Solution of the 3-D equations of elasticity in prolate spheroidal coordinates

Abstract

The 3-D equations of elasticity in prolate spheroidal coordinates are solved using a Helmholtz decomposition for the displacement vector. The displacement vector is written in terms of a scalar potential (dependent on the dilatational wave number) and a vector potential (dependent on the shear wave number). These wave potentials are solutions, respectively, of the scalar and vector wave equations cast in prolate spheroidal coordinates. An eigenfunction expansion of spheroidal wave functions is employed to solve the scalar wave equation. The solution of the vector wave equation is facilitated using expansions of prolate spheroidal vector wave functions. These vector wave functions are constructed by applying certain vector differential operators to the scalar wave functions. The canonical problem of free vibration of a traction-free confocal elastic layer (shell) is considered. The shape of the prolate spheroid is defined by the major-axis length L and minor-axis length D. Since the sphere is a limiting form of the prolate spheroid, the analytical approach is validated by comparing the frequency spectra of a prolate spheroidal shell with L/D∼1 to those of a spherical shell using a 3-D solution in spherical coordinates. [Work supported by the NAVSEA Newport ILIR Program.]