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Phase-Field theory and modeling of dislocations in
ductile single crystals
M. Koslowski 1, M. Ortiz 1 , A.M. Cuitiño 2
1 California Institute of Technology, Pasadena, CA2 Rutgers University, Piscataway, NJ
MotivationThe aim of this study is to develop a
phase-field theory of dislocation dynamics, strain hardening andhysteresis 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 and anisotropy and thermal effects.
Rate effects are observed at finite temperatures.
Ortiz,1999
Overview
(Ortiz, 1999)
Effective Dislocation Energy • Core Energy
• Dislocation Interaction
• Irreversible Obstacle Interaction
Macroscopic Averages >∝< ξγ >∇∝< ξρ
Equilibrium configurations
• Closed form solution at zero temperature.
• Metropolis Monte Carlo algorithm and mean field approximation at finite temperatures.
Effective Energy
∫∫∫ −+= iiekl
eijijkl dStdSxdcuE δδφββ )(
2][ 3
SS
1
Elastic interaction Core energy External field
where withpij
eijjiu ββ +=,
)(Sm Djipij δδβ =
Slip Surface
S
Displacement jump across Sm
Elastic interaction
xdcE ekl
eijijkl
3int
21∫= ββ
( ) pklj
pmnijmnlki
ekl cG βββ −∗−=
,,Elastic distortion:
kdkkKbE 22
21
2
2int ˆ),(
4)2(1][ ζµπ
ζ ∫=
with bxxxx /),(),( 2121 δζ =
22
21
22
22
21
21
21 11),(
kk
k
kk
kkkK+
++−
=ν
External Field
∑= −
+=Ndis
n n
n
xxCxs
1)( τ
applied
shear stress forest dislocations∫=S
ext dSbsE ζ
∫=S
iiext dStE δ
ii bxxxx ),(),( 2121 ζδ =
ii bxxtxxs ),(),( 2121 =
with:
Core Energy
ZS→:ξ
δ0 b/2 b-b/2-b-3b/2
φ(δ)
( )( )jjiiijZbbC ξδξδδφ
ξ−−=
∈ 21min)(
ijij dC δµ
2=
22
4inf)( ξζµζφξ
−=∈ db
Z Ortiz and Phillips, 1999
∫=S
core dSE )(ζφ
Phase-Field Energy
[ ]( ) ∫
−+−=
∈kdsbKb
dbE
Y
2*2222
2ˆˆˆ
4ˆˆ
221inf ζζµξζµπ
ξζ
Minimization with respect to ζ gives:
2/1
ˆ
2/1ˆˆ
KdKds
bd
++
+=
ξµ
ζ
core regularization factor
02
*222
2 2/1
ˆˆˆ2/14)2(
1][ EkdKdsbkd
KdKbE +
+−
+= ∫∫
ξξµπ
ξ
elastic energy
Phase-Field Energy
ζξ
d
δPeierls
( )kd
KsdE
d
2
2
2
20 1ˆ
221
+= ∫ µπ
ξϕζζ ∗+= d0
211)(ˆ
dd Kk
+=ϕ
dxb
Peierls)1(2tan
21 ν
πδ −
−= −
Elastic energy
Core regularization
Core regularization factor
Irreversible Process and Kinetics
Irreversible dislocation-obstacle interaction may be built into a variational framework, we introduce the incremental work function:
xdxfEEW nnnnnn 2111 )(][][]|[ ∫ −+−= +++ ξξξξξξ
incremental work dissipated at the obstacles
πµ4
2bf ∝Primary and forest dislocations react to form a jog:
]|[inf 1
1
nn
XW
n
ξξξ
+
∈+
∑=
−+=N
iidi
P xxfbxf1
)()( ϕτ
Updated phase-field follows from:
Short range obstacles:
Irreversible Process and Kinetics
( ) xdxgxdxf nnn
fg
nn
n
21121 )(sup)(1∫∫ −=− ++
≤
+
+
ζζζζ
Kuhn-Tucker optimality conditions:ζ
τg g
−++ −=− iini
ni λλξξ 1
01 ≤−+i
ni fg 01 ≤−− +
ini fg
0≥+iλ 0≥−
iλ
( ) 01 =− ++ii
ni fg λ ( ) 01 =+ −+
