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.
Keywords
random media, scale effects, plasticity, RVE,
homogenization
