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Multiscale analysis of dislocations
Adriana Garroni
Sapienza, Universita di Roma
”Mathematical challenges motivated by multi-phase materials:Analytic, stochastic and discrete aspects”
Anogia, CreteJune 22 - 26, 2009
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 1 /21
Elastic vs Plastic deformations
Single crystal Elastic deformation (reversible)
Elasto-plastic deformation Permanent deformation
The plastic deformation is due to slips on slip planes
In terms of the displacement u we can write
Du = ∇uL3 + ([u]⊗ n) dH2 Σ
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 2 /21
Elastic vs Plastic deformations
Single crystal Elastic deformation (reversible)
Elasto-plastic deformation Permanent deformation
The plastic deformation is due to slips on slip planes
In terms of the displacement u we can write
Du = ∇uL3 + ([u]⊗ n) dH2 Σ
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 2 /21
DISLOCATIONS
NOTE: The slip in general is not uniform ⇐⇒ DEFECTS (dislocations)
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
DISLOCATIONS
NOTE: The slip in general is not uniform ⇐⇒ DEFECTS (dislocations)
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Dislocation coreBurgers circuit
Burgers vector
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
DISLOCATIONS
NOTE: The slip in general is not uniform ⇐⇒ DEFECTS (dislocations)
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Dislocation coreBurgers circuit
Burgers vector
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
DISLOCATIONS
NOTE: The slip in general is not uniform ⇐⇒ DEFECTS (dislocations)
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Dislocation coreBurgers circuit
Burgers vector
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
DISLOCATIONS
NOTE: The slip in general is not uniform ⇐⇒ DEFECTS (dislocations)
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Dislocation coreBurgers circuit
Burgers vector
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
DISLOCATIONS
NOTE: The slip in general is not uniform ⇐⇒ DEFECTS (dislocations)
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Dislocation coreBurgers circuit
Burgers vector
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21
TOPOLOGICAL SINGULARITIES OF THE STRAIN
We can identify dislocations using the decomposition of the deformationgradient
Du = ∇uL3 + ([u]⊗ n) dH2 Σ = βe + βp
- where [u] is the jump of the displacement along the slip plane Σ- ∇u is the absolutely continuous part of the gradient
In presence of dislocations
Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0
µ is the dislocations density
Then dislocations can be understood as
I singularities of the Curl of the elastic strain
I regions where the slip is not uniform
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 4 /21
TOPOLOGICAL SINGULARITIES OF THE STRAIN
We can identify dislocations using the decomposition of the deformationgradient
Du = ∇uL3 + ([u]⊗ n) dH2 Σ = βe + βp
- where [u] is the jump of the displacement along the slip plane Σ- ∇u is the absolutely continuous part of the gradient
In presence of dislocations
Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0
µ is the dislocations density
Then dislocations can be understood as
I singularities of the Curl of the elastic strain
I regions where the slip is not uniform
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 4 /21
TOPOLOGICAL SINGULARITIES OF THE STRAIN
We can identify dislocations using the decomposition of the deformationgradient
Du = ∇uL3 + ([u]⊗ n) dH2 Σ = βe + βp
- where [u] is the jump of the displacement along the slip plane Σ- ∇u is the absolutely continuous part of the gradient
In presence of dislocations
Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0
µ is the dislocations density
Then dislocations can be understood as
I singularities of the Curl of the elastic strain
I regions where the slip is not uniform
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 4 /21
Why dislocations are importantDislocations in crystals favor the slip =⇒ Plastic behaviour
(Caterpillar, Lloyd, Molina-Aldareguia 2003)
(Crease on a carpet, Cacace 2004)
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 5 /21
DIFFERENT SCALES ARE RELEVANT
Microscopic- Atomistic description
Mesoscopic- Lines carrying an energy
- Interaction, LEDS, Motion...
Macroscopic- Plastic effect
- Dislocation density, Strain gradient theories...
Objective: 3D DISCRETE −→ CONTINUUM POSSIBLE DISCRETE MODELS
Ariza - Ortiz, ARMA 2005
Luckhaus - Mugnai, preprint.
