Linear-periodic Hertz contact problems for an elastic half-space

Spatial contact problems are considered for an elastic half-space, when one or two endless periodic chains of rigid punches are lined up along a straight line. The problems generalize the classical Hertz contact problem for a unit elliptic stamp on an elastic half-space that has an exact solution. In the linearly periodic case, the superposition of kernels of the integral equation of the Hertz contact problem yields a divergent series of harmonic type. Following the idea of E.A. Kuznetsov on the introduction of additional (relative) displacements, regularization of the kernels is carried out by adding given periodic systems of concentrated forces (with the same period as the punch systems), symmetric with respect to the punch systems. The vector sum of the forces applied to the punches and the additional forces outside the contact area must be zero. This leads to converging series in the kernels of integral equations. For unknown contact areas, the method of B. A. Galanov is used for numerical solution with the introduction of special nonlinear operators, which allows us to determine the contact area and contact pressures simultaneously by iterations according to the Newtonian scheme for nonlinear integral equations of the Hammerstein type. The nonlinearity of the problems is due to the fact that the contact area is not known in advance and is determined from the condition that the contact pressure at its boundary is equal to zero, which even in the Hertz problem leads to a nonlinear relationship between the characteristics of the contact. Punch systems in the form of elliptical paraboloids are considered. Contact areas are calculated, as well as the integral characteristics of the contact (pressing forces) depending on the settlement of the punches and on the distance between the chains of punches and additional forces. To debug the program, an exact solution to the Hertz problem is used. It has been shown that percolation (merging of discrete contact zones) with the formation of a continuous infi nitely long contact zone is observed for suffi ciently elongated along the axis of the chain of punches when the contact is strengthened (increase in the settlement of punches).