Statistical methods for mechanical characterization of randomly reinforced media

One of the main problems of materials science is establishing connection between the physical and mechanical properties of materials, their microstructural features and parameters influenced by manufacturing process. For many centuries, the main option for investigation of this link remained direct experiments. Gathering the vast amount of empirical information was the only way to progress in creation of advanced materials. New models and numerical simulation techniques which had started to emerge in twentieth century significantly increased possibilities of production of materials with tailored parameters of microstructure to suit needs of the specific engineering applications. That allowed to substitute resource consuming experimental identification with mathematical modeling and gave a boost to creation of new classes of materials. Among the most highly potential are heterogeneous materials that consist of several phases with distinguished properties combined together. Examples of materials belonging to this class are particle and fibre reinforced composites, polycrystalline metal and ceramic alloys, poly-phase amorphous materials and some classes of polymers (Fullwood et al. 2010). In order to consider specifics of their microstructure, which frequently contains randomly placed elements, the special simulation techniques were developed.

The problem of modeling of heterogeneous structures is historically based on the mechanics of liquids and is tightly connected with the problem of statistical description of behavior of particles systems (Torquato 2000; Binder and Heermann 2010). The most common approach is to investigate heterogeneous media at different scales. The microstructural peculiarities are usually being studied within the concept of representative volume element (RVE). This element can be defined as the smallest material’s volume at which the properties of phases can be distinguished and for which effective (homogenized) properties can be measured.

RVE is usually being characterized at two scales. At the lowest, which is yet by several orders larger than the atomic scale, constitutive behavior at the material point is described using traditional continuum equations. Local properties and fields at this scale oscillate when moving from one heterogeneity to another – that could be phases in composite materials or grains in polycrystals.

To avoid computationally expensive models involving a large number of structural elements at the microscale, the higher (macroscopic) scale RVE description is usually being used. It operates effective properties, which can be constant or varying smoothly, and are defined basing on the microstructural specifics.

Morphology of RVE plays an important role in modelling of effective thermomechanical and physical properties of random heterogeneous materials (Kanit et al. 2003; Khisaeva and Ostoja-Starzewski 2006). During the latest decades, a large number of models, theories and approaches were developed for characterization of inhomogeneous materials. Their description is given, for instance, in (Kamiński 2005; Buryachenko 2007; Kanouté et al. 2009). One of the direction is to attract statistics and theory of random function to describe random systems, which in most cases heterogeneous materials can be considered as.

There are several assumptions about the statistical properties of the microstructure that are commonly being taken. First of all, the statistical homogeneity hypothesis presumes that each RVE sampled from the material will have the same statistical properties (such as, for instance, volume fraction). This assumption is usually followed by the ergodicity hypothesis, which means that ensemble average of values over a set of materials’ RVEs is equal to averaging over a single RVE.

This paper offers a review of techniques which are based on or utilize the statistical instruments for characterization of internal structure, effective behavior and local stress and strain fields of heterogeneous media. It is primarily focused on the approaches for characterization of randomly reinforced media, although some of the instruments are also suitable for the other classes of heterogeneous materials (the respective remarks are given in the text).

The section 2 of the paper is related to application of statistical descriptors to obtaining formalized information about microstructural morphology. Such information can be used both for stochastic mechanical models or as a ground for geometrical operations like media reconstruction. The most commonly used descriptors are introduced, their development in terms of various applications is discussed. Methods of geometrical modelling of random structures are also addressed.

The section 3 contains a summary of the homogenization approaches which in certain ways rely on statistical information about the microstructure. Recent advances, which justify involvement of multipoint and high order correlation functions, are presented.

The section 4 continues discussion of the methods of stochastic mechanics in application to investigation of local fields of stress and strain in components and interfaces between them. This is determined by necessity of rigorous estimation of the critical zones when it comes to analysis of damage and non-linear behavior. Statistical moments for fields of stress and strain can be introduced for particular elements of the microstructure and give opportunity to indicate local changes and the respective response of microscopic fields.

In addition to solving the problems of morphological analysis, statistical descriptors can be used as an instrument for establishing the relationship between microstructure, local and effective properties of heterogeneous media. Finding suitable relationship between distribution, shape and properties of each constituent and macroscopic response of the heterogeneous material is one of the most investigated questions of micromechanics. Among the variety of the homogenization methods there is a broad class of those rely on statistical approach. Some advantages in these methods are introduced below.

The perturbation method is usually suitable for characterization of weakly inhomogeneous media (Lomakin 1970; Beran and McCoy 1970), which means that they are expected to give correct results in case of small differences between the components properties (Shermergor 1977) or small concentration of inhomogeneities (Jeffrey 1973; Lu and Song 1996).

The mean field approach facilitates the calculation of the averaged field quantities in the constituent phases and resultant determination of the effective properties (Budiansky 1965; Hill 1965; Willis 1992; Benveniste 2008).

