RVE Problem: Mathematical aspects and related stochastic mechanics

The paper examines (i) formulation of field problems of mechanics accounting for a random material microstructure and (ii) solution of associated boundary value problems. The adopted approach involves upscaling of constitutive properties according to the Hill–Mandel condition, as the only method yielding hierarchies of scale-dependent bounds and their statistics for a wide range of (non)linear elastic and inelastic, coupled-field, and even electromagnetic problems requiring (a) weakly homogeneous random fields and (b) corresponding variational principles. The upscaling leads to statistically homogeneous and isotropic mesoscale tensor random fields (TRFs) of constitutive properties, whose realizations are, in general, everywhere anisotropic. A summary of most general admissible correlation tensors for TRFs of ranks 1, …, 4 is given. A method of solving boundary value problems based on the TRF input is discussed in terms of the torsion of a randomly structured rod. Given that many random materials encountered in nature (e.g., in biological and geological structures) are fractal and possess long-range correlations, we also outline a method for simulating such materials, accompanied by an application to wave propagation.