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
For special copyright notice, see page 1362.
MODELLING OF LASER SPALL EXPERIMENTS ON ALUMINIUM
C. M. Robinson
AWE, Aldermaston, Reading, Berks. RG74PR. U.K.
Recently a series of shots have been fired on the AWE HELEN 2TW high power laser in order to study
the spall of aluminium at high strain rates (1). In the first shot a radiograph was taken which showed a
spall layer had formed. Further shots were fired and the free surface velocity of the aluminium was
obtained using interferometry. Several of these shots showed that spall had occurred. This paper
attempts to model these shots using the extended Johnson spall model (2,3). Previously determined
spall parameters (3), which model low strain rate plate impact experiments, are found not to model the
spall well, so new spall parameters are determined that match the laser results.
of spall parameters are determined which fit these
experiments.
INTRODUCTION
Recently a series of shots have been fired on the
AWE HELEN 2TW high power laser in order to
study the spall of aluminium at high strain rates
(~106 s"1) (1). The laser irradiates the aluminium,
producing a triangular shaped attenuating pressure
pulse which travels through the aluminium and is
reflected as a tensile wave when it reaches the free
surface. If the tension is sufficiently large the
aluminium will fail causing the formation of a spall
layer. This may be observed in the free surface
velocity which will show a sudden jump when the
shock wave arrives followed by a gradual fall as the
rarefaction wave travels back into the aluminium. If
spall occurs the free surface velocity will reach a
minimum and then oscillate around a constant
value. The oscillation, or ringing, is caused by
pressure waves in the spall layer.
This paper considers whether it is possible to
model these high strain rate experiments with the
extended Johnson spall model (2,3) with spall
parameters (3) which model low strain rate
(~104 s"1) plate impact experiments, such as those of
Kanel et al (4). The model is found not to model the
spall observed in the laser experiments, so a new set
EXPERIMENTAL DETAILS
The aluminium sample, SOO^im thick, was
directly irradiated using a nominal 200ps Gaussian
pulse and the laser energy was adjusted for each
experiment in order to apply different pressures. In
the first experiment an X-ray backlighting
technique was used to obtain a radiograph which
showed that spall had occurred (3). For the
remaining experiments a Michelson interferometer
was used to obtain the free surface velocity as a
function of time. The lower energy shots showed no
signs of spall, spall was observed for the higher
energy shots.
CALCULATED PRESSURE PROFILES
A one dimensional radiation-hydrodynamics
code including a laser light absorption model was
used to calculate the laser induced pressure. Since
the nominal laser energy was subject to a large error
the laser energy used in the calculation was chosen
to match the experimental shock arrival time at the
free surface. Absolute timings were not available
for all of the shots. However from the shots where
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absolute timings were available it was observed that
the applied pressure was proportional to the
maximum free surface velocity. Hence the applied
pressure profile for shots where no absolute timings
were available were obtained by scaling. An
example of a calculated pressure profile 10 jam
from the irradiated surface is shown in fig. 1.
THE SPALL MODEL
The spall was accounted for by using the Johnson
void growth model (2), with the addition of the
failure criterion described by Giles & Maw (3). The
model is described briefly here.
The voids in the material are described by a
single dependent variable, the distension a,
defined as
FIGURE ^Calculated Pressure Profile 10 [im From the
Irradiated Surface for a Shot With a Maximum Free Surface
Velocity of 1.29 km/s.
equal two thirds of the yield strength, but suggested
that it be treated as a free parameter in order to
account for localised temperature and strain
hardening effects. This approach is taken here. The
spall viscosity rj determines the rate at which voids
grow. The model is extended as suggested by Giles
& Maw (3) to include a failure distension ac . Once
the distension reaches this value the voids are
assumed to coalesce and the material fails, so for
distensions above ac the material can support no
stresses and the pressure is zero.
