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 CONTRIBUTION OF THE EXPANDING SHELL TEST TO
THE MODELING OF ELASTOPLATICITY AT HIGH STRAIN
RATES
Fabrice Llorca, Francois Buy
CEA/Valduc, DRMN, 21120 Is/Tille, France.
Abstract. The expanding shell test allows to load a material in the domain of high strain levels while
strain rate is about 10Y1. This test submits an hemisphere to a radial expanding free flight, using a
pyrotechnic device. The experiment (experimental apparatus, measurements...) is described with the
difficulties encountered for the interpretation of the experimental data. Under some assumptions, the
. numerical transformation of radial velocities gives indications about the evolution of the strain, stress,
strain rate and temperature rise, this last one being related to plastic work. We show how it is possible
to associate both analytical and numerical approaches. Numerical simulation of the test is presented in
a companion paper (see [BuyOl]). Results obtained for copper, tantalum and TA6V4 are presented.
The contribution of this test to the modeling of elastoplastic behavior is discussed and further works
are proposed.
INTRODUCTION
Many problems setting in the design of some
pyrotechnic devices are connected to the mechanical
behavior of materials in the area of large strains and
high strain rates. When the loading is highly
dynamic, the prediction of dynamic structural
response by means of numerical calculations needs
optimized constitutive relations to describe plastic
behavior of metallic materials. These models are
determined from mechanical tests conducted in a
wide range of thermomechanical conditions. Usually,
tests are performed on classical hydraulically driven
machines in the domain of low strain rates whereas
for higher levels, the well - established Split
Hopkinson Bar technique is often used. In order to
extend the domain of strain rate and strain, free
expanding structure tests are particularly well suited
to model evaluation by virtue of their inherently
homogeneous stress state. In this work, we propose a
review of the state of the art concerning an
expanding spherical shell test developed at CEA
since 1997. In this paper, we focus on the
experimental aspects of the research and we propose
results about three materials : copper, tantalum and
TA6V4. The design by numerical simulation is
presented in a companion paper [1].
GOVERNING EQUATIONS
The test is derived from the classical expanding ring
test: it consists to impose a free dynamic expansion
to a thin object. Kinetic energy is then consumed in
plastic work so that expanding velocity decreases.
Through mathematical operations, time velocity data
give the stress - strain - strain rate state within the
sample. Under the assumptions of the free radial
expansion, thin shell spherical geometry, the
fundamental dynamic relation simply resolved itself
and leads to the following equations :
a -
rr , s = 2 li
Th material is subjected to an equibiaxial tension
state of stress reduced to a scalar value according to
the Von Mises criteria.
EXPERIMENTAL DEVICE
TESTS CONFIGURATION
The principle of the experimental procedure has
already been presented (see [2,3]). The free radial
expansion is imposed by means of an explosive setup. A shock wave is generated by the detonation of a
high explosive and transmitted outward through a
steel driver. Therefore, the shock wave propagates
within the sample which expands at high speed, by
virtue of its own inertia, i.e. without the action of any
external forces. Under this motion, the thin sphere
undergoes large plastic deformations as its initial
kinetic energy is dissipated into plastic work.
Technical difficulties associated with explosively
driven devices in spherical geometry (initiation,
complete sphere machining processes) have led us to
restrict the test to the expansion of hemispherical
shells. The experimental set up is constructed as
shown in figure 1.
Three materials have been chosen for this study in
order to put in evidence the capabilities of the test to
reveal plastic behavior. There were copper, tantalum
and TA6V4.
TABLE 1. Characteristics of the tests.
Material Explosive (mm)
Copper
40
Tantalum
40
TA6V4
36
Sphere thickness (mm)
3
2
3
In all calculations, initial shock pressure levels have
been found lower than 10 GPa. It is well known that
application of a shock wave can lead to large
modifications of the mechanical properties. These
effects
are material dependent. In first
approximation, we choose to neglect these
consequences, considering the shock wave
amplitude to low to induced large microstructural
modifications.
INTERFEROMETRIC MEASUREMENTS
Interferometers are investigated by means of a semi
automatic procedure using numerization and
imaging analysis in order to obtain optimized radial
velocity curves. Figure 2 is a plot of the radial
velocity versus times for the three material tests.
They show the measurements realized at the
different optical heads locations. The initial velocity
is of the same order for the three tests, between 250
m/s and 300 m/s. The decreasing of the radial
velocity is clearly observed, putting in evidence the
plastic deformation of the materials under the
equibiaxial tension. The deceleration is higher for
TA6V4 than for copper and tantalum. The DLI
measurements at locations 30° and 45° put in
evidence the effects of the 2D geometry (non
spherical expansion all over the sphere). The elastic
rebound observed for TA6V4 is due to
bidimensional effects which propagate along the
object. Observation of elastic oscillation depends on
FIGURE 1. Cross section of the experimental set up.
The driver material is XC38 steel with a 330 MPa
yield strength. The driver size and the explosive
diameter are chosen to launch the sphere at an initial
strain rate of 104s"1. Dimensions of the principal
elements are : explosive diameter from 38 to 44 mm,
sphere inner diameter 100 mm and thickness from
1.5 to 3 mm. The radial sphere velocity is measured
with the Doppler Laser Interferometry technique
(DLI) for different locations (called 0°, 30° 45°) in
order to record bidimensionnal effects. Furthermore,
expansion of the material sample is observed by
means of ultra high speed cameras.
