Finite element modeling simulation of in-plane forming limit diagrams of sheets containing finite defects
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I.
INTRODUCTION
FLOWlocalization
during sheet stretching limits metal formability. A sheet necks and eventually fails in locations where critical limit strains are exceeded. A representation of all combinations of such critical major and minor strains gives rise to a forming limit diagram (FLD). The concept of forming limits was first introduced by Keeler and Backofen m and Keeler, t2,31 and the standard form of the FLD was later presented by Goodwin. [4] Hecker tS] presented a detailed procedure to measure the FLD from large negative to large positive minor strains. Azrin and Backofen [6] introduced a new testing technique to determine the in-plane FLD. The current state of art on both the experimental and theoretical FLDs is reviewed in a recent book edited by Wagoner et al.[7] Theoretical calculations of the FLDs were initially based on Hill's criterion for localized necking along a direction of zero extension, t8] Hill's criterion does not allow for localized necking of materials with smooth yield surfaces under biaxial stretching (e2 > 0) conditions. Marciniak and Kuczynski (M-K) [91 and Marciniak et al. ,[10] by introducing a thickness imperfection of infinite length normal to the principal stress, developed the first analytical model to predict localized necking in biaxial stretching of sheets. They showed that the presence of even slight intrinsic inhomogeneities in load bearing capacity throughout a deforming sheet can lead to unstable growth of strain in the weaker regions and subsequently lead to localized necking and failure. Since then, several researchers have used M-K analysis to predict localized necking during biaxial stretching (with e2 > 0). The K. NARASIMHAN, Postdoctoral Researcher, and R.H. W A G O N E R , Professor, are with the Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210. Manuscript submitted November 2, 1990. METALLURGICAL TRANSACTIONS A
variants of M - K analysis for predicting localized necking have been reviewed in the past. [11,L2,t3] In establishing the original theory, M - K [91 considered a material element with uniform mechanical properties but with a notch in thickness extending across it, along the minor principal stress direction, X2, as represented in Figure 1. The geometrical notch serves as a mechanical analog for a hypothesized initial local weakness. The ratio of the thickness of the notch to that of the bulk was defined as the weakness factor, f. thickness of notch f =
thickness of bulk
[1]
The source of an intrinsic inhomogeneity in a real material is not clear, but suggestions have been made relating it to material property v a r i a t i o n s , [l~ local prestrains, thermal notches, and thickness v a r i a t i o n s . B4] In real sheet forming applications, strain localization can be initiated and developed without material or thickness inhomogeneities. Friction and contact conditions existing during the sheet forming control the development of nonuniform strain distribution and the eventual strain localization pr
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