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
COMPUTATIONAL MODELING OF THE SHOCK COMPRESSION OF
POWDERS
David J. Benson*, lan Do1^ and Marc A. Meyers*
""University of California, San Diego, CA, U. S. A.
^Livermore Software Technology Corporation, Livermore, CA, U. S..A.
Abstract. The modeling of both inert and reactive materials at the meso-scale with an Eulerian finite element
program is discussed. Issues that have an effect on the calculated response, including mixture theory and interface
tracking, are briefly presented with recent calculations modeling shock initiated chemical reactions (SICR).
INTRODUCTION
many years for ballistic impact and shaped charge calculations [1, 2, 3]. Eulerian and arbitrary LagrangianEulerian (ALE) formulations permit material to move
relative to the mesh, and therfore permit arbitrarily
large material flows. Individual elements, or zones,
may contain several materials, and numerical interface
reconstruction methods [4, 5, 6] calculate the location
and orientation of the material interfaces within the
elements. Phase changes, chemical reactions, and material failure are relatively easy to include in an Eulerian formulation by simply changing the composition
within the elements. Williamson [7] pioneered the discrete modeling of particles in powders with the Eulerian hydrocode CTH [2], and the Eulerian formulatinon has proven effective for simulating a variety of
powder processes, e.g., [8, 9, 10, 11, 12, 13, 14, 15,
16,17].
Research on the shock compression of powders is very
challenging because it is currently impossible to experimentally monitor the response of individual particles. Experimentalists can only measure the bulk response of the powder and characterize the specimen
before and after the experiment. The details of the
transient response on the particle level are therefore
largely inferred. Analytical models which generate
closed-form solutions must necessarily introduce simplifying assumptions so that the powder is treated as
a continuum or as a periodic structure. Computational
modeling, which permits arbitrarily complicated models limited only by the available computer resources,
has therefore become increasingly important as researchers try to bridge the gap between experimental
investigations and our understanding of the response
on the mesoscopic level.
Hydrocodes have been the most productive computational tool for investigating the shock compression
of powders. For periodic arrays with closest packing at
modest pressures, the Lagrangian formulation is both
accurate and effective. However, real powders have
random packing and are usually at a density significantly below their theoretical packing density. Large
scale jetting occurs with localized melting, and the Lagrangian mesh can no longer track the flow of the material.
The Eulerian formulation has been popular for
THE EULERIAN FINITE
DIFFERENCE/ELEMENT METHOD
Eulerian hydrocodes were originally developed by the
finite difference community. In many instances, including the research presented here, the same algorithm can be described with either a finite difference
or finite element formalism. A brief overview of the
Eulerian finite element method is presented in this section to introduce the essential computational concepts.
The detailed description [18] of the numerical meth1087
ods used in this paper, and the review of the methods
currently in general use [19], provide additional information on multi-material Eulerian formulations.
Operator splitting permits the sequential solution in
two steps of the Eulerian conservation equations,
dpe
+ V • (peu)
=
0
(1)
=
V-
(2)
= a :8
simple, however, the algorithmic details are very complicated. Within an element, the material interface is
approximated as a straight line or plane. The orientation of the interface is determined by a difference stencil or least squares algorithm, and the location of the
interface is determined by volume conservation.
Adjacent particles must have individual material
numbers to prevent them from blending into each
other. When the material interfaces of particles blend
together, the particles have effectively bonded mechanically, which may adversely affect the accuracy
of the calculation.
The spatial resolution of the interface reconstruction scheme may limit the resolution of material jetting or exhibit break-up in the jet which is not physically real. The orientation of the material interfaces
may enter into the physics of the calculation, e.g., contact mixture theories [20]. At the high pressure associated with shocks, the shear stress at a contact point
is limited by the material flow stress, which is likely
to be very low relative to the shock pressure. Therefore, the precise orientation of the material interface
has little influence on the particle interactions, but it is
important for the transport.
(3)
where p is the density, u is the velocity, x is the
spatial coordinate, a is the Cauchy stress tensor, 8 is
the strain rate tensor, b is the body force, and e is
the internal energy. The Lagrangian step, performed
first, advances the solution in time, while the Eulerian
step accounts for the transport between the elements.
An Eulerian formulation uses a spatially fixed mesh,
while an arbitrary Lagrangian-Eulerian (ALE) adds
the option of the mesh evolving with time, but in a
manner that is independent of the material motion.
The Eulerian and ALE formulations include algorithms that aren't in the Lagrangian formulation and
which have a potential to affect the accuracy of a
mesoscale analysis of powders:
MIXTURE THEORIES
The mixture theory in an element containing multiple
materals partitions the strain rate during each time step
among the materials, and calculates the mean stress in
the element from the stresses in the individual materials. The mixture theory therefore has a large potential for affecting the accuracy of the solution since it
governs the interactions of the material interfaces of
adjacent particles.
The mean stress, a is calculated as the volumeweighted average of the individual stresses in the materials, am,
• The interface reconstruction defines the material
boundaries.
• The mixture theory for elements containing a
mixture of materials governs the distribution of
strain between the materials, and therefore the
material response between adjacent powder particles.
• Both the mixture theory [20] and the interface
reconstruction may require information about the
ordering of the material interfaces [21].
a=
INTERFACE RECONSTRUCTION
(4)
where Vm is the volume fraction of material m in
the element. This average, which can be justified by
homogenization theory, is used by virtually all mixture
theories.
The two most popular mixture theories for hydrocodes are the mean strain rate ("springs in parallel") and the mean pressure ("springs in series") mix-
Modern Eulerian hydrocodes use either volume of
fluid (VOF) [22] or level set [23] methods to calculate the location of the material interfaces. For
problems in high pressure physics, volume of fluid
methods are currently more popular. Most codes use
Young's method [4], or a method derived from it, e.g.,
[5, 6, 24]. On a conceptual level, the approach is very
1088
ture theories. In the former, the mean strain rate of
the element is used to update the stress in every material. The latter mixture theory equilibrates the pressure
between the materials by adjusting the partitioning of
the volumetric strain rate. These two mixture theories
bound the volumetric response of the mixture, with the
mean strain rate giving the stiffer response. One serious flaw with the mean strain rate mixture theory is
that empty space, usually referred to as the "void material", is compressed at the same rate as all the other
materials in the element, and the void between two
material interfaces is never completely closed. The
mixture theory is therefore usually modified to preferentially compress out void before compressing the
remaining materials.
In a series of computational experiments [12], the
shock compression of a powder was simulated with
both mixture theories. To explore the interaction of
the material models with the mixture theories, the
simulations were run with and without a viscous term
in the material model. Contrary to expectations, the
simulations with the viscous term were more sensitive
to the mixture theory than the inviscid simulations in
terms of the temperature profiles and the collapse of
the voids between the particles. Overall, the pressure
equilibration mixture theory gives better results.
Imposed
Velocity
FIGURE 1. The initial state of a metallic powder subjected to shock compression.
THE MODEL BOUNDARY VALUE PROBLEM
A typical model for a metallic powder [14], idealized
as having spherical particles, contains approximately
50 to 250 particles, see Figure 1. The number of particles is chosen so that the particle size distribution is
represented with a reasonable statistical accuracy and
to eliminate boundary effects.
The compression of the powder is modeled by imposing a velocity on one boundary while leaving the
remainder fixed. These boundary conditions idealize
the sample holder and piston as rigid bodies, but
they are typically stiffer than the material being compressed. Boundary effects are observed in the calculations, namely a different packing geometry due to the
rigid walls, and for strong shocks, there are some temperature anomalies at the piston face due to the shock
viscosity.
Although three-dimensional calculations would be
best, two-dimensional calculations show excellent
qualitative correlation with experiments. For example,
FIGURE 2. Comparison of the predicted morphology of
a steel powder with the experiment. The contours in the
solution represent temperature, with the light regions being
the hottest.
in Figure 2, the calculation predicts the characteristic
shapes of the deformed particles and regions of localized melting.
Accurate microstructural models are required for
accurately modeling the response of a powder. Many
metallic powder particles are adequately approximated by circles in two dimensions. Other powders,
however, have particles with irregular shapes, and it
is difficult to generate realistic model microstructures.
To circumvent this problem, digitized micrographs
have been used to generate the computational model.
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3. The microstructure is mapped to the computational mesh pixel by pixel. Based on a pixel's
gray level, a material number is associated with
it, and the pixel's volume is distributed to the appropriate overlapping elements.
