Rolling Contact Phenomena

In this paper, we treat the rolling contact phenomena of linear elasticity, with special emphasis on the elastic half-space. Section 1 treats the basics; rolling is defined, the distance between the deformable bodies is calculated, the slip velocity between the bodies is defined and calculated; a very brief recapitulation of the theory of elasticity follows, and the boundary conditions are formulated. Section 2 treats the half-space approximation. The formulae of Boussinesq-Cerruti are given, and the concept of quasiidentity is introduced. Then follows a brief description of the linear theory of rolling contact for Hertzian contacts, with numerical results, and of the theory of Vermeulen-Johnson for steady-state rolling. Finally, some examples are given. Section 3 is devoted to the simplified theory of rolling contact. In Section 4, the variational, or weak theory of contact is considered. First, we set up the virtual work inequality, and it is shown that it is implied by the boundary conditions of contact. Then the complementary virtual work inequality is postulated, and it is shown that it implies the boundary conditions of contact. Elasticity is introduced into both inequalities, and the potential energy and the complementary energy follow. Finally, surface mechanical principles are derived. In Section 5, we return to the exact half-space theory. The problem is discretized, and solved by means of the CONTACT algorithm. Finally, results are shown in Section 6.