IHPC-IMS Program on Advances & Mathematical Issues in Large Scale Simulation (Dec 2002 - Mar 2003 & Oct - Nov 2003) Tutorial I: Mechanics of Ductile Crystalline Solids Alberto M. Cuitiño Mechanical and Aerospace Engineering Rutgers University Piscataway, New Jersey [email protected]Institute of High Performance Computing Institute for Mathematical Sciences, NUS
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IHPC-IMS Program on Advances & Mathematical Issues in Large Scale Simulation (Dec 2002 - Mar 2003 & Oct - Nov 2003) Tutorial I: Mechanics of Ductile Crystalline.
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IHPC-IMS Program onAdvances & Mathematical Issues
in Large Scale Simulation(Dec 2002 - Mar 2003 & Oct - Nov 2003)
Tutorial I:Mechanics of Ductile Crystalline Solids
Alberto M. CuitiñoMechanical and Aerospace Engineering
• Metropolis Monte Carlo algorithm and mean field approximation at finite temperatures.
Singapore 2003 cuitiño@rutgers
Effective Energy
where with
S
ii
S
ekl
eijijkl dStdSxdcuE )(
2
1][ 3
pij
eijjiu ,
)(Sm Djip
ij Elastic interaction Core energy External field
Slip Surface
S
Displacement jump across Sm
Singapore 2003 cuitiño@rutgers
Elastic interaction
j
pmnijmnkik cGu
, Displacement field:
Elastic distortion:
Elastic interaction:
Green function for an isotropic crystal:
pklj
pmnijmnlki
ekl cG
,,
kdAE p
uvp
mnmnuv3*
21
3int ˆˆˆ
2
1
ki
kiki xx
rrG
22
28
1
Singapore 2003 cuitiño@rutgers
Elastic Interaction
kdkkKb
E 22
21
2
2int ˆ),(
4)2(
1][
bxxxx /),(),( 2121 with
22
21
22
22
21
21
21 1
1),(
kk
k
kk
kkkK
b
RA2
A1
21
2121
1
1
2int
10
01
8 AA
dSdSR
bE
(Hirth and Lothe,1969)
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External Field
Ndis
n n
n
xx
Cxs
1
)(
applied
shear stressforest dislocations
S
ext dSbsE
S
iiext dStE
with: ii bxxxx ),(),( 2121
||/),(),( 2121 bbxxtxxs ii
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Core Energy
ZS :
0 b/2 b-b/2-b-3b/2
() jjiiijZ
bbC
2
1min)(
ijij dC
2
22
4inf)(
d
bZ
S
core dSE )(
Ortiz and Phillips, 1999
d: inter-planar distance
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Phase-Field Energy
kdsbK
b
d
bE 2*
2222
2ˆˆˆ
4ˆˆ
22
1,
2/1
ˆ
2/1
ˆˆKdKd
s
b
d
02
*2
22
2 2/1
ˆˆˆ2/14)2(
1][ Ekd
Kd
sbkd
Kd
KbE
Minimization with respect to gives:
core regularization factor
elastic energy
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Phase-Field Energy
d
Peierls
kd
K
sdE
d
2
2
2
20 1
ˆ
22
1
Core regularization
Elastic energy
d0
Core regularization factor 21
1)(ˆ
dd Kk
d
xbPeierls
)1(2tan
21
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Energy minimizing phase-field
][inf
EY
0s CsG
Unconstrained minimization problem: sKb
ˆˆ2
if
with )1/(
1
2
2)(
22
21
22
212
2
xx
xx
bkd
K
e
bxG
xik
CsGPXd PX
d
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Irreversible Process and Kinetics
xdxfEEW nnnnnn 2111 )(][][]|[
Irreversible dislocation-obstacle interaction may be built into a variational framework, we introduce the incremental work function:
incremental work dissipated at the obstacles
4
bf Primary and forest dislocations react to form a jog:
Updated phase-field follows from: ]|[inf 1
1
nn
XW
n
Short range obstacles:
N
iidi
P xxfbxf1
)()(
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Irreversible Process and Kinetics
g g
Kuhn-Tucker optimality conditions: ii
ni
ni 1
01 i
ni fg
0i
01 i
ni fg
0i
01 ii
ni fg 01
iini fg
Equilibrium condition:
N
i
ni
n gb1
11 1
xdxgxdxf nnn
fg
nn
n
21121 )(sup)(1
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Solution Procedure
1. Stick predictor. Set
2. Reaction projection
3. Phase-field evaluation
ni
ni 1~ and compute the reactions:
)~(~ 1
1
11
ni
N
iij
nj CGg
ini
ni fgg 11 ~~
1
1
11
nN
j
njij
ni CgG
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Closed-form solution
Dislocation loops
1
1
11
nN
j
njij
ni CgG
)( jiij xxGG
db
Gii
1
2
122
Calculations are gridless and scale with the number
of obstacles
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Macroscopic averages
Slip
Dislocation density
Obstacle concentration
Shear stress
01
N
iilN
b
b0
c
f
sb0 42
0 1010 cF obs
bl 310
216
0
110
1
mbl
21712 1
1010m
c
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Forest Hardening
Obstacle distribution
BOUNDARY CONDITIONSPeriodic
OBSTACLE STRENGTH Uniform, f = 10 G b2
PEIERLS STRESS p= 0
Parameters
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Monotonic loading
Stress-strain curve. Evolution of dislocation density with strain.
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Dislocation Patterns
Evolution of dislocation patternas a function of slip strain
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Interaction with Obstacles
Detail of the evolution of the dislocation patternshowing dislocations bypassing a pair of obstacles
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Fading memory
a
c d
b
e f
Stress-strain curve.
Three dimensional view of the evolution of the phase-field, showing the the switching of the cusps.
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Stress-strain curve.
a
Cyclic loading
b c
d e f
g h iEvolution of dislocation density with strain.
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Irreversibility/Cyclic Loading
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Poisson ratio effects
b
-50 0 50-1
-0.5
0
0.5
1/0
/o
= 0.0 = 0.3 = 0.5
-50 0 500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
/0
/o
= 0.0 = 0.3 = 0.5
Stress-strain curve Dislocation density vs.strain
0.0 3.0 5.0
Stress-strain curve. Evolution of dislocation density with strain.
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Obstacle density
22 10bc
82 10bc62 10bc
42 10bc
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Multiple Glide
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Physics Capturing Capabilities
Ortiz,1999
The aim of this study is to develop a phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals.
This representation enables to identify individual dislocation lines and arbitrary dislocation geometries, including tracking intricate topological transitions such as loop nucleation, pinching and the formation of Orowan loops.
This theory permits the coupling between slip systems, consideration of obstacles of varying strength, anisotropy, thermal and strain rate effects.
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Summary
1. Phase-field model.
2. Closed form solution at zero temperature.
3. Temperature effects.
4. Strain rate effects.
5. Dislocation structures in grain boundaries.
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Another Study Case: FerroElectrics
ab initio QMEoS of various phasesTorsional barriersVibrational frequencies
Force Fields and MDElastic, dielectric constantsNucleation BarrierDomain wall and interface mobilityPhase transitionsAnisotropic Viscosity