Thermoelastic effective properties and stress concentrator factors of composites reinforced by heterogeneities of noncanonical shape

We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of heterogeneities of arbitrary shape. The general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure (Buryachenko, 2010a) where the notion of perturbator is introduced and statistical averages are obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. Effective properties (such as compliance and thermal expansion) as well as the first statistical moments of stresses in the phases are estimated for statistically homogeneous composites with the general case of both the shape and inhomogeneity of the thermoelastic heterogeneities properties. The explicit new representations of the effective thermoelastic properties and stress concentration factor are expressed through some building blocks described by numerical solutions for one heterogeneity inside the infinite medium subjected to the homogeneous remote loading. Numerical results are obtained for some model statistically homogeneous composites reinforced by aligned identical homogeneous heterogeneities of noncanonical shape. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.     

Calculation of electro and thermo static fields in matrix composite materials of regular or random microstructures

In the work, a numerical method for calculation of electro and thermo static fields in matrix composite materials is considered. Such materials consist of a regular or random set of isolated inclusions embedded in a homogeneous background medium (matrix). The proposed method is based on fast calculation of fields in a homogeneous medium containing a finite number of isolated inclusions. By the solution of this problem, the volume integral equations for the fields in heterogeneous media are used. Discretization of these equations is carried out by Gaussian approximating functions that allow calculating the elements of the matrix of the discretized problem in explicit analytical forms. If the grid of approximating nodes is regular, the matrix of the discretized problem proves to have the Toeplitz structure, and the matrix-vector product with such matrices can be calculated by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. In the case of an infinite medium containing a homogeneous in space random set of inclusions, our approach combines a self-consistent effective field method with the numerical solution of the conductivity problem for a typical cell. The method allows constructing detailed static (electric or temperature) fields in the composites with inclusions of arbitrary shapes and calculating effective conductivity coefficients of the composites. Results are given for 2D and 3D-composites and compared with the existing exact and numerical solutions.     

Combining self-consistent and numerical methods for the calculation of elastic fields and effective properties of 3D-matrix composites with periodic and random microstructures

The work is devoted to the calculation of static elastic fields in 3D-composite materials consisting of a homogeneous host medium (matrix) and an array of isolated heterogeneous inclusions. A self-consistent effective field method allows reducing this problem to the problem for a typical cell of the composite that contains a finite number of the inclusions. The volume integral equations for strain and stress fields in a heterogeneous medium are used. Discretization of these equations is performed by the radial Gaussian functions centered at a system of approximating nodes. Such functions allow calculating the elements of the matrix of the discretized problem in explicit analytical form. For a regular grid of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and matrix-vector products with such matrices may be calculated by the fast fourier transform technique. The latter accelerates significantly the iterative procedure. First, the method is applied to the calculation of elastic fields in a homogeneous medium with a spherical heterogeneous inclusion and then, to composites with periodic and random sets of spherical inclusions. Simple cubic and FCC lattices of the inclusions which material is stiffer or softer than the material of the matrix are considered. The calculations are performed for cells that contain various numbers of the inclusions, and the predicted effective constants of the composites are compared with the numerical solutions of other authors. Finally, a composite material with a random set of spherical inclusions is considered. It is shown that the consideration of a composite cell that contains a dozen of randomly distributed inclusions allows predicting the composite effective elastic constants with sufficient accuracy.     

The problem of two plastic and heterogeneous inclusions in an anisotropic medium

We demonstrate an integral equation for the total local strain ε^T in an anisotropic heterogeneous medium with incompatible strain ε^p and which is at the same time submitted to an exterior field. The integral equation is solved in the case of an heterogeneous and plastic pair of inclusions, for which we calculate the average fields in each inclusion as well as the different parts of the elastic energy stocked in the medium.The solution is applied to the case of two isotropic and spherical inclusions in an isotropic matrix loaded in shear. The results are compared with those deduced from a more approximate method based on Horn's approximation of the integral equation. In appendix we give a numerical method for calculating the interaction tensors between anisotropic inclusions in an anisotropic medium as well as the analytic solution in the case of two spherical inclusions located in an isotropic medium.     

