Post on 30-Apr-2019
Introduction to Quantitative Texture AnalysisIntroduction to Quantitative Texture Analysis
L. Kestens
IntroductionIntroduction
Material PropertiesProcessingConditions ?
! !• hot rolling• cold rolling• annealing
• chemistry
• mechanical properties(e.g.: yield strength, tensile strength,elongation, drawability,...)
• electrical properties• magnetic properties
Physical Material Parameters:• grain size• second phase, inclusions, precipitates,...• texturetexture
IntroductionIntroduction
Texture forming transformations in steels:
1. Solidification
2. Phase transformation from austenite to ferritic phase(during hot rolling or subsequent annealing).
3. Deformation(during cold or warm rolling, deep drawing, stretching,...)
4. Recrystallization and grain growth.(during annealing)
Polycrystalline material = aggregate of single crystalliteswith individual orientation w.r.t. sample reference system.
Textured MaterialTextured Material = Material in which the individualcrystallites occupy preferrential orientations
>< TexturelessTextureless or Random TexturedRandom Textured material
x1
x2
x3
Kc
ND
TD
RDKS
Kc = crystal reference systemKs = sample reference system
Crystallographic OrientationCrystallographic Orientation
Representation of single crystal orientationsRepresentation of single crystal orientations
1. Orientation Matrices
Transformation from sample to crystal reference system= Three degrees of freedom
[ ]g =g g gg g gg g g
11 12 13
21 22 33
31 32 33
RD TD NDg gik jk ij
k=∑ δ
g gki kj ijk
=∑ δ
2. Miller Indices (for sheet materials)
(h k (h k l)[ul)[u v w]v w]
crystallographic plane (hkl) // rolling plane
crystallographic direction [uvw] // rolling direction
(h k l)(h k l)
[[uvwuvw]]
Euler SpaceEuler Space
ϕ1
ϕ2
Φ
g(ϕ1,Φ,ϕ2)
Properties:
cyclic: 0 < ϕ1 < 2π0 < Φ < π0 < ϕ2 < 2π
non-bijective
non-Euclidean:dg = sinΦ dϕ1 dΦ dϕ2
The Orientation Distribution Function (ODF)The Orientation Distribution Function (ODF)
dVV
f g dg= ( )
dV/V is the volume fraction of orientation in an infinitesimalenvironment of g (g+dg)
g2, V2
g3, V3
g1, V1
g4, V4
Iso-intensity lines on equidistant sections
ϕ2 = constant sections∆ϕ2 = 5°
Φ
ϕ1
units = x random intensity
Crystal SymmetryCrystal Symmetry
Definition of object symmetrySymmetry operators
“Object has not changed after being subjected tosymmetry operators”
Symmetry axes: 2-fold, 3-fold, 4-fold,….
Crystal SymmetryCrystal Symmetry
Cubic crystal (BCC, FCC, Primitive)
24 equivalent ways of attaching right-handedorthogonal reference system to a cube
24 symmetry elements in cubic symmetry group
24 symmetrical equivalent points to represent one singlecubic orientation in Euler space
Crystal SymmetryCrystal Symmetry
Cubic crystal (BCC, FCC, Primitive)
Total Euler Space
0 < ϕ1 < 2π0 < Φ < π0 < ϕ2 < 2π
V d d d= =∫∫∫ sinφ ϕ φ ϕ ππππ
0
2
00
2
1 228
Fundamental Zone:
VV
'=24
No Linear Boundaries
Convenient Zone:
0 < ϕ1 < 2π0 < Φ < π/20 < ϕ2 < π/2
Sample SymmetrySample Symmetry
E.g. : rolled sample before rolling
after rolling
RD
ND
TD
3 two-fold axes: RD, TD, ND orthorombic samplesymmetry
Further reduction in fundamental zone of Euler Spacewith factor 4
Orthorombic Sample Symmetry = 4 symmetry elements
(Convenient) Fundamental zone for orthorombic sampleand cubic crystal symmetry:
0 < ϕ1 < π/20 < Φ < π/20 < ϕ2 < π/2
For BCC materials (e.g. low carbon steels) :All important rolling and recrystallization componentsare represented in ϕ2 = 45° section (with boundaries 0-90°)
(001)[110](001)[110] (001)[010]
(113)[110]
(112)[110](223)[110]
(110)[110] (110)[001]
(111)[112]
(554)[225]
( )[ ]111 110 ( )[ ]111 121
( )[ ]111 011
α fibre
γ fibre
Sample SymmetrySample Symmetry
E.g. : rolled samplenear surface
before rolling
after rolling
RD
ND
TD
1 two-fold axes: TD monoclinic sample symmetry
Reduction in fundamental zone of Euler Spacewith factor 2
Monoclinic Sample Symmetry = 2 symmetry elements
(Convenient) Fundamental zone for monoclinic sampleand cubic crystal symmetry:
-π/2 < ϕ1 < π/2 0 < Φ < π/20 < ϕ2 < π/2
The Pole FigureThe Pole Figure
♦ Distribution of <hkl> crystallographic poles w.r.t.to sample reference system
♦ Sample reference system + crystal pole <hkl> must berepresented in the pole figure
♦ Displays the sample symmetry(orthorombic vs. monoclinic symmetry)
♦ Cannot represent the complete texture
Measured Pole FigureMeasured Pole Figure (orthorombic symmetry)
1.6
4.0
110
3
.8 2.0
5.0
1.0
2.5 6.4
1.3
3.2
1.6
4.0
200
3
.8 2.0
5.0
1.0
2.5 6.4
1.3
3.2
(110) (200)TD
RD
TD
RD
Measured Pole FigureMeasured Pole Figure (monoclinic symmetry)
4.0
16
110
3
1.4 5.6
22
2.0
8.0 32
2.8
11
3.2
20
200
3
.8 5.0
32
1.3
8.0 50
2.0
13
(110) (200)TD TD
RD RD
The Inverse Pole FigureThe Inverse Pole Figure
♦ Distribution of sample direction (e.g. RD, TD or ND) w.r.t.to crystal reference system
♦ Crystal reference system + sample direction must berepresented in the pole figure
♦ Displays the crystal symmetry
♦ Cannot represent the completetexture
ND
Measuring Pole Figures by X-ray DiffractionMeasuring Pole Figures by X-ray Diffraction
Bragg diffraction
2d nBsinθ λ=
θB
λ
d
Pole Figures InversionPole Figures Inversion
f C Tl
N lM l
ll( , , ) && ( , , )
( )( )ϕ φ ϕ ϕ φ ϕµν
νµ
µν1 2
1101 2=
===
∞∑∑∑
(Harmonic Method)
p F kln
ln
N l
l( , ) & ( , )
( )χ ϕ χ ϕ
ν=
==
∞∑∑
10
&&Tlµν
= generalized spherical harmonics
&kln
= symmetrized spherical harmonicsF = pole figure coefficients (known)
C = ODF coefficients (unknown)
( )p f dhkl( ) ( , ) , ,χ ϕπ
ϕ φ ϕπ
= ∫1
2 1 20
2Γ
Γ denotes path through E.S corresponding to rotation about (hkl)
Pole Figures InversionPole Figures Inversion
Fl
C kl l
M l
lν µν
µ
µπξ η=
+ =∑
42 1 1( )
& ( , )( ) *
Transformation texture of a low carbon steelTransformation texture of a low carbon steel
FCC BCC
Bain(001)γ // (001)α[001]γ // [001]α
Kurdjumow-Sachs(111)γ // (110)α[101]γ // [111]α
<001>45deg <112>90deg
Transformation texture of a low carbon steelTransformation texture of a low carbon steel
Ferrite
DeformedAustenite
0 90
90
0
PHI2= 0
0 90
90
0
PHI2= 45
0 90
90
0
PHI2= 65
0 90
90
0
PHI2= 45
0 90
90
0
PHI2= 45
Simulation Experimental
K-S
6.4 3.2
4.0
2.0
Transformation texture of a low carbon steelTransformation texture of a low carbon steel
Ferrite
RecrystallizedAustenite
Simulation Experimental
0 90
90
0
PHI2= 45
0 90
90
0
PHI2= 45
0 90
90
0
PHI2= 45
K-S
4.0
4.02.0
1.0
Deformation texture of a low carbon steelDeformation texture of a low carbon steel
(110)
[11-1]Dislocation Glide
Slip planes: {110} + {211}
Slip directions: <111>
24 slip systems
Deformation texture of a low carbon steelDeformation texture of a low carbon steel
Taylor TheoryTaylor Theory
Basic Assumptions:• macroscopic strain = microscopic strain• dissipated plastic power is minimized
Imposed displacement tensor Eaccommodated by combination of 5 slip systems (out of 24)
Crystal rotation:initial orientation gi → final orientation gf
Deformation texture of a low carbon steelDeformation texture of a low carbon steel
0 90
90
0
PHI2= 45
measured
simulated
Hot Band Text. Cold Rolling Text.
2.0
0 90
90
0
PHI2= 45
8 5
3
6.4
0 90
90
0
PHI2= 45
11
45.6