Condensed Matter Physics - Dislocationsmetal.elte.hu/~groma/mater_phys/materphys7.pdf ·...
Transcript of Condensed Matter Physics - Dislocationsmetal.elte.hu/~groma/mater_phys/materphys7.pdf ·...
-
Condensed Matter PhysicsDislocations
István Groma
ELTE
April 10, 2019
István Groma, ELTE Condensed Matter Physics, Dislocations 1/32
-
Plastic deformation
Ideal shear
István Groma, ELTE Condensed Matter Physics, Dislocations 2/32
-
Plastic deformation
Ideal shear
F (x) = F0
∣∣∣∣sin(πa x)∣∣∣∣ = σa2
Small deformationγ =
xa, σ = µ
xa
So
F0 = µa2
π
From thisσf ∼=
µ
π∼= 30GPa
Experimentσf ∼= 100MPa
István Groma, ELTE Condensed Matter Physics, Dislocations 3/32
-
MDD
Polányi, Orován, Taylor (1934)
Vito Volterra (1905)
István Groma, ELTE Condensed Matter Physics, Dislocations 4/32
-
MDD
Polányi, Orován, Taylor (1934)
Vito Volterra (1905)
István Groma, ELTE Condensed Matter Physics, Dislocations 4/32
-
MDD
Polányi, Orován, Taylor (1934)
Vito Volterra (1905)
István Groma, ELTE Condensed Matter Physics, Dislocations 4/32
-
Dislocation
Glide: the number of atom is conservedClimb: the number of atom is not conserved, requires high temperature.
István Groma, ELTE Condensed Matter Physics, Dislocations 5/32
-
Dislocation
Field theory of dislocations
∂i uj := βij = βpij + β
eij
dudr
:= β̂ = β̂e + β̂p
Dislocation density tensor
αij = eikl∂kβplj = −eikl∂kβ
elj α̂ = ∇× β̂
p = −∇× β̂e
bj =∫
Aαij dAi
bj = −∫
Aeikl
∂
∂rkβelj dAi = −
∮βeij dsi = −
∮duj .
István Groma, ELTE Condensed Matter Physics, Dislocations 6/32
-
Dislocation
Individual dislocation
αij = li bjδ(ξ) (α̂ = l ◦ bδ(ξ))Plastic distortion
βpij (~r) = ni bkδ(ζ)
Multiple with the elastic constants and take its div:
∂
∂riCijkl
∂uk∂rl
=∂
∂riCijklβ
pkl = fj
(div Ĉ
dudr
= div Ĉβ̂p = f)
Displacement field of a screw dislocation
u3 =b
2πϕ
Stress field of a straight dislocation σ ∝ b/r
σ11 =µb
2π(1− ν)y(3x2 + y2)(x2 + y2)2
σ22 =µb
2π(1− ν)y(x2 − y2)(x2 + y2)2
σ12 =µb
2π(1− ν)x(x2 − y2)(x2 + y2)2
István Groma, ELTE Condensed Matter Physics, Dislocations 7/32
-
Energy of a dislocation
σ ∝ b/r ı́gy E ∝ b2
2(
b2
)2< b2
b = aiFcc b = 12 (1, 1, 0)Bcc b = 12 (1, 1, 1)
István Groma, ELTE Condensed Matter Physics, Dislocations 8/32
-
Dislocation-dislocation interaction
Elastic energy
E =12
∫σij∂i uj dV
Let us
σij = σextij + σ
dislij
ui = uexti + udisli
leads to
E =12
∫ [σextij ∂i u
extj + σ
extij ∂i u
dislj + σ
dislij ∂i u
extj + σ
dislij ∂i u
dislj
]dV
Since
σij = Cijkl∂i uj
E = Eext + Edisl +∫σextij ∂i u
dislj dV
István Groma, ELTE Condensed Matter Physics, Dislocations 9/32
-
Dislocation dislocation interaction
Eint =∫σextij ∂i u
dislj dV
Partial integration
Eint =∫∂i (σ
extij u
dislj )dV −
∫(∂iσ
extij )u
dislj dV
Since
∂iσextij = 0
Eint =∫∂i (σ
extij u
dislj )dV =
∫(σextij u
dislj )dni
István Groma, ELTE Condensed Matter Physics, Dislocations 10/32
-
Dislocation-dislocation interaction
u-nak b has a jump on the cut surface!
