Extensional and Transversal Wave Motion in Transversely Isotropic Thermoelastic Plates by Using Asymptotic Method

Abstract

The present investigation is concerned with the study of extensional and transversal wave motions in an infinite homogenous transversely isotropic, thermoelastic plate by using asymptotic method in the context of coupled thermoelasticity, Lord and Shulman (1967, “The Generalized Dynamical Theory of Thermoelasticity,” J. Mech. Phys. Solids, 15, pp. 299–309), and Green and Lindsay (1972, “Thermoelasticity,” J. Elast., 2, pp. 1–7) theories of generalized thermoelasticity. The governing equations for extensional, transversal, and flexural motions have been derived from the system of three-dimensional dynamical equations of linear thermoelasticity. The asymptotic operator plate model for extensional motion in a homogeneous transversely isotropic thermoelastic plate leads to sixth degree polynomial secular equation that governs frequency and phase velocity of various possible modes of wave propagation at all wavelengths. It is shown that the purely transverse motion (SH mode), which is not affected by thermal variations, gets decoupled from rest of the motion. The Rayleigh–Lamb frequency equation for the plate is expanded in power series in order to obtain polynomial frequency equation and velocity dispersion relations. Their validation has been established with that of asymptotic method. The special cases of short and long wavelength waves are also discussed. The expressions for group velocity of extensional and transversal modes have been derived. Finally, the numerical solution is carried out for homogeneous transversely isotropic plate of single crystal of zinc material. The dispersion curves of phase velocity and attenuation coefficient are presented graphically.