Effective Conductivity Of Composite Research Articles

Estimating the effective conductivity for ellipse-inclusion model with Kapitza thermal resistance

The ellipse assemblage model with imperfect interface has quite complex microstructure, that can be considered an extension of the circle assemblage model with imperfect interfaces. The paper introduces an approximate method for computing the effective conductivity of isotropic composites with imperfect interfaces in two-dimensional space. Based on the coated-ellipse assemblage model and the equivalent inclusion approximation, one can determine the effective thermal conductivity of the composites. The polarization approximation is given in an explicit form (PEK) and this method will be applied to calculate the effective conductivity of the composite with Kapitza thermal resistance model. The PEK result will have compared with the Fast Fourier Transform (FFT) simulation and Hashin-strikman bounds (HS).

Local fields and effective conductivity of composites with anisotropic components

A homogeneous anisotropic conductive medium with a set of anisotropic heterogeneities of arbitrary shapes is considered. Calculation of local fields in the medium subjected to arbitrary external fields is reduced to systems of volume integral equations. For numerical solution, these equations are discretized using Gaussian approximating functions concentrated at the nodes of a regular grid. The elements of the matrices of the discretized problems have form of 1D-integrals that can be tabulated. For regular node grids, these matrices have Teoplitz’ structures, and fast Fourier transform algorithms can be used for iterative solution of the discretized problems. The method is applied to calculation of fields around isolated anisotropic spherical and cylindrical inclusions in an anisotropic homogeneous host medium. The results are used for calculation of the tensor of effective conductivity of the medium containing random sets of cylindrical inclusions. The self-consistent effective field method is used for solution of the homogenization problem. Dependencies of the components of the tensor of the effective conductivity on the volume fraction and orientations of the inclusions are presented.

Comparison of effective conductivities calculated by the effective medium approximation and the self consistent approximation for core-shell particulate composites

The effective conductivity of composites containing simple or core-shell particles has been estimated in the literature using the Mean Field Approximation (MFA) and the Self-Consistent Approximation (SCA) among other techniques. It is shown here that for both simple and core-shell particles the two approximations agree to first order in the particle volume fraction but differ at the second order term. For simple particles the coefficient of the second order term calculated by SCA is at much better agreement with previous exact results than the coefficient calculated by MFA. For core-shell particles the results of the two approximations are almost identical up to particle volume fraction 0.20 but diverge with increasing volume fraction and particle-to-matrix conductivity ratio.

Dipole moments, property contribution tensors and effective conductivity of anisotropic particulate composites

The paper addresses the homogenization problem for a particulate composite with anisotropic constituents. Its specific goal is to bridge the gap between two different approaches to the problem of homogenization focusing on the anisotropic materials and to identify and discuss the key microstructural parameters affecting overall conductivity of heterogeneous materials. The basic concepts of the homogenization theory including a consistent way of introducing the macroscopic field parameters are discussed and clarified. The exact explicit relations have been obtained between the dipole moments, property contribution tensors and effective conductivity of composite with phase anisotropy of the general type. A detailed comparison of the analytical expressions for the dipole moments obtained by the multipole expansion method and the independently derived expressions for the conductivity contribution tensors has been made between and their equivalence is shown for the matrix type composites with transversely isotropic constituents and spheroidal inhomogeneities. The numerical examples illustrate effect on the overall conductivity of particulate composite of the properties of constituents, shape, volume content, spatial arrangement and orientation of particles.

Model for predicting effective thermal conductivity of composites with aligned continuous fibers of graded conductivity

Governing differential equations in both transverse and longitudinal directions for predicting the effective thermal conductivities of composites with aligned, graded continuous fibers are derived. It is shown that the effective conductivities of composites with graded fibers are predicted by solving the equations. The results by the present approach are applicable to both dilute and non-dilute cases without additional procedures unlike other approaches. The results are compared with those in the literature, and the applicability of the present approach is justified. A solution by the present approach is obtained analytically or numerically as long as thermal conductivity profile of fibers is given.

Effective conductivities of two-phase composites with a singular phase

We calculate the effective conductivity of a two-phase composite with a periodic array of inhomogeneities. The shape of the inhomogeneities is assumed to be a periodic E-inclusion. The effective conductivity is expressed in terms of the volume fraction of the inhomogeneities and a matrix, which characterizes the shape of the periodic E-inclusion. This solution is rigorous, closed-form, and applicable to situations that the conductivity of the inhomogeneities is singular, i.e., zero or infinite. Further, when the periodic E-inclusion degenerates to a periodic array of slits with vanishing volume fraction, we give explicit solutions to local fields and effective conductivity of the composite with singular inhomogeneities.