The material strength is reduced due to the effect
of voids as follows
Y
—, a<a,
~ —, a<ac
Y = a
, C
a
0, a > ac
0, a > a
where v is the specific volume of the material
excluding voids and v is the mean specific volume
of the material including voids. The equation of
state of the material is given by
- = 1 (L
a \a'
where p is the mean pressure in the material, E is
the internal energy and the function p(v,E) is the
equation of state of the material without voids. The
distension is described by the following differential
equation
where Y and G are respectively the yield strength
and shear modulus of the material with no voids
and Y and G are respectively the mean yield
strength and mean shear modulus of the material
with voids.
a
<? logI —&— I , a < a
when p— < -—
c
a
\a-l)
a
~ when
,
s ,log/ ——
a \, a<a
a. = 0
p- > - —
a
\a-\)
SPALL PARAMETERS
p\ — , E = 0 when a > a ,
Spall parameters for the extended Johnson spall
model have been determined for aluminium by
Giles & Maw (3), see table 1. These spall
parameters predict the low strain rate plate impact
spall experiments of Kanel et al (4).
where a 0 , as, rj and ac are all parameters. The
initial distension is a0 . as is the spall strength
parameter, which determines how large a tension is
required before the voids begin to grow.
Johnson (2) noted that theoretically as should
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RESULTS
The laser spall experiments have been modelled
with a one dimensional Lagrangian hydrocode. The
predicted free surface velocity profiles are
compared with the interferometer results for two
typical shots in figs. 2 and 3. In fig. 2 no spall is
observed and the model predicts no void growth
and matches the observed pullback. In fig. 3 spall is
observed, the calculation predicts the pull back but
does not predict the subsequent ringing. At late
times the free surface velocity is greater than the
calculation made with no spall indicating that void
growth has occurred, however the growth is not
large enough to cause ringing. This is the case for
all shots fired, the model correctly predicts the free
surface velocity when there is no spall (the model
predicts no void growth), but fails to predict spall
and ringing when it occurs (the model predicts void
growth that is insufficient to reach the failure
distension).
In order to improve the simulation the spall
parameters were adjusted to give a better match to
the timing and velocity of the first minimum in the
free surface velocity profile. The initial distension
and spall strength parameter were not changed.
Then the tension required for void growth to occur
remains unchanged and this ensures that the
simulation will still correctly predict the shots
where no spall occurred. The spall viscosity and
failure distension were adjusted, which allows the
voids to grow at a faster rate and allows failure to
occur at lower distensions. The new parameters are
shown in table 1. The simulation shown in fig. 2
remains unchanged. The simulation with the new
spall parameters is shown in fig. 3. It can be seen
that the new parameters give an improved spall
signature, the first minimum in the velocity is
correctly predicted and the period of the ringing is
roughly correct. The amplitude of oscillation is too
large however.
Tune(ns)
FIGURE 2.Model and Experimental Results. Time Axis of the
Experimental Results Shifted to Match the Shock Arrival Times.
No Spall is Observed.
1. No spall
2.
Giles & Maw (3)
parameters
3.
New parameters
4.
Experiment
FIGURE 3.Model and Experimental Results. Time Axis of the
Experimental Results Shifted to Match the Shock Arrival Times.
Spall is Observed.
spall layer thickness are respectively defined as
1 Aw
rarefaction strain rate = ———
2c0 Ar
spall strength = -—p 0 c 0 Aw
spall layer thickness = — c{T
where c0 is the bulk sound speed (5.41 km/s),
PQ is the density (2.70 g/cc) and cl is the
longitudinal sound speed (6.55 km/s). Aw is the
difference between the maximum free surface
velocity and the free surface velocity of the first
minimum, A/ is the difference between the time of
the first minimum in the free surface velocity
profile and the shock arrival time and T is the
The rarefaction strain rate, spall strength and
TABLE 1.
Parameter
CC0
as
V
<*c
Spall Parameters for Aluminium.
