Sheet orientation and forming limits under diffuse necking

Abstract

This paper examines the influence of orientation (θ) between the principal stress axes and the material axes when modelling the forming limits for a rolled sheet. Stress and strain transformation equations are coupled with orthotropic plasticity theory 1 when the stress ratio remains constant. Three limit strains, two normal and one shear, appear in the material axes at diffuse instability. These strains define a forming limit diagram (FLD) which rotates, with increasing θ, about its pole. Shear strains reach a maximum for θ = 45 ° but are absent for θ = 0 and 90 °. A simpler plane presentation of the FLD appears in axes of major and minor principal plastic strains. This shows that the sensitivity of the FLD to orientation depends upon the r-values. Examples are given of 1. (i) a CR steel whose r-values exceed unity but where r 1 ∼- r 2, i.e., where only a slight deviation exists between the principal axes of stress and strain and 2. (ii) a 6000 aluminium alloy whose r values are different and less than unity. The observations made provide a mathematical model of the experimental Keeler-Goodwin FLD 2,3. The use of thickness strain for one axis of the FLD is also considered for when ultrasonic thickness monitoring is more convenient than surface strain measurement. Fracture strain data from bulge forming, punch indentation, and Ericksen tests fall around the limit lines predicted by a diffuse instability model in the region of most interest to users.