Spatially localized nonlinear magnetoelastic waves in an electrically conductive micropolar medium

Abstract

AbstractA nonlinear model of an electrically conductive micropolar medium interacting with an external magnetic field is proposed. The deformable state of such a medium is described by two asymmetric tensors: tensor of deformations and bending‐torsion tensor. Both tensors consider both linear and nonlinear terms in rotation and displacement gradients (geometric nonlinearity). The components of the bending‐torsion tensor which have the same indices, describe torsional deformations, while the rest describe bending deformations. The stress state of the medium is described by two asymmetric tensors: stresses tensor and moment stresses tensor. It is assumed, as is customary in magnetoelasticity, that the action of an electromagnetic field on a strain field occurs through Lorentz forces. From the system of Maxwell's equations follow the equations for electric and magnetic induction, which together with electromagnetic equations of state, should be added to the group of equations which describe the dynamics of a micropolar medium. In the frames of the proposed model, a one‐dimensional nonlinear magnetoelastic shear‐rotation wave is considered. In the equations of dynamics, a nonlinear term is considered. This term makes the most significant contribution to wave processes. It is shown that two factors will influence the wave propagation: dispersion and nonlinearity. Nonlinearity leads to the emergence of new harmonics in the wave, which contributes to the appearance of sharp drops in the moving wave profile. Dispersion, on the contrary, smoothed out differences due to differences in phase velocities of harmonic component waves. The combined action of these factors can lead to the formation of stationary waves that propagate at a constant speed without changing shape. Only those cases where a constant component is absent in the deformation wave are physically realizable. Stationary waves can be either periodic or aperiodic. The latter are spatially localized waves—solitons. It is shown that the behavior of “subsonic” and “supersonic” solitons will be qualitatively different.