Research Article Multipoint Approximation of Statistical Descriptors of Local Strain and Stress Fields in Heterogeneous Media Using Integral Equation Method Mikhail Tashkinov Faculty of Applied Mathematics and Mechanics, Perm National Research Polytechnic University, Komsomolsky Ave. 29, Perm 614990, Russia Correspondence should be addressed to Mikhail Tashkinov; m.tashkinov@pstu.ru Received 30 July 2017; Accepted 28 March 2018; Published 6 May 2018 Academic Editor: Ricardo Weder Copyright © 2018 Mikhail Tashkinov. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tis paper is devoted to derivation of analytic expressions for statistical descriptors of stress and strain felds in heterogeneous media. Multipoint approximations of solutions of stochastic elastic boundary value problems for representative volume elements are investigated. Te stress and strain felds are represented in the form of random coordinate functions, for which analytical expressions for the frst- and second-order statistical central moments are obtained. Such moments characterize distribution of felds under prescribed loading of a representative volume element and depend on the geometry features and location of components within a volume. Te 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 ffh 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. Teir analysis and comparison with previously obtained results are performed. 1. Introduction Heterogeneous materials nowadays are the essential link of advanced engineering solutions. Te advantages of such materials are underpinned by an ability to obtain tailored properties by combining constituents and their characteris- tics on microscale level. 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 infuence of inhomogeneities. As a rule, modeling of the efective response of materials is performed within the concept of representative volume ele- ment (RVE). In these frameworks, wide range of mechanical approaches can be used to establish connection between diferent scales of materials with complex microstructure. Randomly reinforced media can be distinguished in a sepa- rate class of materials where local mechanical characteristics can be estimated using statistical instruments and the theory of random functions. Tus, moments of the stress and strain felds in components can be used as the statistical measures of local mechanical behavior [1]. Te morphological properties of the microstructure can be formalized using correlation functions that describe interaction of the inhomogeneities with each other and can sense geometrical efects, such as shape, distribution, clustering, and percolation [2]. Statistical assessment of mechanical behavior can be done analytically for each phase of media. Kroener [3] and Beran [4] developed statistical mathematical formulations in order to connect geometrical correlation functions and properties of heterogeneous materials. Explicit relations for second moments of stresses were obtained in [5] and utilized second- and third-order interactions between inclusions. Several exact solutions for second-order moment of stress felds were ofered for some cases of deterministic structures. One of such models is media with regular structure [6–8]. Te method of integral equations can be implemented for obtain- ing analytical expressions for local statistics. It presumes that mechanical properties of microstructural components Hindawi Advances in Mathematical Physics Volume 2018, Article ID 5785193, 9 pages https://doi.org/10.1155/2018/5785193