The contact problem of the mathematical theory of elasticity with stick and slip areas. The theory of rolling and tribology

The contact problem of the mathematical theory of elasticity, taking adhesion of the contact into account, is considered as a topic in fracture mechanics. An exact solution of the general contact problem of fracture mechanics under plane deformation conditions with stick and slip areas of two contacting half-spaces with dissimilar elastic properties is given. In fact, this problem is the basis of theoretical tribology. The solution is obtained in closed form for a class of inhomogeneous materials. The problem of the pressure of an absolutely rigid punch on an elastic body under plane deformation conditions, taking adhesion on the stick and slip areas into account, is also solved in closed form, when Poisson's ratio is 1/2. The initial mathematical problem also covers problems in fracture mechanics of composites concerning the propagation of cracks along the interfaces of two different elastic materials, taking into account superposition/slip areas of the faces of the cracks. The analytic continuation method is used to reduce the problems to a single generalized Riemann boundary value problem, the solution of which is obtained in closed form. Using the example of the solution of typical contact problems of fracture mechanics, a rigorous quantitative theory of the fundamental rolling modes and the stick-slip phenomenon is given and analysed. It is shown that when there is no slip and adhesion the rolling friction coefficient in Coulomb's law is directly proportional to (NRP)1/2 for wheels and cylinders, and (NRP)1/3 for spheres, where N is the normal force (the weight of the sphere or the weight per unit length of the cylinder), R is the radius of the wheel or sphere and P is the elastic compliance of the system. The effect of adhesion and roughness of the materials on the rolling, and also wear of the materials during rolling are characterized by two material constants of fracture mechanics. By a decision of the editorial board of the journal, the last section is added as an answer to critical comments on the paper, published after this paper.