AN ANALYSIS OF LOCALIZED NECKING IN BIAXIALLY STRETCHED SHEETS

Localized necking of sheet metals subject to out-of-plane punch stretching is studied. Attention is directed to the effects of various constitutive laws on the development of a localized neck. An axisymmetric finite element analysis, within the framework of membrane theory, is carried out of the hemispherical punch stretching of a circular sheet. Three rate-independent material models are considered, namely, classical plasticity theory with a smooth yield surface of von Mises kind, a deformation theory model of a solid with a vertex on its yield surface, and a plastically dilating constitutive relation with pressure dependent yielding, that models, approximately, ductile rupture on the microscale. Based on the numerical results, forming limit curves are determined for each of these constitutive models. In contrast to in-plane stretching, the geometric effects of out-of-plane stretching makes the formation of a localized thickness trough possible, even if the classical plasticity theory is employed. For the deformation theory model, the out-of-plane forming limit curves are shown to coincide with those for in-plane stretching. However, in out-of-plane stretching the loading path in the sheet can differ substantially from proportional loading so that the appropriateness of a deformation theory model of a more sophisticated flow theory that develops a vertex on its yield surface is questionable. The shape of the forming limit curves obtained with the model of a progressively cavitating dilational solid depends on the localized necking criterion employed. If the development of a thickness trough is the only criterion, then the forming limit curves obtained are virtually identical to the corresponding flow theory curves. If a ductile rupture criterion, which limits the maximum volume fraction of voids, is adopted the forming limit curves can differ significantly, in shape, from those for the other two models.; The effects of void nucleation occurring during the deformation history on forming limit curves are considered for both in-plane and punch stretching employing a constitutive model of a porous plastic solid. Both plastic strain controlled and stress controlled nucleation processes are simulated by a two parameter void nucleation criterion. For in-plane stretching, plastic strain controlled nucleation can have, in certain circumstances, a significantly destabilizing effect on the forming limit curve. However, within the framework of plane stress theory which neglects the enhancement of the hydrostatic stress due to necking, a stress controlled nucleation process is not found to be significantly destabilizing. In punch stretching a ductile rupture criterion, which limits the maximum volume fraction of voids, as well as the appearance of a well defined thickness trough, is adopted as a localized necking criterion. Only plastic strain controlled void nucleation is considered here in out-of-plane stretching. The resulting forming limit curves have the same shape as those obtained previously with void nucleation neglected.