1 Tri-Lab Short Course on Dislocations in Materials Pleasanton, CA June 8-10, 1998 Lecture on: 3D Dislocation Dynamics: Numerical Treatment H.M. Zbib, M. Rhee & J.P. Hirth School of Mechanical and Materials Engineering Washington State University
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)
15
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
41
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
21
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:
23
• 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
29
( )
−−=
−−=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
33
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