Localization of deformation and loss of macroscopic ellipticity in microstructured solids

Abstract

Localization of deformation, a precursor to failure in solids, is a crucial and hence widely studied problem in solid mechanics. The continuum modeling approach of this phenomenon studies conditions on the constitutive laws leading to the loss of ellipticity in the governing equations, a property that allows for discontinuous equilibrium solutions. Micro-mechanics models and nonlinear homogenization theories help us understand the origins of this behavior and it is thought that a loss of macroscopic (homogenized) ellipticity results in localized deformation patterns. Although this is the case in many engineering applications, it raises an interesting question: is there always a localized deformation pattern appearing in solids losing macroscopic ellipticity when loaded past their critical state?In the interest of relative simplicity and analytical tractability, the present work answers this question in the restrictive framework of a layered, nonlinear (hyperelastic) solid in plane strain and more specifically under axial compression along the lamination direction. The key to the answer is found in the homogenized post-bifurcated solution of the problem, which for certain materials is supercritical (increasing force and displacement), leading to post-bifurcated equilibrium paths in these composites that show no localization of deformation for macroscopic strain well above the one corresponding to loss of ellipticity.