The elastic moduli of simple two-dimensional isotropic composites: Computer simulation and effective medium theory

Abstract

An algorithm, combining digital‐image with spring network techniques, has been developed that enables computation of the elastic moduli of random two‐dimensional multiphase composites. This algorithm is used to study the case of isotropic, randomly centered, overlapping circular inclusions in an isotropic elastic matrix. The results of the algorithm for the few‐inclusion limit, as well as the case where both phases have the same shear moduli, agree well with the exact results for these two problems. The case where the two phases have the same Poisson’s ratio, but different Young’s moduli, is also studied, and it is shown that the effective medium theory developed by Thorpe and Sen agrees well with the numerical results. A surprising result is that the effective moduli of systems with nonoverlapping circular inclusions are almost identical with the overlapping inclusion case, up to an inclusion area fraction of 50%. Using the validated effective medium theory, we illustrate how the effective Poisson’s ratio ν behaves as a function of ν0, the pure phase Poisson’s ratio, and the stiffness ratio between the two components. Two distinct regions of behavior are found, defined by ν0≳ν* and ν0<ν*, where ν* is approximately 1/3 for the geometry of this model.