Fomenko M. Homogeneous solutions of problems of equilibrium and vibrations of three-layer isotropic plates of symmetric structure

The thesis is devoted to the development of analytical methods of the reduction of three-dimensional boundary problems of equilibrium and steady-state vibration of three-layer isotropic plates of symmetric structure to two-dimensional problems. Variability of boundary-value problems is caused by various kinds of boundary conditions on flat faces and contacting planes of plate in aggregate with equations of equilibrium or motion. Homogeneous boundary conditions are observed on plate ends: rigid clamping, free stresses, sliding fixing, diaphragm. Layers of plates are under conditions of ideal or sliding contact. Using the method of homogeneous solutions the considered problem classes were reduced to countable sets of metaharmonic equations and spectral problems in respect to transverse coordinate function. Eigenfunctions of spectral problems, transcendental and dispersion equations for finding eigenvalues are obtained in an explicit analytical form. On the basis of dispersion equations, It was possible to are obtain and analyze equations for finding values of cutoff frequencies of modes of travelling waves. The limit relations for phase velocities of traveling waves at short-wave high-frequency (Stoneley's and Rayleight's waves) and long-wave low-frequency ranges are obtained using asymptotic analysis of the dispersion equation. Localization character (limit behaviour) of phase velocities is found in each problem. The procedure for numerically-analytical finding real, imaginary and complex roots of transcendental equations in problems of statics and dynamics is developed. Influence of the kind of boundary conditions at flat faces and contact condition on plate layers, (influence) of relative stiffness, layers thickness and Poisson ratio on character of boundary layer penetration inside region and change characteristics of wave process (cutoff frequencies, phase and group velocities) are shown.