Collaborations: S. Cacace, P. Cermelli, S. Conti, M. Focardi, C. Larsen, G. Leoni,
S. Muller, M. Ortiz, M. Ponsiglione.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 6 /21
DIFFERENT SCALES ARE RELEVANT
Microscopic- Atomistic description
Mesoscopic- Lines carrying an energy
- Interaction, LEDS, Motion...
Macroscopic- Plastic effect
- Dislocation density, Strain gradient theories...
Objective: 3D DISCRETE −→ CONTINUUM POSSIBLE DISCRETE MODELS
Ariza - Ortiz, ARMA 2005
Luckhaus - Mugnai, preprint.
Collaborations: S. Cacace, P. Cermelli, S. Conti, M. Focardi, C. Larsen, G. Leoni,
S. Muller, M. Ortiz, M. Ponsiglione.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 6 /21
DIFFERENT SCALES ARE RELEVANT
Microscopic- Atomistic description
Mesoscopic- Lines carrying an energy
- Interaction, LEDS, Motion...
Macroscopic- Plastic effect
- Dislocation density, Strain gradient theories...
Objective: 3D DISCRETE −→ CONTINUUM POSSIBLE DISCRETE MODELS
Ariza - Ortiz, ARMA 2005
Luckhaus - Mugnai, preprint.
Collaborations: S. Cacace, P. Cermelli, S. Conti, M. Focardi, C. Larsen, G. Leoni,
S. Muller, M. Ortiz, M. Ponsiglione.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 6 /21
DIFFERENT SCALES ARE RELEVANT
Microscopic- Atomistic description
Mesoscopic- Lines carrying an energy
- Interaction, LEDS, Motion...
Macroscopic- Plastic effect
- Dislocation density, Strain gradient theories...
Objective: 3D DISCRETE −→ CONTINUUM POSSIBLE DISCRETE MODELS
Ariza - Ortiz, ARMA 2005
Luckhaus - Mugnai, preprint.
Collaborations: S. Cacace, P. Cermelli, S. Conti, M. Focardi, C. Larsen, G. Leoni,
S. Muller, M. Ortiz, M. Ponsiglione.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 6 /21
We have an almost complete analysis under different scales (mesoscopicand macroscopic) for special geometries.
MESOSCOPICI Cilindrical geometry (dislocations are points)
I Screw dislocations - Burgers vector parallel to the dislocation line -(Ponsiglione, ’06)
I Edge dislocations - Burgers vector orthogonal to the dislocation line -(Cermelli and Leoni ’05)
I Only one slip plane (dislocations are lines on a given slip plane)I A phase field approach for a generalized Nabarro-Peierls model (the
phase is the jump along the slip plane and the energy is a Cahn-Hilliardtype energy with non-local singular perturbation and infinitely manywells potential)(G.- Muller ’06, Cacace-G ’09, Conti-G.-Muller preprint)
All the results above are based on the analysis of a ”semi-discrete” model.
El Hajj, Ibrahim and Monneau for the 1D multiscale analysis for the dynamics.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 7 /21
THE DISCRETE MODEL (Ariza-Ortiz, ARMA 2005)
For simplicity we consider the cubic lattice.
E(u, βp) =3X
i,j=1
Xl, l′∈lattice bonds
1
2Bij(l − l ′)(dui (l)− βp i (l))(duj(l ′)− βpj(l ′))
- u = displacements of the atoms;- du(l) = discrete gradient along the bond l ;- βp = eigen-deformation induced by dislocations (defined on bonds).
βp = b ⊗m
where b ∈ Z3 (Burgers vectors) and m ∈ Z3 (normal to the slip plane)
Four-point interaction energy with interaction coefficients Bij(l − l ′) with finite range.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 8 /21
PARTICULAR CASE: Anti-plane problem (screw dislocations)
Scalar (vertical) displacement u : Z2 ∩ Ω→ R. Take atwo-point interaction discrete energy
Ediscr(u, βp) =X<i,j>
|u(i)− u(j)− βp(< i , j >)|2
Dislocations are introduced through the plastic strainβp : bonds → Z.
Minimizing w.r.t. βp
minβp
Ediscr(u, βp) = Ediscr(u) =X<i,j>
dist2(u(i)− u(j),Z)
Note: βp corresponds to the projection of du on integers.
