Computing overall elastic constants of polydisperse particulate composites from microtomographic data

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 Greenʹs 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.