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
Al AND Cu DYNAMIC STRENGTH AT A STRAIN RATE OF 5-l(r8 s-1
M. Werdiger, S. Eliezer, E. Moshe, Z. Henis, E Dekel, Y. Horovitz and B. Arad
Plasma Physics Department, Soreq NRC, Yavne, 81800, Israel.
Abstract. Strain rates in the range (1.5-5)-10s s"1 were achieved in Al and Cu. A pico-second laser
(20-100 ps) beam that was focused on thin foils, created shock waves in them. An optically recording
velocity interferometer system (ORVIS) was used to measure the free-surface-velocity history by a
nano-second laser. The spall strengths that were reached in the experiments are very close to the
ultimate strength given by the EOS.
INTRODUCTION
metals. Mechanical properties of solids are
controlled by lattice imperfections and dislocations,
which weaken the materials. The usual spall process
occurs due to the gathering and grouping of those
imperfections.
The hypothesis is, that at high strain rates (larger
then 107 s"1), the process mentioned above does not
have enough time to cause a spall, so that the atomic
process remains the main spall-producing process.
Therefore, under these conditions, the spall strength
of metals should closely match the theoretically
calculated values from the EOS.
When a shock wave is reflected from the rear
surface of a material, dynamic tensile stress is
developed. If this stress exceeds the strength of the
material, spallation occurs in a plain parallel to the
rear surface (1-4).
The experimental spall strength of metals at
strain rates smaller than 106 s"1, differs from the
theoretical value obtained by the equation of state,
by as much as an order of magnitude. These
differences are explained by the imperfection of the
ImJ
10ns
X=0.532 urn
Grating
FIGURE 1. A schematic description of the experimental setup
583
accelerated. The interference pattern was imaged by
a cylindrical lens, to a set of bright points on the
entrance slit of a streak camera. The interference
pattern was analyzed with an image processing
system, including a cooled charged-coupled device
camera, a frame grabber and a PC. The time
resolution of the experiments was 10 ps.
Before performing each experiment, the
oscillator of the main beam induced a hole in an
aluminized Mylar foil that was mounted side by side
with the target. Then the main and the diagnostic
beams were aligned to coincide. Next, the target
moved to the hole position and the experiment was
executed. The trigger for the streak camera was
taken from an avalanche photodiode that was
located at the beginning of the main beam.
A typical fringe pattern representing the free
surface velocity in an experiment with a 100 ps, 31
mJ pulse focused onto a 10 urn thick aluminum
target, is shown in Fig. 2. Time increases from left
to right. The full time scale of this figure is 1.14 ns
and the fringe constant is 0.872 km/s. The solid line
denotes the free-surface velocity.
Laser generated shock wave experiments,
causing strain rates up to -5-108 s"1 on targets of
aluminum and copper, have been performed. The
results confirmed the hypothesis: at these strain
rates, the experimental values indeed matched the
theoretical ones.
THE EXPERIMENTS
The experimental setup is shown in fig. 1. A
Nd:YAG oscillator-amplifier system yielding 20 ps
pulses, with an output energy of 600 UJ per pulse,
was used. This pulse was split, using a half wave
plate and a polarizer. Part of the beam was left as it
was, while the other part was stretched to duration
of 100 ps, using a pair of gold gratings (1740
grooves/mm). Each partial beam was farther
amplified, using a double pass Nd:glass amplifier,
to yield final energies of 10-50 mJ per beam. This
arrangement enabled us to switch alternately
between 20 ps and 100 ps pulses. The laser beam
was focused with a 50 cm focal length lens, to a
spot diameter in the range of 60-400 |im. The laser
intensity varied between 5-1011 and 3-1013 W/cm2.
This main beam induced a shock wave in the
target. The experiments were performed with Al and
Cu targets 1-10 um thick. The pressure of the shock
wave reaching the rear surface of the target was
from tens up to hundreds of kilobars in aluminum,
and up to 1.3 Mbar in copper.
