Prediction of Forming Limit Diagrams in Sheet Metals Using Different Yield Criteria
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INTRODUCTION
THE forming limit diagram (FLD) is a useful tool in the analysis of metal forming in press shops. Its application for determining the materials stretchability and troubleshooting in the sheet metal stamping processes has caused a great deal of interest in both academia and industry. Over the last 40 years, since Keeler[1] introduced the concept of the FLD, the experimental procedures for constructing FLDs have become well established.[1–3] However, different approaches have been used in modeling the stamping process to predict the FLDs analytically and numerically. The models can be considered in two main categories: macro- and microstructural views. Although they are not totally independent from each other, they are distinguishable from deformation localization and failure mode view points. Different models such as Marciniak–Kuczynski (MK)[4] and crystal plasticity[5–10] are considered in the literature. In the MK model, the assumption of the existence of an initial imperfection is necessary. In the crystal plasticity models, however, reflections of crystals to plastic deformation are related to final failure in sheet metal forming. One of the most common approaches to analyze the limit strain and H. NOORI, Research Assistant, and R. MAHMUDI, Professor, are with the School of Metallurgical and Materials Engineering, University of Tehran, Tehran, Iran. Contact e-mail: [email protected] Manuscript submitted January 10, 2007. Article published online July 26, 2007. 2040—VOLUME 38A, SEPTEMBER 2007
plastic instability has been proposed by Jones and Gillis (JG).[11–13] Their analysis is based on the modeling of the easily observable features from a tensile test of a typical sheet metal. The capability of this model to predict the FLDs has been discussed in some articles. Jones and Gillis[14] and Choi et al.[15,16] applied the JG model using a generalized anisotropic yield criterion.[14] Later, Pishbin and Gillis[17] used the Hills nonquadratic flow law (case iv) for sheets having planar isotropy to calculate the FLDs. More recently, Aghaie et al.[18] applied the Hosford criterion as a special case of the Hills nonquadratic flow law in the prediction of the FLDs for materials obeying the power law and Voce constitutive equations. In the present work, Hills 1948,[19] Hills 1979,[20] and Hosords 1979[21] yield criteria are used in conjunction with the JG model. These yield criteria have different approaches to the anisotropy phenomenon, and investigation of their capabilities in the prediction of the FLDs based on the JG analysis is worthwhile. A new formulation is proposed that makes the calculation easier and introduces the JG model in a more usable fashion than before.
II.
JG ANALYSIS
According to this model, the plastic deformation is approximated by three phases, as schematically shown in Figure 1: METALLURGICAL AND MATERIALS TRANSACTIONS A
Fig. 1—Schematic representation of the three deformation phases. Points H and J are, respectively, denoted as the end of homogeneous deformation (phase I) and the d
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