Forming severity concept for predicting sheet necking under complex loading histories

A new method of predicting neck formation in sheets under non-proportional loading is proposed, based on the concept of “cumulative forming severity”. This concept is borrowed from a macroscopic model of ductile fracture where the crack initiation is governed by the accumulated equivalent plastic strain modified by the stress triaxiality and the Lode angle parameter. Such an approach necessitates a representation of the forming limit diagram (FLD) in the space of the equivalent strain to neck and the Lode angle parameter.Another new factor is the assumption of the non-linear accumulation of forming severity for non-proportional and complex loading histories. A class of non-linear weighting function is proposed with only one free parameter. A starting point in the derivation is the known FLD corresponding to proportional loading. This can be determined from Hill's and Stören and Rice analytical solutions, from numerical simulation, or else taken directly from experiments. In the case of proportional loading, necking depends on the final state of stress or strain, so it does not matter if necking severity index is accumulated in a linear or non-linear way. For non-proportional loading, the unknown free parameter of the non-linear accumulation rule must be determined from a test.Experimental data on FLDs under complex strain paths for two types of material, aluminum alloy 6111-T4 [Graf A, Hosford W. The influence of strain-path changes on forming limit diagrams of A1 6111 T4. International Journal of Mechanical Sciences 1994;36(10):897–910.] and aluminum-killed sheet steel [Muschenborn W, Sonne HM. Influence of the strain path on the forming limits of sheet metal. Archiv fur das Eisenhuttenwesen 1975;46:597–602], found in the literature are revisited by the proposed model. Calibrated from only one test with non-proportional loading condition, the model is able to predict the remaining tests of complex loading paths with good accuracy.