The velocity time evolution of the shocked
material free surface was measured with an optically
recording velocity interferometer system - ORVIS
(5,6). The diagnostic system included a Continuum
NY-80 oscillator, operating at 532nm, with pulse
duration of 10 ns and energy of ImJ. This laser was
focused to a spot diameter of 40 urn on the rear
surface of the shocked target. The oscillator was
synchronized with the main beam so that the two
beams hit the target simultaneously from both sides.
The light reflected from the moving free surface
underwent a Doppler shift proportional to the
velocity of the free surface. The light was collected,
collimated and directed into a Michelson
interferometer. The light leaving the interferometer
produced an interference pattern of parallel fringes,
which shifted when the rear surface of the target was
FIGURE 2. The interference pattern and the free surface
velocity deduced from it by the ORVIS in an experiment with a
10 |im thick aluminum foil, irradiated by a 100 ps, 31 mJ pulse.
Time increases from left to right. The full time scale of fig. 1 is
1.14ns and the fringe constant is 0.872 km/s.
RESULTS
The velocity history of the moving rear surface ufs(t)
was determined from the displacement y(t) of the
interference pattern as a function of time t (5,6,7):
584
by the averaging effect over the various
imperfections, e.g. voids, cracks etc. For aluminum,
this logarithmic dependence seems to hold up to a
strain rate of 108 s"1. From there on there is a quick
rise toward the theoretical value. At a strain rate of
107 s"1 the averaging seems to become less affective,
and to depend strongly on the local kind and density
of the imperfections.
y(t)
U f s (t) =
4L e (n --)(! + 8)
n
(l)
y(t)
X is the diagnostic laser wavelength, Le is the length
of the etalon, n is its index of refraction, c is the
speed of light in vacuum, 8 is a correction term due
to the wavelength dependence of the refractive
index of the etalon material (7), d is the fringe
spacing and v0 is termed the fringe constant.
The spall pressure or the material strength is
determined from the measurement of the free
surface velocity time history. In the acoustic
approximation the spall strength is (8,9,10):
EOS&i«S
100-
•
S
8
^
60-
0
a
2
m
40-
50
'S
cx
U
0
rain)
1
T
ai
^
a
oa
(2)
a
'Ifc
20-
CO
*spall ~~
present experiments with 100 ps laser
present experiments with 20 ps laser
Fortovet. al. [ref. 11]
Moshe et. al. [ref. 14]
T
103
T »
104
YT
10s
T TTT
o
D
107
106
108
10s
Strain rate [sec" ]
c0 is the sound velocity, p0 is the initial density of
the target, umax is the peak velocity of the free
surface, and umin is the first minimum in the
free-surface velocity profile.
The strain rate was calculated by:
£= •
Au
1
At 2c
FIGURE 3. The spall strength as a function of the strain rate in
aluminum.
Cu
EOS
(3)
Au is the velocity difference between the first
maximum and the first minimum in the free surface
velocity profile and At is the difference between the
corresponding times.
The spall strength as a function of the strain rate
is shown in Fig. 3 for aluminum and in Fig. 4 for
copper, together with the ultimate strength predicted
by the EOS (see the next section). Additional data
measured in experiments using impact of a
projectile on a target (11,12) and in experiments of
nanosecond laser induced shock waves (13,14) is
also shown in figs. 3 and 4.
A straight line can represent quite accurately the
spall strength values in the figures, up to a strain
rate of about 107 s"1. An almost linear relationship
on a semi-log representation means a weak
dependence. This weak dependence may be caused
a
10*
10s
JE,
o
106
°i
107
108
109
Strain rate [sec ]
FIGURE 4. The spall strength as a function of the strain rate in
copper.
Probably, this is also the reason for the large
scattering of the data in that stain rate interval.
Unfortunately, for copper, we do not have
enough experimental data to follow the exact
pattern. However, even from two experimental
points, it is clear that above the strain rate value of
108 s"1, the experimental spall strength value
585
approaches the ultimate strength predicted by the
EOS.
REFERENCES
DISCUSSION
1. Grady D.E., J. Mech. Phys, Solids 36, 353-384
(1988).
2. Bushman A.V., Kanel G.L, Ni A.L., Fortov V.E.,
Intense Dynamic Loading of Condensed Matter
Taylor&Francis, London, 1993, Ch. 5,8.
