Abstract :
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. © 1998 Elsevier Science Ltd. All rights reserved