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the appearance of damage and fracture phenomena.
Flat curves are observed at the end of the
interferograms for copper and tantalum, showing a
typical fragment behavior : these two materials have
reached their fracture limits and the spheres are then
composed of undeformable pieces which move at
constant flying velocity.
FIGURE 2. Expanding velocity curves for the three
tests.
ULTRA HIGH SPEED CINEMATOGRAPHY
Figure 3, 4 and 5 present three pictures obtained by
ultra high speed cinematography for the different
materials. The film shows the clean and apparent
axisymetrical expansion of the hemispheres until the
time when first damages appear for copper and
tantalum.
FIGURE 5. Expansion of the sphere for TA6V4.
DATA PROCESSING AND ANALYSIS
For all the configurations, the application of
dynamic equations presented in part 2 of this paper
at the 0° location (called the pole) is justified
because due to the axisymetrical geometry of the
test, the state of stress at this point staying always
equibiaxial. Transformation of the experimental
velocity curves into stress-strain curves requires
integrating
and
differentiating
operations.
Considerable difficulty is usually encountered with
the single differentiation of raw data. Various
methods are then applied in order to reduce scatter.
In this study, we simply choose to fit the velocity
time data within the usable data window to a linear
or a quadratic function. Acceleration is then
obtained by differentiating the analytical formula
and is consequently a constant or a linear function
of time, depending of the tested materials. Figures 6
and 7 present plots of the effective stress and strain
rate. The high flow stress of TA6V4 stops rapidly
the test while more ductile materials like tantalum
and copper reach high levels of homogeneous
plastic strains levels (more than 0.5). Strain rate is
always less than 10000 /s. Concerning TA6V4, the
stress - strain curve shows a classical behavior, with
the increasing of flow stress with plastic
deformation without any saturation effects.
FIGURE 3. Expansion of the sphere for copper.
^^
FIGURE 4. Expansion of the sphere for tantalum.
FIGURE 6. Stress strain curves.
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of elastoplastic behavior of various materials.
Nevertheless, the aim of this work is to validate
elastoplastic models. The correlation could not be
immediate. The reasons are the next . Elastoplastic
models are generally optimized under compression
tests and this experiment subjects the materials to
equibiaxial tension loadings, various materials are
characterized by tension -compression asymetries of
behavior. Even if shock effects have been neglected
in this work, it is well known (see [4-5] for copper)
that the post shock elastoplastic behavior is always
modified and the consideration of a simple residual
plastic strain is not realistic for a fine analysis. The
effect of manufacturing operations have also to be
taken into account because of their strong hardening
effect on materials behavior [6]. For all these
reasons, the interpretation of this test with classical
elastoplastic models is very risked. Further works
are engaged at CEA in order to improve our
capabilities for comparing expanding sphere results
with optimized elastoplastic models.
FIGURE 7. Strain rate as a function of strain.
For TA6V4, the experiment does not allow to extend
the classical range of study in terms of plastic strains
(limited to 0,2). The sphere is not fractured so it
could be possible to modify the experimental
apparatus (increasing of the explosive diameter) in
order to increase the shock pressure and
consequently the homogenous plastic deformation
range. The modification needs to limit spalling
effects by playing on the sphere thickness. The
tantalum results are more interesting because they
put in evidence the thermal softening effect on the
dynamic behavior of the material. This phenomena is
the result of the coupling of saturation and heat
effects. Saturation stress is reached around 0,3 in
plastic deformation. Large strains are reached during
this experiment and the increase of temperature is
about 170 K at 0,6 strain level (this rising is due to
the adiabatic nature of the loading). The heat effect
leads to the dramatic fall of stress at high strains
levels. This phenomena is very difficult to observe
under classical compression tests (Hopkinson bares)
because of the limited levels of homogeneous
strains. The conclusions are the same for copper but
saturation is reached at more than 0,5. This
mechanism is the same as tantalum but is simply
rejected at higher strain levels. This is typical
behavior of fee materials compared to bcc.
ACKNOWLEDGMENTS
The authors gratefully acknowledge MM Patrice
ANTOINE, Jacques MATHIAS, Ph MARTINUZZI
for their contributions to the conducting of the
expanding tests.
REFERENCES
[1] F. Buy, F. Llorca this conference.
[2] F. Llorca, A. Juanicotena, C. Dambrun, CP505,
Shock Compression of Condensed Matter, pp 455458, 1999.
[3] A. Juanicotena, Thesis of the University ofMetz,
France, 1998.
[4] G.T. Graylll, P. S. Follansbee, C.E. Frantz
Materials Science and Engineering, Alll, pp 916,1989.
[5] F. Llorca, J. Farre, F. Buy, this conference
[6] F. Llorca, J. Farre, J. Physique Vol IV, 10, pp
93-98, 2000.
CONCLUSION
Based on the results presented in this paper, it is
clearly shown that the test is efficient for comparison
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