Micro-fracture instabilities in granular solids

We study the aspect of unstable behavior (like strain localization bands) in elastic solids as a consequence of micro-fracturing. A two-scale approach of computational homogenization is considered. The macroscopic behavior is investigated by finite element computations on a unit cell. At the micro-level, we consider a granular structure with elastic grains. The inter-granular boundaries are modeled with cohesive laws, friction and unilateral contact. We show that decohesion between grains gives rise to macro-instabilities, indicated by the loss of ellipticity, typical for deformation localization bands. The relation between the microscopic softening on inter-granular boundaries and the onset of macro-instabilities is studied through numerical examples. The influence of the cohesive law and friction parameters is analyzed. For periodic distributions of granular structures, we prove the loss of periodicity by failure and the corresponding size dependence effect in the homogenized response. We present numerical examples of bifurcation of solutions for granular cell structures and of particular solutions specific to elementary volumes with periodic cell distribution. Size dependence appears in the unstable regime and is strongly influenced by cohesion and friction parameters.