Comparison of the effective conductivity between composites reinforced by graphene nanosheets and carbon nanotubes

Both graphene nanosheets and carbon nanotubes have exceptional electric and thermal conductivities, rendering them excellent candidates as second-phase fillers in composite materials to substantially enhance their effective conductivities. Their markedly different geometries, however, can have significant effect on the effective conductivities of composites, which we investigate using an effective medium approximation. It is demonstrated that graphene nanosheet is more effective in conductivity enhancement than carbon nantubes, and both fillers lead to substantially higher conductivity and much reduced percolation threshold in composites. The effects of conductivity anisotropy and interfacial resistance are also discussed.

Effective conductivity of composites of graded spherical particles

We have employed the first-principles approach to compute the effective response of composites of graded spherical particles of arbitrary conductivity profiles. We solve the boundary-value problem for the polarizability of the graded particles and obtain the dipole moment as well as the multipole moments. We provide a rigorous proof of an ad hoc approximate method based on the differential effective multipole moment approximation (DEMMA) in which the differential effective dipole approximation (DEDA) is a special case. The method will be applied to an exactly solvable graded profile. We show that DEDA and DEMMA are indeed exact for graded spherical particles.

The Effective Conductivities of Composites with Cubic Arrays of Spheroids and Cubes

A boundary element method is developed for computing the effective conductivity of a composite material containing periodic arrays of particles of arbitrary shape. Utilizing the periodic nature of the problem, steady conduction through a unit cell is analyzed by solving the integral representation of the Laplace equation. The accuracy of the method is validated by comparing the numerical results with analytic solutions for periodic arrays of spherical and spheroidal particles, in particular at high particle volume fractions. The utility of the boundary element method is demonstrated by determining the effective conductivities of simple-cubic, body-centered-cubic, and face-centered-cubic arrays of spheroids and cubes over the entire range of the particle volume fraction.

Effect of Cylindrically Orthotropic Carbon Fibers with Transversely Isotropic Core on Effective Thermal Conductivity of Unidirectional Composites

Carbon fibers can be considered cylindrically orthotropic. Depending on the arrangement of basal planes, predominant configurations are circumferential and radial, but in the core of the fiber, the alignment is undetectable, and consequently it is considered as transversely isotropic. In composites reinforced with these types of fibers, the effective thermal conductivity transverse to the fiber direction will be affected by the core and by the thermal resistance between the fiber and the matrix. The analytical models available are not adequate because they assume that the fiber is totally homogeneous. In this paper, a model of finite elements is presented, which allows combination of the effects of core with the cylindrical orthotropic of the fiber and the thermal contact resistance between the fiber and the matrix. The effective conductivity of composites with orthotropic and inhomogeneous fibers are compared to those with orthotropic and homogeneous fibers. This allows evaluation of the effect for the different morphology of the fibers. The model reveals a significant effect for radially orthotropic fibers (pitch-based). The model also contributes to a better understanding of the heat flux paths in cylindrically orthotropic and inhomogeneous fibers. Finally, this numerical simulation can be of interest first, in the microstructural studies focused on the relationship between the transverse thermal conductivity of carbon fibers and some microstructural parameters, and secondly, in the studies designed to estimate the interfacial contact resistance.

Effective conductivity of loaded granular materials by numerical simulation

Thermal (or electrical) conduction through a stationary, random close packed granular material is studied by numerical simulation. The particles are smooth uniform spheres with either point contact...

Effect of interfacial characteristics on effective conductivities of composites containing randomly distributed aligned long fibers

The effect of interfacial characteristics on the effective thermal conductivity, k eff, of composites containing randomly distributed aligned long fibers is studied. An expression for the reduced effective thermal conductivity, k eff k 1 , is derived with pair interactions rigorously taken into account. Two types of interfaces are treated: one with finite thickness and one with no thickness but possessive of certain thermal barriers. The expressions of k eff k 1 for these two types of interfaces are found to be identical but with different definitions for involved multipole polarizabilities, θ n . The effect of interfacial characteristics, quantified as the relative interfacial thickness, δ a , reduced interfacial thermal conductivity, σ 3, and Biot number, Bi, on k eff k 1 is thoroughly investigated. It is further found that interfacial characteristics may be well represented by the dipole polarizability, θ 1, although higher orders of multipole polarizabilities appear in the expression of k eff k 1 . Comparison of the present results with those obtained from three less rigorous models, the equivalent inclusion model, modified effective medium theory, and modified Hashin and Shtrikman's bounds, is also presented. The present results stay well within the bounds for the combined area fraction of interface and inclusion, f, up to 0.5. Finally, the effect of microstructure on k eff k 1 is examined by a comparison of present results with those for square and hexagonal arrays. The randomness in configuration tends to intensify the thermal interactions between the effective inclusion and the matrix such that, under the same condition, k eff k 1 values of the random array deviate farther from unity than those of the regular arrays.

