Microscopic and macroscopic instabilities of periodic composite solids

This contribution proposes a theoretical and numerical investigation on the relations between microscopic instability mechanisms in composite solid with periodic microstructure and the onset of macro-instabilities. To this end a stability analysis on the micro and macro scales is here carried out in order to investigate macroscopic stability conditions able to predict microscopic instability phenomena. 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. Numerical applications, devoted to unidirectional fiber-reinforced and particle reinforced composite microstructures with hyperelastic constituents, are developed by implementing a one-way coupled finite element approach. The ability of the proposed macroscopic measures is examined by means of comparisons between the exact microscopic stability region in the macro-strain space, obtained by taking into account microstructural details, and the macroscopic stability regions, determined by investigating the homogenized material properties. In fact, results show the hierarchy of critical loads for both macroscopic and microscopic stability measures and show how the macroscopic stability region (determined by using a macroscopic constitutive stability measure) can be included within the exact microscopic stability region (obtained by including microstructural details).