1

Tri-Lab Short Course onDislocations in Materials

Pleasanton, CAJune 8-10, 1998

Lecture on:3D Dislocation Dynamics:

Numerical Treatment

Tri-Lab Short Course onDislocations in Materials

Pleasanton, CAJune 8-10, 1998

Lecture on:3D Dislocation Dynamics:

Numerical Treatment

H.M. Zbib, M. Rhee & J.P. Hirth

School of Mechanical and Materials EngineeringWashington State University

2

ContentsContents

– Discretization of dislocation curves

– Identification of Slip geometry (bcc)– Long range interaction

– Equation of Motion: Glide, climb, cross-slip, multiplication

– Short-range interactions: Annihilation - Production andFrank-Read sources, Junction formation: Co-planar and non-coplanar, Jogs, and Dipoles

•Basic Structure of Dislocations Dynamics

•Numerical issues:

•Long-range Interactions: Superdislocations•Time step and Segment length•Parallel processing: Family decomposition

•Critical Issues

•Movie (Typical simulations)

3

FLOW CHART OF Dislocation DynamicsFLOW CHART OF Dislocation Dynamics

4

I. Basic Geometry (bcc)I. Basic Geometry (bcc)

Simulation Cell(5-20 )

[100] [010]

[001]

Slip plane

(101) b

dislocation

>>

b=Burgers vector>>=line sense vector

[ ]111

µm

5

DiscretizationDiscretization

• Stress Field of a 3DStraight dislocationsegment is known explicitly(Hirth & Lothe, 1982).

• Discretize each curve intoa set of mixed segments.

b

b

x

y

z

Identification of basic geometryIdentification of basic geometry

i(x,yz)

j

kFor each node identify:• Coordinates, Burgers vector• slip plane index• neighboring nodes (k & j)• Node type (free, fixed, junction, jog,boundary,etc.

b

6

Slip Systems & Cross-slip planesSlip Systems & Cross-slip planes

7

Initial Configuration

*Move Nodes*Maintain continuity Nodal Velocity = average velocity “V” of adjacent segments“V” is in the glide plane and normal to the dislocation segment

x

y

z

V

ggglide FTMV ),(θ=

bθ Glide Mobility

Net Glide Force/unit length

II. Equation of MotionII. Equation of Motion

8

Macroscopic StrainMacroscopic Strain

)( iii1

p nbbn2

D ⊗+⊗= ∑=

i

N

i

gii

V

vl

)( iii1

p nbbn2

W ⊗−⊗= ∑=

i

N

i

gii

V

vl

Cell volume

Strain rate tensor:

Spin Tensor:

Segment length

9

0=+++ ai FFMvvm /* &

Effective mass = dv

dW

v

1

For screw dislocation: ( )3120 −− +−= γγ

v

Wm*

( ) 21221/

/ Cv−=γ

=

0

2

0 4 r

Rn

bW l

πµ

(Hirth, Zbib and Lothe, 1998)

For edge dislocation:

[ ]531314

20 622501484016 −−−−− +−+++−−= γγγγγγγ lllv

CWm*

( ) 21221/

/ ll Cv−=γIn the limit of small velocity, they reduce to standard forms, e.g. Gilman(1997), Beltz (1968), Weertman (1961)

v

Equation of Motion: InertiaEquation of Motion: Inertia

10Inertia effect is very small for V< 0.5 C

Rise-time to for V to reach steady state MD calculations by Shastry, 1998

Effect of InertiaEffect of Inertia τ

11

III. Driving Force: Peach-Koehler ForceIII. Driving Force: Peach-Koehler Force

• Average stress is calculated at the center of eachsegment and includes 5 contributions:

• a) self-force due to adjacent segments

• b) force due to other remote segments,

• c) force due to the applied stress,

• d) force due to the Peierls stress

( ) ,, ,,

,

11

11

1−+

−≠+≠

≠=

++×

⋅+= ∑ iiiii

N

ijijij

ji

aDji FFbF ξσσ

12

Force from a remote segmentForce from a remote segment

x

y

z

A

B

C

D

b

b

ABABCDAB bF ζσ ×= ).(

Stress field of segment CDEvaluated at the Center of AB. (Variation over AB is very small)

ζ AB

13

z

x

y A

B

b

x,y,zp

)()( AB ijijPij σσσ −=

( )( )

( )( )

zzzRyx

RR

xb

R

xb

R

y

Rb

RR

yb

R

x

Rb

R

xb

x

R

x

R

yb

x

R

x

R

xb

R

x

R

xb

R

y

R

yb

y

R

y

R

xb

y

R

y

R

yb

x

R

x

R

xb

x

R

x

R

yb

zyxyz

zyxxz

yxxy

yxzz

yxyy

yxxx

−′=+=+=

+−−+

−=

+−+

+−+−=

+−−

−−=

−−+

+−=

+++

−−=

+−−

++−=

λρρ

λνλλν

σσ

λνλνλ

σσ

ρρλ

ρρλ

σσ

λρνλ

ρν

σσ

ρρλ

ρρλ

σσ

ρρλ

ρρλ

σσ

,, 222222

33

2

0

3

2

30

2

2

2

2

22

2

2

2

20

32320

2

2

2

2

22

2

2

2

20

2

2

2

2

22

2

2

2

20

1

1

21

21

22

21

21

21

21

Other forms are given inHirth and Lothe (1982, p. 134)This form is most convenient to use

