variational models for dislocations lecture 1...
TRANSCRIPT
Variational models for dislocations
Lecture 1 (Introduction and phase field)
Adriana Garroni
Sapienza, Universita di Roma
Ile de Re, May 23-27, 2011.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 1 /24
Elastic vs Plastic deformations
Single crystal Elastic deformation (reversible)
Elasto-plastic deformation Permanent deformation
PLASTIC SLIP: slip on slip planes is the main mechanism for plasticdeformation
OBSERVATION:I Measured yield stress lower than the theoretical oneI Hardening effects
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 2 /24
Elastic vs Plastic deformations
Single crystal Elastic deformation (reversible)
Elasto-plastic deformation Permanent deformation
PLASTIC SLIP: slip on slip planes is the main mechanism for plasticdeformation
OBSERVATION:I Measured yield stress lower than the theoretical oneI Hardening effects
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 2 /24
DISLOCATIONSNOTE: The slip in general is not uniform ⇐⇒ DEFECTS (dislocations)
b
Screw dislocation
b
Edge dislocation
b
Dislocation loop
Continuum definition of dislocations:Lines on slip planes separating regions undergoing different slips (Volterra 1905).
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 3 /24
MICROSCOPIC DESCRIPTION OF DISLOCATIONS
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 4 /24
MICROSCOPIC DESCRIPTION OF DISLOCATIONS
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Dislocation coreBurgers circuit
Burgers vector
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 4 /24
MICROSCOPIC DESCRIPTION OF DISLOCATIONS
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Dislocation coreBurgers circuit
Burgers vector
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 4 /24
MICROSCOPIC DESCRIPTION OF DISLOCATIONS
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Dislocation coreBurgers circuit
Burgers vector
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 4 /24
MICROSCOPIC DESCRIPTION OF DISLOCATIONS
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Dislocation coreBurgers circuit
Burgers vector
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 4 /24
MICROSCOPIC DESCRIPTION OF DISLOCATIONS
Dislocations are line defects in crystals (topological defects)At the microscopic level:
Dislocation coreBurgers circuit
Burgers vector
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 4 /24
TOPOLOGICAL SINGULARITIES OF THE STRAIN
Considering a displacement including slips (discontinuities) we can identifydislocations using the decomposition of the deformation gradient
Du = ∇u L3︸ ︷︷ ︸diffuse elastic distorsion
+ ([u]⊗ n) dH2 Σ︸ ︷︷ ︸slip along slip planes
= 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 ∇u is not a gradient and
Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0
µ is the dislocations density (Nye ’53)
Then dislocations can be understood as
I singularities of the Curl of the elastic strain
(D-D model)
I regions where the slip is not uniform
(Peierls-Nabarro)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 5 /24
TOPOLOGICAL SINGULARITIES OF THE STRAIN
Considering a displacement including slips (discontinuities) we can identifydislocations using the decomposition of the deformation gradient
Du = ∇u L3︸ ︷︷ ︸diffuse elastic distorsion
+ ([u]⊗ n) dH2 Σ︸ ︷︷ ︸slip along slip planes
= 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 ∇u is not a gradient and
Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0
µ is the dislocations density (Nye ’53)
Then dislocations can be understood as
I singularities of the Curl of the elastic strain
(D-D model)
I regions where the slip is not uniform
(Peierls-Nabarro)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 5 /24
TOPOLOGICAL SINGULARITIES OF THE STRAIN
Considering a displacement including slips (discontinuities) we can identifydislocations using the decomposition of the deformation gradient
