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.     

On the effective elastic properties of matrix composites: Combining the effective field method and numerical solutions for cell elements with multiple inhomogeneities

The work is devoted to calculation of effective elastic constants of homogeneous materials containing random or regular sets of isolated inclusions. Our approach combines the self-consistent effective field method with the numerical solution of the elasticity problem for a typical cell. The method also allows analysis of detailed elastic fields in the composites. By the numerical solution of the elasticity problem for a cell, integral equations for the stress field are used. Discretization of these equations is carried out by Gaussian approximating functions. For such functions, elements of the matrix of the discretized problem are calculated in explicit analytical forms. If the lattice of approximating nodes is regular, the matrix of the discretized problem proves to have the Toeplitz structure. The matrix-vector products with such matrices may be carried out by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. Results are given for 2D-media with regular and random sets of circular inclusions, and compared with existing exact solutions.     

Interactions of cracks and inclusions in homogeneous elastic media

The work is devoted to static problems of elasticity for an infinite homogeneous medium containing planar parallel cracks and heterogeneous inclusions of arbitrary shapes. Cracks and inclusions occupy a finite region of the medium that is subjected to arbitrary external forces. The problem is reduced to a system of surface integral equations for crack opening vectors and volume integral equations for the stress tensor in the region. Gaussian approximating functions are used for discretization and efficient numerical solution of this system. Such functions are centered at the nodes of a regular node grid that covers all the inclusions and the crack surfaces. For Gaussian functions, the elements of the matrix of the discretized system have forms of standard integrals that can be tabulated and calculated fast. The matrix of the discretized system is not sparse but it has Teoplitz’s structure, and the number of independent matrix elements is much smaller than the total number of the elements. In addition, fast Fourier transform technique can be used for calculation matrix-vector products with such matrices. It accelerates substantially the process of iterative solutions of the discretized system. The method is mesh free. Examples of numerical solutions of the problems for planar circular cracks and spherical inclusions are presented and compared with analytical and numerical solutions available in the literature.     

Fast Solution of 3D-Elasticity Problem of a Planar Crack of Arbitrary Shape

A planar crack of an arbitrary shape in a homogeneous elastic medium is considered. The problem is reduced to integral equation for the crack opening vector. Its numerical solution utilizes Gaussian approximating functions that drastically simplify construction of the matrix of a linear algebraic system of the discretized problem. For regular grids of approximating nodes, this matrix turns out to have the Teoplitz structure. It allows one to use the Fast Fourier Transform algorithms for calculation of the matrix-vector products in the process of iterative solution of the discretized problem. The method is applied to a crack bounded by the curve for 0.2 ≤ p ≤ 4. The contribution of a crack to the overall effective elastic constants is calculated.     

Coupled heat conduction and thermal stress analyses in particulate composites

This study introduces two micromechanical modeling approaches to analyze spatial variations of temperatures, stresses and displacements in particulate composites during transient heat conduction. In the first approach, a simple micromechanical model based on a first order homogenization scheme is adopted to obtain effective mechanical and thermal properties, i.e., coefficient of linear thermal expansion, thermal conductivity, and elastic constants, of a particulate composite. These effective properties are evaluated at each material (integration) point in three dimensional (3D) finite element (FE) models that represent homogenized composite media. The second approach treats a heterogeneous composite explicitly. Heterogeneous composites that consist of solid spherical particles randomly distributed in homogeneous matrix are generated using 3D continuum elements in an FE framework. For each volume fraction (VF) of particles, the FE models of heterogeneous composites with different particle sizes and arrangements are generated such that these models represent realistic volume elements “cut out” from a particulate composite. An extended definition of a RVE for heterogeneous composite is introduced, i.e., the number of heterogeneities in a fixed volume that yield the same expected effective response for the quantity of interest when subjected to similar loading and boundary conditions. Thermal and mechanical properties of both particle and matrix constituents are temperature dependent. The effects of particle distributions and sizes on the variations of temperature, stress and displacement fields are examined. The predictions of field variables from the homogenized micromechanical model are compared with those of the heterogeneous composites. Both displacement and temperature fields are found to be in good agreement. The micromechanical model that provides homogenized responses gives average values of the field variables. Thus, it cannot capture the discontinuities of the thermal stresses at the particle-matrix interface regions and local variations of the field variables within particle and matrix regions.     

