An analytical solution of a 3d problem of magnetoelasticity

A behavior of the soft ferromagnetic solid in a magnetic field is conventionally investigated with the framework of the developed by Brown, Pao and Yeh linear theory of magnetoelasticity. Beside of the general two-dimensional magnetoelastic problems have been successfully studied by Shindo and other researchers, the three-dimensional problems of magnetoelasticity have not been explored with some exceptions. For example the case of infinite solid with an ellipsoidal inclusion when its shape is essentially compressed in the direction of magnetic field propagation was considered using the series of simplifying assumptions. In the present paper a new approach to finding the solution for the solid with an inclusion of arbitrary ellipsoidal shape was offered. The problem was solved using a general method of constructions the exact analytical solutions of the static problem, which has been developed by Podil'chuk on the basis of Fourier method. This method implies the using of curvilinear coordinates and separating of variables in the governing equations. As a result magnetoelastic stresses was obtained in the closed form. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

State-space approach for an infinite medium with a spherical cavity based upon two-temperature generalized thermoelasticity theory and fractional heat conduction

This paper is concerned with the determination of the thermoelastic displacement, stress, conductive temperature, and thermodynamic temperature in an infinite isotropic elastic body with a spherical cavity. A general solution to the problem based on the two-temperature generalized thermoelasticity theory (2TT) is introduced. The theory of thermal stresses based on the heat conduction equation with Caputo’s time-fractional derivative of order α is used. Some special cases of coupled thermoelasticity and generalized thermoelasticity with one relaxation time are obtained. The general solution is provided by using Laplace’s transform and state-space techniques. It is applied to a specific problem when the boundary of the cavity is subjected to thermomechanical loading (thermal shock). Some numerical analyses are carried out using Fourier’s series expansion techniques. The computed results for thermoelastic stresses, conductive temperature, and thermodynamic temperature are shown graphically and the effects of two-temperature and fractional-order parameters are discussed.     

A hollow ellipsoidal needle in an orthotropic elastic medium

The problem of the stress distribution on the surface of a hollow ellipsoidal needle in an orthotropic elastic medium and a homogeneous external field is solved. Explicit expressions are obtained for the stresses on the needle surface in terms of the elastic constants of the medium and parameters of the ellipsoid in a local system of coordinates connected to the normal to the surface at each point of the needle. The general solution of the problem of the stress concentration on an ellipsoidal inhomogeneity /1/ and the passage to the limit cases of an ellipsoidal cavity based on the presence of small parameters /2/ is used.     

Thermo-magneto-dynamic stresses and perturbation of magnetic field vector in a non-homogeneous hollow cylinder

An analytical method is presented to investigate thermo-magneto-elastic stresses and perturbation of the magnetic field vector in a conducting non-homogeneous hollow cylinder under thermal shock. The interaction between the deformation and the magnetic field vector in a non-homogeneous hollow cylinder is considered by adding a Lorentz’s electro-magneto-force into the equation of thermo-elastic motion of the non-homogeneous hollow cylinder in an axial magnetic field. The exact solution for magneto-thermo-dynamic stresses and perturbation responses of an axial magnetic field vector in a conducting non-homogeneous hollow cylinder was obtained by using finite integral transforms. From numerical calculations, the dynamic characteristics on both thermo-magneto-stresses and perturbation of the axial magnetic field vector in the conducting non-homogeneous hollow cylinder is revealed and discussed.     

The interaction of a magnetoelastic shear wave with longitudinal cavities in a conducting layer

We study the influence of a strong magnetic field on the interaction of a shear wave with longitudinal cylindrical cavities in an elastic ideally conducting layer. The resulting singular integral equation of the boundary-value problem under consideration is implemented numerically for the case of a single cavity. We present the results of computation of the stresses on the edge of a circular cavity and an elliptical cavity. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 74–78.     

Analysis of Direct and Inverse Cavity Scattering Problems

Consider the scattering of a time-harmonic electromagnetic plane wave by an arbitrarily shaped and filled cavity embedded in a perfect electrically conducting infinite ground plane.A method of symmetric coupling of finite element and boundary integral equations is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases.Given the incident field,the direct problem is to determine the field distribution from the known shape of the cavity; while the inverse problem is to determine the shape of the cavity from the measurement of the field on an artificial boundary enclosing the cavity.In this paper,both the direct and inverse scattering problems are discussed based on a symmetric coupling method.Variational formulations for the direct scattering problem are presented,existence and uniqueness of weak solutions are studied,and the domain derivatives of the field with respect to the cavity shape are derived.Uniqueness and local stability results are established in terms of the inverse problem.     

