Electrical Characteristics of Three‐Component Dielectric Composites with Close‐Packed Inclusions

The electric field and effective permittivity are calculated for a twodimensional threecomponent dielectric material reinforced by cylindrical fibers. A composite material with a square close packing of inclusions is considered. The field in the periodic system is investigated using the exact solution of the model problem of interaction of two dissimilar cylindrical inclusions in an external homogeneous electric field. A diagram of the relative effective permittivity is constructed.     

Analytical Solution of the Problem of Displacement of a Gas Dissolved in a Melt by Plane and Spherical Crystallization Fronts

Selfsimilar solutions of the problem of displacement of a gas dissolved in a melt by plane and spherical crystallization fronts are found for the case where the crystal growth rate is inversely related to the square root of time. A criterion of the absence of gas displacement due to segregation is found. The problem for a plane crystallization front moving with a constant velocity is analytically solved by means of the Laplace transform method.     

Solution of the plane Hertz problem

The paper considers the problem of onesided frictionless compression of plane elastic bodies that are initially in contact with each other at a point. The first terms of an asymptotic solution of the problem are constructed by the method of joined asymptotic expansions. Determination of the approach of the bodies as a function of the pressing force reduces to calculating socalled of local compliance. The problems of contact of an elastic ring and elastic circular disks with punches and an elastic disk compressed between two elastic strips are considered. An asymptotic model for the quasistatic collision of plane elastic bodies is proposed.     

Small parameter method for determining motion of a viscous incompressible fluid in a thrust bearing

We consider the plane stationary motion of a viscous incompressible fluid between two surfaces. The fixed surface is given by the equation y=h[1+f(x/h)], where the functionf(x/h=h) characterizes the deviation of the fixed surface from the plane y=h(h and , are constants). The moving surface is a plane which moves with constant velocity along the x axis and remains parallel to the plane y=h. The small parameter method is used to solve the problem. The problem formulation is presented in the first section, the solvability of the linear equations obtained using the small parameter method is investigated in the second section, and the third section studies the convergence of the method and finds the radius of convergence of the constructed series.     

The Plane Wedge Problem Loaded at Its Apex – the Self-similarity Property and the Characteristic Vector

In the present paper the elastostatic problem of a generally anisotropic and angularly inhomogeneous plane wedge loaded at its apex by a concentrated force, is studied in linear elasticity. At first the self-similarity property is formulated and the stress field of the inhomogeneous anisotropic self-similar wedge problem, is deduced. The wedge is radially separated and the plane wedge problem is reformulated by the introduction of a characteristic vector. Furthermore, the angular distribution of the load is determined. The multi-material wedge problem in terms of a formulation based on the isotropic angularly inhomogeneous wedge, is confronted, and necessary conditions that ensure the self-similarity property, are found. Finally, the similar elastostatic wedge problems and the involution between stresses, are studied. Mathematics Subject Classifications (2000) 74B05, 74K30, 34B05, 51N15.     

Hypersonic flow-past of plane blunt bodies by a nonviscous radiating gas

An analytic solution is obtained in the work in a Newtonian approximation [1] for the flow-past problem for a plane blunt body by a steady-state uniform hypersonic inviscous space-radiating gas flow. The hypersonic flow-past problem for axisymmetrical blunt bodies by a nonviscous space-radiating gas has been previously considered [2–4]. In this case a satisfactory solution of the problem was obtained even in a zero-th approximation by decomposing the unknown values in terms of a parameter equal to the ratio of gas densities before and after passage of the shock wave. The solution of the problem in a zero-th approximation with respect to in the case of flow-past of plane blunt bodies does not turn out to be satisfactory, since the departure of the shock and the radiant flux to the body as gas flows into the shock layer turns out to be strongly overstated under nearly adiabatic conditions. Freeman [5] demonstrated that results may be significantly improved for flow-past of a plane blunt body by a nonradiating gas if a more precise expression is used for the tangential velocity component expressed in a new approximation with respect to the parameter . This refinement is applied in this work for solving the flow-past problem for a plane blunt body by a space-radiating gas. The distribution of the gasdynamic parameters in the shock layer, the departure of the shock wave, and the radiant heat flux to the surface of the body are found. The solution obtained is analyzed in detail for the example of flow-past regarding a circular cylinder.Translated from Zhurnal Prikladnoi Mekhanikii Tekhnicheskoi Fiziki, No. 3, 68–73, May–June, 1975.     