SHOCK INITIATED CHEMICAL REACTIONS
The roles of the material properties, the geometry,
and the configuration on the shock-induced reaction
threshold (SICR) and the extent of reaction (EOR) in
reactions of the type A + B ^ C are investigated in
[25, 26,27]. The model reaction, Nb + 2Si ^ NbSi2,
was chosen for the research because it has been studied extensively. Modeling the process turned out to be
very challenging because of the binary reaction. The
material models are independently evaluated for every
element in the computational mesh. Using this strategy in our early calculations, when a multi-material element exhausted one of its reactants, the reaction was
quenched in the element, thereby limiting the reaction
zone to less than one element width. An operator splitting method was introduced to permit nonlocal reactions to occur [25, 27]. Further development of this
approach is planned.
The chemical kinetics are modeled with an Arrhenius rate equation. There are many methods for constructing the species evolution equations for chemical reactions. The method chosen here closely follows
the approach taken by [28] in CHEMKIN, a software
package for treating chemical kinetics, developed at
Sandia National Laboratory.
The general species balance equation for the k-th
reaction can be written as
B
FIGURE 3. The three stages of image processing for importing an experimentally acquired microstructure: A) original micrograph, B) after enhancing the contrast, and C) after
the gray scale levels are assigned to indicate the material
number.
(5)
1=1
The steps for building a model from an experimentally acquired microstructure, shown in Figure 3, are:
1=1
where [jc/] is the molar concentration (mole per unit
volume) of the z-th species; and v'ik and v"k are, respectively, the forward and backward stoichiometric
coefficients of the z'-th species and the k-th reaction.
The corresponding molar production (or destruction)
rate of the /-th species (concentration per unit time,
summing over K reactions) is given as
1. Generate the digital image of the microstructure.
CCD cameras directly provide a digitized image,
while photographs must be scanned.
2. Use image processing techniques so that each
material element (ME) has a unique gray level.
When MEs of the same gray level are adjacent,
one of them is assigned a new gray level to preserve their individual identities.
K
(6)
k=l
1090
where r# is the reaction rate for the k-th reaction,
which can be expressed as
(7)
As indicated, r^ is a function of the species concentrations and the instantaneous thermodynamic state of
the reacting species. The forward and backward reaction rate coefficients, kfk and &^, respectively, contain
the physical state information. The general forms of
the Arrhenius rate coefficients are
RT
(8)
and A^k are the forward and backward frequency
factors, respectively, which represent the collision
probabilities among the species. Efk and E^ are the
forward and backward activation energies which represent the energy barriers that the system must overcome in order for the reaction to occur. R is the universal constant and T is the reacting temperature.
The morphology evolution in Figure 4 shows that
the reaction front essentially coincides with the shock
front. Severe plastic flow and jetting of the solid particles can be seen. The silicon reacts almost completely
within this duration, and the unreacted material is the
excess niobium in the non-stoichiometric powder. Experiments [29, 30, 31, 32] have shown that shockinduced chemical reactions can occur over a time
range from hundreds of nanoseconds to microseconds
for various solid powder mixtures. Some results support the hypothesis that the reactions initiated during
the shock compression are dominated by processes occurring during the stress pulse rise time or throughout
the high pressure state before the expansion release,
i.e. on the scale of the mechanical equilibration time.
The present model produces results that approximate
those observations. The pressure pulse first arrived at
the right hand side boundary at about 0.34 /*s, and all
the voids have collapsed by 0.36/^s.
Comparing the left one-third of the domain at times
of 0.24 and 0.36 ,ws, it is observed that reactions in
this region went to completion before the shock had
passed through the specimen. For example, notice the
deformed particle shapes in the lower left hand corner
FIGURE 4. Material evolution of a baseline model during
a SICR simulation at 0.00, 0.24, and 0.36 ps.
of the computational domain. The reaction consumes
the reactant particles quickly. The unreacted niobium
particle shapes deform further, but the niobium content stays the same, indicating that the local reaction
rate is very high, and that possibly the entire reaction
occurs during the pressure rise time.
CONCLUSIONS
The multi-material Eulerian finite element method
has proven to be an effective tool for modeling the
shock processing of powders, including processes
with chemical reactions. Although the present calcu1091
lations are limited to two dimensions, they compare
well in a qualitative sense to the available experimental data, and in some cases, the quantitative agreement
is also very good.
ACKNOWLEDGEMENTS
Funding for this research was provided by the NSF
DMI grant 9612017, LANL grant UC94610017-3L,
and the ARO MURI grant for lightweight armor.
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