Localized necking predictions for an imperfect sheet using a porous plastic constitutive relation with two porosity parameters

Abstract

The role of void nucleation and void growth in triggering localized necking in biaxially stretched sheets is investigated using a rate independent porous plastic constitutive relation with two porosity parameters; one associated with the void volume fraction and the other associated with the weakening effect of void shape changes in shear dominated stress states. Proportional straining plane stress calculations are carried out for ratios of imposed in-plane principal strain rates ranging from $\approx -1$ (shear dominated) to 1 (equal biaxial tension). The framework for the localized necking calculations is that in which an imperfection band triggers the onset of localized necking as defined by a loss of ellipticity of the governing equations in the imperfection band. The imperfection band is taken to be either an increase in initial void volume fraction or an increase in the volume fraction of void nucleating particles. The predicted forming limit curves are compared with predictions for a localized necking bifurcation of a rigid-plastic solid. For negative values of the imposed strain ratio, except for near in-plane shear where the second porosity reduces the critical strains, the predicted critical localization strains and the predicted critical localization band orientations differ little from the corresponding critical values predicted by a rigid plastic bifurcation analysis. For biaxial tensile states the critical localization strains are sensitive to the nature and magnitude of the imperfection. When void nucleation occurs over a very narrow range of strain or stress, void nucleation and the onset of localized necking can coincide.