An assessment of the role of the third stress invariant in the Gurson approach for ductile fracture

• We introduce consistently Lode angle in Gurson's analysis.
• We improve predictions of the Gurson approach at Small triaxialities by using a new velocity trial field.
• We assess the effects of Lode angle in the yield criterion.
• We compare all results to simulations based on limit analysis.

The present paper is concerned with the effects of the Lode angle (or the third stress invariant) in the yielding of porous materials. This is addressed in the framework of Gurson's analysis of voided materials. It is shown first that without the approximations operated by Gurson, the Lode angle of the macroscopic strain rate is naturally involved in the analysis and consequently the third stress invariant affects the yield criterion. Pushing forward Gurson's analysis without his approximations and still considering his axisymmetric trial velocity field, changes on the predicted yield locus are observed but only in the intermediate range of triaxialities. The same predictions are obtained for very small and very large stress triaxialities. A more carefull inspection of the results shows in fact that most of the changes are second order effects and direct effects of the Lode angle are hardly visible.Pursuing the analysis with a new class of trial velocity fields due to Si et al. (2007) (containing the Gurson field) and including non-axisymmetric contributions, changes on the prediction of the yielding behaviour are observed at the low triaxiality regime. Again, effects of the Lode angle are seen to be rather small at least in the range of porosities pertinent to ductile fracture. However, inclusion of Lode angle brings changes on the shape but also on the symmetries of the yield domain. The exact hydrostatic prediction obtained in the Gurson model is also maintained here. Comparison with the numerical simulations presented by Thoré et al. (2011) using optimization algorithms in the context of limit analysis show a very good agreement with the results and in particular that these results fall well between the static and kinematic limit analysis bounds that they furnished.