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(1)

결정미소역학 (Crystal Mechanics)

Lecture 9 – Theory of Crystal Plasticity

R f C ti Th f Pl ti it A S Kh d S H 1995

Ref : Continuum Theory of Plasticity, A.S. Khan and S. Huang, 1995

Heung Nam Han

Associate Professor

Department of Materials Science & Engineering College of Engineering g g g

Seoul National University Seoul 151-744, Korea

Tel : +82-2-880-9240

2010-11-15

Tel : +82 2 880 9240 Fax : +82-2-885-9647 email : hnhan@snu.ac.kr

Jump to first page

(2)

Kinematics of Single Crystal Kinematics of Single Crystal

Deformation

Deformation of metal ;

Elastic Deformation + Rigid Body + Plastic Deformation g y

Elastic Deformation ;

No dislocation movement , volume change

Rigid Body Rotation ;

Rigid Body Rotation ;

No dislocation movement , no volume change

Plastic Deformation ;

Dislocation movement , no volume change

(3)

3

Kinematics of Single Crystal Kinematics of Single Crystal

Deformation

ee

F

e

X F

x d

d

li l l

2010-11-15

n :slip plane normal

s : slip direction

Total deformation = Plastic Deformation + Other Deformation.

(4)

Kinematics of Single Crystal Kinematics of Single Crystal

Deformation

1. Plastic Deformation ; F

p

X F

p d

d 

p

d p F d X

Thus F

p

can be given

0

0

p

I n s

F  

(5)

Kinematics of Single Crystal

5

Kinematics of Single Crystal Deformation

2. Elastic Deformation and Rigid Body Rotation ; F e

dx Fe dp

Finally by multiplicative decomposition of deformation gradient, total deformation gradient F is given by.

p

e F

F F

fi d l i d f i di p

To find pure plastic deformation gradient Fp F

F Fp ( e)1

2010-11-15

(6)

Kinematics of Single Crystal Kinematics of Single Crystal

Deformation

Definition of slip system vector at current state s , n due to Fe

s0

F

ses : Direction 0

1 0 ( )

n Fe

n n0 : Area (plane normal)

0 0 )

(

α 0 α

α α

1 e α

0 α

0 e

s F )

(F n

s n

F n

s F n

s

e 1

e

0

( )

0

0

Velocity Gradient L at Current Configuration;y g ;

e 1 p 1

p 1 e

e e

1

     

F F F F F F F F

L   

L F F F FF F F F

(7)

Kinematics of Single Crystal

7

Kinematics of Single Crystal Deformation

W Here

D, can be written as

p

e D

D D

Here,

0 0

0 I s n

Fp D D D

p

e W

W

W

0 0 p

0 s n

F since s0 n0 0

e 1 e e e

e

D W F F

L

e 1 p 1

p e p p

p

D W F F F F

L p e 1

0

e

F L F

Insert above into Lp0 F0p F0p1

1

p

p L

L

0 0 0 0 0

0

0 0 0

0 p 1

0 p

0

n s n s n

s

} n

s { n

s I

F F

0

0 L

L

p 1 0 p

0 p

0

F F

L Using normality of slip vectors

p

2010-11-15

0 0

p

0 s n

L

(8)

Kinematics of Single Crystal g y Deformation

p 0 p

0 p 1

p p

0 F F D W

L

p pT

p 1

L L

D

p0T

Wp 1

Lp LpT

p 0 p

0 2

1 L L

D

p0T

p 0 p

0 2

1 L L

W

n s n

Lp0 0 0

At Reference Configuration

s n L

1

0 0 0

symm. : Dp0

n P0  here 0

s0 n0 n0s0

2

P 1

 1 2

anti. :

n W

1 0 p

0 here 0

s0 n0 n0s0

2 1

 1

(9)

Kinematics of Single Crystal g y

9

Deformation

At Current Configuration

e 1

0 0

e

s and n n

s

F

F

n

L

p

s n 

 1

n

p P

D  here P

s n ns

2 1

 1 2

n Wp

1

 here

s n ns

2 1

2010-11-15

 1

(10)

