About the mechanical data required to describe the anisotropy of thin sheets to correctly predict the earing of deep-drawn cups Stefan Soare 1 , Dorel Banabic 1 1 Technical University of Cluj-Napoca - C. Daicoviciu 15, 400020 Cluj-Napoca, Romania e-mail: Stefan.Soare@tcm.utcluj.ro; Banabic@tcm.utcluj.ro ABSTRACT: The nonuniform height profiles of deep-drawn cylindrical cups is caused by the plastic anisotropy of the blank sheet. Which locations on the yield surface determine uniquely the earing profile is still an unsolved problem. Present (phenomenological) theories take into consideration only the directional properties of the sheet (r-value and yield stress). We show here that these directional properties are not enough and that for a unique characterization of the earing profile, information about the pure shear yielding point must also be considered. KEYWORDS: deep drawing, earing profile, orthotropic polynomial yield function 1 INTRODUCTION One of the most interesting manifestations of the plas- tic anisotropy of metals is the phenomenon of earing of deep drawn cylindrical cups. The problem is well known: a disc shaped thin sheet is deformed into a cylindrical cup by pushing it into a die cavity with the help of a cylindrical punch with flat bottom, the die and the punch being coaxial. The geometry of the problem and the loading conditions have spherical symmetry. Yet, the height (profile) of the final deep- drawn cup is not uniform. The profile shows a series of maxima and minima commonly known as ears and troughs (hollows). To explain the location of the ears (and hollows) on the rim of the cup in the deep drawing prob- lem, Hill, [1], advances the hypothesis that, assum- ing isotropic hardening and plane stress conditions, the ears begin to develop at those locations on the rim where the strain increment has the radial direction as principal direction. Since at the rim only the hoop stress is nonzero (compressive), this hypothesis im- plies that ears begin to grow at those locations on the rim where the stress and strain increment tensors have the same principal directions. Hill then proves that those radial directions where the stress and strain in- crement are collinear are completely determined by the locations of the stationary points of the direc- tional yield strength. We note here that, due to the orthotropic symmetry of the blank, ears or hollows will always develop at 0 o , 90 o , 180 o , 270 o , 360 o from the rolling direction. For the intermediary locations, Hill’s theory will predict ears/hollows at 90 o - θ, etc, where θ ∈ (0 o , 90 o ) is a stationary point of the di- rectional yield stress. A small deviation from this for- mula, of 2.5 o is featured by the profile of AA2090-T3, see [2], where the intermediary stationary point of the directional yield stress is located at 57.5 o , whereas the intermediary ear is located at exactly 45 o . This indi- cates that the directional r-value (which in this case has θ = 45 o as stationary point) also might influence the earing profile. A more striking example in this di- rection is the earing profile of a hypothetic material with uniform directional yield stress and nonuniform directional r-value. In this case, Hill’s theory cannot specify the location, whereas the simulated profile has six ears, [2]. After careful examinations of many experimen- tal profiles, Yoon and his coworkers reach in [3] the conclusion that the earing profile is approximately the mirror image of the directional r-value. Based on this observation they propose an approximate formula for the height of the profile, formula in which the plas- tic properties of the sheet are modelled only through its r-value. Taking into account its simplicity, the for- mula seems to predict with reasonable accuracy the shape (topology) of the the profile in many cases of practical importance. However, a counterexample has been constructed in [2] where a hypothetical material with uniform directional r-value and nonuniform di- rectional yield stress is considered. FE simulations of the deep drawing of a cup from such material showed 1