Du = ∇u L3︸ ︷︷ ︸diffuse elastic distorsion
+ ([u]⊗ n) dH2 Σ︸ ︷︷ ︸slip along slip planes
= 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 ∇u is not a gradient and
Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0
µ is the dislocations density (Nye ’53)
Then dislocations can be understood as
I singularities of the Curl of the elastic strain
(D-D model)
I regions where the slip is not uniform
(Peierls-Nabarro)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 5 /24
TOPOLOGICAL SINGULARITIES OF THE STRAIN
Considering a displacement including slips (discontinuities) we can identifydislocations using the decomposition of the deformation gradient
Du = ∇u L3︸ ︷︷ ︸diffuse elastic distorsion
+ ([u]⊗ n) dH2 Σ︸ ︷︷ ︸slip along slip planes
= 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 ∇u is not a gradient and
Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0
µ is the dislocations density (Nye ’53)
Then dislocations can be understood as
I singularities of the Curl of the elastic strain
(D-D model)
I regions where the slip is not uniform
(Peierls-Nabarro)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 5 /24
TOPOLOGICAL SINGULARITIES OF THE STRAIN
Considering a displacement including slips (discontinuities) we can identifydislocations using the decomposition of the deformation gradient
Du = ∇u L3︸ ︷︷ ︸diffuse elastic distorsion
+ ([u]⊗ n) dH2 Σ︸ ︷︷ ︸slip along slip planes
= 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 ∇u is not a gradient and
Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0
µ is the dislocations density (Nye ’53)
Then dislocations can be understood as
I singularities of the Curl of the elastic strain (D-D model)
I regions where the slip is not uniform (Peierls-Nabarro)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 5 /24
These defects favor the slip =⇒ Plastic behaviour
(Caterpillar, Lloyd, Molina-Aldareguia 2003)
(Crease on a carpet, Cacace 2004)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 6 /24
First conjectured: 1905 Volterra, 1934 Orowan/Polanyi/Taylor
Then observed: ∼ ’50
Transmission Electron Micrograph of Dislocations
Hull and Bacon, “Introduction to Dislocations”, 1965Nabarro, “Theory of crystal dislocations”, Oxford University Press, London 1967Hirth and Lothe, “Theory of Dislocations”, Wiley. 1982Phillips, “Crystals, Defects and Microstructures”, Cambridge University Press, 2001
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 7 /24
DIFFERENT SCALES
Microscopic- Atomistic description
Mesoscopic- Lines carrying an energy
- Interaction, LEDS, Motion...
Macroscopic- Plastic effect
- Dislocation density, Strain gradient theories...
PROJECT: DISCRETE −→ CONTINUUM POSSIBLE DISCRETE MODELS
Ariza - Ortiz, ARMA 2005(Lattice reference configuration)
Luckhaus - Mugnai, Continuum Mech.Thermodyn. 2010(Periodic multi-body interaction potential)
Collaborations: S. Cacace, P. Cermelli, S. Conti, M. Focardi, C. Larsen, G. Leoni,
S. Muller, M. Ortiz, M. Ponsiglione.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 8 /24
DIFFERENT SCALES
Microscopic- Atomistic description
Mesoscopic- Lines carrying an energy
- Interaction, LEDS, Motion...
Macroscopic- Plastic effect
- Dislocation density, Strain gradient theories...
PROJECT: DISCRETE −→ CONTINUUM POSSIBLE DISCRETE MODELS
Ariza - Ortiz, ARMA 2005(Lattice reference configuration)
Luckhaus - Mugnai, Continuum Mech.Thermodyn. 2010(Periodic multi-body interaction potential)
Collaborations: S. Cacace, P. Cermelli, S. Conti, M. Focardi, C. Larsen, G. Leoni,
S. Muller, M. Ortiz, M. Ponsiglione.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 8 /24
DIFFERENT SCALES
Microscopic- Atomistic description
Mesoscopic- Lines carrying an energy
- Interaction, LEDS, Motion...
Macroscopic- Plastic effect
- Dislocation density, Strain gradient theories...