Constitutive Equation of Single q g Crystal

• Macroscopic

F F I

E*  1 *T *T * F *1

 

det F *

F *T

 

*

T

*

C

E

E

F F I

E  

2 T F

 

det F

F

F* F Fp 1 F : Deformation Gradient

p P

0

p L F

F 

0

P

0 m

L

LP : Plastic Velocity Gradient : Plastic Shearing Rate

0

0 m

L g

on Slip System 

m0 s0 n0

( )

/

1 m

sign m : Rate Sensitivity

• Microscopic

 

(11)

Basic Considerations

11

for polycrystalline material

Sachs Model:

- All single-crystal grains with aggregate or polycrystal

i h f

experience the same state of stress;

- Equilibrium condition across the grain boundaries satisfied;

satisfied;

- Compatibility conditions between the grains violated;

T l M d l Taylor Model:

- All single-crystal grains within the aggregate experience the same state of deformation (strain);

experience the same state of deformation (strain);

- Equilibrium condition across the grain boundaries violated;

2010-11-15

violated;

- Compatibility conditions between the grains assured;

(12)

Sachs and Taylor Model

Sachs and Taylor Model

(13)

Example 1 p

- Texture Evolution of Torsion-Reverse Torsion

School of Material Science and Engineering School of Material Science and Engineering

Seoul National University S.-J. Park et. al

Jump to first page

(14)

Torsion Reverse Torsion Torsion-Reverse Torsion



 = tan

 = 0  = tan  = 0

 = 0  = 0

Restoration of macroscopic shape after simple shear deformation (zero net shear strain)

( )

(15)

15

B k d

Backgrounds

• Backofen (1950) : Cu,  = 5.25 0

residual shear texture, restoration to initial grain shape

• Rollet et al. (1987) : Al-1Mg,  = 3.5 0

residual shear texture, restoration to initial grain shape residual shear texture, restoration to initial grain shape Taylor model restoration to initial texture

• Cho et al. (1998) : Cu,  = -0.5 0

similar to initial texture, restoration to initial grain shape

small  similar to initial texture large resid al shear te t re

Cause

1. Grain division into subgrains large residual shear texture 2. Grain interaction

(16)

Experimental Procedures Experimental Procedures

Torsion

Torsion Drilling Drilling Etching Etching

Diameter : 6mm Gauge Length :

10mm

Shear strain rate : 0.016/sec

• Initial

 = -2.5

 = -2.5 x 2.5

Flattening Flattening Pole Figure

Pole Figure Flattening Flattening Pole Figure

Pole Figure

(17)

17

Experimental Photographs Experimental Photographs

150m

(a)  = 0 (b)  = -2.5 (c)  = -2.5 x 2.5

2010-11-15

(18)

(111) pole figure

Ideal orientations

A fiber B fiber Shear

plane Shear direction

A A (1 1 1) [1 1 0]

A A1*

A A *

A (1 1 1) (1 1 1) (1 1 1) (1 1 1)

[1 1 0]

[2 1 1]

[1 0 1]

[1 1 2]

A2* A

B

(1 1 1) (1 1 1) (1 1 2)

[1 1 2]

[0 1 1]

[1 1 0]

C B A

(0 0 1) (1 1 2) (1 1 1)

[1 1 0]

[1 1 0]

[1 1 0]

(19)

19

(111) Pole Figures (Experimental Data) (111) Pole Figures (Experimental Data)

SPN SPN SPN

SD

Contour Lines : 1.0 1.5 2.0 2.5 3.0

( ) I iti l (b) 2 5 ( ) 2 5 2 5

2010-11-15

(a) Initial (b)  = -2.5 (c)  = -2.5 x 2.5

(20)

P l t l M d l Polycrystal Model

• Sachs : the same stress state in single crystal grain and aggregate equilibrium (O) compatibility (X)

equilibrium (O) , compatibility (X)

• Taylor : the same strain state in single crystal grain and aggregate equilibrium (X) , compatibility (O)

