An introduction to computations in crystallographic textures · Preferred Orientations in...
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An introduction to computations in crystallographic textures Institute of Metallurgy and Materials Science A.Morawiec Polish Academy of Sciences
Transcript of An introduction to computations in crystallographic textures · Preferred Orientations in...
An introduction to computations
in crystallographic textures
Institute of Metallurgy and Materials Science
A.Morawiec
Polish Academy of Sciences
Wikipedia.com
Motivation
Examples:
• Orientation relationships
• Crystal deformation mechanisms
• Some phase transformation mechanisms
• Orientation mapping
Comprehensive understanding of description of orientations
is crucial for research on polycrystalline materials.
=20 µm; Euler3 + 3/ 10/ 20; Step=1 µm; Grid100x100
http://www.museumwales.ac.uk
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Outline
• Rotations and rotation parameterizations
• Crystal orientation and crystal symmetry
• Statistics in the orientation space
• Standard (mis)orientation distributions
• Example of texture application: effective elastic properties of polycrystals
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Suggested reading
• H.J. Bunge,
Texture Analysis in Materials Science,
Butterworths, London, 1982.
• U. F. Kocks, C. N. Tome, H.R. Wenk,
Texture and Anisotropy,
Preferred Orientations in Polycrystals and their Effect on Materials Properties,
Cambridge University Press, Cambridge, 1998.
• A. Morawiec,
Orientations and Rotations,
Computations in Crystallographic Textures,
Springer Verlag, Berlin, 2004.
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Rotations and rotation parameterizations
Institute of Metallurgy and Materials Science
A.Morawiec
Polish Academy of Sciences
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Wikipedia.com
Outline
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• Basics: orientations vs. rotations
• Numerical representation of rotations and orientations
• Composition of rotations
• Parameterizations of rotations and orientations
Basics
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Basics
Rotation about a line – a displacement in which points of a line
retain their locations.
Rotation about a point is equivalent to a rotation about a line
Euler theorem:
Rotation about a point – a displacement in which the location of
the point is not changed.
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An orientation – the equivalence class
of all displacements which differ by a translation.
object's orientations rotations about an axis
With a universal reference orientation,
an orientation of an object is determined by the rotation
from the reference orientation to that orientation.
one-to-one correspondence
Orientation – state Rotation – process (displacement)
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Basics
Effects of proper rotations i.c
Mirror images
‘Mirror’ transformation with a fixed point = improper
rotation
Improper rotations change handedness
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Basics
Effects of proper and improper rotations i.c
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Inversion = half-turn about a line followed by reflection
with respect to a plane perpendicular to the line.
improper rotation = proper rotation composed with inversion
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Basics
Inversion
Numerical representation
of rotations and orientations
To refresh memory …
Matrix – an array of m n numbers
mnmm
n
n
AAA
AAA
AAA
A
...
............
...
...
21
22221
11211
ijA mi ,...,2,1 nj ,...,2,1
An m n matrix A is: square matrix if m = n
symmetric if Aij = Aji
anti-symmetric if Aij = -Aji
zero if Aij = 0
A square matrix A is: unit if Aij = Iij = dij
0
1ijd
ji
ji if
if
Matrix algebra
Matrix product: C = AB kjiknjinjiji
n
k
kjikij BABABABABAC
...2211
1
Trace of square matrix Aij: Tr(A)= Aii = A11 + A22 + … + Ann
Det of 3 3 matrix Aij: det(A)= eijk Ai1 Aj2 Ak3
ijkif
0
1
1
ijke if ijk
even permutation of (123)
odd permutation of (123)
in other cases
Square matrix A is invertable if det(A) is non-zero; AA-1=I
transpose of B=AT if Aij = Bji
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Matrix representation of orientations
p dim object immersed in N dim Euclidean space
N=3, p=2
Orientation is determined by p
linearly independent vectors
321
321
bbbb
aaaa
O
bbb
aaa
321
321
p
T IOO
With orthonormal bases
10
01
bbab
baaaOOT
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An orientation of a p dim object immersed in N dim Euclidean space
can be represented numerically by a p x N matrix O satisfying OOT=I.
