Far-field dependence on the end conditions in a semi-infinite elastic rod of circular cross-section

Abstract

The propagation of elastic waves in a semi-infinite rod of circular cross-section having a stress-free lateral surface is considered for two different sets of boundary conditions imposed at the input end. The first set of boundary conditions prescribes the normal stress and requires the radial displacement to be zero. The second set prescribes the normal stress and requires the tangential stress across the end to be zero. In both cases the normal stress is a step function. The first set is an example of “xed” boundary conditions and corresponds to striking the end while permitting no lateral motion. The second set is an example of “pure” boundary conditions and corresponds to the pressure shock problem. The purpose of the paper is to show that far-field results for the displacements, stresses and strains do not depend in a significant way on whether the boundary conditions are “mixed” or “pure”. In fact, we find the initial pulse is similar (to within 4 %) for both pure and mixed boundary conditions. For distances as close as two diameters from the end of the rod, the subsequent oscillations differ, while for distances greater than five diameters, they become similar. Our conclusions concerning the far field are in agreement with those held by other research workers. The linearized equations of elasticity are used in this paper and solved using finite difference methods.