Abstract
At present, continuum damage mechanics is in good progress, studying both the stress-strain state of structures and the accumulation of damages in their materials. Treatment of a number of problems is associated with the necessity of taking into account the fact that structural elements operate under the action of external magnetic field, which affects the formation and propagation of elastic waves. In this work, for an electrically conducting rod performing longitudinal oscillations, we formulate a self-consistent system, which includes the equation of rod dynamics, the equation of variation of the external magnetic field strength, and the kinetic equation of accumulation of damage in the material of the rod. Here we assume that damage is uniformly distributed in the rod material, the magnetic field is stationary and use the classical model of the homogeneous Bernoulli rod as the rod model. The linearized system and the system of equations, including geometrical and physical elastic nonlinearities, are considered sequentially. In the first case, it is shown that the characteristic features of the waves described by this system are the dispersion and frequency-dependent damping due to the presence of two types of dissipation, one of which is caused by damage of the material and the other by a magnetic field. In the second case, an evolution equation for the function of longitudinal deformation is derived as a generalization to the Burgers equation. Its approximate solution is found and analyzed. Depending on the ratio of the damage to conductivity parameters it allows us to estimate the possibility of the existence of stationary waves that retain their shape and velocity during propagation in space. Moreover, the limiting cases of the evolution equation are considered, in particular, in the absence of conductivity of the electromagnetic field and damage in the material. For these cases exact solutions of the stationary profile are obtained.
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