PROJECT: DISCRETE −→ CONTINUUM POSSIBLE DISCRETE MODELS
Ariza - Ortiz, ARMA 2005(Lattice reference configuration)
Luckhaus - Mugnai, Continuum Mech.Thermodyn. 2010(Periodic multi-body interaction potential)
Collaborations: S. Cacace, P. Cermelli, S. Conti, M. Focardi, C. Larsen, G. Leoni,
S. Muller, M. Ortiz, M. Ponsiglione.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 8 /24
DIFFERENT SCALES
Microscopic- Atomistic description
Mesoscopic- Lines carrying an energy
- Interaction, LEDS, Motion...
Macroscopic- Plastic effect
- Dislocation density, Strain gradient theories...
PROJECT: DISCRETE −→ CONTINUUM POSSIBLE DISCRETE MODELS
Ariza - Ortiz, ARMA 2005(Lattice reference configuration)
Luckhaus - Mugnai, Continuum Mech.Thermodyn. 2010(Periodic multi-body interaction potential)
Collaborations: S. Cacace, P. Cermelli, S. Conti, M. Focardi, C. Larsen, G. Leoni,
S. Muller, M. Ortiz, M. Ponsiglione.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 8 /24
THE DISCRETE MODELFor simplicity we consider the cubic lattice.
Four-point interaction quadratic energy with interaction coefficientsBij(`− `′) with finite range.
E(u) =3∑
i,j=1
∑`, `′∈lattice bonds
1
2Bij(`− `′)dui (`)duj(`′)
- u = displacements of the atoms;- du(`) = u(i`)− u(i` − 1) discrete gradient along the bond `;
We assume that the rescaled energy εE(u) Γ-converges to a linear elastic continuumenergy ∫
Ω
C∇u∇u dx
where Cξξ = E(ξx) for every ξ ∈ M3×3, with Cξξ ≥ |ξsym|2.
Some abstract assumptions on the coefficients will guarantee this convergence (e.g.
Alicandro-Cicalese ’04)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 9 /24
THE EIGEN-DEFORMATION (Ariza-Ortiz, ARMA 2005)
E(u, βp) =3∑
i,j=1
∑`, `′∈lattice bonds
1
2Bij(`− `′)(dui (`)− βp i (`))(duj(`′)− βpj(`′))
- u = displacements of the atoms;- du(`) = u(i`)− u(i` − 1) discrete gradient along the bond `;- 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)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 10 /24
PARTICULAR CASE: Anti-plane problem (screw dislocations)
Scalar (vertical) displacement u : Z2 ∩ Ω→ R. Take atwo-point interaction discrete energy
Ediscr(u, βp) =∑<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) =∑<i,j>
dist2(u(i)−u(j),Z)
Note: βp corresponds to the projection of du onintegers.
Remark: βp in general is not a discrete gradient. Wecan define a discrete Curl of βp, denoted by dβp, and
α = dβp
is the discrete dislocation density
Generalization of Frenkel Kontorowa ’38 (for the analysis see e.g. Bonilla Carpio ’05, Fino-Ibrahim-Monneau ’11)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 11 /24
PARTICULAR CASE: Anti-plane problem (screw dislocations)
Scalar (vertical) displacement u : Z2 ∩ Ω→ R. Take atwo-point interaction discrete energy
Ediscr(u, βp) =∑<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) =∑<i,j>
dist2(u(i)−u(j),Z)
Note: βp corresponds to the projection of du onintegers.
Remark: βp in general is not a discrete gradient. Wecan define a discrete Curl of βp, denoted by dβp, and
α = dβp
is the discrete dislocation densityGeneralization of Frenkel Kontorowa ’38 (for the analysis see e.g. Bonilla Carpio ’05, Fino-Ibrahim-Monneau ’11)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 11 /24
PARTICULAR CASE: Anti-plane problem (screw dislocations)
Scalar (vertical) displacement u : Z2 ∩ Ω→ R. Take atwo-point interaction discrete energy
Ediscr(u, βp) =∑<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) =∑<i,j>
dist2(u(i)−u(j),Z)
Note: βp corresponds to the projection of du onintegers.
