 Review
 Open Access
 Published:
Statistical methods for mechanical characterization of randomly reinforced media
Mechanics of Advanced Materials and Modern Processes volume 3, Article number: 18 (2017)
Abstract
Advanced materials with heterogeneous microstructure attract extensive interest of researchers and engineers due to combination of unique properties and ability to create materials that are most suitable for each specific application. One of the challenging tasks is development of models of mechanical behavior for such materials since precision of the obtained numerical results highly depends on level of consideration of features of their heterogeneous microstructure. In most cases, numerical modeling of composite structures is based on multiscale approaches that require special techniques for establishing connection between parameters at different scales. This work offers a review of instruments of the statistics and the probability theory that are used for mechanical characterization of heterogeneous media with random positions of reinforcements. Such statistical descriptors are involved in assessment of correlations between the microstructural components and are parts of mechanical theories which require formalization of the information about microstructural morphology. Particularly, the paper addresses application of the instruments of statistics for geometry description and media reconstruction as well as their utilization in homogenization methods and local stochastic stress and strain field analysis.
Introduction
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, polyphase 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 OstojaStarzewski 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 nonlinear 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.
Microstructural description
Heterogeneous media characterization
Among the important problems of modelling of heterogeneous materials is consideration of peculiarities of their random microstructure to the most possible extent. The morphology of random media microstructure can be studied as a complex system of many interacting components. Mathematical instruments that are used for description of such systems were developed within the individual discipline – mathematical morphology (Matheron 1975; Serra 1982).
Mathematical morphology was initially introduced for image analysis and was primarily aimed on formalization of microscopy images by extracting quantitative information (Matheron 1971, 1975; Najman and Talbot 2010). It is mostly based on set theory, lattice theory, topology and random functions. Capabilities of the mathematical morphology were significantly expanded from its emerging in 1964 during the last decades of twentieth century, from processing of 2D binary black and white images to 3D geometrical structures.
Particular instruments of mathematical morphology are widely used in materials science for characterization of complex internal structures of heterogeneous materials, such as experimentally obtained data from scanning electron microscopy (SEM), transmission electron microscopy (TEM), Xray tomography (see Fig. 1). According to the methodology, images and structures are subject to discretization and transformation prior to quantification. For instance, microstructure image data in a form of 2D and 3D arrays can be reduced by thresholding to a binary array, which is mathematically described as a random set (Chiu et al. 2013). Later, automatic procedures can be applied to determine desired measurements and various statistical estimates.
Calculating the average values (or first (lower)order statistics) of local descriptors was the traditional way for characterizing and quantifying microstructure (Fullwood et al. 2010). Such metrics are often introduced as distribution functions which deliver probability of presence of a particular local state in materials point. The simplest characteristic, which is volume fraction of each instance, can be calculated as proportion of the area or volume covered by one of the binary values. Another example could be the orientation distribution function (ODF) for polycrystalline microstructures.
More complicated descriptors originate from random field theory and are introduced with the functions applied over the random sets of data. Among the most commonly applied statistical descriptor functions are npoint probability functions, surface correlation functions, poresize functions, cluster functions, nearest neighbor functions, linear path functions and others (Matheron 1975; Torquato 2002).
One of the most widely used statistical instrument is correlation functions. These functions show correlation between random values, which, for instance, are spatial positions of heterogeneities in matrix. They are one of the simplest statistical measures that contain spatial information and yet are much more informative than onepoint statistics (Fullwood et al. 2010). The correlation functions themselves can be constructed for any random field inherent to heterogeneous media. In general, geometrical characterization is performed with random indicator function field, for which npoint spatial correlation function for the RVE can be defined as:
where \( {\lambda}_{\alpha}\left(\overrightarrow{r}\right) \) is the indicator function, which is equal 1 if the position of the radiusvector \( \overrightarrow{r} \) indicates phase α and 0 in any other case; \( \left.{\lambda}_{\alpha}\Big(\overrightarrow{r}\right) \) is the mean value of the function averaged over the RVE, which is equal to the volume fraction of phase α. The latter representation applied to other random fields will be discussed in section 3 within the perturbation approach. The higherorder correlation functions for the indicator function of twophase RVE with polydisperse spherical inclusions are presented on Fig. 2.
The other definition of the correlation functions is suggested by (Torquato 2002) who introduced the npoint correlation function as:
where p _{ S } is probability that n points represented by radiusvector are in local state S. These functions were used by many authors in expressions for determination of effective properties, such as conductivity, elastic moduli, fluid permeability.
There are several methods of obtaining values of correlation functions for plain images or digitized threedimensional models. The traditional way is sampling the image via set of randomly placed vectors and calculating local state at the points (Yeong and Torquato 1998a). The specific image processing algorithms were applied by Berryman (Berryman 1985). Cule and Torquato as well as Fullwood (Cule and Torquato 1999; Fullwood et al. 2008) used fast Fourier transform techniques. The rigorous description of the procedures and algorithms of obtaining values and applications of correlation functions can be found, for example, in works of S. Torquato (Torquato 2002) and D.T. Fullwood (Fullwood et al. 2010). Layered fast spherical harmonics expansion were proposed for representation of the correlation functions in (Li et al. 2010).
Application of correlation functions in engineering analysis had started with development of Xray scattering and diffraction. One of the fundamental works (Debye et al. 1957) studied application of correlation functions to characterization of Xray scattering data from porous materials with random distribution of microstructural properties. The paper (Corson 1974) delivered methodologies that link properties of twophase structures to the experimentally calculated two, and threepoint probability functions. Small angle Xray scattering technique has been used to get information on the distribution of inclusions and dispersion of particles (Walenta 1985). Being used in the analysis of effective properties or microstructural stress and strain fields, correlation functions can sense the influence of microstructural morphology parameters on the studied values (Saheli et al. 2005; Fullwood et al. 2010). A number of works were devoted to implementation of highorder multipoint correlation functions in characterization of the microstructure. Highorder correlation functions can be approximated using lower order functions. Such approach was implemented in (Remond et al. 2016).
The technique based on npoint correlation functions is applicable not only to randomly reinforced types of materials. The concept of npoint orientation correlation function applied to polycrystals is presented in (Huang 2005). Npoint correlations for multiphase materials and polycrystals are also discussed in (Groeber et al. 2008). Spatial clustering of 3D microstructures obtained from the vertical metallographic planes was analyzed by means of twopoint correlation functions in (Tewari et al. 2004). The paper (Hansen et al. 2003) incorporates results of the exhaustive branching technique, which is an alternative approach for reconstructing discrete microstructures from twopoint correlation functions, applied to polycrystalline microstructures. This technique showed some advantages before annealing methods. Gaussian random fields were used by Roberts (Roberts and Teubner 1995; Roberts 1999). Principles of genetic algorithms for media reconstruction with correlation functions were applied by Basanta and Kumar (Basanta et al. 2003; Kumar et al. 2008). Research presented in paper (Fullwood et al. 2008) proposes novel algorithms based on phaserecovery methods used in signal processing. More case studies were discussed in (Jeulin 2000; Bochenek and Pyrz 2004; AlOstaz et al. 2007). The authors (Baniassadi et al. 2012) investigated the minimum required size of RVE basing on convergence of the independent correlation functions.
In general, several authors have mentioned limitations of the lowerorder correlation functions that led, for example, to unallowable error estimate for the original and reconstructed media. It was also shown that utilizing 2D sections as a reference for 3D reconstruction puts limits on possibilities to consider certain morphological features intrinsic specifically to 3D structures. This has required introducing the supplement functions, capable to capture specifics of the internal geometry in order to perform more sophisticated analysis. Some of these functions are introduced below.
