Homogenization for random micropolar composites. The case of masonry-like materials.

The worldwide concern for the stewardship of old structures and gems of ancient architecture calls for mechanics models of walls built of stones with random properties and shapes. Among different typologies according to the common constructive technologies, we focus our attention on two revealing random patterns: concrete masonry, in which aggregates of small size are surrounded by a mortar matrix, and “rubble” masonry where randomly distributed stones are held together by small amounts of mortar. The aim of this work is to study the constitutive behavior of such materials adopting homogenization techniques in order to obtain the overall constitutive coefficients directly related to the actual structure. Unlike the periodic homogenization in which the size of the Representative Volume Element(RVE) is a datum of the problem, in this case the minimal size of RVE is a-priori unknown. We adopt a statistical homogenization procedure, well established for standard continua, in order to obtain information to detect the minimal size of RVE and to estimate the effective constitutive properties of such materials. Under the assumption that microstructure’s statistics is spatially homogeneous and ergodic, a so-called Statistical Volume Element(SVE) can be set up on a mesoscale, i.e. any finite scale relative to the microstructural length scale, and, on that basis, by solving Dirichlet and Neumann boundary value problems, two scale dependent hierarchies of mesoscale bounds for the effective material properties can been obtained. The convergence trend, as the SVE increases, allow one to approximate the RVE size to adopt for performing the homogenization process. We here propose an extension of the aforementioned statistical homogenization procedure for composites perceived as Cosserat continua, both at the micro and macro level. The choice of micropolar continua modeling is related to the possibility of taking into account effects of material internal lengths and the non-symmetries of strain and stress tensors. This procedure exploits the macrohomogenity condition generalized to micropolar continua, which holds also in the case of spatial nonperiodicity. The linear-elastic case is taken into account, providing a computational estimate of the bounds for the classical and micropolar elastic coefficients for the two kind of random media considered. The numerical results, holds for several kind of composite materials, among which masonry is just one example.