¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

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Fig. Gii.3. Diagram of the “Goss” component, showing the how the crystal can be thought of as being rotated 45° about one edge of the cube. The Euler angles for this component are simple, and yet other variants exist, just as for the cube component. Only one rotation of 45° is needed to rotate the crystal from the reference position (i.e. the cube component) to (011)[100]; this happens to be accomplished with the 2nd Euler angle. (φ 1, Φ, φ 2) = (0°,45°,0°). Other variants will be shown when symmetry is discussed. ⎛ 1 0 0 ⎞ ⎜ ⎟ Matrix: ⎜ 0 1/ 2 1/ 2 ⎟ ⎜ ⎟ ⎝ 0 −1/ 2 1/ 2⎠ Rodrigues vector: [ tan(22.5°), 0 , 0 ] Unit quaternion: [ sin(22.5°) , 0, 0, cos(22.5°)] Note that, since there is only € one non-zero Euler angle, the rotation axis is obvious by inspection, i.e. the x-axis. For more general cases, the rotation axis has to be calculated. 8/27/09 22
Fig. Gii.4. Schematic {110} pole figure for the Goss component, showing strong intensity aligned with the ND and TD sample axes as a consequence of the 45° rotation about the RD. AS above, one could not measure such a pole figure unless reflection and transmission methods were combined. 3.Gii.2 The “Brass” Texture Component The {110} component is known as the Brass Texture and occurs as a rolling texture component for materials such as Brass, Silver, and Stainless steel. Its origin is still mildly controversial because it can be obtained in models of plastic deformation but the physical basis for the boundary conditions that produce it are not obvious. Fig. Gii.5. Diagram of the “Brass” component. 8/27/09 23
Page 1 and 2: 3. Mathematical Representation of C
Page 3 and 4: 3.A Orthogonal Tensors of Rank 2. R
Page 5 and 6: Fig. 3.B.1. Diagram showing the rel
Page 7 and 8: 3.C.1 Bunge Euler angles We will us
Page 9 and 10: Figure 3.C.1 Diagrams showing the s
Page 11 and 12: Fig. 3.C.2. Kocks Euler angles illu
Page 13 and 14: If the second angle is near to zero
Page 15 and 16: esult that all vectors correspondin
Page 17 and 18: sinθ 2 = 1 2 3− tr a ( ) (3.F.6)
Page 19 and 20: xi = (q4 2 -q1 2 -q2 2 -q3 2 )Xi +
Page 21: Fig. Gii.2. Schematic {100} pole fi
Page 25 and 26: physically distinguishable objects.
Page 27 and 28: ⎛ −1 0 0⎞ ⎜ ⎟ ⎜ 0 1 0
Page 29 and 30: 3.K Effect of Symmetry on Orientati
Page 31 and 32: ∆a = a2.a1 T (3.M.3.i) Note that
Page 33 and 34: 3.L.9 Disorientation The Disorienta
Page 35 and 36: 3.L.11 Misorientation (Active Rotat
Page 37 and 38: 3.L.13 Range of Parameters for Rodr
Page 39 and 40: Now, at last, we can apply this bis
Page 41 and 42: q’=q•S, where S is the symmetry
Page 43 and 44: To project the point w onto the pag
Page 45 and 46: Pole Figure Orientation Distributio
Page 47 and 48: ( ) = f h,ζ,w P h w ζ = 2π ∫

¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

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