Effective conductivity of composites containing aligned spheroidal inclusions of finite conductivity

The effective conductivity of composites containing aligned spheroids of finite conductivity is determined with the pair interaction rigorously taken into account. The pair interaction is evaluated by solving a boundary-value problem involving two aligned spheroids with a boundary collocation scheme. Our results are in the form of virial expansion in the inclusion volume fraction f, truncated at the O(f2) term. Effective conductivities obtained from this study agree well with Willis’s [J. Mech. Phys. Solids 25, 185 (1977)] bounds and Kim and Torquato’s [J. Appl. Phys. 74, 1844 (1993)] first-passage-time simulation results. An expression for the effective conductivity tensor based on an equivalent inclusion method originated by Hasselman and Johnson [J. Compos. Mater. 21, 508 (1987)] is also derived, from which a criterion for testing the virial expansion validity is constructed. The relevant series ratio is found to be a good measure of intensity of the system thermal interaction. The smaller the value of the series ratio, the better agreement of the virial expansion with the simulation results. Explicit expressions for the second-order virial expansion for the parallel and perpendicular components of the effective conductivity tensor are tabulated for inclusion aspect ratios 1, 10/9, 2, 5, 10 and phase conductivity ratios σ=0, 0.1, 2, 10, 100, 10 000, and ∞.

Effective conductivity of composites with spherical inclusions: Effect of coating and detachment

An expression for the reduced effective thermal conductivity, keff/k1, of a random array of coated or debonded spherical inclusions with pair interactions rigorously taken into account is derived. Pair interactions are evaluated through solution of a boundary value problem involving two coated or debonded spheres with twin spherical expansions. The resulting keff/k1 is of O(f2) accuracy, where f is the combined volume fraction of the inclusion and interface. The effect of interfacial characteristics manifested as the reduced thermal conductivity, σ3, and relative thickness, δ/a, of the interfacial layer is thoroughly investigated. It is found that keff/k1 can be approximately viewed as a function of f and the dimensionless dipole polarizability, θ1, over a large parameter domain, despite the existence of higher order polarizabilities in the expression of keff/k1. The value of θ1 alone determines whether the effective inclusion is enhancing (θ1≳0), neutral (θ1=0), or impairing (θ1<0) to the matrix. Furthermore, the evaluation of keff/k1 for the present model system can be approximately replaced with that for composites containing inclusions of no interface but possessive of a reduced thermal conductivity of (1+2θ1)/(1−θ1). A contour plot of keff/k1 on the θ1−f domain that is useful in estimating keff/k1 for interfacial properties characterized by an arbitrary combination of σ3 and δ/a, is constructed.

Effective conductivity of composites containing spheroidal inclusions: Comparison of simulations with theory

We determine, by first-passage-time simulations, the effective conductivity tensor σe of anisotropic suspensions of aligned spheroidal inclusions with aspect ratio b/a. This is a versatile model of composite media, containing the special limiting cases of aligned disks (b/a=0), spheres (b/a=1), and aligned needles (b/a=∞), and may be employed to model aligned, long- and short-fiber composites, anisotropic sandstones, certain laminates, and cracked media. Data for σe are obtained for prolate cases (b/a=2, 5, and 10) and oblate cases (b/a=0.1, 0.2, and 0.5) over a wide range of inclusion volume fractions and selected phase conductivities (including superconducting inclusions and perfectly insulating ‘‘voids’’). The data always lie within second-order rigorous bounds on σe due to Willis [J. Mech. Phys. Solids 25, 185 (1977)] for this model. We compare our data for prolate and oblate spheroids to our previously obtained data for spheres [J. Appl. Phys. 69, 2280 (1991)].

A scalar inequality which bounds the effective conductivity of composites

New bounds on the eigenvalues of the effective conductivity tensor in spatial dimension d = 3 are proved using the compensated compactness method. The inequalities are obtained using the nine quadratic null-lagrangians given by the 2 x 2 minors of a jacobian matrix ∂ ϕ i /∂ x j . It is proved that in general the bounds depend at most on four parameters.