Random elastic media: why zero mean stress does not imply zero mean strain

Abstract

In this note we are concerned with (linearized) elastic media that are heterogeneous on the microscale and homogeneous on the macroscale. We assume the validity of an ergodic hypothesis so that we can form ensemble averages such as mean stress ⟨σ⟩ and mean strain ⟨ε⟩. We argue that ⟨σ⟩ = 0 does not, in general, imply that also ⟨ε⟩ = 0 (end of Sec. 2). This is the case, for instance, when the distribution of the stress sources (external forces or incompatibilities) are correlated with the spatial distribution of the local elastic moduli. It is shown how problems of this type can be treated.