Electrophysical properties of the rayleigh three-dimensional model

A new approach to solving the problem of conductivity of composites with a regular structure is suggested that provides a fundamental possibility of analyzing media with arbitrarily shaped inclusions. The conductivity, its derivatives with respect to both arguments, and the two-parameter function entering into the expression for the effective Hall constant are considered using the Rayleigh three-dimensional model, a periodic system with spherical inclusions, as an example. The results of the numerical analysis of general equations obtained in this way are represented in graphic form.     

On the theory of galvanomagnetic properties of composites

The conductivity of composites in the presence of a magnetic field H is considered. The galvanomagnetic characteristics for a weakly inhomogeneous medium are determined in explicit form in an approximation quadratic in the deviations of conductivity tensor $\hat \sigma $ (r) from its mean value 〈 $\hat \sigma $ 〉. The contribution to the effective conductivity tensor $\hat \sigma _e $ linear in concentration c of inclusions for a composite with a small value of c is expressed in terms of the dipole polarizability of an individual inclusion, which is defined in the transformed system in which it is surrounded by an isotropic matrix with a scalar conductivity. Transition to this system is performed using a symmetry transformation that does not change the dc equations. An approximate approach proposed for describing the galvanomagnetic properties of composites in the wide range of parameters appearing in the problem generalizes the standard theory of an effective medium to the case of anisotropic systems with inclusions of arbitrary shape in field H ≠ 0.     

Increase and Decrease of the Effective Conductivity of Two Phase Composites due to Polydispersity

We present a two-dimensional mathematical model of a composite material with conducting inclusions (fibers) embedded in a matrix. Our main objective is to study how polydispersity (two different sizes of particles) affects the overall conductivity of the composite. If the conductivity of inclusions is higher than the conductivity of the matrix, then previous studies suggest an increase of the effective conductivity due to polydispersity. We show that for high volume fraction when inclusions are not well-separated and percolation effects play a significant role, polydispersity may result in either an increase or decrease of the effective conductivity. Our proof is based on the method of functional equations and it provides sufficient conditions for both the increase and the decrease of the effective conductivity.     

On the theory of conductivity of anisotropic composites: A linear-concentration approximation of inclusions

A general approach is proposed for calculating the conductivity of anisotropic composites with a low concentration of inclusions of an arbitrary shape. The contribution to effective conductivity [^(s)] _e\hat \sigma _e, which is linear in concentration, is expressed in terms of the polarizability of the inclusion defined in a certain transformed system in which the inclusion is surrounded by an isotropic matrix. A transition to this system is performed using a symmetry transformation that does not change the equations for direct current.     

On the conductivity of composites with two-dimensional periodic structure

A method is suggested of successive solution of the problem on the conductivity of two-dimensional periodic systems with inclusions of arbitrary shape. The complex potential outside of the inclusions is expressed in terms of the Weierstrass zeta function and its derivatives. The field induced on a separate inclusion is described using the matrix of multipole polarizabilities. The “joining” of potentials is performed at a distance such that R < ρ < a, where R is the characteristic dimension (maximum “radius”) of the inclusion and a is the half-period of the lattice. The approach suggested enables one to find exact virial expansions for the conductivity of other effective characteristics of similar systems as well.     

The theory of conductivity of ternary composite films with a two-sublattice structure

A general method is suggested for solving problems on the conductivity and other effective characteristics of two-dimensional ternary two-sublattice models with inclusions of arbitrary shape. The complex potential outside of the inclusions is expressed in terms of the Weierstrass zeta function and its derivatives. The electric field induced on a separate inclusion is described using the matrix of multipole polarizabilities, for which the symmetry relation is found. The suggested approach enables one to find exact virial expansions for the basic electrophysical characteristics of such systems.     

Effect of macroscopic inclusions on the electrical constants of solids

The problem of streamline current flow in an anisotropic medium with an ellipsoidally shaped macroscopic inclusion having a conductivity tensor different from that of the medium is solved. The solution obtained allowed us to find an expression for the effective conductivity tensor of the anisotropic medium with anisotropic ellipsoidal inclusions in a magnetic field. The problem of the effective values of the magnetoresistance coefficients of a medium with spherical dielectric inclusions is examined as a special case of the application of the general formula. It is shown how one may analyze galvanomagnetic effects in composites with the aid of the results obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 32–37, May, 1978.The authors are deeply grateful to V. M. Lenchenko for his continuous attention to this paper and for valuable comments.     