N=3, p=3
N=p = 3
1det2O
1OOT3IOOT
O – an orthogonal matrix
3IOOT 1det O
O – a special orthogonal matrix
Arbitrary rotations (including improper rotations)
Proper rotations
Matrix representation of orientations
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N=3, p=3
N=p = 3
1det O
Matrix representation of orientations
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3/23/13/2
O
3
100
010
001
3/23/13/2
3/23/23/1
3/13/23/2
3/23/23/1
3/13/23/2
3/23/13/2
IOOT
Example (special) orthogonal matrix
N=3, p=3
N=p = 3
1det2O
1OOT3IOOT
O – an orthogonal matrix
3IOOT 1det O
O – a special orthogonal matrix
Arbitrary rotations (including improper rotations)
Proper rotations
Matrix representation of orientations
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Wikipedia.comGroups of special orthogonal matrices – SO(3)
orthogonal matrices – O(3)
Composition of rotations
Wikipedia.com
Groups O(3)
SO(3)
– all rotations
– proper rotationsone-to-one correspondence to
Wikipedia.com
Composition of rotations
O, O’ and O’’ – orthogonal matrices
O 'O ''O
TOOR '' TOOR '''''
TOOR ''
''')')('''()''('''''' RROOOOOOOOIOOOOR TTTTTT
''' RRR
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Composition of rotations corresponds to multiplication
of representing them orthogonal matrices.
Wikipedia.com
Composition of rotations
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Composition of rotations corresponds to multiplication
of representing them orthogonal matrices.
3/23/23/1
3/13/23/2
3/23/13/2
'R
RRR
3/23/23/1
3/23/13/2
3/13/23/2
3/23/23/1
3/13/23/2
3/23/13/2
100
001
010
'''
100
001
010
''R
''' RRR
Wikipedia.com
Composition of rotations − algebra of quaternions
e - basis of a 4 dimensional vector space, 3,2,1,0
Standard multiplication plus
the quaternion multiplication rule
33221100 exexexexexx
0eeee ijkijkji de
eeeee 00
3,2,1,, kji
yxxy
{
exx eyy
kjiijkkkii eyxyxyxeyxyxxy e 00000
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Unit quaternions and rotations
3,2,1,03,2,1,, kji
0
2
0 22 qqqqqqqA kijkjiijkkij ed
A is a special orthogonal matrix
22
010
001
100
A
]2/1,2/1,2/1,2/1[],,,[ 3210 qqqqq
q - components of a unit quaternion
12
3
2
2
2
1
2
0 qqqqqq
Wikipedia.com
Unit quaternions and rotations
O, O’ – special orthogonal matrices
'' OOqq
Oq '' Oq
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q, q’ – unit quaternions
There is a two-to-one corespondence between
unit quaternions and special orthogonal matrices.
There is a two-to-one corespondence between unit quaternions and proper rotations.
Unit quaternion sphere in 4D
12
3
2
2
2
1
2
0 qqqq
Quaternions q and –q represent the same orientations
q
q
Unit quaternions and rotations
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qOqqqqqqqqO ijkijkjiijkkij 0
2
0 22 ed
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Summary
(Proper) rotations can be represented (special) orthogonal matrices.
Composition of rotations is represented by the product of orthogonal matrices.
There is two-to-one correspondence between unit quaternions
and special orthogonal matrices.
Euler theorem.
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Composition of rotations is represented by the product of unit quaternions.
Wikipedia.com
Transformation of vector components
j
ijij
ji xRaaxx ''
ik
k
j
j aaxaxx '''
ijji Raa '
Rxx '
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ijji aa d ''ijji aa dx and
Parameterizations
• Rodrigues parameters
• Axis and angle
• Rotation vector
• Euler angles
• Miller indices
Wikipedia.com
Independent orientation parameters
Special orthogonal matrix – 9 parameters
Number of independent parameters - 3
Is it possible to have a „nice” global parameterization
(one-to-one, continuous, with continuous inverse),
which would map rotations into the 3 dimensional
Euclidean space?
987
654
321
No!