Surface correlation functions capture information about the spatial distribution of interfaces in the material and are based upon the surface indicator function \( {\chi}_{\alpha}\left({\overrightarrow{r}}_1\right) \), which is nonzero only at the interface of phase α (Torquato 2002; Fullwood et al. 2010). Such twopoint functions can contain both surface and phases indicator functions:
Some functions were introduced due to needs of taking into account the features of the specific types of media. Thus, pore size distribution function P(δ)dδ was introduced to characterize porous media and shows probability that a randomly chosen point in RVE lies at the distance interval [δ, δ + dδ] to the nearest point belonging to poresolid interface.
In case when the microstructure of heterogeneous materials is not statistically homogeneous, the particles in RVE may form groups and clusters. The cluster function \( {C}^{\left(\alpha \right)}\left({\overrightarrow{r}}_1,{\overrightarrow{r}}_2\right) \) gives the probability that two points \( {\overrightarrow{r}}_1 \) and \( {\overrightarrow{r}}_2 \) are in the same cluster of the phase α. This function was described and used in (Torquato 2002; Jiao et al. 2009; Liu et al. 2013).
The lineal path function L ^{(α)}(z) defines probability that a randomly sampled line segment of the specified length lies inside the particular phase. The same approach is used to introduce chords  all of the line segments which are formed by intersections of an infinitely long line with the twophase interface. The chord length distribution function C ^{(α)}(z)dz outputs probability of finding a chord of length between z and z + dz in phase α. The chord length distribution and the lineal path function are two common measures that contain information on particle (or grain) shape that is independent from spatial distribution. The applications examples of these two measures are transport problems, fluid permeability (including fluid flow through porous media), stereology.
The group of the nearest neighbor functions is associated with particlereinforced media and analyses probability of finding the nearest neighbor at some given distance from a reference particle. A number of variations of these functions that differ by the way of definition of the sampled distance and reference points, which can be distance between centers of the particles, between boundaries of the particles, between arbitrary points in RVE and particle’s center. Reverse functions may be introduced, for instance, to estimate spaces in matrix which are free of the particles.
Lineal path function and nearestneighbor function were used for accurate statistical media reconstruction (Quintanilla and Torquato 1997; Yeong and Torquato 1998a; Hahn et al. 1999; Davis et al. 2011).
As was mentioned above, some specific statistical descriptors can be introduced for characterization of polycrystalline materials (Groeber et al. 2008). Thus, the orientation distribution function is a measure of the macroscopic texture of a material. The misorientation distribution function provides information on the local arrangement of grains. The microtexture function is a measure of the amount of clustering of orientations and misorientations.
The summary of major morphological descriptors is presented in Table 1. Order of the function determines whether the function is basic for the microstructural property or is a derivative of basic descriptor. For instance, the indicator function is lower order since it extracts particular feature of the microstructure, while the correlation functions are of higher order since they are constructed from the lowerorder indicator functions. The «Measure» column gives brief description of main characteristic that the function delivers. There is also information about the type of material for RVE of which the function can be applied as well as the dimension of sample required to experimentally obtain the values of the function (threedimensional data or plain twodimensional image).
The mathematical properties and characteristics of descriptors that have been introduced for microstructure of heterogeneous media and materials are comprehensively studied in (Serra 1982; Sobczyk and Kirkner 2001; Torquato 2002; Fullwood et al. 2010).
Application of the mathematical morphology instruments for microstructural characterization of heterogeneous media allows to establish quantitative microstructureproperty link and evaluate impact of important inherent features such as clustering, percolation, dispersion, orientation on overall mechanical properties using the specific type sensitive descriptors. This theoretically underpins the concept of design of materials with prescribed properties by controlling microstructure during manufacturing and processing.
For the materials with predominantly periodic microstructure, such as textile composites with woven and braided microstructure, some of the above described statistical correlation instruments can be used to evaluate deviations from the periodic trends (Cox et al. 2014). They also help to understand how randomness in microstructure affects the properties of composites.
Heterogeneous media reconstruction
An important practical application of the statistical descriptors is mathematical reconstruction of RVEs based on the reference sample experimental data, formalized in a form of statistical descriptors. Such procedure helps to avoid repeating of imaging techniques each time when the microstructure model is required. Created microstructure is assumed to correspond to all relevant microstructural features.
S. Torquato offered the method which involved latticetype representation of the discretized structure and application of correlation functions (Yeong and Torquato 1998a). It is aimed on minimizing difference between actual and target correlation functions by simulated annealing. In order to tackle high computational demands of the method, the discrete fast Fourier transform is used (Fullwood et al. 2008). The other reconstruction scheme relies on correlated random Gaussian field and assigns a region the material phase property if the value of a Gaussian random variable is within the specified range. This approached was mathematically developed in further works. Its limitations relate to inconsistency when processing more than twophase media and Gaussian character of field requirement (Rozman and Utz 2001).
Correlation functions, obtained from the experimentally gathered data, allows to create geometrical models of microstructure that are statistically corresponding to the reference real ones. The layout of this procedure is presented on Fig. 3.
The goal of the reconstruction algorithms is to minimize functional:
where \( {S}_2^{\alpha}\left(\overrightarrow{r}\right) \) is target twopoint correlation function, \( {\widehat{S}}_2^{\alpha}\left(\overrightarrow{r}\right) \) is twopoint correlation function of the reconstructed microstructure.
Some techniques allow to restore 3D structure from information extracted from 2D sections (Liu et al. 2013), which is helpful when threedimensional imaging is unavailable.
S. Torquato investigated the problem of reconstruction of one and twodimensional microstructures based on stochastic optimization involving lower order correlation functions (Yeong and Torquato 1998a; Torquato 2002). Particularly, annealing and MonteCarlo techniques are used to minimize an error between correlation functions of model and sample. C. Yeong and S. Torqauto offered a procedure that incorporates a set of statistical correlation functions obtained from a twodimensional image of the heterogeneous medium to reconstruct the medium in 3D (Yeong and Torquato 1998b). The artificial anisotropic effect which appears in this case and the ways of its consideration were discussed in (Sheehan and Torquato 2001). Reconstruction of random media using lower order correlation functions was later improved by Y. Jiao and S. Torquato in (Jiao et al. 2007, 2008), where the latticepoint procedure for sampling the digitized reference data as well as the modification of the optimization algorithm was described. Such approach was also implemented and investigated in (Manwart and Hilfer 1999; Rozman and Utz 2001; Talukdar et al. 2002). The other suggested variations of the reconstruction procedure included introducing different optimization techniques such as simulated annealing and maximum entropy. Necessary conditions on the twopoint correlation functions were derived in (Torquato 2006).
Further developments of the above described method in the part of MonteCarlo procedure are presented in works of Baniassadi et al. (Garmestani et al. 2009; Li et al. 2010). The latter study was particularly applied to reconstruction of waviness, geometry and preferential distribution of carbon nanotubes (CNTs). These authors also investigated solid oxide fuel cell materials, for which threedimensional reconstruction was performed using planar section of multiphase composites (Baniassadi et al. 2011). 2D realization of the microstructure obtained from SEM images of carbon black particle fillers dispersed in rubber was studied in (Deng et al. 2012). The work (Sheidaei et al. 2013) was devoted to 3D reconstruction of nanotube polypropylene composite from focused ion beam (FIB) and SEM images. It was shown that twopoint correlation functions can only produce an approximate reconstruction while the other groups of statistical descriptors are required to obtain better results. The problems of experimental measurement of the correlation functions are also discussed and studied in (Remond et al. 2016). Threedimensional reconstruction technique using threepoint third order correlation functions was introduced in (Baniassadi et al. 2012).