Remark: βp in general is not a discrete gradient. We candefine a discrete Curl of βp, denoted by dβp, and
α = dβp
is the discrete dislocation density
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 9 /21
PARTICULAR CASE: Anti-plane problem (screw dislocations)
Scalar (vertical) displacement u : Z2 ∩ Ω→ R. Take atwo-point interaction discrete energy
Ediscr(u, βp) =X<i,j>
|u(i)− u(j)− βp(< i , j >)|2
Dislocations are introduced through the plastic strainβp : bonds → Z.
Minimizing w.r.t. βp
minβp
Ediscr(u, βp) = Ediscr(u) =X<i,j>
dist2(u(i)− u(j),Z)
Note: βp corresponds to the projection of du on integers.
Remark: βp in general is not a discrete gradient. We candefine a discrete Curl of βp, denoted by dβp, and
α = dβp
is the discrete dislocation density
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 9 /21
PARTICULAR CASE: Anti-plane problem (screw dislocations)
Scalar (vertical) displacement u : Z2 ∩ Ω→ R. Take atwo-point interaction discrete energy
Ediscr(u, βp) =X<i,j>
|u(i)− u(j)− βp(< i , j >)|2
Dislocations are introduced through the plastic strainβp : bonds → Z.
Minimizing w.r.t. βp
minβp
Ediscr(u, βp) = Ediscr(u) =X<i,j>
dist2(u(i)− u(j),Z)
Note: βp corresponds to the projection of du on integers.
Remark: βp in general is not a discrete gradient. We candefine a discrete Curl of βp, denoted by dβp, and
α = dβp
is the discrete dislocation density
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 9 /21
PARTICULAR CASE: Anti-plane problem (screw dislocations)
Scalar (vertical) displacement u : Z2 ∩ Ω→ R. Take atwo-point interaction discrete energy
Ediscr(u, βp) =X<i,j>
|u(i)− u(j)− βp(< i , j >)|2
Dislocations are introduced through the plastic strainβp : bonds → Z.
Minimizing w.r.t. βp
minβp
Ediscr(u, βp) = Ediscr(u) =X<i,j>
dist2(u(i)− u(j),Z)
Note: βp corresponds to the projection of du on integers.
Remark: βp in general is not a discrete gradient. We candefine a discrete Curl of βp, denoted by dβp, and
α = dβp
is the discrete dislocation density
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 9 /21
SEMI-DISCRETE ANALYSIS FOR SCREW DISLOCATIONS - Ponsiglione ’07
One consider a cylindrical symmetry and we fix Ω ⊂ R2 the cross section
Ω
• xi = cross section of a dislocation• ε = core radius ∼ lattice spacing
• Fix a distribution of dislocations
µ =X
i
ξiδxi
with ξi ∈ Z.• Consider a strain field β satisfyingZ
∂Bε(xi )
β · t ds = ξi
andCurlβ = 0 in Ω \ ∪iBε(xi )
• Elastic Energy (Linearized)
Eε(µ, β) =
ZΩ
|β|2 dx
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 10 /21
SEMI-DISCRETE ANALYSIS FOR SCREW DISLOCATIONS - Ponsiglione ’07
One consider a cylindrical symmetry and we fix Ω ⊂ R2 the cross section
Ω
xi
• xi = cross section of a dislocation
• ε = core radius ∼ lattice spacing
• Fix a distribution of dislocations
µ =X
i
ξiδxi
with ξi ∈ Z.
