The effective conductivity equation for a highly heterogeneous periodic medium

We consider the homogenization of a conductivity equation for a medium made up of a set ${F_\varepsilon}$ ( ${\varepsilon}$ being the size of the period of the medium) of highly conductive vertical fibers surrounded by another material (the matrix) assumed to be a poor conductor. The conductivity coefficients in the fibers behave as ${\frac{1}{\varepsilon^2}}$ while whose of the matrix behave as ${\varepsilon^2}$ . We show that the homogenized problem consists of an equality of the kind u(x) = m(x) f (x) where u denotes the macroscopic temperature, f the source term and m(x) a coefficient given by solving some cell equation.     

Wave scattering in a randomly inhomogeneous medium with long-range noise correlation function ∼1/r

Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 84, No. 2, pp. 250–261, August, 1990.     

The conductivity of a medium interrupted by voids

Résumé On calcule la conductivité d'un milieu en forme de parallelépipède (ou rectangle) percé par une répartition de trous et cadré par des surfaces de conductivité infini ou nul. On emploie un modèle stochastique où on associe une variable aléatoire au processus physique. Les résultats obtenus sont comparés avec les théories de Maxwell, Rayleigh, Wiener et Hashin-Shtrikman.     

The oscillations of a rod in an inhomogeneous elastic medium

The local length-dependence of the natural frequencies and forms of plane transverse oscillations of a thin inhomogeneous rod in an elastic medium with a variable stiffness and arbitrary elastic-fastening boundary conditions is investigated. It is established that the presence of an external elastic medium, described by the Winkler model, can lead to an anomalous effect – an increase in the natural frequencies of lower oscillation modes as the length of the rod increases continuously. The extremely fine properties of this change as a function of the length, the mode number and the method of fastening are revealed. The oscillations in the case of standard methods of fastening are investigated separately. Simple examples, which illustrate the anomalous dependence of the natural oscillation frequencies of the rod in an extremely inhomogeneous elastic medium with different boundary conditions are calculated.     

Wave propagation in a randomly inhomogeneous medium with strongly developed fluctuations. IV. Light wave in a uniaxial liquid crystal

Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 78, No. 2, pp. 200–214, February, 1989.     

The statistical characteristics of the displacement front in a randomly heterogeneous medium

The problem of the displacement of oil by water in a randomly hetrogeneous medium with specified statistical characteristics is considered. Using an improved perturbation theory within the framework of the Buckley – Leverret two-phase model, a statistical solution is constructed which enables the average shape of the displacement front to be obtained, and also the variance of the saturation and the longitudinal velocities on the displacement front, together with a variogram of the fluctuations of the front shape.     

Model of a spatially inhomogeneous one-dimensional active medium

We investigate the dynamics of one-dimensional discrete models of a one-component active medium analytically. The models represent spatially inhomogeneous diffusively concatenated systems of one-dimensional piecewise-continuous maps. The discontinuities (the defects) are interpreted as the differences in the parameters of the maps constituting the model. Two classes of defects are considered: spatially periodic defects and localized defects. The area of regular dynamics in the space of the parameters is estimated analytically. For the model with a periodic inhomogeneity, an exact analytic partition into domains with regular and with chaotic types of behavior is found. Numerical results are obtained for the model with a single defect. The possibility of the occurrence of each behavior type for the system as a whole is investigated. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 506–519, September, 2000.     

The energy conservation law in acoustics of a moving nonstationary inhomogeneous medium

Expressions for the energy density of a sound wave and the sound energy flow vector in a moving nonstationary inhomogeneous medium are obtained. The energy conservation law is stated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 52–57, 1989.     

Wave propagation in a fluid-saturated inhomogeneous porous medium

A closed system of constitutive equations for the dynamical and geometric quantities in a fluid- saturated inhomogeneous elastic porous medium is constructed within the framework of the three-dimensional theory of elasticity. The geometrical characteristics of the wave front and of the ray in a fluid-saturated inhomogeneous medium are obtained from the Fermi's principle.     

A three-dimensional inverse problemfor a physically non-linear inhomogeneous medium

A three-dimensional linearly elastic (viscoelastic) domain (finite or infinite) containing a physically non-linear inclusion of arbitrary shape is considered. The possibility of creating a prescribed uniform stress-strain state in the inclusion by a suitable choice of loads on the outer boundary of the domain is considered. A solution is constructed in closed form. Some examples are considered, including, in particular, the case of an ellipsoidal inclusion with the property of non-linear creep.     

The effective conductivity of an infinitely interchangeable mixture

We are concerned with the estimation of the effective electrical conductivity of random heterogenous materials. The purpose of this paper is to discuss a property of “statistical symmetry” verified by the symmetric cell materials of Miller. This property will be referred to as infinite interchangeability. The usual way to approach cell materials is through n-point correlation functions. The property of infinite interchangebility permits us to approach cell materials from a completely different point of view. Our main result is a simple algorithm, based on this symmetry property, for computing any coefficient of the perturbation expansion in terms of information from the dilute limit. Specifically, knowledge of the coefficients of the expansion in powers of the volume fraction up to order r allows for computation of the perturbation expansion coefficients up to order (2r + 1). This result, which was previously known for r = 2 in the isotropic case and for r = 1 in the anisotropic case, can also be obtained from the standard correlation function approach, as pointed out by Milton.     

The effective thermal conductivity of carbon nanotube composites

The effective thermal conductivity of Carbon Nanotube (CNT)-polymer composites has been estimated using analytical and computational models. The analytical approach is based on the Cascade Continuum Micromechanics (CCM) model formulated within the framework of mean-field homogenization and the computational approach is based on numerical homogenization of the composite microstructure using image based Voxel-FEM (Finite Element Method). Comparison of the analytical and computational model predictions with experimental data show that the interfacial thermal resistance is overestimated by the analytical model as a consequence of not taking into account the CNT fiber tortuosity (curviness). (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)