Generation of random microstructures
Types of microstructure of heterogeneous materials can be grouped into two main classes. First of them can be defined as a statistical mixture and is represented by complex interpenetrating frameworks of constituents, each of which has its own individual bearing capacity. The second group consists of materials that are formed by matrix reinforced with inclusions of spherical, lamellar, fibrous, elliptic and other forms. Some multicomponent composites may also have mixed types of structures. Parameters influencing the RVE geometry are the shape and size of the inclusions, the volume fraction and the morphological details like the spatial orientation and spatial distribution of the inclusions (Bailakanavar et al. 2012). To be statistically representative, RVEs should contain sufficient information about these features.
While now modern simulation methods allow to create geometrical models of RVEs with wide range of forms of phases and overall complexity, the simulation of random microstructures had begun with modelling of random arrangement of spheres (Fig. 4). Such microstructures initially were used in models for simple liquids, concentrated suspensions, amorphous and powder materials. Further a brief description for development of models for RVE random packing is given.
The advances in modeling of dense disordered structures are mostly connected with the two groups of approaches: random sequential adsorption (RSA) and dynamic (or concurrent construction) methods.
Sequential approaches presume determined or random generation of microstructural particles. The first inclusion is placed with its center at the chosen position inside a RVE. Next inclusion and center point is randomly chosen from the diminished volume in the RVE. Likewise, the process of sequentially and randomly positioning an inclusion is continued till the desired volume fraction is achieved or till the jamming limit is encountered. This method generates unit cells with nonintersecting inclusions wherein the gap between the inclusions is userdefined, typically of the order of inclusion size (Bailakanavar et al. 2012).
Sequential placement was studied by Bennett (Bennett 1972), who describes a consistent method when each new inclusion is placed at the point closest to the first so that it comes into contact with the existing inclusions. Intersection is controlled by testing the distance between the centers. Later, this method was modified by introduction of a parameter characterizing the filling of the area bounded by the tetrahedron (Lu et al. 1994). The algorithm that allows to control clustering of inclusions was proposed in (Kansal et al. 2000). The variation when boundary point of each new inclusion coincide with an arbitrary point on the surface of existing inclusions located at the bottom of the virtual box is used in (Nolan and Kavanagh 1992; He et al. 1999; Yang et al. 2000; Furukawa et al. 2000).
Typical particles in heterogeneous materials are often have different shape and size (see Figs. 4, 5 and 6). The RSA algorithm was used to create geometry of RVE with disks (Hinrichsen et al. 1986), spheres and spheroids (Sherwood 1999; Segurado and Llorca 2002; Ghossein and Levesque 2012), cylinders and spherocylinders (Böhm et al. 2002; Williams and Philipse 2003; Iorga et al. 2008; Redenbach and Vecchio 2011; Zhao et al. 2012), thick fibers and ellipsoids (Evans and Ferrar 1989; Parkhouse and Kelly 1995; Man et al. 2005; Bezrukov and Stoyan 2007).
The RSA algorithm is generally limited by the volume fraction it can reach (Evans and Gibson 1986; Parkhouse and Kelly 1995; Toll 1998; Williams and Philipse 2003). Methods aimed on increasing this limit up to 35% volume fraction were proposed by (Pan et al. 2008). The hierarchical random sequential adsorption introduced by (Bailakanavar et al. 2012) allowed to reach 45%. The cylindrical inclusions can be packed with the volume of 50% using the approach that is described in (Islam et al. 2016), while the same volume fraction can be obtained for spherical inclusions with the denser packing algorithm proposed by (Segurado and Llorca 2002).
The distinct class of the random heterogeneous materials are cellular materials with voids, such as foams, wood, bones, etc., and, consequently, low matrix volume fraction. In this case, voids can be either topologically connected or unconnected. The geometrically simplest topologically connected cellular materials are regular honeycombs, the inplane behavior of which can be modeled with planar hexagonal cell models. However, unconnected voids require complex threedimensional geometry modeling: inclusions take forms of cubes (Santosa and Wierzbicki 1998) and polyhedrons (Grenestedt 1998; Sihn and Roy 2004; Daxner et al. 2007; Su et al. 2014; Zhang et al. 2014; Böhm and Rasool 2016). Voronoi tesselations have become a common tool in modeling irregular foams, and recently random Laguerre tessellations have been proposed for generating periodic multicell models (Böhm 2004; Redenbach 2009).
Inclusions of complex forms are essential part of digitized geometrical representation of polycrystalline materials (see Fig. 7). Thus, for instance, within the monodispersive grain size model microstructural grains are replaced by polyhedral inclusions. Voronoi tesselations are a widespread approach for the geometrical models of polycrystals, where they are used for partitioning subregions that correspond to individual grains (Lazar et al. 2012). The grains are constructed around the centers that are distributed randomly in space. (Gross and Li 2002) combined the MonteCarlo and the Voronoi tessellation methods for graingrowth to generate nanocrystalline structures with specified distributions of grain size. Molecular dynamics simulations and equilibration were applied within analysis of nanocrystalline materials in (Tomar and Zhou 2007).
The methods described above belong to a class of static methods and presumes that positions of inclusions are fixed after synthesis. Dynamic models allow reorganization of the structure depending on a way of interaction of inclusions with each other (Buryachenko and Pagano 2005). Various modifications of this method can be found, for example, in (Clarke and Wiley 1987; Oger et al. 1999; He and Ekere 2001; Knott et al. 2001).
Method for generation dense structures with nonintersecting inclusions is proposed in (Bennett 1972). Each consequently placed inclusion “moves” in the direction of the others along a straight line connecting their centers until the contact condition is met. In more complex models a system of differential equations is solved for specifying movements and detecting intersections (Knott et al. 2001).
There is also a method of achieving the desired density of inclusions by changing the radii of spheres (see Fig. 4df). In some models, each inclusion is subject to small random movements, regardless of position of the neighbor inclusions, until the best possible position for denser packing of inclusions is found (Berryman 1983; Cheng et al. 2000). The HardCore model, also called random sequential adsorption model, was discussed by (Lotwick 1982; Hinrichsen et al. 1986).
In addition, the models of inhomogeneous structures can be obtained by Monte Carlo methods. Some models of random structures are based on periodic lattices. Two of them are described in (Kroener 1986): according to the first, inclusion centers are aligned with randomly selected lattice nodes, while each cell were assigned to some phase forming what is now known as voxel model.
Another type of synthesis algorithms with presumed particles movement is based on molecular dynamics (Lubachevsky and Stillinger 1990; Lubachevsky et al. 1991). Each inclusion is assigned a random velocity vector and then are set in motion. Simultaneously, their volumes gradually increase. When a binary collision occurs, the velocities of the two concerned particles are updated according to the kinetic energy conservation principle. To meet the periodicity conditions, if particle leaves the cell through a face during movement, it appears from the opposite side. This approach allows to create the densest packing structures – up to 74% volume fraction for polydisperse spheres (Ghossein and Lévesque 2013). The similar method was used for nonspherical particles as well (Donev et al. 2004).
Effective properties
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.
First of all, one of the ground concepts uses perturbation (or fluctuation) representation, which presumes decomposition of the local stress and strain fields as well as local stiffness tensor into an averaged value and an addition:
where \( {\sigma}_{ij}^{\hbox{'}}\left(\overrightarrow{r}\right)={\varepsilon}_{ij}^{\hbox{'}}\left(\overrightarrow{r}\right)={C}_{ij kl}^{\hbox{'}}\left(\overrightarrow{r}\right)=0 \), \( \overrightarrow{r} \) is radiusvector with components (x _{1}, x _{2}, x _{3}). The angled brackets indicate ensemble averaging, which is equal to volume averaging when the hypothesis of ergodicity for statistically homogeneous material is assumed. The averaging procedure for any coordinatedependent value \( f\left(\overrightarrow{r}\right) \) is determined as:
where integration is performed over the representative volume V.
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).
Models which are built in terms of the mean field approach introduce strain concentration tensor as the functional connecting volume averaged strain for RVE and each phase:
where ε ^{∗} is homogeneous strain, ε _{ C } is averaged strain in a phase C.