Remark: βp in general is not a discrete gradient. Wecan define a discrete Curl of βp, denoted by dβp, and
α = dβp
is the discrete dislocation density
Generalization of Frenkel Kontorowa ’38 (for the analysis see e.g. Bonilla Carpio ’05, Fino-Ibrahim-Monneau ’11)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 11 /24
PARTICULAR CASE: Anti-plane problem (screw dislocations)
Scalar (vertical) displacement u : Z2 ∩ Ω→ R. Take atwo-point interaction discrete energy
Ediscr(u, βp) =∑<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) =∑<i,j>
dist2(u(i)−u(j),Z)
Note: βp corresponds to the projection of du onintegers.
Remark: βp in general is not a discrete gradient. Wecan define a discrete Curl of βp, denoted by dβp, and
α = dβp
is the discrete dislocation densityGeneralization of Frenkel Kontorowa ’38 (for the analysis see e.g. Bonilla Carpio ’05, Fino-Ibrahim-Monneau ’11)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 11 /24
SEMI-DISCRETE MODELS
Hybrid models with coexistence of different scales:
Continuum relevant variables with constraints of microscopical nature:far from the dislocations we approximate the discrete interactions with acontinuum elastic interaction
Examples of these hybrid models in the classical literature
I D-D model(elastic strain with topological singularities: the Curl is concentrated in points,
with multiplicity given by the Burgers vectors)
I Peierls-Nabarro model(slip, on a single slip plane, which prefers values that are multiples of the Burgers
vector - i.e. compatible with an underlying lattice)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 12 /24
2-DIMENSIONAL GEOMETRIESCylindrical geometry - screw dislocations
Cylindrical geometry - edge dislocations
Dislocations on a slip plane
These are the special geometries considered in the classic literature andone should start from these to understand the complexity of the problem.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 13 /24
In the variational case (stationary)We have an almost complete analysis (in terms of Γ-convergence) under different scales.
MESOSCOPIC (Line tension)
I Cylindrical geometry (D-D model: dislocations are points singularities)
I Screw dislocations - (Ponsiglione, ’06)I Edge dislocations - (Cermelli-Leoni ’05, Scardia-Zeppieri ’11)
I Only one slip plane (dislocations are lines on a given slip plane)
I A phase field approach for a generalized Peierls-Nabarro model -
(Garroni-Muller ’06, Cacace-Garroni ’09, Conti-Garroni-Muller ’10)
I 3D Core radius approach (Conti-Garroni-Ortiz, in progress)
MACROSCOPIC (Strain gradient theories)
I Cylindrical geometry (dislocations are points)
I 1D - (Focardi-Garroni ’07)I Edge dislocations - (Garroni-Leoni-Ponsiglione ’10)
I 3D Core radius approach (Garroni-Ponsiglione, in progress)
All the results above are based on “semi-discrete” (micro) models.Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 14 /24
Multiscale analysis for dislocation dynamics
I El Hajj-Ibrahim-Monneau for the full multiscale (1D) analysis for the dynamics
(’09).
I FK to Peierls Nabarro (Fino-Ibrahim-Monneau, preprint)I Peierls Nabarro to Discrete Dislocation Dynamics (Gonzalez-Monneau,
preprint)I Discrete Dislocation Dynamics to Dislocation Density model
(Forcadel-Imbert-Monneau ’09)
I Related asymptotics(Da Lio-Forcadel-Monneau ’08, Briani-Monneau ’09, Forcadel-Imbert-Monneaupreprint, Caffarelli-Souganidis ’10, Imbert-Souganidis ’10)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 15 /24
THE ENERGY OF A STRAIGHT DISLOCATIONBasic facts of the “continuum” (semi-discrete) setting (Volterra)
The elastic strain β0 in the presence of a dislocation decays as 1r.
