Non-local modeling with asymptotic expansion homogenization of random materials

The aim of this study is to build a non-local homogenized model for three-dimensional composites with inclusions randomly embedded within a matrix according to a stochastic point process w = (w(i))(i is an element of N) in a bounded open set of R-3 associated with a suitable probability space ((sic), A, P) as defined in Nait-ali (2017) and Michaille a al. (2011). Both phases were linear elastic. Asymptotic expansion homogenization (AM) was revisited by taking into account the stochastic parameter (w) representing the inclusion centers distribution. The macroscopic behavior was then studied by combining the variational approach with the mean-ergodicity. At the end, the advanced approach makes naturally emerge non-local terms (involving the second displacement gradient) as well as a strong microstructural content through the presence of the characteristic tensors in the expression of the homogenized elastic energy. Microstructures with a high contrast between constituents Young's modulus leading to non-local effects were considered to test the model. Virtual microstructures were first generated with a fixed, simple, pattern before considering real microstructures of Ethylene Propylene Dien Monomer (EPDM) containing cavities in order to envision morphological situations with increasing complexity.