On the dynamic contact problem with two deformable stamps

The problem of the time-harmonic behavior of two deformable semi-infinite stamps lying on a deformable base is considered. It is assumed that the stamps converge with parallel ends in such a way that they form a crack, defect, or tectonic fault in the convergence zone. The deformable die material has a simple rheology described by the Helmholtz equation. To consider the cases of deformable stamps of complex rheologies, a new universal modeling method can be used. It allows solutions of vector boundary value problems for systems of partial differential equations describing materials of complex rheologies to be represented as decomposed by solutions of individual scalar boundary value problems. A high-precision solution to the boundary value problem is constructed, which makes it possible to obtain a dispersion equation describing resonant frequencies. The existence of resonant frequencies for deformable stamps was predicted in the works of I. I. Vorovich. The result remains valid for the case of absolutely solid semi-infinite stamps. Earlier, it was shown that resonances arise in the contact problem of the oscillation of two absolutely rigid stamps of finite dimensions on a deformable layer. However, the dynamic contact problem for the case of two semi-infinite stamps acting on a multilayer medium has not been studied before. The study is based on the block element method, which makes it possible to construct exact solutions to boundary value problems for partial differential equations. In addition, factorization methods are used and some subtle properties of the Wiener – Hopf equations are used, in particular, those belonging to the famous mathematician M. G. Crane. The proposed methods make it possible to conduct research for the entire frequency range and an arbitrary distance between the ends of semi-infinite plates. The results of the study can be used to evaluate the strength properties of structures with contact joints made of different types of materials in dynamic modes.