**2. Texture and anisotropy of cubic metals**

Plastic anisotropy of polycrystals arises from crystallographic texture. In a material with a plasticity-induced texture, anisotropy at the microscopic level is determined by the different ways in which the material is deformed. In metals, plastic deformation occurs by crystallographic slip, due to the movement of dislocations within the lattice. In general, slip takes place on the planes which possess the highest atomic density, *slip planes*, and in the most densely packed directions, *slip directions*. The slip plane is characterized by the unit vector **n**, which is normal to the plane, and the slip direction, represented by the unit vector **b** (Burger's vector). The combination of both vectors, which are perpendicular to each other, defines a *slip system*.

Since crystallographic slip is limited to certain planes and directions, the applied stress required to initiate plastic flow depends on the orientation of the stress relative to the crystallographic axis of the crystal. If the plane is either normal or parallel to the applied stress, the shear stress on the plane is zero and no plastic deformation is possible. Slip begins when the shear stress on a slip system reaches a critical value *<sup>c</sup>* τ . This yield criterion is called *Schmid´s Law*. In most crystals slip can occur either in the **b** or –**b** direction.

176 Metal Forming – Process, Tools, Design

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. 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.

Plastic anisotropy of polycrystals arises from crystallographic texture. In a material with a plasticity-induced texture, anisotropy at the microscopic level is determined by the different ways in which the material is deformed. In metals, plastic deformation occurs by crystallographic slip, due to the movement of dislocations within the lattice. In general, slip takes place on the planes which possess the highest atomic density, *slip planes*, and in the most densely packed directions, *slip directions*. The slip plane is characterized by the unit vector **n**, which is normal to the plane, and the slip direction, represented by the unit vector **b** (Burger's vector). The combination of both vectors, which are perpendicular to each other,

Since crystallographic slip is limited to certain planes and directions, the applied stress required to initiate plastic flow depends on the orientation of the stress relative to the crystallographic axis of the crystal. If the plane is either normal or parallel to the applied stress, the shear stress on the plane is zero and no plastic deformation is possible. Slip begins

**2. Texture and anisotropy of cubic metals** 

defines a *slip system*.

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:

$$
\pi\_{\mathbf{r}} = \sigma\_{12}' = b\_{\mathbf{i}} \quad n\_{\mathbf{j}} \quad \sigma\_{\mathbf{i}\mathbf{j}} \tag{1}
$$

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

$$
\tau\_{\mathbf{r}} = \sigma \cos \mathcal{A} \cos \phi,\tag{2}
$$

where λ is the angle between the slip direction and the tensile axis, and φ is the angle between the tensile axis and the normal to the slip plane.

**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 planes meet the crystal surface.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 179

ϕ φϕ, , , is

ϕ

. This

(3)

y

, ii) a rotation about the new X-

 ϕφ

 ϕφ

φ

ϕ1

φ

 φ

> φ

> > z

ϕ1

z' y'

ϕ2 ϕ2

φ

x'

ϕ2

A polycrystal is composed of crystals, each with a particular crystallographic orientation. Several parameters are involved in characterizing a polycrystal, such as the shape, size, crystallographic orientation and position of each grain inside the sample. The orientation of each crystal in the polycrystal can be defined by a rotation from the sample coordinate system to the crystal coordinate system. The sample coordinate system is referenced to the sample, and it can be chosen arbitrarily. For an example, the *Rolling Direction* (RD), the *Transverse Direction* (TD) and the *Normal Direction* (ND) are typically chosen as sample coordinate system for a rolled sheet. The orientation relation between a single crystal and the sample coordinate systems may be thought of as rotating one frame into the other. Euler angles are useful for describing one frame in term of the other, or vice versa. Several different notations have been used to define these angles, but that of Bunge is most common and will be used in this chapter (Bunge, 1982). These three angles represent three consecutive rotations that must be given to each grain to bring its crystallographic <100> axes into coincidence with the sample axes. This is equivalent to saying that any orientation can be obtained by conducting three elemental rotations (rotations around a single axis). Consequently, any rotation matrix can be decomposed into a product of three elemental rotation matrices. The matrix rotation (Eq. 3), written in terms of Euler angles ( ) 1 2

obtained by multiplication of the elementary matrices defining the three successive Euler

1 2 12 1 2 12 2 12 1 2 12 1 2 2

y'

ϕ1

φ

 − + −− −+ <sup>−</sup>

cos cos sin sin cos sin cos cos sin cos sin sin cos sin sin cos cos sin sin cos cos cos cos sin sin sin cos sin cos

 ϕϕ

 ϕϕ

1 1

z

φ

ϕ1

 φ

**Figure 3.** Definition and sequence of rotation through the different Euler angles.

x' ϕ1

x

φ

z'

 φ ϕ

 ϕϕ

 ϕϕ

y

x

φ

ϕ1

 φ

ϕ

and iii) a rotation about the last Z-axis through an angle 2

rotations: i) a rotation about the Z-axis through the angle <sup>1</sup>

φ

gives the crystal coordinate system (see Figure 3).

 ϕϕ

 ϕϕ

> φ

y

ϕ

y'

ϕ1

**2.1. Crystal orientation** 

axis through the angle

z

x

x' ϕ1

z'

ϕ1

ϕϕ

ϕϕ

**Table 1.** Slip systems of FCC and BCC cubic metals.

#### **2.1. Crystal orientation**

178 Metal Forming – Process, Tools, Design

**Table 1.** Slip systems of FCC and BCC cubic metals.

A polycrystal is composed of crystals, each with a particular crystallographic orientation. Several parameters are involved in characterizing a polycrystal, such as the shape, size, crystallographic orientation and position of each grain inside the sample. The orientation of each crystal in the polycrystal can be defined by a rotation from the sample coordinate system to the crystal coordinate system. The sample coordinate system is referenced to the sample, and it can be chosen arbitrarily. For an example, the *Rolling Direction* (RD), the *Transverse Direction* (TD) and the *Normal Direction* (ND) are typically chosen as sample coordinate system for a rolled sheet. The orientation relation between a single crystal and the sample coordinate systems may be thought of as rotating one frame into the other. Euler angles are useful for describing one frame in term of the other, or vice versa. Several different notations have been used to define these angles, but that of Bunge is most common and will be used in this chapter (Bunge, 1982). These three angles represent three consecutive rotations that must be given to each grain to bring its crystallographic <100> axes into coincidence with the sample axes. This is equivalent to saying that any orientation can be obtained by conducting three elemental rotations (rotations around a single axis). Consequently, any rotation matrix can be decomposed into a product of three elemental rotation matrices. The matrix rotation (Eq. 3), written in terms of Euler angles ( ) 1 2 ϕ φϕ , , , is obtained by multiplication of the elementary matrices defining the three successive Euler rotations: i) a rotation about the Z-axis through the angle <sup>1</sup> ϕ , ii) a rotation about the new Xaxis through the angle φ and iii) a rotation about the last Z-axis through an angle 2 ϕ . This gives the crystal coordinate system (see Figure 3).

**Figure 3.** Definition and sequence of rotation through the different Euler angles.
