Reinforced elastomers: Homogenization, macroscopic stability and relaxation

Abstract

This paper deals with the stability and post-bifurcation response of reinforced hyperelastic composites under general loading conditions. It has long been known that these types of materials can undergo both microscopic and macroscopic instabilities. In the latter case, when the instability does not result in material failure, the behavior of the composite after the onset of the instability is less well understood. Recent work (Avazmohammadi and Ponte Castañeda, 2016) indicates that it is possible for the “principal” solution, i.e. the solution before the onset of any instability, to bifurcate into a lower energy solution via the formation of domains. These domains form on a scale much larger than that of the heterogeneity, but still smaller than that of the macroscopic specimen. This work is concerned with such domain formation in neo-Hookean laminates under general three-dimensional loading. In order to obtain the post-bifurcation behavior, the quasiconvexification or relaxation of the principal solution is computed explicitly. In addition, it is shown that the macroscopic instabilities are triggered not by the loss of strong ellipticity, but rather by the loss of global rank-1 convexity of the principal solution, which, in general, happens first. The calculation also reveals that the relaxation requires, in general, a rank-2 “laminate-within-a-laminate” microstructure and allows for multiple “perfectly soft” modes of deformation.