Two-dimensional shear waves scattering by a large rigidity strip with interface crack

A two-dimensional problem of shear horizontal (SH) waves scattering by a finite width planar elastic (piezoelectric) inclusion partially debonded from its surrounding elastic matrix is investigated using the effective boundary conditions and singular integral equations technique. The case of large rigidity inclusions with blunted tips is considered, in which the upper face of the inclusion is perfectly bonded to the matrix. The debonding region is modeled as interface crack with non-contacting faces. Using the Green theorem the mixed boundary value problem is reduced to a system of the hypersingular integral equations. Numerical results of the scattering fields characteristics are presented. The effects of incidence direction, various material parameters of the strip on the scattering field are discussed and phenomenon of the non-specular reflection of SH waves is considered. The accuracy of the numerical results is confirmed by the use of analytical approximate problem solution of high-frequency SH waves scattering on a finite hard/soft inclusion.     

Effective dynamic properties of 3D composite materials containing rigid penny-shaped inclusions

The propagation of time-harmonic plane elastic waves in infinite elastic composite materials consisting of linear elastic matrix and rigid penny-shaped inclusions is investigated in this paper. The inclusions are allowed to translate and rotate in the matrix. First, the three-dimensional (3D) wave scattering problem by a single inclusion is reduced to a system of boundary integral equations for the stress jumps across the inclusion surfaces. A boundary element method (BEM) is developed for solving the boundary integral equations numerically. Far-field scattering amplitudes and complex wavenumbers are computed by using the stress jumps. Then the solution of the single scattering problem is applied to estimate the effective dynamic parameters of the composite materials containing randomly distributed inclusions of dilute concentration. Numerical results for the attenuation coefficient and the effective velocity of longitudinal and transverse waves in infinite elastic composites containing parallel and randomly oriented rigid penny-shaped inclusions of equal size and equal mass are presented and discussed. The effects of the wave frequency, the inclusion mass, the inclusion density, and the inclusion orientation or the direction of the wave incidence on the attenuation coefficient and the effective wave velocities are analysed. The results presented in this paper are compared with the available analytical results in the low-frequency range.     

含成层杂质的复合媒质的电导性质

本文用瑞利模型研究含成层杂质的复合媒质的直流电导性质,推导了适用于此类复合媒质的广义瑞利恒等式,并定义系统的有效电导率。对于具有立方对称性的系统,给出复合媒质有效电导率的解析计算公式。 关键词：     

Conductivity of the 3D model of a composite with spheroidal inclusions

The solution to the problem of the conductivity of the 3D model of a composite with inclusion having a shape of oblate ellipsoids of revolution (spheroids) has been obtained in a wide range of concentrations. The entire range of variation of the shape of inclusions (from a sphere to an infinitely thin circular disk) has been considered. The relation between the percolation thresholds in this model and the quantities characterizing its effective conductivity in the limit of low concentrations of inclusions has been established.     

Derivation of governing equation for predicting thermal conductivity of composites with spherical inclusions and its applications

A governing differential equation for predicting the effective thermal conductivity of composites with spherical inclusions is shown to be simply derived by using the result of the generalized self-consistent model. By applying the equation to composites including spherical inclusions such as graded spherical inclusions, microballoons, mutiply-coated spheres, and spherical inclusions with an interphase, their effective thermal conductivities are easily predicted. The results are compared with those in the literatures to be consistent. It can be stated from the investigations that the effective thermal conductivity of composites with spherical inclusions can be estimated as long as their conductivities are expressed as a function of their radius.     

Electrical characteristics of three-component dielectric media

We study the effective parameters of a three-component plane dielectric medium with a doubly periodic arrangement of circular inclusions. The problem is solved in the one-dipole approximation. The computational results are compared with a two-component Rayleigh model. The general structure of the formulas for the effective parameters is discussed and the reciprocity relations for a three-component matrix system are determined. Explicit expressions are given for the dielectric permittivity when the concentration of circular inclusions is low, and their domain of applicability is determined. Under certain polarization conditions, the effective conductivity in a three-component medium is exactly equal to the dielectric permittivity of the matrix. Zh. éksp. Teor. Fiz. 114, 1121–1136 (September 1998)     