One of the homogenization schemes is represented by the selfconsistent 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.
Under the ergodic assumption, statistical measures such as Npoint statistics can be used in equations of the averaging schemes (Adams et al. 2005; Fullwood et al. 2007) that combine decomposition of the perturbation method and mean field homogenization relation by means of localization tensor \( \boldsymbol{a}\left(\overrightarrow{r}\right) \), which can be obtained from the information regarding the local response of a material (Fullwood et al. 2010):
The solutions based on Green’s functions, that are used in these methods, include correlation functions of different orders. Thus, localization tensor \( \boldsymbol{a}\left(\overrightarrow{r}\right) \) can be defined with the Green’s function as follows:
where G is the Green’s function, \( {C}^{\hbox{'}} \) is fluctuation of stiffness tensor (7) that can be expressed via statistics and structural correlation functions for the RVE, χ denotes order of expansion of the series. The Green’s function type is determined by the properties of medium, while the order of the series expansion χ defines the order of required statistical descriptors presumed by fluctuation tensor \( {C}^{\hbox{'}} \).
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 twophase 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 HashinShtrikman bounds, depend on high order correlation functions rather than volume fractions. Threepoint 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 strongcontrast formulation to predict the effective electrical and thermal conductivities of a twophase composites using twopoint and threepoint 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 fractaltype media based on spherical or cylindrical coated inclusions (Hashin and Shtrikman 1962; Benveniste and Milton 2003), twophase 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 nonelliptical shape in the context of homogenization method: meanfield methods used in combination with analytical expressions for the Eshelby tensor, MoriTanaka 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).
Statistical characterization of local stress and strain fields
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 secondorder moments of the random elastic fields in the case of a composite of inclusionmatrix 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.
Method of integral equations allows estimation of second moments of stresses in the components. Considering both binary and triple interaction of the inclusions, explicit relations for second moments of stresses can be obtained (Buryachenko 2011). The method implies that description of heterogeneous medium can be implemented via structuralphenomenological approach when 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. Fluctuations of displacements can be obtained by integration with the Green’s function method:
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; PhanThien and Pham 2000), MoriTanakaEshelby 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 twopoint 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 onepoint moments. Such modified problem is solved in twopoint 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 shortrange 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 singlepoint 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 onepoint statistical characteristics of a random structure can be taken into account. For the composite with deterministic properties of the structural elements all onepoint 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 MonteCarlo simulation, the perturbationbased 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.
Conclusions
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 highorder interactions and open possibility to enhance the existing mechanical theories with more precise microstructural statistical instruments.
Abbreviations
 2D:

Twodimensional
 3D:

Threedimensional
 CNTs:

Carbon Nanotubes
 FIB:

Focused Ion Beam
 RSA:

Random Sequential Algorithm
 RVE:

Representative Volume Element
 SBVPs:

Stochastic Boundary Value Problems
 SEM:

Scanning Electron Microscopy
 TEM:

Transmission Electron Microscopy
References
Adams BL, Gao X, Kalidindi SR (2005) Finite approximations to the secondorder properties closure in single phase polycrystals. Acta Mater 53:3563–3577. 10.1016/j.actamat.2005.03.052
Agoras M, Ponte Castañeda P (2011) Homogenization estimates for multiscale nonlinear composites. Eur J Mech A/Solids 30:828–843. 10.1016/j.euromechsol.2011.05.007
AlOstaz A, Diwakar A, Alzebdeh KI (2007) Statistical model for characterizing random microstructure of inclusionmatrix composites. J Mater Sci 42:7016–7030. 10.1007/s1085300611171
Anoshkin AN, Sokolkin YV, Tashkinov AA (1991) Microstress fields and the mechanical properties of disordered fiber composites. Mech Compos Mater 26:628–633. 10.1007/BF00616643
Babuška I, Andersson B, Smith PJ, Levin K (1999) Damage analysis of fiber composites part I: statistical analysis on fiber scale. Comput Methods Appl Mech Eng 172:27–77. 10.1016/S00457825(98)002254
Bailakanavar M, Liu Y, Fish J, Zheng Y (2012) Automated modeling of random inclusion composites. Eng Comput 30:609–625. 10.1007/s003660120310x
Baniassadi M, Ahzi S, Garmestani H et al (2012) New approximate solution for Npoint correlation functions for heterogeneous materials. J Mech Phys Solids 60:104–119. 10.1016/j.jmps.2011.09.009
Baniassadi M, Garmestani H, Li DS et al (2011) Threephase solid oxide fuel cell anode microstructure realization using twopoint correlation functions. Acta Mater 59:30–43. 10.1016/j.actamat.2010.08.012
Baniassadi M, Laachachi A, Makradi A et al (2011) Statistical continuum theory for the effective conductivity of carbon nanotubes filled polymer composites. Thermochim Acta 520:33–37. 10.1016/j.tca.2011.02.037
Basanta D, Bentley PJ, Miodownik MA, Holm EA (2003) Evolving cellular automata to grow microstructures. Springer, Berlin, Heidelberg, pp 1–10
Bennett CH (1972) Serially deposited amorphous aggregates of hard spheres. J Appl Phys 43:2727–2734. 10.1063/1.1661585
Benveniste Y (1986) On the effective thermal conductivity of multiphase composites. ZAMP Zeitschrift fur Angew Math und Phys 37:696–713. 10.1007/BF00947917
Benveniste Y (2008) Revisiting the generalized selfconsistent scheme in composites: clarification of some aspects and a new formulation. J Mech Phys Solids 56:2984–3002. 10.1016/j.jmps.2008.06.006
Benveniste Y, Milton GW (2003) New exact results for the effective electric, elastic, piezoelectric and other properties of composite ellipsoid assemblages. J Mech Phys Solids 51:1773–1813. 10.1016/S00225096(03)000747
Beran M (1965) Statistical continuum theories. Trans Soc Rheol 9:339–355. 10.1122/1.548991
Beran MJ (1971) Application of statistical theories to heterogeneous materials. Phys Status Solidi 6:365–384. 10.1002/pssa.2210060202
Beran MJ, McCoy JJ (1970) Mean field variations in a statistical sample of heterogeneous linearly elastic solids. Int J Solids Struct 6:1035–1054. 10.1016/00207683(70)900466
Bergman DJ (1978) The dielectric constant of a composite materiala problem in classical physics. Phys Rep 43:377–407
