*2.2.1. Pole figures*

Texture measurements are used to determine the orientation distribution of crystalline grains in a polycrystalline sample. There are several experimental methods that can be used to measure texture. The most popular is X-ray diffraction. A pole figure – which is a projection that shows how the specified crystallographic directions of grains are distributed in the sample reference frame – results from an X-ray diffraction texture measurement. This representation must contain some reference directions that relate to the material itself. Generally these directions refer to the forming process.

The inverse pole figure is a particularly useful way to describe textures produced from deformation processes. In this case only a single axis needs to be specified. An inverse pole figure shows how the selected direction in the sample reference frame is distributed in the reference frame of the crystal. The frequency with which a particular crystallographic direction coincides with the sample axis is plotted in a single triangle of the stereographic projection.

### *2.2.2. ODF and Euler space*

The ODF specifies the probability density for the occurrence of particular orientations in the Euler space. This space is defined by the three Euler angles, which are required to fully describe a single orientation. Mathematical methods have been developed that allow an ODF to be calculated from the numerical data obtained from several pole figures. The most widely adopted notations employed for these description were those proposed independently by Bunge and Roe. They used generalized spherical harmonic functions to represent crystallite distributions (A detailed description of the mathematics involved can be found in Bunge, 1982). ODF analysis was developed originally for materials with cubic crystallography and orthorhombic sample symmetry, i.e. for sheet products. In the Bunge notation, for cubic/orthotropic crystal/sample symmetry, a three dimensional orientation volume may be defined by using three orthogonal axes for 1 ϕ , φ and 2 ϕ , with each ranging from 0º to 90º. The value of the orientation density at each point in Euler space is the strength or intensity of that orientation in multiples of random units.

Most of the texture data available in the literature and almost all of the ODF data refer to rolled materials. The information contained in a three-dimensional ODF can be expressed in terms of typical components and fibers for cubic symmetry materials. A fiber is a range of orientations limited to a single degree of freedom about a fixed axis, which appears as a line that may or may not lie entirely in one section of ODF. The ideal components and fibers are associated with more or less constant intensity for a group of orientations related to one another by rotations around a particular crystallographic direction.

During cold rolling of FCC metals, two crystallographic fibers arise: the α-fiber containing <110>//ND orientations and extending from *Goss* {110}<001> to *Brass* {110}<112>; and the βfiber which starts at *Brass*, runs through *S* {123}<634>, and finally reaches C*opper* {112}<111>. The β-fiber contains the most stable components of the rolling texture (Humphreys & Hatherly, 2004). Considering recrystallized rather than rolled material, the typical texture components are C*ub*e {001}<100> and *Goss*. Table 2 shows a schematic representation of the rolling texture characteristic of the {111} pole figure (left) and the main texture components for FCC (right). The nature of the FCC rolling texture is such that the data are best displayed in 2 ϕ sections, while the typical {100} and {111} pole figures best represent these orientations.


**Table 2.** FCC rolling components.

180 Metal Forming – Process, Tools, Design

aggregate.

projection.

*2.2.2. ODF and Euler space* 

*2.2.1. Pole figures* 

**2.2. Crystallographic texture** 

Texture refers to a non-uniform distribution of crystallographic orientations in a polycrystal. The textures of rolled or rolled and recrystallized sheets have been most widely investigated in metallurgy. Crystallographic orientations in rolled sheets are generally represented as being of the type {hkl}<uvw>, where {hkl} are the grain planes that lie parallel to the plane of the sheet. On the other hand, the <uvw> directions lie parallel to the rolling direction. Conventionally, the standard method of representing textures was by means of pole figures. However, while pole figures provide a useful description of texture, the information they contain is incomplete. A complete description can be obtained by the Orientation Distribution Function (ODF), which describes the orientation of all individual grains in the

Texture measurements are used to determine the orientation distribution of crystalline grains in a polycrystalline sample. There are several experimental methods that can be used to measure texture. The most popular is X-ray diffraction. A pole figure – which is a projection that shows how the specified crystallographic directions of grains are distributed in the sample reference frame – results from an X-ray diffraction texture measurement. This representation must contain some reference directions that relate to the material itself.

