Computational homogenization

Abstract

AbstractThis part of the CISM course addresses basics and advanced topics on the computational homogenization of the mechanics of highly non-linear solids with (possibly evolving) microstructure under complex non-linear loading conditions. The key components of the computational homogenization scheme, i.e. the formulation of the microstructural boundary value problem and the coupling between the micro and macrolevel based on the averaging theorems, are addressed. The numerical implementation of the framework, particularly the computation of the macroscopic stress tensor and extraction of the macroscopic consistent tangent operator based on the total microstructural stiffness, are treated in detail. The application of the method is illustrated by the simulation of pure bending of porous aluminum. The classical notion of a representative volume element is introduced and the influence of the spatial distribution of heterogeneities on the overall macroscopic behaviour is investigated by comparing the results of multi-scale modelling for regular and random structures. Finally, an extension of the classical computational homogenization scheme to a framework suitable for multi-scale modelling of macroscopic localization and size effects is briefly discussed.