A novel concept to develop composite structures with isotropic negative Poisson’s ratio: Effects of random inclusions

Materials with negative Poisson’s ratio (NPR) effects have been studied for decades. However, the studies have mainly focused on 2D periodic structures which only have NPR effects in certain in-plane directions. In this paper, a novel concept is proposed to develop composite structures with isotropic NPR effects using NPR random inclusions. The study starts from a finite element analysis of deformation mechanisms of two 2D representative cells which are embedded with a re-entrant square and a re-entrant triangle, respectively. Based on the analysis results, the re-entrant triangles are selected as random inclusions into a matrix to form 2D composite structures. Four such composite structures are built with different numbers of inclusions through a parametric model, and their NPR effects and mechanical behaviors are analyzed using the finite element method. The results show that the isotropic NPR effects of composites can be obtained with high random re-entrant inclusions. Thus, the novel concept proposed is numerically proved by this study.     

Effective conductance of a high-contrast,random-structure composite. Numerical simulation

This paper considers a high-contrast, two-component composite of random structure, for whose simulation a two-dimensional network model is used. The dependence of the medium conductance on the volume content and composition of the filler that is characteristic of percolation theory has been obtained: up to some volume content, the effective conductance is small and then it grows rapidly. The results are based on statistical modeling (solving a large number of problems at various random distributions of inclusions and with subsequent statistical processing). __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 78, No. 6, pp. 170–177, November–December, 2005.     

On the thermo-elastostatics of heterogeneous materials: II. Analysis and generalization of some basic hypotheses and propositions

One considers linearly thermoelastic composite media, which consist of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated inclusions. Effective properties (such as compliance and thermal expansion) 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. At first, one shortly reproduces both the basic assumptions and propositions of micromechanics used in most popular methods, namely: effective field hypothesis, quasi-crystallite approximation, and the hypothesis of “ellipsoidal symmetry”. The explicit new 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 both the homogeneous and inhomogeneous remote loading. The method uses as a background the new general integral equation proposed in the accompanying paper and makes it possible to abandon the basic concepts of micromechanics mentioned above. The results of this abandonment are quantitatively estimated for some modeled composite reinforced by aligned continuously inhomogeneous fibers. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.     

Incremental and differential Maxwell Garnett formalisms for bi-anisotropic composites

We present, compare and contextualize two approaches to the homogenization of bi-anisotropic-in-bi-anisotropic particulate composite medias: (i) the incremental Maxwell Garnett (IMG) formalism, in which the composite medium is built incrementally by adding the inclusions in N discrete steps to the host medium; and (ii) the differential Maxwell Garnett (DMG) formalism, which is obtained from the IMG in the limit N→∞ . Both formalisms are applicable to arbitrary inclusion concentration and are well-suited for computational purposes. Either of the two formalisms may be used as an alternative to the well-known Bruggeman formalism. Numerical results for the homogenization of a uniaxial dielectric composite medium and of a chiroferrite are presented.     

Equivalent inclusion approach and effective medium approximations for the effective conductivity of isotropic multicomponent materials

Most effective medium approximations for isotropic inhomogeneous materials are based on dilute solutions of some typical inclusions in an infinite matrix medium, while the simplest approximations are those for the composites with spherical and circular inclusions. Practical particulate composites often involve inhomogeneities of more complicated geometry than that of the spherical (or circular) one. In our approach, those inhomogeneities are supposed to be substituted by simple equivalent spherical (circular) inclusions from a comparison of their respective dilute solution results. Then the available simple approximations for the equivalent spherical (circular) inclusion material can be used to estimate the effective conductivity of the original composite. Numerical illustrations of the approach are performed on some 2D and 3D geometries involving elliptical and ellipsoidal inclusions.     

