On the problem of non-stationary contact problems

For the first time, the paper considers the dynamic contact problem of nonstationary impact in the contact zone of a flexible die on the surface of a deformable multilayer medium. The contact problem is considered in a two-dimensional formulation. The stamp is accepted semi-infinite and is valid for a semi-infinite time interval. Traditionally, in such dynamic contact problems, the decomposition into a double series or Fourier integral of a time-varying function describing the behavior of the stamp sole was used. After that, a contact problem with excluded exponential functions containing a time parameter is considered. However, it is known that such Fourier series do not effectively describe processes accompanied by a violation of the smoothness of functions, in particular, if there are features in them. The problem was solved sequentially, first in geometric parameters, then the time dependence was investigated. This did not allow us to reveal the subtle features of the dependence of solutions on the time parameter. In this paper, this shortcoming is eliminated. The case of a two-dimensional problem in which geometric and time parameters are equally included is considered. The contact problem is reduced to the two-dimensional Wiener-Hopf integral equation, the solution method of which has been developed recently. The resulting solution, depending on the geometric and time parameters, allowed us to identify the previously undescribed effect of a temporary surge at the initial moment of contact stresses under the stamp. The result allows, by adjusting the time of impact of the stamp on the medium, to choose the optimal modes of emerging contact stresses. The method allows generalization to contact problems of higher dimensions.