Application of the kinematical shakedown theorem to rolling and sliding point contacts

In the plane-strain conditions of a long cylinder in rolling line contact with an elastic-perfectly-plastic half-space an exact shakedown limit has been established previously by use of both the statical (lower bound) and kinematical (upper bound) shakedown theorems. At loads above this limit incremental strain growth or “ratchetting” takes place by a mechanism in which surface layers are plastically sheared relative to the subsurface material.In this paper the kinematical shakedown theorem is used to investigate this mode of deformation for rolling and sliding point contacts, in which a Hertz pressure and frictional traction act on an elliptical area which repeatedly traverses the surface of a half-space. Although a similar mechanism of incremental collapse is possible, the behaviour is found to be different from that in two-dimensional line contact in three significant ways: (i) To develop a mechanism for incremental growth the plastic shear zone must spread to the surface at the sides of the contact so that a complete segment of material immediately beneath the loaded area is free to displace relative to the remainder of the half-space, (ii) Residual shear stresses orthogonal to the surface are developed in the subsurface layers, (iii) A range of loads is found in which a closed cycle of alternating plasticity takes place without incremental growth, a condition often referred to as “plastic shakedown”.Optimal upper bounds to both the elastic and plastic shakedown limits have been found for varying coefficients of traction and shapes of the loaded ellipse. The analysis also gives estimates of the residual orthogonal shear stresses which are induced.