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
EXPERIMENT AND THEORY FOR THE CHARACTERIZATION
OF POROUS MATERIALS
A.D. Resnyansky1, N.K. Bourne2, and J.C.F. Millett2
2
Weapons Systems Division, AMRL, DSTO, PO Box 1500, Salisbury SA 5108, Australia
Royal Military College of Science, Cranfield University, Shrivenham, Swindon, SN6 8LA, UK
Abstract. Two basic experimental methods are generally used for constructing the Hugoniot of
porous materials. The first allows measurement of the shock versus the particle velocity in the
medium whilst the second emphasises measurement of the profiles with embedded gauges. The
first technique uses the jump conditions across the shock to deduce the pressure and density behind
the front. This paper demonstrates that the state behind the shock observed experimentally with the
embedded gauges may deviate significantly from that deduced from the jump conditions.
Theoretical analysis using a strain-rate sensitive model of porous material shows the range of shock
loads where a constitutive description is essential and where the jump conditions may be used.
INTRODUCTION
front. A representative of a porous material
considered in the present paper is sand with a size
distribution from 100 to 300 um We observed the
shock behaviour of the sand experimentally with
embedded manganin gauges placed in surrounding
anvils. Analysing the data, deviations in pressure
were observed behind the shock front from that
calculated from the jump conditions. Using a strainrate sensitive model for the non-equilibrium state of
porous materials, we deduce conditions when the
D-U data may provide a reasonable assessment of
pressure behind the shock. The theoretical analysis
demonstrates that two sources may generate
misinterpretation of the Hugoniot of the porous
material at moderate and low velocities behind the
shock. These are shortening of the pins by a
precursor and deviation of the pressure behind
shock due to the constitutive behaviour of the
loading wave.
In conclusion one may say that in a low range of
loadings of the sand, the embedded pressure gauges
and constitutive description alone may provide a
reliable characterisation of the porous material.
Experimental shock-wave physics extensively
employs the jump conditions based on mass and
momentum conservation for the deduction of the
pressure and density behind a shock front. This
analysis is very attractive because it only requires
knowledge of the shock front velocity D versus the
particle velocity U obtained in experiment. These
data could be obtained in relatively simple set-ups
with pins or shock-sensitive gauges and knowledge
of the Hugoniots of standard materials. There are
compendia of data for various materials based on
this approach (see for example 2). Strictly speaking
the theory is only valid for an ideal jump connecting
the states in front of and behind the shock. In reality
this wavefront is a zone of complex loading even
for conventional materials. However the jump
conditions are also successfully used for substances
whose shock behaviour is far from ideal. The
approach is also extensively used for porous
materials.
The present paper is a study to understand where
the ideal approach and the D-U data may be used
for the description of the state behind the shock
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aluminium alloy (6082-T6) discs and were mounted
onto a polycarbonate sabot with a relieved front
surface in order that the rear of the flyer plate
remained unconfined.
EXPERIMENTAL
Plate impact experiments were performed on 50
and 75 mm bore single stage-gas guns (1). The
target was constructed by placing a cover plate with
an embedded manganin sensor ahead of a ring filled
with a known density sand target. To the rear of the
sand was a further anvil with an embedded sensor
so that the stress rung up within the target during
loading. By this means the stress in the target, and
additionally the time taken to cross the sand, was
recorded for each increment of the stress. The flyer
was 10 mm thick and impacted a 2 mm thick cover
plate on a 4 mm thick sand target. There was an
11 mm thick anvil to the rear with gauges 1 mm
from impact and 1 mm into the anvil. The whole
assembly was built from either copper or an
aluminium alloy of known Hugoniot (2). A
schematic X-t diagram illustrating the geometry is
shown in Fig. 1.
MODELLING
A model of a porous material (4) is employed for
the description of the behaviour of the sand.
Kinetics of the compression are governed by a
parameter p* (reference density), which is the
density of porous material after unloading from a
given state. The system of equations of the model
includes the conservation laws, a constitutive
equation for the shear stress of compacted material,
and the kinetics for the parameter p* (the porosity
kinetics):
ln(p/p*)
dt
here TV is a function of state, and p is current
density. Functional dependencies within the
constitutive equations are fit from the yield stress
data for the compacted material at a variety of strain
rates and from the pressure-density curves for the
porous material at fixed strain rates. For the latter
the Hugoniot may be used if the strain-rate within
the shock wave is supposedly known.
-10
0
5
10
15
/
o>
Distance (mm)
GPa
FIGURE 1. A schematic X-t diagram showing the geometry. The
sand layer is of thickness 4 mm with a 2 mm cover plate. The
vertical dotted lines represent the positions of manganin stress
sensors 1 mm from impact and 1 mm into the anvil.
8
I
4
The gauges used were MicroMeasurements
manganin gauges (LM-SS-125CH-048) and the
calibration data of Rosenberg et al (3) were used in
reducing the voltage data collected. The signals
were recorded using a fast (1 GS s"1) digital storage
oscilloscope and transferred onto a micro-computer
for data reduction. Impact velocity was measured to
an accuracy of 0.5% using a sequential pin-shorting
method and tilt was adjusted to be less than 1 mrad
by means of an adjustable specimen mount.
Impactor plates were made from lapped copper and
;
3 */
—
ty -4
^ -*"1.6 2.2 2.8
/
p,g/cm*
FIGURE 2. Calculated pressure-density curves for sand. Curves
1, 2, and 4 correspond to a strain-rate of 104s"1, curve 2 - to
strain rate of 10~2 s"1. Initial density for curve 1 is 1.51 g cm-3, for
curves 2 and 3, 1.64 g cm"3, and for curve 4, 2.65 g cm"3. Points
are low strain-rate data (5).
