On the homogenization and the simulation of random materials. (English) Zbl 0766.73008

Due to the increasing capacities of computers, it has become possible to compute the homogenized behaviour of a random medium by the simulation of a representative volume element submitted to homogeneous stress and strain boundary conditions. The present work is devoted to the analysis of the approximations adopted in such a numerical method. The equivalence between stress and strain boundary conditions in the computation of the homogenized properties of an elastic medium is proved without any statistical or periodic assumption on the microstructure. Then, it is established with the aid of a recent ergodic theorem [M. A. Akcoglu and U. Krengel, J. Reine Angew. Math. 323, 53-67 (1981; Zbl 0453.60039)] that the overall properties of a sufficiently large domain in an ergodic random medium are deterministic. Further, if this medium is suitably discretized, then, the finite element computations of these overall properties are also deterministic.


74E05 Inhomogeneity in solid mechanics
74A40 Random materials and composite materials
74S05 Finite element methods applied to problems in solid mechanics


Zbl 0453.60039