Effective elastic moduli of a soft medium with hard polygonal inclusions and extremal behavior of effective Poisson's ratio

We consider two-dimensional, two-phase, elastic composites consisting of a soft isotropic medium into which hard elastic inclusions have been placed, requiring that the inclusions be interconnected only at corner points. Denoting by the ratio of Young's modulus for the soft and hard phases, we show that the leading term in the asymptotic expansion as 0 for the effective moduli can be calculated from a finite-dimensional algebraic minimization problem. For several composites with either hexagonal symmetry or orthotropic symmetry, we explicitly solve this algebraic problem. In particular, from the above constituents we construct an isotropic material with maximal positive Poisson's ratio, as well as an orthotropic material with Poisson's ratio less than –1. We also recover in a simple way, Milton's isotropic composite with Poisson's ratio close to –1.     

Network Approximation in the Limit of¶Small Interparticle Distance of the¶Effective Properties of a High-Contrast¶Random Dispersed Composite

We consider a high-contrast two-phase composite such as a ceramic/polymer composite or a fiberglass composite. Our objective is to determine the dependence of the effective conductivity (or the effective dielectric constant or the effective shear modulus) of the composite on the random locations of the inclusions (ceramic particles or fibers) when the concentration of the inclusions is high. We consider a two-dimensional model and show that the continuum problem can be approximated by a discrete random network (graph). We use variational techniques to provide rigorous mathematical justification for this approximation. In particular, we have shown asymptotic equivalence of the effective constant for the discrete and continuum models in the limit when the relative interparticle distance goes to zero. We introduce the geometrical interparticle distance parameter using Voronoi tessellation, and emphasize the relevance of this parameter due to the fact that for irregular (non-periodic) geometries it is not uniquely determined by the volume fraction of the inclusions. We use the discrete network to compute numerically. For this purpose we employ a computer program which generates a random distribution of disks on the plane. Using this distribution we obtain the corresponding discrete network. Furthermore, the computer program provides the distribution of fluxes in the network which is based on Keller's formula for two closely spaced disks. We compute the dependence of on the volume fraction of the inclusions V for monodispersed composites and obtaine results which are consistent with the percolation theory predictions. For polydispersed composites (random inclusions of two different sizes) the dependence is not simple and is determined by the relative volume fraction V _r of large and small particles. We found some special values of V _r for which the effective coefficient is significantly decreased. The computer program which is based on our network model is very efficient and it allows us to collect the statistical data for a large number of random configurations.     

Permeable plate at angle of attack in steady flow of a viscous fluid

An approximate solution is presented for the problem of the resistance of a permeable plate of widthl at an angle of attack in a steady plane flow of an incompressible viscous fluid for the case of both small and very large Reynolds numbers with different permeability laws. The results obtained in the case of large Reynolds numbers are compared with the corresponding results for flow past plane rod grids.     

Stresses in an elastic body under nonlinear antiplane deformation

The stress field in a cylindrical elastic body under antiplane deformation and certain constraints imposed on volume and surface forces is studied in a nonlinear formulation in actualstate variables. A boundaryvalue problem for independent stress components is formulated in Cartesian and complex variables, sufficient ellipticity conditions for this problem are indicated, and constraints on surface loading are imposed. Analytical solutions are given for linear and weak nonlinear elastic potentials. Similarity to a plane subsonic idealgas flow is established. An approximate method for the solution of the problem is developed.     

Problem concerning a plane explosion in a heterogeneous atmosphere with account taken of the post-breakthrough stage

The problem is considered concerning a plane explosion in an exponential and standard atmosphere. The heterogeneity of the medium exerts an extremely marked influence on the gas flow. As shown in [1], under the conditions of an exponential atmosphere the upper part of the shock-wave front recedes to an infinite distance, after a finite time. This phenomenon has received the name breakthrough of the atmosphere. A numerical investigation of a powerful plane explosion in an exponential atmosphere at the stage before breakthrough is contained in [2]. In [3], asymptotic boundary conditions are proposed which permit the gas flow after breakthrough to be calculated. In the present paper, a numerical solution of this problem is obtained at an interval of time which exceeds by a factor of 10–15 the time of break-through. The effect of counterpressure and gravity is studied. Some results are given for a plane explosion in a standard atmosphere.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 124–131, May–June, 1976.The authors thank L. A. Chudov for attention to the project and for useful advice.     

Application of complex path-independent integrals to the solution of the problem of a straight crack in a finite plane isotropic elastic medium

The classical problem of a straight crack in a finite, plane, isotropic, elastic medium of arbitrary shape is reconsidered by the well-known method of Muskhelishvili for such a crack (but in an infinite medium). Both the crack and the boundary of the medium are assumed loaded in an arbitrary way. It is shown that this problem can be completely solved if the numerical values of the first complex potential (z) of Muskhelishvili are known along a closed contour surrounding the crack, probably along the boundary of the medium. To this end, complex path-independent integrals associated with (z) and Chebyshev polynomials have been used. Numerical results for the stress intensity factors are displayed in an application. Generalizations of the method are also proposed and the second fundamental crack problem, the problem of a crack in an anisotropic medium and the problem of an interface crack between two isotropic media are considered in some detail.     

