Limit distributions of the states and homogenization in random media

Abstract

This paper aims at formulating a new approach to handle some homogenization problems in random media. First, an asymptotic definition of a statistically homogeneous (S.H.) medium is obtained by comparing the probability measures of the local “states” in different samples: a definition of “representative” samples is given. If arbitrarily representative samples exist, the measures of the states tend towards a limit measure as the size of the sample increases. The “state” is a data-carrier for the local behaviour, it belongs to a general “space of states”. It may not depend continuously on the position, as in aggregates. The proposed approach consists in a transport of the homogenization problem from the physical space to the space of states. In operational models for inhomogeneous media, the fields of local solicitation and response are often replaced by corresponding functions of the state. This is justified here under a precise definition of S.H. fields. Moreover, the nocorrelation condition between macrohomogeneous S.H. fields of solicitation and response is proved to pass to the corresponding functions of the state. Finally, a variational model is proposed for S.H. media with potential local response, generalizing a model for plastically deformed polycrystals. It is based on the fact that one single numerical parameter of the average local inhomogeneity is sufficient to set the exact (statistical) solution between the two extreme bounds of the average potential. A general way is given to take into account the effect of spatial correlations.