Precisely
β0 =1
rΓ0(θ)
and satisfies
Div(Cβ0) = 0 Curlβ0 = b ⊗ tH1 γ in R3
Remark: Since Curlβ0 = 0 in R3 \ γ, then there existsu : R3 \ Σ→ R3 such that
β0 = ∇u in R3 \ Σ and [u] = b on Σ
Σ
ε = core radius
R
CR
Cε
bh
Straight dislocation
The linear elastic energy outside the core behaves as the logarithm of the core radius
E
h∼ |b|2 log
R
ε.
Precisely
limε→0
1
| log ε|
∫CR (γ)\Cε(γ)
〈Cβ0, β0〉 dx = limε→0
1
| log ε|
∫CR (γ)\Cε(γ)
〈Cβ0, β0〉 dx =
∫S1
〈CΓ0, Γ0〉 ds
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 16 /24
CONSEQUENCES
1. Using a semi-discrete setting (i.e. continuum strain fields) the energyof the core is infinite.
2. One has to regularize the problem in order to avoid singularities.
3. In the fully discrete setting we don’t have this problem (the energy ofthe core is given by a finite number of interactions).
OVERVIEW OF THE REGULARIZED “SEMI-DISCRETE” MODELS
1. Peierls Nabarro(smooth slips)
2. Core Radius approach(linear elastic energy far from the core)
3. Non quadratic energies(the strain induced by a dislocation has finite energy)
Note: Connection with the Ginzburg-Landau model for superconductors
(Alicandro-Cicalese-Ponsiglione ’11)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 17 /24
CONSEQUENCES
1. Using a semi-discrete setting (i.e. continuum strain fields) the energyof the core is infinite.
2. One has to regularize the problem in order to avoid singularities.
3. In the fully discrete setting we don’t have this problem (the energy ofthe core is given by a finite number of interactions).
OVERVIEW OF THE REGULARIZED “SEMI-DISCRETE” MODELS
1. Peierls Nabarro(smooth slips)
2. Core Radius approach(linear elastic energy far from the core)
3. Non quadratic energies(the strain induced by a dislocation has finite energy)
Note: Connection with the Ginzburg-Landau model for superconductors
(Alicandro-Cicalese-Ponsiglione ’11)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 17 /24
CONSEQUENCES
1. Using a semi-discrete setting (i.e. continuum strain fields) the energyof the core is infinite.
2. One has to regularize the problem in order to avoid singularities.
3. In the fully discrete setting we don’t have this problem (the energy ofthe core is given by a finite number of interactions).
OVERVIEW OF THE REGULARIZED “SEMI-DISCRETE” MODELS
1. Peierls Nabarro(smooth slips)
2. Core Radius approach(linear elastic energy far from the core)
3. Non quadratic energies(the strain induced by a dislocation has finite energy)
Note: Connection with the Ginzburg-Landau model for superconductors
(Alicandro-Cicalese-Ponsiglione ’11)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 17 /24
CONSEQUENCES
1. Using a semi-discrete setting (i.e. continuum strain fields) the energyof the core is infinite.
2. One has to regularize the problem in order to avoid singularities.
3. In the fully discrete setting we don’t have this problem (the energy ofthe core is given by a finite number of interactions).