Effective thermal conduction in composite materials

The problem of determining the bounds and/or estimating the effective thermal conductivity (λ _eff) of a composite (multiphase) system given the volume fractions and the conductivities of the components has been investigated. A comparison between the measured data and the results predicted by theoretical models has been made for seven heterogeneous samples. The tested models include those of the effective medium theory (EMT), Hashin and Shtrikman (HS) bounds, and Wiener bounds. These models can be used to characterize macroscopic homogeneous and isotropic multiphase composite materials either by determining the bounds for the effective thermal conductivity and/or by estimating the overall conductivity of the random mixture. It turns out that the most suitable one of these models to estimate λ _eff is the EMT model. This model is a mathematical model based on the homogeneity condition which satisfies the existence of a statistically homogeneous medium that encloses inclusions of different phases. Numerical values of thermal conductivity for the samples that satisfy the homogeneity condition imposed by the effective medium theory are in best agreement with the experimentally measured ones.     

The conductivity of a 2D system with a doubly periodic arrangement of circular inclusions

We present a scheme for the evaluation of the conductivity and other effective properties of a model composite with a regular anisotropic structure, namely, a 2D system with circular inclusions forming a rectangular array. Exact expressions for the electric potential and the effective conductivity tensor [^(s)] _e\hat \sigma _e were obtained in the form of infinite series. For small inclusion densities, a virial expansion for [^(s)] _e\hat \sigma _e was derived from the general formulas and its applicability conditions were found. The first terms of this expansion yield the well-known Rayleigh result for the isotropic model (square array).     

Electrical conductivity of a two-dimensional model for a composite material with structural anisotropy

The electrical conductivity of a structurally anisotropic two-dimensional model for a composite material is considered. The model represents an isotropic matrix with a system of nonconducting inclusions in the form of mutually perpendicular scratches of various lengths. The centers of the scratches are chaotically distributed in plane (x, y). The approximate effective medium method is used to derive an expression for effective conductivity tensor $\hat \sigma _e$ that satisfactorily describes the electrical conductivity of this model over a wide concentration range. The model conductivity in the critical region is considered in terms of the similarity hypothesis.     

Theory of the residual electrical resistivity of binary actinide alloys

A system of self-consistent equations has been proposed for the coherent potential approximation of the multiband conductivity model for the case of conduction electron scattering from chaotic electric fields of ions of disordered binary alloy components at zero temperature. It has been qualitatively demonstrated that the deviation of the concentration dependence of the residual electrical resistivity of actinide alloys with multiband conductivity from the Nordheim rule is caused by the explicit dependence of the electrical resistivity of the alloy on the magnitude and sign of the real part of the Green’s function at the Fermi level. The derived system of equations for the multiband coherent potential approximation has been used to calculate the concentration dependence of the density of states and the residual electrical resistivity of the alloys of neptunium and plutonium. The results of the calculations have been compared with the available experimental data.     

Nanocomposite-based ultrathin polarization beamsplitter

In terms of the effective medium approximation, the optical properties of heterogeneous composite materials with disordered nanosized metal inclusions of spheroidal shape with plasmon resonances lying in the visible spectral range are studied. The conditions are analyzed under which the optical characteristics of the composite with axial symmetry of the inclusion orientation distribution with respect to a certain preferential direction significantly depends on the electric field direction in the electromagnetic wave. It is shown that, in the vicinity of the plasmon resonance of inclusions, the composite layer of subwavelength thickness may exhibit a high polarization selectivity in both reflection and transmission with moderate absorption. The effect of the orientational disorder of inclusions on the optical characteristics of a polarization splitter of this kind is discussed. The predictions based on equations of quasi-electrostatic effective-medium model agree well with the results of exact numerical solution of the Maxwell equations.     

Conductivity of structurally anisotropic 2D composites

The character of electric conduction in a two-dimensional structurally anisotropic composite containing elliptic inclusions with a large ratio of the semiaxis. Dielectric and ideally conducting inclusions were considered. It was demonstrated that the approximation linear in concentration of inclusions fails beginning from rather small concentrations. The region of intermediate concentrations was studied using approximate methods based on the analogy between electric conductivity and the diffusion of a single particle in a nonuniform medium. The conductivity of the model system under study was analyzed over a wide concentration range with the use of the effective medium theory; the results are in close agreement with those obtained by qualitative methods. The behavior of conductivity near the percolation threshold (metal-to-dielectric phase transition) was examined using the similarity. The conductivity of thin composite films with inclusions in the form of carbon nanotubes was examined, as an example.