dissipated into plastic work so that the velocity of
the shell gradually decreases with time.
The measurement associated with the experiment
leads to the knowledge of the particle velocity r
at the free surface. From this signal,we can, under
the assumption of free radial expansion and of
infinitely small thickness of the shell, estimate the
equivalent stress and the equivalent strain [2].
0 5 10 15 20 25 30
Time (u,s)
FIGURE 8. Analysis of the expansion of the TA6V4 shell.
The values of velocity have been collected on the
free surface. Thanks to the volume conservation
hypothesis, we can calculate the velocity an the
position of a point located at the mid-thickness of
the sample.
329
and
CONCLUSION
The freely expanding spherical test proves to be
an interesting device for the characterization of
metals at very high strain rates.
In a first part, we proposed a setting which
allows high strain rates characterization without
spalling the sample. The results of our simulations
are in good agreement with the experimental results
whatever material we tested (copper, tantalum an
titanium alloy). This can be a global validation of
the calculations from the equation of state to the
plastic behavior. Zerilli-Armstrong model exhibits
interesting results for all three materials all the more
since these results are extrapolated from the
characterization domain in terms of strain rate.
In a second part we focus on the validation of the
analytical treatment of the experimental data. When
the deformation and stress levels are homogeneous
in the sample, the analytical treatment of the
experimental data is validated. In order to improve
the diagnostic of the test, we have to transpose the
treatment at the mid-thickness of the sample. In all
cases, the crucial point is the derivation of the
velocity signal which goes through an interpolation
stage.
(w£ and./? stand for mid-tickness and free surface).
The difference is tiny but leads to a significant
change in the determination of the acceleration
(fig. 8.c) and thus of the stress (fig 8.d). We can
notice a very good fit between the data of flow
stress determined by the analytical treatment of the
mid-thickness velocity curve and the data of flow
stress calculated by the Hesione code. The error
caused by a direct analysis of the free surface
velocity curve is about 2.5%.
However a linear fit may not be sufficient as
shown in the next simulation conducted with a
tantalum shell (Ta2).
Particle velocity (m.s )
300
Acceleration (10 m.s"")
-1
Flow stress (MPa)
800
600
- model resu ts
• mid thickness
«freesurfac<!
200
0 20 40 6Q 80.1,00
Acceleration(10 m.s")
-1 pi i • i i i i
0
0 20 40 60 80 100
Flow stress (MPa)
800
600
400
REFERENCES
200
0 20 40 60 80 100
0 20 40 60 80 100
Time ((is)
0
0 20 40 60 80 100
[ 1 ] A. Juanicotena, Thesis of the University of Metz, France,
1998.
[2] F. Llorca, F. Buy, The expanding shell test:: its contribution
to the modeling of elastoplastic behavior at high strain rates,
this conference.
[3] F. J. Zerilli, R. W. Armstrong, Dislocation—mechanicsbased constitutive relations for material dynamic
calculations, J. Appl. Phys. 61 (5), (1987), pp 1816-1825 .
[4] J. Farre , F. Llorca Private communication.
[5] D. J. Steinberg, Equation of state and strength properties of
selected metals, UCRL-MA-106439, (1996).
FIGURE 9. Analysis of the expansion of the tantalum shell.
Figures 9.d. and 9.f. come from a linear
interpolation of the velocity signal whereas figures
9.c. and 9.e. come from a quadratic interpolation.
We can notice an excellent correlation between the
model and the analysis in the case of a reinterpreted
signal at mid-thickness (fig 9.e) with a quadratic fit
when all the other analysis may give disappointing
results specially when we use a linear interpolation.
330