The propagation of bending waves in a beam, the material of which accumulates damage during its operation

Abstract

The linear and nonlinear formulations of a self-consistent material model are developed based on the equation of the bending vibrations of the beam and the kinetic equation of damage accumulation in its material. The beam is considered endless. Such idealization is permissible if optimal damping devices are located at its boundaries, that is the parameters of boundary fixing are such that disturbances acting on it will not be reflected. This allows us to exclude boundary conditions from the beam model and consider the vibrations propagating along the beam as traveling bending waves. As a result of analytical studies and numerical modeling, it was found that the damage of the material involves a frequency-dependent attenuation and significantly changes the nature of the dispersion of the phase velocity of a bending elastic wave. Note that in a classical Euler-Bernoulli beam there is one dispersion branch for bending waves at any frequency value, whereas for a beam with accumulated material damage in the entire frequency range there are two dispersion branches, characterizing wave propagation, and two dispersion branches, characterizing its attenuation. The problem of the formation of intense bending waves of a stationary profile is considered in the framework of a geometrically nonlinear model of a damaged beam.. It is shown that such essentially non-sinusoidal waves can be either periodic or solitary (localized in space).The dependencies relating the parameters of the waves (amplitude, width, wavelength) with the damage to the material are determined. It is shown that the amplitude of the periodic wave and the amplitude of the solitary wave increase with increasing material damage parameter, while the length of the periodic wave and the width of the solitary wave decrease with increase of this parameter.