iini fg λ
∑=
++
Ω=
N
i
ni
n gb1
11 1τEquilibrium condition:
Energy minimizing phase-field
sKb ˆˆ2
=ηµ][inf ηηW
Y∈Unconstrained minimization problem:
CsG +∗=η0=sif
with( ) ( ) )1/(
12
2)( 22
21
22
212
2 νπµπµ −++
== ∫⋅
xxxx
bkd
Ke
bxG
xik
( ) CsGPXd +∗∗=ϕζPX
ϕd
Closed-form solution
Dislocation loops
1
1
11 +
=
++ +−= ∑ nN
j
njij
ni CgGη
)( jiij xxGG −=
( )db
Gii1
212
2 µπνν −−
=
Macroscopic averages
ξγξγ 01
== ∑=
N
iilN
bSlip bl 310≈
216
01101mbl
≈=ρξρρ ∇= b0Dislocation density
21712 11010m
c −≈Obstacle concentration η=c
obsFf
0ττ =Shear stress µτ 420 1010 −− −≈= cF obs
Monotonic loading
00.0/ 0 =ττ
60.0/ 0 =ττ
Stress-strain curve.
20.0/ 0 =ττ
80.0/ 0 =ττ
Evolution of dislocation density with strain.
40.0/ 0 =ττ
99.0/ 0 =ττ
Evolution of dislocation density with shear stress.
Fading memory
a
c d
b
e fThree dimensional view of the evolution of the phase-field, showing the the switching of the cusps.
Stress-strain curve.
Stress-strain curve.
a
Cyclic loading
b c
d e f
g h iEvolution of dislocation density with strain.
Cyclic loading
Poisson ratio effects
b
-50 0 50-1
-0.5
0
0.5
1τ/τ0
γ/γo
ν = 0.0ν = 0.3ν = 0.5
-50 0 500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45ρ/ρ0
γ/γo
ν = 0.0ν = 0.3ν = 0.5
Stress-strain curve Dislocation density vs.strain
0.0=ν 3.0=ν 5.0=ν
Stress-strain curve. Evolution of dislocation density with strain.
Obstacle density
210−=bρ410−=bρ
610−=bρ 810−=bρ
Temperature effects
• Incremental work function
∑=
+++ −+−=Nobs
i
nnobsi
nnnn FEEW1
111 ][][]|[ ξξξξξξ
incremental work dissipated at the obstacles
• Metropolis Monte Carlo algorithm with hysteresis.
• Dislocation interaction is calculated in Fourier transformed space.
Pa10106.5 ⋅=µ mb 10105.2 −⋅=
Temperature Effects
-6 -4 -2 0 2 4 6-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
γ/γ0
ρ/ρ0
T = 0 KT = 300 KT = 600 K
-6 -4 -2 0 2 4 60
0.2
0.4
0.6
0.8
1
γ/γ0
ρ/ρ0
T = 0 KT = 300 KT = 600 K
Evolution of dislocation density with strain.
Stress-strain curve.
Mean field approximation
[ ]nnnnnnnn bw γγτγτγτγγ −++−=∆ ++++ 10113
1 ),(
)(2
21
20
31
1+
++ −
+=n
nnn b ττβ
τγγ
Mean field approximation
−
=
kTUAr
n
exp0ττ
(Kocks, Argon, Ashby, 1975)
Flow stress vs. temperature Deformation rate vs. flow stress
Conclusions
• The phase-field representation furnishes a simple and effective means of tracking the motion of large numbers of dislocations within discrete slip planes through random arrays of obstacles (forest dislocations, defects) under the action of an applied shear stress.
• The theory predicts a range of behaviors which are in qualitative agreement with observation, Bauschinger effect, formation of Orowan loops.
• Softening and yield stress dependence on the temperature are observed.
• Rate effects are observed at finite temperature.