**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 aggregate.

Self-Consistent Homogenization Methods for Predicting Forming Limits of Sheet Metal 181

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

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

sections, while the typical {100} and {111} pole figures best represent these

Typical texture components in rolled FCC metals

*Copper* 112 111 90º 35º 45º

*S* 123 634 59º 37º 63º

*Goss* 011 100 0º 45º 0/90º

*Brass* 011 211 35º 45º 0/90º

*Cube* 001 100 0º 0º 0º

ϕ φ 2 ϕ

Component {hkl} <uvw> <sup>1</sup>

another by rotations around a particular crystallographic direction.

*Cube Goss Brass S Copper*

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

TD

in 2 ϕ

orientations.

RD

{111} Pole figure

**Table 2.** FCC rolling components.
