On Nonstationary Contact Problems for Anisotropic Composites in Nonclassical Areas

Abstract

For the first time, an exact solution is given to the contact problem of the non-stationary action of a wedge-shaped, right-angled stamp occupying the first quadrant, which act on a deformable multilayer base. The base, which is affected by a rigid stamp in the shape of a quarter plane, can be a multilayer anisotropic composite material. It is assumed that it is possible to construct a Green’s function for it, which makes it possible to construct an integral equation of the contact problem. The geometric Cartesian coordinates of the first quadrant and the time parameter, which varies along the entire axis, are taken as parameters describing the integral equation. It is assumed that time in the boundary value problem under consideration follows from negative infinity, crosses the origin and grows to infinity, covering the entire time interval. Thus, there is no requirement in the formulation of the Cochet problem when it is necessary to set initial conditions. In this formulation, the problem is reduced to solving the three-dimensional Wiener–Hopf integral equation. The authors are not aware of any attempts to solve this problem analytically or numerically. The investigation and solution of the contact problem was carried out using block elements in a variant applicable to integral equations. It is proved that the constructed solution exactly satisfies the integral equation. The properties of the constructed solution are studied. In particular, it is shown that the solution of the non-stationary contact problem has a higher concentration of contact stresses at the edges of the stamps and at the angular point of the stamp, compared with a static case. This corresponds to the observed in practice more effective non-stationary effect of rigid bodies on deformable media, for their destruction, compared with static. The results may be useful in engineering practice, seismology, in assessing the impact of incoming waves on foundations, in the areas of using Wiener–Hopf integral equations in probability theory and statistics, and other areas.