A theoretical and numerical stability analysis for composite micro-structures by using homogenization theory

Abstract

A theoretical and numerical non-linear analysis of the homogenized response of composite solids with periodic micro-structure subjected to macroscopically uniform strain is here developed, by including the effects of instabilities occurring at both macroscopic and microscopic levels. The theory, formulated for incrementally linear materials, provides an original closed-form representation of homogenized material response which puts in evidence the competing effects of local constitutive response and of microstructural heterogeneity. Macroscopic constitutive stability measures are introduced corresponding to the positivity of the homogenized moduli tensors relative to a class of conjugate stress–strain pairs, and their relations with microscopic stability are investigated by using the proposed theoretical framework.These results are then illustrated and verified by analyzing numerically the in-plane stability problem of hyperelastic models of composites with continuous and discontinuous reinforcements and defected cellular solids. To this end a coupled finite element approach is implemented for a micro-structure driven along prescribed monotonic macro-strain paths. Numerical applications provide the hierarchy of critical loads for both macroscopic and microscopic stability measures and show how the proposed macroscopic constitutive stability region (determined by simple unit cell calculations) can be included within the exact microscopic stability region (obtained by including microstructural details and thus more computational expensive) in accordance with the theoretical developments given before. This gives the possibility of obtaining a conservative estimation of the microscopic critical load, essential to determine the validity of homogenization procedure and to predict instability-induced material failure, and thus makes the paper significant from the engineering’s point of view.