A two-dimensional nonlinear magnetoelastic model of a current-carrying orthotropic shell of revolution is constructed taking into account finite orthotropic conductivity, permeability, and permittivity. It is assumed that the principal axes of orthotropy are aligned with the coordinate axes and that the orthotropic body is magnetically and electrically linear. The coupled nonlinear differential equations derived describe the stress-strain state of flexible current-carrying orthotropic shells of revolution that have an arbitrarily shaped meridian and orthotropic conductivity and are in nonstationary mechanical and electromagnetic fields. A method to solve this class of problems is proposed Keywords: current-carrying orthotropic shell of revolution, magnetic field, magnetoelasticityIntroduction. The issues of motion of a continuum with electromagnetic effects fill a highly important place in the mechanics of coupled fields. Studies on coupled fields in solids may be either fundamental or applied, which makes them especially important. Modern engineering employs structural materials that are anisotropic (in particular, orthotropic) even if unstrained, their anisotropy being due to various technological processes. Recently, materials with new electromagnetic properties have been created. These materials can be made effective use of in various areas of modern engineering. Similar issues were examined in [11,12,14,19].An electromagnetic field and an elastic body often interact in the presence of a foreign electric current (e.g., an elastic current-carrying body) in a magnetic field. In this case, we have an electromagnetoelastic problem. Problems with foreign currents are generally very complicated. However, they are much simpler for thin-walled bodies subject to small strains. It is such problems that are dealt with here.1. Magnetoelastic Equations in Lagrangian Variables. Let a body be in a magnetic field created by both an electric current in the body and a source beyond it. Assume that the body is an electric conductor (a current-carrying body). The current from the external source is conveyed to its ends. In the unperturbed state, the foreign electric current is uniformly distributed over the body (current density is independent of the coordinates). The body has finite conductivity and cannot spontaneously polarize and magnetize.Let us define quantities and write equations that describe the electromagnetic field. In an Eulerian coordinate system, the electromagnetic field of the body is characterized by electric-field intensity r e, magnetic-field intensity r h, electric-flux density r d, and magnetic-flux density r b. In a Lagrangian coordinate system, the respective quantities are denoted by r E, r H, r D, and r B. The electromagnetic effects can be analyzed using Maxwell's equations together with the constitutive equations relating the vectors r d and r e, r b and r h, r j and r e. For linear isotropic media, the equations have the following form [18]:

A problem of magnetoelasticity for a flexible conical shell in a nonstationary magnetic field is solved. The effect of conicity on the stress-strain state of the shell is analyzed Keywords: shell, magnetic field, magnetoelasticity Introduction. The mechanics of coupled fields devotes particular attention to the motion of continua in electromagnetic fields. Research on the mechanics of coupled fields in solids is both fundamental and applied and as such is of high priority. The state-of-the-art technology employs structural materials that are anisotropic when not deformed, their anisotropy being caused by various technological processes. Recent developments include isotropic materials with new electromagnetic properties. These materials may efficiently be used in various state-of-the-art technologies. Similar issues were studied in [1, 2, 5, 11, 12, 14, 17, 18, etc.].In most cases, an electromagnetic field interacts with an elastic body through an extraneous electric current, and we deal with electromagnetoelasticity. However, problems related to extraneous currents are rather complicated. They are considerably simplified when dealing with thin bodies that weakly change their shape during deformation. We will study the deformation of a flexible orthotropic electroconductive cone with an extraneous current in an external magnetic field. Problem Formulation. Equations of Magnetoelasticity for an Orthotropic Cone (Axisymmetric Case).Consider a flexible shell of revolution having variable thickness and circumferentially closed coordinate surface. The shell is subject to nonstationary mechanical and electromagnetic loads. A variable electric current from an extraneous source is applied to the end of the shell. Polarization, magnetization, and thermal stresses are neglected. The material of the shell is orthotropic, its principal axes of elasticity are aligned with the coordinate lines, and its electromagnetic properties are characterized by the tensors of conductivity s ij , permeability m ij , and permittivity e ij ( , , , ) i j = 1 2 3 . According to [3, 6, 8 19], the tensors s ij , m ij , and e ij of such conductive orthotropic materials with orthorhombic crystal structure are diagonal.The undeformed coordinate surface is described by curvilinear orthogonal coordinates s and q, where s is the meridional arc length measured from some fixed point, q is the azimuth angle measured from some plane. The coordinate lines s = const and q =const are the lines of principal curvature of the coordinate surface. The shell will be described in a spatial coordinate system

A nonlinear two-dimensional model of a magnetoelastic flexible current-carrying ring plate is developed. A system of nonlinear equations describing the stress-strain state of flexible current-carrying plates in nonstationary mechanical and electromagnetic fields is derived. The stress state of a flexible plate of variable stiffness in a magnetic field is determined

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The effect of the tangential components of magnetic-flux density on the stress state of a circular cylindrical shell of variable stiffness is studied following a geometrically nonlinear problem statement. The cylindrical shell is subject to extraneous electric current and nonstationary mechanical loading Keywords: flexible circular cylindrical shell, magnetic field, magnetoelasticity Introduction. The interaction between mechanical and electromagnetic fields is intensively studied in solid mechanics. The physics behind this interaction is detailed in a number of courses on classical electrodynamics and solid state physics [4, 5, 7, 8, etc.]. Magnetic fields in metal members generate electrodynamic volume forces that may cause large strains at certain field parameters. A rigorous analysis of electromagnetic interaction involves solving the equations of motion and the equations of electrodynamics for an element (internal problem) and the equations of electrodynamics for vacuum (external problem). The general problem can be simplified in special cases, including the case where it is sufficient to solve the internal problem only.Among these classes of problems are problems for thin-walled current-carrying plates and shells in a strong external magnetic field. In nonlinear magnetoelasticity, it is of considerable interest to determine the stress-strain state of current-carrying plates and shells in variable electromagnetic fields.Expanding upon the studies [15][16][17], the present paper analyzes the effect of the tangential components of magnetic-flux density on the stress state of a current-carrying flexible circular cylinder with variable stiffness. According to the