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.

1. Introduction

Heterogeneous materials nowadays are the essential link of advanced engineering solutions. The advantages of such materials are underpinned by an ability to obtain tailored properties by combining constituents and their characteristics 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 influence of inhomogeneities. 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. Randomly 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. Thus, moments of the stress and strain fields in components can be used as the statistical measures of local mechanical behavior [1]. The morphological properties of the microstructure can be formalized using correlation functions that describe interaction of the inhomogeneities with each other and can sense geometrical effects, 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 fields were offered for some cases of deterministic structures. One of such models is media with regular structure [6–8]. The method of integral equations can be implemented for obtaining analytical expressions for local statistics. It presumes that mechanical properties of microstructural components are defined with conventional phenomenological equations and criteria while microscopic strain and stress fields are computed using the solutions of stochastic boundary value problems (SBVPs) with rapidly oscillating coefficients. The numerous ways of resolving and closing the integral equations were offered [9–21].

The commonly applied simplification when constructing statistical metrics for heterogeneous RVEs is that only lower order interaction between the microstructural elements is being considered. 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.

One of the ground concepts used in statistical analysis of random structures and also applied in this work is based on perturbation approach [4, 16, 22, 23] and presumes that local stress and strain fields as well as local stiffness tensor depend on position of the radius vector and can be decomposed into an averaged value and random addition:where , is radius vector with components . The averaging procedure for any coordinate-dependent value is determined aswhere integration is performed over the representative volume .

2. Approximation of Stochastic Boundary Value Problem Solution

Stochastic boundary value problem (SBVP) of elasticity theory for RVEs of heterogeneous materials with random microstructural elements can be defined with the following equations:where is stiffness tensor, are components of tensor of small strains which define applied load, and are coordinates of the surface points. In this formulation it is assumed that conditions of ideal contact are satisfied on the interphase surface. The boundary value problem is statistically nonlinear since its physical equations (5) contain multiplication of random fields.

Due to ergodicity hypothesis volume averaging is equal to statistical averaging: , .

For two-phase heterogeneous media, containing matrix and one type of inclusions, the stiffness tensor is defined aswhere and are constant stiffness tensors of inclusions and matrix, respectively. The indicator function is defined aswhere is spatial volume of inclusions and is volume of matrix.

After averaging (7), the constant tensor, depending on volume fraction, is obtained:where is volume fraction of inclusions.

Going back to solution of the boundary value problem, (3)–(6) can be transformed into the following representation:

Due to symmetrical properties of the stiffness tensor it can be reduced to

Applying the perturbation approach, (11) can be expressed through mean and fluctuation:

Here, averaged values and are expressed via (9) and boundary conditions (6).

Expanding (12) gives the following:

Denoting the right part as :

Solution of this differential equation in displacements can be found using Green’s function method. According to definition of Green’s functions [24], for the constant tensor Green’s function and its derivatives are equal 0 in infinity and satisfy

Here, is Dirac’s delta function and is Kronecker symbol.

Using Green’s function, fluctuations of displacements are obtained as the following integral equation:

Such representation of fluctuations of the local displacement field can be used in expressions for statistical descriptors of random fields-multipoint moments of various orders.

3. Statistical Descriptors of Local Stress and Strain Fields

In the general case of a multicomponent material, an indicator function takes the value 1 if the radius vector is in component and is equal to 0 in all other cases. Then the local statistical moments of the first and second order in the component of heterogeneous media are determined with the following relations [20]:

Random fields of fluctuations of stresses and strains with regard to the statement of the boundary value problem are defined as

Thus, the moments for the unknown statistical characteristics depend on the fluctuations of the field of the indicator function , the displacements , and strains .

Returning to two-phase materials, taking into account the latest transformations, the moments containing stress fluctuations can be expressed in terms of the moments containing strain fluctuations:

Here, the tensor is the difference of the stiffness tensors of the inclusions and the matrix.

In order to calculate the local moments of the strain and stress fields, it is necessary to know the analytical form of the moment and also mixed unconditional moments and . These moments are not related to certain phase and characterize the RVE as homogeneous medium. All moments necessary for calculating the statistical characteristics depend on the geometry of the microstructure, its physical properties, and loading conditions. The physical properties of the components of the structure are determined by the tensors of the moduli of elasticity of inclusions and the matrix , . The geometry of the structure is characterized by the volume fraction , the central moments of the indicator function , and multipoint high order correlation functions.

4. Integral Equations for Statistical Descriptors

After differentiating and expanding, expression (16) can be rewritten as

The displacement fluctuations , with respect to which the equation is being solved, are both on the left and on the right side of the expression, so the successive approximations are used for the solution. The recurrence sequence has the following form:where is order of approximation.

This paper utilizes the second approximation, which contributes to the accuracy of the solution by taking into account multipoint structural correlation functions:

The first term in expression (22) is the solution of the problem in the first approximation.

Integrals in (21) and (22) as well as expressions for the moments , , can be converted to the second derivative of Green’s function using Stieltjes convolution. The Stieltjes convolution of the two functions and over the interval is defined as function :

Application of this definition to the integrals containing the first derivative of Green’s function and the derivative of the indicator function in the solutions of the boundary value problem (22) gives the next form of the integrals:

For a two-phase inhomogeneous material, the solution using the second derivatives of the Green’s functions takes the form

The product of the fluctuations of the indicator function is the correlation function , which characterizes the relative position of the components in microstructure.

By definition, Green’s function satisfying the expression (15) has a singularity at points . To apply numerical integration methods, the second derivative of Green’s function can be represented as the sum of the singular and formal components .