Eigenstrain and Fourier series for evaluation of elastic local fields and effective properties of periodic composites

The elastic stress and strain fields and effective elasticity of periodic composite materials are determined by imposing a periodic eigenstrain on a homogeneous solid, which is constrained to be equivalent to the heterogeneous composite material through the imposition of a consistency condition. To this end, the variables of the problem are represented by Fourier series and the consistency condition is written in the Fourier space providing the system of equations to solve. The proposed method can be considered versatile as it allows determining stress and strain fields in micro-scale and overall properties of composites with different kinds of inclusions and defects. In the present work, the method is applied to multi-phase composites containing long fibers with circular transverse section. Numerical solutions provided by the proposed method are compared with finite element results for both unit cell containing a single fiber and unit cell with multiple fibers of different sizes.     

A self-consistent approach to dynamic effective properties of a composite reinforced by distributed spherical particles

Summary A self-consistent approach to dynamic effective properties of a composite reinforced by randomly distributed spherical inclusions is studied. The coherent plane waves propagating through the particle-reinforced composite are of attenuation nature. It implies that there is an analogy between the particle-reinforced composite and the effective medium with complex-valued elastic constants from the viewpoints of wave propagation. A composite sphere consisting of the inclusion, the matrix and the interphase between them is assumed embedded in the effective medium. The effective wavenumbers of the coherent plane waves propagating through the particle-reinforced composite are obtained by the dynamic self-consistent conditions which require that the forward scattering amplitudes of such a composite sphere embedded in the effective medium are equal to zero. The dynamic effective properties (effective phase velocity, effective attenuation and effective elastic constants) obtained by the present dynamic self-consistent approach for SiC-Al composites are compared numerically with that obtained by the effective field approach at various volume concentrations. It is found that there is a good agreement between the two approaches at a relatively low frequency and low volume concentration but the numerical results deviate from each other at a relatively high frequency and high volume concentration.     

Propagation of low-frequency elastic waves in double-porositymedia containing viscoelastic component in the pores

A self-consistent scheme named the effective field method (EFM) is applied for the calculation of the velocities and quality factors of elastic waves propagating in double-porosity media. A double-porosity medium is considered to be a heterogeneous material composed of a matrix with primary pores and inclusions that are represent by flat (crack-like) secondary pores. The prediction of the effective viscoelastic moduli consists of two steps. First, we calculate the effective viscoelastic properties of the matrix with the primary small-scale pores (matrix homogenization). Then, the porous matrix is treated as a homogeneous isotropic host where the large-scale secondary pores are embedded. Spatial distribution of inclusions in the medium is taken into account via a special two-point correlation function. The results of the calculation of the viscoelastic properties of double-porosity media containing isotropic fields of crack-like inclusions and double-porosity media with some non-isotropic spatial distributions of crack-like inclusions 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.     

A polarization‐based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast

It is recognized that the convergence of FFT‐based iterative schemes used for computing the effective properties of elastic composite materials drastically depends on the contrast between the phases. Particularly, the rate of convergence of the strain‐based iterative scheme strongly decreases when the composites contain very stiff inclusions and the method diverges in the case of rigid inclusions. Reversely, the stress‐based iterative scheme converges rapidly in the case of composites with very stiff or rigid inclusions but leads to low convergence rates when soft inclusions are considered and to divergence for composites containing voids. It follows that the computation of effective properties is costly when the heterogeneous medium contains simultaneously soft and stiff phases. Particularly, the problem of composites containing voids and rigid inclusions cannot be solved by the strain or the stress‐based approaches. In this paper, we propose a new polarization‐based iterative scheme for computing the macroscopic properties of elastic composites with an arbitrary contrast which is nearly as simple as the basic schemes (strain and stress‐based) but which has the ability to compute the overall properties of multiphase composites with arbitrary elastic moduli, as illustrated through several examples. Copyright © 2011 John Wiley & Sons, Ltd.     

