Revisiting stress propagation in a three-dimensional elastic sphere under diametric loading

Abstract

The stress propagation for a three-dimensional elastic sphere under a diametric loading condition in the framework of the linear elastodynamics is revisited. By describing displacements in terms of scalar and vector potentials using the Helmholtz theorem, the Navier-Cauchy equation, the time evolution equation for displacements, is converted to the wave equation. The wave equation is Laplace transformed and further solved in spherical coordinates to develop the tabular expressions for displacement and stress in terms of modified spherical Bessel functions, and the coefficients are determined to satisfy the initial and boundary conditions. The obtained solutions are given in the form of an inverse Laplace transform. For the steady solution, the long-time limit of the obtained solutions is derived by using the final value theorem of the Laplace transform. The unsteady solutions are obtained by applying the residue theorem of complex analysis by adding a path with zero contribution in the complex plane. The obtained solution includes longitudinal and transverse waves, and also Rayleigh waves propagating on the surface. The origin of the von Schmidt wave is discussed as a reflected wave produced by the longitudinal wave travelling on the surface. It is also discussed that the wave is bent, unlike in the case of semi-infinite systems.