Controlled accuracy finite element estimates for the effective stiffness of composites with spherical inclusions

Mixed form finite element method was used to study the microstructural effects in the effective stiffness of composites reinforced or weakened by regular and random dispersions of non-overlapping identical spheres. Controlled accuracy stiffness estimates were achieved by using the unstructured mesh polynomial extension and asymptotic nested grid extension of the finite element method. Periodic regular and random microstructure computer models with up to a hundred spheres were considered. The significance of the elastostatic bridges separating near-to-touch spheres was studied and their impact on the effective stiffness was assessed. It was found that for a variety of practically relevant composites with moderate stiffness contrast between the phases, the effective stiffness estimates were remarkably insensitive to the underlying composite microstructure. The obtained finite element estimates were compared with predictions of the generalized self-consistent (GSC) model (Christensen and Lo, 1979) and it was shown that the model was accurate over the entire range of practically relevant sphere fractions. We studied a high stiffness contrast composite consisting of a near incompressible rubber matrix filled with rigid spherical inclusions. In this case strong microstructural effects were observed but, nevertheless, for the random composite obeying the hard sphere Percus-Yevick statistics the GSC model remained strikingly accurate over the whole range of practically relevant inclusion fractions. The asymptotic Einstein limit GSC solution was derived and it was demonstrated that it gave accurate micromechanics-based solution of the classical Einstein homogenization problem of a random dispersion of perfectly rigid spherical inclusions in a fully incompressible matrix.