13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- Elastic-plastic constitutive equations with isotropic and non-linear kinematic hardening, identified from the stress-strain loops obtained from push-pull tests were used. Cyclic loading with R=0.1 was simulated. The stress and strain fields computed at maximum and minimum load were used for a local application, ahead of each node of the front, of a fatigue criterion derived from that identified by Zhao and Jiang from an extensive multiaxial fatigue database on 7075 T6 [12]. Their criterion successfully captured the transition in fracture mode observed in torsion as well as in push-pull or combined loading: when the loading range increases, fatigue damage changed from normal-stress-driven to shear-driven, yet with an assistance of an opening stress. Their damage function (DF) was thus: γ τ σ ε Δ Δ − + = Δ 2 1 2 max b DF b n n (4a) eq b a a σ = − Δ2 1 (4b) in which <x> denotes the positive part of x, a1 and a2 two fitted constants (0.862 and 0.00125), Δεn, Δγ, Δτ, σnmax, respectively the normal strain, shear strain and stress range and peak opening stress, all computed along the critical plane. The latter is that for which the damage function, DF is maximum. As a consequence of Eq. 4b, the normal stress and strain play a major role at low stress range, but their influence decreases as the stress range increases, down to a transition stress range above which damage is merely driven by shear (b=0). In the present study, this criterion was slightly modified. The transition in fracture mode occurs in the low-cycle fatigue regime, where fatigue tests are usually strain-controlled, while the equivalent stress range evolves due to cyclic hardening. The value of Δσeq which enters equation 4b was thus not clearly defined. The parameter b was thus considered to be more clearly related to the applied equivalent strain range, rather than to the stress range. Equation 4b was therefore turned into: 2) ( 2. 1 eq trans eq eq trans eq b ε ε ε ε Δ Δ + Δ Δ = − (5) in which Δεeq,trans corresponds to the equivalent strain range for which the transition from one fracture mode to the other is observed. A smoother evolution of b is predicted, compared to Eq. 4b. In addition, b tends toward 1 when the strain range vanishes, so that fracture is controlled merely by the normal stress and strain, which was not the case in the original criterion, for which the maximum value of b was 0.862. To apply the criterion ahead of each node of the crack front, Δεeq was first averaged over a 90μm-long segment parallel to the x axis, comprising three elements. The local value of b was then computed using Eq. 5 with Δεeq,trans=0.92% for the aluminium alloy and 0.45% for steel. For each node, the damage function was computed, using the local value of b, along all potential twisted planes, for a twist angle θ ranging from 0 to 45°. The value of θ corresponding to the maximum of DF was considered as the local direction of crack extension. Here again stresses and strains were averaged over a distance of 90μm. This arbitrary distance, which influences the predicted crack paths should be considered as an adjustable parameter. 5. Numerical analysis of the experimental results
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