• Self-Consistent Model Single crystal plasticity by FEM

• Self-Consistent Model, Single crystal plasticity by FEM equilibrium (O) , compatibility (O)

(21)

21

Finite Element Calculation Finite Element Calculation

ABAQUS/Standard (Version 5.7) User Subroutine UMAT

Ti I t ti * *tr

• Time Integration :

D

*( ) *tr

Newton Iterative Method

 

*n1( ) *n( ) Hn1 Gn

H  J

D Hn  J

D

n( ) Gn *n( ) *tr

D

• Jacobian Matrix : Perturbation Method

(Numerical Differentiation)

2010-11-15

(Numerical Differentiation)

Virtual work principle equilibrium and compatibility

(22)

Initial Finite Element Mesh t a te e e t es

SPN

(23)

23

(111) Pole Figure of Initial Random Texture

SPN SPN

SD

2010-11-15

Number of crystals = 400

(24)

Deformed Meshes Deformed Meshes

 = -1  = -1 x 1  = -4 x 4

(25)

25

(111) Pole Figure (by FEM) (111) Pole Figure (by FEM)

 = -1  = -1 x 1  = -4  = -4 x 4

2010-11-15

(26)

(111) Pole Figure (by Taylor model) (111) Pole Figure (by Taylor model)

 = -1  = -1 x 1  = -4  = -4 x 4

(27)

27

Summary Summary

Si l t l l ti it b FEM

• Single crystal plasticity by FEM

Small  similar to initial texture Large residual shear texture Large  residual shear texture

• Taylor model Taylor model restoration to initial texture restoration to initial texture

• Grain Interaction residual shear texture

2010-11-15

(28)

- Prediction of Rolling Texture Evolution i FCC P l t lli M t l

in FCC Polycrystalline Metals

Using Finite Element Method of Crystal Plasticity

School of Material Science and Engineering Seoul National University

Seoul National University

S.-J. Park et. al

(29)

29

단결정의 구성식 단결정의 구성식

 

• Macroscopic

 

E* 1 F F*T * I * F*1

 

detF*

F*T

 

*

*CE E

 

E F F I

2

   

F* F Fp 1 F : Deformation Gradient

Fp L Fp p Lp

m

LP : Plastic Velocity Gradient : Plastic Shearing Rate

on Slip System g m0 s0 n0

  ( )

/

  

 

1 m

sign m : Rate Sensitivity

• Microscopic

 

F F*T * *:m0  *:m0

(30)

유한요소 계산 유한요소 계산

• 평면변형 압축 (Plane Strain Compression) 가정 ( p )

E = 200GPa,  = 0.3, 

0

= 100 MPa, 

0

= 0.01sec

-1

m = 0.05, 0.2, 0.5

(31)

31

초기 무질서 방위의 (111) 극점도 초기 무질서 방위의 (111) 극점도

RD RD

TD

2010-11-15

Number of crystals = 400

(32)

초기 유한요소망과 경계조건기 유한 망과 경계 건

ND

(33)

33

변형된 요소망 ( = -1) 변형된 망 (

)

m = 0.05

m = 0.2

2010-11-15

m = 0.5

(34)

평면변형 압축시 계산된 (111) 극점도 평면변형 압축시 계산된 (111) 극점도

m = 0 05 m = 0 2 m = 0 5

m = 0.05 m = 0.2 m = 0.5

(35)

35

평면변형 압축시 주요 집합조직 성분 평면변형 압축시 주요 집합조직 성분

RD

TD

{4 4 11}<11 11 8>

{ }

{1 1 0}<1 1 2>

(36)

평면변형 압축시 계산된 ODF 평면변형 압축시 계산된 ODF

Fiber A

Fiber C

(37)

37

계산된 방위밀도 (섬유A) 계산된 방위밀도 (섬유A)

30

(1 1 0) [0 0 1]

(1 1 1) [1 1 2]

(4 4 11) [11 11 8]

(0 0 1) [1 1 0]

m=0.05 m=0.2 20

m 0.2 m=0.5

f(g)

