Pole instability theory for ellipsoidal bulging of rolled sheet

Abstract

Abstract The circular bulge test has long been used for assessing sheet metal formability under in-plane biaxial tension. The limiting formability is identified with pole failure under a pressure maximum. Theoretical solutions, which apply to bulging isotropic sheet, identify the sub-tangent to an equivalent stress-strain curve with an instability point corresponding to the maximum pressure. When this pressure is coupled with suitable description of the equivalent flow curve, it is possible to determine the in-plane limiting strains. A theory of instability is outlined for ellipsoidal bulging of rolled orthotropic sheet metals, e.g. the CR steels and 6000 series aluminium alloys used in the forming industry. The theory provides the limiting strains, height and peak pressure for a pole failure when a diffuse instability condition applies to these materials. Predictions to the pressure versus bulge height curves are compared with experimental results for an automotive, zinc clad, steel sheet. A plane strain rim failure, where it occurs, is also identified with a rising pressure along this curve. A general approach is adopted in terms of the die axis length ratio, the sheet's r -values, the Hollomon hardening constants an any inclination of the die's minor axis to the material's rolling direction. The theory of instability requires equivalence in the flow behaviour and this has been found to apply more consistently than was previously reported by employing larger dies within a modified experimental procedure.