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
IMPACT RESPONSE OF PBX 9501 BELOW 2 GPA
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2
K. Kline , Y. Horie , J. J. Dick , and W. Wang
Los Alamos National Laboratory, Los Alamos, NM 87545
North Carolina State University, Raleigh, NC 27695
Abstract. DDEM2, a discrete element code, is used to model plane shock-wave profiles in unreacted
PBX 9501. The constitutive properties of the material are modeled in terms of hydrostatic pressure
and viscoelastic shear stress. The former is modeled using Hugoniot data. The latter is represented by
a relaxation functional, which is approximated by a three-term Prony series. Material parameters were
found to give a satisfactory fit to the measurements. The relaxation times are lxlO" 7 s, lxlO~ 6 s, and an
elastic assumption. A single shear modulus is reintroduced at the beginning of unloading. The
unloading elastic component is 5x1010 dynes/cm2 and the relaxation time is 4xlO~ 7 s.
stress relaxation. Second, the code has a capability
to model two-dimensional heterogeneous materials
using elastic and viscoelastic elements. In this
exploratory study, calculations are limited to shock
propagation in a one-dimensional chain.
DDEM2 uses, as in molecular dynamic
simulations, a force-and-particle method to
calculate stress wave propagation in solids.
However, particles in DDEM2 have a finite size
and internal variables representing the state; e.g.,
composition and internal energy.
Particle
trajectories are determined by solving Newton's
equations of motion. This force includes a damping
force, which is a function of relative particle
velocity, to control high-frequency oscillations.
This force is equivalent to artificial viscosity in
hydrocode calculations of shock propagation. For
the weak shocks of current interest, bulk
temperature changes have been ignored.
INTRODUCTION
This study covers the numerical modeling of
wave profile measurements [1] using a nonlinear
viscoelastic constitutive model in a discrete element
code called DDEM2. The focus of this paper is to
investigate the plane impact response of the
material known as PBX 9501 from 0.2 to 2 GPa and
to improve through DDEM2 understanding of the
viscoelastic response of PBX 9501 to mild
mechanical shock loading where no reaction is
expected. For this discussion and in the regime of
the authors' interest, the mechanical response of the
material is modeled as viscoelastic and is attributed
to the particulate behavior in a plastic binder.
A recent effort [2] integrated available data from
Hugoniot or ultrasonic measurements for modeling
non-shock ignition of PBX 9501 using a
generalized Maxwell model. However, the focus
of the parametric evaluation was placed on low
strain-rate regimes.
Constitutive Model
For a one-dimensional application of the
DDEM2 code to simulate shock wave events on a
chain, uniaxial strain conditions are imposed
between an interacting element pair.
The
longitudinal stress, a, is resolved into hydrostatic
DISCRETE ELEMENT METHOD
DDEM2, under development over the last few
years at North Carolina State University, is chosen
for two reasons. First, the code contains a
viscoelastic model with a convolution integral for
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pressure, P , and deviatoric stress, 5 . The
hydrostatic response is determined by the Hugoniot
and is assumed to be a function of volume only. [3]
If Hugoniot data are described by a linear shock
velocity/particle velocity relation, Us=C0+kup,
then the pressure is given by P0 =
where £ is volume strain and CQ = B0 I p0 .
The deviatoric stress is described by a linear
viscoelastic model through use of a convolution
integral that signifies fading memory. [4]
5(0=
where chf jdl is the deviatoric strain rate and M
is the relaxation modulus. Because the strain is
assumed to be uniaxial, the physical meaning of M
is 4/3 G , where G represents the material shear
modulus. Mis represented by a sum of n
exponentials (called the Prony series).
The deviatoric stress can be thought of as a
system of n differential equations such that
^L-M^L^
dt ~ ' dt \ '
n(2)
}
where Mf is the instantaneous elastic modulus and
AY is the relaxation time for each of the differential
equations.
The meaning of the differential
equations can be depicted by mechanical models
composed of springs, signifying elastic response
( S f = Mfl f) and dashpots, describing viscous
effects (Sj = J]l dy/dt, with r\. = Mfa
as
viscosity) in parallel, basically a generalized
Maxwell model. The code does not account for any
volumetric influence in viscosity. If the number of
functions is limited to three, this model is similar to
one used for polymers [3], but DDEM2 does not
contain a relaxation time as a function of strain rate.
