The class of generalized standard materials is not relevant to model the nonassociative constitutive equations. The bipotential approach, based on a possible generalization of FenchelÕs inequality, allows the recovery of the flow rule normality in a weak form of an implicit relation. This defines the class of implicit standard materials. For such behaviours, this leads to a weak extension of the classical bound theorems of the shakedown analysis. In the present paper, we recall the relevant features of this theory. Considering an elastoplastic material with nonlinear kinematic hardening rule, we apply it to the problem of a sample in plane strain conditions under constant traction and alternating torsion in order to determine analytically the interaction curve bounding the shakedown domain. The aim of the paper is to prove the exactness of the solution for this example by comparing it to step-by-step computations of the elastoplastic response of the body under repeated cyclic loads of increasing level. A reliable criterion to stop the computations is proposed. The analytical and numerical solutions are compared and found to be closed one of each other. Moreover, the method allows uncovering an additional Ô2 cycle shakedown curveÕ that could be useful for the shakedown design of structure.

International audienceThis paper proposes a numerical approach based on a steady-state algorithm to predict the rolling contact fatigue crack initiation in railway wheels in practical conditions. This work suggests taking into account the cyclic hardening of the wheel's material and one of its originality is to conduct a complete numerical approach whatever the loading level. The main stages are the characterization and modelling of the material behaviour, the determination of the stress-strain fields using a numerical steady-state method and the application of a high cycle fatigue criterion. Computations were made with the Abaqus FE commercial software. Three cases are studied: rolling with or without sliding and skating. The numerical results give several types of mechanical responses: elastic or plastic shakedown. Otherwise, the results show that the location where the shear stress is maximal is not the same as where the risk of crack is the highest

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