Surface waves in an elastic medium in the presence of a magnetic field

Waves in a perfectly conducting elastic medium are considered in two cases: 1) at the free surface of a medium occupying an infinite halfspace and located in a homogeneous constant magnetic field; 2) at the interface of two media one of which is located in a magnetic field. At a certain relative velocity instability develops. A similar problem was investigated in [1] for zero magnetic field and a critical velocity was obtained. This paper examines the instability due to vibrations propagating at right angles to the magnetic field.The author thanks M. I. Kiselev for his helpful suggestions.     

Regular oblique interaction of shock waves coming from the same direction in a conducting medium with a magnetic field

The regular interaction of two plane MHD shock wave fronts overtaking one another at an arbitrary angle is investigated within the framework of the ideal magnetohydrodynamic model. The self-similar solution in which all the discontinuities are explicitly distinguished is obtained by means of a computer. The solution is analyzed over a wide range of the key parameters.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 127–143, November–December, 1995.     

Antiplane strain in a nonlinearly elastic incompressible body

The stress-strain state of an incompressible cylindrical elastic body with antiplane strain under the action of potential forces and surface loading constant along the body is considered in a nonlinear formulation in actual variables. The stresses are expressed via the pressure and independent strains, the pressure is expressed via the force and elastic potentials, and nonlinear boundary-value problems are posed for strains (and displacements). Various methods for solving these problems are developed. For the nonlinear equations obtained, some analytical solutions containing free parameters are given, which can be used as a basis for solving particular problems. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 93–101, November–December, 2006.     

Sphere scattering of nonlinearly interacting acoustic waves

Sphere scattering of the field of nonlinearly interacting plane acoustic waves when the sphere is located in the region of nonlinear interaction between the primary pumping waves of a parametric antenna is considered. An analytic expression for the secondary field pressure at the difference frequency is obtained. This expression describes the process of nonlinear interaction of the incident and scattered waves. The secondary-field total pressure components, which characterize the interaction between the incident plane waves and scattered spherical waves are analyzed. The numerical results and experimental data are given.Taganrog. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 4–12, March–April, 1995.     

Equilibrium states of an elastic conducting rod in a magnetic field

Our principal concern is an analysis of the equilibrium states of a nonlinearly elastic conducting rod in a magnetic field. We assume hyperelasticity so the equilibria formally appear as critical points of a potential energy functional on the strains. Fairly standard methods give existence of a minimum (not necessarily unique) with e.g., L₂-regularity. The assumptions imposed on the functional preclude the use of the usual techniques for justification of the formal necessary conditions for optimality. A new general technique is developed to justify these conditions; it then follows that minimizers satisfy the equilibrium conditions in the classical sense. (A feature of this technique is that the variations considered are homotopies so one can consider minimization within a homotopy class.) In the symmetric case, which admits trivial (straight and untwisted) solutions, we show that nontrivial solutions also exist if the field is strong enough.     

Dynamic strain of a conducting half space with a cavity in a strong magnetic field

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 6, pp. 16–20, November–December, 1991.     

Propagation of elastic waves in a rod in a longitudinal magnetic field

The propagation of harmonic waves in discussed for an ideally conducting continuous elastic cylindrical rod within an ideally conducting cylindrical rube. The annulus contains a steady homogeneous longitudinal magnetic field. The dispersion equation is derived. The case of bending vibrations is considered.     

Electromagnetically conducting elastic plates in a magnetic field: modeling and dynamic implications

The basic field equations and boundary conditions necessary for the dynamic approach of electromagnetically conducting flat plates subjected to an external magnetic field are derived.Whereas the structural equations include the geometrical non-linearities of elastic isotropic plates, the electromagnetic equations are used in a linearized form that is obtained from their non-linear counterpart by applying the small disturbance concept. In this context, it was shown that the governing equations involve the bending-stretching coupling arising from both the geometrical non-linear approach of the problem and the inclusion of the Lorentz ponderomotive forces that are reduced to a 2-D plate counterpart.A number of special cases are considered, implications of the external magnetic field on non-linear/linear eigenfrequencies are highlighted, and pertinent conclusions are outlined.     