Binary inclusion model for the overall elasticity of imperfectly bonded composites

Summary. A micromechanical approach is developed to estimate the overall elastic moduli of composite materials with imperfectly bonded spherical fillers. The randomly dispersed particles are assumed to satisfy linear interfacial conditions where both tangential and normal interface displacement discontinuities are linearly related to the respective surface tractions. Using the generalized version of Eshelbys equivalent inclusion method proposed by Furuhashi et al. [6] the analysis of the heterogeneous medium reduces to the study of a corresponding homogeneous medium containing spherical inclusions with a proper distribution of eigenstrain and Somigliana dislocation fields. Based on the estimated pair-wise average of strain fields in two interacting imperfect fillers embedded in the homogeneous infinite matrix, the ensemble phase volume average of field quantities has been evaluated within a representative volume element containing a finite number of imperfect particles. For the case of a constant radial distribution function, results are in reasonable agreement with those based on the generalized self-consistent method and composite sphere assemblage proposed by Hashin [11].     

The problem of thermal conduction for two ellipsoidal inhomogeneities in an anisotropic medium and its relevance to composite materials

Summary We consider the problem of thermal conduction for an unbounded medium containing two ellipsoidal inhomogeneities subjected to a remote homogeneous boundary condition of temperature. The constituents are anisotropic and the ellipsoids could be at arbitrary orientations. In the formulation we first introduce some appropriate transformations into the heterogeneous medium and transform the problem into an isotropic matrix consisting of two analogous ellipsoidal inhomogeneities. Next, we replace the effect of inhomogeneities by some polynomial types of equivalent eigen-intensities by the concept of equivalent inclusion. These procedures allow us to write the local fields in terms of harmonic potentials and their derivatives. Numerical results show that linear approximations of eigen-fields yield accurate results in comparison with existing solutions by Honein et al. [2] for moderately separated inhomogeneities. Solutions of this type are used to estimate the overall thermal conductivity of composites with periodic microstructure. Finally, we present results for composites consisting of spherical inclusions with body-centered cubic, face-centered cubic, body-centered orthorhombic, and face-centered orthorhombic arrays.     

OPTIMAL DISTRIBUTION OF FILLER MATERIAL IN A CIRCULAR PLATE MADE OF PARTICULATE COMPOSITE

Optimal distribution of reinforcing particles in a continuous matrix is determined in order to maximize the stiffness of a simply supported and symmetrically loaded circular plate. The elastic constants are computed from a dilute suspensions model which relates the concentration of inclusions to material properties by assuming a low volume fraction of spherical particles within the matrix. A linear relationship between the concentration of inclusions and the Young's modulus is obtained by neglecting the higher order concentration terms in the expression for the elastic modulus. This procedure leads to a closed form solution of the design problem which is accurate for small volumes of the filler material. Efficiency of the design is assessed by comparing the maximum deflections of the optimal (nonhomogeneous) and the homogeneous plates. Numerical results are given in graphical form     

Two-dimensional and axisymmetric unit cell models in the analysis of composite materials

Unit cell models have been widely used for investigating fracture mechanisms and mechanical properties of composite materials assuming periodically arrangement of inclusions in matrix. It is desirable to clarify the geometrical parameters controlling the mechanical properties of composites because they usually contain randomly distributed particulate. To begin with a tractable problem this paper focuses on the effective Young’s modulus E of heterogeneous materials. Then, the effect of shape and arrangement of inclusions on E is considered by the application of FEM through examining three types of unit cell models assuming 2D and 3D arrays of inclusions. It is found that the projected area fraction and volume fraction of inclusions are two major parameters controlling effective elastic modulus of inclusions.     

Effective elastic moduli of three-phase composites with randomly located and interacting spherical particles of distinct properties

A micromechanical analytical framework is presented to predict effective elastic moduli of three-phase composites containing many randomly dispersed and pairwisely interacting spherical particles. Specifically, the two inhomogeneity phases feature distinct elastic properties. A higher-order structure is proposed based on the probabilistic spatial distribution of spherical particles, the pairwise particle interactions, and the ensemble-volume homogenization method. Two non-equivalent formulations are considered in detail to derive effective elastic moduli with heterogeneous inclusions. As a special case, the effective shear modulus for an incompressible matrix containing randomly dispersed and identical rigid spheres is derived. It is demonstrated that a significant improvement in the singular problem and accuracy is achieved by employing the proposed methodology. Comparisons among our theoretical predictions, available experimental data, and other analytical predictions are rendered. Moreover, numerical examples are implemented to illustrate the potential of the present method.     

