Multiparticle Effective Field Method Research Articles

Efficient homogenization techniques for elastic composites: Maxwell scheme vs. effective field method

The work is devoted to the comparative analysis of four homogenization techniques (the original and generalized Maxwell schemes and the one-particle and multi-particle effective field methods) in application to the problem of calculation of the effective elastic stiffness tensor of matrix composites. The generalized Maxwell scheme and multi-particle effective field method reduce the problem to calculation of elastic fields in a finite volume of the composite embedded in the infinite homogeneous matrix medium and subjected to a constant external stress or strain field. For the solution of this problem, an efficient numerical method based on Gaussian approximating functions and fast Fourier transform technique is used. The results of the methods are compared for composites with ellipsoidal inclusions which elastic moduli are much smaller or much larger than the moduli of the matrix. The case of hybrid composites with two different families of ellipsoidal inclusion is also considered. It is shown that for various shapes and elastic properties of inclusions, the generalized Maxwell scheme and multi-particle effective field method give close results. But in the case of hard inclusions of large volume fractions the results of these methods deviate substantially from the original Maxwell scheme that does not take into account interactions between inclusions.

Effective thermoelastic properties of random structure composites reinforced by the clusters of deterministic structure (application to clay nanocomposites)

Polymer/clay nanocomposites consisting of an epoxy matrix reinforced by silicate clay plates have been observed to exhibit enhanced mechanical properties at low volume fraction of clay. The matrix and embedded nanoelements are modeled in the framework of continuum mechanics with known mechanical properties previously evaluated by, e.g., molecular dynamic simulation. Nanoclay composite is modeled by the aligned, uniformly distributed in the matrix stacks of parallel clay sheets separated from one another by interlayer matrix galleries of nanometer scale. Interaction of a finite number of oblate spheroidal inclusions modeling an individual stack inside the infinite matrix is carried by the multipole expansion technique. The obtained accurate numerical solution was incorporated into the multiparticle effective field method [5] for the estimation of effective thermoelastic properties. Detailed parametric analyses demonstrate the influence on the effective elastic moduli and stress concentrator factors of such key factors as the shape of nanoelements, interlayer distance, and the number of nanoelements in the stacks of deterministic structure.

Generalization of the multiparticle effective field method to random structure matrix composites

In this paper linearly thermoelastic composite media are treated, which consist of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated inclusions. Effective properties (such as compliance, thermal expansion, stored energy) as well as the first statistical moments of stresses in the phases are estimated for the general case of nonhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on the generalization of the ``multiparticle effective field'' method (MEFM, see [7] for references), previously proposed for the estimation of stress field averages in the phases. The refined version of the MEFM takes into account both the variation of the effective fields acting on each pair of fibers and inhomogeneity of statistical average of stresses inside the inclusions. One considers in detail the connection of the method proposed with numerous related methods. The explicit representations of the effective thermoelastic properties and stress concentration factor are expressed through some building blocks described by numerical solutions for both the one and two inclusions inside the infinite medium subjected to the homogeneous loading at infinity. Just with some additional assumptions (such as an effective field hypothesis) the involved tensors can be expressed through the Green's function, Eshelby tensor and external Eshelby tensor. The dependence of effective properties and stress concentrator factors on the radial distribution function of the inclusion locations is analyzed.

Effective elastic moduli and stress concentrator factors in random structure aligned fiber composites

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically uniform random set of aligned fibers. Effective elastic moduli as well as the stress concentrator factors in the components are estimated. The micromechanical approach is based on the Green’s function technique as well as on the generalization of the “multiparticle effective field method” (MEFM, see for references, Buryachenko [1]). The refined version of the MEFM takes into account the variation of the effective fields acting on each pair of fibers. The dependence of effective elastic moduli and stress concentrator factors on the radial distribution function of the fiber locations is analyzed.

Effective elastic moduli of triply periodic particulate matrix composites with imperfect unit cells

