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

摘要

Investigation of inhomogeneities has wide applications in different areas ofmechanics including the study of composite materials. Here, we analyticallystudy an arbitrarily-shaped isotropic inhomogeneity embedded in a finite-sizedheterogeneous medium. By modal decomposition of the influence of theinhomogeneity on the deformation of the composite, a relation is presented thatdetermines the variation of effective elastic stiffness caused by the presenceof the inhomogeneity. This relation indicates that the effective elasticstiffness of a composite is always a concave function of the properties of theinhomogeneity, embedded inside the composite. Therefore, as the heterogeneityof elastic random composites increases, the rate of increase in effectivestiffness caused by the stiffer constituents is smaller than the rate of itsdecrease due to the softer constitutions. So, weakly heterogeneous randomcomposites become softer and less conductive with increasing heterogeneity atthe same mean of constituent properties. We numerically evaluated the effectiveproperties of about ten thousand composites to empirically support theseresults and extend them to conductive materials. This article presents ageneralization of our recent theoretical study on the influence of thestiffness of a single fiber on the elastic stiffness of a network of fibers toarbitrarily-shaped inhomogeneities and different transport phenomena.