Calculations of forming limit
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I.
INTRODUCTION
S I N C E the article by Marciniak and Kuczynski t~} (M-K), there have been many analytical predictions of forming limit diagrams (FLDs) following their approach.t2-1u The M-K analysis assumes that an initial defect in the sheet, in the form of a long groove, grows and eventually fails during stretching along a linear strain path in the surrounding material. The M-K calculations have been quite successful in predicting the left-hand side of the FLDs. However, there are discrepancies between the shapes of the experimental and calculated FLDs for the right-hand side (RHS), with most predictions showing a strong dependence on the value of the strain ratio R, which is not observed experimentally, j~2j Sowerby and Duncan, t6i and later Chan, tS) explained the influence of the yield function used on the shape of the predicted FLDs. Lian et al. 19j showed that the shape of the FLD is affected by the ratio p = ~p/crb, where o-r and o'b are the major principal stresses for plane-strain and biaxial tension, respectively. Hill's 1948 yield criterion I]3}predicts that p changes quite drastically with R. Recent articles by the authors I~0.J]l showed good agreement of calculations using a high-exponent yield criterion []41 in the M-K analysis with experimental FLDs for different materials. During an actual forming operation, a material element may undergo considerably large changes in strain path, and these changes can significantly alter the forming limits. An experimental study of the effects of such path changes on the FLDs of A1 2008 T4 was reported in a companion article. The ability to include path changes in FLD calculations is important, because the number of potentially significant changes is too great to be thoroughly covered by experiments and because calculations allow general trends to be explored over a large range of variables. There have been a number of calculations using the M-K analysis of the effect of strain path changes on forming limits. 12'3"71 Unfortunately,
ALEJANDRO F. GRAF, is with Automated Analysis Corp., Ann Arbor, MI 48104-6767, and WILLIAM F. HOSFORD, Professor, is with the Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109-2136. Manuscript submitted December 10, 1992. METALLURGICAL TRANSACTIONS A
these were all done using Hill's yield criterion, and the results exhibit the same R-value dependence mentioned before. This article discusses the incorporation of abrupt as well as gradual strain-path changes into FLD calculations using a high-exponent yield function. The calculations are applied to AI 2008 T4 alloy and compared with experimental FLDs.
II. A.
Calculation
T H E O R E T I C A L ANALYSIS Method
The RHS of the FLDs was calculated with the M-K analysis, modified to incorporate strain-path changes. It is assumed that there is an initial imperfection in the form of a long groove oriented perpendicular to the direction of the largest stress (Figure 1). Although an infinitely long trough is unrealistic, the results of finite elem
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