理想导体中具有两个松弛时间的电磁热弹性波

研究处于均布磁场中的理想导体的二维电磁热弹性耦合问题，引入势函数使控制方程转化为3个偏微分方程．运用Laplace变换和Fourier变换得到该问题在变换域内的精确表达式，再通过级数展开和Laplace逆变换法求得在时间较短时的逆变换，得到时间-空间域内问题的解．运用此方法研究了表面受到热冲击的半无限空间问题．给出了电磁热弹性波、膨胀波和横向波传播的速度，并通过数值计算，给出了各个场量的分布图．所得结论与已有的结论一致．     

Internal waves in a two-layer electrically conducting fluid in the presence of a magnetic field

The propagation of linear and nonlinear internal waves along the interface between two weakly conducting media differing in density and electrical conductivity is investigated and the influence of MHD interaction effects on their characteristics is analyzed. It is shown that in this system the waves propagate with dispersion and dissipation, and for harmonic waves of infinitesimal amplitude there exists a range of wave numbers on which propagating modes do not exist. For waves of finite amplitude a nonlinear Schrödinger equation with a dissipative perturbation is obtained and its asymptotic solution is found. It is established that the presence of electrical conductivity and an applied magnetic field leads to a decrease in the amplitude and the frequency of the envelope of the wave train.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 104–108, September–October, 1990.     

Instability of the quiescent state of an ideal conducting medium in a magnetic field

The linear stability of the quiescent states of an ideal compressible medium with infinite conductivity in a magnetic field is studied. It is shown by Lyapunov’s direct method that these quiescent states are unstable relative to small spatial perturbations, which decrease the potential energy (the sum of the internal energy of the medium and the energy of the magnetic field in this case). Two-sided exponential estimates of perturbation growth are obtained; the exponents in these estimates are calculated using the parameters of the quiescent states and the initial data for perturbations. A class of the most rapidly growing perturbations is separated and an exact formula to determine the rate of their increase is derived. An example is constructed of the quiescent states and the initial perturbations whose linear stage of evolution in time occurs in correspondence with the estimates. From the mathematical viewpoint, our results are preliminary, because the existence theorems for the solutions of the problems considered are not proved. Deceased. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 148–155, March–April, 1999.     

Influence of fluid viscosity on waves in an initially deformed,compressible, elastic layer interacting with a fluid medium

The propagation of harmonic waves along the interface of an initially stressed, compressible layer and a viscous, compressible fluid half-space is investigated. A dispersion relation that does not depend on the form of the elastic potential is derived on the basis of the three-dimensional linearized elasticity equations for elastic bodies with uniform initial deformations and on the linearized Navier-Stokes equations for a viscous Newtonian fluid at rest. The phase velocities and attenuation coefficients of the elastic modes are determined numerically as functions of the thickness of the elastic layer using a Murnaghan-type three-invariant elastic potential.S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 9, pp. 3–9, September, 1994.     

Channel flow of an anisotropically conducting medium in a zone of entry into a magnetic field

A considerable number of papers are devoted to the problem of the deformation of a plane flow of a conducting liquid moving through a channel |x| < , 0 y h=const in a zone of entry into a magnetic field B=(0, 0, B. (x)), where (x) is the Heaviside unit function((x)=0 for x < 0 and (x)=i for x < 0). Apparently the first paper in this direction was that of Shercliff [1, 2] in which the asymptotic (for x .o- )profile of a perturbed velocity was. determined for a flow of an isotropic conducting liquid in a channel with nonconducting walls. The flow considered by Shemliff takes place in magnetohydrodynarnic flowmeters. Complete calculation of the perturbed flow of an isotropie conducting liquid in the channel of a magnetohydrodynamic generator is carried out in [3]. Asymptotic velocity profiles in the channel of a magnetohydrodynamic generator, with ideally segmented electrodes and the flow of an anisotropically conducting medium along them, were found in [4]. General formulas for the calculation of the asymptotic velocity profile, from the known distribution of the perturbing forces along the channel, are presented in [5]. In [6, 7] the Green function is proposed for the solution of the equation for the stream function of the perturbed flow. Finally, in [8], the solution for the perturbed flow of an anisotropically conducting liquid in a channel with continuous electrodes is described by means of the Green function, and the asymptotic profiles of the velocity are calculated.In this paper the flow of anauisotropically conducting liquid is determined in a channel with ideally segmented electrodes. The solution is set up with the aid of the Fourier method. The resulting series, in which the slowly converging part can be related to the asymptotic profile [4] calculated from the solution of an ordinary differential equation, make it possible to determine the velocity field rapidly. A detailed deformation pattern of the velocity profile is set up. Certain general properties of the flow in a zone of entry into a magnetic field are noted; with the aid of these it is possible to discover the error in the calculations [8].     

Plane strain problem for an incompressible nonlinearly elastic solid

The plane strain of an incompressible body is studied with geometrical and physical nonlinearity and potential forces taken into account. A nonlinear system of equations for strains is obtained in actual variables, and conditions of its ellipticity are derived in terms of the elastic potential. Boundary conditions for strains are found from specified loads. Analytical solutions of the boundary problem in strains and their corresponding stress fields are found for the case of identical elongations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 217–225, March–April, 2009