Quantitative Texture Analysis - PMT - Escola Politécnica da USPpmt.usp.br/pagina no site do...

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]]

RD

RDTD

(110)

[001]

Goss orientation = (110)[001]

Euler Angles:

Bunge notation: ϕ1, Φ, ϕ2

Roe notation: Ψ, Ξ, Φ(2nd rotation around y’)

Axis Angle Pair

[abc]ωx1

x2

x3

Ks

x’1

x’2

x’3

Kc

Rodrigues-Frank representation

N

R = N . tan(ω/2)

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 surfaces

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,….


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


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

Stereographic projectionof three <100> poles


♦ 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

The Texture Goniometer

Diffraction vector = fixedSample rotates

χϕ

RD

TD

Diffraction vector k describes concentric circles onstereographic projection

(hkl)

ϕ

χ

0 360

0 360

0 360

0 360

0 5 10 15

0 360

90

random level

Intensity

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


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


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


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


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

Recrystallization texture of a low carbon steelRecrystallization texture of a low carbon steel

T

t

800°C

60s 0 90

90

0

PHI2= 45

0 90

90

0

PHI2= 45

6.4

8

6.4

Quantitative Texture Analysis - PMT - Escola Politécnica da USPpmt.usp.br/pagina no site do...

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