In many problems the material may possess a periodic microstructure formed by the spatial repetition of small microstructures, or unit cells. Such a perfectly regular distribution, of course, does not exist in actual cases, although the periodic modeling can be quite useful, since it provides rigorous estimations with a priori prescribed accuracy for various material properties. Triply periodic particulate matrix composites with imperfect unit cells are analyzed in this paper. The multiparticle effective field method (MEFM) is used for the analysis of the perfect and imperfect periodic structure composites. The MEFM is originally based on the homogeneity hypothesis (H1) (see for details [Buryachenko, V.A., 2001. Multiparticle effective field and related methods in micromechanics of composite materials. Appl. Mech. Rev. 54, 1–47]) of effective field acting on the inclusions. In this way the pair interaction of different inclusions is taken directly into account by the use of analytical approximate solution. For perfect periodic structures the hypothesis (H1) is enough for estimation of effective properties. Imperfection of packing necessitates exploring some additional assumption called a closing hypothesis. The next imperfections are analyzed. (A) The probability of location of an inclusion in the center of a unit cell below one (missing inclusion). (B) Some hard inclusions are randomly replaced by the porous (modeling the complete debonding) with some probability. At first, one obtains general explicit integral representations of the effective elastic moduli and strain concentrator factors depending on three numerical solutions: for the perfect periodic structure, for the infinite periodic structure with one imperfection, and for the infinite periodic structure with two arbitrary located imperfections. The method proposed is general; it is not limited by concrete numerical scheme. No restrictions were assumed on both the concrete microstructure and inhomogeneity of stress fields in the inclusions. The inclusions of one kind are assumed to be aligned. The problem (A) is solved at the level of numerical results obtained in the framework of the hypothesis (H1). For the problem (B) the numerical results are obtained if the elastic inclusions (for example hard inclusions) are randomly replaced by another inclusion (for example by the voids modeling the complete debonding). The mentioned problems are solved by three methods. The first one is a Monte Carlo simulation exploring an analytical approximate solution for the binary interacting inclusions obtained in the framework of the hypothesis (H1). The second one is a generalization of the version of the MEFM proposed for the analysis of the perfect periodic particulate composites and based on the choice of a comparison medium coinciding with the matrix. The third method uses a decomposition of the desired solution on the solution for the perfect periodic structure and on the perturbation produced by the imperfections in the perfect periodic structure. All three methods lead to close results in the considered examples; however, the CPU times expended for the solution estimation by Monte Carlo simulation differ by a factor of 1000.

Effective elastic and failure properties of fiber aligned composites

A digital image processing technique is used for measurement of centroid coordinates of fibers with forthcoming estimation of statistical parameters and functions describing the stochastic structure of laminated fiber composites. Comparative statistical analysis of the experimentally measured and numerically simulated fiber distributions are performed. We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous set of ellipsoidal inclusions. The multiparticle effective field method (MEFM) [Appl. Mech. Rev. 54 (2001) 1] based on the theory of functions of random variables and Green's functions is used to demonstrate the dependence of effective elastic moduli of fiber reinforced composites on the fiber radial distribution functions as estimated from measured experimental data as well as from the ensembles generated by the proposed method. The MEFM is applied for the estimations of second statistical moments of stresses in both the constituents and the interfaces between the matrix and fibers. These estimations are used in turn for the prediction of the effective envelope for failure initiation. The dependence of the effective failure envelope on the elastic, geometrical, and failure parameters of the constituents and the interphase matrix/fibers are analyzed.

Quantitative description and numerical simulation of random microstructures of composites and their effective elastic moduli

A digital image processing technique is used for measurement of centroid coordinates of fibers with forthcoming estimation of statistical parameters and functions describing the stochastic structure of a real fiber composite. Comparative statistical analysis of the real and numerically simulated structure are performed. Accompanying of known methods of the generation of random configurations by the random shaking procedure allows creating of the most homogenized and mixed structures that do not depend on the initial protocol of particle generation. We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous set of ellipsoidal inclusions. The multiparticle effective field method (see for references, Buryachenko, Appl. Mech. Rev., (2001a), 54, 1–47) based on the theory of functions of random variables and Green’s functions is used for demonstration of the dependence of effective elastic moduli of fiber composites on the radial distribution functions estimated from the real experimental data as well as from the ensembles generated by the method proposed.

A Local Theory of Elastoplastic Deformation of Two-Phase Metal Matrix Random Structure Composites

A two-phase material is considered, which consists of a homogeneous elastoplastic matrix containing a homogeneous statistically uniform random set of ellipsoidal inclusions with the same form, orientation, and mechanical properties. The multiparticle effective field method (used in this paper) in the original form assumes constant plastic strains in the matrix. This assumption is replaced by the following micromechanical model: Each inclusion consists of an elastic core and a thin coating. The mechanical properties of both the matrix and the coating are the same but with different plastic strains. Homogeneous plastic strains are assumed inside the matrix and in each of separate subdomains of the coating. A general theory of plasticity is developed for arbitrary loading based on incremental elastoplastic analysis. The consideration of inhomogeneity of plastic strains in the coating enables to obtain some principally new effects of elastoplastic deformation.

Multiparticle Effective Field and Related Methods in Micromechanics of Random Structure Components

In this paper, linearly thermoelastic composite media are treated, which consist of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated inclusions. Effective properties (such as compliance, thermal expansion, stored energy) as well as the first statistical moments of stresses in the components are estimated for both the general case of nonhomogeneity of the thermoelastic inclusion properties and arbitrary choice of comparison medium. The micromechanical approach is based on Green's function techniques as well as on the generalization of the "multiparticle effective field" method (MEFM), previously proposed for the estimation of stress field averages in the components for the case of coinciding elastic moduli of the matrix and comparison medium. The author considers in detail the connection of methods proposed with numerous related methods and demonstrates that the MEFM includes, as particular cases, the well-known methods of mechanics of strongly heterogeneous media (such as the effective medium and the mean field methods and some others).

