Crystallographic analysis of the influence of stress state on earing behavior in deep drawing of face-centered cubic met
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I.
INTRODUCTION
It has been well established that earing behavior in deep drawing of a circular blank is initially caused by crystallographic texture; therefore, many researchers have attempted to predict earing behavior by use of crystallographic methods in order to effectively suppress earing.[1–13] These methods provide more satisfactory predictions than the continuum approaches, which are based on some phenomenological plastic potentials.[14–19] With the advantage of explicit mathematic form and less time needed for calculation compared with the methods based on Taylor/Bishop-Hill’s theory,[1–6] the Tucker method has been exploited in detail and widely used in both single crystals and polycrystals.[7–13] In these investigations, the predicted ear profiles generally were in good agreement with those observed. By introducing some adjustable parameters, the predicted cup profiles also agreed well with those measured. However, these parameters were not taken from experimental data and used just for a best fit between the predictions and measurements.[10–13] These agreements, therefore, seem to be unrealistic. Moreover, all of these predictions were carried out by assuming that the stress state in the flange, another important factor which may influence the earing behavior, is a plane stress state with the ratio of radial to circumferential stress simply to be unity. Recently, Inagaki[8] has studied the influence of stress state on earing and concluded that with increasing stress ratio, the earing extent increases significantly, while ear positions alter slightly. However, the total radial strain Inagaki calculated and concluded from is related with the stress level, which cannot be determined theoretically by use of S.Y. LI, Doctoral Candidate, and X.M. ZHANG, Professor, are with the Department of Materials Science and Engineering, Central-South University of Technology, Changsha, 410083 China. H.Z. ZHOU, Senior Engineer, is with the Southwest Aluminum Fabrication Plant, Chongqing, 631326 China. G. GOTTSTEIN, Professor, is with the Institut fu¨r Metallkunde und Metallphysik, 52074 Aachen, Germany. Manuscript submitted August 2, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS A
the crystallographic method (cf. Eq. [6] in Reference 8). As a result, the variation of total radial strain with different stress ratio in Inagaki’s investigation includes the interference of stress level. Therefore, the validity of his conclusion is extremely questionable. In this article, taking the uncertainties as previously stated into account, a more realistic representation of the anisotropic deformation along the radial direction of the blank to be deep drawn is introduced into the original Tucker method. Then, the influence of stress state on earing behavior is analyzed in detail. Finally, a simple method for industrial application is proposed and used to predict the cup profiles for several aluminum single crystal and polycrystal sheets. II.
THEORY
A. Basic Assumptions In the calculation of radial strain induced by each slip sys
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