• Consider a strain field β satisfyingZ∂Bε(xi )
β · t ds = ξi
andCurlβ = 0 in Ω \ ∪iBε(xi )
• Elastic Energy (Linearized)
Eε(µ, β) =
ZΩ
|β|2 dx
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 10 /21
SEMI-DISCRETE ANALYSIS FOR SCREW DISLOCATIONS - Ponsiglione ’07
One consider a cylindrical symmetry and we fix Ω ⊂ R2 the cross section
Ω
Bε(xi )
• xi = cross section of a dislocation• ε = core radius ∼ lattice spacing
• Fix a distribution of dislocations
µ =X
i
ξiδxi
with ξi ∈ Z.• Consider a strain field β satisfyingZ
∂Bε(xi )
β · t ds = ξi
andCurlβ = 0 in Ω \ ∪iBε(xi )
• Elastic Energy (Linearized)
Eε(µ, β) =
ZΩ
|β|2 dx
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 10 /21
SEMI-DISCRETE ANALYSIS FOR SCREW DISLOCATIONS - Ponsiglione ’07
One consider a cylindrical symmetry and we fix Ω ⊂ R2 the cross section
Ω
Bε(xi )
• xi = cross section of a dislocation• ε = core radius ∼ lattice spacing
• Fix a distribution of dislocations
µ =X
i
ξiδxi
with ξi ∈ Z.• Consider a strain field β satisfyingZ
∂Bε(xi )
β · t ds = ξi
andCurlβ = 0 in Ω \ ∪iBε(xi )
• Elastic Energy (Linearized)
Eε(µ, β) =
ZΩ
|β|2 dx
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 10 /21
Γ-convergence result(in the flat norm for the dislocation density µ =
∑ξiδxi )
1
| log ε|Eε(µ) =
1
| log ε|min
”Curlβ=µ”
∫Ω|β|2dx
Γ−→ 1
2π
∑i
|ξi |
Note: the semi-discrete analysis provides the limit of the fully discretemodelIf one consider the discrete energy as above it is also true that
1
| log ε|∑<i ,j>
dist2(u(εi)− u(εj),Z)Γ−→ 1
2π
∑i
|ξi |
where βi ,j = arg mins∈Z|u(εi)− u(εj)− s|2 and dβi ,j =∑ξiδxi
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 11 /21
DISLOCATIONS VS THE GINZBURG-LANDAU FUNCTIONALBoth have topological singularities with logarithmic scaling.
In the 2D case it can be shown that the ”Screw dislocation energy” isvariationally equivalent to the Ginzburg-Landau energy for vortices(Alicandro, Cicalese, Ponsiglione, to appear) .
EssentiallyI They have the same Γ-limitI From the convergence of one it can be deduced the convergence of
the other
In the 3D case a general model for dislocations has the samephenomenology: this suggests that it can be formulated as aGizburg-Landau type energy.
• We start with the analysis of a 3D model at a continuum level(”semi-discrete”)• This will show similarity with Ginzburg-Landau models, but also morecomplexity
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 12 /21
DISLOCATIONS VS THE GINZBURG-LANDAU FUNCTIONALBoth have topological singularities with logarithmic scaling.
In the 2D case it can be shown that the ”Screw dislocation energy” isvariationally equivalent to the Ginzburg-Landau energy for vortices(Alicandro, Cicalese, Ponsiglione, to appear) .
EssentiallyI They have the same Γ-limitI From the convergence of one it can be deduced the convergence of
the other
In the 3D case a general model for dislocations has the samephenomenology: this suggests that it can be formulated as aGizburg-Landau type energy.
• We start with the analysis of a 3D model at a continuum level(”semi-discrete”)• This will show similarity with Ginzburg-Landau models, but also morecomplexity
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 12 /21
3D SEMI-DISCRETE ANALYSIS (Conti-G.-Ortiz, in preparation)
• Dislocations are identified with measures concentrated on curves γPrecisely µ ∈MB(Ω) = dislocation denities such that
µ = b ⊗ tH1 γ
with
