The aim of this work is to take into account the effect of the stress gradient as a beneficial effect in fatigue, since its presence within a material systematically leads to an increase in the endurance limit in terms of local maximum stress. We have therefore proposed that the stress gradient be taken into account in the fatigue criterion using a statistical approach based on the variance method. This method assumes that the fracture plane is the one on which the variance of the equivalent stress is maximum. This method defines three types of equivalent stress on a plane: normal stress, shear stress and combined normal and shear stress. The results obtained using the simple variance method and the stress gradient method were compared for a cyclic biaxial loading condition. We found that in terms of prediction of the fracture plane, there is not much difference between the two formulations (the simple variance method and the one taking into account the stress gradient), but the maximum values of the variance for the variance method taking into account the stress gradient vary greatly compared to the case of the simple variance method.

This paper focuses on the improvement of the nonlinear fatigue law of B. Tikri from 2011, which was born from the old laws of Chaboche according to the Basquin model. Several parameters can influence the life of structures in service. It is a question in this work to take into account the influence of the residual stress on the damage law proposed in the literature. Two values of stress relaxation coefficients of 0.5 and 1 have been considered to test the proposed model with two different overload ratios in order to evaluate the influence of overload on steels used in the automotive industry. The ratios were 1.4 and 2.3. Two findings were made, the first is that when the relaxation coefficient is 0.5 the residual stress improves the life but it is far from the experimental case contrary to the case of relaxation coefficient equal to the unit. For the second case, the model is much more satisfactory for overloads of ratio 1.4 than for overloads of ratio 2.3 for HE360D materials. For future designs, the consideration of its residual stresses over the life of parts or structures in service is paramount.