Vibrations of a cylinder in a concentric vessel filled with a two-layer fluid

A hybrid vibrational system containing a solid (a cylinder) with an elastic connection to a coaxial cylindrical cavity, completely filled with a heavy ideal stably stratified two-layer fluid, is considered. The combined self-consistent vibrations of the body and the fluid (of the internal waves) are studied. An explicit solution of the internal boundary value problem of an inhomogeneous liquid in an annular domain for a specified motion of the body is obtained. An integrodifferential equation of the Newton type is constructed on the basis of this. This equation describes the self-consistent oscillations of the cylinder. In the case of weak coupling of the interaction between the solid and the medium, an approximate solution is obtained using asymptotic methods and an analysis is carried out. Qualitative effects of the mutual effect of the motions of the cylinder and the fluid are found.     

Propagation of Rayleigh waves in generalized magneto-thermoelastic orthotropic material under initial stress and gravity field

In this paper the influence of the gravity field, relaxation times and initial stress on propagation of Rayleigh waves in an orthotropic magneto-thermoelastic solid medium has been investigated. The solution of the more general equations are obtained for thermoelastic coupling by Helmoltz’s theorem. The frequency equation which determines Rayleigh wave velocity have been obtained. Many special cases are investigated from the present problem. Numerical results analyzing the frequency equation are obtained and presented graphically. Relevant results of previous investigations are deduced as special cases from these results. The results indicate that the effect of initial stress, magnetic field and gravity field are very pronounced.     

Radial deformation of a nonhomogeneous spherically isotropic elastic sphere with a concentric spherical inclusion

The elasticity of a spherically isotropic medium bounded by two concentric spherical surfaces subjected to normal pressures is discussed. The material of the structure is spherically isotropic and, in addition, is continuously inhomogeneous with mechanical properties varying exponentially as the square of the radius. An exact solution of the problem in terms of Whittaker functions is presented. The St. Venant’s solution in the case of homogeneous material and Lamé’s solution in the case of homogeneous isotropic material are derived from the general solution. The problem of a solid sphere of the same medium under the external pressure is also solved as a particular case of the above problem. Finally, the displacements and stresses of a composite sphere consisting of a solid spherical body made of homogeneous material and a nonhomogeneous concentric spherical shell covering the inclusion, both of them being spherically isotropic, are obtained when the sphere is under uniform compression.     

Wave function and the probability current distribution for a bound electron moving in a uniform magnetic field

We study the effects of electromagnetic fields on nonrelativistic charged spinning particles bound by a short-range potential. We analyze the exact solution of the Pauli equation for an electron moving in the potential field determined by the three-dimensional δ-well in the presence of a strong magnetic field. We obtain asymptotic expressions for this solution for different values of the problem parameters. In addition, we consider electron probability currents and their dependence on the magnetic field. We show that including the spin in the framework of the nonrelativistic approach allows correctly taking the effect of the magnetic field on the electric current into account. The obtained dependences of the current distribution, which is an experimentally observable quantity, can be manifested directly in scattering processes, for example.     

On the analytic solution of nonlinear flow problem involving Oldroyd 8-constant fluid

The problem dealing with the steady flow of an Oldroyd 8-constant fluid over a suddenly moved plate is considered. The fluid is electrically conducting and a uniform magnetic field is applied in the transverse direction. An analytical solution of the nonlinear boundary value problem is obtained using homotopy analysis method (HAM). The behavior of the material constants and the magnetic field is seen on the velocity distribution. It is noted that the boundary layer thickness decreases by increasing the magnetic parameter.     

The axisymmetric contact problem taking into account transient heat production due to sliding friction

The contact problem of the sliding of a solid heat insulator with a plane surface along the boundary of an axisymmetric elastic body is considered, taking into account heat release and the thermal distortion of the boundary of the deformable body due to friction. It is assumed that the shear stresses have no effect on the value of the contact pressures, which enables the problem to be investigated in an axisymmetric formulation. The solution is constructed in two stages: first the form of the thermally distorted surface is determined using known expressions, obtained by Carslaw and Jaeger and also by Barber, and then the contact condition is considered taking into account the elastic displacements and distortion of the form of the surface due to heating, and the integral equation of the problem for determining the unknown contact pressures is derived. The latter equation is solved numerically by approximating the unknown contact pressures by a piecewise-constant function.     

Micropolar fluid flow through the membrane composed of impermeable cylindrical particles coated by porous layer under the effect of magnetic field

In the present work, the magnetohydrodynamic flow of a micropolar fluid through the membrane composed of impermeable cylindrical particles coated by porous layer is considered. The flow of a fluid is taken parallel to an axis of cylinder and a uniform magnetic field is applied in transverse direction of the flow. The problem is solved by using the cell model technique for the flow through assemblage of cylindrical particles. The solution of the problem has been obtained by using no-slip condition, continuity of velocity and stresses at interfaces along with Happle's no-couple stress condition as the boundary conditions. The expressions for the linear velocity, micro-rotational velocity, flow rate and hydrodynamic permeability of the membrane are achieved in this work. The obtained solution for velocities is used to plot the graph against various transport parameters such as, Hartmann number, coupling parameter, porosity, scaling parameter etc. The effect of these transport parameters on the flow velocity, micro-rotational velocity, and the hydrodynamic permeability of the membrane have been presented and discussed in this work.     