Effective rheological parameters of a suspension with polyfractionated filler

We examine a microscopically inhomogeneous medium consisting of a homogeneous nonlinear viscoplastic matrix and a random set of rigid ellipsoidal inclusions of various sizes. A variant of the effective field method is used to calculate the macroscopic rheological constants of the medium.Translated from Inzhenerno-Fiz-icheskii Zhurnal, Vol. 61, No. 6, pp. 928–933, December, 1991.     

Multivariate Spartan spatial random field models

This paper introduces a family of stationary multivariate spatial random fields with D scalar components that extend the scalar model of Gibbs random fields with local interactions (i.e., Spartan spatial random fields). We derive permissibility conditions for Spartan multivariate spatial random fields with a specific structure of local interactions. We also present explicit expressions for the respective matrix covariance functions obtained at the limit of infinite spectral cutoff in one, two and three spatial dimensions. Finally, we illustrate the proposed covariance models by means of simulated bivariate time series and two-dimensional random fields.     

Stress Concentration in Microstructural Elements of Viscoelastic Composite Materials

We present solution of the problem of estimating the stress concentration in the constituents (matrix and inclusions) of a multiconstituent viscoelastic composite material, depending on the shape of inclusions, properties of the matrix and the whole composite. The matrix material is isotropic and viscoelastic. A wide range of properties of inclusions (e.g., pores, solid and viscous particles) is considered. To solve this problem, we use the method of integral transformations. __________ Translated from Problemy Prochnosti, No. 5, pp. 138 – 149, September – October, 2005.     

Numerical modeling of elastohydrodynamic lubrication in point or line contact for heterogeneous elasto-plastic materials

A semi-analytic solution is developed for heterogeneous elasto-plastic materials with inhomogeneous inclusions under elastohydrodynamic lubrication in point contact or line contact. The inhomogeneous inclusions within a material are homogenized as homogeneous inclusions with properly determined eigenstrains based on the equivalent inclusion method, and the surface displacements induced by these eigenstrains are then introduced into the gap between the contact bodies to update surface geometry. The accumulative plastic deformation is iteratively obtained by a procedure involving a plasticity loop and an incremental loading process. The model takes into account the interactions among the contact bodies, the embedded inclusions, and the plastic zones, thus leading to a solution of the surface pressure distributions, film thickness profiles, plastic zones, and subsurface stress fields. This solution is of great importance for the analysis of elasto-plasto damage of heterogeneous materials subject to lubricated contact.     

COMPLEX VARIABLE BOUNDARY ELEMENT METHOD FOR TORSION OF COMPOSITE SHAFTS

The problem of torsion of composite shafts consisting of a cylindrical matrix surrounding a finite number of inclusions is solved by using the complex variable boundary element method. The method consists in reducing the problem to the solution of a singular integral equation in terms of an analytic function of a complex variable using the Cauchy integral. The resulting integral equation is then solved numerically by discretizing the boundaries into segments called complex boundary elements and replacing the analytic function on the boundaries by interpolating function. Numerical examples are given for a square shaft with a circular inclusion, and for an elliptic shaft with two elliptic inclusions. © 1997 by John Wiley & Sons, Ltd.     

Reconstruction of spatially inhomogeneous dielectric tensors through optical tomography

A method to reconstruct weakly anisotropic inhomogeneous dielectric tensors inside a transparent medium is proposed. The mathematical theory of integral geometry is cast into a workable framework that allows the full determination of dielectric tensor fields by scalar Radon inversions of the polarization transformation data obtained from six planar tomographic scanning cycles. Furthermore, a careful derivation of the usual equations of integrated photoelasticity in terms of heuristic length scales of the material inhomogeneity and anisotropy is provided, resulting in a self-contained account about the reconstruction of arbitrary three-dimensional, weakly anisotropic dielectric tensor fields.     

