Asymptotics of Extensional Waves in Isotropic Elastic Plates

The long and short-wave asymptotics of order $O(η6),η=kh$⁠, for free extensional vibrations of an infinite isotropic elastic plate are studied. The asymptotic model for flexural and extensional wave motion applicable for both long and short-wave approximations and for any materials is developed. The velocity and frequency dispersion relations for extensional waves are derived in analytical form from the system of three-dimensional dynamic equations of linear elasticity. All dispersion equations and the group velocity formula are presented as explicit functions in material parameter $γ=cs2/cL2$ (the ratio of the velocities squared of the flexural and extensional waves) without any correction factors as in the Reissner-Mindlin theory. Variations of the velocity and frequency spectra depending on Poisson’s ratio ν are illustrated graphically. The results are discussed and compared to those obtained and summarized by Mindlin (1960), Mindlin and Medick (1959), Tolstoy and Usdin (1953, 1957), Achenbach (1973), and Graff (1991).