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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]
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Crystallographic Coordinates
• position: fractional multiples of the unit cell edge lengths) P:ex) P: q,r,s
cubic unit cell
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Crystallographic Directions
• a line between two points or a vector• [uvw] square bracket, smallest integer
• families of directions: <uvw> angle bracket
cubiccubic
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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
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Crystallographic Planes
A B C DA B C D
E F
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{110} Family
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{110} Family
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Definite of crystal orientationDefinite of crystal orientation
Specimen coordinate KA =KS and crystal coordinate KB =KC
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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
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Crystal orientation Euler anglesCrystal orientation g
g ={1 2}
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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 ]
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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>
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Definition of the Pole Figure andDefinition of the Pole Figure and Inverse Pole Figure
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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.
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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.
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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
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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
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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.
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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.
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How Texture is represented?
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(110) Pole Figure
ODF :OrientationDistributionFunctionFunction
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Random orientation distributionRandom orientation distribution