Eint = bj
∫fσextij dni
For the moving line
4Eint = bj∮
vσextij (4r × dl)i
4Eint =∮
v(dl × (σ̂ext b))i 4 ri
Peack Koehler force
f = (σ̂ext b)× l
István Groma, ELTE Condensed Matter Physics, Dislocations 11/32
-
Dislocation-dislocation interaction
For glide only, we need
fglide = [(σ̂ext b)× l]bb
= nσ̂ext b = τb ahol n =bb× l
Edge dislocation
f = b1b2µ
2π(1− ν)x(x2 − d2)(x2 + d2)2
István Groma, ELTE Condensed Matter Physics, Dislocations 12/32
-
Dislocation multiplication
Frank-Read source
Flow stress (Taylor relation)
τf = αµb√ρ
István Groma, ELTE Condensed Matter Physics, Dislocations 13/32
-
Work hardening by precipitate
Line tension τl = El/R
Critical stress
τc =∝1l
István Groma, ELTE Condensed Matter Physics, Dislocations 14/32
-
Dislocation patterning
Cell structure PSB structure
István Groma, ELTE Condensed Matter Physics, Dislocations 15/32
-
Size effect
Torsion test Microhardness
István Groma, ELTE Condensed Matter Physics, Dislocations 16/32
-
Size effect
Torsion test Microhardness
Local plasticity
τclass(γ, γ̇, ...)
István Groma, ELTE Condensed Matter Physics, Dislocations 16/32
-
Size effect
Torsion test Microhardness
Nonlocal plasticity
τ(γ, γ̇, ...) = τclass(γ, γ̇, ...) + l2µd2
d~r2γ
István Groma, ELTE Condensed Matter Physics, Dislocations 16/32
-
Pillar compression
10-2
10-1
100
S=s L (Γ+Θ)/(Eb)
10-5
10-4
10-3
10-2
10-1
100
101
102
P(S
)
Data from Dimiduk et al.L=1.5µm, bending
L=0.5µm, load control
L=0.5µm, strain control
L=0.8µm, strain control
exp(-(S/0.6)2)/S
3/2
István Groma, ELTE Condensed Matter Physics, Dislocations 17/32
-
Pillar compression
10-2
10-1
100
S=s L (Γ+Θ)/(Eb)
10-5
10-4
10-3
10-2
10-1
100
101
102
P(S
)
Data from Dimiduk et al.L=1.5µm, bending
L=0.5µm, load control
L=0.5µm, strain control
L=0.8µm, strain control
exp(-(S/0.6)2)/S
3/2
István Groma, ELTE Condensed Matter Physics, Dislocations 17/32
-
Pillar compression
10-2
10-1
100
S=s L (Γ+Θ)/(Eb)
10-5
10-4
10-3
10-2
10-1
100
101
102
P(S
)Data from Dimiduk et al.L=1.5µm, bending
L=0.5µm, load control
L=0.5µm, strain control
L=0.8µm, strain control
exp(-(S/0.6)2)/S
3/2
István Groma, ELTE Condensed Matter Physics, Dislocations 17/32
-
Relevant quantities
Plastic deformation
d~Rd~r
= 1̂ +d~ud~r
= F̂ e · F̂ p
Definitions
F̂ e,p = 1̂ + β̂e,p
F̂ e · F̂ e∗ = 1̂ + 2�̂e
CurlF̂ p = α̂
István Groma, ELTE Condensed Matter Physics, Dislocations 18/32
-
Relevant quantities
Plastic deformation
d~Rd~r
= 1̂ +d~ud~r
= F̂ e · F̂ p
Definitions
F̂ e,p = 1̂ + β̂e,p
F̂ e · F̂ e∗ = 1̂ + 2�̂e
CurlF̂ p = α̂
István Groma, ELTE Condensed Matter Physics, Dislocations 18/32
-
Small deformations
Deformations
∂i uj := βij = βpij + β
eij
d~ud~r
:= β̂ = β̂e + β̂p
Nye’s dislocation density tensor
αij = eikl∂kβplj = −eikl∂kβ
elj α̂ = ∇× β̂
p = −∇× β̂e
Is it that we learned in school?