10 1 = 90o

(g)

2 = 45o

2010-11-15

0 30 60 90

0

(38)

계산된 방위밀도 (섬유C) 계산된 방위밀도 (섬유C)

(1 1 0) [0 0 1]

(1 1 0) [1 1 2]

(1 1 0) [1 1 1]

(1 1 0) [1 1 0]

30

[ ]

[ ]

m=0 05

20

m 0.05 m=0.2 m=0.5

f(g)

10 1 = 90o 45o

f(g)

2 = 45

(39)

39

요 약 요 약

대변형 탄소성 속 의 성방정식을 이용 유한 소해석

• 대변형, 탄소성, 속도 의존 구성방정식을 이용 유한요소해석

• 소성 이방성의 차이 불균일한 변형

• 소성 이방성의 차이 불균일한 변형

• 평면변형 압축 압축

low m : Dillamore {4 4 11}<11 11 8>

high m : Brass {1 1 0}<1 1 2>

(40)

- Finite Element Simulation of Grain

Fragmentation during a Cold Rolling Process g g g of BCC Metals

Technical Research Laboratories, POSCO S -J Park et al

S.-J. Park et. al

(41)

41

Orientation Stability Orientation Stability

Metastable

Unstable Unstable

Goss {011}<100>

under Plane Strain Compression

Stable

(011)[100]

(111)[2-1-1] (-111)[211]

(42)

Objects Objects

To simulate orientation fragmentation

To simulate orientation fragmentation numerically. y

To investigate the effect of grain interaction on orientation fragmentation

on orientation fragmentation.

(43)

43

Initial Finite Element Mesh Initial Finite Element Mesh

2010-11-15

(44)

Lattice Rotation Rate Map Lattice Rotation Rate Map

1

=0

o

section of Euler space

Center

Surrounding

Surrounding

(45)

Center and Surrounding Orientations

45

Ce e a d Su ou d g O e a o s

Normal Rolling Euler Angles

Location Normal Direction

Rolling

Direction Stability

1oo2o

Center

0 35 45 Stable

0 18 73 Unstable

  1 1 2

1 3 4 4 2

  1 1 0

4 1 3 0

Center 0 18 73 Unstable

0 45 0 Metastable

1 3 4 4 2

  0 1 1

4 1 3 0

1 0 0

0 0 45 Stable

0 0 1

4 1 3 4 2

  1 1 0

 

Surrounding 0 18 17 Unstable

0 90 45 Metastable

4 1 3 4 2

  1 1 0

13 4 0

  1 1 0

2010-11-15

   

All 9 combinations for bicrystal

(46)

Deformed Meshes (= -0.69) Deformed Meshes (= 0.69)

Initial Center

 

112

Orientation

   

11 0

 

143143402

 

 

011

  

100110

I iti l Initial

(47)

(111) Pole Figures for   1 1 2   1 1 0

47

( ) g    

Initial Center Orientation

 

112

 

11 0

 

Initial

2010-11-15

0 01

  

11 0

413 42

 

13 4 0

  

110

 

11 0

Surrounding Orientation

(48)

(111) Pole Figures for 1 3 4 4 2   4 1 3 0  

Initial Center Orientation

1 3 4 4 2   4 1 3 0

 

Initial

           

(49)

(111) Pole Figures for   0 1 11 0 0

49

( ) g    

Initial Center Orientation

 

011

10 0

 

Initial

2010-11-15

0 01

  

11 0

413 42

 

13 4 0

  

110

 

11 0

Surrounding Orientation

(50)

in Center Crystal   0 1 11 0 0

0 01

  

11 0

413 42

13 4 0

  

110

 

11 0

Initial Surrounding

Orientation O e tat o

(51)

51

Summary Summary

• The orientation fragments in the metastable  

011

10 0

initial orientation.

• Grain interaction with neighbor grains does not affect

• Grain interaction with neighbor grains does not affect the orientation fragmentation tendency of the

the orientation fragmentation tendency of the metastable initial orientation a lot.  

011

10 0

2010-11-15

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