Berryman JG (1983) Random close packing of hard spheres and disks. Phys Rev A 27:1053–1061. 10.1103/PhysRevA.27.1053
Berryman JG (1985) Measurement of spatial correlation functions using image processing techniques. J Appl Phys 57:2374–2384. 10.1063/1.334346
Bezrukov A, Stoyan D (2007) Simulation and statistical analysis of random packings of ellipsoids. Part Part Syst Charact 23:388–398. 10.1002/ppsc.200600974
Bhadeshia HKDH (1999) Neural networks in materials science. ISIJ Int 39:966–979. 10.2355/isijinternational.39.966
Binder K, Heermann DW (2010) Monte Carlo simulation in statistical physics. Springer, Berlin Heidelberg, Berlin, Heidelberg
Bobeth M, Diener G (1986) Field fluctuations in multicomponent mixtures. J Mech Phys Solids 34:1–17. 10.1016/00225096(86)900025
Bochenek B, Pyrz R (2004) Reconstruction of random microstructures  a stochastic optimization problem. Comput Mater Sci 31:93–112. 10.1016/j.commatsci.2004.01.038
Böhm HJ (2004) A short introduction to continuum micromechanics. In: Mechanics of microstructured materials. Springer Vienna, Vienna, pp 1–40
Böhm HJ, Eckschlager A, Han W (2002) Multiinclusion unit cell models for metal matrix composites with randomly oriented discontinuous reinforcements. Comput Mater Sci 25:42–53. 10.1016/S09270256(02)002483
Böhm HJ, Rasool A (2016) Effects of particle shape on the thermoelastoplastic behavior of particle reinforced composites. Int J Solids Struct 87:90–101. 10.1016/j.ijsolstr.2016.02.028
Budiansky B (1965) On the elastic moduli of some heterogeneous materials. J Mech Phys Solids 13:223–227
Buryachenko VA (1996) The overall elastoplastic behavior of multiphase materials with isotropic components. Acta Mech 119:93–117. 10.1007/BF01274241
Buryachenko VA (2007) Micromehcanics of heterogenous materials. Springer US, Boston, MA
Buryachenko VA (2011) Inhomogeneity of the first and second statistical moments of stresses inside the heterogeneities of random structure matrix composites. Int J Solids Struct 48:1665–1687. 10.1016/j.ijsolstr.2011.02.015
Buryachenko VA, Kreher WS (1995) Internal residual stresses in heterogeneous solidsa statistical theory for particulate composites. J Mech Phys Solids 43:1105–1125. 10.1016/00225096(95)00029I
Buryachenko VA, Pagano NJ (2005) The multiscale analysis of multiple interacting inclusions problem: finite number of interacting inclusions. Math Mech Solids 10:25–62. 10.1177/1081286504031174
Buryachenko VA, Rammerstorfer FG (1998) Thermoelastic stress fluctuations in randomstructure coated particulate composites. Eur J Mech  A/Solids 17:763–788. 10.1016/S09977538(98)800041
Buyevich YA (1992) Heat and mass transfer in disperse mediaI. Averaged field equations. Int J Heat Mass Transf 35:2445–2452. 10.1016/00179310(92)900879
Castaneda P (1998) Threepoint bounds and other estimates for strongly nonlinear composites. Phys Rev B 57:12077–12083. 10.1103/PhysRevB.57.12077
Chen CH, Wang YC (1996) Effective thermal conductivity of misoriented shortfiber reinforced thermoplastics. Mech Mater 23:217–228. 10.1016/01676636(96)000105
Chen Y, Schuh CA (2006) Diffusion on grain boundary networks: percolation theory and effective medium approximations. Acta Mater 54:4709–4720. 10.1016/j.actamat.2006.06.011
Cheng YF, Guo SJ, Lai HY (2000) Dynamic simulation of random packing of spherical particles. Powder Technol 107:123–130. 10.1016/S00325910(99)001783
Chiu SN, Stoyan D, Kendall WS (2013) Mecke J. Stochastic Geometry and Its Applications, Third Edition
Clarke AS, Wiley JD (1987) Numerical simulation of the dense random packing of a binary mixture of hard spheres: amorphous metals. Phys Rev B 35:7350–7356. 10.1103/PhysRevB.35.7350
Corson PB (1974) Correlation functions for predicting properties of heterogeneous materials. II Empirical construction of spatial correlation functions for twophase solids J Appl Phys 45:3165–3170. 10.1063/1.1663742
Cox BN, Bale HA, Begley M et al (2014) Stochastic virtual tests for hightemperature ceramic matrix composites. Annu Rev Mater Res 44:479–529. 10.1146/annurevmatsci122013025024
Cule D, Torquato S (1999) Generating random media from limited microstructural information via stochastic optimization. J Appl Phys 86:3428–3437. 10.1063/1.371225
Davis MA, Walsh SDC, Saar MO (2011) Statistically reconstructing continuous isotropic and anisotropic twophase media while preserving macroscopic material properties. Phys Rev E  Stat Nonlinear, Soft Matter Phys 83:1–11. 10.1103/PhysRevE.83.026706
Daxner T, Bitsche RD, Bohm HJ (2007) Micromechanical models of metallic sponges with hollow struts. Mater Sci Forum 539–543:1857–1862. 10.4028/www.scientific.net/MSF.539543.1857
Debye P, Anderson HR, Brumberger H (1957) Scattering by an inhomogeneous solid. II. The correlation function and its application. J Appl Phys 28:679–683. 10.1063/1.1722830
Deng H, Liu Y, Gai D et al (2012) Utilizing real and statistically reconstructed microstructures for the viscoelastic modeling of polymer nanocomposites. Compos Sci Technol 72:1725–1732. 10.1016/j.compscitech.2012.03.020
Donev A, Torquato S, Stillinger FH (2004) Neighbor list collisiondriven molecular dynamics simulation for nonspherical particles. I. Algorithmic details II. Applications to ellipses and ellipsoids. J Comput Phys 202:45. 10.1016/j.jcp.2004.08.014
Dykhne AM (1970) Conductivity of a twodimensional twophase system. Eksp Teor Fiz 32:110–115
Evans AG (1978) Microfracture from thermal expansion anisotropy: I. Single phase systems. Acta Metall 26:1845–1853
Evans KE, Ferrar MD (1989) The packing of thick fibres. J Phys D Appl Phys 22:354–360. 10.1088/00223727/22/2/020
Evans KE, Gibson AG (1986) Prediction of the maximum packing fraction achievable in randomly oriented shortfibre composites. Compos Sci Technol 25:149–162. 10.1016/02663538(86)900400
Frary M, Schuh CA (2005) Grain boundary networks: scaling laws, preferred cluster structure, and their implications for grain boundary engineering. Acta Mater 53:4323–4335. 10.1016/j.actamat.2005.05.030
Fu Y, Evans AG (1985) Some effects of microcracks on the mechanical properties of brittle solidsI. Stress, strain relations. Acta Metall 33:1515–1523. 10.1016/00016160(85)900525
Fullwood DT, Adams BL, Kalidindi SR (2007) Generalized Pareto front methods applied to secondorder material property closures. Comput Mater Sci 38:788–799. 10.1016/j.commatsci.2006.05.016
Fullwood DT, Kalidindi SR, Niezgoda SR et al (2008) Gradientbased microstructure reconstructions from distributions using fast Fourier transforms. Mater Sci Eng A 494:68–72. 10.1016/j.msea.2007.10.087
Fullwood DT, Niezgoda SR, Adams BL, Kalidindi SR (2010) Microstructure sensitive design for performance optimization. Prog Mater Sci 55:477–562. 10.1016/j.pmatsci.2009.08.002
Fullwood DT, Niezgoda SR, Kalidindi SR (2008) Microstructure reconstructions from 2point statistics using phaserecovery algorithms. Acta Mater 56:942–948. 10.1016/j.actamat.2007.10.044
Furukawa K, Imai K, Kurashige M (2000) Simulated effect of box size and wall on porosity of random packings of spherical particles. Acta Mech 140:219–231. 10.1007/BF01182512
Garmestani H, Baniassadi M, Li DS et al (2009) Semiinverse Monte Carlo reconstruction of twophase heterogeneous material using twopoint functions. Int J Theor Appl Multiscale Mech 1:134. 10.1504/IJTAMM.2009.029210