bj =∫
Aαij dAi
bj = −∫
Aeikl
∂
∂rkβlj dAi = −
∮βij dsi = −
∮dui .
For a single dislocation
αij = li bjδ(ξ)(α̂ =~l ◦ ~bδ(ξ)
)
István Groma, ELTE Condensed Matter Physics, Dislocations 19/32
-
Small deformations
Deformations
∂i uj := βij = βpij + β
eij
d~ud~r
:= β̂ = β̂e + β̂p
Nye’s dislocation density tensor
αij = eikl∂kβplj = −eikl∂kβ
elj α̂ = ∇× β̂
p = −∇× β̂e
Is it that we learned in school?
bj =∫
Aαij dAi
bj = −∫
Aeikl
∂
∂rkβlj dAi = −
∮βij dsi = −
∮dui .
For a single dislocation
αij = li bjδ(ξ)(α̂ =~l ◦ ~bδ(ξ)
)
István Groma, ELTE Condensed Matter Physics, Dislocations 19/32
-
Small deformations
Deformations
∂i uj := βij = βpij + β
eij
d~ud~r
:= β̂ = β̂e + β̂p
Nye’s dislocation density tensor
αij = eikl∂kβplj = −eikl∂kβ
elj α̂ = ∇× β̂
p = −∇× β̂e
Is it that we learned in school?
bj =∫
Aαij dAi
bj = −∫
Aeikl
∂
∂rkβlj dAi = −
∮βij dsi = −
∮dui .
For a single dislocation
αij = li bjδ(ξ)(α̂ =~l ◦ ~bδ(ξ)
)
István Groma, ELTE Condensed Matter Physics, Dislocations 19/32
-
Small deformations
Deformations
∂i uj := βij = βpij + β
eij
d~ud~r
:= β̂ = β̂e + β̂p
Nye’s dislocation density tensor
αij = eikl∂kβplj = −eikl∂kβ
elj α̂ = ∇× β̂
p = −∇× β̂e
Is it that we learned in school?
bj =∫
Aαij dAi
bj = −∫
Aeikl
∂
∂rkβlj dAi = −
∮βij dsi = −
∮dui .
For a single dislocation
αij = li bjδ(ξ)(α̂ =~l ◦ ~bδ(ξ)
)István Groma, ELTE Condensed Matter Physics, Dislocations 19/32
-
Incompatibility
Incompatibility operator
(IncÂ)ij = −eikmejln∂k∂l Amn Inc = −∇× Â×∇
Inc(∂i fj + ∂j fi ) ≡ 0 Inc[
d~fd~r
]sim
≡ 0
Incompatibility field
ηij = −eikmejln∂
∂rk
∂
∂rl�emn
(inc �̂e = η̂
)
ηij =12
(ejln
∂
∂rlαim + eiln
∂
∂rlαjm
) (η̂ = [α̂×∇]sym
)
István Groma, ELTE Condensed Matter Physics, Dislocations 20/32
-
Incompatibility
Incompatibility operator
(IncÂ)ij = −eikmejln∂k∂l Amn Inc = −∇× Â×∇
Inc(∂i fj + ∂j fi ) ≡ 0 Inc[
d~fd~r
]sim
≡ 0
Incompatibility field
ηij = −eikmejln∂
∂rk
∂
∂rl�emn
(inc �̂e = η̂
)
ηij =12
(ejln
∂
∂rlαim + eiln
∂
∂rlαjm
) (η̂ = [α̂×∇]sym
)
István Groma, ELTE Condensed Matter Physics, Dislocations 20/32
-
Incompatibility
Incompatibility operator
(IncÂ)ij = −eikmejln∂k∂l Amn Inc = −∇× Â×∇
Inc(∂i fj + ∂j fi ) ≡ 0 Inc[
d~fd~r
]sim
≡ 0
Incompatibility field
ηij = −eikmejln∂
∂rk
∂
∂rl�emn
(inc �̂e = η̂
)
ηij =12
(ejln
∂
∂rlαim + eiln
∂
∂rlαjm
) (η̂ = [α̂×∇]sym
)
István Groma, ELTE Condensed Matter Physics, Dislocations 20/32
-
Stress
Linear elasticity
σij = Lijkl�ekl(σ̂ = L̂ : �̂e
)
Elastic deformation↔ Stress
ηij = −eikmejln∂
∂rk
∂
∂rlL−1mnopσop
(inc(
L̂−1σ̂)= η̂
)
Is it enough?
