Calculations of forming limit diagrams
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INTRODUCTION
SINCE Keeler ",2j introduced the concept of forming limit diagrams (FLDs), they have been a valuable tool in the analysis of metal forming in press shops. The FLD separates the strains which lead to failure from those which do not. The experimental methods of constructing FLDs are well established, ~,2,3J although there are differences between in-plane loading (in which there is no tool contact) and forming over tools where there is both friction and bending J 4~ Beginning with the early paper of Marciniak and Kuczynski (MK)/51 numerous attempts have been made to predict FLDs analytically, taking into account the strainhardening exponent, n, the strain-rate sensitivity, m, and the strain ratio, R, value. For the left-hand side of the FLD (negative minor strains), the analyses have been q u i t e s u c c e s s f u l , [6,71 with failure occurring at a critical thickness strain, e3, that depends on m and n but is independent of R and the yield criterion. However, on the right-hand side of the FLD (positive minor strain), there are large discrepancies between the shapes of calculated FLDs and those obtained experimentally, except in a few special cases. Marciniak and Kuczynski tS~ pointed out that for localized necking to occur in biaxial stretching, the local stress state must change from biaxial stretching to plane strain. Sowerby and Duncan ts] argued that the difference between these two stress states depends on the shape of the yield locus and the corresponding yield criterion. They used Hill's 1948 yield criterion t91 in their calculations, and their results show a very strong dependence of the right-hand side of the FLD on the R value. Others I~~ have calculated a similar R dependence of the FLD. Figure 1, calculated here, shows this effect and is almost identical to an FLD calculated by Parmar and Mellor. II~ It should be noted that such a dependence of FLD on the R value has not been observed experimentally. I131 Calculations based on the von Mises criterion also tend to overestimate the limit strains under biaxial tension. In fact, it has been suggested II41 that "flow theory" should
ALEJANDRO GRAF, Graduate Student, currently on leave from the Instituto Argentino de Siderurgia, Carlos Maria Della Paolera 226, Buenos Aires 1104, Argentina, and WILLIAM F. HOSFORD, Professor, are with the Department of Materials Science and Engineering, Dow Building, University of Michigan, Ann Arbor, MI 48109-2136. Manuscript submitted March 21, 1989. METALLURGICAL TRANSACTIONS A
be abandoned in favor of "deformation theory" in these calculations. Other yield criteria have been used in the calculations of FLDs. Parmar and Mellor tl~ used case 4 of Hill's 1979 yield criterion; t~51however, the influence of R cannot be evaluated with this criterion, because as the R value is varied, the exponent in the yield criterion must also be changed to maintain reasonable yield locus shapes. Barlat tl61 has calculated FLDs for isotropic materials using a modified von Mises criterion tlT,~8,191 with a high exponent and
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