On the applicability of replacement relations to tetrahedron-like inhomogeneities

Abstract

In this paper, we consider the problem of elastic materials containing tetrahedron-like inhomogeneities with spherical polygons as faces. The overall properties are expressed in terms of compliance contribution tensors calculated for inhomogeneities of different face curvature and elastic properties. The calculations were performed by two numerical techniques: Finite Element Method (FEM) and Volume Integral Equation Method (VIEM) combined with mesh-free discretization by Gaussian approximating functions. Consistency of the results obtained by these methods is observed in most cases. Applicability of replacement relations that predict the contribution to the overall elastic properties of inhomogeneities with different elastic properties, but the same shape, was analyzed. The replacement relations provide considerably good approximation for “soft” inclusions (Young's modulus lower than the matrix’) only, in certain cases wrong approximation trends were observed. It was possible to propose an adjustment tensor for the replacement relations which components depend linearly on the face curvature.