One of the homogenization schemes is represented by the self-consistent methods, according to which analysis is performed for a single particle of a typical constituent embedded into the effective medium instead of studying the whole RVE embedded in a homogeneous medium.

The Green’s function in (11) has a singularity in point \( \overrightarrow{r}={\overrightarrow{r}}_1 \). The approaches that helps to overcome it and some exact expressions are obtained in (Shermergor 1977; Kroener 1986; Torquato 1997, 2002). The results of application of the first and second derivatives of the Green’s function to resolve this issue is discussed in (Tashkinov 2015).

Series of works by S. Torquato were devoted to application of perturbation method and integral equations with Green’s function to derive effective conductivity and elastic properties for two-phase media. Some exact solutions for the specific types of media are presented in (Torquato 2002).

Some other statistical models that focus on local response are based on the percolation theory and Weibull analysis (Jayatilaka and Trustrum 1977; Frary and Schuh 2005; Chen and Schuh 2006), and may also be used to connect micro- and macro- properties depending on statistical information about the microstructure. Alternative approaches rest upon neural networks, which improve fitting models for modeled and observed microstructural data (Bhadeshia 1999).

Spectral methods, incorporating microstructural correlation functions, use spectral representations of distribution functions that statistically characterize the internal structure of representative volumes and connect these representations and macroscale effective properties using existing homogenization schemes (Kalidindi and Houskamp 2007).

The known fact is that it is impossible to achieve exact prediction for effective properties of random media, so rigorous assessment can be done in the form of bounds. One of the most commonly used bounds are based on variational principle according to which the effective parameter is expressed in terms of functional for which the variational (extremum) principle is being formulated. Primarily, variational methods for homogenization purposes analyze approximate fields substituted into energy bounds by averaging the energy density (Hashin and Shtrikman 1963), while classical variational methods employ minimum energy principles to generate bounds on effective properties. The microstructural information is considered via different types of correlation functions (Willis 1981; Torquato 1991; Ponte Castaneda and Suquet 2001; Milton 2002). The correlations which are required for calculation of the bounds of effective values using variational methods express, for instance, probability of finding n points with position \( {r}^n=\left\{{\overrightarrow{r}}_1,\dots, {\overrightarrow{r}}_n\right\} \) in phase i for statistically inhomogeneous media, finding a point with position \( {\overrightarrow{r}}_1 \) in the exterior to the phase particles and any n particles with coordinates r ⁿ , finding points at the interface, finding a nearest neighbor at radial distance r from the origin. Each type of these functions depends upon the relative position of the n points inside the RVE. The bounds of the effective properties are usually characterized by the number of points involved in the correlation functions (Torquato 1991).

Improved bounds, which were developed later and are tighter than the traditional Hashin-Shtrikman bounds, depend on high order correlation functions rather than volume fractions. Three-point bounds were developed in (Beran 1971; Torquato 1980, 1991; Milton 1982; Castaneda 1998; Agoras and Ponte Castañeda 2011; Ponte Castañeda 2012). Mikdam and Baniassadi (Mikdam et al. 2009, 2010, Baniassadi et al. 2011, a) have applied the strong-contrast formulation to predict the effective electrical and thermal conductivities of a two-phase composites using two-point and three-point correlation functions. The current limitation is that in the case of strong inhomogeneity of microstructure these bounds might not be helpful because they tend to diverge with increasing contrast between the properties of the constituents (Torquato 2000; Buryachenko 2007).

Large amount of work on developing of the variational estimates of effective properties of heterogeneous media has been done by P.P. Castaneda, the recent developments in that area as well as review of the previous research can be found in (Ponte Castaneda 2016).

In order to simplify modeling, the real stochastic structure can be replaced by some specific one, which is then being analyzed. Among such substituends are fractal-type media based on spherical or cylindrical coated inclusions (Hashin and Shtrikman 1962; Benveniste and Milton 2003), two-phase composites with statistically equivalent phases (Dykhne 1970), regular structures (Keller 1963; McKenzie et al. 1978; McPhedran and McKenzie 1978), unit cell with randomly dispersed inclusions (Kushch 1997; Guseva and Lusti, 2004) or percolation models (Sahimi 1998).

There are many other homogenization techniques exist that utilize different instruments for characterization of specifics of microstructure as well as linear and nonlinear behavior of materials. For instance, Rasool and Boehm (Rasool and Böhm 2012) compared inhomogeneity shape effects on the linear elastic, thermoelastic and thermal conduction effective responses of composites. Klusemann (Klusemann et al. 2012) studied three strategies for dealing with inhomogeneities of non-elliptical shape in the context of homogenization method: mean-field methods used in combination with analytical expressions for the Eshelby tensor, Mori-Tanaka method in combination with the replacement tensor approach and direct Finite Element discretization of microstructures. The review of multiscale methods for modelling mechanical and thermomechanical responses of composites can be found, for example, in (Kanouté et al. 2009).

