An effective medium theory for multi-phase matrix-based dielectric composites with randomly oriented ellipsoidal inclusions

An effective medium approximation is formulated for multi-phase matrix-based dielectric composites with randomly oriented ellipsoidal inclusions. The main idea is based on considering a homogenized effective medium which is subjected to a uniform electric field, embedding in it a finite group of representative sub-elements of the composite, and then demanding that the dominant part of the far-field correction to the uniform field which prevailed in it vanishes. This condition results in an algebraic equation for the sought effective property. The calculation of the dominant part of the far-field correction is achieved in an approximate manner. That disturbance is assumed to be the sum of the disturbances caused individually by the each of the embedded elements that consist of a particle of the inclusion phases, surrounded first by some matrix material and then embedded separately in the effective medium. The volumetric ratio of the matrix shell surrounding a particle of the inclusion phase to the total volume of the embedded entity is determined according to the following strategy: the particle of an inclusion phase is assigned an amount of matrix, in proportion to the volume that this specific phase occupies relative to the total volume of all the inclusion phases in the actual composite. Numerical results are produced for a three-phase composite with randomly oriented ellipsoidal inclusions, and compared with the predictions of the average field approximation, the Mori-Tanaka mean field method, and the differential scheme. It is shown that the predictions of the average field approximation and the Mori-Tanaka model violate the multi-phase Hashin-Shtrikman bounds in several circumstances, whereas those of the effective medium approximation and the differential scheme obey those bounds.     

Effective conductive and dielectric properties of matrix composites with inclusions of arbitrary shapes

The effective field method is applied to the calculation of overall dielectric permittivities, and electro- or thermo-conductivities of composite materials consisting of a homogeneous matrix and a set of isolated inclusions. The problem is reduced to the solution of the one particle problem for a typical inclusion subjected to a constant external field. An original numerical method is proposed for the solution of the one particle problem for an inclusion of an arbitrary shape. As an example, the effective dielectric properties of composites with cylindrical inclusions of various sizes and properties are calculated.     

Acoustical and optical branches of wave propagation in random particulate composites

The work is dedicated to the analysis of acoustical and optical branches of longitudinal elastic wave propagation in the medium with a random set of spherical inclusions. The effective field method and quasicrystalline approximation are used for the construction of the dispersion equation for the wave number of the mean (coherent) wave field propagating in the composite. This dispersion equation serves for all frequencies of the incident field, properties and volume concentrations of inclusions. Different branches of the solutions of this equation are obtained and analyzed. Each of these branches may be interpreted as a specific mode of wave propagation, and its input in the mean (coherent) wave field is essential only in a certain frequency region. The predictions of the method are compared with some experimental data existing in the literature.     

考虑周期微结构分布特征的Mori-Tanaka方法

利用两点间应变Green函数张量概念所建立的应变场积分方程, 推导了两相复合材料中夹杂的应变集中张量。该张量较之传统Mori-Tanaka (MT)法采用的由稀疏法导出的应变集中张量, 增加了一个与夹杂体积分数和分布相关的项, 并由此发展了考虑周期微结构分布特征的MT法。传统的MT法虽然能很好地预测正六角形分布圆截面纤维增强复合材料等的有效模量, 但不能反映正方形分布时的四方对称性特征, 本文作者所发展的方法弥补了这方面的不足, 并且所预报的有效刚度和柔度仍然保持了原MT方法所具有的自洽特性。最后通过与双周期有限元计算结果的对照验证了本文方法的精度。      

Effective permittivity of layered dielectric sphere composites

A straightforward model is presented for analysing the effective permittivities of layered dielectric sphere composites. Using the present model, the effective permittivity, _eff, of layered dielectric sphere composites can be deduced using classical two-phase dielectric mixture formulae in two steps: first, the effective permittivity, _incl, of the inclusions is calculated by taking the layered dielectric sphere inclusions as sub-composites; and second, the effective permittivity, eff, of the composites is found by substituting the layered dielectric sphere inclusions with homogeneous spheres whose permittivity is equal to _incl. The present model is applicable to multi-layer sphere composites. Experiments on resin-based hollow bead composites show that the present model accurately predicts the effective permittivity of layered dielectric sphere composites.     

Electrical properties of multiwalled carbon nanotube reinforced fused silica composites

Multiwalled carbon nanotube (MWCNT)-fused silica composite powders were synthesized by solgel method and dense bulk composites were successfully fabricated via hot-pressing. This composite was characterized by XRD, HRTEM, and FESEM. MWCNTs in the hot-pressed composites are in their integrity observed by HRTEM. The electrical properties of MWCNT-fused silica composites were measured and analyzed. The electrical resistivity was found to decrease with the increase in the amount of the MWCNT loading in the composite. When the volume percentage of the MWCNTs increased to 5 vol%, the electrical resistivity of the composite is 24.99 omega cm, which is a decrease of twelve orders of value over that of pure fused silica matrix. The electrical resistivity further decreases to 1.742 omega. cm as the concentration of the MWCNTs increased to 10 vol%. The dielectric properties of the composites were also measured at the frequency ranging from 12.4 to 17.8 GHz (Ku band) at room temperature. The experimental results reveal that the dielectric properties are extremely sensitive to the volume percentage of the MWCNTs, and the permittivities, especially the imaginary permittivities, increase dramatically with the increase in the concentration of the MWCNTs. The improvement of dielectric properties in high frequency region mainly originates from the greatly increasing electrical properties of the composite.     

