Physical interpretation of the general Poynting's theorem for shell vibrations

Abstract

An improved derivation involving tensor analysis and covarient differentiation, as well as Hamilton's principle and a realization of Noether's theorem, is given for the energy corollary of the basic shell dynamic equations for a thin shell of arbitrary shape, and with position-dependent elastic modulus E, Poisson's ratio v, and shell thickness h. This corollary states that the sum of (i) the time derivative of an energy density and (ii) the divergence (within the shell middle surface) of a structural intensity vector is identically zero in the absence of external sound radiation and internal damping. The requirement that the intensity transform as a two-dimensional tensor removes the intrinsic ambiguity (with which intensity component does one associate a double derivative term?) that the author had previously reported in an analysis of shells of revolution. The compact tensorial form of this “Poynting's theorem” is seemingly simple, although its explicit unraveling for specific cases can lead to cumbersome expressions. The two tensorial groupings in the structural intensity reduce in the flat plate case to (i) energy flux associated with in-plane (e.g., stretching) motions and (ii) energy flux associated with bending (flexural) motions. Further clarifications of the derived expressions are found in the limiting cases of cylindrical and spherical shells, and in appropriate “high-” and “low-frequency” limits. The latter part of the paper discusses how the corollary may be extended (although with some approximations) to incorporate fluid loading and sound radiation. [Work supported by ONR.]