The uniform motion of rectangular and parabolic punches in a viscoelastic layer

The plane contact problem of the interaction of a rigid moving punch with a viscoelastic layer is considered. It is assumed that the punch moves at a constant speed and is forced into the layer by a constant normal force. There is no friction in the contact area between the stamp and the surface of the layer. The displacement of the boundary of the layer, due to the normal load applied to it, is determined. The integral equation of the contact problem for determining the contact pressure is then derived and an approximate solution of this integral equation is constructed using the Multhopp–Kalandiya method for a rectangular punch and one of the modifications of the method of orthogonal polynomials for a parabolic punch.