Unit quaternion – 4 parameters 4321
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Cayley transformation
1))(( RIRIO
)()( 1 OIOIR
R
O - special orthogonal matrix such that
- an antisymmetric 3x3 matrix
)( OI non-singular
3 independent parameters
0
0
0
12
13
23
rr
rr
rr
R
)( RI non-singular
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Cayley transformation, example
]3/1,3/1,3/1[],,[ 321 rrrr
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3/13/23/2
3/23/13/2
O
0
0
0
03/13/1
3/103/1
3/13/10
)()(
12
13
23
1
rr
rr
rr
OIOIR
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Rodrigues parameters
ll
kjijkii
irr
rrrrrr
'1
'')'(
e
jkijk
ll
i OO
r e
1
1 kijkjiijkk
ll
ij rrrrrrr
O ed 2211
1
kijkij rR ejkijki Rr e
2
1
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'' OOrr
Or '' Or
one-to-one corespondence
0
0
0
12
13
23
rr
rr
rr
R
0q
qr ii Rodrigues parameters / unit quaternion: Rodrigues space
Wikipedia.com
Axis and angle
Euler theorem rotation axis
magnitude of rotation – rotation angle
21cos iiO
coskOk0nk
rrn The axis is represented by vector n of unit magnitude
rOr - Rodrigues vector is parallel to rotation axis
1kk
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k
Ok
n
ed sin)cos1(cos),( kijkjiijij nnnnO
2Sn
0
0
20
2
Sn
2
0
Axis and angle
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)2,(),( nOnO
ii nr )2tan( ii nq )2sin(
Axis and angle, example
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]3/1,3/1,3/1[],,[ 321 rrrr
3/23/23/1
3/13/23/2
3/23/13/2
O
3/]1,1,1[]3/1,3/1,3/1[
rr
rn
2/121cos iiO 603
3],1,1,1[
3
1),(
n
Wikipedia.com
Rotation vector
ii nf
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]1,1,1[33
n
]1,1,1[3
1]1,1,1[
3
1)6/tan()2/tan(
nr
3],1,1,1[
3
1),(
n
Wikipedia.com
Rotation vector
ii nf
Rf ],0[: 00 f
parametric ball
strictly increasing,0
f(w) n
3
1
2
: ii
2/sin f
f
2/tan f
3/124/sin3 f
0
0.5
1
1.5
2
2.5
3
0 20 40 60 80 100 120 140 160 180
in degrees
f( )
)sin()2/sin(
)2/tan(
)4/tan(
3/12 ))]sin())(4/(3[(
– isochoric parameters 36
Euler angles
),( nOO x-convention
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),"(),'(),(),,( 2121 zxz eOeOeOO
),(),(),(),,( 1221 zxz eOeOeOO
ex
ey
e’x
ez
e’x
e”z
e”ye”x
ez
Euler angles
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),(),(),(),,( 1221 zxz eOeOeOO
20 1 0 20 2
The domain of all proper rotations is covered when
1
2
20 1
20 2 0
Euler angles, example
60
),( nOO
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901 902
),(),(),(),,( 1221 zxz eOeOeOO
100
001
010
)90,( zeO
2/12/30
2/32/10
001
)60,( xeO
2/102/3
010
2/302/1
100
001
010
2/12/30
2/32/10
001
100
001
010
)90,60,90(O
,2/]3,0,1[),( n
Euler angles
),(),(),(),,( 1221 zxz eOeOeOO
),0,(),0,( 2121 OO
),,(),,( 2121 OO
Lattman angles: 221 221
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Singularity:
Wikipedia.com
Miller indices
hkl uvw
001110.,.ge
131211 ,, iiiiii OBOBOBuvw
0 lwkvhu
332313 OOOhkl 312111 OOOuvw
1 AB
333231 ,, iiiiii OAOAOAhkl
jiij aeA
Cubic system
ie – i-th external basis vector
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ja – j-th basis vector of crystal direct lattice
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3/23/23/1
3/13/23/2
3/23/13/2
O
Miller indices, example
332313 OOOhkl 312111 OOOuvw Cubic system
)212(332313 OOOhkl
]122[312111 OOOuvw
]122)[212(uvwhkl
hkl uvw
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Summary
• There is one-to-one correspondence between object's orientations and
rotations about a fixed point.
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• An orientation of an object can be represented an orthogonal matrix.
Composition of rotations corresponds to multiplication of representing
them orthogonal matrices.
Composition of rotations corresponds to multiplication of representing
them unit quaternions.
Summary
• Euler theorem: Rotation about a point is equivalent to a rotation about a line.
An orientation of an object can be represented a unit quaterion.
(Proper) rotations are represented (special) orthogonal matrices.
• There is two-to-one correspondence between unit quaternions
and special orthogonal matrices.
Summary
• There are a number of orientation parameterizations.
• Axis and angle
• Euler angles
• Rodrigues parameters
• Rotation vector
• Miller indices
„Nice” parameterizations involve more than 3 numbers.
There is a relationship between antisymmetric matrices
and special orthogonal matrices (Cayley transformation).
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Cayley transformation is a good starting point for deriving
3-number rotation parameterizations.
Unit quaternions and rotations
3,2,1,03,2,1,, kji
O – special orthogonal matrix
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q - components of a unit quaternion
12
3
2
2
2
1
2
0 qqqqqq
e 2jkijki Oq e 2kjijki Oq
2/11 llO
0
2
0 22 qqqqqqqO kijkjiijkkij ed
