A longitudinal magnetoelastic wave in a rod with account of the damage of its material

For an electrically conducting rod that performs longitudinal oscillations, a self-consistent system is formulated that includes the equation of rod dynamics, the equation for the variation of the strength of the external magnetic field, and the kinetic equation for the accumulation of damages in the material of the rod. The linearized system and the system of equations, including geometrical and physical elastic nonlinearities, are consecutively considered. It is shown that the waves described by linearized system have dispersion and frequency-dependent damping due to the presence of two types of dissipation, one of which is caused by the damage of the material and the other by a magnetic field. In nonlinear case, an evolution equation is derived that generalizes the Burgers equation, known in nonlinear wave dynamics. Its solutions are found and analyzed.