Value determined
Value deter mined
byGiles&Maw(3)
in this paper
1.0003
1.0003
O.lSGPa
O.lSGPa
5xlO'4 GPa us
5.4xlO'4 GPa \is
1.3
1.0045
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period of oscillation of the ringing. The observed
strain rates, spall strengths and spall layer
thicknesses are compared with the calculated values
in table 2. In most cases the calculated rarefaction
strain rates are within 10% of the observed values
and the calculated spall strengths are within 5% of
the observed values. The calculated spall layer
thickness are about 20% too small compared to the
observed values.
ultimate strength of the material has been reached,
as observed by Moshe et al (5)
ACKNOWLEDGMENTS
I would like to thank Peter Graham of AWE for
producing the pressure profile calculations.
REFERENCES
DISCUSSION
1. Evans, A. M.,
Rothman, S. D.,
Graham, P.,
Robinson, C, "Laser Driven Spallation Experiments." in
Shock Compression of Condensed Matter-2001.
submitted.
2. Johnson, J. N., J.Appl. Phys. 52(4), 2812-2825
(1981).
3. Giles, A. R., and Maw, J. R., "Modelling the
Temperature and Strain Rate Dependence of Spallation in
Metals," in Shock Compression of Condensed
Matter-1997, edited by S. C. Schmidt et al., AIP
Conference Proceedings 429, New York, 1998,
pp. 243-246.
4. Kanel, G. I., Razorenov, S. V., Utkin, A. V., and
Baumung, K., Shock Wave Profile Data, Publisher,
Scientific Association IVTAN of Russian Academy of
Siences, 1996, pp. 68-71.
5. Moshe, E., Eliezer, S., Henis, Z., Werdiger, M.,
Dekel, E., Horovitz, Y., Maman, S., Goldberg, I. B.,
Eliezer, D., Appl. Phys. Letters. 76(12), 1555-1557
(2000)
The spall parameters of Giles & Maw (3) and the
spall parameters derived here have identical initial
distensions and spall strength parameters and
similar spall viscosities. The only significant
difference between the two sets of parameters is the
failure distension. The spall parameters of Giles &
Maw (3) predict that in the low strain rate
experiments there is significant growth of voids
which relieves the tension in the aluminium before
the failure distension is reached. For the new set of
spall parameters the failure distension is very small.
Therefore almost no void growth occurs before the
material is assumed to have failed, and this failure
reduces the tension in the material. This suggests
that at high strain rates the spall of the material is
not caused by the growth of voids. Possibly the
TABLE 2.
Comparison of Experiment and Simulations for Shots Where Spall was Observed.
Observed Max.
Calculated Max.
Rarefaction Strain Rate
Spall Strength
Spall Thickness
Pressure 10 \im
Free Surface
(GPa)
(|im)
(I/Ms)
Velocity (km/s)a
From Irradiated
Observed Calculated Observed Calculated
Observed Calculated
Surface (GPa)
225.1
1.50
4.60
3.71
4.4
3.6
26
27
183.1
1.22
3.28
3.32
3.4
3.4
44
29
1.25
187.6
3.29
3.36
3.4
3.4
28
100.4
0.669
2.70
2.55
3.1
3.1
32
0.541
81.2
2.34
2.31
3.2
3.0
34
0.500b
75.0
2.17
2.20
3.1
3.0
52
37
90.3
0.602
2.68
2.43
3.0
3.1
44
35
98.7
0.658
2.85
2.54
3.3
3.1
35
98.1
0.654
2.84
2.52
3.2
3.1
47
34
91.8
0.612
2.22
2.45
2.9
3.1
41
33
0.727
109.1
2.65
2.82
3.1
3.2
33
42
a
A further six shots with maximum free surface velocities in the range 0.104 km/s-0.299 km/s did not spall.
b
Observed and calculated free surface velocity shown in fig. 3.
© British Crown Copyright 2001/MOD
Published with the permission of the Controller of Her Britannic Majesty's Stationery Office.
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