Analytical solution of the contact problem for pre-stressed two half-spaces and an ring stamp

The article presents an analytical solution to the contact problem for two elastic half-spaces with initial stresses and a prestressed ring stamp, without considering frictional forces. We will assume that the surfaces outside the contact boundary of the ring stamp and the half-spaces remain free from the influence of external forces, and at the contact boundary displacements and stresses are continuous. The problem is solved in the case of equal roots of the defining equation. The study is presented in a general form for the theory of large initial strains and two variants of the theory of small initial strains within the linearized theory of elasticity with an arbitrary structure of the elastic potential. It is assumed that the initial states of the elastic ring stamp and the elastic bases are homogeneous and equal. The research is carried out in the coordinates of the initial deformed state, which are related to the Lagrangian coordinates. In addition, the influence of the ring stamp causes small perturbations of the corresponding values of the basic stress-strain state. It is also assumed that the elastic ring stamp and the elastic half-spaces are made of different isotropic, transversally isotropic or composite materials. In the case of orthotropic bodies, we will assume that the elastically equivalent directions coincide with the direction of the coordinate axes in the deformed state. As a result, the solutions of the given problem are presented in the form of infinite series, the coefficients of which are determined from an infinite quasi-regular system of algebraic equations. Several fundamental results, such as the Hankel transformation, triple integral equations, and other methods of the theory of contact problems of the linearized theory of elasticity, are used to study this problem. In the article also establishes the connection between sinking and the corresponding burden. Therefore, with the help of the obtained solutions, it is possible to study the influence of initial (residual) stresses on the distribution of contact stresses and displacements in two elastic half-spaces and an elastic ring stamp.