A least squares approach for effective shear properties in an n-layered sphere model

This work presents the derivation of the effective shear modulus for a heterogeneous material composed of multilayered composite spheres embedded in a linear elastic matrix. It is based on the composite spheres model known from the literature. In contrast to Herve and Zaoui (Int J Eng Sci 31:1–10, 1993 ), the effective shear modulus is obtained by equating the results of two models: In the first model, a heterogeneous sphere is embedded in an equivalent homogeneous material, whereas in the second model, the heterogeneous sphere is replaced by an equivalent homogeneous sphere. In the context of both, a shear stress approach and a shear deformation approach, this results in an overdetermined system of equations which is solved with the least squares method. In a numerical study, our results are compared to effective moduli and bounds from the literature. Furthermore, a convincing agreement with experimental data for glass microspheres embedded in a polyester matrix is demonstrated.