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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 : hnhan@snu.ac.kr

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