- γ = ∪iγi , γi Lip. curves in Ω- t is the tangent vector of γ- b : γ → Z3
- Divµ = 0
loops
b
t
b1+ b2
b1 b2
• The elastic strain associate to µ, β ∈ Adε(µ) = admissible strains, such that
β : Ω ⊆ R3 → R3×3 β ∈ L2(Ω) ”Curlβ = µ” (Curlβ = µ ? ϕε)
The Elastic Energy
Eε(β, µ) =
ZΩ
〈Cβ, β〉 dx µ ∈MB(Ω) β ∈ Adε(µ)
with C the elastic tensor (〈Cβ, β〉 ≥ C |βsym|2)
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 13 /21
3D SEMI-DISCRETE ANALYSIS (Conti-G.-Ortiz, in preparation)
• Dislocations are identified with measures concentrated on curves γPrecisely µ ∈MB(Ω) = dislocation denities such that
µ = b ⊗ tH1 γ
with
- γ = ∪iγi , γi Lip. curves in Ω- t is the tangent vector of γ- b : γ → Z3
- Divµ = 0
loops
b
t
b1+ b2
b1 b2
• The elastic strain associate to µ, β ∈ Adε(µ) = admissible strains, such that
β : Ω ⊆ R3 → R3×3 β ∈ L2(Ω) ”Curlβ = µ” (Curlβ = µ ? ϕε)
The Elastic Energy
Eε(β, µ) =
ZΩ
〈Cβ, β〉 dx µ ∈MB(Ω) β ∈ Adε(µ)
with C the elastic tensor (〈Cβ, β〉 ≥ C |βsym|2)
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 13 /21
3D SEMI-DISCRETE ANALYSIS (Conti-G.-Ortiz, in preparation)
• Dislocations are identified with measures concentrated on curves γPrecisely µ ∈MB(Ω) = dislocation denities such that
µ = b ⊗ tH1 γ
with
- γ = ∪iγi , γi Lip. curves in Ω- t is the tangent vector of γ- b : γ → Z3
- Divµ = 0
loops
b
t
b1+ b2
b1 b2
• The elastic strain associate to µ, β ∈ Adε(µ) = admissible strains, such that
β : Ω ⊆ R3 → R3×3 β ∈ L2(Ω) ”Curlβ = µ” (Curlβ = µ ? ϕε)
The Elastic Energy
Eε(β, µ) =
ZΩ
〈Cβ, β〉 dx µ ∈MB(Ω) β ∈ Adε(µ)
with C the elastic tensor (〈Cβ, β〉 ≥ C |βsym|2)
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 13 /21
THE GOAL: Study the asymptotics in terms of Γ-convergence for theenergy
Fε(β, µ) =1
| log ε|
∫Ω〈Cβ, β〉 dx µ ∈MB(Ω) β ∈ Adε(µ)
Subject to a diluteness condition (big loops and well separated)
γ = ∪iγi
with- γi are closed segments of length ≥ ρε >> ε ( | log ρε|
| log ε| → 0)
- If γi ∩ γj = ∅ =⇒ dist(γi , γj) > ηρε- If γi ∩ γj 6= ∅ the angle is larger than θ0 > 0.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 14 /21
THE CLASSICAL STRAIN FOR INFINITE STRAIGHT DISLOCATIONS
The elastic strain is given by
β0 =1
rΓ0(θ)
and satisfiesDiv(Cβ0) = 0 Curlβ0 = b ⊗ tH1 γ in R3
1) Since Curlβ0 = 0 in R3 \ γ, then there exists u : R3 \ Σ→ R3
such that
β0 = ∇u in R3 \ Σ and [u] = b on Σ
Σ
ε
R
CR
Cε
b
γ
2)
limε→0
1
| log ε|
ZR3\Cε(γ)
〈Cβ0, β0〉 dx = limε→0
1
| log ε|
ZCR (γ)\Cε(γ)
〈Cβ0, β0〉 dx
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 15 /21
THE CLASSICAL STRAIN FOR INFINITE STRAIGHT DISLOCATIONS
The elastic strain is given by
β0 =1
rΓ0(θ)
and satisfiesDiv(Cβ0) = 0 Curlβ0 = b ⊗ tH1 γ in R3
1) Since Curlβ0 = 0 in R3 \ γ, then there exists u : R3 \ Σ→ R3
such that
β0 = ∇u in R3 \ Σ and [u] = b on Σ
Σ
ε
R
CR
Cε
b
γ
2)
limε→0
1
| log ε|
ZR3\Cε(γ)
〈Cβ0, β0〉 dx = limε→0
1
| log ε|
ZCR (γ)\Cε(γ)
〈Cβ0, β0〉 dx
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 15 /21
THE CLASSICAL STRAIN FOR INFINITE STRAIGHT DISLOCATIONS
The elastic strain is given by
β0 =1
rΓ0(θ)
and satisfiesDiv(Cβ0) = 0 Curlβ0 = b ⊗ tH1 γ in R3
1) Since Curlβ0 = 0 in R3 \ γ, then there exists u : R3 \ Σ→ R3
such that
β0 = ∇u in R3 \ Σ and [u] = b on Σ
Σ
ε
R
CR
Cε
b
γ
2)
limε→0
1
| log ε|
ZR3\Cε(γ)
〈Cβ0, β0〉 dx = limε→0
1
| log ε|
ZCR (γ)\Cε(γ)
〈Cβ0, β0〉 dx
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 15 /21
LOWER BOUNDIdea: Optimal configurations will have the same decay, also for generaldislocations lines.
