Theoretical model for forming limit diagram predictions without initial inhomogeneity

We report on our attempts to build a theoretical model for determining forming limit diagrams (FLD) based on limit analysis that, contrary to the well-known Marciniak and Kuczynski (M-K) model, does not assume the initial existence of a region with material or geometrical inhomogeneity. We first give a new interpretation based on limit analysis for the onset of necking in the M-K model. Considering the initial thickness defect along a narrow band as postulated by the M-K model, we show that incipient necking is a transition in the plastic mechanism from one of plastic flow in both the sheet and the band to another one where the sheet becomes rigid and all plastic deformation is localized in the band. We then draw on some analogies between the onset of necking in a sheet and the onset of coalescence in a porous bulk body. In fact, the main advance in coalescence modeling has been based on a similar limit analysis with an important new ingredient: the evolution of the spatial distribution of voids, due to the plastic deformation, creating weaker regions with higher porosity surrounded by sound regions with no voids. The onset of coalescence is precisely the transition from a mechanism of plastic deformation in both regions to another one, where the sound regions are rigid. We apply this new ingredient to a necking model based on limit analysis, for the first quadrant of the FLD and a porous sheet. We use Gurson's model with some recent extensions to model the porous material. We follow both the evolution of a homogeneous sheet and the evolution of the distribution of voids. At each moment we test for a potential change of plastic mechanism, by comparing the stresses in the uniform region to those in a virtual band with a larger porosity. The main difference with the coalescence of voids in a bulk solid is that the plastic mechanism for a sheet admits a supplementary degree of freedom, namely the change in the thickness of the virtual band. For strain ratios close to the plane-strain case the limit-analysis model predicts almost instantaneous necking but in the next step the virtual band hardens enough to deactivate the localization condition. In this case we apply a supplementary condition for incipient necking similar to the one used in Hill's model for the second quadrant. We show that this condition is precisely the one for incipient bifurcation inside the virtual (and weaker) band. Finally we discuss some limitations, extensions and possible applications of the new necking model based on limit analysis.