Slow vertical motions of an elliptic punch on an elastic half-space

The dynamic contact problem for a homogeneous, isotropic elastic half-space and a punch of an arbitrary base shape is studied. During the motion of the punch it is assumed that the contact area is fixed and there is no friction under the punch. An approximate solution to the problem is obtained under the assumption that the contact pressure under the punch is slightly varied during the time of travel of the Rayleigh wave along the distance equal to the diameter of the contact area. The solution of the problem of slow motions of a punch on an elastic half-space is reduced to the solution of the recurrent system of integral equations of the static contact problem. Asymptotic models of vertical motions of a flat-ended punch with an arbitrary base are constructed. The case of an elliptic punch is considered in detail.