Dynamics of multilayers: elastic waves in an anisotropic graded or stratified plate

A theory is developed for the description of elastic wave propagation in an arbitrarily layered plate (piece-wise homogeneous or continuously inhomogeneous) of unrestricted anisotropy. The formalism is based on the propagator (transfer) matrix approach. The boundary-value problems of the plate with free, clamped or free/clamped faces are formulated in terms of the propagator and impedance matrices and reduced to real dispersion equations. These equations are analyzed both in general terms, on the basis of some known properties of the impedance matrices, and approximately, in the long- and short-wavelength limits. It is proved that three fundamental wave branches do exist in an arbitrary inhomogeneous plate but only in so far as both of its faces are free of tractions. The corresponding dispersion dependences for the tracing speeds vl(k) (l = 1, 2, 3) found analytically are similar to those for the case of a homogeneous plate. A plate, which has at least one clamped surface, in principle cannot carry fundamental waves. On the contrary, a wave-guide part of the plate spectrum exists for all considered types of boundary conditions. The found high-frequency asymptotics of dispersion curves relate to three kinds of eigenwaves: localized modes at surfaces and interfaces (Rayleigh- and Stoneley-type modes) and bulk wave-guided modes. They are characterized by different dependencies vl(k), different numbers of curves and asymptotic levels and different sensitivity to boundary conditions. Branch repulsion, arising due to anisotropy when a curve crossing is forbidden, and a resulting formation of terracing patterns in spectra of inhomogeneous plates are also discussed.