Take a sequence µε ∈MB(Ω) and βε ∈ Adε(µε)
1
| log ε|
∫Ω
〈Cβε, βε〉 dx ≥ 1
| log ε|∑
i
∫Cηρε (γ i
ε)\Cε(γ iε)
〈Cβε, βε〉 dx
≥ 1
| log ε|∑
i
minCurlβ=µε
∫Cηρε (γ i
ε)\Cε(γ iε)
〈Cβ, β〉 dx
=∑
i
ϕε(biε, γ
iε) ≥ c
∑i
|biε|H1(γ i
ε)
Compactness: If Fε(βε, µε) ≤ C =⇒ (up to subseq.) µε∗∑
i bi ⊗ tiH1 γi
(in the sense of 1-currents)
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 16 /21
LOWER BOUNDIdea: Optimal configurations will have the same decay, also for generaldislocations lines.
Take a sequence µε ∈MB(Ω) and βε ∈ Adε(µε)
1
| log ε|
∫Ω
〈Cβε, βε〉 dx ≥ 1
| log ε|∑
i
∫Cηρε (γ i
ε)\Cε(γ iε)
〈Cβε, βε〉 dx
≥ 1
| log ε|∑
i
minCurlβ=µε
∫Cηρε (γ i
ε)\Cε(γ iε)
〈Cβ, β〉 dx
=∑
i
ϕε(biε, γ
iε) ≥ c
∑i
|biε|H1(γ i
ε)
Compactness: If Fε(βε, µε) ≤ C =⇒ (up to subseq.) µε∗∑
i bi ⊗ tiH1 γi
(in the sense of 1-currents)
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 16 /21
LOWER BOUNDIdea: Optimal configurations will have the same decay, also for generaldislocations lines.
Take a sequence µε ∈MB(Ω) and βε ∈ Adε(µε)
1
| log ε|
∫Ω
〈Cβε, βε〉 dx ≥ 1
| log ε|∑
i
∫Cηρε (γ i
ε)\Cε(γ iε)
〈Cβε, βε〉 dx
≥ 1
| log ε|∑
i
minCurlβ=µε
∫Cηρε (γ i
ε)\Cε(γ iε)
〈Cβ, β〉 dx
=∑
i
ϕε(biε, γ
iε) ≥ c
∑i
|biε|H1(γ i
ε)
Compactness: If Fε(βε, µε) ≤ C =⇒ (up to subseq.) µε∗∑
i bi ⊗ tiH1 γi
(in the sense of 1-currents)
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 16 /21
LOWER BOUNDIdea: Optimal configurations will have the same decay, also for generaldislocations lines.
Take a sequence µε ∈MB(Ω) and βε ∈ Adε(µε)
1
| log ε|
∫Ω
〈Cβε, βε〉 dx ≥ 1
| log ε|∑
i
∫Cηρε (γ i
ε)\Cε(γ iε)
〈Cβε, βε〉 dx
≥ 1
| log ε|∑
i
minCurlβ=µε
∫Cηρε (γ i
ε)\Cε(γ iε)
〈Cβ, β〉 dx
=∑
i
ϕε(biε, γ
iε) ≥ c
∑i
|biε|H1(γ i
ε)
Compactness: If Fε(βε, µε) ≤ C =⇒ (up to subseq.) µε∗∑
i bi ⊗ tiH1 γi
(in the sense of 1-currents)
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 16 /21
CELL PROBLEM FORMULA
Let γ be a segment of length 1, t its tangent vector and b ∈ Z3 we canshow that
ϕ0(b ⊗ t) := limε→0
1
| log ε|min
Curlβ=µ
∫Cηρε (γ)\Cε(γ)
〈Cβ, β〉 dx
= limε→0
1
| log ε|min
Curlβ=µ
∫C1(γ)\Cε(γ)
〈Cβ, β〉 dx
= minβ= 1
rΓ(θ) Curlβ=b⊗tH1 γ
∫S1
〈CΓ, Γ〉 ds =
∫S1
〈CΓ0, Γ0〉 ds
This is a variational characterization of what is called the pre-logarithmicfactor.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 17 /21
CELL PROBLEM FORMULA
Let γ be a segment of length 1, t its tangent vector and b ∈ Z3 we canshow that
ϕ0(b ⊗ t) := limε→0
1
| log ε|min
Curlβ=µ
∫Cηρε (γ)\Cε(γ)
〈Cβ, β〉 dx
= limε→0
1
| log ε|min
Curlβ=µ
∫C1(γ)\Cε(γ)
〈Cβ, β〉 dx
= minβ= 1
rΓ(θ) Curlβ=b⊗tH1 γ
∫S1
