Propagation of natural waves in extended viscoelastic plates of variable thickness

Abstract

Wedge waves are waveguide modes localized near the edge of an elastic wedge and propagating at a speed lower than the speed of the bulk and surface modes existing in the material. These waves are of some interest to practice. The purpose of this paper is to develop calculation methods based on the analysis of the problem for a viscoelastic plates, calculate the relative phase velocities of elastic waves propagating along the edge of the wedge-shaped plate. The wedge is represented as plates of variable thickness. The basic equations of oscillatory processes are obtained on the basis of the principle of possible displacements. For plates of variable thickness, the hypotheses are fulfilled Kirchhoff - Love. The relationship between efforts and deformations satisfies the relationship between the theory of hereditary mechanics. It is shown that the square of the eigenwave number for an infinite strip of variable thickness is valid for any combination of boundary conditions. Using the Lagrange formula, biorthogonality conditions are obtained for wave propagation in a plate of variable thickness. The solution to the problem is sought in the form of a traveling wave along the longitudinal axis. After that, a spectral problem is formulated that describes the propagation of flexural plane waves in a waveguide made in the form of a strip with an arbitrary law of thickness variation along the longitudinal coordinate. The numerical solution of spectral problems was carried out on a computer using a software package based on the orthogonal sweep method of S.K. Godunov in combination with Muller’s method. It is found that, in contrast to the band of constant cross-section, in the case of a wedge-shaped waveguide with a small angle at the base of the wedge, there is a finite limit to the phase velocity of the mode propagation. The analysis of the obtained data shows that the area of applicability of the Kirchhoff-Love theory to a plate of constant thickness is limited to the low frequency range. In the region of high frequencies, when the wavelength of the mode is comparable to or less than the thickness of the plate, the Kirchhoff-Love theory does not allow us to obtain reliable results.