Ghossein E, Levesque M (2012) A fully automated numerical tool for a comprehensive validation of homogenization models and its application to spherical particles reinforced composites. Int J Solids Struct 49:1387–1398. 10.1016/j.ijsolstr.2012.02.021
Ghossein E, Lévesque M (2013) Random generation of periodic hard ellipsoids based on molecular dynamics: a computationallyefficient algorithm. J Comput Phys 253:471–490. 10.1016/j.jcp.2013.07.004
Grenestedt JL (1998) Influence of wavy imperfections in cell walls on elastic stiffness of cellular solids. J Mech Phys Solids 46:29–50. 10.1016/S00225096(97)000355
Groeber M, Ghosh S, Uchic MD, Dimiduk DM (2008) A framework for automated analysis and simulation of 3D polycrystalline microstructures. Part 1: statistical characterization. Acta Mater 56:1257–1273. 10.1016/j.actamat.2007.11.041
Gross D, Li M (2002) Constructing microstructures of poly and nanocrystalline materials for numerical modeling and simulation. Appl Phys Lett 80:746–748. 10.1063/1.1432448
Guseva O, Lusti HR, Gusev AA (2004) Matching thermal expansion of mica–polymer nanocomposites and metals. Model Simul Mater Sci Eng 12:101–105. 10.1088/09650393/12/3/S04
Hahn U, Micheletti A, Pohlink R et al (1999) Stereological analysis and modelling of gradient structures. J Microsc 195:113–124. 10.1046/j.13652818.1999.00487.x
Hansen BL, Adam BL, Lyon ME, Henrie AJ (2003) On the reconstruction of polycrystalline microstructures from twopoint correlation statistics. J Comput Mater Des 10:163–173. 10.1007/s1082000508843
Hashin Z (1968) Assessment of the self consistent scheme approximation: conductivity of particulate composites. J Compos Mater 2:284–300. 10.1177/002199836800200302
Hashin Z, Shtrikman S (1962) A Variational approach to the theory of the effective magnetic permeability of multiphase materials. J Appl Phys 33:3125–3131. 10.1063/1.1728579
Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11:127–140. 10.1016/00225096(63)900607
Hatta H, Taya M (1985) Effective thermal conductivity of a misoriented short fiber composite. J Appl Phys 58:2478–2486. 10.1063/1.335924
He D, Ekere N, Cai L (1999) Computer simulation of random packing of unequal particles. Phys Rev E 60:7098–7104. 10.1103/PhysRevE.60.7098
He D, Ekere NN (2001) Structure simulation of concentrated suspensions of hard spherical particles. AICHE J 47:53–59. 10.1002/aic.690470108
Hill R (1965) A selfconsistent mechanics of composite materials. J Mech Phys Solids 13:213–222. 10.1016/00225096(65)900104
Hinrichsen EL, Feder J, Jøssang T (1986) Geometry of random sequential adsorption. J Stat Phys 44:793–827. 10.1007/BF01011908
Hori M, Kubo J (1998) Analysis of probabilistic distribution and range of average stress in each phase of heterogeneous materials. J Mech Phys Solids 46:537–556. 10.1016/S00225096(97)000884
Huang M (2005) The npoint orientation correlation function and its application. Int J Solids Struct 42:1425–1441. 10.1016/j.ijsolstr.2004.06.037
Iorga L, Pan Y, Pelegri A (2008) Numerical characterization of material elastic properties for random fiber composites. J Mech Mater Struct 3:1279–1298. 10.2140/jomms.2008.3.1279
Islam M, Tudryn GJ, Picu CR (2016) Microstructure modeling of random composites with cylindrical inclusions having high volume fraction and broad aspect ratio distribution. Comput Mater Sci 125:309–318. 10.1016/j.commatsci.2016.08.051
Jayatilaka ADS, Trustrum K (1977) Statistical approach to brittle fracture. J Mater Sci 12:1426–1430. 10.1007/BF00540858
Jeffrey DJ (1973) Conduction through a random suspension of spheres. Proc R Soc A Math Phys Eng Sci 335:355–367. 10.1098/rspa.1973.0130
Jeulin D (2000) Random texture models for material structures. Stat Comput 10:121–132. 10.1023/A:1008942325749
Jiao Y, Stillinger FH, Torquato S (2007) Modeling heterogeneous materials via twopoint correlation functions: basic principles. Phys Rev E  Stat Nonlinear, Soft Matter Phys 76:1–15. 10.1103/PhysRevE.76.031110
Jiao Y, Stillinger FH, Torquato S (2008) Modeling heterogeneous materials via twopoint correlation functions. II. Algorithmic details and applications. Phys rev E  stat nonlinear. Soft Matter Phys 77:1–15. 10.1103/PhysRevE.77.031135
Jiao Y, Stillinger FH, Torquato S (2009) A superior descriptor of random textures and its predictive capacity. Proc Natl Acad Sci 106:17634–17639. 10.1073/pnas.0905919106
Kalidindi SR, Houskamp JR (2007) Application of the spectral methods of microstructure design to continuous fiberreinforced composites. J Compos Mater 41:909–930. 10.1177/0021998306067256
Kamiński MM (2005) Computational mechanics of composite materials : sensitivity, randomness, and multiscale behaviour. SpringerVerlag London
Kanit T, Forest S, Galliet I et al (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40:3647–3679. 10.1016/S00207683(03)001434
Kanouté P, Boso DP, Chaboche JL, Schrefler BA (2009) Multiscale methods for composites: a review. Arch Comput Methods Eng 16:31–75. 10.1007/s1183100890288
Kansal AR, Truskett TM, Torquato S (2000) Nonequilibrium harddisk packings with controlled orientational order. J Chem Phys 113:4844–4851. 10.1063/1.1289238
Keller JB (1963) Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders. J Appl Phys 34:991–993. 10.1063/1.1729580
Khisaeva ZF, OstojaStarzewski M (2006) On the size of RVE in finite elasticity of random composites. J Elast 85:153–173. 10.1007/s106590069076y
Khoroshun LP, Maslov BP, Shikula EN, Nazarenko LV (1993) Statistical mechanics and effective properties of materials. Naukova Dumka, Kiev
Klusemann B, Böhm HJ, Svendsen B (2012) Homogenization methods for multiphase elastic composites with nonelliptical reinforcements: comparisons and benchmarks. Eur J Mech A/Solids 34:21–37. 10.1016/j.euromechsol.2011.12.002
Knott GM, Jackson TL, Buckmaster J (2001) Random packing of heterogeneous propellants. AIAA J 39:678–686. 10.2514/2.1361
Koelman JMVA, de Kuijper A (1997) An effective medium model for the electric conductivity of an Ncomponent anisotropic and percolating mixture. Phys A Stat Mech its Appl 247:10–22. 10.1016/S03784371(97)003853
Kreher W (1990) Residual stresses and stored elastic energy of composites and polycrystals. J Mech Phys Solids 38:115–128. 10.1016/00225096(90)90023W
Kroener E (1986) Statistical Modelling. In: Modelling small deformations of polycrystals. Springer Netherlands, Dordrecht, pp 229–291
Kumar NC, Matouš K, Geubelle PH (2008) Reconstruction of periodic unit cells of multimodal random particulate composites using genetic algorithms. Comput Mater Sci 42:352–367. 10.1016/j.commatsci.2007.07.043
Kushch VI (1997) Conductivity of a periodic particle composite with transversely isotropic phases. Proc R Soc A Math Phys Eng Sci 453:65–76. 10.1098/rspa.1997.0004
Lazar EA, Mason JK, MacPherson RD, Srolovitz DJ (2012) Complete topology of cells, grains, and bubbles in threedimensional microstructures. Phys Rev Lett 109:10–14. 10.1103/PhysRevLett.109.095505
Lee H, Gillman AS, Matouš K (2011) Computing overall elastic constants of polydisperse particulate composites from microtomographic data. J Mech Phys Solids 59:1838–1857. 10.1016/j.jmps.2011.05.010
Li D, Khaleel M, Sun X, Garmestani H (2010) Representation of correlation statistics functions in heterogeneous materials using layered fast spherical harmonics expansion. Comput Mater Sci 48:133–139. 10.1016/j.commatsci.2009.12.019
