On the theory of contact problems for composite media with anisotropic structure

For the first time, an exact solution to the contact problem of the action of a strip rigid stamp of finite width on a composite layered material having an anisotropic structure is constructed. Problems of this kind have been studied in sufficient depth for isotropic materials. Contact problems for non-classical shaped stamps acting on composite materials have been poorly studied. The applied numerical methods for composite materials do not take into account the contact stress concentrations occurring at the boundary, which are characteristic of contact problems, do not fully reveal the malleability of the stamp insertion into an anisotropic medium when the stamp size changes, and are difficult to analyze in dynamic cases. In contrast to the isotropic case, when the symbol of the kernel of the integral equation is described by a meromorphic function, in the anisotropic case one has to meet with an analytical function of two complex variables of complex structure. Contact problems for anisotropic materials arise in many areas when creating various engineering equipment and products, in construction, when creating an electronic element base, as well as in the mechanics of natural processes.. In this paper, using the example of the effect of a strip rigid stamp of finite width on a composite laminated material, an exact solution of a static problem for one type of anisotropy is constructed using the block element method. The practice of constructing exact solutions to boundary value problems shows that with their help it is possible to capture and identify properties of solutions, the study of which is inaccessible to numerical methods. Examples are the identification of new types of earthquakes, starting ones, a new type of cracks not previously described, new types of earthquake precursors and resonances of structures. On the basis of exact solutions, it is possible to build high-precision approximations, the application of numerical methods to which already turns out to be more effective than as a result of direct inversion of volumetric and boundary differential operators of boundary problems. The result of this article can be useful both in engineering practice and in geophysics in describing the behavior of a mountain range on an anisotropic bedrock. In addition, the method opens up the possibility to investigate anisotropic cases in a dynamic formulation using contour integrals in the representation of solutions.