About dynamic contact points problems with stamps of complex rheologies in the quarter plane of an anisotropic composite

In this paper, for the first time, a solution is constructed to the dynamic contact problem of the time-harmonic effect of a deformable die on a layer of anisotropic composite material. It is assumed that the stamp occupies the region of the first quadrant and has a complex rheology, in particular, the linear theory of elasticity. The paper uses a universal modeling method developed by the authors, which makes it possible to apply the block element method to both differential and integral equations. The solutions of boundary value problems for deformable dies of complex rheology are constructed in the form of decompositions according to the solutions of boundary value problems for materials of simple rheology described, for example, by Helmholtz equations. This possibility was previously established for materials of a wide range of rheologies using Galerkin transformations. The solution of the two-dimensional Wiener–Hopf integral equation is obtained both in coordinate form and in Fourier transforms. This makes it particularly convenient to further study it using analytical or numerical methods using standard computer programs. They will make it possible to identify certain properties of composites used as structural materials in various engineering technologies dictated by types of anisotropies, as well as in issues of seismology in the study of seismicity in mountainous areas. The constructed integral representation of the solution of the contact problem, which makes it possible to identify terms describing the concentrations of contact stresses under the stamp, makes it possible to select the soles of deformable stamps or the properties of the materials used to get rid of undesirable concentrations of contact stresses or enhance them. Since Vorovich resonances can occur during vibration in contact problems with a deformable die, systems of equations are constructed in the work that allow, when solved, to obtain a dispersion equation for finding resonant frequencies.