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Averaged equations for bi-continuum material in the long-wavelength approximation

Fatigue failure is known to occurs in two stages. In the first stage the damage is slowly accumulated, whereas the second stage is characterised by a rapid destruction with formation of surface cracks. The hydrogen, contained both in the environment and inside the metal, reduces duration of the first stage by a hundred times. In order to describe the phenomenon of dynamic destruction, the bi-continual material model is applied. The long-wave approximation allows one to investigate the wave solutions of the equation for a slowly varying concentration amplitude of bound hydrogen, which is averaged over the period of the external load. Analysis of the dispersion relation shows that there are two different types of the wave processes depending upon the initial conditions. They are (i) slowly decaying wave processes that model the behavior of the material at the stage of accumulation of damages, and (ii) fast developing processes that lead to destruction.