Mean Field Homogenization of Soft Materials with a Periodic Microstructure

The present thesis investigates the homogenization of soft materials with a periodic microstructure subjected to large strains, particularly those with a doubly periodic pattern of holes (voids). The idea of mean field homogenization is to determine material parameters of an “effective” material such that its stress response coincides with the (spatially) averaged stress response of the heterogeneous material. As opposed to well-established approaches to computational homogenization, in which tangent stiffness moduli are identified from representative volume elements, the present thesis aims at parametric homogenization. The key idea is to find a representation of the stress response of the effective material based on strain energy functions, where the material parameters are functions of microstructural quantities such as the void ratio in the present case. For this purpose, representative volume elements are identified and subjected to elementary load cases, i.e., tensile and (simple) shear tests, to identify parameters of various kinds of strain energy functions of finite-strain hyperelasticity using the averaged stress response. In particular, the hyperfoam model, which was originally developed for the modeling of elastomeric foams, proves as suitable for the kind of materials studied in this thesis. Different orders of the material model are investigated and an extension to describe the anisotropy introduced by the periodic pattern is proposed. By means of several numerical examples, accuracy and efficiency of the proposed homogenization scheme are analyzed. For small void ratios, for which scales are clearly separated, accurate results are obtained. As the void ratio increases, the accuracy decreases, as expected, but the stress response can still be captured from a qualitative point of view.