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
INERTIA AND TEMPERATURE EFFECTS IN VOID GROWTH
L. Seaman and D. R. Curran
SRI International, 333 Ravenswood Avenue, Menlo Park, California 94025-3493, U.S.A.
Abstract The usual solution for void growth in a ductile metal is based on the assumption that the
velocity of growth is proportional to the current void size and inversely proportional to the material
viscosity. In this study we examine the additional effects of inertia and temperature on void growth.
The solution of Poritsky (1) includes the large effect of inertia on the growth rate of large voids. As
the voids rapidly expand during high-rate fracture, the material near the void surface becomes hot and
its strength and viscosity may both reduce because of this heating. We performed finite-element
simulations to study the growth of a single spherical void under tensile loading in a viscous material,
thereby accounting for viscous, thermal, and inertia effects. In aluminum the thermal strength
reduction effect allows small voids to expand about 35% more rapidly than without the effect. We
propose a simple approximate method for treating the viscous, thermal, and inertia effects in void
growth.
INTRODUCTION
This study began with an effort to determine the
importance of the thermal effects which must occur
near the surface of a void growing at high rate in a
ductile metal. Johnson (2) suggested that the thermal effect is unimportant, yet Wang (3) concluded
that the thermal effect is dominant. In addition we
wanted to pursue the inertial effect which arises for
large voids, according to the equation of Poritsky
(1) for a void growing in an infinite block of
viscous material:
d2R
3 f d R f , 4ii dR = T
2Rl dt I PR2 dt pR
From this equation we solved for the growth rates
for aluminum, copper, and tantalum for a viscosity
coefficient of 20 Pa-s using a fourth order RungeKutta method. The resulting curves are shown in
Fig. 1. The inertial effect (reduction in velocity for
larger voids) is enhanced for larger densities.
(D
Here R is the radius, t is the time, r| is the
coefficient of viscosity, p is the density, and T is
the applied tension. The third and fourth terms
represent the usual conditions for small voids and
their solution leads to the constant value of V/R
equal to T/4r|, where V is the void growth rate
dR/dt. The first two are acceleration terms; they
lead to a reduced growth velocity for larger voids.
2.E+06
l.E+06
O.E+00
0.0001
0.001
0.1
Radius - cm
Figure 1. Void growth velocity for Al, Cu, and Ta, assuming
TI = 20 Pa-s.
607
We used the larger 0.02 Pa-s for the liquid case to
minimize oscillations in our simulations.
The simulations showed that void growth velocity varies essentially as 1/R2 to maintain a nearly
constant density throughout the block. The circumferential deviator stresses are proportional to the
circumferential strain rate V/R, where V is the void
growth rate.
CONDITIONS FOR OUR SIMULATIONS
To study both the thermal and inertia! effects we
performed dynamic explicit one-dimensional
finite-element simulations of a spherical void in the
center of a spherical block of material. The inner
void surface was given an initial velocity and a
constant tensile stress was applied to the outer
boundary. The material was treated as a fluid with
a linear viscous behavior.
To verify the simulation method, we made short
simulations with several initial void radii and no
thermal strength reduction effect. The internal
energy in these short cases never approached the
melting energy. The computed growth rates were
close to the curve for aluminum given in Fig. 1;
therefore, we assumed our method was sound.
We performed the next finite-element simulations with a thermal effect which modified the
viscosity as a function of the current internal energy as shown in Fig. 2. Our intention was to
approximate the model of Steinberg et al (4), and
also data of Razorenov et al (5) and Hanim and
Klepaczko (6). We assume that this form is reasonable for very high rate loading as in our case
near the rim of the void. Then as the energy
approaches melting there is a sudden drop in the
visosity from 20 Pa-s (from our earlier work with
ductile fracture in aluminum (7)) to 0.02 Pa-s, or
about 16 times the measured value (1.2 mPa-s (8))
for liquid aluminum near the melt temperature.
= 2t|V/R
The radial deviator stress is -2c'c to balance the
two circumferential deviator stresses, and hence is
given by
r = -4t|V/R
(3)
This compressive deviator stress is usually (without the thermal reduction effect) balanced by the
large tensile pressure which exists throughout the
block. But when the viscosity reduces because of
the thermal effect, the pressure must also approach
zero at the void surface so that the zero-stress
boundary condition is maintained. The variation of
the deviator stress, pressure, and total stress are
shown in Fig. 3 at some time in a simulation
(starting radius of 5 micrometers) after a steady
condition has been achieved for an applied tensile
stress of 0.8 GPa. The computation was initialized
with the internal energy at melting, about 586 J/g.
