Yield of random elastoplastic materials

When separation of scales in random media does not hold, the representative volume element (RVE) of deterministic continuum mechanics does not exist in the conventional sense, and new concepts and approaches are needed. This subject is discussed here in the context of microstructures of two types-planar random chessboards, and planar random inclusion-matrix composites-with microscale behavior of the elastic-plastic-hardening (power-law) variety. The microstructures are assumed to be spatially homogeneous and ergodic. Principal issues under consideration are yield and incipient plastic flow of statistical volume elements (SVE) on mesoscales, and the scaling trend of SVE to the RVE response on the macroscale. Indeed, the SVE responses under uniform displacement (or traction) boundary conditions bound from above (or below, respectively) the RVE response. We show through extensive simulations of plane stress that the larger the mesoscale, the tighter are both bounds. However, mesoscale flows under both kinds of loading do not generally display normality. Also, within the limitations of currently available computational resources, we do not recover normality (or even a trend towards it) when studying the largest possible SVE domains.