Earing in cup drawing face-centered cubic single crystals and polycrystals
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I.
INTRODUCTION
S E V E R A L attempts have been made to predict earing tendency during cup drawing from known properties of the initial sheet material. Hill tl] first proposed an explanation for ear and trough positions based on his anisotropic yield function t2] and on the assumption of no hold-down pressure (plane stress) or no thickening of the blank (plane strain). Sowerby and Johnson ]3] used Hill's yield function with the plane strain assumption to predict earing using the slip-line field theory. Lee and Kobayashit4] and Gotoh and Ishis6 tSl used t-mite element methods based on Hill's yield function and a fourth-degree polynomial yield criterion, respectively. But all of these methods rely too much on postulated yield conditions which are not very accurate and consequently lead to somewhat crude results. In order to take better account of the microstructure of metals, models based on the plastic slip of a single crystal or of a polycrystal have been developed. Tucker's approach, t6] based on the Schmid law for a face-centered cubic (fcc) single crystal, is well known and is in reasonable agreement with experiments. Kanetake et al. ]7] applied Tucker's work to the polycrystal case. Numerous studies have used the TaylortS]/Bishop and Hill r9] (TBH) polycrystal model to describe the material behavior and some assumptions concerning the state of stress under the blank holder to predict the earing p r o f i l e . [ 1~
Recently, Panchanadeeswaran e t a l . tls] proposed a simple model of earing based on the TBH theory. They used this model to predict the earing profile in cups of high-purity aluminum processed with various amounts of cold rolling. They obtained theoretical earing profiles in good agreement with experimental ones for sheets with a high reduction. However, the agreement was not very good for sheets with a low reduction. In the last section of the present article, we show that these predictions can F. BARLAT, Staff Scientist, S. PANCHANADEESWARAN, Senior Engineer, and O. RICHMOND, Corporate Fellow, are with Alcoa Laboratories, Alcoa Center, PA 15069. Manuscript submitted March 19, 1990. METALLURGICAL TRANSACTIONS A
be substantially improved when suitable assumptions concerning the stress and strain state under the blank holder are made. But we first describe the model and the associated assumptions employed and show the relationship between yield surface shape and earing tendency. II. S I M P L E M O D E L FOR EARING PREDICTION
In this work, the material is assumed to possess orthotropic symmetry with principal axes, (rolling, transverse, and normal directions) denoted by x, y, and z. The plastic behavior of the material is described by the yield surface and the associated flow rule. Now, considering the cup drawing operation, we assume that ears start to form in the flange and that the final positions of ears and troughs coincide with initial positions. This assumption is supported by many experimental observations. Therefore, one needs only to analyze the state of stress and strain at the begi
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