  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, a 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 properties 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 and extend them to conductive materials. This article presents a
generalization of our recent theoretical study on the influence of the
stiffness of a single fiber on the elastic stiffness of a network of fibers to
arbitrarily-shaped inhomogeneities and different transport phenomena.