Remote stress FieldRemote stress Field

•Intrinsic coordinate system ººRequires matrix transformation•More numerically efficient form has been developed byDevincre (1995)

14

Self-Force Self-Force

1ld

2ld

∫ ×= 22221 dlbF self ξσ . 2

1C

A

B

D

ldForce at sub-segment ld

= Force from segment CA+ Force from segment BD+ Force from segment AB

(see Hirth and Lothe, 1982, p. 131)

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Self-Force per unit length Self-Force per unit length

CA

B

D

∞ λ

∞

gF

( ) ( ) ( )

( ) ∞−∞− −−

−−

−

+−

+=

DCAB

BDBACAg

FFbL

bfL

bfF

ααλλνπ

µν

θλπ

µθπλµ

cossin

,,

211

14

44

Aθ

αABb

zx

Bθ

Explicitexpression, moreefficient

16

( )

νν

θθ

ν

−+

−

+

−+=

1

1

1

1

ABz

CAx

A

AABy

CAy

ABx

CAx

ABz

CAzCA

bb

bbbbbbfsin

cos

Average force per unit length :

( ) ( )[ ] ∞−∞− −−

+= DCBDBACAg FF

Lnbfbf

LF

ρθθ

πµ

l,,4

Cut-off parameter;numerical parameterwhich can be adjustedto account for core energy

•Similar expressions are obtained forthe normal force.•These expressions reduce to thosegiven in Hirth and Lothe (1982, p.138)for

e.g.

CAAB bb =

17

A

B

b

For pure edge pure screw dislocations, this reduces toe.g.

b

gF

gF

(force per unit length)

=

ρπµ L

nL

bFg l

4

2

( )

−

=ρπ

µν

Ln

L

bFg l

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1 2

18

Bow out of edge and mixed dislocations

Examples:Prismatic loop

19

Example: Stress field

Demir, Zbib and Hirth (1992)Exact Approximate

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IV. Short Range InteractionsIV. Short Range Interactions

BCC:4 Burgers vectors, without regard to slip planesºº 8 possible distinct reactions,

-4 repulsive, 3 attractive, and -one annihilation.

When slip planes are considered: For the {110} and {112} planes ºº 420 attractive reactions all of which are sessile (Baird and Galye, 1965)

d

A short-range interaction occurs when thedistance “d” between two dislocations becomescomparable to the size of the core *annihilation, *formation of dipoles, *jogs, and *junctions.

Detailed investigation of each possible interaction can become very cumbersome

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1. Critical distance criterion (Essman and Mughrabi, 1979)

Implicitly takes into account the effect of the local fields arising from allsurrounding dislocations.

Criteria for determining interaction:

Rule 1: If c

PK F≥F and ↑PKF or changes sign as

the segment is advanced, short range interaction is possible provided that local interaction between segmentAB and its closest neighbor CD, CDAB−F is attractive

A

B

C

D

2. Force-based criterion:

22

MPAbL

Fc

,τ+×= 8102

This captures interactions for adistance varying from 10b to 100b

Effective applied stress

2. Force-based criterion:

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• Rule 2: If Rule 1 is satisfied and

1≈CDAB î.î and 0=+ CDAB bb (or 1−≈CDAB î.î and 0=− CDAB bb ),

AB and CD will annihilate by either

i. Glide (if they are on the same plane) or

ii. Cross slip (if they are on intersecting planes).

Annihilation

Free nodes

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T h e p r o b a b il i ty t h a t t w o a t t r a c t iv e d is lo c a t io n sf o r m a j o g o r a j u n c t io n is im p lic it ly d e t e r m i n e db y t h e ir in t e r a c t i o n f o r c e s .

F o r e x a m p le , i f th e y w e r e t o f o r m a j u n c t i o n , t h ei n t e r a c t i n g s e g m e n t s r o t a t e r e l a t i v e t o e a c h o t h e rt o a lig n t h e m s e l v e s i n t o a c o n f i g u r a t i o n m o r es u ita b le fo r j u n c t i o n f o r m a t i o n , y i e l d i n g ar e d u c t i o n i n t h e e n e r g y .

I f n o t , th e y f o r m a j o g s i n c e i t i s e n e r g e t i c a l l ym o r e f a v o r a b le

A

B

C

D

25

• Rule 3: If Rule 1 is satisfied, andcjnCDAB θθ ≤− with 0≠+ CDAB bb ,

then a junction is formed.Segments AB and CD are combined toform a new segment at the junctionwhose Burgers vector is equal to .bb CDAB +

Junction nodeJunction

22

21

2bbbb 21 +<+

In a crude approximation, this implies that this reaction isenergetically favorable.

