On the Contact Problem with a Deformable Die in a Quadrant

This paper is the first ever rigorous investigation of a two-dimensional dynamic contact problem of the action of a deformable die on a quadrant of a multilayer medium. Compared to an absolutely rigid die, its deformable counterpart introduces additional features. They consist in the potential occurrence of discrete resonances predicted by Academician I. I. Vorovich. The paper shows that a method based on using brick-level elements makes it possible to derive an equation describing resonant frequencies to study contact problems with a deformable die made of complex-rheology materials, including smart materials. First of all, the case of a deformable die made of a simple-rheology material described by the Helmholtz equations is considered. Solutions of the boundary problems for complex-rheology dies are represented as a combination of solutions of boundary problems for simple-rheology dies.