Computational Physics Group

Karel Matous










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 Greenís 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. Matousí research group at the University of Illinois at Urbana-Champaign.

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© 2011 Notre Dame and Dr. Karel Matous