A simple explicit formula for the effective dielectric constant of binary 0-3 composites

We have derived a simple, analytic formula for the prediction of the effective dielectric constant of binary 0-3 composite materials. In comparison with a popular formula in the field (Jayasundere and Smith [1]), it gives the expected asymptotic value for the internal electric field of the inclusions when their total volume fraction tends to one. It is also applicable to the whole range of the volume fraction of the inclusions and is reasonably good for all values of the dielectric constants of the constituents. For non-conductive constituents, it gives effective complex permittivity prediction that fits well to experimental data, out-performs the Jayasundere-Smith formula and a linearized version [2, 3] of the well-known Bruggeman formula.     

Modeling of Effective Properties of Multiphase Magnetoelectroelastic Heterogeneous Materials

In this paper an N-phase Incremental Self Consistent model is developed for magnetoelectroelastic composites as well as the N-phase Mori-Tanaka and classical Self Consistent. Our aim here is to circumvent the limitation of the Self Consistent predictions for some coupling effective properties at certain inclusion volume fractions. The anomalies of the SC estimates are more drastic when the void inclusions are considered. The mathematical modeling is based on the heterogeneous inclusion problem of Eshelby which leads to an expression for the strain-electric-magnetic field related by integral equations. The effective N-phase magnetoelectroelastic moduli are expressed as a function of magnetoelectroelastic concentration tensors based on the considered micromechanical models. The effective properties are obtained for various types, shapes and volume fractions of inclusions and compared with the existing results.     

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.     

Cross-property relations for two-phase planar composites

The overall property of a composite material is dictated by parameters that characterize its microstructure. Theoretically, cross-links between different physical properties of the same material have been established by eliminating all or partially these microstructural parameters. Practically, such a correlation may be used to determine one property from another once the latter is measured or calculated: the success of this approach depends on whether the correlation is insensitive to the detailed material microstructure. In the present paper, cross-property relations for planar two-phase composites are examined using both analytical approaches and the digital-based finite element method. Both isotropic and transversely isotropic two-phase planar composites are studied. Focus is placed on studying how the microstructure (e.g., shape, size, distribution and volume fraction of inclusions) affects the correlation between two different overall properties of the composite. At a fixed volume fraction, questions on whether the correlation is one-to-one and whether it is sensitive to large material contrast (e.g., voids or rigid inclusions) or how the inclusions are distributed in the matrix will be answered.     

Tight bounds for three-dimensional nonlinear incompressible elastic composites

Tighter variational bounds, in the whole range of inclusion volume fraction, that is to say, even near percolation, for the effective energy of nonlinear composites, in the special case of 3D two-phase incompressible elastic composites with isotropic constituents are presented. Following the methodology of Talbot, Willis and Ponte Castañeda, a linear comparison material with the same microgeometry as the nonlinear composite is employed. The asymptotic homogenization method (AHM) combined with a finite element analysis (FEM), is used to find the displacement field as well as the effective properties for the comparison material. An elastic composite with periodically distributed spherical inclusions in a cubic array is considered as an example. Various numerical examples are performed. Comparisons with others theories (i.e. variational bounds, self-consistent estimates, etc.) are shown. Coincidence of the AHM-FEM results with the universal bounds of Nemat-Nasser, Yu and Hori serves as a useful check to the numerical calculation.     

Interdigital pair bonding for high frequency (20-50 MHz) ultrasonic composite transducers

Interdigital pair bonding is a novel methodology that enables the fabrication of high frequency piezoelectric composites with high volume fractions of the ceramic phase. This enhancement in ceramic volume fraction significantly reduces the dimensional scale of the epoxy phase and increases the related effective physical parameters of the composite, such as dielectric constant and the longitudinal sound velocity, which are major concerns in the development of high frequency piezoelectric composites. In this paper, a method called interdigital pair bonding (IPB) is used to prepare 1-3 piezoelectric composite with a pitch of 40 microns, a kerf of 4 microns, and a ceramic volume fraction of 81%. The composites prepared in this fashion exhibited a very pure thickness-mode resonance up to a frequency of 50 MHz. Unlike the 2-2 piezoelectric composites with the same ceramic and epoxy scales developed earlier, the anticipated lateral modes between 50 to 100 MHz were not observed in the current 1-3 composites. The mechanisms for the elimination of the lateral modes at high frequency are discussed. The effective electromechanical coupling coefficient of the composite was 0.72 at a frequency of 50 MHz. The composites showed a high longitudinal sound velocity of 4300 m/s and a high clamped dielectric constant of 1111 epsilon 0, which will benefit the development of high frequency ultrasonic transducers and especially high frequency transducer arrays for medical imaging.     