inc
[d~fd~r
]sym
≡ 0
Equilibrium condition
∂
∂riσij = 0 (div σ̂ = 0)
István Groma, ELTE Condensed Matter Physics, Dislocations 21/32
-
Stress
Linear elasticity
σij = Lijkl�ekl(σ̂ = L̂ : �̂e
)Elastic deformation↔ Stress
ηij = −eikmejln∂
∂rk
∂
∂rlL−1mnopσop
(inc(
L̂−1σ̂)= η̂
)
Is it enough?
inc
[d~fd~r
]sym
≡ 0
Equilibrium condition
∂
∂riσij = 0 (div σ̂ = 0)
István Groma, ELTE Condensed Matter Physics, Dislocations 21/32
-
Stress
Linear elasticity
σij = Lijkl�ekl(σ̂ = L̂ : �̂e
)Elastic deformation↔ Stress
ηij = −eikmejln∂
∂rk
∂
∂rlL−1mnopσop
(inc(
L̂−1σ̂)= η̂
)Is it enough?
inc
[d~fd~r
]sym
≡ 0
Equilibrium condition
∂
∂riσij = 0 (div σ̂ = 0)
István Groma, ELTE Condensed Matter Physics, Dislocations 21/32
-
Stress
Linear elasticity
σij = Lijkl�ekl(σ̂ = L̂ : �̂e
)Elastic deformation↔ Stress
ηij = −eikmejln∂
∂rk
∂
∂rlL−1mnopσop
(inc(
L̂−1σ̂)= η̂
)Is it enough?
inc
[d~fd~r
]sym
≡ 0
Equilibrium condition
∂
∂riσij = 0 (div σ̂ = 0)
István Groma, ELTE Condensed Matter Physics, Dislocations 21/32
-
Stress
Linear elasticity
σij = Lijkl�ekl(σ̂ = L̂ : �̂e
)Elastic deformation↔ Stress
ηij = −eikmejln∂
∂rk
∂
∂rlL−1mnopσop
(inc(
L̂−1σ̂)= η̂
)Is it enough?
inc
[d~fd~r
]sym
≡ 0
Equilibrium condition
∂
∂riσij = 0 (div σ̂ = 0)
István Groma, ELTE Condensed Matter Physics, Dislocations 21/32
-
Second order stress function
σij = −eikmejln∂
∂rk
∂
∂rlχmn (σ̂ = inc χ̂)
for isotropic materials
χ′ij =1
2µ
(χij −
ν
1 + 2νχkkδij
)χij = 2µ
(χ′ij +
ν
1− νχ′kkδij
)
∇4χ′ij = ηij(∇4χ̂′ = η̂
).
gauge condition
∂
∂riχ′ij = 0
(div χ̂′ = 0
)
István Groma, ELTE Condensed Matter Physics, Dislocations 22/32
-
Second order stress function
σij = −eikmejln∂
∂rk
∂
∂rlχmn (σ̂ = inc χ̂)
Due to the identity
div inc ≡ 0 div σ̂ = 0
for isotropic materials
χ′ij =1
2µ
(χij −
ν
1 + 2νχkkδij
)χij = 2µ
(χ′ij +
ν
1− νχ′kkδij
)
∇4χ′ij = ηij(∇4χ̂′ = η̂
).
gauge condition
∂
∂riχ′ij = 0
(div χ̂′ = 0
)
István Groma, ELTE Condensed Matter Physics, Dislocations 22/32
-
Second order stress function
σij = −eikmejln∂
∂rk
∂
∂rlχmn (σ̂ = inc χ̂)
ηij = eikmejlneoqv epuw L−1mnop
∂
∂rk
∂
∂rl
∂
∂rq
∂
∂ruχvw
(η̂ = inc
(L̂−1inc χ̂
))
for isotropic materials
χ′ij =1
2µ
(χij −
ν
1 + 2νχkkδij
)χij = 2µ
(χ′ij +
ν
1− νχ′kkδij
)
∇4χ′ij = ηij(∇4χ̂′ = η̂
).
gauge condition
∂
∂riχ′ij = 0
(div χ̂′ = 0
)
István Groma, ELTE Condensed Matter Physics, Dislocations 22/32
-
Second order stress function
σij = −eikmejln∂
∂rk
∂
∂rlχmn (σ̂ = inc χ̂)
for isotropic materials
χ′ij =1
2µ
(χij −
ν
1 + 2νχkkδij
)χij = 2µ
(χ′ij +
ν
1− νχ′kkδij
)
∇4χ′ij = ηij(∇4χ̂′ = η̂
).
gauge condition
∂
∂riχ′ij = 0
(div χ̂′ = 0
)
István Groma, ELTE Condensed Matter Physics, Dislocations 22/32
-
Second order stress function
σij = −eikmejln∂
∂rk
∂
∂rlχmn (σ̂ = inc χ̂)
for isotropic materials
χ′ij =1
2µ
(χij −
ν
1 + 2νχkkδij
)χij = 2µ
(χ′ij +
ν
1− νχ′kkδij
)
∇4χ′ij = ηij(∇4χ̂′ = η̂
).
gauge condition
∂
∂riχ′ij = 0
(div χ̂′ = 0
)
István Groma, ELTE Condensed Matter Physics, Dislocations 22/32
-
2D case
Stress function→ stress (∂/∂z = 0)
σ11 = −∂2χ
∂y2, σ22 = −
∂2χ
∂x2, σ12 =
∂2χ
∂x∂y, χ ≡ χ33
σ23 = −∂φ
∂x, σ13 =
∂φ
∂y, φ = −
∂χ23
∂x+∂χ31
∂y
Field equationsEdge dislocation
∇4χ =2µ
1− ν
(b1
∂
∂y− b2
∂
∂x
)(ρd+ − ρd−)
Shrew dislocation
∇2φ = µb3(ρd+ − ρd−)
István Groma, ELTE Condensed Matter Physics, Dislocations 23/32
-
2D case
Stress function→ stress (∂/∂z = 0)
σ11 = −∂2χ
∂y2, σ22 = −
∂2χ
∂x2, σ12 =
∂2χ
∂x∂y, χ ≡ χ33
σ23 = −∂φ
∂x, σ13 =
∂φ
∂y, φ = −
∂χ23
∂x+∂χ31
∂y
Field equationsEdge dislocation
∇4χ =2µ
1− ν
(b1
∂
∂y− b2
∂
∂x
)(ρd+ − ρd−)
Shrew dislocation
∇2φ = µb3(ρd+ − ρd−)
István Groma, ELTE Condensed Matter Physics, Dislocations 23/32
-
2D case
Stress function→ stress (∂/∂z = 0)
σ11 = −∂2χ
∂y2, σ22 = −
∂2χ
∂x2, σ12 =
∂2χ
∂x∂y, χ ≡ χ33
σ23 = −∂φ
∂x, σ13 =
∂φ
∂y, φ = −
∂χ23
∂x+∂χ31
∂y
Field equationsEdge dislocation
∇4χ =2µ
1− ν
(b1
∂
∂y− b2
∂
∂x
)(ρd+ − ρd−)
Shrew dislocation
∇2φ = µb3(ρd+ − ρd−)
István Groma, ELTE Condensed Matter Physics, Dislocations 23/32
-
2D stress
∇4G = δ(r) G(~r) =1
8πr2 ln(r)
σ11 =µb
2π(1− ν)y(3x2 + y2)(x2 + y2)2
σ22 =µb
2π(1− ν)y(x2 − y2)(x2 + y2)2
σ12 =µb
2π(1− ν)x(x2 − y2)(x2 + y2)2
István Groma, ELTE Condensed Matter Physics, Dislocations 24/32
-
Variational approaches I.
Classical elasticity F (�̂) = F ([d~u/d~r
]sim
σ̂ =δFδ�
δFδui
= ∂jδFδ�ij
= ∂jσij = 0
Plasticity
Q[�̂e, χ̂] = F (�̂)−∫
[χ̂ : (Inc�̂e − η̂)]dV
δQδ�̂e
= σ̂ − Incχ̂ = 0,δQδχ̂
= Inc�̂e − η̂ = 0
István Groma, ELTE Condensed Matter Physics, Dislocations 25/32
-
Variational approaches I.
Classical elasticity F (�̂) = F ([d~u/d~r
]sim
σ̂ =δFδ�
δFδui
= ∂jδFδ�ij
= ∂jσij = 0
Plasticity
Q[�̂e, χ̂] = F (�̂)−∫
[χ̂ : (Inc�̂e − η̂)]dV
δQδ�̂e
= σ̂ − Incχ̂ = 0,δQδχ̂
= Inc�̂e − η̂ = 0
István Groma, ELTE Condensed Matter Physics, Dislocations 25/32
-
Variational approaches II.
Gibbs free energy G(σ̂)
−δGδσ̂
= �̂e
Consider
P[χ̂] = G(Incχ̂ = σ̂) +∫
[χ̂ : η̂)]dV
One obtains
δPδχ̂
= IncδGδσ̂
+ η̂ = −Inc�̂e + η̂ = 0,
Dislocation dislocation interaction (Peach-Koehler force)
Fi = eikl tkσlmbm = −dP
dr ji
István Groma, ELTE Condensed Matter Physics, Dislocations 26/32
-
Variational approaches II.
Gibbs free energy G(σ̂)
−δGδσ̂
= �̂e
Consider
P[χ̂] = G(Incχ̂ = σ̂) +∫
[χ̂ : η̂)]dV
One obtains
δPδχ̂
= IncδGδσ̂
+ η̂ = −Inc�̂e + η̂ = 0,
Dislocation dislocation interaction (Peach-Koehler force)
Fi = eikl tkσlmbm = −dP
dr ji
István Groma, ELTE Condensed Matter Physics, Dislocations 26/32
-
Variational approaches II.
Gibbs free energy G(σ̂)
−δGδσ̂
= �̂e
Consider
P[χ̂] = G(Incχ̂ = σ̂) +∫
[χ̂ : η̂)]dV
One obtains
δPδχ̂
= IncδGδσ̂
+ η̂ = −Inc�̂e + η̂ = 0,
Dislocation dislocation interaction (Peach-Koehler force)
Fi = eikl tkσlmbm = −dP
dr ji
István Groma, ELTE Condensed Matter Physics, Dislocations 26/32
-
Plain strain (2D) case
P[χ] =∫ [−
D2(4χ)2 + bχ∂yκ)
]dV
with D = (1− ν)/2µ, and
κ =∑
j
±δ(~r −~rj )
Leading to
D42 χ = b∂y
∑j
±δ(~r −~rj )
István Groma, ELTE Condensed Matter Physics, Dislocations 27/32
-
Core regularization
Nonlocal term
Gnonlocal[σ] = G0 − b2∫
Nijklmn(∂iσjk )∂lσmndV ,
2D isotopic material
P[χ] =∫ {−
1− ν4µ
[|4χ|2 + a2|∇4χ|2
]+ bchi∂2δ(r)
}d2r
Leading To
42χ− a243χ =2bµ
1− ν∂2δ(r)
χ =ba
4πDyr
[−2K1
(ra
)+ 2
ar+
raln
(ra
)]
István Groma, ELTE Condensed Matter Physics, Dislocations 28/32
-
Core regularization
Nonlocal term
Gnonlocal[σ] = G0 − b2∫
Nijklmn(∂iσjk )∂lσmndV ,
2D isotopic material
P[χ] =∫ {−
1− ν4µ
[|4χ|2 + a2|∇4χ|2
]+ bchi∂2δ(r)
}d2r
Leading To
42χ− a243χ =2bµ
1− ν∂2δ(r)
χ =ba
4πDyr
[−2K1
(ra
)+ 2
ar+
raln
(ra
)]
István Groma, ELTE Condensed Matter Physics, Dislocations 28/32
-
Core regularization
Nonlocal term
Gnonlocal[σ] = G0 − b2∫
Nijklmn(∂iσjk )∂lσmndV ,
2D isotopic material
P[χ] =∫ {−
1− ν4µ
[|4χ|2 + a2|∇4χ|2
]+ bchi∂2δ(r)
}d2r
Leading To
42χ− a243χ =2bµ
1− ν∂2δ(r)
χ =ba
4πDyr
[−2K1
(ra
)+ 2
ar+
raln
(ra
)]
István Groma, ELTE Condensed Matter Physics, Dislocations 28/32
-
Core regularization
-0.4
-0.2
0
0.2
0.4
0.6
-20 -15 -10 -5 0 5 10 15 20
τxy
(x,0
) [
Gb
/a]
x [a]
Classical stress fieldStress field with core regularisation
0.02
0.1
1
1 10 50
István Groma, ELTE Condensed Matter Physics, Dislocations 29/32
-
Role of elastic anharmonicity
σij = Lijkl�kl + Kijklmn�kl�mn
Ganharm[σ] := G0 +∫ [
Cijklmnσijσklσmn +O(σ4ij )]
dV
2D edge dislocation
G[χ] :=∫ [−
D2(4χ)2 + �a
D2
3(4χ)3
]d2
Extra energy
V1(r) =(b1 − b2)b2
16π2D
[y(3x2 + y2)
r4+ 4
yr2
ln
(rr0
)].
István Groma, ELTE Condensed Matter Physics, Dislocations 30/32
-
Role of elastic anharmonicity
Ganharm[σ] := G0 +∫ [
Cijklmnσijσklσmn +O(σ4ij )]
dV
2D edge dislocation
G[χ] :=∫ [−
D2(4χ)2 + �a
D2
3(4χ)3
]d2
Extra energy
V1(r) =(b1 − b2)b2
16π2D
[y(3x2 + y2)
r4+ 4
yr2
ln
(rr0
)].
István Groma, ELTE Condensed Matter Physics, Dislocations 30/32
-
Role of elastic anharmonicity
Ganharm[σ] := G0 +∫ [
Cijklmnσijσklσmn +O(σ4ij )]
dV
2D edge dislocation
G[χ] :=∫ [−
D2(4χ)2 + �a
D2
3(4χ)3
]d2
Extra energy
V1(r) =(b1 − b2)b2
16π2D
[y(3x2 + y2)
r4+ 4
yr2
ln
(rr0
)].
István Groma, ELTE Condensed Matter Physics, Dislocations 30/32
-
Role of elastic anharmonicity
Ganharm[σ] := G0 +∫ [
Cijklmnσijσklσmn +O(σ4ij )]
dV
2D edge dislocation
G[χ] :=∫ [−
D2(4χ)2 + �a
D2
3(4χ)3
]d2
Extra energy
V1(r) =(b1 − b2)b2
16π2D
[y(3x2 + y2)
r4+ 4
yr2
ln
(rr0
)].
István Groma, ELTE Condensed Matter Physics, Dislocations 30/32
-
Time evolution
Dislocation density tensor
αij = eikl∂
∂rkβ
plj
(α̂ = ∇× β̂p
)
Evolution equation
α̇ij + eikl∂
∂rkjlj = 0
(˙̂α+∇× ĵ = 0
)where
ĵ = − ˙̂βp
Physical meaning
dbjdt
= −∮
Ljij dsi
For individual dislocation (Orowan’s equation)
jik = eilm ll vmbkδ(ξ)(̂
j = (~l ◦ ~bδ(ξ))× ~v)
István Groma, ELTE Condensed Matter Physics, Dislocations 31/32
-
Time evolution
Evolution equation
α̇ij + eikl∂
∂rkjlj = 0
(˙̂α+∇× ĵ = 0
)where
ĵ = − ˙̂βp
Physical meaning
dbjdt
= −∮
Ljij dsi
For individual dislocation (Orowan’s equation)
jik = eilm ll vmbkδ(ξ)(̂
j = (~l ◦ ~bδ(ξ))× ~v)
István Groma, ELTE Condensed Matter Physics, Dislocations 31/32
-
Time evolution
Evolution equation
α̇ij + eikl∂
∂rkjlj = 0
(˙̂α+∇× ĵ = 0
)where
ĵ = − ˙̂βp
Physical meaning
dbjdt
= −∮
Ljij dsi
For individual dislocation (Orowan’s equation)
jik = eilm ll vmbkδ(ξ)(̂
j = (~l ◦ ~bδ(ξ))× ~v)
István Groma, ELTE Condensed Matter Physics, Dislocations 31/32
-
Time evolution
Evolution equation
α̇ij + eikl∂
∂rkjlj = 0
(˙̂α+∇× ĵ = 0
)where
ĵ = − ˙̂βp
Physical meaning
dbjdt
= −∮
Ljij dsi
For individual dislocation (Orowan’s equation)
jik = eilm ll vmbkδ(ξ)(̂
j = (~l ◦ ~bδ(ξ))× ~v)
István Groma, ELTE Condensed Matter Physics, Dislocations 31/32
-
Displacement field
Plastic deformation
∂uj∂ri
= βeij + βpij
(d~ud~r
= β̂e + β̂p)
Multiply with L̂ and take its div
∂
∂riLijkl
∂uk∂rl
=∂
∂riLijklβ
pkl
(div L̂
d~ud~r
= div L̂β̂p)
Taking the time derivative
∂
∂riLijkl
∂u̇k∂rl
=∂
∂riLijkl jkl
(div L̂
d~̇ud~r
= div L̂̂j
)
István Groma, ELTE Condensed Matter Physics, Dislocations 32/32
-
Displacement field
Plastic deformation
∂uj∂ri
= βeij + βpij
(d~ud~r
= β̂e + β̂p)
Multiply with L̂ and take its div
∂
∂riLijkl
∂uk∂rl
=∂
∂riLijklβ
pkl
(div L̂
d~ud~r
= div L̂β̂p)
Taking the time derivative
∂
∂riLijkl
∂u̇k∂rl
=∂
∂riLijkl jkl
(div L̂
d~̇ud~r
= div L̂̂j
)
István Groma, ELTE Condensed Matter Physics, Dislocations 32/32
-
Displacement field
Plastic deformation
∂uj∂ri
= βeij + βpij
(d~ud~r
= β̂e + β̂p)
Multiply with L̂ and take its div
∂
∂riLijkl
∂uk∂rl
=∂
∂riLijklβ
pkl
(div L̂
d~ud~r
= div L̂β̂p)
Taking the time derivative
∂
∂riLijkl
∂u̇k∂rl
=∂
∂riLijkl jkl
(div L̂
d~̇ud~r
= div L̂̂j
)
István Groma, ELTE Condensed Matter Physics, Dislocations 32/32