DDEM2 material parameters can be cast into their
framework for multidimensional applications.
As shown in Fig. 1, the model is best described
by the viscoelastic shear-yielding model for a solid.
[5] The model doesn't include any material
damage effects. The material loading path is
initially bound by the Elastic Instantaneous
modulus at zero strain, EL(0), which also
determines the initial wave speed. The viscoelastic
material follows a loading path to the terminal state,
T. At the steady state, the loading path will
collapse to the Rayleigh line, R. At the terminal
point, the unloading path wave speed is initially
determined by the Elastic Instantaneous tangent,
EL(e), but deviates from this slope due to viscous
effects. The unloading elastic path is bound by the
hydrostatic Hugoniot, H, and is tangent to the
Isentropic Compression, S^, at the final state.
DISCUSSION OF RESULTS
Three symmetric impact test shots, [1]
shotsl!16, 1049, and 1058, are analyzed. In these
experiments, wave profiles of time vs. particle
velocity were obtained using magnetic velocity
gauges at five locations. Impact velocity and
induced maximum stress are in the range of 120 to
440 m/s and 300 MPa to 1.2 GPa, respectively.
The best fit parameters are based on the
parameters determined in the preliminary report. [6]
Table 1 lists the parameters used in this best fit
case. The results of the best fit are displayed along
with the experimental data in Figs. 2-4. The
loading and unloading arrival times as well as the
dispersion and relaxation agree fairly well with the
experimental data for shots 1116 and 1049. But for
shot 1058, the computational unloading wave
consistently arrives approximately 100 ns slower
than the experimental wave. Also, the relaxation
doesn't satisfactorily describe the measured profile.
The authors observed that the relaxation for shot
1058 could be matched to the measured data with a
p
«ad
Figure 1. Shear-yielding model that does not include damage.
Volume^ V0 (1-e). The dotted lines are the loading and
unloading paths. The OD line is an overdriven loading path.
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Table 1.Computational Best Fit Material Parameters
Best Fit Case
Parameters
£<0
3
£ >0
1.82dO
1.82dO
9.0dlO
9.0dlO
2.5dlO
2.5dlO
l.d-7
l.d-7
1.121dlO
1.121dlO
l.d-6
l.d-6
G3 (dynes/cm )
LdO
7.0dlO
X3 (s)
I.d20
4.d-7
P0 (g/cm )
B
o
2
(dynes/cm )
2
Gl (dynes/cm )
XI
(s)
2
G2 (dynes/cm )
X2 (s)
2
FIGURE 3. Best fit case compared to shot 1049 for PBX 9501
at 560 MPa. Half-impact velocity is 116 m/s.
(assumed to be 1/3).
pressure-sensitive modification. This suggests that
the shear modulus is pressure sensitive, but the
results are not presented in this paper.
For gauges 3-5 in all three shots, the
computational peak pressures are roughly 10%
lower than the experimental steady state pressures.
Also, the computational results show decay in the
steady state peak pressures. This is puzzling. For
shot 1116, the decay could be attributed to the
rarefaction wave overtaking the initial wave before
it reached the final state. But for the two other
shots, the initial wave should be traveling faster
then rarefaction wave in the time frame analyzed.
The longitudinal sound speed at zero pressure,
CLQ, is calculated using Eqs. 3 and 4 where Cs is
the shear sound speed, and v is Poisson's ratio
2
Time (us)
Po
BQ = p0C0 =
(3)
2(1 -v)
—
(4)
For the computation, CLQ is 2.75 mm/(is, which
corresponds to an experimental value of 2.95
mm/jis. The discrepancy in the longitudinal sound
speed, which allows the computational and
measured loading arrival times to agree, is unclear.
The bulk modulus is evaluated using the Hugoniot
[1], but the Hugoniot intercepts a slightly slower
sound speed at zero pressure than expected.
