Prediction of the forming limit diagrams of anisotropic sheets in linear and non-linear loading

Abstract

The formability of sheet metals is often evaluated from strain analysis using the concept of forming limit diagrams (FLDs). Numerous experimental results show that the formability of the material strongly depends on the loading history and on the plastic anisotropy of the rolled sheet. The limiting strains at the onset of localized necking can be predicted from a plastic instability analysis introduced by Marciniak and coworkers. However, the hypothesis of proportional loading is no longer valid for complex industrial stampings. Multistage forming operations often involve non-linear strain paths, and abrupt changes in the strain ratio can be observed. In the present work the occurrence of plastic instabilities in the general case of a rate-sensitive anisotropic material with orthotropic symmetry under non-proportional loading is analysed. Theoretical FLDs are determined for proportional loading, for a sequence of two proportional loadings, for a sequence of two proportional loadings and for non-proportional loading (curved strain paths). The simulations are carried out for a wide range of loading conditions, from uniaxial tension in the rolling direction to uniaxial tension in the transverse direction through biaxial stretching. The influence of the anisotropic behaviour of the sheet is studied using Hill's theory of plastic anisotropy for both linear and non-linear strain paths in terms of the anisotropic coefficient measured during uniaxial tensile stretching in three different directions referred to the rolling direction (r o, r 45 and r 90). A good correlation is obtained between the theoretically and experimentally determined FLDs, and it is shown that important increases in formability may be achieved through careful specification of strain paths, blank-holder pressure and anisotropic blank orientation.