Limit analysis and numerical modeling of spherically porous solids with Coulomb and Drucker–Prager matrices
Résumé
The first goal of this work was to develop efficient limit analysis (la) tools to investigate the macroscopic criterion of a porous material on the basis of the hollow sphere model used by Gurson, here with a Coulomb matrix. Another goal was to give the resulting rigorous lower and upper bounds to the macroscopic criterion to enable comparisons and validations with further analytical or numerical studies on this micro–macro problem. In both static and kinematic approaches of la, a quadratic formulation was used to represent the stress and displacement velocity fields, in triangular finite elements. A significant improvement of the quality of the results was obtained by superimposing, on the fem fields, analytical fields which are the solutions to the problem under isotropic loadings. The final problems result in conic optimization, or linear programming after linearization of the criterion, so as to determine the “Porous Coulomb” criterion. A fine iterative post-analysis strictly restores the admissibility of the static and kinematic solutions. After presenting the results for various values of the porosity and internal friction angle, a comparison with a heuristic Cam–Clay-like criterion shows that this criterion cannot be considered a precise general approximation. Then a comparison with the “Porous Drucker–Prager” criterion treated by specific 3D codes is presented. With the same numerical tools, a final analysis of recent results in the literature is detailed, and tables of selected numerical data are presented in the appendices.