The Semi-Infinite Body Subjected to a Concentrated Impact Load on the Surface

Abstract

The problem of the semi-infinite body subjected step-wise to a concentrated impact load on the surface is treated by using the dynamical theory of elasticity. By using integral transformation, the results are obtained in forms of dual integrals which involve Laplace inverse-transforms and infinite integrals. The infinite integrals are evaluated by introducing a new variable. Laplace inverse-transforms which are separated from the static results are integrated numerically along the Bromwich integral path. As the result, the variations with time of the displacements and stresses are presented. The displacements and stresses begin to fluctuate suddenly upon arrival of a dilatational wave. In the neighbourhood of the symmetric axis, their absolute values increase till the arrival of a distortional wave. After this arrival, they decrease discontinuously and tend gradually to static results with an increasing time. On the symmetric axis, the radial stress is always compression and becomes maximum just before the arrival of the distortional wave. The circumferential stress is tension except for a short time just after the arrival of the dilatational wave, and becomes also maximum just before the arrival of the distortional wave. The magnitude of the maximum radial stress is about 3 times as large as the static one. The ratio of this stress to the static one in this case is considerably greater than the corresponding ratio in the plane stress state.