Interestingly, the unloading shear modulus is
twice as big as the total loading modulus. This may
be due to compression of the material after impact
(porosity influence), compensation for the decrease
in the computational bulk modulus, or a change in
material properties.
Figure 5 shows the fifth-gauge records for all
three shots. Notice that shots 1116 and 1049 have
roughly the same arrival time of the loading wave
in the computation as well as in the experimental
data. This unexpected behavior of the traveling
wave is examined further.
If the material were elastic-plastic, then the wave
speed for shot 1058 can be explained as an
overdriven case, in which the slope of the Rayleigh
line is greater than the longitudinal elastic slope.
Because the material is assumed to be viscoelastic
in the present study, the overdriven situation occurs
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FIGURE 2. Best fit case compared to shot 1116 for PBX 9501
at 303 MPa. Half-impact velocity is 66.2 m/s.
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>100
I
50
\ \ l \ \ \ l \
0
1
gp....l.,....l......l,..., .... ........ ........ ,...,...,,
fj,. . . . . .j. . . . . . . . liq:]::: :):::[::):
2
Time (MS)
:J
4
FIGURE 4. Best fit case compared to shot 1058 for PBX 9501
at 1.18 GPa. Half-impact velocity is 220 m/s.
FIGURE 5. Comparison of shots 1116, 1049, and 1058 at the
fifth gauge location.
because the Elastic Instantaneous modulus at zero
strain is greater than the relaxed modulus, as shown
in Fig. 1. In the region where compression is less
than the threshold stress (0 <0 r / ? ), the precursor
wave speed is the longitudinal speed at zero stress,
as observed in shots 1116 and 1049. However, for
any terminal load above the threshold stress
(0 >0 f / l ), the wave speed is controlled by the
magnitude of the terminal. An estimate for alh is
1.103 GPa, corresponding to a particle velocity of
roughly 220 m/s. This result is consistent with the
stress data from the experimental gauge records.
The viscoelastic explanation of overdriven loading
does not contradict the existence of Hugoniot
Elastic Limit (HEL) below Gth. In fact, some HEL
experimental evidence shows its existence at about
140 Mpa [1], but additional work is needed to
resolve this issue.
The unloading shear modulus is twice as big as
the loading shear modulus. This could be due
changes in material properties, porosity influence,
or compensation for a decrease in the initial
computational bulk modulus. At present, the
parameters are estimated from the Hugoniot
Us - up relation, where up = 0 is interpreted as
the solid bulk sound speed.
ACKNOWLEDGMENTS
We would like to thank Kazushige Yano, Los
Alamos National Laboratory, for the contribution of
his thoughts and ideas in our discussions.
REFERENCES
1. Dick, J. J., Martinez, A. R., and Hixson, R, S., Los
Alamos National Laboratory report LA-13426-MS,
Los Alamos, NM 87545, April 1998. Also found in
Dick, J. J., Martinez, A. R., and Hixson, R. S., "Plane
Impact Response of PBX 9501 below 2 GPa," in
Proc. 11th Int. Det Symp., 1998 (in press), p. 317.
2. Bennett, J. G., Haberman, K. S., Johnson, J. N., Asay,
B. W., and Henson, B. F., J. Mech. Phys, Solids 46,
2303-2322(1998).
3. Johnson, J. N., Dick, J. J., and Hixson, R. S., J. Appl
Phys. 84, 2520-2529 (1998).
4. Lakes, R. S., Viscoelastic Solids, CRC Press, Boca
Raton, 1999.
5. Band, W., J. Geop. Res. 65(2), 695-719 (1960).
6. Kline, K., Horie, Y., Dick, J. J., and Wang, W., Los
Alamos National Laboratory report, Los Alamos, NM
87545, In press.
CONCLUSIONS
A discrete element code, DDEM2, is used to
numerically
investigate
symmetric
impact
experiments [1] and to explore the viscoelastic
response of PBX 9501 to mild mechanical shock.
This model is similar to the viscoelastic shear-yield
model [5] for solid materials. This best fit case
shows satisfactory agreement with the experimental
data in all aspects of the wave profile. But the
numerical results at the steady state are about 10%
lower than the measured peak pressures for the
latter gauges for all three shots.
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