Propagation of Bending Waves in a Beam the Material of Which Accumulates Damage During Its Operation.

A self-consistent mathematical model is stated in linear and nonlinear formulations, which includes the equation of bending vibrations of a beam and the kinetic equation of damage accumulation in its material. The beam is assumed to be infinite. Such idealization is permissible if its boundaries are attached to optimal damping devices, i.e., the parameters of the boundary fixation are such that incident perturbations will not be reflected. This makes it possible to consider the beam model without taking into account the boundary conditions and regard vibrations propagating along the beam as traveling bending waves. As a result of analytical studies and numerical modeling, it is shown that material damage introduces frequency-dependent attenuation and significantly changes the character of the dispersion of the phase velocity of a bending elastic wave. In a classical Bernoulli–Euler beam, bending waves have one dispersive branch at any frequency, while, for a beam made of a damage-accumulating material, there are two pairs of dispersive branches in the entire frequency range, with one pair describing the wave propagation and the other, the wave attenuation. Within a geometrically nonlinear model of a damaged beam, the formation of intense bending waves of a stationary profile is studied. It is shown that such essentially nonsinusoidal waves can be both periodic and solitary (localized in space). Dependences have been determined relating the parameters of the waves (amplitude, width, and wave number) with the material damage. It was found that, with an increase in the material damage parameter, the amplitudes of the periodic and solitary waves as well as wave number of the periodic waves increase, while the width of the solitary wave decrease. [ABSTRACT FROM AUTHOR]