Spatially-Localized Nonlinear Magnetoelastic Waves in a Micropolar Electrical Conducting Medium

Abstract

A nonlinear model of an electrically conducting 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. In both tensors, linear and nonlinear terms are taken into account in rotation gradients and displacement gradients (geometric nonlinearity). The components of the bending-torsion tensor, which have identical indices, describe torsional deformations, and the rest - bending deformations. The stress state of the medium is described by two asymmetric tensors: stress tensor and moment stress tensor. It is assumed, as it is usual in magnetoelasticity, that the action of the electromagnetic field on the deformation field occurs through the Lorentz forces. From the system of Maxwell equations follow the equations for electrical and magnetic inductions, which, together with the electromagnetic equations of state, must be added to the equations of the dynamics of a micropolar medium. Within the framework of the proposed model, a one-dimensional nonlinear shear-rotation magnetoelastic wave is considered. The nonlinear term is selected and taken into account in the equations of dynamics, making 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 a sharp drop in the moving profile. The dispersion, on the contrary, smoothes the differences due to the difference in the phase velocities of the harmonic components of the waves. The combined effect of these factors can lead to the formation of stationary waves that propagate at a constant speed without changing the shape. Only those cases are physically feasible when there is no constant component in the deformation wave. Stationary waves can be both periodic and aperiodic. The latter are spatially localized waves - solitons. It is shown that the behavior of "subsonic" and “supersonic” solitons will be qualitatively different.