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.     

Hybrid effective medium approximations for random elastic composites

Several popular effective medium approximations for elastic constants of random composites are reformulated in terms of a pair of canonical functions and their transform variables. This choice of reformulation enables easier comparisons of the results of all these methods with rigorous bounds. Furthermore, insight into the various methods gained by taking this point of view suggests a number of new effective medium approximations that, in some cases, are natural variants and/or combinations (i.e., hybrids) of the existing ones, and in other cases are new ones based in part on the bounds themselves. Numerical comparisons are given for several standard inclusion models — including spherical, needle, and penny-shaped inclusions — as well as the penetrable sphere model. Of the various alternatives considered, a new method called the split-step differential (SSD) scheme is one of the more useful ones, as it simplifies the differential scheme by replacing half of this scheme’s integration routines with a simple update formula for the bulk modulus.     

Bounds on the effective thermal conductivity of composites with imperfect interface

A new model is developed to bound the effective thermal conductivity of composites with thermal contact resistance between spherical inclusions and matrix. To construct the trial temperature and heat flux fields which satisfy the necessary interface conditions, the transition layer for each spherical inclusion is introduced. For the upper bound, the trial temperature field needs to satisfy the thermal contact resistance conditions between spherical inclusions and transition layers and the continuous interface conditions between transition layers and remnant matrix. For the lower bound, the trial heat flux field needs to satisfy the continuous interface conditions between different regions. It should be pointed out that the continuous interface conditions mentioned above are absolutely necessary for the application of variational principles, and the thermal contact resistance conditions between spherical inclusions and transition layers are suggested by the author. According to the principles of minimum potential energy and minimum complementary energy, the bounds of the effective thermal conductivity of composites with imperfect interfaces are rigorously derived. The effects of the size and distribution of spherical inclusions on the bounds of the effective thermal conductivity of composites are analyzed. It should be shown that the present method is simple and does not need to calculate the complex integrals of multi-point correlation functions. Meanwhile, the present method provides an entirely different way to bound the effective thermal conductivity of composites with imperfect interface, which can be developed to obtain a series of bounds by taking different trial temperature and heat flux fields. In addition, the present upper and lower bounds are finite when the thermal conductivity of spherical inclusions tends to ∞ and 0, respectively.     

Material properties of piezoelectric composites by BEM and homogenization method

Applications of boundary element method (BEM) to piezoelectric composites in conjunction with homogenization approach for determining their effective material properties are discussed in this paper. The composites considered here consist of inclusion and matrix phases. The homogenization model for composites with inhomogeneities is developed and introduced into a BE formulation to provide an effective means for estimating overall material constants of two-phase composites. In this model, a representative volume element (RVE) is used whose volume average stress and strain are calculated by the boundary tractions and displacements of the RVE. Thus BEM is suitable for performing calculations on average stress and strain fields of the composites. Numerical results for a piezoelectric plate with circular inclusions are presented to illustrate the application of the proposed micromechanics––BE formulation.     

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.     

Neutral spherical inhomogeneities in a twisted circular cylindrical bar

We consider the torsion problem of a circular cylindrical bar which is filled up with composite spherical inclusions. The composite inclusions consist of a core and coating both of which are spherically orthotropic with the volume fractions of the core being the same in every composite inclusion. The center points of the spherical inhomogeneities are on the axis of revolution of the circular cylinder. The neutral inhomogeneity in the considered problem of elastic equilibrium is defined as a foreign body (inclusion) which can be introduced in a host body without disturbing the elastic field (displacements, stresses) in it. The conditions of the neutral inhomogeneity for the twisted circular cylindrical bar are derived, and some special cases of inhomogeneity are analyzed. The present paper gives a new example for neutral inhomogeneity in the field of elasticity.     

The effects of surface elasticity and surface tension on the transverse overall elastic behavior of unidirectional nano-composites

The effects of surface elasticity and surface tension on the transverse overall behavior of unidirectional nano-scale fiber-reinforced composites are studied. The interfaces between the nano-fibers and the matrix are regarded as material surfaces described by the Gurtin and Murdoch model. The analysis is based on the equivalent inhomogeneity technique. In this technique, the effective elastic properties of the material are deduced from the analysis of a small cluster of fibers embedded into an infinite plane. All interactions between the inhomogeneities in the cluster are precisely accounted for. The results related to the effects of surface elasticity are compared with those provided by the modified generalized self-consistent method, which only indirectly accounts for the interactions between the inhomogeneities. New results related to the effects of surface tension are presented. Although the approach employed is applicable to all transversely isotropic composites, in this paper we consider only a hexagonal arrangement of circular cylindrical fibers.     

