Transient responses of a thick-walled layered spherical shell by the ray theory

Abstract

In this work, the transient response of a thick-walled layered spherical shell subjected to radially symmetric loadings is studied using the ray theory. Normal mode solution in the Fourier transform space is expanded into a series where each term represents a spherical harmonic wave. The inverse transform of these terms which are called ray integrals can be obtained in closed form. Since each ray reaching a receiver point has a unique arrival time, only a finite number of them should be considered once the time interval of interest is specified. Summation of these rays up to a specific time gives the exact solution of the transient response. Since the number of rays to be considered increases geometrically as the time interval increases, the method loses its advantage in calculating the long time responses of the medium. Similar problem will arise in those cases where the shell is a thin one. A computer program is developed in order to investigate the transient displacements and radial and tangential stresses, and numeiical results are given for a suddenly applied uniform internal pressure case. The peak value for tangential stress is found to be 142 per cent of applied pressure in the two-layered shell, although this value was 165 per cent for a single layered shell having the same material properties with the first layer. The reason for radial stress changes from compression to tension due to dynamic pressure can be found in the result of multiple reflection of waves. As the radial displacement case is invesvi tigated, it is found that the peak value of 1.31 unit of displacement is reached for the two-layered shell although it was 1.70 for the single layered case.