OVERVIEW OF THE REGULARIZED “SEMI-DISCRETE” MODELS
1. Peierls Nabarro(smooth slips)
2. Core Radius approach(linear elastic energy far from the core)
3. Non quadratic energies(the strain induced by a dislocation has finite energy)
Note: Connection with the Ginzburg-Landau model for superconductors
(Alicandro-Cicalese-Ponsiglione ’11)
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 17 /24
1. GENERALIZED PEIERLS-NABARRO (Peierls 1940, Nabarro 1947, Koslowski-Ortiz 2004)
Slip only on one single slip plane Q = (0, 1)2 ⊆ R2
(a domain on the slip plane)
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 Variational models for dislocations Ile de Re, 2011 18 /24
1. GENERALIZED PEIERLS-NABARRO (Peierls 1940, Nabarro 1947, Koslowski-Ortiz 2004)
slip plane
Bulk elastic energy Q = (0, 1)2 ⊆ R2
(a domain on the slip plane)
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 Variational models for dislocations Ile de Re, 2011 18 /24
1. GENERALIZED PEIERLS-NABARRO (Peierls 1940, Nabarro 1947, Koslowski-Ortiz 2004)
slip plane Interfacial energy
Q = (0, 1)2 ⊆ R2
(a domain on the slip plane)
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 Variational models for dislocations Ile de Re, 2011 18 /24
1. GENERALIZED PEIERLS-NABARRO (Peierls 1940, Nabarro 1947, Koslowski-Ortiz 2004)
slip plane Interfacial energy
Bulk elastic energy
Q = (0, 1)2 ⊆ R2
(a domain on the slip plane)
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
Eε(v) =
∫Q
∫Q
(v(x)− v(y))tK(x − y)(v(x)− v(y)) dx dy +1
ε
∫QW (v) dx
K is a matrix valued singular kernel
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 18 /24
1. GENERALIZED PEIERLS-NABARRO (Peierls 1940, Nabarro 1947, Koslowski-Ortiz 2004)
slip plane Interfacial energy
Bulk elastic energy
Q = (0, 1)2 ⊆ R2
(a domain on the slip plane)
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
Eε(v) =
∫Q
∫Q
(v(x)− v(y))tK(x − y)(v(x)− v(y)) dx dy +1
ε
∫QW (v) dx
K is a matrix valued singular kernel
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 18 /24
The interfacial EnergyWe write v in a basis given by two normalized Burgers vectors v = v1b1 + v2b2
Emisfit(v) =1
ε
∫QW (v) dx
W = Peierls potential (multi-well potential with zeros on the Z2)
EXAMPLE: W (t) = dist2 (t,Z2)
Note: The Burgers vectors are determined by the crystalline structure
b1b1
b 2 b 2
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 19 /24
The Long-Range Elastic Interaction
NOTE: In order to understand the properties of the singular kernel, consider thefollowing elementary example
π minu|x3=0
=v
∫R3
+
|∇u|2dx =
∫R2
∫R2
|v(x)− v(y)|2
|x − y |3dx dy =: [v ]2
H12
Similarly the term Eelast.(v) is obtained minimizing the bulk elastic energygiven the slip on x3 = 0
Eelast.(v) =
∫Q
∫Q
(v(x)− v(y))tK(x − y)(v(x)− v(y)) dx dy
Then: K(t) ∈M2×2
I c1
|t|3 |ξ|2 ≤ ξtK(t)ξ ≤ c2
|t|3 |ξ|2 H
12 -type of kernel
I For simplicity we assume K(λt) = |λ|−3K(t)
Note: The matrix K depends on the boundary conditions and the crystalline structure
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 20 /24
The rescaled functional
Fε(v) =1
| log ε|
∫Q
∫Q
(v(x)−v(y))tK(x−y)(v(x)−v(y)) dx dy+1
ε| log ε|
∫Q
dist2 (v ,Z2) dx
This is a multi-well potential with a singular perturbation (non local,singular and anisotropic): SHARP INTERFACE LIMIT.
Ancestor: Van der Waals free energy for fluid-fluid phase transitions (1863);Cahn-Hilliard ’58
Γ-convergence results:
I (local perturbation)
Modica-Mortola ’77, Modica ’87, Fonseca-Tartar ’89, Bouchitte ’90,Owen-Sternberg ’91, Barroso-Fonseca ’94...
I (non local, regular perturbation)
Alberti-Bellettini-Cassandro-Presutti ’96, Alberti-Bellettini ’98
I (non local, singular perturbation)
Alberti-Bouchitte-Seppecher ’94-’98, Kurzke ’06, G.-Muller ’06, Cacace-G. ’07,Conti-G.-Muller ’11
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 21 /24
A Simple Nonlocal 1D case(Alberti-Bouchitte-Seppecher ’94 - Model for fluid-fluid phase transition at the boundary)
Eε(v) =
∫I
∫I
|v(x)− v(y)|2
|x − y |2dx dy +
1
ε
∫IW (v) dx
With W a double well potential (with zeros α and β).
