A stress-based variational model for ductile porous materials

The main objective of this paper is to formulate a very new statically-based model of ductile porous materials having a von Mises matrix. In contrast to the Gurson’s well known kinematical approach applied to a hollow sphere, the proposed study proceeds by means of a statical limit analysis procedure. Its development and implementation require the choice of an appropriate trial stress field. The starting point is Hill’s variational principle for rigid plastic matrix. The use of a lagrangian multiplier allows to satisfy the plastic criterion in an average sense. The proposed trial stress field, complying with internal equilibrium, is composed of an heterogeneous part (exact solution for the stress field in the hollow sphere under pure hydrostatic loading) to which is added a uniform deviatoric stress field. Owing to this choice, the stress vector conditions on the void boundary are relaxed. By solving the resulting Saddle point problem, we derive closed form formula which depends not only on the first and second invariant of the macroscopic stress tensor but also on the sign of the third invariant of the stress deviator. The obtained results are fully discussed and compared to existing models, available numerical data and to Finite Elements results obtained from cell calculation carried out during the present study. Finally, we provide for the new model the macroscopic flow rule as well as the porosity evolution equations which also show very original features. Some of these features are illustrated.