A Stress-Based Variational Model for Ductile Porous Materials and Its Extension Accounting for Lode Angle Effects

Abstract

The aim of this work is to derive by homogenization techniques a macroscopic plastic model for porous materials with von Mises matrix. In contrast to the Gurson’s well known kinematical approach [19] applied to a hollow sphere, the proposed study proceeds by means of a statical limit analysis procedure, for which a suitable trial stress field is proposed. In the first part, the formulation of the stress variational model is developed, by considering the Hill’s variational principle, and introducing a Lagrange’s multiplier to solve the resulting saddle-point minimization problem. This methodology being opposite to the Gurson’s kinematical approach, complements the limit analysis methods for porous materials. The second part is devoted to an application of the proposed approach to the porous materials with von Mises matrix. To this end, an axisymmetric model is first studied by adopting a suitable trial stress field, which is composed by a heterogeneous part corresponding to the exact solution of hydrostatic loading and a homogeneous part for capturing the shear effects. 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. Moreover, an extension of the above axisymmetric model to the general case of non-axisymmetric loadings by introducing a more general trial stress field is studied. The established new yield locus explicitly depends on the effect of the third invariant (equivalently the Lode angle). 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.