Inertial phase separation in rotating self-gravitating media

The centrifugal separation of foreign inclusions (particles) in a rotating spherical volume of a self-gravitating medium is considered in the hydrodynamic approximation. Using the full Lagrangian approach, the particle trajectories and radial concentration profiles are studied for a rigid-body velocity distribution in the carrier phase. The regimes of continuum and free-molecular flow around the particles are considered. The cases of a heavy (with density greater than that of the carrier phase and traveling toward the center) and a light-weight (traveling toward the periphery) admixture are investigated. Analytical and numerical solutions corresponding to steady-state spherically symmetric boundary conditions for the dispersed phase are found. It is shown that the presence of rotation may result in a significant angular anisotropy of the radial particle concentration distributions and, in particular, in the formation of ring-shaped accumulation zones of heavy inclusions in the equatorial plane. The solutions obtained can be used to explain the mechanisms of onset of density nonuniformities in planet cores, the formation of planetary systems from gas-particle clouds, and the behavior of aerosol particles in atmospheric vortices.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 86–100.Original Russian Text Copyright © 2004 by Ahuja, Belonoshko, Johansson, and Osiptsov.     

One case of the explosion of a symmetric variable-thickness surface charge

Within the framework of the model of explosion phenomena proposed by M. A. Lavrent'ev, the plane problem of determining the shape of the excavation during the explosion of a constant-thickness surface charge is solved (when the pulse pressure is constant over the width of the charge). The problem of determining the shape of the excavation during the explosion of a surface charge whose section thickness varies linearly is considered below for the same model of explosion phenomena called solid-liquid [2]. (The problem reduces to a homogeneous linear Hilbert boundary-value problem with discontinuous coefficients. The solution is obtained in closed form, and recommendations are given for its practical realization.)Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 154–161, May–June, 1976.     

Scattering of Surface Waves by the Edge of a Floating Elastic Plate

The diffraction of plane surface waves by a semiinfinite floating plate in a fluid of finite depth is studied. An explicit analytical solution of the problem is obtained using the Wiener–Hopf technique. Simple exact formulas for reflection and transmission coefficients and their asymptotic expressions are derived. Results of numerical calculations using the obtained formulas are presented.     

Some exact solutions of the equations of a stationary monoenergetic beam of charged particles

We consider the class of invariant solutions which can describe only vortex flows (curl P 0, P is the generalized momentum) and show that they contain solutions corresponding to flows from a plane or cylindrical emitter with a linear voltage drop across it (direct heating) in the temperature-limited regime^*. The solution is obtained in analytic form for emission from a plane in a uniform magnetic field perpendicular to the flow plane. It also (for=0) defines a plane magnetron in the T-regime. The solution of the problem for a cylindrical emitter reduces to considering equations describing a cylindrical diode or magnetron in the T-regime, where the shape of the collector is given by the potential distribution curve for these cases. We can extend the results to a relativistic beam if restrictions are imposed on its relative dimensions which permit us to ignore the magnetic self-field. Brillouin type flows (including irrotational ones) are studied in which particles move without intersecting the equipotential surfaces along three-dimensional spirals on the surface of cones. An analytic solution is given for relativistic Brillouin flow in a conical diode when strict allowance is mede for the magnetic self-field.     

Unsteady motion of rotating ring of viscous incompressible liquid with free boundary

We examine the plane problem of rotationally symmetric motion of a rotating ring of viscous incompressible liquid with free boundary. The theorem for the existence and uniqueness of the problem solution is obtained. The qualitative properties of the solution and its asymptotic behavior as t are studied.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 3, pp. 82–88, May–June, 1970.This study was made under the guidance of V. V. Pukhnachev.     

Free vibrations of a thin rectangular plate

An asympototic method developed by V.V. Kucherenko and the author is applied to the problem of free vibrations of a thin plate strip in a state of a plane strain. The first few terms of an asymptotic expansion of the natural frequencies are calculated. The results are compared with those obtained by using the classical and refined theories.     

The problem of a dihedral piston

The problem of the plane nonstationary motion of a gas behind a dihedral piston is considered. The problem is linearized on the assumption that the piston angle is small. The mixed problems and the Goursat problem are solved for the linearized double-wave equation in the region of hyperbolicity and then the mixed boundary value problem is solved in the region of ellipticity. The solutions are obtained in elementary functions and quadratures.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 74–81, May–June, 1969.The author thanks L. V. Ovsyannikov for his interest and helpful advice.     

Stress–strain state of a composite anisotropic plate with curvilinear cracks and thin rigid inclusions

A complexpotential solution of a mixed problem of the linear theory of elasticity is given for an infinite plate composed of two anisotropic halfplanes. The plate contains cuts and thin undeformable inclusions shaped like arbitrary open smooth curves that do not intersect each other and the interface between the halfplanes.