Multiscale analysis of dislocations Adriana Garroni Sapienza, - - PowerPoint PPT Presentation

Multiscale analysis of dislocations

Adriana Garroni

Sapienza, Universit` a di Roma

”Mathematical challenges motivated by multi-phase materials: Analytic, stochastic and discrete aspects” Anogia, Crete June 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

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:

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:

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:

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:

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 core Burgers 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 deformation gradient 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 = 0 µ is the dislocations density Then dislocations can be understood as

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 deformation gradient 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 = 0 µ is the dislocations density Then dislocations can be understood as

◮ singularities of the Curl of the elastic strain

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 deformation gradient 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 = 0 µ is the dislocations density Then dislocations can be understood as

◮ singularities of the Curl of the elastic strain ◮ regions where the slip is not uniform

Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 4 /21

Why dislocations are important

Dislocations 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

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...

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...

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. M¨

uller, 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 (mesoscopic and macroscopic) for special geometries. MESOSCOPIC

◮ Cilindrical geometry (dislocations are points)

◮ Screw dislocations - Burgers vector parallel to the dislocation line -

(Ponsiglione, ’06)

◮ Edge dislocations - Burgers vector orthogonal to the dislocation line -

(Cermelli and Leoni ’05)

◮ Only one slip plane (dislocations are lines on a given slip plane)

◮ 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-Hilliard type energy with non-local singular perturbation and infinitely many wells 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) =

3

X

i,j=1

X

l, l′∈lattice bonds

1 2Bij(l − l′)(dui(l) − βpi(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)

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 a two-point interaction discrete energy E discr(u, βp) = X

<i,j>

|u(i) − u(j) − βp(< i, j >)|2 Dislocations are introduced through the plastic strain βp : {bonds} → Z.

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 a two-point interaction discrete energy E discr(u, βp) = X

<i,j>

|u(i) − u(j) − βp(< i, j >)|2 Dislocations are introduced through the plastic strain βp : {bonds} → Z. Minimizing w.r.t. βp min

βp E discr(u, βp) = E discr(u) =

X

<i,j>

dist2(u(i) − u(j), Z) Note: βp corresponds to the projection of du on integers.

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 a two-point interaction discrete energy E discr(u, βp) = X

<i,j>

|u(i) − u(j) − βp(< i, j >)|2 Dislocations are introduced through the plastic strain βp : {bonds} → Z. Minimizing w.r.t. βp min

βp E discr(u, βp) = E discr(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 can define 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

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
• Fix a distribution of dislocations

µ = X

i

ξiδxi with ξi ∈ Z.

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 satisfying

Z

∂Bε(xi )

β · t ds = ξi and Curlβ = 0 in Ω \ ∪iBε(xi)

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 satisfying

Z

∂Bε(xi )

β · t ds = ξi and Curlβ = 0 in Ω \ ∪iBε(xi)

• Elastic Energy (Linearized)

Eε(µ, β) = Z

Ω

|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 discrete model If 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 FUNCTIONAL Both have topological singularities with logarithmic scaling. In the 2D case it can be shown that the ”Screw dislocation energy” is variationally equivalent to the Ginzburg-Landau energy for vortices

(Alicandro, Cicalese, Ponsiglione, to appear) .

Essentially

◮ They have the same Γ-limit ◮ 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 same phenomenology: this suggests that it can be formulated as a Gizburg-Landau type energy.

Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 12 /21

DISLOCATIONS VS THE GINZBURG-LANDAU FUNCTIONAL Both have topological singularities with logarithmic scaling. In the 2D case it can be shown that the ”Screw dislocation energy” is variationally equivalent to the Ginzburg-Landau energy for vortices

(Alicandro, Cicalese, Ponsiglione, to appear) .

