Effective characteristics and stress concentrations in materials with self-similar microstructure

It is proposed to model materials with self-similar structure by a continuum sequence of continua of increasing scales each determined by its own size of the averaging volume element. The scaling is represented by power laws with the exponents determined by the microstructure, but not necessarily by the material fractal dimension. The scaling laws for tensors are shown to be always isotropic (the same exponent for all non-zero components) with the prefactors accounting for anisotropy. For materials with self-similar distributions of pores, cracks and rigid inclusions the scaling laws for elastic characteristics were determined using the differential self-consistent method. Stresses are defined in each continuum (and are measured in conventional units of stress) with the scaling law controlling the transition from one continuum to another, i.e. from one stress field to another. In the case of strong self-similarity the scaling exponent for the stress field is uniform, coincides with the one for the average (nominal) stress and is controlled by the sectional fractal dimension of the material. Within each continuum the stress concentrators--point force, dislocation, semi-infinite crack--produce conventional stress singularities. However, as the point of singularity is approached, the transition to finer continua is necessary, resulting, in some cases, in apparent non-conventional exponent of the stress increase.