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MATHEMATICAL MODELING OF MAGNETOELASTIC VIBRATIONS OF A ROD IN A MAGNETIC FIELD.

When solving geometrically nonlinear problems of magnetoelasticity in the theory of plates and shells, it is difficult, in a general case, to evaluate the effect of nonlinearity in determining their stress state. To evaluate such a process, nonlinear oscillations of an isotropic rod of constant cross section under the influence of the electromagnetic Lorentz force are considered. The obtained estimates for the rod also characterize the qualitative side of the behavior of flexible plates and shells in a magnetic field.


Introduction
In the mechanics of conjugate fields, an important place is occupied by the study of a continuous medium motion taking into account electromagnetic effects. Analysis of electromagnetic processes is possible only on the basis of a system of electrodynamic equations, together with material relationships.In recent decades, considerable attention in special literature has been devoted to the study of strain processes in electrically conductive bodies placed in an external constant magnetic field under the influence of force, thermal, and electromagnetic loads [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25].
Interest in research in this area is associated with the importance of a quantitative study and assessment of the observed effects of the relationship of mechanical, thermal and electromagnetic processes and their practical application in various fields of modern technology in the development of new practices, as well as in the field of radio electronics, electrical engineering, modern measuring systems and etc. Most of the known works on elastic conductive body strain have been performed for a linearized system of equations. However, the solution of a number of applied problems, which include nonstationary problems of determining the stress state of flexible current-carrying anisotropic plates and shells, requires a more complete study of mechanical

I. STATEMENT OF THE PROBLEM. THE EQUATIONS OF MAGNETOELASTICITY.
Suppose that an electrically conductive body is in a magnetic field formed by an electric current in the body itself (self-magnetic field) and a source located at a distance from the body (external magnetic field).
The body has finite electrical conductivity  and does not have the property of spontaneous polarization and magnetization. Let's assume that, in the general case, there are no surface currents and foreign charges.
Magnetoelasticity equations for similar bodies in the Euler coordinates are presented in the form [3,9,10]: equations of motion: In relations (1)-(4) the following notation is introduced: It is necessary to add kinematic equations for the processes of electrical and thermal conductivity and the governing equations to the equations of magnetoelasticity (1)-(4). Later, we will neglect the temperature effects, and take the kinematic equations for the processes of electrical conductivity in the form of Ohm's law: and where  is the coefficient of electrical permeability,  is the coefficient of magnetic permeability When setting the boundary value problems, it is necessary to formulate the boundary conditions for mechanical and electromagnetic characteristics. In spatial variables for the full stress tensor, it can be written: Here  ki is the Maxwell tensor

II. NUMERICAL EXAMPLE. ANALYSIS OF RESULTS.
Consider a rectilinear rod made of aluminum, l long with a hinged fixing in its ends. We believe that the flexible rod is in a constant external magnetic field and serves as a conductor of electric current, supplied to the ends of the rod from an external source and is a function of time t . As a result of current interaction with the magnetic field, the Lorentz volume forces arise in the rod [3]: The current density is set by the formula , sin (10) and the magnetic induction vector is taken as constant where −  is the circular frequency. In this case, the ponderomotive force is Equation (13) is a differential equation of forced bending vibrations of a rod of constant cross section. Consider the physical meaning of various terms of equation (13). Their signs depend on the chosen sign rule, and do not have a special physical meaning. The first term of equation is the deflection resistance, calculated as a variation of transverse force, the moment of which balances the variation of the bending moment, which occurs due to a change in curvature, i.e. we have the bending resistance to deflection proportional to the bending rigidity of the rod.
The second term is the transverse component caused by the curvature of a certain axial force Then, if the rod is bent by transverse forces, the axial line will extend, and since it bends, and therefore becomes longer, than it was originally, the supports will create tensile force x N acting on the rod, which will increase in proportion to the square of deflection. The second term in this case will increase in proportion to the third degree of deflection, and the equation will become non-linear with respect to w .
The third term presents the action of inertial volume load. The fourth term of equation is a transverse load, tending to cause deflection. Representing the electromagnetic load as Substituting expression (18) and (19) Given that The first term of relation (21) characterizes the load resistance due to bending; the second term characterizes the resistance due to the force action This is due to reaching or exceeding the critical value according to Euler, and the rod loses stability.   Note that everything said above regarding the considered rod characterizes qualitatively the behavior of flexible plates and shells located in an electromagnetic field.

III. CONCLUSIONS.
When a material body moves in an electromagnetic field, force interaction and energy exchange occur between the body and the field, due to conduction currents and body polarization and magnetization phenomena. If to ignore the body polarization and magnetization, and consider only the conduction currents, then the force interaction of the body and the field occurs only due to the Lorentz forces, and the energy exchange is caused by the Joule heat only. To assess the influence of nonlinearity in determining the stress-strain state of flexible currentcarrying plates and shells, nonlinear oscillations of an isotropic rod of constant cross section under the Lorentz electromagnetic force are considered. The obtained estimates for the rod also characterize the qualitative side of flexible plates and shells behavior in a magnetic field.