A novel treatment of a direct procedure for elastic-plastic analysis and shakedown is presented and its application to problems in three-dimensional rolling contact with or without case-hardened layers is demonstrated. The direct approach consists of an operator split technique, which transforms the elastic-plastic problem into a purely elastic problem and a residual problem with prescribed eigenstrains. These eigenstrains are determined using an incremental projection method based on the purely elastic solution and a special representation of the yield condition for a linear-kinematic hardening material. The three-dimensional residual problem is then further split into a plane problem and an anti-plane problem which are readily solved using the finite element method. A significant advantage of the present analysis over the alternative approach of simulating repeated rolling until shakedown occurs is that in the present analysis, the final shakedown solution is obtained directly by solving three elasticity problems. Results are compared with full elastic-plastic finite element calculations available from the literature and good agreement is observed. The effects of surface hardened layers on the distributions of residual stress and displacement are investigated for both two- and three-dimensional contact. The direct approach is shown to be a straightforward and efficient method for obtaining the steady state solution in the analysis of three-dimensional problems in rolling and/or sliding contact.

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