On the construction of effective constitutive relations for porous elastic materials subjected to finite deformations including the case of their superposition

Abstract

This study deals with a new approach to the construction of effective constitutive relations for porous nonlinearly elastic materials subjected to finite deformations. The approach allows for superimposing the deformations when pores are formed after a preliminary loading. The effective constitutive relations based on this approach are shown to remain unchanged in the case of a superimposed rigid motion. The construction of the effective constitutive relations is performed on the basis of well-known general principles [1, 2]. We isolate in a material a certain representative volume (for the two-dimensional case, this is a certain area) and, studying its mechanical behavior under loading, we can determine properties of the material as a whole. For this volume (area), the static problem of nonlinear elasticity is solved for a given stress at the boundary. Furthermore, deformations and stresses are averaged over the representative volume (area), and the effective constitutive relations are constructed as a dependence between averaged deformations and averaged stresses. When solving problems on effective properties of inhomogeneous materials, a question arises: how to define an effective material in order that this definition would have both a clear physical meaning (for use in experiments) and a sufficiently simple mathematical representation. In addition, in the process of seeking average deformations of porous media, a problem arises associated with the fact that the deformations are not defined in the interior of pores (this is the difference between porous materials and those with elastic inclusions). In the case of small deformations, these issues are solved in a rather simple manner [1, 3, 4]. For example, the deformation tensor of an effective material is