Energy flow associated with fundamental elastic wave types on inhomogeneous cylindrical shells

Abstract

The basic shell dynamic equations for an inhomogeneous cylindrical shell (slowly varying radius, thickness, and material properties) result from Hamilton's principle. The version adopted here is one which yields the Donnell equations in the limiting case of a homogeneous thin shell, although the same technique applies for more sophisticated shell theories. There are three wave types that result from the theory that can be identified in the high-frequency limit as compressional (longitudinal), in-plane shear, and flexural. The propagation of these waves is directionally dependent, frequency dependent, and dispersive. The general ray acoustics formulation developed in previous papers by the author, by Felsen and Lu, and by Norris is extended to include the proper invariants for propagation along ray tubes. The energy density and energy flux vectors for each of the three wave types are derived with the aid of Hamilton's principle and a Poynting's theorem expressing conservation of wave energy is obtained for each wave type. This identification allows the discussion of the energy transfer when waves are partially reflected and converted to other wave types at ribs or at abrupt transitions of shell radius. [Work supported by ONR and by the William E. Leonhard endowment to Pennsylvania State University.]