Note: The nonlocal term is the H12 1 dimensional seminorm
α β
“Cost” of a jump in the H12 norm.
α
βε
vε
A B
Eε(vε) ∼∫Q
∫Q
|vε(x)− vε(y)|2
|x − y |2 dx dy + l.o.t.
= 2
∫A
∫B
(α− β)2
|x − y |2 dx dy + l.o.t.
= 4(α− β)2| log ε| + l.o.t.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 22 /24
A Simple Nonlocal 1D case(Alberti-Bouchitte-Seppecher ’94 - Model for fluid-fluid phase transition at the boundary)
Eε(v) =
∫I
∫I
|v(x)− v(y)|2
|x − y |2dx dy +
1
ε
∫IW (v) dx
With W a double well potential (with zeros α and β).
Note: The nonlocal term is the H12 1 dimensional seminorm
α β
“Cost” of a jump in the H12 norm.
α
βε
vε
A B
Eε(vε) ∼∫Q
∫Q
|vε(x)− vε(y)|2
|x − y |2 dx dy + l.o.t.
= 2
∫A
∫B
(α− β)2
|x − y |2 dx dy + l.o.t.
= 4(α− β)2| log ε| + l.o.t.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 22 /24
A Simple Nonlocal 1D case(Alberti-Bouchitte-Seppecher ’94 - Model for fluid-fluid phase transition at the boundary)
Eε(v) =
∫I
∫I
|v(x)− v(y)|2
|x − y |2dx dy +
1
ε
∫IW (v) dx
With W a double well potential (with zeros α and β).
Note: The nonlocal term is the H12 1 dimensional seminorm
α β
“Cost” of a jump in the H12 norm.
α
βε
vε
A B
Eε(vε) ∼∫Q
∫Q
|vε(x)− vε(y)|2
|x − y |2 dx dy + l.o.t.
= 2
∫A
∫B
(α− β)2
|x − y |2 dx dy + l.o.t.
= 4(α− β)2| log ε| + l.o.t.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 22 /24
A Simple Nonlocal 1D case(Alberti-Bouchitte-Seppecher ’94 - Model for fluid-fluid phase transition at the boundary)
Eε(v) =
∫I
∫I
|v(x)− v(y)|2
|x − y |2dx dy +
1
ε
∫IW (v) dx
With W a double well potential (with zeros α and β).
Note: The nonlocal term is the H12 1 dimensional seminorm
α β
“Cost” of a jump in the H12 norm.
α
βε
vε
A B
Eε(vε) ∼∫Q
∫Q
|vε(x)− vε(y)|2
|x − y |2 dx dy + l.o.t.
= 2
∫A
∫B
(α− β)2
|x − y |2 dx dy + l.o.t.
= 4(α− β)2| log ε| + l.o.t.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 22 /24
A Simple Nonlocal 1D case(Alberti-Bouchitte-Seppecher ’94 - Model for fluid-fluid phase transition at the boundary)
Eε(v) =
∫I
∫I
|v(x)− v(y)|2
|x − y |2dx dy +
1
ε
∫IW (v) dx
With W a double well potential (with zeros α and β).
Note: The nonlocal term is the H12 1 dimensional seminorm
α β
“Cost” of a jump in the H12 norm.
α
βε
vε
A B
Eε(vε) ∼∫Q
∫Q
|vε(x)− vε(y)|2
|x − y |2 dx dy + l.o.t.
= 2
∫A
∫B
(α− β)2
|x − y |2 dx dy + l.o.t.
