Meso-Scale Simulation and Measurementof Dislocation/Grain Boundary Interactions

AFOSR Grant Number: FA9550-05-0068, 0088

Robert H. Wagoner, PI, Myoung-Gyu Lee, Hojun Lim

Department of Materials Science and Engineering

Ohio State University

B. L. Adams, PI, Colin Landon, Josh Kacher

Department of Mechanical Engineering

Brigham Young University

(Lee, Wagoner, OSU)

Fundamental Role of Grain Boundaries: Meso-scale Simulation and Measurement

Tensile Specimen

Grain Orientations

Single crystal properties

Two-scale Simulation

Grain ScaleGrain Scale

Dislocation ScaleDislocation Scale

Predictions

(Lee, Wagoner, OSU) Verification

(Lee, Wagoner, OSU) (Homer, Adams, BYU)

Slip Activity

Stress

Choice of Materials

0

100

200

300

400

500

0 1 2 3 4 5 6 7 8

0.2

% Y

ield

Str

ess

(MP

a)

d-0.5 (um-0.5)

Minimum Alloy Steel

Stainless Steel 439

Fe-Si 3%

0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30

En

g.S

tre

ss (

MP

a)

Eng.Strain (%)

Fe-Si 3%

Stainless Steel 439

Minimum Alloy Steel

Stress/strain, Hall-Petch relations(Lim, Wagoner, OSU)

Fundamental Role of Grain Boundaries: Meso-scale Simulation and Measurement

3-D Curvature Recovery viaOblique Double Sectioning (ODS)

Opacity Limitations on Curvature Recovery

Verification of ODS RecoveryTrue (Left), Recovered (Right)

Verification of Experimental Resolution

0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Experimental Data

.5/dx

ij ij 12ijkk eilk jk,l

e

x1

x2

x3

),(ˆ n

gc

gb

ga

3

23

2

23

1

23

3

31

2

31

1

31

3

32

2

32

1

32

dx

gd

dx

gd

dx

gddx

gd

dx

gd

dx

gddx

gd

dx

gd

dx

gd

3-D only

j

iij dx

d

(Adams, Homer, Lemmon, and Landon, BYU)

Summary of Results as of AFOSR Review, Nov. 3, 2006

Simulation

6

Procedure

Two-Scale ModelTwo-Scale ModelInput (OIM)Input (OIM)

Tensile Specimen

Grain Orientations

Slip Activity

Lattice Curvature

BYU AFOSR FA9550-05-1-0088

PredictionsPredictions

Single-Crystal Properties

OSU AFOSR FA9550-05-0068

• Superdislocations at the center of elements• Generalized pileup configuration

Grain ScaleGrain Scale

Dislocation ScaleDislocation Scale

7

Numerical Tests of Simulation Procedure

1-D Pileup

0

1

2

3

4

5

0 5 10 15 20 25 30

Analytical Solution [10]

200 nm (every 5th point)

10,000 nm

2,000 nm

Dis

loca

tion D

ensi

ty (10

9 cm

-2)

Distance from Pileup (microns)

Element Size

2-D Pileup

• CPU: < 2min. (2.8 GHz PC) • Mesh independent• Reproduce analytical solutions• Numerically stable

Dislocation pileup with Superdislocation concept

8

Constitutive Equations: SCCE-T

Slip activities(Asaro & Needleman, 1985)

Hardening of slip systems(Peirce et al., 1982)

sign

1

0

m

g

β

αβα γhg

1.

4.11

sym

hh

SCCE-T: Single Crystal Constitutive Equations - Texture

• Arbitrary parameters: ≥ 6 (m, hii, hij, h1, h2

, h3

9

Constitutive Equations: SCCE-D

sign

1

0

m

g

0

94.047.00

H

)(

1

1

b

k

kbb

n

a

Slip activities(Asaro & Needleman, 1985)

Hardening of slip systems

SCCE-D: Single Crystal Constitutive Equations - Dislocation

• Arbitrary parameters: 4 (m, 0, ka, kb

10

SCCE-D: Orowan hardening modelForest dislocation

Active (moving) dislocation

Slip plane

l

)()(

l

bg )(

)(

1

l

Effective forest dislocation density

)()()()()( ξncos H

Orowan model [ E. Orowan, 1948]

n()

11

Results: Constitutive Equations

0

30

60

90

120

150

0 0.05 0.1 0.15 0.2 0.25Strain

SCCE-D Predicted

SCCE-TPredicted

Stre

ss (

MP

a)

[001]

SCCE-T: Texture-based Const. Eq.Measured Fe Single Crystal (Keh, 1964)

SCCE-D: Dislocation-based Const. Eq. Fit Curves

[-123]

Iron (BCC) Copper (FCC)

SCCE-T vs. SCCE-D SCCE-T vs. SCCE-D

0

30

60

90

120

150

0 0.02 0.04 0.06 0.08 0.1S

tres

s (M

Pa)

Strain

[001]

SCCE-D Predicted

SCCE-TPredicted

Fit CurvesSCCE-D: Dislocation-based

Const. Eq.

SCCE-T: Texture-based Const. Eq.Measured Cu Single Crystal [18]

[-123]

Measurement

13

Characterization of complete curvature tensor

κ - 6 of 9 lattice curvatures

1 of 2 boundary inclination parameters,

Full orientation characterization, g

ˆ n ()

3-D Curvature(Under Development)

2-D Curvature(Currently Available)

gc

gb

Lattice Curvature

ga

3

23

2

23

1

23

3

31

2

31

1

31

3

32

2

32

1

32

dx

gd

dx

gd

dx

gddx

gd

dx

gd

dx

gddx

gd

dx

gd

dx

gd

3-D only

j

iij dx

d

ga

gbgc

x1

x2

x3

x1

x2

x3

)(ˆ n

),(ˆ n

κ - All 9 lattice curvatures

Full boundary inclination description,

Full orientation characterization, g

),(ˆ n

14

Experimental resolution limits for lattice curvature

0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Experimental Data

.5/dx

ij ij 12ijkk eilk jk,l

e

Exp. data

.5/dx

15

Oblique Double-Sectioning

• Oblique Double-Sectioning– Combination of serial

sectioning and stereology

– 2 parallel section-cuts for direct measurement of grain boundary character

16

Registry and interpolation

• Alignment of layers– Reference marks

– Grains

– Triple-Junction Distribution

• Interpolation of boundaries to obtain GBCD– Meshing Algorithm

17

Application: Fe-3%Si Multi-crystal

Input (OIM)Input (OIM)

Verification Verification Lattice Curvatures

0

0.02

Fe-3%Si Tensile SpecimenFe-3%Si Tensile Specimen

Simulated (CPU=7h) Measured

(rad/m)

(BYU)

Results since AFOSR Review, Nov. 3, 2006

Simulations

20

Parametric Tests: Bi-crystal

B A

• Simple bi-crystal structure• Iron single crystal properties • Dislocation=mobile + immobile• Only mobile density can be piled up

near the grain boundary• Apply grain boundary strength

A B

45o320

340

360

380

400

420

0 2 4 6 8 10

En

g.S

tres

s (M

Pa)

Mat. + Mech. (Infinite )

Mech.

Strain (%)

Mat.+Mech. (*=5y

)

total = app+defect

Fsuper= total · (b x )

Force on Superdislocation

where

Obstacle force Fobs= = nY·A

Slip transmission Fsuper ≥ Fobs

Slip

tran

smissio

n

21

Parametric Tests: Dislocation DensityD

islo

ca

tio

n d

en

sit

y (

1/m

m2)

= 1% = 5%= 10%

Von Mises Stress at 10% strain

(121)[111] (211)[111] 1]11)[10(1

Dislocation density on various slip systems

Total dislocation density at different strain levels

22

Parametric Tests: Size dependence

398

400

402

404

406

408

410

0 200 400 600 800 1000

Hall-Petch Slopes

Str

ess

at

10

% s

tra

in (

MP

a)

d-0.5(m-0.5)

1 um10 um100 um

Stress at infinite grain boundary strength

= 1.5 GPa398

400

402

404

406

408

410

10-5 0.0001 0.001 0.01 0.1 1 10 100 1000

Stress at 10% strain (MPa) vs. grain size (mm)

Str

ess

at

10

% s

trai

n(M

Pa)

Grain size d (mm)

Stress at infinite grain boundary strength

= 1.5 GPa

Stress vs. grain size (d) Stress vs. grain size-1/2 (d-1/2)

• Constant grain boundary strength: 5*300 MPa• Different grain sizes with same grain configuration

*=5y=1,500 MPa *=5y=1,500 MPa

23

Parametric Tests

Crystal orientation

(grain boundary = 45°)

CrystalOrientation

Simulation Method

Mech. Mech.+Mat.Infinite τ*

Mech.+Mat. Finite τ*

0° 377.92 409.76 403.33

15° 377.58 410.14 404.40

30° 378.82 413.77 408.12

45° 379.62 415.91 409.90

Δσ 2.04 6.15 5.57

Grain boundary orientation

(crystal = 45°)

Grain boundaryorientation

Simulation Method

Mech. Mech.+Mat.Infinite τ*

Mech.+Mat. Finite τ*

0° 380.23 408.63 399.69

15° 380.01 414.42 406.80

30° 379.20 414.47 407.25

45° 379.62 415.91 409.90

Δσ 1.03 7.28 10.21

Eng.Stress at 10% strain (MPa)

Experiments

25

• Measuring shifts to 1/20 of a pixel increase resolution of rotation by at least a factor of ten

• The correlation based method is also sensitive to lattice strains

Cross Correlation Technique: Promising New Method

Ref: Angus Wilkinson (Oxford University)

26

Cross Correlation Technique

• Reference Image• Comparison image at

adjacent scan point

A region in the reference image is placed over the comparison image and progressively shifted. The correlation intensity is recorded and forms the correlation image.

27

Cross Correlation Technique: Correlation image

The peak intensity in the correlation image shows the x and y shift of the image to the pixel level. The center of the image correlates to a zero shift.

Shifts can be measured to 1/20 of a pixel using a surface fitting scheme and the intensities.

x

y

28

Cross Correlation Technique: Algorithm

• This results in a system of 2 independent equations for each region of interest with 8 unknowns

31132

321

1

321

3

123

2

132

3

3

1

131 r qrq

x

urr

x

ur

x

ur

x

urr

x

u

x

urr

32232

322

1

321

3

223

1

231

3

3

2

232 qrqr

x

ur

x

urr

x

ur

x

urr

x

u

x

urr

29

• Using the deformation gradient tensor you can find the strain and rotation gradients

i

j

j

iij x

u

x

u

2

1

i

j

j

iij x

u

x

u

2

1

Cross Correlation Technique: Algorithm

30

• After analyzing a line scan any component of the strain or rotation gradient tensors can be displayed

Components of Rotation

Point Number

Rot

atio

n (R

ad)

100 105 110 115 120

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

w12

w23w31

Cross Correlation Technique: Line scan

31

• An area scan can be analyzed to show the variation of any component of the strain or rotation tensor.

11

1 1.5 2 2.5 3 3.5 41

1.5

2

2.5

3

3.5

4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

x 10-4

The x and y axis indicate the position in the scan (This example was a 4 point x 4 point grid)

Strain in the 1 1 direction

Cross Correlation Technique: Area Scan

Plans: 2007 (or 2008?)

33

2007 Plans

• Incorporate slip transmission criteria, determine physical * (many more specimens)

• Ratio () between mobile/immobile dislocation density

f (dislocation density), current model: constant

• Improving cross section technique

34

2007 work : New material-Minimum Alloy Steel

C 0.001 Cr 0.014

Mn 0.13 Mo 0.003

P 0.006 Sn 0.002

S 0.005 Al 0.038

Si 0.004 Ti 0.001

Cu 0.023 N 0.003

Ni 0.007 Nb 0.001

K11

Measured Lattice CurvatureInitial Grain Orientations

Composition

1) High Hall-Petch Slopes2) Good Ductility / Hardening3) Grain Size4) Good OIM imaging/polishing

Desirable Material CharacteristicsDesirable Material Characteristics

Choice: Minimum Alloy SteelChoice: Minimum Alloy Steel

0

100

200

300

400

500

0 1 2 3 4 5 6 7 8

0.2

% Y

ield

Str

ess

(MP

a)

d-0.5 (um-0.5)

Minimum Alloy Steel

Stainless Steel 439

Fe-Si 3%

Stress- Strain Curves Hall-Petch Slopes

0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30

En

g.S

tre

ss (

MP

a)

Eng.Strain (%)

Fe-Si 3%

Stainless Steel 439

Minimum Alloy Steel

Grain size attained (OSU) : 80m ~ 1500m

35

2007 work: New specimen/OIM

Total dislocation density(simulated)

*Grain Boundary at 5°

Plans: 2008~2011

37

2008-2011 Plans

• Recover elastic strain gradient by cross correlation (+ adaptive OIM)

• Develop high resolution OIM technique, couple with new adaptive OIM

• Parallel mesh refinement at grain boundaries and triple junctions

(FEA, OIM)

• Parallelize Mech.+Mat. Simulation (Suitable for many grains)

• Grain boundary transmission criteria and Hall-Petch slopes for wide range of grain sizes

*=f (slip transmission), current model: *=constant.

• Compare H-P slope: simulation, measurement (Use range of real grain size)

• Extend to HCP materials

38

2007 work : Grain boundary transmission

Curvature plot with infinite GB Exp. curvature

Obstacle strength with slip transmissivity*)1( NN CRSSobstacle

39

2007~ work : Adaptive or Other Mesh refinement (FEA, OIM)

10m 5m 1m 1.4m

Extra Slides

41

Results: Constitutive Equations

TCCE DCCE

Cu

[-112] 59 (177) 10 (30)

[-123] 64 (410) 6 (46)

Fe

[011] 5 (12) 3 (7)

[-348] 8 (35) 2 (3)

Average 34 (159) 5 (22)

Units: MPa (%)

Standard DeviationStandard Deviation (Average % errors) (Average % errors)

42

Results: Constitutive Equations

Iron (BCC)Copper (FCC)

Uni-axial Compression of PolycrystalsUni-axial Compression of Polycrystals

43

• The shift in the EBSD pattern q at a region of interest (ROI) centered at point x with crystallographic direction r is related to the displacement gradient tensor a

rrQQq ˆ)ˆ(

3

2

1

3

3

2

3

1

3

3

2

2

2

1

2

3

1

2

1

1

1

ˆ

r

r

r

x

u

x

u

x

ux

u

x

u

x

ux

u

x

u

x

u

raQ

High Resolution Strain and Rotation Measurement

If the region of interest were centered here then r would be a unit vector in the (-1,1,-3) direction

44

Summary I: single Xl constitutive equations

• Novel two-scale simulation model based on Finite Element Method was developed

• Superdislocation concept is well validated with analytical pileup solution

• SCCE-D (4 parameters) fits real single Xl (no gb effects)

• SCCE-T (≥6 parameters) does not match single Xl (gb effects)

• Stress-strain response and texture evolution are similar with different single Xl models

45

Summary II: Parametric tests• Flow stress

– Mech. Simul. < Mech+Mat. Simul. w/ finite boundary strength < Mech.+Mat. Simul. w/ infinite boundary strength

• High dislocation density is observed near the grain boundary at low strain level

• High dislocation density increases crystal hardness

• Dislocation density of grain interior becomes higher as the deformation proceeds due to the high slip activity

• Hall-Petch relation is observed with two-scale simulation model with finite grain boundary strength

• Bi-crystal analysis showed that grain boundary orientation is more sensitive to the dislocation pileup than crystal orientation

46

• Preliminary result with reasonable distribution:

• Meso model is CPU efficient (7hr/10 grains)

mrad/0.017~κvs0.013~κ maxmeasured

maxpredicted

Summary III: Verification

47

• Measuring shifts to 1/20 of a pixel increase resolution of rotation by at least a factor of ten

• The correlation based method is also sensitive to lattice strains

Summary VI: High Resolution Strain Measurement