An Efficient Numerical Method for Calculation of Elastic and Thermo-Elastic Fields in a Homogeneous Medium with Several Heterogeneous Inclusions

Sergey Kanaun

doi:10.4236/wjm.2011.12005

Abstract

The work is devoted to calculation of static elastic and thermo-elastic fields in a homogeneous medium with a finite number of isolated heterogeneous inclusions. Firstly, the problem is reduced to the solution of inte-gral equations for strain and stress fields in the medium with inclusions. Then, Gaussian approximating func-tions are used for discretization of these equations. For such functions, the elements of the matrix of the dis-cretized problem are calculated in explicit analytical forms. The method is mesh free, and only the coordi-nates of the approximating nodes are the geometrical information required in the method. If such nodes compose a regular grid, the matrix of the discretized problem obtains the Toeplitz properties. By the calcula-tion of matrix-vector products with such matrices, the Fast Fourier Transform technique may be used. The latter accelerates essentially the process of the iterative solution of the disretized problem. The results of calculations of elastic fields in 3D-medium with an isolated spherical heterogeneous inclusion are compared with exact solutions. Examples of the calculation of elastic and thermo-elastic fields in the medium with several inclusions are presented.

Highlights

Calculation of elastic fields in homogeneous materials with isolated heterogeneous inclusions is an important problem of stress analysis of composites and materials with defects
The problem should be solved only in the region occupied by the inclusions. This is the main advantage of the Integral Equation Method (IEM) over the Finite Element Method (FEM), where the fields in the medium as well as in the inclusions are equivalent unknowns, and the solution has to be found in the whole region
The method may be applied to the solution of other problems of mathematical physics that can be reduced to volume integral equations

Summary

Introduction

Calculation of elastic fields in homogeneous materials with isolated heterogeneous inclusions is an important problem of stress analysis of composites and materials with defects. Efficient numerical methods of the solution of this problem are based on the volume integral equations for the fields in heterogeneous media (see, e.g., [1,2,3]). By the use of these equations, the fields inside the inclusions become principal unknowns of the problem. If the fields inside the inclusions are known, the fields in the medium are reconstructed from the original integral equations. The problem should be solved only in the region occupied by the inclusions. This is the main advantage of the Integral Equation Method (IEM) over the Finite Element Method (FEM), where the fields in the medium as well as in the inclusions are equivalent unknowns, and the solution has to be found in the whole region. The IEM is preferable if inclusions being smaller than the characteristic sizes of the body are situated far from its boundary

Methods

Results

Conclusion