Li DS, Baniassadi M, Garmestani H et al (2010) 3D reconstruction of carbon Nanotube composite microstructure using correlation functions. J Comput Theor Nanosci 7:1462–1468. 10.1166/jctn.2010.1504
Lifshitz IM, Rosenzweig LN (1946) On the theory of the elastic properties of polycrystals. J Exp Theor Phys 16:967–980
Liu Y, Steven Greene M, Chen W et al (2013) Computational microstructure characterization and reconstruction for stochastic multiscale material design. CAD Comput Aided Des 45:65–76. 10.1016/j.cad.2012.03.007
Lomakin VA (1970) Statistical problems of the mechanics of solid deformable bodies. Nauka, Moscow
Lotwick HW (1982) Simulation of some spatial hard core models, and the complete packing problem. J Stat Comput Simul 15:295–314. 10.1080/00949658208810596
Lu GQ, Ti LB, Ishizaki K (1994) A new algorithm for simulating the random packing of Monosized powder in CIP processes. Mater Manuf Process. 10.1080/10426919408934934
Lu S, Song J (1996) Effective conductivity of composites with spherical inclusions: effect of coating and detachment. J Appl Phys 79:609–618. 10.1063/1.360803
Lubachevsky BD, Stillinger FH (1990) Geometric properties of random disk packings. J Stat Phys 60:561–583. 10.1007/BF01025983
Lubachevsky BD, Stillinger FH, Pinson EN (1991) Disks vs. spheres: contrasting properties of random packings. J Stat Phys 64:501–524. 10.1007/BF01048304
Man W, Donev A, Stillinger FH et al (2005) Experiments on random packings of ellipsoids. Phys Rev Lett. 10.1103/PhysRevLett.94.198001
Manwart C, Hilfer R (1999) Reconstruction of random media using Monte Carlo methods. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 59:5596–5599
Matheron G (1971) The theory of regionalized variables and its applications. Les Cah du Cent Morphlogie Math 5:218
Matheron G (1975) Random sets and integral geometry. Wiley, New York
McKenzie DR, McPhedran RC, Derrick GH (1978) The conductivity of lattices of spheres. II. The body Centred and face Centred cubic lattices. Proc R Soc A Math Phys Eng Sci 362:211–232. 10.1098/rspa.1978.0129
McPhedran RC, McKenzie DR (1978) The conductivity of lattices of spheres. I. The simple cubic lattice. Proc R Soc A Math Phys Eng Sci 359:45–63. 10.1098/rspa.1978.0031
Mikdam A, Makradi A, Ahzi S et al (2009) Effective conductivity in isotropic heterogeneous media using a strongcontrast statistical continuum theory. J Mech Phys Solids 57:76–86. 10.1016/j.jmps.2008.10.001
Mikdam A, Makradi A, Ahzi S et al (2010) Statistical continuum theory for the effective conductivity of fiber filled polymer composites: effect of orientation distribution and aspect ratio. Compos Sci Technol 70:510–517. 10.1016/j.compscitech.2009.12.002
Milton GW (1982) Bounds on the elastic and transport properties of twocomponent composites. J Mech Phys Solids 30:177–191. 10.1016/00225096(82)900229
Milton GW (1985) The coherent potential approximation is a realizable effective medium scheme. Commun Math Phys 99:463–500. 10.1007/BF01215906
Milton GW (2002) The theory of composites. Cambridge University Press, Cambridge
Mishnaevsky L, Derrien K, Baptiste D (2004) Effect of microstructure of particle reinforced composites on the damage evolution: probabilistic and numerical analysis. Compos Sci Technol 64:1805–1818. 10.1016/j.compscitech.2004.01.013
Najman L, Talbot H (2010) Mathematical morphology: from theory to applications
Nolan GT, Kavanagh PE (1992) Computer simulation of random packing of hard spheres. Powder Technol 72:149–155. 10.1016/00325910(92)880219
Oger L, Troadec JP, Gervois A, Medvedev NN (1999) Computer simulations and tessellations of granular materials. In: Foams and emulsions. Springer Netherlands, Dordrecht, pp 527–546
Pan Y, Iorga L, Pelegri AA (2008) Numerical generation of a random chopped fiber composite RVE and its elastic properties. Compos Sci Technol 68:2792–2798. 10.1016/j.compscitech.2008.06.007
Pan’kov AA, Sokolkin YV, Tashkinov AA (1997) Singular approximation of the method of periodic components in statistical mechanics of composite materials. Mech Compos Mater 33:322–331. 10.1007/BF02256282
Parkhouse JG, Kelly A (1995) The random packing of Fibres in three dimensions. Proc R Soc London A Math Phys Eng Sci 451:737–746. 10.1098/rspa.1995.0152
PhanThien N, Pham DC (2000) Differential multiphase models for polydispersed spheroidal inclusions: thermal conductivity and effective viscosity. Int J Eng Sci 38:73–88. 10.1016/S00207225(99)000166
Ponte Castañeda P (2012) Bounds for nonlinear composites via iterated homogenization. J Mech Phys Solids 60:1583–1604. 10.1016/j.jmps.2012.05.004
Ponte Castaneda P (2016) Stationary variational estimates for the effective response and field fluctuations in nonlinear composites. J Mech Phys Solids 96:660–682. 10.1016/j.jmps.2016.06.010
Ponte Castaneda P, Suquet P (2001) Nonlinear composites and microstructure evolution. In: Mechanics for a new Mellennium. Kluwer Academic Publishers, Dordrecht, pp 253–274
Quintanilla J, Torquato S (1997) Microstructure functions for a model of statistically inhomogeneous random media. Phys Rev E 55:1558–1565. 10.1103/PhysRevE.55.1558
Rasool A, Böhm HJ (2012) Effects of particle shape on the macroscopic and microscopic linear behaviors of particle reinforced composites. Int J Eng Sci 58:21–34. 10.1016/j.ijengsci.2012.03.022
Redenbach C (2009) Microstructure models for cellular materials. Comput Mater Sci 44:1397–1407. 10.1016/j.commatsci.2008.09.018
Redenbach C, Vecchio I (2011) Statistical analysis and stochastic modelling of fibre composites. Compos Sci Technol 71:107–112. 10.1016/j.compscitech.2010.10.014
Remond Y, Ahzi S, Baniassadi M, Garmestani H (2016) Applied RVE reconstruction and homogenization of heterogeneous materials
Roberts A (1999) Statistical reconstruction of threedimensional porous media from twodimensional images. 56:3203–3212. doi:10.1103/PhysRevE.56.3203
Roberts AP, Teubner M (1995) Transport properties of heterogeneous materials derived from Gaussian random fields: bounds and simulation. Phys Rev E 51:4141–4154. 10.1103/PhysRevE.51.4141
Rozman MG, Utz M (2001) Efficient reconstruction of multiphase morphologies from correlation functions. Phys Rev E  Stat Nonlinear, Soft Matter Phys 63:1–8. 10.1103/PhysRevE.63.066701
Saheli G, Garmestani H, Adams BL (2005) Microstructure design of a two phase composite using twopoint correlation functions. J Comput Mater Des 11:103–115. 10.1007/s1082000531643
Sahimi M (1998) Nonlinear and nonlocal transport processes in heterogeneous media: from longrange correlated percolation to fracture and materials breakdown. Phys Rep 306:213–395. 10.1016/S03701573(98)000246
Sakata S, Ashida F, Enya K (2012) A microscopic failure probability analysis of a unidirectional fiber reinforced composite material via a multiscale stochastic stress analysis for a microscopic random variation of an elastic property. Comput Mater Sci 62:35–46. 10.1016/j.commatsci.2012.05.008
Santosa S, Wierzbicki T (1998) On the modeling of crush behavior of a closedcell aluminum foam structure. J Mech Phys Solids 46:645–669. 10.1016/S00225096(97)000823
Segurado J, Llorca J (2002) A numerical approximation to the elastic properties of spherereinforced composites. J Mech Phys Solids 50:2107–2121. 10.1016/S00225096(02)000212
