A new analytical theory for earing generated from anisotropic plasticity

Commercial canmaking processes include drawing, redrawing and several ironing operations. It is experimentally observed that during the drawing and redrawing processes earing develops, but during the ironing processes earing is reduced. It is essential to understand the earing mechanism during drawing and ironing for an advanced material modeling. A new analytical approach that relates the earing profile to r-value and yield stress directionalities is presented in this work. The analytical formula is based on the exact integration of the logarithmic strain. The derivation is for a cylindrical cup under the plane stress condition based on rigid perfect plasticity while force equilibrium is not considered. The earing profile is obtained solely from anisotropic plastic properties in simple tension. The earing mechanism is explained from the present theory with explicit formulae. It has been proved that earing is the combination of the contributions from r-value and yield stress directionalities. From a directionality (y-axis) vs. angle from the rolling (x-axis) plot, the earing profile is generated to be a scaled mirror image of the r-value directionality with respect to 90° (x = 90) and also a scaled mirror image of the yield stress directionality with respect to the reference yield stress (y = 1). Three different materials (Al–5% Mg alloy, AA 2090-T3 and AA 3104 RPDT control coil) are considered for verification purposes. This approach provides a fundamental basis for understanding the earing mechanism. In practice, the present theory is also very useful for the prediction of the earing profile of a drawn and iron cup and its related convolute cut-edge design for an earless cup.