Explicit effective elasticity tensors of two-phase periodic composites with spherical or ellipsoidal inclusions

The effective elasticity tensors of two-phase composites are estimated by solving the localization problem in the wave-vector domain for the case of non overlapping spherical or ellipsoidal inclusions. With previous works showing that the effective properties can be computed from lattice sums, we propose a method to compute the sums analytically and obtain the explicit expressions for the effective tensors. In the case of different periodic cells leading to cubic or orthotropic elasticity tensors, the effective elasticity tensors are obtained in closed forms that are in good agreement with the exact solutions for a large range of physical parameters. In the random distribution cases, the statistical connection of the effective tensor to the structure factor is shown and a closed-form expression is obtained in the infinite volume limit.