Random field models of heterogeneous materials

One of the main challenges in solid mechanics lies in the passage from a heterogeneous microstructure to an approximating continuum model. In many cases (e.g. stochastic finite elements, statistical fracture mechanics), the interest lies in resolution of stress and other dependent fields over scales not infinitely larger than the typical microscale. This may be accomplished with the help of a meso-scale window which becomes the classical representative volume element (RVE) in the infinite limit. It turns out that the material properties at such a mesoscale cannot be uniquely approximated by a random field of stiffness/compliance with locally isotropic realizations, but rather two random continuum fields with locally anisotropic realizations, corresponding, respectively, to essential and natural boundary conditions on the meso-scale, need to be introduced to bound the material response from above and from below. We study the first-and second-order characteristics of these two meso-scale random fields for anti-plane elastic response of random matrix-inclusion composites over a wide range of contrasts and aspect ratios. Special attention is given to the convergence of effective responses obtained from the essential and natural boundary conditions, which sheds light on the minimum size of an RVE. Additionally, the spatial correlation structure of the crack density tensor with the meso-scale moduli is studied.