Analysis of the effective conductivity of composite materials in the entire range of volume fractions of inclusions up to the percolation threshold

The analytical interpolation formulas for the effective electrical conductivity of fiber-reinforced and particulate composites for any volume fractions of inclusions are derived in the present paper. The Garnett formulas are used for a limiting case of small volume fractions of inclusions. And the formulas based on the Shklovskii–De Gennes model are adopted for a limiting case of large volume fractions of inclusions approaching the percolation threshold. The derivation is based on application of the method of asymptotically equivalent functions. This approach presents a natural generalization of the two-point Padé approximants to the case, when in one of the limits the interpolated function cannot be represented in the form of power series. The application of this interpolation technique made it possible to derive the formulas for the effective conductivity of fiber-reinforced and particulate composites with very high volume fractions of inclusions, up to the percolation threshold. The numerical results are compared with the known asymptotic expressions, and also with the existing exact expressions in some limiting cases, for example, in the case of statistically equivalent arrangement of the constituent materials. Comparison with the experimental data confirms the satisfactory accuracy of the obtained analytical results.     

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.     

Bounds on effective magnetic permeability of three-phase composites with coated spherical inclusions

An effective model is developed to bound the effective magnetic permeability of three-phase composites with coated spherical inclusions. In the present model, the trial magnetic potential for the upper bound and the trial magnetic induction field for the lower bound are constructed to satisfy continuity interface conditions. According to the variational principle, the upper and lower bounds on the effective magnetic permeability of three-phase composites with coated spherical inclusions are derived. In this paper, trial magnetic potentials with different function forms are taken and the optimal upper bound is obtained for the trail magnetic potential corresponding to the third-order function. When the three-phase model degenerates into the composite spheres assemblage model [1], it is interesting that the optimal upper and lower bounds are the same. The effects of the volume fraction of coated spherical inclusions and the thickness and magnetic permeability of coated layers between the matrix and spherical inclusions on the effective magnetic permeability of composites are analyzed. The upper and lower bounds are finite non-zero values when the magnetic permeability of spherical inclusions tends to ∞$\infty$ and 0, respectively.     

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.     

Rigorous bounds on the effective thermal conductivity of composites with ellipsoidal inclusions

A new method is developed to derive the bounds of the effective thermal conductivity of composites with ellipsoidal inclusions. The transition layer for each ellipsoidal inclusion is introduced to make the trial temperature field for the upper bound and the trial heat flux field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the bounds of the effective thermal conductivity of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective thermal conductivity of composites are analyzed. It should be shown that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present method provides a powerful way to bound the effective thermal conductivity of composites, which can be developed to obtain a series of bounds by taking different trial temperature and heat flux fields. In addition, the present upper and lower bounds still are finite when the thermal conductivity of ellipsoidal inclusions tends to ∞ and 0, respectively.     

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.     

Influence of the inclusions distribution on the effective properties of heterogeneous media

In this paper we investigate the effective conductivity of composite materials by means of the homogenization method. We concentrate on composites with circular or elliptic cylindrical inclusions. In particular, we are interested in the effect of the distribution of the cylinders in the continuous material on the effective properties. We compare rectangular and hexagonal distributions with random distributions for different volume fractions of the inclusions. We also study the effect of the number of inclusions in each periodic cell for the random structure as well as shape influence of the elliptical inclusions.     

3D elastodynamic fields of non-uniformly coated obstacles: Notion of eigenstress and eigenbody-force fields

Based on wave-function expansion, the time harmonic wave scattered by a circular and spherical inhomogeneity has been studied by numerous investigators. This method has also been employed to axisymmetrically coated circular and spherical inhomogeneities by some authors. When the geometry of the obstacle is not axisymmetric, the wave-function expansion is no longer applicable. In this paper, it is proposed to employ the dynamic equivalent inclusion method (DEIM) which is more general than the methods presented in the literature. It will be seen that DEIM may be used to treat a wide range of situations in a unified manner and is not bound to certain symmetries. The DEIM was first proposed by Fu and Mura [Fu, L.S., Mura, T., 1983. The determination of elastodynamic fields of an ellipsoidal inhomogeneity. ASME J. Appl. Mech. 50, 390–396], and no further developments have been done on it ever since. Its original formulation has some shortcomings with regard to the concept of homogenizing eigenstrains, and for usage of polynomial eigenstrains. Moreover, it is limited to single ellipsoidal inhomogeneity without coating. The new viewpoints of homogenizing eigenstress and eigenbody-force fields which are compatible with the physics of the problem are given. Expressing the eigenstress, eigenbody-force fields and the Green’s function associated with the governing Helmholtz equation in terms of the spherical wave-functions is the natural choice and is very effective. Another important task is the development of the three dimensional DEIM for inhomogeneities having homogeneous or functionally graded (FG) coating with variable thickness, which eliminates any possible symmetries.     

Effect of inclusion shape on the effective elastic moduli for composites with imperfect interface

Summary Bounds on the effective elastic moduli for isotropic composites consisting of randomly oriented spheroidal inclusions with imperfect matrix-inclusion interface are proposed based on Hashin's extremum principle. Phenomenally, these bounds are the first-order ones for such composites, and contain the effect of the size and shape of inclusions, and the elastic properties of constituent phases and interfaces. In the limit cases, these bounds reduce to those known ones. The effect of inclusion shape and interface imperfection on the bounds is discussed with some numerical results for a WC/Co metal-matrix composite.     

The influences of soft and stiff inclusions on the mechanical properties of cementitious composites

The embedment of microencapsulated phase change materials (PCMs) is a promising means for improving the thermal inertia of concrete. However the addition of such soft microcapsules degrades the mechanical properties, i.e., the elastic moduli and compressive strength, of cement-based composites. This study experimentally quantifies the effects of stiff quartz inclusions and soft PCM microcapsules, individually, and when added together, on the mechanical properties of cementitious composites. In addition, a variety of effective medium approximations (EMAs) were evaluated for their ability to predict the experimentally measured composite effective moduli. The EMAs proposed by Hobbs and Garboczi and Berryman (G-B) reliably estimate experimental data. The experimental data and the EMAs were applied to develop a design rule for performance equivalence, such that the composite modulus of elasticity can be maintained equivalent to that of the cementitious paste matrix, in spite of the addition of soft PCM microcapsules.     

Effective porothermoelastic properties of transversely isotropic rock-like composites

The present work is devoted to the determination of the overall porothermoelastic properties of transversely isotropic rock-like composites with transversely isotropic matrix and randomly oriented ellipsoidal inhomogeneities and/or pores. By using the solution of a single ellipsoidal inhomogeneity arbitrarily oriented in a transversely isotropic matrix presented by Giraud et al. [A. Giraud, Q.V. Huynh, D. Hoxha, D. Kondo, Effective poroelastic properties of transversely isotropic rocks-like composites with arbitrarily oriented ellipsoidal inclusions, Mechanics of Materials 39 (11) (2007) 1006-1024], it is possible to observe the effect of the shape and orientation distribution of inhomogeneities on the effective porothermoelastic properties. Based on recent works on porous rock-like composites such as shales or argillites, an application of the developed solution to a two-level microporomechanics model is presented. The microporosity is homogenized at the first level, and multiple solid mineral phase inclusions are added at the second level. The overall porothermoelastic coefficients are estimated in the particular context of heterogeneous solid matrix. The present model generalizes to transversely isotropic media a recently developed two-level model in the simpler case of isotropic media (see Giraud et al. [A. Giraud, D. Hoxha, D.P. Do, V. Magnenet, Effect of pore shape on effective thermoporoelastic properties of isotropic rocks, International Journal of Solids and Structures 45 (2008) 1-23]). Numerical results are presented for data representative of transversely isotropic rock-like composites.     

Thermoelastic properties and conductivity of composites reinforced by spherically anisotropic particles

The present paper is concerned with the overall thermoelastic properties and conductivity of composites reinforced by spherically anisotropic particles. Based on the concept of the replacement particle an equivalent bulk modulus, thermal expansion coefficient and thermal conductivity are derived for the spherically anisotropic particle. Such equivalent properties can be employed in the micromechanical models to predict the overall behavior of the composite. In addition to these, the shear loading is considered. The effective shear modulus is evaluated on the basis of Mori and Tanaka's approximations and the dilute phase concentration factors are derived from exact solutions of an auxiliary boundary value problem.