A B S T R A C TThe general idea of this paper is to model the mixed-mode cyclic elastic-plastic behaviour of the crack tip region at the global scale. It should be helpful, for instance, for predicting the effect of mixed mode overloads in fatigue. It is aimed at establishing a model reasonably precise (compared with elastic-plastic finite element (FE) computations) but condensed into a set of partial derivative equations so as to avoid huge elastic-plastic FE computations in the future. For this purpose, the kinematics of the crack tip region is characterized by a set of condensed variables. This is classical in linear elastic fracture mechanics (LEFM), the displacement field is approached by the product of spatial reference fields (u e I and u e II ) and nominal stress intensity factors (K ∞ I and K ∞ II ). Therefore, in LEFM, two condensed variables only, K ∞ I and K ∞ II , fully define the kinematics in the crack tip region. So as to generalize this approach to mixed mode cyclic elastic-plastic conditions, we define first the intensity factors (˙K I and˙K II ) of the elastic spatial reference fields (u e I and u e II ) and we introduce two additional spatial reference fields (u c I and u c II ) and their intensity factors (ρ I andρ II ) to account for plastic deformation within the crack tip region. Such an approximation is shown to be reasonably precise using FE computations. Therefore, the velocity field in the crack tip region is fully defined by only four condensed variables (˙K I ,K II ,ρ I andρ II ). Using the multi-scale approach proposed herein, evolutions of ρ I and ρ II for various mixed-mode loading conditions defined by K ∞ I and K ∞ II were generated using the finite-element method (FEM). Then, it was shown that we can model these evolutions at the global scale through a yield locus, a flow rule and a kinematics hardening rule. It is also suggested how this model could be employed for predicting the effects of mixed mode plasticity on fatigue crack growth. FEM = finite-element method LEFM = linear elastic fracture mechanics K ∞ I , K ∞ II = mode I and mode II nominal stress intensity factors u e I , u e II = mode I and mode II elastic reference displacement fields u c I , u c II = mode I and mode II complementary reference displacement fields K I ,K II = mode I and mode II pseudo-elastic intensity factors ρ I , ρ II = mode I and mode II plastic intensity factors I N T R O D U C T I O NFatigue crack growth under non-proportional mixedmode loading conditions remains an open problem. As a matter of fact, history effects in fatigue were shown to

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Plastic deformation within the crack tip region introduces internal stresses that modify subsequent behaviour of the crack and are at the origin of history effects in fatigue crack growth.Consequently, fatigue crack growth models should include plasticity induced history effects. A model was developed and validated for mode I fatigue crack growth under variable amplitude loading conditions. The purpose of this study was to extend this model to mixed mode loading conditions. Finite element analyses are commonly employed to model crack tip plasticity and were shown to give very satisfactory results. However, if millions of cycles need to be modelled to predict the fatigue behaviour of an industrial component, the finite element method becomes computationally too expensive. By employing a multiscale approach, the local results of FE computations can be brought to the global scale. This approach consists of partitioning the velocity field at the crack tip into plastic and elastic parts. Each part is partitioned into mode I and mode II components, and finally each component is the product of a reference spatial field and an intensity factor. The intensity factor of the mode I and mode II plastic parts of the velocity fields, denoted by dρ I /dt and dρ II /dt, allow measuring mixed-mode plasticity in the crack tip region at the global scale.Evolutions of dρ I /dt and dρ II /dt, generated using the FE method for various loading histories, enable the identification of an empirical cyclic elastic-plastic constitutive model for the crack tip region at the global scale. Once identified, this empirical model can be employed, with no need of additional FE computations, resulting in faster computations. With the additional hypothesis that the fatigue crack growth rate and direction can be determined from mixed mode crack tip plasticity (dρ I /dt and dρ II /dt), it becomes possible to predict fatigue crack growth under I/II mixed mode and variable amplitude loading conditions. To compare the predictions of this model with experiments, an asymmetric four point bend test system was set-up. It allows applying any mixed mode loading case from a pure mode I condition to a pure mode II. Initial experimental results showed an increase of the mode I fatigue crack growth rate after the application of a set of mode II overload cycles.