The inverse pole figure is a particularly useful way to describe textures produced from deformation processes. In this case only a single axis needs to be specified. An inverse pole figure shows how the selected direction in the sample reference frame is distributed in the reference frame of the crystal. The frequency with which a particular crystallographic direction coincides with the sample axis is plotted in a single triangle of the stereographic

The ODF specifies the probability density for the occurrence of particular orientations in the Euler space. This space is defined by the three Euler angles, which are required to fully describe a single orientation. Mathematical methods have been developed that allow an ODF to be calculated from the numerical data obtained from several pole figures. The most widely adopted notations employed for these description were those proposed independently by Bunge and Roe. They used generalized spherical harmonic functions to represent crystallite distributions (A detailed description of the mathematics involved can be found in Bunge, 1982). ODF analysis was developed originally for materials with cubic crystallography and orthorhombic sample symmetry, i.e. for sheet products. In the Bunge notation, for cubic/orthotropic crystal/sample symmetry, a three dimensional orientation

ranging from 0º to 90º. The value of the orientation density at each point in Euler space is the

ϕ , φ  and 2 ϕ

, with each

Generally these directions refer to the forming process.

volume may be defined by using three orthogonal axes for 1

strength or intensity of that orientation in multiples of random units.

Cold rolling and recrystallization textures in BCC metals are commonly described in terms of five ideal orientations: {001}<110>, {112}<110>, {111}<110>, {111}<112> and {554}<225>. The positions of theses orientations in the {100} pole figure are shown at the left in Table 3. In general, BCC metals and alloys tend to form fiber textures. That is most orientations are assembled along two characteristic fibers that run through orientation space: the α-fiber and the γ-fiber. The RD or α-fiber runs from {001}<110> to {111}<110>, containing orientations with the <110> axis parallel to RD, and the γ-fiber runs from {111}<110> to {111}<112>, gathering orientations with a <111> axis parallel to ND. The two fibers intersect at the

{111}<110> component (Ray et al., 1994). The data are best displayed by sections at constant values of 1 ϕ , but the most important texture features can all be found in the 2 ϕ = 45º section (right in Table 3). Table 3 also gives the Miller indices and Euler angles of the typical BCC texture components. The {100} and {110} pole figures best represent the ideal BCC material orientations.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 183

*s s ss s* **m n bn b** = ⊗+⊗ (6)

*s s ss s* **q n bn b** = ⊗−⊗ (7)

*<sup>τ</sup>* <sup>=</sup> **m S m S** (8)

*<sup>c</sup> τ* is the critical resolved shear stress on the slip system

**LDW** = + , (9)

\* \*T <sup>p</sup> **D RDR** = :: , (10)

\* \*T <sup>p</sup> **W** = + **Ω RWR** ::. (11)

\* \*T <sup>p</sup> **Ω** = − **WRWR** ::, (12)

, , , related to the lattice spin as follows:

here *<sup>s</sup> γ* represents the dislocation slip rates, *<sup>s</sup>* **n** and *<sup>s</sup>* **b** are the normal to the system´s or systems´ glide plane and the Burgers' vector, respectively. They define the symmetric *<sup>s</sup>* **m**

( ) <sup>1</sup> , <sup>2</sup>

( ) <sup>1</sup> . <sup>2</sup>

The dislocation slip rates are derived using a viscoplastic exponential law (Hutchinson,

1/

labeled *s*, **S** is the deviatoric tensor stress and *m* is the strain-rate sensitivity exponent. The rate sensitivity *m* is typically quite small, a large value of 1/*m* tends to be almost a rate independent case, *~* 50. As 1/*m* ∝, the plastic constitutive formulation becomes formally

The velocity gradient can be additively decomposed into symmetric and skew-symmetric

where **D** is the distortion rate tensor and **W** is the rotation rate tensor. They can be obtained by evaluating the symmetric and screw-symmetric parts of equation (4),

The rotation rate contains an extra contribution, the lattice spin tensor, defined as \* \*T **Ω** ≡ **R R**: . Rearranging Eq. (11) allows us to obtain the rate of change of the crystal

which is used to determine the re-orientation of the crystal and consequently, to follow the texture evolution. The orientation change during plastic deformation can be described by a

> ϕ φϕ

*<sup>m</sup> <sup>s</sup> s s <sup>0</sup> γ γ sign*

: ( : ).

s c

and the screw-symmetric *<sup>s</sup>* **q** parts of the Schmid orientation tensor:

1976):

rate-independent.

parts

respectively:

orientation matrix:

list of the Euler angle change rates, ( ) 1 2

where *<sup>0</sup> <sup>γ</sup>* is the reference slip rate, *<sup>s</sup>*

**Table 3.** BCC rolling components.