Thereafter the rapid reduction in the viscosity
causes the continuing increase in energy at the rim
to proceed at a very slow pace. Hence, although
the energy reaches incipient melt at the void
100 i
I
(2)
10
0.1
0.01 H—
0
200
400
600
800
-i.o
1000
0.0000
Internal energy, J/g
0.0005
0.0010
0.0015
0.0020
0.0025
Radius - cm
FIGURE 2. Viscosity variation according to the thermal
strength reduction effect
FIGURE 3. Stress and pressure variation with radius.
608
0.0030
The thermal effect appears to increase the velocity
by about 35% for small voids, but much less for
larger voids.
surface, it does not significantly exceed this value.
Therefore, one might not see indications of melting
on the void surface.
RESULTS
AN APPROXIMATE SOLUTION
We made a series of short simulations with
starting void radii of 1, 3, 10, 30, 100, 300, and
1000 micrometers and an imposed radial stress of
0.8 GPa at the outer boundary. The internal
energies were initialized at melting at the void rim,
decreasing with the inverse 6th power of radius.
Velocities were initialized with an estimated
velocity at the rim and then varied inversely with
the square of radius. This initialization was
undertaken so that a steady-state condition could
be achieved rapidly and consistent with the
imposed external stress. Similarly, the deviator
stress and pressure were initialized to represent our
estimate of the steady-state conditions. The
resulting velocity-to-radius ratio and the velocity of
void growth are shown in Figs. 4 and 5. The
velocities from the simulations appear to oscillate
somewhat initially and then reach a steady or
smoothly varying condition until a reflected wave
returns from the outer boundary. The results of the
many small simulations do not fit well together:
this probably indicates that a steady state was not
yet reached. These simulation results are compared with the solution of Eq. (1) in these figures.
We sought a simple approximation to the void
growth equation which can be used economically
to follow the growth of large arrays of voids.
Through our analyses of Eq. (1) we found that the
first term, the second derivative, has a small effect
on the solution over the whole range of radii.
Therefore, we omitted this term and the quadratic
formula gives the growth rate:
This solution is also shown in Figs. 4 and 5. This
equation includes the "adjustment factors" Qv and
Qt which are each equal to one for Poritsky's
equation, but here are added to account for the
thermal effects. The curve is made using Qv =
0.81 and Qt = 1.1. Clearly this solution is an
approxi-mation that can be applied appropriately in
many situations.
1.4E+07
Poritsky
- - -0- - "Summ. of sim.s
— -X — Mg: Qv=.81,Qf=U
*
Simulations
O.OE+00 4
0.0001
0.001
0.01
0.1
Void radius - cm
FIGURE 4 Void growth velocity / radius in aluminum with thermal and inertia effects
609
:-xxx
Poritsky
- - -0- - "Summ. of sim.s
— -X — Alg: Qv=.81,Qf=l.l
•
Simulations
5.0E+03
O.OE+00
0.0001
0.001
0.01
0.1
Void radius - cm
FIGURE 5 Void growth velocity in aluminum with thermal and inertia effects
metals at high temperatures," Shock Compression of
Condensed Matter-1997, edited by S. C. Schmidt et
al, AIP Conference Proceedings 429, New York,
1998,, pp 447-450.
6. Hanim, S., and Klepaczko, J. R., "Effects of initial
temperature on spalling of metals," /. de Physique
IV France, 10(Pr9), 2000, pp. 397-402.
7. Curran, D. R., Seaman, L., and Shockey, D. A.,
Physics Reports, 147(5 & 6), March, 1987, pp 254388.
8. Hutch, J., ed., Aluminum: Properties and Physical
Metallurgy, American Society for Metals, Metals
Park, Ohio, 1984.
CONCLUSIONS
Temperature has an effect on the rate of void
growth. When thermal effects are not accounted
for, there is a misinterpretation of the magnitude of
the viscosity. The effects of inertia become important for larger voids, and especially for higher
density materials. A simple approximation for the
altered void growth process was presented to
facilitate the separate treatment of a range of void
sizes.
REFERENCES
1.Poritsky, H., "The collapse or growth of a spherical
bubble or cavity in a viscous fluid," Proc. of the
First U.S National Congress of Applied Mechanics,
ASME, New York, 1952, pp. 813-821.
2. Johnson, J. N., "Dynamic fracture and spalla-tion in
ductile solids," J. Appl Phys, 52(4), April 1981, pp
2812-2825.
3. Wang, Ze-Ping, "Growth of voids in porous ductile
materials at high strain rate," /. Appl. Phys., 76(3),
lAug 1994, pp. 1535-1542.
4.Steinberg, D. J., Cochran, S. G., and Guinan, M.
W., "A constitutive model for metals applicable at
high-strain rate," J. Appl Phys, 51(3), March 1980,
pp. 1498-1504.
5. Razorenov, S. V., Bogatch, A. A., Kanel, G. I.,
Utkin, A. V., Fortov, V. E., and Grady, D, E.,
"Elastic-plastic deformation and spall fracture of
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