An investigation on microscopic and macroscopic stability phenomena of composite solids with periodic microstructure

An analysis of the effects of microscopic instabilities on the homogenized response of heterogeneous solids with periodic microstructure and incrementally linear constitutive law is here carried out. In order to investigate the possibility to obtain a conservative prediction of microscopic primary instability in terms of homogenized properties, novel macroscopic constitutive stability measures are introduced, corresponding to the positive definiteness of the homogenized moduli tensors relative to a class of conjugate stress-strain pairs. Numerical simulations, addressed to hyperelastic microstructural models representing cellular solids and reinforced composites, are worked out through the implementation of an innovative one-way coupled finite element formulation able to determine sequentially the principal equilibrium solution, the incremental equilibrium solutions providing homogenized moduli and the stability eigenvalue problem solution, for a given monotonic macrostrain path. Both uniaxial and equibiaxial loading conditions are considered. The exact microscopic stability region in the macrostrain space, obtained by taking into account microstructural details, is compared with the macroscopic stability regions determined by means of the introduced macroscopic constitutive measures. These results highlight how the conservativeness of the adopted macroscopic constitutive stability measure with respect to microscopic primary instability, strictly depends on the type of loading condition (tensile or compressive) and the kind of microstructure. (C) 2010 Elsevier Ltd. All rights reserved.