About the contact problem with the system deformable stamps

Contact problems with deformable dies, in order to build high-precision solutions, have not been practically investigated. The problem is the complexity of solving the boundary value problem for a deformable die. One of the approaches for constructing a solution to such a contact problem is to decompose the solution of the boundary value problem for a stamp into eigenfunctions corresponding to the boundary value problem. However, if the domain of the boundary value problem is nonclassical, and the rheology of the material is complex, then such a contact problem has difficulties. Another approach for constructing a high-precision solution to the contact problem with a deformable die is to use new topological research methods and solve boundary problems for stamps - this is the block element method. However, it should be borne in mind here that the nonclassical areas available for the analytical construction of block elements are quite limited. Nevertheless, they can be used as approximating more complex ones, due to the practical importance of contact problems, including dynamic ones, in engineering practice. The block element method has advantages in contact problems with deformable dies. They are as follows. The sum of solutions of individual boundary value problems for deformable stamps of simple rheology, could be solving a boundary value problem for a system of differential equations. However, this is not the only advantage of block elements in contact problems with deformable dies. It is possible to solve contact problems with a deformable die in those non-classical areas in which it is possible to build block elements. These include half planes, quarter planes, half stripes, circles, sectors, wedge-shaped areas, stripes, and rectangles. This is very naturally consistent with the Wiener-Hopf integral equation, which is involved in describing the contact of a deformable die with a deformable base in the contact area. Formulation and method of solving the contact problem with a deformable die, using the approach described above, allow to identify the equations of resonant frequencies.