## Computing overall elastic constants of polydisperse

## particulate composites from microtomographic data

H. Lee^{2}, A.S. Gillman^{1} and K.
Matous^{1,*}

^{1}Department of Aerospace and Mechanical
Engineering

University of Notre Dame

Notre Dame, IN, 46556, USA.

^{2}Department of Mathematics

Florida State University

Tallahassee, FL, 32306, USA.

^{
}
#### Abstract

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.

#### Conclusions

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.

#### Acknowledgment

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 Urbana-Champaign.