Multipoint Approximation of Statistical Descriptors of Local Strain and Stress Fields in Heterogeneous Media Using Integral Equation Method

Mikhail Tashkinov

doi:10.1155/2018/5785193

Abstract

This paper is devoted to derivation of analytic expressions for statistical descriptors of stress and strain fields in heterogeneous media. Multipoint approximations of solutions of stochastic elastic boundary value problems for representative volume elements are investigated. The stress and strain fields are represented in the form of random coordinate functions, for which analytical expressions for the first- and second-order statistical central moments are obtained. Such moments characterize distribution of fields under prescribed loading of a representative volume element and depend on the geometry features and location of components within a volume. The information of the internal geometrical structure is taken into account by means of multipoint correlation functions. Within the framework of the second approximation of the boundary value problem, the correlation functions up to the fifth order are required to calculate the statistical characteristics. Using the method of Green’s functions, analytical expressions for the moments in distinct phases of the microstructure are obtained explicitly in a form of integral equations. Their analysis and comparison with previously obtained results are performed.

Highlights

Heterogeneous materials nowadays are the essential link of advanced engineering solutions
In the context of solving problems connected with evaluation of microstructural behavior of such heterogeneous materials and media, it is necessary to take into account mutual influence of inhomogeneities
Reinforced media can be distinguished in a separate class of materials where local mechanical characteristics can be estimated using statistical instruments and the theory of random functions

Summary

Introduction

Heterogeneous materials nowadays are the essential link of advanced engineering solutions. As a rule, modeling of the effective response of materials is performed within the concept of representative volume element (RVE) In these frameworks, wide range of mechanical approaches can be used to establish connection between different scales of materials with complex microstructure. Several exact solutions for second-order moment of stress fields were offered for some cases of deterministic structures One of such models is media with regular structure [6,7,8]. The method of integral equations can be implemented for obtaining analytical expressions for local statistics It presumes that mechanical properties of microstructural components. This work contains explicit derivation of the expressions for local statistical descriptors of stress and strain in heterogeneous media using multipoint high order approximation of SBVP solution and second derivative expansion of Green’s function approach.

Approximation of Stochastic Boundary Value Problem Solution

Statistical Descriptors of Local Stress and Strain Fields

Integral Equations for Statistical Descriptors

Conclusions