**Acknowledgement**

The author thanks the partial financial supports under the projects from the National Natural Science Foundation of China (NSFC Grant No. 50875161), Program for New Century Excellent Talents in University (Grant No. NCET-10-0567), the Research Fund of State Key Lab of MSV, China (Grant No.MSV-ZD-2010-02), and National Science and Technology Major Projecct, China (Grant No. 2010ZX04004-112).

**Chapter 8** 

© 2012 Signorelli and Bertinetti, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Signorelli and Bertinetti, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Self-Consistent Homogenization Methods for** 

Formability of sheet metals can be characterized by the forming-limit diagram (FLD). This concept has proved to be extremely useful for representing conditions leading to the onset of sheet necking (Hecker, 1975), and now is one of the best tools available to metallurgical engineers to assess a particular steel sheet's ability to be drawn or stretched. In a single diagram, the FLD represents all combinations of critical-limit surface strains corresponding to failure. Within the FLD, a line called forming-limit curve (FLC) separates the region of uniform sheet deformation from the region of slightly greater deformation, where the sheet will likely develop a local deformation instability or neck. Experimental measurement of the FLD is not an easy task, requiring a wide range of sample geometries and even more than one type of mechanical test. Also, many test factors measurably affect the limit-strain determination: friction conditions, small deviations in loading paths due bending effects, and strain-measurement procedures. Similarly, several physical factors related to material properties (e.g. plastic anisotropy, work hardening and strain-rate sensitivity) have an important influence in the development of localized necking or failure. Numerical simulation promotes a better understanding of deformation and failure in polycrystal sheet metal aggregates, by examining issues related to crystal anisotropy and stress / strain heterogeneity. Considerable effort has recently been made to develop theoretical models for predicting the FLD behavior. Most of them are based either on a bifurcation analysis (Storen & Rice, 1975) or a model where the strain instability appears in the deformation process due to an imperfection already present in the material (Marciniak & Kuczynski, 1967). The latter, MK from now on, has probably been the most widely used of the two techniques. Within the MK framework, the influence of various constitutive features on FLDs has been explored using phenomenological plasticity models and crystal plasticity. In recent years, research has shown that the localization of plastic flow is influenced by deformation anisotropy (Asaro & Needleman, 1985; Tóth et al., 1996; Wu et al., 2004a; Lee & Wen, 2006). Thus,

**Predicting Forming Limits of Sheet Metal** 

Javier W. Signorelli and María de los Angeles Bertinetti

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50662

**1. Introduction** 
