About the properties of static contact solutions problems for anisotropic composites in the quarter plane

In this work, for the first time, an exact solution of the static contact problem of the action of a rigid wedge-shaped die occupying the first quadrant on a layer of composite material having arbitrary anisotropy is constructed using the block element method. Unlike numerous, mostly unsuccessful attempts to solve this and similar problems by analytical or numerical methods, which allowed us to identify only partial properties of the solution to this problem, the block element method made it possible to reveal a richer set of properties of its solutions. The solution is obtained in both coordinate and Fourier transforms. This makes it especially convenient to further study it by numerical analysis using standard computer programs. They will allow us to identify certain properties of composites as structural materials dictated by different types of anisotropies. It is shown that the obtained solution exactly satisfies the two-dimensional Wiener–Hopf equation for an arbitrary right-hand side. A number of previously unknown properties of the solution have been revealed. In particular, the obtained representation of the solution of the contact problem in a wedge gave it a general appearance. In comparison with strip stamps, it contains an additively additional term describing the concentration of contact stresses at the angular point, that is, at the top of the stamp. The calculation of the indicator of the peculiarity of the concentration of contact stresses at this point is close to the values performed by numerical methods in a number of works. The paper shows that the zone near the top of the stamp has superior malleability when the stamp is inserted into the medium, compared with remote zones. This corresponds to the estimates obtained by the example of the introduction of strip stamps narrowing in width into the layer. In the zone considered away from the top of the stamp, the exact solution turns into a solution for the case of a semi-infinite stamp. The developed method is applicable to composites of arbitrary anisotropies arising in linearly elastic materials and crystals of any cross-sections that allow the construction of the Green function, and hence the two-dimensional Wiener–Hopf integral equations. The establishment of a general type of solution to the considered contact problem opens up the possibility of studying the precursors of increased seismicity in mountainous areas, as well as improving numerical methods to obtain more accurate solutions to complicated contact problems in engineering practice.