Conditionally Averaged Response Formulations for Two-Phase Random Mixtures

Abstract

A technique known as a projection or a smoothing, which has been used successfully to derive formulations on the mean (or unconditionally averaged) field response of specimens with a random substructure, is extended to obtain formulations on conditionally averaged response measures for two-phase mixtures. The condition in the averaging refers to the location of the field point, in one or the other of the phases. The obtained formulation has the structure of a theory for interacting mixtures of nonlocal continua. The formulations are then investigated in a two-scale microscale/macroscale limit; specifically we determine the conditions necessary for the obtained formulations to reduce to local formulations which can be interpreted to provide bases for physical theories. It is argued that for weakly coupled, two-phase mixtures for which both phases are connected over distances that are measured on the macroscale, the mixture theory-type formulation is local in the limit whereas the mean-field formulation is not. In the presence of strong coupling, or for mixtures in which one of the phases is connected only for distances measured on the microscale, both type formulations are local in the two-scale limit.