Effective-Medium Theory for Two-Phase Random Composites with an Interfacial Shell

According to the Bruggeman theory and Maxwell-Garnett theory, the effective dielectric constant of a two-phase random composite with an interfacial shell is presented. The nonlinearity of the theory is obvious. Especially, the theory is suited to study the dielectric properties of two-phase random composites with a spherical interfacial shell. The theoretical results on dielectric properties of polystyrene-barium titanate composites with an interfacial shell are in good agreement with experimental data.     

A stepping scheme for predicting effective properties of the multi-inclusion composites

A stepping scheme is developed in the present paper to predict the effective properties of composites with high inclusion volume fraction and/or several kinds of inclusions. In this method the inclusions are treated step by step and the effective stiffness coefficients are calculated for the resulting composite in each step. The numerical results of the effective properties of multi-inclusion composites indicate that materials with a high volume fraction of inclusions or multiple inclusions can be dealt with precisely by the stepping scheme. For the case of a binary composite consisting of a matrix and one kind of inclusion, the present results agree with those from a differential scheme.     

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.     

A new local meshless method for steady-state heat conduction in heterogeneous materials

In this paper a truly meshless method based on the integral form of energy equation is presented to study the steady-state heat conduction in the anisotropic and heterogeneous materials. The presented meshless method is based on the satisfaction of the integral form of energy balance equation for each sub-particle (sub-domain) inside the material. Moving least square (MLS) approximation is used for approximation of the field variable over the randomly located nodes inside the domain. In the absence of heat generation, the domain integration is eliminated from the formulation of presented method and the computational efforts are reduced substantially with respect to the conventional MLPG method. A direct method is presented for treatment of material discontinuity at the heterogeneous material in the presented meshless method. As a practical problem the heat conduction in fibrous composite material is studied and the steady-state heat conduction in unidirectional fiber–matrix composites is investigated. The solution domain includes a small area of the composite system called representative volume element (RVE). Comparison of numerical results shows that the presented meshless method is simple, effective, accurate and less costly method for micromechanical analysis of heat conduction in heterogeneous materials.     

Evaluation of dielectric behaviour of particulate composites consisting of polymeric matrix and conductive filler

Abstract

Extension of the logarithmic law of mixtures enables equations to be formulated which describe the dielectric behaviour of particulate polymeric composites containing conductive particles. These equations give the permittivity and dielectric loss of the composite as a function of the permittivity of the polymeric matrix, the volume fraction and aspect ratio of the inclusions, and the frequency of the applied field. The proposed equations were tested with experimental data obtained over a wide range of frequencies and temperatures from composites consisting of epoxy resin and aluminium powder. Satisfactory agreement was observed when the volume fraction of the inclusions was small, but at higher values discrepancies appeared which are attributed to the intrinsic weakness of the logarithmic law of mixtures, on which the proposed equations are based.

MST/3167     

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

Summary 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.     

On the thermo-elastostatics of heterogeneous materials: I. General integral equation

We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of inclusions, when the concentration of the inclusions is a function of the coordinates (so-called functionally graded materials). The composite medium is subjected to essentially inhomogeneous loading by the fields of the stresses, temperature, and body forces (e.g. for a centrifugal load). The general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained for the random fields of inclusions. The method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions, such as effective field hypothesis implicitly exploited in the known centering methods. In so doing the size of a region including the inclusions acting on a separate one is finite, i.e. the locality principle takes place.     

Analysis on dielectric loss characteristics of polyvinylidene fluoride and its composites

In this study, barium calcium zirconate titanate nanoparticles and nanofibers (denoted as BZT-BCT NPs and BZT-BCT NFs, respectively) were prepared by the sol–gel method and electrospinning, respectively. Under different temperatures and frequencies, the dielectric spectra of polyvinylidene fluoride (PVDF), BZT-BCT NPs, and BZT-BCT NFs composites were measured. On the basis of the experimental data, the polarisation activation energies of the polymer matrix interfacial polarisation and the dipole turn polarisation were calculated, and the basic polarisation characteristic parameters of the polymer matrix materials and fillers were obtained. Moreover, the effects of the filling phase and filling ratio on the dielectric properties of the composites were studied through applying BZT-BCT NPs and BZT-BCT NFs as the filling phases of the PVDF matrix and PVDF matrix composites. Furthermore, the double-layer low-density polyethylene (LDPE)/PVDF composites as well as the LDPE/PVDF composites uniformly mixed at a volume fraction of 1:1 were prepared, and the interfacial polarisation behaviours of the two materials were studied by dielectric spectroscopy to establish an effective analytical method so as to characterize interfacial polarisation established. The experimental results revealed as follows: interfacial polarisation was a significant mechanism of the polarisation behaviour of the composite materials; the fillers with different shape factors had varying effects on the dielectric constant of composites; meanwhile, the dielectric constant of the composite conformed to the predictions of the effective medium theory model.