Several calculated pressure-density curves are
shown in Fig. 2. For initial density 1.64 g/cm3 the
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slow and comes onto the gauge later. Higher
pressure recorded in the tests (curve El over Cl in
Fig. 4 after 5 jus) is likely of the same nature as has
been mentioned above.
Fig. 5 does not demonstrate any unusual
behaviour. The sand layer is apparently fully
compacted after the first passage of the shock S W3
through the material.
calculated curve correlates with the corresponding
one published in (5).
The model has the potential to describe the
anomalous behaviour of a porous material with
relation to the pressure increase with the density
drop and release of thermal energy. However, the
porosity kinetics for sand was designed in a
traditional way for the present case with a slow
temperature influence in the Arrenius form.
ANALYSIS OF THE RESULTS
Result of the experimental records and
corresponding calculations are shown in Figs 3-5.
2.8
4.2
t,jusec
FIGURE 4. Experimental records from the gauge Gl (curve El)
and calculated curves Cl and C2 (Cl is correspondent to the
curve El). The copper assembly with a copper flyer plate at a
velocity of 509ms"1.
t,jusec
FIGURE 3. Experimental recordings by the gauges Gl and G2
(curves El and E2, respectively) and corresponding calculated
curves (Cl and C2). The aluminium assembly was impacted with
an aluminium flyer plate at a velocity of 493 m s"1.
The sound velocity in the aluminium alloy is 6.4
kms" 1 . Therefore, as can be seen from Fig. 1, the
rarefaction wave RW3 comes onto the first gauge
zone before the reflected shock SW4 increases the
signal level of the gauge. This results in a pressure
drop near 3.6 u,s (curves El and Cl in Fig. 3). Later
on an increase in pressure for El is likely to be
caused by thermal energy release, which is not
taken into account with the present kinetics.
For the copper assembly (Figs. 4 and 5) shock
and rarefaction wave-speeds (CL = 4.76 mm jis"1)
are relatively slow compared with the aluminium so
that the pressure at the first gauge is at a nearly
constant level until the reflected shock SW4 comes
onto the gauge. The rarefaction wave RW3 is too
0
FIGURE 5. Experimental recordings by the gauges Gl and G2
(curves El and E2, respectively) and corresponding calculated
curves (Cl and C2). The copper assembly with a copper flyer
plate impact at a velocity of 781 m s"1.
Comparison of the data in Figs. 3 and 4 allows
one to conclude that the states at the metal-sand
interface are close for both tests. Taking into
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Due to the constitutive behaviour of sand in the
present model the shock velocity is not a constant
and attenuates slowly during its propagation to a
stationary value. The constitutive behaviour of a
material, which is associated with noninstantaneous rearrangement within the shock front,
has an important typical feature: shock wave may
be preceded by a precursor. A numerical example is
shown in Fig. 7. The precursor amplitude is
relatively small compared with the shock amplitude
of more than 1 GPa but this is still enough to cause
the shortening of pins in an experimental
determination of the shock wave velocity.
account the difference in the shock velocities in the
aluminium alloy and the copper, the time of travel
for the first shock over the porous sand is
approximately 2.85 jis that corresponds to a shock
velocity of order 1.4 km s"1 with the particle
velocity U behind the shock front slightly lower
than 0.5 km s"1. Calculation of the pressure behind
the front according to the momentum conservation
law across the front P=po U D gives a value close to
1 GPa that is slightly lower than that observed in
the tests.
•
s0
CONCLUSIONS
u
6
Experiments on the shock wave compression of
sand have demonstrated that this material requires a
well-developed constitutive model. The shock jump
conditions may result in inaccuracies associated
with errors in determination of the shock wave
velocity especially in the range of low and moderate
amplitudes of loading. The manganin gauge
technique is necessary to record the pressure-time
histories as well as the arrival dependence in such a
complex material as sand. Constitutive modelling is
necessary for a description of the behaviour of the
material and the full scale stress measurement
technique should be employed to design both
equations of state and constitutive equations for the
models.
^ o
Q
2
0.1
0.3
U, km/s
FIGURE 6. Calculated DC-DH deviations for the shock wave
entering the sample (1) and stationary shock wave (2).
0.8
REFERENCES
1. Bourne, N.K. and Stevens, G.S., Rev. ScL Instrum.
72,2214-2218(2001).
2. Marsh, S.P., LASL Shock Hugoniot Data, University of
California Press, Berkeley, California, 1980.
3. Rosenberg, Z., Yaziv, D. and Partom, Y., J. Appl
Phys. 51, 3702-3705 (1980).
4. Romenskii, E.I., J. of Appl Mech. and Techn. Physics
(translfrom Zh. Prikl Mekh. Tekh. Fiz.), no. 5, 145-149
(1988).
5. Morishita, M., Ando, T., Yamaguchi, H., "Centrifugal
and Numerical Simulations of a Projectile Penetrating
Sand", in Proc. 9th Int. Symp. on Interaction of the Effects
of Munitions with Structures, Berlin-Strausberg, May 3-7,
1999, pp. 449-456.
0
0
Distance (mm)
5
FIGURE 7 Numerical calculation of a shock wave in the porous
material (U = 500 m s"1); fragments of the stress profiles at
ascending moments of time (1-5).
This fact was a subject of theoretical study. We
calculated the shock wave propagation problem
over the sand for a range of loads. The calculated
shock velocity DC was compared with the shock
velocity £>H found from the conservation relation
DH=p/(p0 U). Their deviation Dev=(Dn-Dc)/Dc was
plot for a number of particle velocities U in Fig. 6.
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