Spectral analysis of inhomogeneities shows that the elastic stiffness of random composites decreases with increasing heterogeneity

Abstract

Investigation of inhomogeneities has wide applications in different areas of mechanics including the study of composite materials. Here, we analytically study an arbitrarily-shaped isotropic inhomogeneity embedded in a finite-sized heterogeneous medium. By modal decomposition of the influence of the inhomogeneity on the deformation of the composite, an exact relation is presented that determines the variation of effective elastic stiffness caused by the presence of the inhomogeneity. This relation indicates that the effective elastic stiffness of a composite is always a concave function of the elastic modulus of the inhomogeneity, embedded inside the composite. Therefore, as the heterogeneity of elastic random composites increases, the rate of increase in effective stiffness caused by the stiffer constituents is smaller than the rate of its decrease due to the softer constitutions. So, weakly heterogeneous random composites become softer and less conductive with increasing heterogeneity at the same mean of constituent properties. We numerically evaluated the effective properties of about ten thousand composites to empirically support these results, characterize the nonaffinity of the displacement field in random composites, characterize the sample to sample variability of the effective properties and extend the results to the thermal conductivity of composites.