Effective behaviour of porous ductile solids with a non-quadratic isotropic matrix yield surface

Abstract

In this study, we examine the macroscopic yielding of isotropic porous ductile solids having a matrix yield function dependent on the second and third deviatoric stress invariants. Numerical limit analyses using a three-dimensional finite element model of a hollow sphere with a Hershey-Hosford matrix yield function are conducted for different shapes of the matrix yield surface and porosity levels. These numerical results are then used to elucidate first-order effects of the third deviatoric stress invariant on the macroscopic yielding and further used as reference data to assess the performance of two porous plasticity models that incorporate effects of the third deviatoric stress invariant using the isotropic non-quadratic Hershey-Hosford yield function. The first model is derived from an upper-bound limit analysis of the hollow sphere representative volume element using the Gurson-Rice trial velocity field, but with a rather general isotropic matrix yield function. The second model is a simple, heuristic extension of the Gurson model incorporating the equivalent stress measure of the Hershey-Hosford yield function. From the numerical limit analyses, it is found that the contours of the macroscopic yield surface in the deviatoric plane transform from the hexagonal shape of the underlying matrix yield surface to a rounded triangular shape that converges to the circular shape of the Gurson model as the macroscopic stress triaxiality ratio increases. This shape transformation is dependent upon the porosity level. The upper-bound model was found to be in very good agreement with the numerical data for all stress states, shapes of the matrix yield surface, and porosity levels. The heuristic model provides good predictions for low and moderate levels of porosity pertinent to ductile fracture, but the predictions deteriorate when the stress triaxiality ratio and the porosity level increase. We also address the issue of how representative the spherical unit cell is for the description of real porous solids. To that end, we make comparisons between a space-filling representative volume element in the form of a cubic unit cell with a centric spherical void and the hollow sphere model. These results show that the hollow sphere model generally provides slightly higher values for the yield limits. The shape of the yield loci is similar for the two models in the case of non-quadratic matrix yield surfaces, while the cubic model gives a different shape of the yield loci for low and intermediate stress triaxiality ratios when the von Mises yield function is used for the matrix.