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]:
