多尺度位错动力学框架(MDDP)

(101) [100] Simulation Cell (5-20 µm ) [010]
>
Slip plane
b
b=Burgers vector 1 11 >=line sense vector
4
[ ]
Discretization Discretization
• Stress Field of a 3D Straight dislocation segment is known explicitly (Hirth & Lothe, 1982). • Discretize each curve into a set of mixed segments.
[
In the limit of small velocity, they reduce to standard forms, e.g. Gilman(1997), Beltz (1968), Weertman (1961)
µb 2 R W0 = ln 4π r0
γ = 1− v / C
ν x2 σ xz xλ yλ (1 − ν ) = − bx 3 + b y − + 3 + bz R R R (R + λ ) σ0 R σ yz σ0 ν y2 xλ xλ (1 − ν ) = bx − 3 + b y 3 − b z R R R (Rentificationof ofbasic basicgeometry geometry Identification
k For each node identify: • Coordinates, Burgers vector • slip plane index • neighboring nodes (k & j) • Node type (free, fixed, junction, jog, boundary,etc.
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Contents Contents
•Basic Structure of Dislocations Dynamics
– – – – –
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 and Frank-Read sources, Junction formation: Co-planar and noncoplanar, Jogs, and Dipoles
m
*
& + v / M + Fi + Fa = 0 v
Effective mass =
1 dW (Hirth, Zbib and Lothe, 1998) v dv
m =
*
For screw dislocation:
W0 v2
(−γ
)
−1
+γ
−3
)
2 2 1/ 2 Cl
9
v
For edge dislocation: 2 W C m* = 0 4 − 16γ l − 40γ l−1 + 8γ l− 3 + 14γ + 50γ −1 − 22γ −3 + 6γ −5 v
Lecture on: on: Lecture 3D Dislocation Dislocation Dynamics: Dynamics: 3D Numerical Treatment Treatment Numerical
H.M. Zbib, M. Rhee & J.P. Hirth School of Mechanical and Materials Engineering Washington State University
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III.Driving DrivingForce: Force:Peach-Koehler Peach-KoehlerForce Force III. • Average stress is calculated at the center of each segment 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
Vglide = M g (θ , T ) Fg
Initial Configuration
V
θ
y x z
b
Glide Mobility
Net Glide Force/unit length
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MacroscopicStrain Strain Macroscopic
Strain rate tensor:
σ yy σ0
yλ y2 2 y2 = bx 2 1 − 2 − 2 ρ R ρ R
2 2 + b y xλ 1 + y + 2 y 2 ρ 2R ρ2 R

2νy yλ σ zz + b y − 2νx − xλ = bx − + Rρ 2 R 3 Rρ 2 R 3 σ0 σ xy σ0
•Movie (Typical simulations)
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FLOWCHART CHARTOF OFDislocation DislocationDynamics Dynamics FLOW
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I. Basic Basic Geometry Geometry (bcc) (bcc) I.
[001] dislocation
Self-Force Self-Force
F1self =
∫
dl 2
σ 2 .b2 × ξ 2 dl 2
2
C Force at sub-segment dl
A
1 dl 1
dl
= Force from segment CA+ Force from segment BD+ Force from segment AB (see Hirth and Lothe, 1982, p. 131)
L µ [ f CA (θ A ,b) + f DB (θ B ,b)]ln − FC −∞ − FD −∞ Fg = 4πL ρ
•Similar expressions are obtained for the normal force. •These expressions reduce to those given in Hirth and Lothe (1982, p.138) for b AB = bCA Cut-off parameter; numerical parameter which can be adjusted to account for core energy
ρ 2 = x2 + y2 ,
R2 = ρ 2 + z2 ,
λ = z′ − z
•Intrinsic coordinate system ºRequires matrix transformation •More numerically efficient form has been developed by 13 Devincre (1995)
FAB = (σCD.bAB )×ζ AB
Stress field of segment CD Evaluated at the Center of AB. (Variation over AB is very small) z y
b
A B
ζ AB
b
D x C
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Remote stress stress Field Field Remote
x
z A
∞
C
θ A Fg
λ
α
b AB
θB
B
D
∞
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e.g.
f CA
CA AB 1 CA AB CA AB cos θ A − 1 = bz bz + bx b x + by by (1 − ν ) sinθ A CA AB ν + bx b z 1 −ν
Average force per unit length :
B
D
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Self-Force per per unit unit length length Self-Force
µ µ Explicit θ A , b )+ Fg = f CA ( f DB (θ B , b ) + 4πλ 4π (L − λ ) expression, more efficient µν 1 2 1 b AB sin α cos α − FC − ∞ − F D − ∞ − π ( − ν ) λ − λ L 4 1
(
2
2 1/ 2
γ l = 1− v /
(
)
]
Effect of of Inertia Inertia Effect
τ
Rise-time to for V to reach steady state
MD calculations by Shastry, 1998
Inertia effect is very small for V< 0.5 C
D
p
=
∑
N
N
l i v gi 2V l i v gi 2V
( ni ⊗ bi + bi ⊗ ni ) ( ni ⊗ bi − bi ⊗ ni )
i =1
Spin Tensor:
W
p
=
∑
i =1
Cell volume Segment length
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Equationof ofMotion: Motion:Inertia Inertia Equation
P σ ij = σ ij ( B ) − σ ij ( A )
σ xx yλ x xλ x 2x 2x = −b x 2 1 + 2 + 2 − b y 2 1 − 2 + 2 σ0 R ρ R ρ ρ R ρ R
2 2 2 2
x y A b B p z
Fi = ⋅ b i × ξ i , + F i ,i + 1 + F i ,i − 1 ,
j =1 j≠i, j ≠ i +1 j ≠ i −1
∑ (σ
N
D j
+σ
a
)
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Forcefrom froma aremote remotesegment segment Force
i(x,yz)
z
y x
5
j
SlipSystems Systems& &Cross-slip Cross-slipplanes planes Slip
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II.Equation Equationof ofMotion Motion II.
*Move Nodes *Maintain continuity Nodal Velocity = average velocity “V” of adjacent segments “V” is in the glide plane and normal to the dislocation segment
Tri-Lab Short Short Course Course on on Tri-Lab Dislocations in in Materials Materials Dislocations Pleasanton, CA CA Pleasanton, June 8-10, 8-10, 1998 1998 June
•Numerical issues:
•Long-range Interactions: Superdislocations •Time step and Segment length •Parallel processing: Family decomposition •Critical Issues
xλ x 2 2x 2 yλ x 2 2x 2 = bx 2 1 − 2 − 2 − b y 2 1 − 2 + 2 ρ R ρ ρ R ρ R R
x,y,z
Other forms are given in Hirth and Lothe (1982, p. 134) This form is most convenient to use