Local effective thermoelastic properties of graded random structure matrix composites

We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of ellipsoidal uncoated or coated inclusions, where the concentration of the inclusions is a function of the coordinates (functionally graded material). Effective properties, such as compliance and thermal expansion coefficient, as well as first statistical moments of stresses in the components are estimated for the general case of inhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on the Green function technique as well as on the generalization of the multiparticle effective field method (MEFM), previously proposed for the research of statistically homogeneous random structure composites. The hypothesis of effective field homogeneity near the inclusions is used; nonlocal effects of overall constitutive relations are not considered. Nonlocal dependences of local effective thermoelastic properties as well as those of conditional averages of the stresses in the components on the concentration of the inclusions are demonstrated.

On the thermoelasticity of random structure particulate composites

We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing statistically uniform random set of ellipsoidal coated inclusions. Effective properties (such as compliance, thermal expansion, stored energy) as well as first and second statistical moments of stresses in the components are estimated for the general case of nonhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on the Green's function technique as well as on the generalization of the “multiparticle effective field method” (MEFM), previously proposed for the analysis of composites with homogeneous inclusions, see e.g. Buryachenko and Kreher (J. Mech. Phys. Solids (1995) 43, 1105-1125). In the framework of the effective field hypothesis one obtains the generalization of the classical formula by Rosen and Hashin (Int. J. Engng. Sci. (1970), 8, 157-173) for composites with identical coated inclusions, which is exact for two-component composites. The application of the theory is demonstrated by the prediction of the overall yield surface of composite materials.

Thermoelastic stress fluctuations in random-structure coated particulate composites

In this paper linearly thermoelastic composite media are considered, consisting of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated inclusions. This study constitutes an extension of the theory regarding the purely isothermal elastic case and uncoated ellipsoidal inclusions. Effective properties (such as compliance, thermal expansion, stored energy) and both first and second statistical moments of stresses in the components are estimated for the general case of nonhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on Green's function techniques and on the generalization of the ‘multiparticle effective field’ method (MEFM), previously proposed for the estimation of stress field averages in the components. The application of this theory is demonstrated by calculating this overall yield surfaces of composite materials. The influence of the coating is analyzed both by the assumption of homogeneity of the stress field in the inclusion core and the thin-layer hypothesis.

Modeling of layer-structured high speed steel

High Speed Steels (HSSs) produced by electro slag remelting can be viewed as metal matrix composites containing alternating layers of high and low inclusion volume fraction. To account for such specific phase arrangements a hierarchical micro-meso-macro approach for elastoplastic analyses of HSSs is presented. It combines an extended lamination theory for modeling the layered structure at the mesoscale with the Multi-particle Effective Field Method for describing the matrix-inclusion topology at the microscale. The overall elastoplastic behavior and the microfields relevant for microscale damage initiation are studied with respect to their dependence on meso- and microtopology parameters. The results are compared with predictions from a two-dimensional finite element micromechanics model, which treats the HSS as a matrix-inclusion composite with a graded microstructure.

Hierarchical Modeling of the Mechanical Behavior of High Speed Steels as Layer-Structured Particulate MMCs

High Speed Steels (HSS) produced by electro-slag remelting can be viewed as particle reinforced Metal Matrix Composites (MMCs) consisting of alternating layers of high and low inclusion volume fraction These phase arrangements require specific models to be used in analytical and numerical studies of the elastoplastic response of HSS. In the present study, a hierarchical micro-meso-macro approach is discussed, which combines the Multi-Particle Effective Field Method (MIEFM) for describing the matrix-inclusion topology at the microscale with an extended lamination theory for handling the layered geometry at the mesoscale. In addition, a Finite Element based two-dimensional method is presented, in which HSS is modeled as a material with a graded microstructure. The obtained results are discussed in terms of overall elastoplastic behavior and of damage relevant microscale fields.

The overall elastopIastic behavior of multiphase materials with isotropic components

A multiphase material is considered, which consists of a homogeneous elastoplastic matrix containing a homogenous statistically uniform random set of ellipsoidal elastic inclusions. An approach based on the multiparticle effective field method is introduced for determining the overall elastoplastic behavior of the material under monotonic loading. A secant modulus concept is employed, and linearized problems are solved at each step of an iterative procedure. Physically consistent assumptions are used for linearizing nonlinear functions which depend on the phase averages of the second invariant of stress and on the stress deviator. Exact expressions for the second moments of the microstresses are employed.