Hypervelocity impact on isotropic composites with metal or ceramic inclusions

Experimental results of studying the hypervelocity impact on isotropic heterogeneous composites consisting of an epoxy or aluminum matrix containing fine-grained metal (Al, Pb) or ceramic (SiO₂) inclusions are given. The aim of the study is to develop composite materials offering higher penetration resistance to a high-velocity projectile than the component material. This resistance is characterized by the magnitude of the ratio of the crater depth in a thick target to the diameter of spherical projectile. In the case of two particulate composites studied it is shown that the crater depth from impact of steel projectiles is lower about by one projectile diameter than for homogeneous lead or aluminum over the impact velocity ranged from 3 up to 11 km/s.     

Eshelby tensors for a spherical inclusion in microelongated elastic fields

Eshelby tensors are found for a spherical inclusion in a microelongated elastic field. Here, a special micromorphic model is introduced to describe the damaged material which defines the damage as the formation and the growth of microcracks and microvoids occurred in the material at the microstructural level. To determine the new material coefficients of the model, an analogy is established between the damaged body and the composite materials and then Mori–Tanaka homogenization technique is considered to obtain overall material moduli. Following this idea, the determination of the Eshelby tensors which establish the relation between the strains of the matrix material and of the inclusion becomes the first task. Introducing the concept of eigenstrain and microeigenstrain, the general constitutive theory is given for a homogeneous isotropic centrosymmetric microelongated media with defects. Then by the use of Green’s functions, micro and macro elastic fields are presented for the case of spherical inclusions embedded in an infinite microelongated material. Thus, the Eshelby tensors are obtained for a microelongated elastic field with a spherical inclusion and it is also shown that the classical Eshelby tensors can be obtained as a limit case of the microelongation.     

A new homogenization method based on a simplified strain gradient elasticity theory

Homogenization methods utilizing classical elasticity-based Eshelby tensors cannot capture the particle size effect experimentally observed in particle–matrix composites at the micron and nanometer scales. In this paper, a new homogenization method for predicting effective elastic properties of multiphase composites is developed using Eshelby tensors based on a simplified strain gradient elasticity theory (SSGET), which contains a material length scale parameter and can account for the size effect. Based on the strain energy equivalence, a homogeneous comparison material obeying the SSGET is constructed, and two sets of equations for determining an effective elastic stiffness tensor and an effective material length scale parameter for the composite are derived. By using Eshelby’s eigenstrain method and the Mori–Tanaka averaging scheme, the effective stiffness tensor based on the SSGET is analytically obtained, which depends not only on the volume fractions and shapes of the inhomogeneities (i.e., phases other than the matrix) but also on the inhomogeneity sizes, unlike what is predicted by the existing homogenization methods based on classical elasticity. To illustrate the newly developed homogenization method, sample cases are quantitatively studied for a two-phase composite filled with spherical, cylindrical, or ellipsoidal inhomogeneities (particles) using the averaged Eshelby tensors based on the SSGET that were derived earlier by the authors. Numerical results reveal that the particle size has a large influence on the effective Young’s moduli when the particles are sufficiently small. In addition, the results show that the composite becomes stiffer when the particles get smaller, thereby capturing the particle size effect.     

Failure modes and effective strength of two-phase materials determined by means of numerical limit analysis

Summary In the most general case, composites composed of two materials, exhibiting a matrix-inclusion morphology, can be described by three phases: the matrix phase, the inclusion phase, and the interface between the inclusion and the matrix. In order to relate effective material properties to the matrix-inclusion behavior and the morphology, i.e., to the arrangement of the inclusions within the matrix phase, analytical and/or numerical schemes may be employed. Regarding effective strength properties, averaging schemes used, e.g., for upscaling elastic and viscoelastic properties are not able to capture the localized mode of material failure and do not provide information about the failure modes within the material. In this paper, the application of limit analysis to two-phase materials subjected to uniaxial/biaxial loading is proposed, giving access to the respective material strength and the corresponding failure modes. Based on a discretized form of limit analysis, different strength properties are assigned to the matrix, the inclusion, and the interface. The solution of the underlying optimization problem arising from the respective upper- and lower-bound formulation is based on second-order-cone-programming (SOCP). The presented upscaling scheme is used to illustrate the finer-scale origin of frequently observed failure and degradation scenaria in matrix-inclusion materials, highlighting the effect of strength properties, morphology, and interface degradation on the effective strength of the composite material. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday