a practical meso-scale polycrystal model to predict dislocation...

Center forAdvancedMaterials andManufacturing ofAutomotiveComponents

A Practical Meso-Scale Polycrystal Model to Predict Dislocation Densities, Lattice

Curvatures, and the Hall-Petch EffectR. H. Wagoner1, H. Lim1, J. H. Kim2,

M.G. Lee3, B. L. Adams4

1Ohio State University, 2Korean Institute of Materials Research

3Pohang University of Science and Technology4Brigham Young University

April 23, 2012

Dept. Mat. Sci. and Eng. Seoul National University

R. H. Wagoner

Acknowledgements

Funding– National Science Foundation

– Air Force Office of Scientific Research

Discussions– J. P. Hirth, C. S. Hartley

– BYU: E. Homer, C. Landon, J. Kacher, J. Parker

2

R. H. Wagoner

Highlights

3

Practical meso-scale method linking disloc, slip, gbobstacles, back stress.

disloc - based single crystal constitutive equations without tacit gb effects.

Formulation of gb slip transmission stress.

First prediction of H-P slopes with realistic model. Agreement with experiment, pure Fe.

First prediction of disloc(x. Agreement with experiment, Fe-3%Si.

R. H. Wagoner

Outline

• Background

• Two- Scale Model- Grain-Scale Simulation (GSS)- Meso-Scale Simulation (MSS)

• Multicrystal Testing

• Hall-Petch Tests

• Conclusions

4

R. H. Wagoner

Goals

Objective: Prediction of the Hall-Petch effect w/o arbitrary length scales or unrealistic basis

Approach: Two- scale model

5

• Fewest arbitrary parameters (simplest model)

• Predictive (no arbitrary length scales) (stra

• No unrealistic basis (pileup)

• Computational efficiency (≥ 100 grains)

• Local / real grain boundary properties

R. H. Wagoner

Hall-Petch Models

6

Dislocation pileup model[Hall, 1951; Petch, 1953]

-Generally not observed-Dimensionality, disloc. density?

Composite model[Kocks, 1970; Meyers, 1982]

-GB area/thickness (arb. Length) -Disloc. density?

1/21/2

0obsbM D

k

BULKf

GBff ff )1(

R. H. Wagoner

Hall-Petch Models (2)

7

Work hardening model[Conrad, 1961; Li. 1963)

[Evers et al. 2002]-Arbitrary division of crystals-Grain structure not considered

[Arsenlis et al., 2002, 2004]-Computationally intensive-Idealized single crystal-Simplified single slip geometry

Strain gradient approach[Fleck et al., 1994; Gurtin, 2000, 2002)

1/ D Mb -Arbitrary length scale-Ignore crystal structure, gb structure, slip systems

R. H. Wagoner

Experimental Hall- Petch Slopes

8

FCC BCCMaterials ky (MN/m3/2) Reference Materials ky (MN/m3/2) Reference

Cu 0.15 Hansen (1982) Mild Steel 0.74 Meyers (1998)

Cu-30% Zn 0.22 Phillips (1972) Fe-3% Si 1.08 Hull (1975)

Al 0.11 Abson (1970) Spheroidized Steel 0.41-0.58 Anand (1976)

Ag 0.07 Meyers (1998) Carbon Steel (0.03%) 0.81 Chang (1985)

Ni 0.30 Suits (1961) Carbon Steel (0.07%) 0.88 Chang (1985)

HCP Carbon Steel (0.17%) 1.21 Chang (1985)

Materials ky (MN/m3/2) Reference Carbon Steel (0.23%) 1.58 Chang(1985)

Zn 0.22 Meyers (1998) Fe-3% Si 0.70 OSU

Mg 0.28 Meyers (1998) Stainless Steel 439 0.44 OSU

Ti 0.40 Meyers(1998) Minimum alloy steel 0.88 OSU

R. H. Wagoner

Dislocation Transmission: SWC Criteria

9

• SWC1: Livingston and Chalmers’ criterion

•

•SWC2: Intersection line and slip direction criterion

•

•SWC3: Stress criterionSlip system chosen which the force on the head dislocation is maximized

•

•SWC4: Combined geometric and stress criterionSlip plane chosen by SWC2 + slip direction by SWC3

))(())(( 11111 iiii eggeggeeN

)()( 112 ii ggLLN

Z. Shen, R. H. Wagoner, and W. A. T. Clark: Dislocation and Grain Boundary Interactions in Metals, Acta Metall., 1988, Vol. 36, pp. 3231-3242.

R. H. Wagoner

Two-Scale Model

10

M. G. Lee, H. Lim, B. L. Adams, J. P. Hirth, R. H. Wagoner: A Dislocation Density-Based Single Crystal Constitutive Equation, Int. J. Plasticity, 2010, Vol. 26, pp. 925-938

H. Lim, M. G. Lee, J. H. Kim, B. L. Adams, R. H. Wagoner: Simulation of Polycrystal Deformation with Grain and Grain Boundary Effects, Int. J. Plasticity, 2011, vol. 27, pp. 1328-1354.

R. H. Wagoner

Two-Scale Simulation Procedure

11

Texture Analysis

Two-Scale Model

2nd level: Meso-Scale Simulation (MSS)

1st level: Grain-Scale Simulation (GSS)

,b

, ,ij FE, output:

Dislocation based, output:

Taylor, Sachs, etc.:

R. H. Wagoner

Two-Scale Model

Grain-Scale Simulation (GSS)

12

M. G. Lee, H. Lim, B. L. Adams, J. P. Hirth, R. H. Wagoner: A Dislocation Density-Based Single Crystal Constitutive Equation, Int. J. Plasticity, 2010, Vol. 26, pp. 925-938

R. H. Wagoner

Peirce, Asaro, Needleman (“PAN”) Framework

13

• Deformation gradient

• Rate of change of deformation gradient by dislocation glide

• Slip activities

peFFF

ppp FLF )(0

1

)(0

)(

nsL

NS

p

)(

1

)(

)(

0)( sign

m

g

R. H. Wagoner

Constitutive Eqns.: SCCE-T (“PAN”) vs. SCCE-D

14

Hardening Law(Asaro, 1985)

Hardening of slip systems(Peirce, 1982; Brown, 1989)

αβhg

SCCE-T: Single Crystal Constitutive Equations - Texture

4 arbitrary parameters (g0, gs, h0 ,a

aβ

β 0 βs

gh h 1g

SCCE-D: Single Crystal Constitutive Equations – Dislocation densityn

ααβ

β 1

g hb

α βαβh n ξ

αb

a

1 k γk

n

b

Evolution of dislocation density

(Kocks, 1976)

Hardening Law

3 arbitrary parameters (ka, kb, ρ0

αβ β

1 1.4h =h

. 1sym

0.4

Lee IJP 2010

R. H. Wagoner

Constitutive Equations: SCCE-D

15

l

)()(

lbg

)(

)(

1

l

Effective forest dislocation density

)()()()()( cos ξn

h

Orowan model [ E. Orowan, 1948]

Forest dislocation

Active (moving) dislocation

Slip plane

n()

( ) ( ) ( )

1h

nαg μb

R. H. Wagoner

SCCE-T vs. SCCE-D: Copper Single Crystal

16

Engineering Strain

0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200

250

Measured SCCE-T (Fit)SCCE-D (Fit)

Cu [001] (Takeuchi, 1975)(8 equal slip systems)

Engineering Strain

0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200

250Cu [-111] (Takeuchi, 1975)(6 equal slip systems)

SCCE-D

SCCE-T

Measured

Engineering Strain

0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200


SCCE-D

SCCE-T

Measured

Engineering Strain

0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200

250Cu [-123] (Takeuchi, 1975)(single slip system)

SCCE-D

SCCE-T

Measured

Fit Predicted

PredictedPredicted

R. H. Wagoner

Comparison: SCCE-T and SCCE-D

17

Fit direction Tensile axis direction

SCCE-T SCCE-D

Cu[001]

[111] 8 (6%) 23 (16%)[-112] 54 (150%) 13 (35%)[-123] 58 (451%) 10 (79%)

Fe [001] [011] 5 (8%) 7 (11%)[-348] 31 (66%) 15 (31%)

Avg. (Multiple slip fit) 31 (136%) 14 (34%)Cu

[-123][001] 62 (82%) 38 (50%)[111] 113 (78%) 79 (55%)[-112] 19 (54%) 7 (20%)

Fe [-348] [001] 30 (42%) 23 (32%)[011] 24 (36%) 20 (30%)

Avg. (Single slip fit) 50 (58%) 33 (37%)

2error percentage(%) = standard deviation/averaged flow stress×100

Standard deviations1 and error percentage2 between predicted and measured stress-strain curves

1standard deviation = 2( ( ) )measured

n ( 0,0.01,...,0.1)

R. H. Wagoner

Polycrystal Simulation: Texture Evolution

18

50% Compression

50% Tension

SCCE-T SCCE-D

Initial Random Orientations

{110} {111}

SCCE-T SCCE-D

Cu {110} Fe {111}

SCCE-T SCCE-D SCCE-T SCCE-D

Cu {110} Fe {111}

Initial Mesh

R. H. Wagoner

Two-Scale ModelMeso-Scale Simulation (MSS)

19

H. Lim, M. G. Lee, J. H. Kim, B. L. Adams, R. H. Wagoner: Simulation of Polycrystal Deformation with Grain and Grain Boundary Effects, Int. J. Plasticity, 2011, vol. 27, pp. 1328-1354.

R. H. Wagoner

MSS Assumptions, Simplifications

• Dislocation glide only

• Superdislocation lumping

• No elastic image effects, all gb effects into obs

• Edge dislocations only

• Dislocations interact within 1 slip system, 1 grain

20

R. H. Wagoner

MSS Overview

21

Steps

1) Redistribution of dislocations

2) Back stress calculation (τb)

3)Transmission criterion (τobs)

Superdislocation Lumpingα-th slip plane

, B

n

s

( ) ( ) ( )( ) ( ) ( )

( )

V bB n bL

R. H. Wagoner

1) Redistribution of Dislocations

inid out

id

1id id 1id

1il il 1il

netid

1

1( )in i ii

i i

d ddb l l

1

1( )out i ii

i i

d ddb l l

in outi i id d d

1passd d

bl

md b dx Orowan’s Equation (Orowan, 1940)

22

Net dislocation density for i-th element

Net dislocation density that passes through the element

R. H. Wagoner

2) Back Stress Calculation

ith superdislocation

jth superdislocation

r1

r2

r3dlj

dli

ith superdislocation

Y

X

Z slip direction

slip normal direction

y2

y1x2

x1

Bi

Bj

jth superdislocation

iξ

ˆjξ

2 2 22 1 2

2 21 2

2 2 21 2

( )

( )

ij j ijij

ij i j

r r rg x RR r r

R r r y x

111 22 12 212 2

2 1 1 2

F 1F4 (1 ) ( )

glide i jij

i

B B r g g g gdl x x r r

23

1

1 FN

bi ij

jij i

b

Back Stress:

where

R. H. Wagoner

3) GB Obstacle Stress

24

)()( 112 ii ggLLN (SWC-2 Model)

• Obstacle strength with slip transmissivity

*)1( Nobs

yN 5*10

stress obstacle minimum:1stress obstacle maximum:0

NN

• Dislocation and grain boundary interactions

iL

ig

: intersection lines between grain boundary and slip plane

: slip directions

Boundary (MPa) Transmissivity (N)1 380 0.5882 280 0.9153 870 0.4724 400 0.785

obs

Measured τobs for SS304 (Shen et al., 1986) and calculated transmissivity.

(Shen et al., 1986)

R. H. Wagoner

MSS-GSS Update

25

1/

0 ( )m

b obsb obssign

g

For non-GB elements,

1/

0 ( )m

bbsign

g

For GB elements,

b obs 0

b obs

R. H. Wagoner

Test: 1-D Stressed Pileup

26

122(1 )( ) v l xn x

b x

ˆ 0i ij i ij

F F B

R. H. Wagoner

Predicted Evolution of Dislocation Densities

27

Test geometry

Mises stress at 10% strain

R. H. Wagoner

Predicted Evolution of Dislocation Densities

28

Dislocation densities w/ strain

Dislocation densities for two slip systems

2( )m 2( )m

211 111 112 111

R. H. Wagoner

Accumulated Absolute Strain

0.00 0.02 0.04 0.06 0.08 0.10

Abs

. Tru

e St

ress

(MPa

)

0

50

100

150

200 Compression

Tension

Compression-Tension

Predicted Bauschinger Effect

29

Compression- tension test

R. H. Wagoner

Predicted Bauschinger Effect


0.00 0.02 0.04 0.06 0.08 0.10

Abs

. Tru

e St

ress

(MPa

)

0

50

100

150

200

Compression

Tension

Tension-Compression


0.00 0.02 0.04 0.06 0.08 0.10

Abs

. Tru

e St

ress

(MPa

)

0

50

100

150

200 Compression

Tension

Compression-Tension

30

Tension-compression test Compression- tension test

Grain φ1 Φ φ2

A 56.7 63.0 185.4

B 130.7 35.2 10.3

C 259.9 25.3 357.5

D 353.9 131.7 271.8

Initial grain orientations

R. H. Wagoner

Multicrystal Testing

31

R. H. Wagoner

Multicrystal Testing

32

3mm

15.42

62.00

0.00

R16.00

2.00

10.00

R. H. Wagoner

Material Selection

33

Stress- strain curves (as received) Hall-Petch slopesD-0.5 (m-0.5)

0 50 100 150 200 250

Eng.

Str

ess

(MPa

)

0

100

200

300

400

500

600

0.88 MN/m3/2

0.70 MN/m3/2

Minimum Alloy Steel

Fe-3% Si

Eng. Strain

0.0 0.1 0.2 0.3 0.4 0.5

Eng.

Str

ess

(MPa

)

0

100

200

300

400

Fe-3% Si

Minimum Alloy Steel

Grain size: 10m-30mm

Grain size: 60m-1.4mm

R. H. Wagoner

Tensile Tests: MAS

Grain map (9-39 grains)

34

Eng. Strain

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Eng.

Str

ess

(MPa

)

0

20

40

60

80

100

120

140

160

180

Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6

Minimum alloy steelStrain rate=5x10-4s-1

Measured stress-strain response

R. H. Wagoner

Meshing Procedure

35

Measured OIM figure in bitmap data

Discretized into elements

Grain information assigned for each element from bitmap data

R. H. Wagoner

Eng. Strain

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Eng.

Str

ess

(MPa

)

60

80

100

120

140

160

180

Measured Two- scalePANTaylor (SCCE-T)Taylor (SCCE-D)

Sample 6

Fit Procedure

36

Two-scale(SCCE-D)

PAN(SCCE-T)

Taylor(SCCE-T)

Taylor (SCCE-D)

1.1 1.1 2.3 0.9

Std. error of fit (MPa)

Fitted: Sample 6: 39 grains

SCCE-T: 4 fitting parameters (g0, gs, h0 ,a

SCCE-D: 3 fitting parameters (ka, kb, ρ0

R. H. Wagoner

Predicted Hardening, Sample 1 (9 grains)

37

(9 grains)

R. H. Wagoner

Prediction Summary: Multicrystal Hardening

Standard deviations between predicted and measured hardening curves (Unit: MPa)

38

Samples # of grains

Taylor (PAN)

Taylor(SCCE-D)

CPFEM(PAN)

CPFEM(SCCE-D)

Two-scale (SCCE-D)

Sample 1 9 22.6 18.6 19.8 4.9 4.6

Sample 2 12 16.8 13.6 16.3 11.1 3.7

Sample 3 18 6.6 19.1 6.0 12.1 4.4

Sample 4 32 25.1 9.5 22.3 8.0 4.2

Sample 5 34 21.8 25.3 18.4 9.2 11.1

Average 18.6 17.2 16.6 9.1 5.6

R. H. Wagoner

Lattice Curvature, Fe-3% Si

39

Inverse pole figure Surface image (optical)

Measured lattice curvature Predicted lattice curvatureMax= 9.2×10-3 rad/μm Avg.= 7.3×10-5 rad/μm

Max= 9.5×10-3 rad/μm Avg.= 5.8×10-4 rad/μm

R. H. Wagoner

Hall-Petch Tests

40

R. H. Wagoner

Size-Dependent Simulation

41

32 grains 64 grains16 grains

L=1mm

4 grains

8 grains 16 grains 64 grains 125 grains

L=1mm

L=1mm

2D grain assemblies

3D grain assemblies

R. H. Wagoner

Size-Dependent Simulation

42

Measured and simulated Hall-Petch slopes (0.2% offset, MN/m3/2)Measured H-P model Two-scale (2D) Two-scale (3D)0.9 ± 0.1 0.2

(4.5×)1.2 ± 0.3

(1.3×)1.5 ± 0.3

(1.7×)

D-0.5 (m-0.5)

0 20 40 60 80 100 120 140

Yiel

d St

ress

(MPa

)

40

60

80

100

120

140

160

180

200

220

240

Measured

Two-scale model (2D)Two-scale model (3D)

ky(YS)=0.9 0.1 MN/m3/2

Minimum Alloy Steel

ky(YS)=1.2 0.3 MN/m3/2ky(YS)=1.5 0.3 MN/m3/2

Pileup model (2D)ky(YS)=0.03 0.01 MN/m3/2

R. H. Wagoner

Effect of τ* on ky

43

Effect of τ* on Hall-Petch slope for 3D grain arrays.* (MPa)

0 200 400 600

Hal

l-Pet

ch S

lope

, ky (

MN

/m3/

2 )

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Measured ky (YS)

*=375 MPa (5 x YS)

Minimum alloy steel3D grain assemblies

R. H. Wagoner

Conclusions

1. First realistic, quantitative prediction of Hall-Petch.

2. First quantitative prediction of distributions with- , - gb interactions.

3. New practical meso-scale method for , strain. Enable new dimension of material design (gb)?

4. Extension of SWC gb transmission criterion to predict obs.

44

R. H. Wagoner

Thank you!

45

R. H. Wagoner

Additional Slides

46

R. H. WagonerEngineering Strain

0.00 0.05 0.10 0.15 0.20

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200

250

kb=20bkb=30bkb=40b

Engineering Strain

0.00 0.05 0.10 0.15 0.20

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200

250

300

350

ka=10 ka=20 ka=30

Engineering Strain

0.00 0.05 0.10 0.15 0.20

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200

0=109m-2

0=1011m-2

0=1013m-2

SCCE-D: Effect of Variables

47

Increasing ρ0

Increasing ka

Increasing kb


0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200


SCCE-D

SCCE-T

Measured

Engineering Strain

0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200


SCCE-D

SCCE-T

Measured

Engineering Strain

0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200

250


Cu [-123] (Takeuchi, 1975)(single slip system)

SCCE-T vs. SCCE-D: Copper Single Crystal

48

Engineering Strain

0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200

250Cu [001] (Takeuchi, 1975)(8 equal slip systems)

SCCE-D

SCCE-T

Measured

Parameters fit to measured Cu single crystal [-123]


0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200Fe [011] (Keh, 1964)(2 equal slip systems)

SCCE-D

SCCE-T

Measured

Engineering Strain

0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200Fe [-348] (Keh, 1964)(single slip system)

SCCE-DSCCE-T

Measured

SCCE-T vs. SCCE-D: Iron Single Crystal

49

Engineering Strain

0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200


Fe [001] (Keh, 1964)(4 equal slip systems)

Parameters fit to measured Fe single crystal [001]


0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150


SCCE-D

SCCE-T

Measured

Engineering Strain

0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

20

40

60

80

100

120


SCCE-D

SCCE-T

Measured

SCCE-T vs. SCCE-D: Iron Single Crystal

50

Engineering Strain

0.00 0.02 0.04 0.06 0.08 0.10

Engi

neer

ing

Stre

ss (M

Pa)

0

50

100

150

200


Fe [-348] (Keh, 1964)(single slip system)

Parameters fit to measured Fe single crystal [-348]

R. H. Wagoner

Polycrystal Simulation: Texture Evolution

51

50% Compression

50% Tension

SCCE-T SCCE-D

Initial Random Orientations

{110} {111}

SCCE-T SCCE-D

Cu {110} Fe {111}

SCCE-T SCCE-D SCCE-T SCCE-D

Cu {110} Fe {111}

Initial Mesh

R. H. Wagoner

Superdislocation lumping

52

Slip plane

ElementDiscrete dislocation pileup

Continuous pileup

Finite element discretization into superdislocations

Dislocation density, ρ

( ) ( ) ( )( ) ( ) ( )

( )

V bB n bL

R. H. Wagoner

Dislocation Transmission: SWC Criteria

53

• SWC1: Livingston and Chalmers’ criterion

• SWC2: Intersection line and slip direction criterion

• SWC3: Stress criterionSlip system chosen which the force on the head dislocation is maximized

• SWC4: Combined geometric and stress criterionSlip plane chosen by SWC2 + slip direction by SWC3

))(())(( 11111 iiii eggeggeeN

)()( 112 ii ggLLN

Comparison of the predicted and observed slip systems in five experiments

exp1 exp2 exp3 exp4 exp5

SWC1 0/5 X X X X XSWC2 3/5 X O O O XSWC3 3/5 O O X X OSWC4 5/5 O O O O O

R. H. Wagoner

Crystal Orientations vs. Dislocation Densities

Total dislocation density at 10% strainφ1=15° φ1=30° φ1=45°

6×1012

2( )m

4.5×1012

3×1012

54

Misorientation ( ) Stress at 10% (MPa)

0° 111.90

15° 111.60

30° 112.77

45° 113.65

A B y

xz

xy

zGrain A

Grain B

11

R. H. Wagoner

Material Selection

55

Desired material properties

• High Hall- Petch Slopes• Good Ductility / Hardening• Grain Size• Good OIM imaging/polishing

0

100

200

300

400

0 5 10 15 20 25 30

Eng.

Stre

ss (M

Pa)

Eng.Strain (%)

Fe-Si 3%

Stainless Steel 439

Minimum Alloy Steel

Stress- strain curves (as received) Hall-Petch slopes

Material ky [MN/m3/2] Ductility Grain sizeFe-Si 3% 0.70 8% 10µm~30mm

SS 439 0.44 13% 30µm~70µm

Minimum Alloy Steel 0.88 25% 60µm~1.4mm

D-0.5 (m-0.5)

0 50 100 150 200 250

Yiel

d St

ress

(0.2

% o

ffset

) (M

Pa)

0

100

200

300

400

500

600

0.44 MN/m3/2

0.88 MN/m3/2

0.70 MN/m3/2

Minimum alloy steel

Stainless steel 439

Fe-3% Si

100D (m)

500 50 30 202000

R. H. Wagoner

Heat Treatment of Minimum Alloy Steel

56

50um

1 mm

As received (D~60 µm)

1000ºC 5h (D~140 µm)

Strain annealing1000ºC 1h → 1~ 2.5% straining→ 1250ºC 10h (D~1350µm)

1250ºC 5h (D~620 µm)

C Mn P S Si Cu Ni Cr Mo Sn Al Ti N Nb0.001 0.13 0.006 0.005 0.004 0.023 0.007 0.014 0.003 0.002 0.038 0.001 0.003 0.001

Composition (wt%)

Eng. Strain

0.0 0.1 0.2 0.3 0.4 0.5

Eng.

Str

ess

(MPa

)

0

50

100

150

200

250

300

D=1350 m

D=620 m

D=140 m

D=60 m50um

Minimum Alloy SteelStrain rate =5x10-4 s-1

Grain Size: 60 ~ 1350mm

R. H. Wagoner

Tensile Tests MAS Specimens

15.42

60.00

10.00

R16.00

1.00

8.00

15.42

62.00

10.00

R16.00

2.00

10.00

Type I

Type II

Type III

Thickness=2.1 mm

Thickness=2.1 mm

Thickness=0.4 mm

15.42

62.00

10.00

R16.00

2.00

10.00

Sample 1, Sample 2

Sample 3, Sample 6

Sample 4, Sample 5

Dimensions of tensile samples (unit: mm) OIM grain map

57Eng. Strain

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Eng.