The problem of prediction of effective properties in some approximation can be solved involving only the averaging statistical descriptors. However, models of failure and nonlinear analysis are tied on the specific features of local stress and strain fields where microscale fluctuations must be taken into account. Special attention should be given to the interfaces between the constituents and their vicinities as those are the most common zones of failure initiation. The random structure of heterogeneous materials leads to necessity of analysis of large number of system’s realizations (Buryachenko 2007). To eliminate that, distribution of fields in random structures can be estimated using the statistical moments of stress and strain fields that can be obtained either for homogeneous RVE or for its constituents separately as well as for the interface between the matrix and inhomogeneities. At this case, the same techniques as was developed for analysis of effective properties can be applied. The importance of the stress fluctuations is discussed in details in (Buryachenko 1996; Ponte Castaneda and Suquet 2001) for a wide class of nonlinear problems of micromechanics such as plasticity, damage, viscosity, or creep. The multipoint statistical moments of the stochastic stress and strain fields are used as the characteristics of the deformation processes in the components of the material.

The above described perturbation approach as well as method of integral equations can be applied for calculation of statistical descriptors for the local stress and strain fields in microstructure.

A general scheme for calculating the second-order moments of the random elastic fields in the case of a composite of inclusion-matrix type using perturbation approach is presented, for example, in (Buryachenko 2007). Kroener (Kroener 1986) and Beran (Beran 1965) developed statistical mathematical formulations to link correlation functions to properties in multiphase materials.

The exact estimation of all components of the second moment tensor of the pure elastic and internal residual stresses using perturbation approach is given in (Buryachenko and Kreher 1995), where it was shown that the second moment of the stress field is constant within the inclusions if a homogeneity of random effective stress fields in the neighborhood of each ellipsoidal inclusion was assumed. Explicit relations for second moments of stresses were obtained in (Buryachenko and Rammerstorfer 1998) and utilized second and third order interactions between inclusions.

Combination of perturbation method and variational principle (Bergman 1978), which is based on the estimation of the perturbation of an energetic function due to a variation of the material properties, was developed for estimation of stress fluctuations for RVEs with isotropic components (Bobeth and Diener 1986; Kreher 1990). Xu (Xu et al. 2009) employed the generalized variational principles to decompose a boundary value problem with random microstructure into a slow scale deterministic problem and a fast scale stochastic one using a Green’s function based multiscale method.

Recursive approach for computing a probability density function of stress fields is described in (Hori and Kubo 1998). Numerical statistical analysis at the inclusion scale level was performed by (Babuška et al. 1999).

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 (Evans 1978; Fu and Evans 1985; TVERGAARD and HUTCHINSON 1988). However, their limitations are connected with lack of consideration of spatial distribution of constituents which may have a significant effect on the local stresses.

Here, \( \left.{u}_i^{\hbox{'}}\Big(\overrightarrow{r}\right) \) is fluctuation of displacements, \( {G}_{ij}\left(\overrightarrow{r},{\overrightarrow{r}}_1\right) \) is the Green’s function. \( \left.{P}_{jn}\Big({\overrightarrow{r}}_1\right) \) is a functional, containing constants from boundary conditions as well as stiffness tensor. The structure of the functional is different depending on the ways of solution.

The study of behavior of composites with random structures in these frameworks was established by works of Lifshitz and Rosenzweig (Lifshitz and Rosenzweig 1946) which were devoted to SBVPs of elasticity theory for polycrystalline media. Depending on the SBVP solution different statistical models can be distinguished. The numerous ways of closing the integral equations were offered – the method of effective medium (Hashin 1968; Buyevich 1992; Koelman and de Kuijper 1997), differential method (Milton 1985; Zimmerman 1996; Phan-Thien and Pham 2000), Mori-Tanaka-Eshelby method of the average fields (Hatta and Taya 1985; Benveniste 1986; Chen and Wang 1996; Weber et al. 2003), the singular approximation method (Shermergor 1977; Shvidler 1985), the strong isotropy hypothesis and method of conditional moments (Khoroshun et al. 1993), correlation approximation and multipoint approximation (Volkov and Stavrov 1978; Tashkinov et al. 2012; Tashkinov 2015). Some of the techniques are described below.

If composite’s constituents are isotropic and RVE can be macroscopically considered an isotropic medium, the strong isotropy hypothesis can be used (Khoroshun et al. 1993). It assumes that for calculation of two-point moments the components that depend on a choice of direction between two points of can be neglected.