3. Grady D.E., Kipp M.E., "Dynamic Fracture and
Fragmentation" in High Pressure Shock Compression
of Solids, edited by J.R. Asay and M. Shahinpoor ,
Springer-Verlag, New York, 1993, Ch.8.
4. Davison L., Grady D.E., Shahimpoor M. (Eds.), High
Pressure Shock Compression of Solids II, Dynamic
Fracture and Fragmentation, Springer-Verlag, New
York, 1996.
5. Bloomquist D.D., Sheffield A.A., J. Appl. Phys, 54,
1717-1722(1983).
6. Moshe E., Dekel E., Henis Z., Eliezer S., Appl. Phys.
Lett. 69, 1379- 1381 (1996).
7. Barker L.M., Schuler K.W., J. Appl. Phys. 45,
3692-3693 (1974).
8. Kanel G.L, Razorenov S.V., Utkin A.V., Fortov V.E.,
Baumung K., Karow K.U., Rush D., Licht V., J.
Appl. Phys. 74, 7162-7165 (1993).
9. Kanel G.L, Razorenov S.V., Utkin A.V., in
High-Pressure Shock Compression of Solids II, edited
by Davison L., Grady D.E., Shahinpoor M.,
Springer-Verlag, New York, 1996, ch. 1.
10.Dekel E., Eliezer S., Henis Z., Moshe E., Ludmirsky
A., Goldberg I.B., J. Appl. Phys. 84, 4851-4858
(1998).
11.Fortov V.E., Kostin V.V., Eliezer S., J. Appl. Phys.
70,4524-4531 (1991).
12.Kanel G.L, Razorenov S.V., Bogatch A, Utkin A.V.,
Fortov V.E., Grady D.E., J. Appl. Phys. 79,
8310-8317(1996).
13.Paisley D.L., Warmes R.H., Kopp R.A., "Laser
driven flat plate impact to 100 Gpa with
sub-nanosecond pulse duration and resolution for
material property studies", in Shock waves in
Condensed Matter, edited by Schmidt S.C., Dick
R.D., Forbes J.W., Tasker D.G., Elsevier Science,
New York, 1992, pp. 825-828.
14.Moshe E., Eliezer S., Dekel E., Ludmirsky A., Henis
Z., Werdiger M., Goldberg I.B., Eliaz N., Eliezer D.,
J. Appl. Phys. 83, 4004-4011 (1998).
15.Moshe E., Eliezer S., Henis Z., Werdiger M., Dekel
E., Horowitz Y., Maman S., Goldberg I.B., Eliezer
D.,Appl. Phys. Lett. 76, 1555-1557-4011 (2000).
The shock pressure near the rear surface of the
target was estimated from the peak velocity of the
free surface, umax. The particle velocity is given by:
=
umax/2
(4)
The shock velocity and the pressure were
determined by Hugoniot relations and the EOS. The
EOS relation is:
u s = c0
(5)
where us is the shock velocity and a is a constant.
The pressure is given by the Hugoniot relation
P = p0usup
(6)
In the experiments reported here, the shock
pressures near the rear surface were up to 430 kbar
in aluminum and up to 1300 kbar in copper. The
corresponding residual temperatures (on the release
isentrope) as calculated by our EOS (14,15) were in
the range (300-660) K in aluminum and (310-1340)
K in copper. For these temperatures, the theoretical
spall strength by the EOS is in the range (101-108)
kbar in aluminum and (172-211) kbar in copper.
These values are shown in figs. 3 and 4 as the EOS
spall strength, together with the spall pressures (for
aluminum and copper), as function of the strain rate.
In the present experiments the highest measured
spall strength is (80±10) kbar in aluminum and
(156120) kbar in copper (the accuracy of the free
surface velocity is 5%). These spall strengths
measured in the experiments reported here, are very
close to the ultimate strength given by the EOS. At
strain rates larger than 108 s"1 the spall strength
increases sharply towards its EOS value. The
present experiments suggest that the ultimate
strength will be achieved at a strain rate > 5-108 s"1.
The generation of such high strain rates is possible
due to the short loading time used in these
experiments.
586