Solving the problems of deformation mechanics of polycrystalline materials on the basis of perturbation theory

Abstract

A method for solving the boundary value problem of highly inhomogeneous polycrystalline solids is offered. The method is based on the original scheme of a perturbation theory used to solve integral equations in mathematical physics. It consists in splitting the initial equation into an unperturbed (zero) part plus a perturbation. The intergrain interaction of deformations is taken as a perturbation to expand the solution. The solution for deformation in each grain is represented as a sum of interaction corrections of different order. These corrections satisfy the infinite chain of interconnected systems of integral equations. The structure of equation chains is so that every equation can be solved by the same method. Neglecting the inhomogeneities of deformation within a separate grain reduces the systems of integral equations to the systems of linear algebraic equations, which are easy to solve by contemporary numerical methods. It is shown that the effects of interactions of grain deformations can be described by fourth rank tensors. For two grains of any types of anisotropy, this tensor has 36 independent components. A solution to the boundary value problem is constructed as a power series in the components of this tensor. The mathematical formalism of the method is stated. Convergence of the solution of intergrain interaction is investigated. The method is applicable to several kinds of problems. One application of the method is the determination of intergrain interaction intensity. It is shown that that the intensity depends to a greater extent on the elastic anisotropy of grains and to a less extent on grain shapes and that it decreases rather slowly with distance. New estimates of representative volume element sizes for polycrystalline materials are given. The overall sizes of a polycrystalline solid when it can be treated as a homogeneous body with effective properties are evaluated.