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.