Multi-surface yield criterion for orthotropic porous materials

Abstract

The yield criterion for rolled sheet metals generally exhibits orthotropic behavior with respect to the rolling, transverse and thickness directions of the sheet. Formability of sheet metals is limited by the onset of a localized necking instability, which depends sensitively on the presence of vertices on the yield surface induced by microstructure changes, such as the evolution of material texture and/or micro-void growth. Prior studies on isotropic porous materials have shown that the transition from diffuse plasticity to localized yielding of the inter-void ligaments at the micro-scale can lead to the appearance of corners on the macroscopic yield surface. In this paper, we use a multi-surface approach to develop an effective yield criterion for plastically orthotropic materials of the Hill type containing a random distribution of equiaxed voids; by combining existing yield criteria from the literature accounting for the two alternative modes of yielding mentioned above. For finite values of the porosity, the resulting yield surface consists of alternating curved and flat segments with sharp corners at their intersection. The predicted shapes of the yield loci are validated by comparison with quasi-exact yield loci obtained from a numerical limit analysis procedure using finite elements. It is shown that the analytical yield loci are in very good agreement with the numerical loci over the experimentally observed range of material anisotropy parameters, particularly for the case of thin sheets loaded under plane stress conditions. The instantaneous void growth rate, computed using the microscopic velocity fields obtained from the numerical limit analysis, exhibits a non-monotonic variation with increasing stress triaxiality under plane stress conditions, which is predicted approximately by the multi-surface model.