Computing overall elastic constants of polydisperse
particulate composites from microtomographic data
H. Lee2, A.S. Gillman1 and K. Matous1,*
1Department of Aerospace and Mechanical Engineering
University of Notre Dame
Notre Dame, IN, 46556, USA.
2Department of Mathematics
Florida State University
Tallahassee, FL, 32306, USA.
In this paper, we use the
well-known Hashin-Shtrikman-Willis variational principle to obtain the
overall mechanical properties of heterogeneous polydisperse particulate
composites. The emphasis is placed on the efficient numerical
integration of complex three-dimensional integrals and on aspects of
the anisotropic material response of real tomographically characterized
packs. For this purpose, we numerically calculate the complete
statistics of real packs, which are numerically or tomographically
generated. We use the parallel adaptive sparse Smolyak integration
method with hierarchical basis to integrate complex singular integrals
containing the product of probability functions and the second
derivative of Green’s function. Selected examples illustrate both the
numerical and physical facets of our work. First, we show the reduction
of integral points for integration in spherical coordinates. Then, we
comment on the parallel scalability of our method and on the numerical
accuracy associated with the integration of a singular function. Next,
we validate the solver against the experimental data and verify the
results by comparing it to a closed-form expression. To investigate the
ability of our scheme to capture the anisotropic nature of packs, we
study a lattice type system. Finally, we report on the elastic
constants computed for the modeled anisotropic particulate system that
is tomographically characterized.
In this manuscript, we propose a computational scheme for evaluation of
mechanical properties of polydisperse particulate composites. The
complex statistical characteristics are obtained from micro-CT data.
The well-known Hashin-Shtrikman-Willis variational principle, that
links directly the statistical descriptors to mechanical properties, is
adopted. Unfortunately, computation of mechanical tensors that are
building blocks of the Hashin-Shtrikman-Willis model is very demanding.
To alleviate this problem, we employ the adaptive sparse Smolyak
integration method with hierarchical basis. Moreover, we extend it to
spherical coordinates and parallelize it for our particular problem. We
show that spatially complex mechanical tensors, based on fully resolved
anisotropic probability spectrum, can be efficiently integrated. Due to
our improved numerics, we capture in detail the anisotropic response of
polydisperse particulate packs that is often hidden. We validate our
numerical method by comparing computer-generated packs to experimental
data, and verify the isotropic model against a closed-form expression.
Finally, we apply the method to a real polydisperse system that is
obtained using microtomography.
Future research directions are to compute both
bounds and to extend this technique to nonlinear media. Application to
ellipsoidal packs and/or other crystalline shapes, where anisotropy is
more pronounced, needs to be investigated. The third-order model with
realistic tomographically characterized probability descriptors that
will tighten the bounds is also of interest.
The authors would like to acknowledge the support from Buckmaster
Research - DoD STTR program, AFOSR: Dr. J. Buckmaster (Buckmaster
Research) and Dr. A. Nachman (AFOSR) program managers.
* The initial work was performed when H. Lee was a postdoctoral scholar
in K. Matous’ research group at the University of Illinois at