〈CΓ, Γ〉 ds =
∫S1
〈CΓ0, Γ0〉 ds
This is a variational characterization of what is called the pre-logarithmicfactor.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 17 /21
CELL PROBLEM FORMULA
Let γ be a segment of length 1, t its tangent vector and b ∈ Z3 we canshow that
ϕ0(b ⊗ t) := limε→0
1
| log ε|min
Curlβ=µ
∫Cηρε (γ)\Cε(γ)
〈Cβ, β〉 dx
= limε→0
1
| log ε|min
Curlβ=µ
∫C1(γ)\Cε(γ)
〈Cβ, β〉 dx
= minβ= 1
rΓ(θ) Curlβ=b⊗tH1 γ
∫S1
〈CΓ, Γ〉 ds =
∫S1
〈CΓ0, Γ0〉 ds
This is a variational characterization of what is called the pre-logarithmicfactor.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 17 /21
CELL PROBLEM FORMULA
Let γ be a segment of length 1, t its tangent vector and b ∈ Z3 we canshow that
ϕ0(b ⊗ t) := limε→0
1
| log ε|min
Curlβ=µ
∫Cηρε (γ)\Cε(γ)
〈Cβ, β〉 dx
= limε→0
1
| log ε|min
Curlβ=µ
∫C1(γ)\Cε(γ)
〈Cβ, β〉 dx
= minβ= 1
rΓ(θ) Curlβ=b⊗tH1 γ
∫S1
〈CΓ, Γ〉 ds =
∫S1
〈CΓ0, Γ0〉 ds
This is a variational characterization of what is called the pre-logarithmicfactor.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 17 /21
LACK OF LOWER SEMICONTINUITY: MICROSTRUCTUREThe line tension energy ∫
γϕ0(b(x)⊗ t(x)) dH1(x)
is not lower semi-continuous w.r.t. the weak convergence of measures(weak convergence of 1-currents).
b1+ b2 b2
b1
b1+ b2
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 18 /21
LACK OF LOWER SEMICONTINUITY: MICROSTRUCTUREThe line tension energy ∫
γϕ0(b(x)⊗ t(x)) dH1(x)
is not lower semi-continuous w.r.t. the weak convergence of measures(weak convergence of 1-currents).
b1+ b2 b2
b1
b1+ b2
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 18 /21
RELAXATION: The H1-elliptic envelope
The lower semicontinuous envelope of the line tension energy above isgiven by ∫
γϕ0(b(x), t(x)) dH1(x)
where ϕ0 is the H1-elliptic envelope of ϕ0 and is given by
ϕ0(b ⊗ t) = inf ∫
γ∩B1(0)ϕ0(b(x)⊗ t(x)) dH1(x) : µ ∈MB(R3) ,
supp(µ− b ⊗ tdH1 (Rt)) ⊂ B1(0).
1. ϕ0 is Lipschitz-continuous in the second argument;
2. ϕ0 is subadditive in its first argument;
Note: Using this formula one can show optimality of the lower bound.
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 19 /21
THE Γ-CONVERGENCE RESULT
Theorem Under the diluteness condition
1. (compactness)If Fε(βε, µε) ≤ C =⇒ up to a subsequence µε
∗ µ = b ⊗ tH1 γ
2. (Γ-convergence)
Fε(β, µ)Γ−→
∫γϕ0(b(x)⊗ t(x)) dH1(x)
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 20 /21
FINAL REMARKS
1. For ”good” discrete energies (for which we have Γ-convergence in thelinear elastic case) in presence of dislocations we obtain the same linetension limit.