Serra J (1982) Image analysis and mathematical morphology
Sheehan N, Torquato S (2001) Generating microstructures with specified correlation functions. J Appl Phys 89:53–60. 10.1063/1.1327609
Sheidaei A, Baniassadi M, Banu M et al (2013) 3D microstructure reconstruction of polymer nanocomposite using FIBSEM and statistical correlation function. Compos Sci Technol 80:47–54. 10.1016/j.compscitech.2013.03.001
Shermergor TD (1977) The theory of elasticity of microinhomogeneous media. Nauka, Moscow
Sherwood JD (1999) Packing of spheroids in threedimensional space by random sequential addition. J Phys A Math Gen 30:L839–L843. 10.1088/03054470/30/24/004
Shvidler MI (1985) Statistical hydrodynamics of porous media. Nauka, Moscow
Sihn S, Roy AK (2004) Modeling and prediction of bulk properties of opencell carbon foam. J Mech Phys Solids 52:167–191. 10.1016/S00225096(03)000723
Sobczyk K, Kirkner DJ (2001) Stochastic modeling of microstructures. Model Simul Sci Eng Technol XX:11–15. 10.1007/9781461201212
Sokolkin YV, Tashkinov AA (1985) Statistical models of deformation and failure of composites. Mech Compos Mater 20:585–590. 10.1007/BF00610607
Sokolkin YV, Volkova TA (1992) Multipoint moment functions of the stress and strain distributions in stochastic composites. Mech Compos Mater 27:429–435. 10.1007/BF00613572
Su Y, Ouyang Q, Zhang W et al (2014) Composite structure modeling and mechanical behavior of particle reinforced metal matrix composites. Mater Sci Eng A 597:359–369. 10.1016/j.msea.2014.01.024
Talukdar MSS, Torsaeter O, Ioannidis MA (2002) Stochastic reconstruction of particulate media from twodimensional images. J Colloid Interface Sci 248:419–428. 10.1006/jcis.2001.8064
Tashkinov M (2014) Statistical characteristics of structural stochastic stress and strain fields in polydisperse heterogeneous solid media. Comput Mater Sci 94:44–50. 10.1016/j.commatsci.2014.01.050
Tashkinov M (2016) Microscale modeling of phaselevel elastic fields of SiC reinforced metal matrix multiphase composites using statistical approach. Comput Mater Sci 116:113–121. 10.1016/j.commatsci.2015.10.047
Tashkinov M, Wildemann V, Mikhailova N (2012) Method of successive approximations in stochastic elastic boundary value problem for structurally heterogenous materials. Comput Mater Sci. 10.1016/j.commatsci.2011.04.025
Tashkinov MA (2015) Methods of stochastic mechanics for characterization of deformation in randomly reinforced composite materials. Springer International Publishing Switzerland, Springer, Cham pp 43–78
Tewari A, Gokhale AM, Spowart JE, Miracle DB (2004) Quantitative characterization of spatial clustering in threedimensional microstructures using twopoint correlation functions. Acta Mater 52:307–319. 10.1016/j.actamat.2003.09.016
Toll S (1998) Packing mechanics of fiber reinforcements. Polym Eng Sci 38:1337–1350. 10.1002/pen.10304
Tomar V, Zhou M (2007) Analyses of tensile deformation of nanocrystalline αFe2O3+fccal composites using molecular dynamics simulations. J Mech Phys Solids 55:1053–1085. 10.1016/j.jmps.2006.10.005
Torquato S (1980) Microscopic approach to transport in twophase random media
Torquato S (1991) Random heterogeneous media: microstructure and improved bounds on effective properties. Appl Mech Rev 44:37. 10.1115/1.3119494
Torquato S (1997) Exact expression for the effective elastic tensor of disordered composites. Phys Rev Lett 79:681–684. 10.1103/PhysRevLett.79.681
Torquato S (2000) Modeling of physical properties of composite materials. Int J Solids Struct 37:411–422. 10.1016/S00207683(99)001031
Torquato S (2002) Random Heterogeneous Materials
Torquato S (2006) Necessary conditions on realizable twopoint correlation functions of random media. Ind Eng Chem Res 45:6923–6928. 10.1021/ie058082t
TVERGAARD V, HUTCHINSON JW (1988) Microcracking in ceramics induced by thermal expansion or elastic anisotropy. J Am Ceram Soc 71:157–166. 10.1111/j.11512916.1988.tb05022.x
Volkov SD, Stavrov VP (1978) Statistical mechanics of composite materials. Izd. Bel. Gos. Univ, Minsk (in Russian)
Walenta E (1985) Small angle xray scattering. Von O. GLATTER und O. KRATKY. London: academic press inc. ltd. 1982. ISBN 0122862805. X, 515 Seiten, geb. £ 43,60; US $ 81.00. Acta Polym 36:296–296. 10.1002/actp.1985.010360520
Weber L, Fischer C, Mortensen A (2003) On the influence of the shape of randomly oriented, nonconducting inclusions in a conducting matrix on the effective electrical conductivity. Acta Mater 51:495–505. 10.1016/S13596454(02)004329
Williams SR, Philipse AP (2003) Random packings of spheres and spherocylinders simulated by mechanical contraction. Phys Rev E 67:51301. 10.1103/PhysRevE.67.051301
Willis JR (1981) Variational and related methods for the overall properties of composites. Adv Appl Mech 21:1–78. 10.1016/S00652156(08)703302
Willis JR (1992) On methods for bounding the overall properties of nonlinear composites: correction and addition. J Mech Phys Solids 40:441–445. 10.1016/S00225096(05)800201
Xu XF, Chen X, Shen L (2009) A greenfunctionbased multiscale method for uncertainty quantification of finite body random heterogeneous materials. Comput Struct 87:1416–1426. 10.1016/j.compstruc.2009.05.009
Yang RY, Zou RP, Yu AB (2000) Computer simulation of the packing of fine particles. Phys Rev E  Stat Physics, Plasmas, Fluids, Relat Interdiscip Top 62:3900–3908. 10.1103/PhysRevE.62.3900
Yeong C, Torquato S (1998a) Reconstructing random media. Phys Rev E 57:495–506. 10.1103/PhysRevE.57.495
Yeong C, Torquato S (1998b) Reconstructing random media. II. Threedimensional media from twodimensional cuts. Phys Rev E 58:224–233. 10.1103/PhysRevE.58.224
Zhang XX, Xiao BL, Andrä H, Ma ZY (2014) Homogenization of the average thermoelastoplastic properties of particle reinforced metal matrix composites: the minimum representative volume element size. Compos Struct 113:459–468. 10.1016/j.compstruct.2014.03.048
Zhao J, Li S, Zou R, Yu A (2012) Dense random packings of spherocylinders. Soft Matter 8:1003–1009. 10.1039/C1SM06487H
Zimmerman RW (1996) Effective conductivity of a twodimensional medium containing elliptical Inhomogeneities
Acknowledgements
This work was supported by the Russian Foundation for Basic Research (projects 160100327_a, 1641590259r_a) and by the Grant of the President of Russian Federation for state support of young Russian scientists (MK2395.2017.1).
Funding
This work was supported by the Russian Foundation for Basic Research (projects 16–0100327_a, 16–41590259r_a) and by the Grant of the President of Russian Federation for state support of young Russian scientists (MK2395.2017.1).
Availability of data and materials
Not applicable
Author information
Affiliations
Contributions
M. Tashkinov prepared this review.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable
Consent for publication
Not applicable
Competing interests
The author declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Tashkinov, M. Statistical methods for mechanical characterization of randomly reinforced media. Mech Adv Mater Mod Process 3, 18 (2017). https://doi.org/10.1186/s4075901700322
Received:
Accepted:
Published:
Keywords
 Heterogeneous media
 Random microstructure
 Stochastic fields
 Correlation functions
 Effective properties
 Local stress fields
 Multipoint statistics
 Media reconstruction
 3D models