A damage model for ductile porous materials with a spherically anisotropic matrix

In the present study, we investigate the macroscopic strength of ductile porous materials having a Hill-type radial anisotropic matrix. The procedure is based on a limit analysis (LA)-based kinematic approach of a rigid plastic hollow sphere. We first established the exact solution (stress and velocity fields) to the problem of the hollow sphere subjected to an external hydrostatic loading. Then, we propose, for general loadings, an appropriate trial velocity field which allows to implement the kinematic LA procedure. The resulting macroscopic criterion, whose closed-form expression is provided, extends the well-known Gurson criterion to materials with radial anisotropy. Numerical limit analyses are provided by performing standard finite elements computations which validate the new criterion. Finally, the yield criterion is supplemented by a plastic flow rule and evolution equations of the internal parameters, allowing to study the predictions of the complete model for axisymmetric proportional loadings at fixed stress triaxiality. A strong influence of the radial anisotropy is observed on the stress softening and the growth of the porosity.