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Crystal Mechanics
Lecture 5 – Orientation of Crystallites
Ref : Texture and Related Phenomena D N Lee 2006Ref : Texture and Related Phenomena, D. N. Lee, 2006Quantitative Texture Analysis, H.J. Bunge & C. Esling, 1979Texture and Anisotropy, U.F. Kocks, C.N. Tome H.-R. Wenk, 1998
Heung Nam HanAssociate Professor
School of Materials Science & EngineeringSchool of Materials Science & EngineeringCollege of Engineering
Seoul National UniversityS 1 1 44Seoul 151-744, KoreaTel : +82-2-880-9240Fax : +82-2-885-9647
2010-09-27email : [email protected]
Crystallographic Coordinates
• position: fractional multiples of the unit cell edge lengths) P:ex) P: q,r,s
cubic unit cell
Crystallographic Directions
• a line between two points or a vector• [uvw] square bracket, smallest integer
• families of directions: <uvw> angle bracket
cubiccubic
Crystallographic Planes (Miller Index)z
p m00 0n0 00p: define lattice plane
c
m00, 0n0, 00p: define lattice plane
m, n, ∞ : no intercepts with axes
a
b yn
xm
x
Plane (hkl)Family of planes {hkl}
Miller indicies ; defined as the smallest integral multiples of the reciprocals of the plane Family of planes {hkl} the reciprocals of the plane
intercepts on the axes
Crystallographic Planes
A B C DA B C D
E F
{110} Family
{110} Family
Definite of crystal orientationDefinite of crystal orientation
Specimen coordinate KA =KS and crystal coordinate KB =KC
Crystal orientation
Specimen reference frame
Crystal orientation
ND Specimen reference frame
Ks
NDNormal direction
Ks
RDTD
Crystal reference frame
Rolling directionTD
Transverse direction
Crystal reference frame
Crystal orientation Euler anglesCrystal orientation g
g ={1 2}
Crystal orientation Euler angles and E lCrystal orientation Euler space
Each point in this space represents a specific choice of the parameters 1 2 and hence a p 1 2specific crystal orientation.
1360 1 [0 -360 ]
g ={1 2}
1 [ ]
[0 -180 ]
2
2
360
180
2 [0 -360 ]
Miller IndicesCrystal orientation
K {110}<001>
Crystal orientation
Ks {110}<001>
Description of the grain orientation
g= (hkl)[uvw]
g={hkl}<uvw>g={hkl}<uvw>
Definition of the Pole Figure andDefinition of the Pole Figure and Inverse Pole Figure
Crystal orientation Matrix RepresentationCrystal orientation p
''' hruRD TD ND
''' ksvg
''' ltw
R t ti f i t ti b ND TD Representation of orientation g by ND, TD, RD parallel to [hkl], [rst], [uvw] respectively.
Crystal orientation Matrix RepresentationCrystal orientation p
Xhrux '''
ZYX
ltksvhru
yx
''' ' ' '
Zltwz '''
X u v w x ' ' 'XY
u v wr s t
xy
' ' ' Z h k l z ' ' '
[XYZ] in specimen coord. [x,y,z] in crystal coord.
Crystal orientationCrystal orientation
0i0010i
'2
Zg 'Xg
'1
Zg
0cossin
0sincossincos0
0010cossin0sincos
11
11
22
22
g 100cossin0100
i i i i i i
cos cos sin sin cos sin cos cos sin cos sin sincos sin sin cos cos sin sin cos cos cos cos sin 1 2 1 2 1 2 1 2 2
1 2 1 2 1 2 1 2 2
sin sin cos sin cos 1 1
Invariant measureInvariant measure
The invariant measure is introduced because the orientation transformation is combined with a distortionorientation transformation is combined with a distortion.
For the Euler angles, the invariant measure isI(1 2)=sin I(1 2) sin
Normalization factorNormalization factor
i d d d2 22 8 sin d d d1 202 2
002 8
1dg d d d1
8 2 1 2 sin
If we add the normalization factor and invariant measure
8
If we add the normalization factor and invariant measure for the orientation element in Euler space, the orientation
can be transformed without a distortion.
How can Orientation DistributionsHow can Orientation Distributions(Texture) be represented?
dggfd )(v
vf(g) : the orientation distribution functionf(g) : the orientation distribution function
f(g)=1 for random distribution.
How Texture is represented?
(110) Pole Figure
ODF :OrientationDistributionFunctionFunction
Random orientation distributionRandom orientation distribution