Macroscopic stability analysis in periodic composite solids

A theoretical and numerical investigation of the effects of microscopic instabilities on the homogenized response for solids with periodic microstructure is here carried out. The theory is formulated for materials characterized by an incrementally linear constitutive law. Novel macroscopic measures of microstructural stability are introduced corresponding to the positive definiteness of the homogenized moduli tensors relative to a class of conjugates tress-strain pairs and their effectiveness to obtain a conservative prediction of microscopic primary instability load is pointed out. Numerical applications, devoted to hyperelastic microstructures representative of cellular solids and reinforced composites, are developed by implementing a one-way coupled finite element approach. Both uniaxial and equibiaxial loading conditions are considered. 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, are shown.Results evidence that an appropriate definition of macroscopic stability measure depending on the type of loading condition(tensile or compressive) and the kind of microstructure may lead to a conservative stability prediction.