Two-dimensional problem of electromagnetoelasticity for multiply connected media

We propose a method for the solution of connected two-dimensional and plane problems of electromagnetoelasticity for multiply connected domains. Basic relations of two-dimensional and plane problems are obtained. Generalized complex potentials of electromagnetoelasticity are introduced and investigated. Boundary conditions for their determination and, using them, expressions of main characteristics of the electromagnetoelastic state (stresses, displacements, electromagnetic field intensity vectors, induction vectors, potentials of the electric and magnetic fields) are obtained. We present a solution of the problem for a plate with an elliptic hole or a crack.     

The stressed state of a transversally isotropic body with inner elliptic crack. A linearly varying pressure on the surface of the crack

An exact solution to the problem of the distribution of stresses in a transversally isotropic elastic body with inner elliptic crack is given. An assumption that the linearly varying pressure acts on the surface of the crack is made. The intensity coefficients of stresses at the vertex of the crack are found. Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 79, 1995, pp. 117–124. Original     

On catastrophe and cavitation for spherical cavity

This work deals with catastrophe of a spherical cavity and cavitation of a spherical cavity for Hooke material with 1/2 Poisson's ratio. A nonlinear problem, which is the Cauchy traction problem, is solved analytically. The governing equations are written on the deformed region or on the present configuration. And the conditions are described on moving boundary. A closed form solution is found. Furthermore, a bifurcation solution in closed form is given from the trivial homogeneous solution of a solid sphere. The results indicate that there is a tangent bifurcation on the displacement-load curve for a sphere with a cavity. On the tangent bifurcation point, the cavity grows up suddenly, which is a kind of catastrophe. And there is a pitchfork bifurcation on the displacement-load curve for a solid sphere. On the pitchfork bifurcation point, there is a cavitation in the solid sphere.     

On perturbations associated with the creation of lift acting on a body in a transonic stream of a dissipative gas PMM vol. 31, no. 6, 1967, pp. 1035–1049

The flow pattern of a viscous imcompressible fluid past a finite body is well known; an approximate solution of the related problem can, for example, be found in the book by Landau and Lifshits [1]. Finn [2] made a rigorous and exhaustive study of plane-parallel flows. No fundamental difficulties arise in passing from the motion of an incompressible fluid to a transonic flow of a compressible gas, however the velocity field is different, when the velocity of particles becomes critical at infinity.

The pattern of a sonic flow past a body of circular cross-section was investigated in paper [3]. This paper deals with perturbations associated with the creation of lift acting on an arbitrary body in a three-dimensional flow. When solving this problem it is necessary to consider not only the external stream, but also the laminar vortex trail because of the velocity vector transverse components becoming infinitely great, if functions defining these are formally extended into the trail area. This difficulty arises in investigations of three-dimensional flows only. The solution defining perturbation damping in an axisymmetric sonic stream of a dissipative gas has in its first approximation one singular point only, and does not contain any other singularities along the axis of symmetry [3].

The external stream pattern is essentially formed by the action of normal viscous stresses and the longitudinal component of the heat flux vector, while the distribution of gas parameters in the laminar trail is defined by tangential stresses. The conjunction of solutions valid for each of these areas makes the closure of the problem, and the determination of all necessary parameters possible.     

Linear and nonlinear problems of active sound control

The problem of active shielding of some domain from the effect of the field generated in another domain is considered. The active shielding is implemented via the placement of additional sources in such a way that the total contribution of all sources leads to the desirable effect. The obtained solution does not require either the knowledge of the particular Green's function or any other information on source distribution and the surrounding medium. It is also important that along with undesirable field to be shielded, a desirable field can be admitted in the analysis. The solution of the problem requires only knowledge of the total field on the perimeter of the shielded domain. The active shielding sources are obtained for both the linear and nonlinear formulations of the problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Large deformations of thermorheologically simple bodies

The relation between the true stresses and true strains associated with the simple finite deformation of a viscoelastic body is modified for the case in which the properties of the material depend on temperature and the body is located in an inhomogeneous nonstationary temperature field. This modification is based on the time-temperature equivalence principle. The validity of this principle for the case in question is confirmed by the experimental data of Andrews, Hofman-Bang, and Tobolsky on the true stress relaxation at large deformations of a polyisobutylene specimen. The modified relations are used to solve the problem of the large deformations of a heated cylinder, made of a viscoelastic material whose properties depend on temperature, under the action of internal and external pressures that are functions of time. The case of polyisobutylene is analyzed as an example of a specific viscoelastic material. The analogous problem in the presence of a reinforcing cylindrical shell is also considered. In the latter problem the solution of the solving nonlinear integrodifferential equation is unique and is obtained in the form of an infinite convergent series.Mekhanika Polimerov, Vol. 3, No. 3, pp. 436–447, 1967