= 4(α− β)2| log ε| + l.o.t.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 22 /24
The rescaled functionalWe rescale by | log ε|
FABSε (v) =
1
| log ε|
∫I
∫I
|v(x)− v(y)|2
|x − y |2dx dy +
1
| log ε|ε
∫IW (v) dx
• FABSε converges to a sharp interface limit
FABSε (v)
Γ−→ 4(α− β)2](jumps of v) v ∈ BV (I , α, β)
• No optimal profile for the transition:any transition between α and β at scale ε is optimal (by rearrangement).
• The limit energy comes only from the nonlocal part of FABSε .
The limit does not depend on the double well potential W .
• No natural scale for the problem:No rescaling for which the two terms balance and the dependence on ε disappears.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 23 /24
The rescaled functionalWe rescale by | log ε|
FABSε (v) =
1
| log ε|
∫I
∫I
|v(x)− v(y)|2
|x − y |2dx dy +
1
| log ε|ε
∫IW (v) dx
• FABSε converges to a sharp interface limit
FABSε (v)
Γ−→ 4(α− β)2](jumps of v) v ∈ BV (I , α, β)
• No optimal profile for the transition:any transition between α and β at scale ε is optimal (by rearrangement).
• The limit energy comes only from the nonlocal part of FABSε .
The limit does not depend on the double well potential W .
• No natural scale for the problem:No rescaling for which the two terms balance and the dependence on ε disappears.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 23 /24
The rescaled functionalWe rescale by | log ε|
FABSε (v) =
1
| log ε|
∫I
∫I
|v(x)− v(y)|2
|x − y |2dx dy +
1
| log ε|ε
∫IW (v) dx
• FABSε converges to a sharp interface limit
FABSε (v)
Γ−→ 4(α− β)2](jumps of v) v ∈ BV (I , α, β)
• No optimal profile for the transition:any transition between α and β at scale ε is optimal (by rearrangement).
• The limit energy comes only from the nonlocal part of FABSε .
The limit does not depend on the double well potential W .
• No natural scale for the problem:No rescaling for which the two terms balance and the dependence on ε disappears.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 23 /24
The rescaled functionalWe rescale by | log ε|
FABSε (v) =
1
| log ε|
∫I
∫I
|v(x)− v(y)|2
|x − y |2dx dy +
1
| log ε|ε
∫IW (v) dx
• FABSε converges to a sharp interface limit
FABSε (v)
Γ−→ 4(α− β)2](jumps of v) v ∈ BV (I , α, β)
• No optimal profile for the transition:any transition between α and β at scale ε is optimal (by rearrangement).
• The limit energy comes only from the nonlocal part of FABSε .
The limit does not depend on the double well potential W .
• No natural scale for the problem:No rescaling for which the two terms balance and the dependence on ε disappears.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 23 /24
The rescaled functionalWe rescale by | log ε|
FABSε (v) =
1
| log ε|
∫I
∫I
|v(x)− v(y)|2
|x − y |2dx dy +
1
| log ε|ε
∫IW (v) dx
• FABSε converges to a sharp interface limit
FABSε (v)
Γ−→ 4(α− β)2](jumps of v) v ∈ BV (I , α, β)
• No optimal profile for the transition:any transition between α and β at scale ε is optimal (by rearrangement).
• The limit energy comes only from the nonlocal part of FABSε .
The limit does not depend on the double well potential W .
• No natural scale for the problem:No rescaling for which the two terms balance and the dependence on ε disappears.
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 23 /24
COMPARISON WITH CAHN-HILLIARD MODEL
Gradient Traceregularization regularization
MAIN FEATURES ∫I|v′|2dx
∫I
∫I
|v(x)− v(y)|2
|x − y|2dx dy
Sharp Interface limit YES YES
Equipartition of energy YES NO
Intrinsic scale YES NO
Optimal profile YES NO
These differences have an effect when we consider more complicated cases as
multidimensional or vector valued problems
Adriana Garroni - Sapienza, Roma Variational models for dislocations Ile de Re, 2011 24 /24