**1. Introduction**

174 Metal Forming – Process, Tools, Design

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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,

© 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.

crystal-plasticity models should provide a framework for better understanding the relation between flow localization and material microstructure. Issues such as yield-surface shape – changes of sharpness – material anisotropy – crystal reorientation – are directly addressed within a polycrystalline model. It is widely recognized that the crystallographic texture strongly affects forming-limit diagrams and the macroscopic anisotropy of polycrystalline sheet metals. Numerous authors have adopted the MK model in conjunction with a crystal plasticity model to describe strain localization in rolled sheets (Kuroda & Tvergaard, 2000; Knockaert et al., 2002; Wu et al., 2004a; Inal et al., 2005; Yoshida et al., 2007; Neil & Agnew, 2009). Based on this strategy, the authors have examined how plastic anisotropy influences limit strains (Signorelli et al., 2009). For the FLD simulations, crystallographic effects were taken into account by combining the MK approach with a viscoplastic (VP) self-consistent (SC) and a Full-Constraint (FC) crystal-plasticity model, MK-VPSC and MK-FC respectively.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 177

τ

r 12 i j ij *τ σ* = = ′ *b n σ* (1)

, (2)

. This yield criterion is

when the shear stress on a slip system reaches a critical value *<sup>c</sup>*

between the tensile axis and the normal to the slip plane.

planes meet the crystal surface.

called *Schmid´s Law*. In most crystals slip can occur either in the **b** or –**b** direction.

Figure 1 shows a slip system represented by the vectors **n** and **b.** Suppose that the crystal has a general state of stress ij *σ* acting on it referenced to the coordinate system **S** (**S** is fixed to the sample). The shear stress 12 *σ*′ acting on the slip system can be obtained by transforming the stress tensor ij *σ* from the **S** to the **S**' system (**S**' is fixed to the slip system). Using the typical equations for tensor transformation, the resolved shear stress acting on the slip system is:

If the crystal is loaded in tension along the X3 axis, the shear stress acting on the slip plane is

<sup>r</sup> *τ σ* = *cos cos* λ

**Figure 1.** A schematic diagram of slip in the direction **b** occurring on a plane with the normal **n**.

In FCC materials, the crystallography of slip is simple, it takes place on the most densely packed planes {111} and in the most densely packed directions <110>. In BCC metals, the most common mode of deformation is {110}<111>, but these materials also slip on other planes: {112} and {123} with the same slip direction. Plastic deformation occurs by 12 crystallographic slip systems of the type {111}<110> for FCC metals and 48 slip systems of the type {110}<111>, {112}<111> and {123}<111> for the BCCs (see Table 1). A slip line is the result of a displacement of the material along a single lattice plane through a distance of about a thousand atomic diameters. The slip lines are visible traces of slip planes on the surface, and they can be observed when a metal with a polished surface is deformed plastically. As an example, in the optical micrograph shown in Figure 2, the slip bands appear as long steps on the surface. The terraced appearance is produced when the slip

where λ is the angle between the slip direction and the tensile axis, and φ is the angle

φ

In this chapter we will analyze the influence that the numerous microstructural factors characterizing metals have on forming-limit strains. Moreover, we will focus on the consequences that selecting either a FC or SC type grain-interaction model has on numerical results. We will start, in the following section, with a brief description of the texture and anisotropy of cubic metals. The representation of crystallographic texture and the determination of the polycrystal texture are addressed. The material´s plastic deformation as a result of crystallographic dislocation motion on the active slip systems is discussed at the end of the section. The single crystal properties and the way in which grains interact in a polycrystal are the subject of Section 3. An outline of the implementation of the VPSC formulation in conjunction with the well-known MK approach for modeling localized necking closes the section. A parametric analysis of the influence of the initial-imperfection intensity and orientation, strain-rate sensitivity and hardening on the limit strains is the content of Section 4. In Section 5 the MK-FC and MK-VPSC approaches will be examined in detail. FLDs will be predicted for different materials in order to clearly illustrate the differences between the FC or the VPSC homogenization schemes, particularly in biaxial stretching.