Method of conditional moments for random composites is based on an assumption that the fluctuations of random fields within a component are quite small. This allows to reduce the problem to a system of linear algebraic equations for one-point moments. Such modified problem is solved in two-point approximation using the statistical information and a number of simplifying hypotheses concerning the nature of the distribution of inclusions in the matrix volume (Khoroshun et al. 1993).

For a wide class of stochastic heterogeneous media models, periodic structure can be regarded as a realization of a random structure. The method of periodic components suggests that averaged parts of fields decomposition correspond to the periodic structure (Sokolkin and Tashkinov 1985; Pan’kov et al. 1997). Such representation allows taking into account fractional content, connectivity and geometric shape of components which are common both to the random and periodic structures.

Local approximation method is based on the specifics of short-range interactions of inclusions in matrix composites, according to which the problem of deformation of heterogeneous media is reduced to the simpler problem of deformation of an unbounded domain with an ensemble of a small number of inclusions. Feature of local interactions is not related to the specific nature of the mutual arrangement of the inclusions as well as to their shape, so the method has been applied for composites with random structures (Anoshkin et al. 1991). The hypothesis of limiting locality of correlation functions allows obtaining a single-point approximation of SBVPs and avoiding computation of the integrals over the field of statistical dependence of the correlation functions.

Many stochastic methods are based on the assumption of statistical independence of random fields of physical and mechanical properties of composites, which means that each geometric point is identified with a grain of heterogeneity and the fluctuations of physical and mechanical properties in the neighboring grains are not correlated, and, thus, only the one-point statistical characteristics of a random structure can be taken into account. For the composite with deterministic properties of the structural elements all one-point structural statistical characteristics are determined by relative volume concentration of elements in the assumption of homogeneity and ergodicity of the random fields.

The fundamental work by T.D. Shermergor (Shermergor 1977) offered a number of techniques for the weak contrast heterogeneous media. In the singular approximation (Shermergor 1977) only the formal component of the second derivatives of the Green’s function for a homogeneous unbounded medium is retained. The correlation approximation also uses the formal component, however, it takes into account only pair interactions, so the correlation functions of a higher order than binary are disregarded. In general, the correlation theory can be applied when the standard deviations of the structural elastic moduli are small in relation to their mathematical expectation. Correlation approximation was developed in (Shermergor 1977; Volkov and Stavrov 1978). Stochastic methods in correlation approximation lead to good results for a small difference in the elastic modulus of the composite or weak anisotropy. The full correlation approximation assumes that all the terms, obtained in solution of the SBVP in the first approximation, are being considered, including those containing correlation functions of order higher than the second. The statistical characteristics in the full correlation approximation and second approximation of the solution of the SBVP were calculated in (Sokolkin and Volkova 1992; Tashkinov et al. 2012; Tashkinov 2014, 2016).

The second order moments of stress can be used in statistical models for nonlinear mechanical behavior and fracture of heterogeneous media. The crucial role of these moments in nonlinear analysis is explained by the fact that the yield surface, inclusions interface failure criterion and the energy release rate are the quadratic functions of the local stress distributions (Buryachenko 2011). Thus, Sakata (Sakata et al. 2012) discussed prediction of microstructural failure probability using Monte-Carlo simulation, the perturbation-based stochastic homogenization method and a stochastic multiscale stress analysis. Mishnaevsky (Mishnaevsky et al. 2004) studied the effect of particle clustering on the effective response and damage evolution in particle reinforced Al/SiC composites and used probabilistic analysis to determine failure of matrix and materials.

While comprehensive analysis of interconnection of multiscale parameters of heterogeneous materials remains a challenging problem of micromechanics, in some instances microstructure can be described statistically. This work reviews the main techniques and methods for multiscale mechanical analysis of random heterogeneous materials by means of statistical descriptors and instruments. Experimental and numerical reconstruction based on statistical descriptors to obtain an accurate structure can be used for optimization of heterogeneous materials. Statistical characterization of local microstructural fields of stress and strain is one of the instruments for estimation of damage nucleation and understanding of nonlinear phenomena. The same numerical techniques and theoretical approaches allow to obtain fast numerical assessment for homogenized properties taking into account multipoint microscale interactions.

Some drawbacks of the statistical methods are that in most cases they are not customized for calculation of precise results. In order to receive more rigorous estimates, the introduced approaches can be used in combination with or within the other analytical and numerical methods of mechanics of composites. Besides, the microstructure of heterogeneous materials in the frameworks of statistical characterization is considered static, although some approaches can be generalized to dynamical problems.

Statistical descriptors and measures are important elements of microstructure sensitive design methods which facilitate inverse design for optimized performance of heterogeneous materials. Effectiveness of a particular statistical metric is connected with ability to capture the most influential features of microstructure. With extension of accessibility of computational capacities, statistical approaches allow to consider high-order interactions and open possibility to enhance the existing mechanical theories with more precise microstructural statistical instruments.