Str

ess

(MPa

)

0

20

40

60

80

100

120

140

160

180

Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6

Minimum alloy steelStrain rate=5x10-4s-1

# of grains Sample type Davg. (mm) YS (MPa) UTS (MPa)

Sample 1 9 I 0.53 84 ‐

Sample 2 13 I 0.38 96 167

Sample 3 18 II 0.82 63 167

Sample 4 32 III 0.09 94 144

Sample 5 34 III 0.11 90 159

Sample 6 39 II 0.46 74 ‐

Measured stress-strain response Material properties for six tensile samples

R. H. Wagoner

Simulation of MAS tensile specimens

Shear modulus (G)

80 242 150 112

C11 C12 C44

PAN model Taylor (SCCE-T)

h0(MPa) 423 402

gs(MPa) 162 240

g0(MPa) 38 40

a 2 2

Two-Scale model Taylor (SCCE-D)

ρ0(mm-2) 9.4×1011 1.1×1012

ka 63 16

kb 7b 25b

Shear modulus and anisotropic elasticity constants (GPa)

Fitting parameters

Meshing procedure

58

Measured OIM figure in bitmap data

Discretized into meshes

Grain information assigned for each element from bitmap data

R. H. Wagoner

0.00 0.02 0.04 0.06 0.08 0.10 0.120

20

40

60

80

100

Eng. Strain

y (M

Pa)

Sample 1

PAN modely= 83 MPastd.dev.=19.8 MPa

Two-scale modely= 72 MPastd. dev.=4.6 MPaMeasured

y= 86 MPa

Taylor Model (SCCE-T)y= 81 MPastd.dev.=22.6 MPa

Taylor Model (SCCE-D)y= 80 MPastd. dev.=18.6 MPa

Prediction of hardening of MAS

59

Sample 1 (9 grains)

0.00 0.02 0.04 0.06 0.08 0.10 0.120

20

40

60

80

100

Eng. Strain

y (M

Pa)

Sample 2 Taylor Model (SCCE-T)y= 78 MPastd.dev.=16.2 MPa


PAN modely= 80 MPastd. dev.=16.3 MPa


y= 101 MPa

Sample 2 (13 grains)

R. H. Wagoner

Prediction- Continued

60

0.00 0.02 0.04 0.06 0.08 0.10 0.120

20

40

60

80

100

Eng. Strain

y (M

Pa)



Two-scale modely= 73 MPastd. dev.= 4.2 MPa

Measuredy= 95 MPa

PAN modely= 82 MPastd. dev.= 22.3 MPa

0.00 0.02 0.04 0.06 0.08 0.10 0.120

20

40

60

80

100

Eng. Strain

y

(MPa

)

Sample 5

Iso-strain (SCCE-T)y= 81 MPastd.dev.=21.2 MPa

Iso-strain (SCCE-D)y= 79 MPastd. dev.=24.2 MPa

PAN modely= 81 MPastd. dev.= 18.4 MPa

Two-scale modely= 75 MPastd. dev.= 11.1 MPa

Measuredy= 90 MPa



0.00 0.02 0.04 0.06 0.08 0.10 0.120

20

40

60

80

100

Eng. Strain

y (M

Pa)



PAN modely= 81 MPastd. dev.=6.0 MPa


y= 74 MPa


R. H. Wagoner

Fe-3% Si Sample Preparation

61

C Mn P S Si Cu Ni Al N0.004 0.09 0.01 0.025 2.95 0.02 0.01 0.03 0.015

Chemical composition (wt.%)

Grain φ1 Φ φ2

1 61 38 2822 266 41 813 74 41 2654 248 30 88

Initial grain orientations (Bunge’s Euler angles, degrees)

Inverse Pole Figure

R. H. Wagoner

Lattice Curvature Calculation

j

i

dxd

ij (Nye, 1953; Sun et al., 2000)

3

1

2

1

3

1

2

1 61

61

i j j

i

i jij dx

d (Adams and Field, 1992; Sun et al., 2000;

