The effective elasticity tensors for dispersed composites

Fundamental Solutions for Periodic Media

“…Lemma 3 (see [14]). The series on the right side of (10) is convergent in the 1 -topology, defining fundamental solution of the class 1 ( , 3 ⊗ 3 ), where 1 is a class of integrable in functions with the zero mean value.…”

“…Lemma 4 (see [14]). Under assumptions stated above, the boundary value problem (12) admits the unique solution.…”

“…Presumably, for the first time periodic fundamental solutions for media with arbitrary anisotropy were developed in [13]. In combination with the boundary integral equation method (BIEM) these fundamental solutions were applied to the cell problem for composites with anisotropic inhomogeneities and porous media in [14,15], while analysis of microstructural stresses in the matrix material was considered in [16]. Problems of wave scattering by pores were studied in [17,18] by the same method.…”

“…For clarity and simplicity it will be assumed further that the considered medium has only one kind of uniformly distributed inhomogeneities placed in nodes of spatial lattice Λ (refer to Figure 3). The region occupied by an individual inhomogeneity in a cell is denoted by Ω. Two-scale asymptotic analyses being applied to such a medium produce the following expression for the corrector [14]:…”

“…Solution of the boundary value problem (12) is constructed by the boundary integral equation method, giving the following representation for the desired solution [14]:…”