Computational homogenization analysis in finite elasticity:: material and structural instabilities on the micro- and macro-scales of periodic composites and their interaction

The paper investigates instability phenomena in the context of homogenization-based micro-to-macro transitions of heterogeneous materials at finite strains. This covers structural instability effects (buckling) and material instability effects (localization) which can occur on both the macro- as well as the micro-scale and may influence each other. We develop a general framework for the theoretical and computational treatment of these instability problems for elastic composites with given periodic fine-scale micro-structures. The key methodology is an investigation of the properties and the interaction of two coupled minimization principles which govern a micro-heterogeneous material in the context of a classical homogenization procedure: The principle of minimum potential energy of the macro-structure and the principle of minimum average energy of the micro-structure. The first variational problem determines the deformation field of the homogenized continuum. The latter yields the fine-scale fluctuation field on a composite micro-structure that is assumed to be attached to each local point of the macro-continuum. Global stability and the existence of solutions are based on weak convexity properties of these variational functionals. The convergence of non-convex homogenization functionals is based on the Gamma-limit of periodic heterogeneous micro-structures. It defines the relevant micro-structure as an a priori unknown critical ensemble of periodic cells that catches a possible minimizing buckling mode. We summarize these basic results and recast them in a consistent notation suitable for numerical implementation. Furthermore, we in detail discuss the discretization of the coupled minimization problems by means of finite element methods and point out numerical concepts for the detection of material and structural instabilities. The treatment provides a comprehensive guide to the classification and computation of instabilities in micro-heterogeneous solids. Focus is put on the interaction of micro- and macro-instability phenomena such as the loss of macroscopic material stability (localization) induced by a microscopic structural instability (buckling). The performance of the proposed computational methods is demonstrated for representative numerical model problems which treat coupled instability phenomena in micro-heterogeneous elastic composites. (C) 2002 Elsevier Science B.V. All rights reserved.