Example:

bb

in

bb

in

A

B

=

=

3111 110

3111 110

[ ] ( )

[ ] ( )

26

• Rule 4: If Rule 1 is satisfied and

cjgCDAB θθ ≥− , jogs are formed as follows:

i. If ABb is not parallel to CDî AB bypasses CD and is pinned at its midpoint.

ii. If CDb is not parallel to ABî CD bypasses AD and is pinned at its midpoint.

Jogs: Formation Example:

222BABA bbbb +>+

bb

in

bb

in

A

B

=

=

3111 110

3111 110

[ ] ( )

[ ] ( )

27

32cos

b

Wvcjgs

µ

θ=

ocjgs 120≈θ

For Ta:

• Rule 5: If cjgsjg θθ ≤ the jog moves forward in

the direction of average velocities of

the two adjacent dislocation segments.

Jogs: Motion

•Vacancy or interstitial generating jogs•Line tension approximation

Vacancy or interstitial formation energy

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Dipole

• Rule 6: If Rule 1 is satisfied and

1≈CDAB î.î , 0=+ CDAB bb

(or 1−≈CDAB î.î and 0=− CDAB bb )

and AD and CD are on parallel planes

they would form a dipole provided that:

( )RcCDAB Vhh ≤−

Dipoles form naturally (numerically) without a need for a “Rule”. However, a rule can be used for numerical efficiency

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( )

−−=

−−=kT

VA

kT

VAP

τττ *

0

*

0 expW

exp∆

Cross-Slip

( )

( )

−

+

++++−−

−

+++−−+=

ρνπµ

νπµ

πµ

e

an

ab

L

aLanaaLL

b

aLL

LnLaLaL

bW

l

l

l

12

22214

2

2

2

2222

2

22

222

∆

Model

Activation energy

bAWG τ−= ∆∆

Junction node

Example:

Initial dislocation sourceon (011) with

bb

M Mgs ge= =3

111[ ],

[ ]1 11 view

30

Short range Interaction: Summary

31

PZo

Q

A: Main Computational CellB,C, D.: Cells/Grains

C, or D Cell with N Dislocations

P: Dislocationin Cell A

For screw Dislocations: Stress at P from Q is obtained from the potential:

Z

ozz

bz

−=)(φ

x

y

z x iy= +

Long Range Interaction: 2D

V. Numerical IssuesV. Numerical Issues

•Long Range Interaction:Computationally most expensive•Use of superdislocation representation

32

cont.

Multipolar Expansion (LeSar et al. 1994)

...)( +++=3

2

2 z

bz

z

bz

z

bz ooφ

For N Dislocations:

φ ( ) . . . .zb

z

b z

z

b z

zi i o i i o i= + + +∑ ∑ ∑

2

2

3

ozzif >>

Field of Single Dislocation with Burgers Vector

bi∑Monopole

Field of Dipoles with an intensity

Bbzoi= ∑

2ε ε2

Quadrupoles

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Superdislocations: 2D

34

Superdislocations: 3D

1 3/ r

35

3D Superdislocations

= ∑

=

N

izixix lb

LB

1

1Monopoles:

= ∑

=

N

iziyiy lb

LB

1

1

= ∑

=

N

iziziz lb

LB

1

1

= ∑

=

N

icixixixz Zlb

LB

1

12 δλDipoles:

= ∑

=

N

icixiyiyz Zlb

LB

1

12 δλ

= ∑

=

N

icixizizz Zlb

LB

1

12 δλ

36

Segment length L ?

Meshing:If Lij > Lmax (200b) sub-dividedIf Lij < Lmin (50b) combine

i j

Time step: Limited by short range interaction and velocity.

max/ vbt 20=δFor strain rate 0.1 s-1 to 1000 s-1 811 1010 −− ≤≤ tδ

Dynamic time step

37

max/ vbt 20=δ

( )t

tM

M

ii

pp

δδεε ∆∆ == ∑

=

,1

1100010 −≤≤ sε&. st 47 1010 −− ≤≤ ∆

Example: Constant Strain Rate

( )ptE εεσ ∆∆∆ −= &

Random distribution of dislocation lines and Frank-Read Sources

Stress increment:

38

Example: Constant Strain Rate

m10862 10−×= .b GPa770.=µν σ µ= = × = =− −0 339 3 10 10 105 4 1. , , / ( . )f gs geM M Pa s

Ta:

)3(450:Pr

000,300

1000

1010,10

/100

5000

/103

10

1097

211

weeksMhzocessorOne

stepstimesubofnumberTotal

incrementsstressofNumber

stst

srateStrain

segmentsofnumberInitial

mdensitynDislocatioInitial

msizeCell

=−=

−==∆

==

×=

=

−−− δ

µ

39

Parallel Processing

•The problem is computationally massive•Use of parallel processing is essential•Computation of long range stress is most time consuming:•The simplest strategy is to use “Familydecomposition”: Distribute segments to Nprocessors.

S1 S9

Sm

S20

S300

S1-S200 º N1S201-S400 º N2 . .

40

Critical Issues

•Treatment of Boundary conditionsFree Boundaries (image stresses)Grain Boundaries

•Computational Effort•Cell size•Time step•Segment length•Parallel processing