El-Dasher et al., 2003)

62

eljkijkkkijijij e ,2

1

NS

s

sss

1

)()()( zbα

NS

s

sj

si

sij zb

1

)()()(or

R. H. Wagoner

Effect of crystal orientations vs. grain orientations

Uniaxial tension (~10%), strain rate=10-3s-1

Crystal orientation Stress at 10%

0° 119.04

15° 119.20

30° 118.94

45° 116.75

Δσ 2.45

x

y

z

0 15

30 45


0° 115.01

15° 115.96

30° 116.73

45° 116.75

Δσ 1.74

Rotation around z axis


0° 115.01

15° 115.25

30° 115.39

45° 115.35

Δσ 0.38

Rotation around x axisGrain A: (φ1,Ф,φ2) = (45,0,0)

Grain B: (φ1,Ф,φ2) = (0,0,0)

63

R. H. Wagoner

Measured Hall-Petch Slope

D-0.5 (m-0.5)

0 20 40 60 80 100 120 140

Stre

ss (M

Pa)

50

100

150

200

250

300

ky(UTS)=0.98 0.13 MN/m3/2

ky(YS)=0.88 0.08 MN/m3/2

ASTM E8 Subsize specimens

Undersizedspecimens

Minimum Alloy Steel

64

R. H. Wagoner

Hall- Petch Slopes for Various Materials

65

FCC BCCMaterials ky (MN/m3/2) Reference Materials ky (MN/m3/2) Reference

Cu 0.15 Hansen (1982) Mild Steel 0.74 Meyers (1998)

Cu-30% Zn 0.22 Phillips (1972) Fe-3% Si 1.08 Hull (1975)

Al 0.11 Abson (1970) Spheroidized Steel 0.41-0.58 Anand (1976)

Ag 0.07 Meyers (1998) Carbon Steel (0.03%) 0.81 Chang (1985)

Ni 0.30 Suits (1961) Carbon Steel (0.07%) 0.88 Chang (1985)

HCP Carbon Steel (0.17%) 1.21 Chang (1985)

Materials ky (MN/m3/2) Reference Carbon Steel (0.23%) 1.58 Chang(1985)

Zn 0.22 Meyers (1998) Fe-3% Si 0.70 OSU

Mg 0.28 Meyers (1998) Stainless Steel 439 0.44 OSU

Ti 0.40 Meyers(1998) Minimum alloy steel 0.88 OSU

R. H. Wagoner

Predicted Hall-Petch Slope Using Pileup Model

Material Measured ky Calculated ky

FCC Al 0.11 0.06Cu 0.15 0.12-0.30Ni 0.30 0.23-0.49

HCP Mg 0.28 0.05-0.09Ti 0.40 0.16-0.25

BCC Fe 0.74 0.18-0.23

1/21/2

0obsbM D

k

1/2obs

ybk M

k

Measured and calculated Hall-Petch slope (Unit: MN/m3/2)

66

R. H. Wagoner

Effect of Grain Boundary Strength on ky

67

* (MPa)

0 200 400 600

Hal

l-Pet

ch S

lope

, ky (

MN

/m3/

2 )

0.0

0.5

1.0

1.5

2.0

2.5

Measured ky (YS)

Measured ky (e=0.1)

Simulated ky (e=0.1)

Simulated ky (YS)

Effect of τ* on Hall-Petch slope for 3D grain arrays.

R. H. Wagoner

Slip activity (FEM) vs. Schmid factor

Specimens # of grains PredictionFeSi sample 1 2 2/2 (100%)FeSi sample 2 12 10/12 (83%)FeSi sample 3 22 20/22 (91%) MAS sample 1 9 5/9 (56%)MAS sample 2 13 8/13 (62%)MAS sample 3 18 11/18 (61%)MAS sample 4 32 26/32 (81%)MAS sample 5 34 24/34 (71%)MAS sample 6 39 24/39 (62%)

Comparison between most active slip systems by(1) FEM (slip activity) and (2) Schmid factor calculation

68

a practical meso-scale polycrystal model to predict dislocation...

Documents