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
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
Kinematics of Single Crystal Kinematics of Single Crystal
Deformation
ee
F
eX F
x d
d
li l l
2010-11-15
n :slip plane normal
s : slip direction
Total deformation = Plastic Deformation + Other Deformation.
Kinematics of Single Crystal Kinematics of Single Crystal
Deformation
1. Plastic Deformation ; F
pX F
p d
d
p
d p F d X
Thus F
pcan be given
0
0
p
I n s
F
Kinematics of Single Crystal
5Kinematics 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
Kinematics of Single Crystal Kinematics of Single Crystal
Deformation
Definition of slip system vector at current state s , n due to Fe
s0
F
s e s : 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
( )
00
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 F F F F F
Kinematics of Single Crystal
7Kinematics 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
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
Kinematics of Single Crystal g y
9Deformation
At Current Configuration
e 1
0 0
e
s and n n
s
F
F
nL
ps 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
Constitutive Equation of Single q g Crystal
• Macroscopic
F F I
E* 1 *T * T * F *1
det F *
F *T
*T
* C
EE
F F I
E
2 T F
det F
FF* 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
Basic Considerations
11for 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;
Sachs and Taylor Model
Sachs and Taylor Model
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
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
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
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
Experimental Photographs Experimental Photographs
150m
(a) = 0 (b) = -2.5 (c) = -2.5 x 2.5
2010-11-15
(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
(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
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
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
Initial Finite Element Mesh t a te e e t es
SPN
23
(111) Pole Figure of Initial Random Texture
SPN SPN
SD
2010-11-15
Number of crystals = 400
Deformed Meshes Deformed Meshes
= -1 = -1 x 1 = -4 x 4
25
(111) Pole Figure (by FEM) (111) Pole Figure (by FEM)
= -1 = -1 x 1 = -4 = -4 x 4
2010-11-15
(111) Pole Figure (by Taylor model) (111) Pole Figure (by Taylor model)
= -1 = -1 x 1 = -4 = -4 x 4
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
- 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
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
mLP : 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
유한요소 계산 유한요소 계산
• 평면변형 압축 (Plane Strain Compression) 가정 ( p )
E = 200GPa, = 0.3,
0= 100 MPa,
0= 0.01sec
-1m = 0.05, 0.2, 0.5
31
초기 무질서 방위의 (111) 극점도 초기 무질서 방위의 (111) 극점도
RD RD
TD
2010-11-15
Number of crystals = 400
초기 유한요소망과 경계조건기 유한 망과 경계 건
ND
33
변형된 요소망 ( = -1) 변형된 망 (
)
m = 0.05
m = 0.2
2010-11-15
m = 0.5
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
평면변형 압축시 주요 집합조직 성분 평면변형 압축시 주요 집합조직 성분
RD
TD
{4 4 11}<11 11 8>
{ }
{1 1 0}<1 1 2>
평면변형 압축시 계산된 ODF 평면변형 압축시 계산된 ODF
Fiber A
Fiber C
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
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
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>
- 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
Orientation Stability Orientation Stability
Metastable
Unstable Unstable
Goss {011}<100>
under Plane Strain Compression
Stable
(011)[100]
(111)[2-1-1] (-111)[211]
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
Initial Finite Element Mesh Initial Finite Element Mesh
2010-11-15
Lattice Rotation Rate Map Lattice Rotation Rate Map
1=0
osection of Euler space
Center
Surrounding
Surrounding
Center and Surrounding Orientations
45Ce e a d Su ou d g O e a o s
Normal Rolling Euler Angles
Location Normal Direction
Rolling
Direction Stability
1o o 2o
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
Deformed Meshes (= -0.69) Deformed Meshes (= 0.69)
Initial Center
112Orientation
11 0
143143402
011
100110
I iti l Initial
(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 0Surrounding Orientation
(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
(111) Pole Figures for 0 1 1 1 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 0Surrounding Orientation
in Center Crystal 0 1 1 1 0 0
0 01
11 0
413 42
13 4 0
110
11 0Initial Surrounding
Orientation O e tat o
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