Analytical solutions to the planar non-axisymmetric elasticity and thermoelasticity problems for homogeneous and inhomogeneous annular domains

This paper presents an analytical approach to solving the plane non-axisymmetric elasticity and thermoelasticity problems in terms of stresses for isotropic, homogeneous or inhomogeneous annular domains. The key feature of this approach is integration of the equilibrium equations in order to: a) express all the stress-tensor components in terms of a governing stress; b) deduce the integral equilibrium conditions, which are vital for the solution. Because the equilibrium equations are insensitive of material properties, the obtained expressions and integral conditions fit both homogeneous and inhomogeneous cases. The governing stress is derived out of the compatibility equation. Regarding complete construction of the solution, the integral compatibility conditions are deduced by integrating the strain-displacement relations. In the case of inhomogeneous material, the governing compatibility equation is reduced to Volterra type integral equation which then is solved by simple iteration method. The rapid convergence of the iterative procedure is established.     

Doubly periodic volume?surface integral equation formulation for modelling metamaterials

A generalised volume-surface integral equation is extended by way of the periodic Green's function to model arbitrarily complex designs of metamaterials consisting of high-contrast inhomogeneous anisotropic material regions as well as metallic inclusions. The unique aspect of the formulation is the integration of boundary and volume integral equations to increase modelling efficiency and capability. Specifically, the boundary integral approach with equivalent surface currents is adopted over regions consisting of piecewise homogeneous materials as well as metallic perfect electric/magnetic conductor inclusions, whereas the volume integral equation is employed only in inhomogeneous and/or anisotropic material regions. Because the periodic Green's function only needs to be evaluated for the equivalent surface currents enclosing an inhomogeneous and/or anisotropic region, matrix fill time is much less as compared to using a volume formulation. Furthermore, the incorporation of curvilinear finite elements allows for greater geometrical modelling flexibility for arbitrarily shaped high-contrast regions found in typical designs of engineered metamaterials     

Inhomogeneous State of the Bi<Subscript>2</Subscript>Te<Subscript>3</Subscript> Doped with Manganese

Basing on electron spin resonance (ESR) data for Bi₂Te₃ doped by Mn ions we argue that this compound can be inhomogeneous and consists of two components with the different structures. Its main phase Bi _2?x Mn _x Te ₃ is intertwined with the microscopical inclusions of MnBi phase. The integral volume of these intermetal clusters is less than 1 % but nevertheless they exert the serious impact on the dynamic magnetic properties of the entire system. These inclusions are ferromagnetic with the Curie temperature of 630 K, while the main bulk phase Bi _2?x Mn _x Te ₃ has x= 0.05 orders at T _c= 10 K (qualitatively this twophase picture is valid not only for this given x). Below this temperature two ferromagnetic phases coexist. Since the integral spontaneous polarization in MnBi phase is averaged out due to its random orientations in different clusters the time-reversal symmetry of Bi ₂Te ₃ doped by Mn ions is violated only at the low-temperature ferromagnetic transition.     

Simulation of local material properties based on moving-window GMC

When analyzing the behavior of composite materials under various loading conditions, the assumption is generally made that the behavior due to randomness in the material can be represented by a homogenized, or effective, set of material properties. This assumption may be valid when considering displacement, average strain, or even average stress of structures much larger than the inclusion size. The approach is less valid, however, when considering either behavior of structures of size at the scale of the inclusions or local stress of structures in general. In this paper, Monte Carlo simulation is used to assess the effects of microstructural randomness on the local stress response of composite materials. In order to achieve these stochastic simulations, the mean, variance and spectral density functions describing the randomly varying elastic properties are required as input. These are obtained here by using a technique known as moving-window generalized method of cells (moving-window GMC). This method characterizes a digitized composite material microstructure by developing fields of local effective material properties. Once these fields are generated, it is straightforward to obtain estimates of the associated probabilistic parameters required for simulation. Based on the simulated property fields, a series of local stress fields, associated with the random material sample under uniaxial tension, is calculated using finite element analysis. An estimation of the variability in the local stress response for the given random composite is obtained from consideration of these simulations.