2. We would like to remove the kinematic constraints (dilutedislocations)
3. There might be a Ginzburg-Landau type formulation that enforcesconcentration on lines with two difficulties:
I The energy density is anisotropic and depends only on the symmetricpart of the strain field
I The line tension limit creates microstructure
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 21 /21
PLANAR DISLOCATIONS(Peierls-Nabarro 1940-1947, Koslowski-Ortiz 2004)
Slip only on one single slip plane Q = (0, 1)2 ⊆ R2
(a domain on the slip plane)
relevant variable v = [u],v : Q → R2 (the slip)
ε = small parameter
∼ lattice spacing
Etot(v) = Eelast.(v)
qLong-range elastic
energy induced by the slip
q
+ Emisfit(v)
qInterfacial energy that
penalizes slips not
compatible with the lattice
q
K is a matrix valued singular kernel
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 22 /21
PLANAR DISLOCATIONS(Peierls-Nabarro 1940-1947, Koslowski-Ortiz 2004)
slip plane
Bulk elastic energy Q = (0, 1)2 ⊆ R2
(a domain on the slip plane)
relevant variable v = [u],v : Q → R2 (the slip)
ε = small parameter
∼ lattice spacing
Etot(v) = Eelast.(v)q
Long-range elastic
energy induced by the slip
q
+ Emisfit(v)
qInterfacial energy that
penalizes slips not
compatible with the lattice
q
K is a matrix valued singular kernel
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 22 /21
PLANAR DISLOCATIONS(Peierls-Nabarro 1940-1947, Koslowski-Ortiz 2004)
slip plane Interfacial energy
Q = (0, 1)2 ⊆ R2
(a domain on the slip plane)
relevant variable v = [u],v : Q → R2 (the slip)
ε = small parameter
∼ lattice spacing
Etot(v) = Eelast.(v)q
Long-range elastic
energy induced by the slip
q
+ Emisfit(v)q
Interfacial energy that
penalizes slips not
compatible with the lattice
q
K is a matrix valued singular kernel
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 22 /21
PLANAR DISLOCATIONS(Peierls-Nabarro 1940-1947, Koslowski-Ortiz 2004)
slip plane Interfacial energy
Bulk elastic energy
Q = (0, 1)2 ⊆ R2
(a domain on the slip plane)
relevant variable v = [u],v : Q → R2 (the slip)
ε = small parameter
∼ lattice spacing
Etot(v) = Eelast.(v)q
Long-range elastic
energy induced by the slip
q
+ Emisfit(v)q
Interfacial energy that
penalizes slips not
compatible with the lattice
qEε(v) =
ZQ
ZQ
(v(x)− v(y))tK(x − y)(v(x)− v(y)) dx dy +1
ε
ZQ
W (v) dx
K is a matrix valued singular kernel
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 22 /21
PLANAR DISLOCATIONS(Peierls-Nabarro 1940-1947, Koslowski-Ortiz 2004)
slip plane Interfacial energy
Bulk elastic energy
Q = (0, 1)2 ⊆ R2
(a domain on the slip plane)
relevant variable v = [u],v : Q → R2 (the slip)
ε = small parameter
∼ lattice spacing
Etot(v) = Eelast.(v)q
Long-range elastic
energy induced by the slip
q
+ Emisfit(v)q
Interfacial energy that
penalizes slips not
compatible with the lattice
qEε(v) =
ZQ
ZQ
(v(x)− v(y))tK(x − y)(v(x)− v(y)) dx dy +1
ε
ZQ
W (v) dx
K is a matrix valued singular kernel
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 22 /21
Theorem (Cacace-G. ’09, Conti-G.-Muller, preprint)
(Compactness)If Eε(vε) ≤ C | log ε|, then (up to a subsequence) ∃ aε ∈ Z2 andv ∈ BV (Q,Z2) such that
vε − aε → v in Lp ∀p < 2
(Γ-convergence)∃ ϕ : Z2 × S1 → R (uniquely determined by the kernel) such that
1
| log ε|Eε(v) Γ-converges to F (v) =
∫Su
ϕ0([v ], tv ) dH1
Svtv
Sv = discontinuity set of v[v ] = jump of v
tv = tangent vector to Sv
Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 23 /21