Essentially

◮ They have the same Γ-limit ◮ 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 same phenomenology: this suggests that it can be formulated as a Gizburg-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 more

complexity

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 γ

Precisely µ ∈ MB(Ω) = {dislocation denities} such that µ = b ⊗ t H1 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

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 γ

Precisely µ ∈ MB(Ω) = {dislocation denities} such that µ = b ⊗ t H1 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)

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 γ

Precisely µ ∈ MB(Ω) = {dislocation denities} such that µ = b ⊗ t H1 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) = Z

Ω

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 the energy 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 = ∅ 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 r Γ0(θ) and satisfies Div(Cβ0) = 0 Curlβ0 = b ⊗ tH1 γ in R3

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 r Γ0(θ) and satisfies Div(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

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 r Γ0(θ) and satisfies Div(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) lim

ε→0

1 | log ε| Z

R3\Cε(γ)

Cβ0, β0 dx = lim

ε→0

1 | log ε| Z

CR (γ)\Cε(γ)

Cβ0, β0 dx

Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 15 /21

LOWER BOUND Idea: Optimal configurations will have the same decay, also for general dislocations lines. Take a sequence µε ∈ MB(Ω) and βε ∈ Adε(µε)

1 | log ε|

• Ω

Cβε, βε dx ≥ 1 | log ε|

• i
• Cηρε(γi

ε)\Cε(γi ε)

Cβε, βε dx

Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 16 /21

LOWER BOUND Idea: Optimal configurations will have the same decay, also for general dislocations lines. Take a sequence µε ∈ MB(Ω) and βε ∈ Adε(µε)

1 | log ε|

• Ω

Cβε, βε dx ≥ 1 | log ε|

• i
• Cηρε(γi

ε)\Cε(γi ε)

Cβε, βε dx ≥ 1 | log ε|

• i

min

Curlβ=µε

• Cηρε(γi

ε)\Cε(γi ε)

Cβ, β dx

Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 16 /21

LOWER BOUND Idea: Optimal configurations will have the same decay, also for general dislocations lines. Take a sequence µε ∈ MB(Ω) and βε ∈ Adε(µε)

1 | log ε|

• Ω

Cβε, βε dx ≥ 1 | log ε|

• i
• Cηρε(γi

ε)\Cε(γi ε)

Cβε, βε dx ≥ 1 | log ε|

• i

min

Curlβ=µε

• Cηρε(γi

ε)\Cε(γi ε)

Cβ, β dx =

• i

ϕε(bi

ε, γi ε) ≥ c

• i

|bi

ε|H1(γi ε)

Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 16 /21

LOWER BOUND Idea: Optimal configurations will have the same decay, also for general dislocations lines. Take a sequence µε ∈ MB(Ω) and βε ∈ Adε(µε)

1 | log ε|

• Ω

Cβε, βε dx ≥ 1 | log ε|

• i
• Cηρε(γi

ε)\Cε(γi ε)

Cβε, βε dx ≥ 1 | log ε|

• i

min

Curlβ=µε

• Cηρε(γi

ε)\Cε(γi ε)

Cβ, β dx =

• i

ϕε(bi

ε, γi ε) ≥ c

• i

|bi

ε|H1(γi 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 can show that ϕ0(b ⊗ t) := lim

ε→0

1 | log ε| min

Curlβ=µ

• Cηρε(γ)\Cε(γ)

Cβ, β dx

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 can show that ϕ0(b ⊗ t) := lim

ε→0

1 | log ε| min

Curlβ=µ

• Cηρε(γ)\Cε(γ)

Cβ, β dx = lim

ε→0

1 | log ε| min

Curlβ=µ

• C1(γ)\Cε(γ)

Cβ, β dx

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 can show that ϕ0(b ⊗ t) := lim

ε→0

1 | log ε| min

Curlβ=µ

• Cηρε(γ)\Cε(γ)

Cβ, β dx = lim

ε→0

1 | log ε| min

Curlβ=µ

• C1(γ)\Cε(γ)

Cβ, β dx = min

β= 1

r Γ(θ) Curlβ=b⊗tH1

γ

• S1CΓ, Γ ds =
• S1CΓ0, Γ0 ds

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 can show that ϕ0(b ⊗ t) := lim

ε→0

1 | log ε| min

Curlβ=µ

• Cηρε(γ)\Cε(γ)

Cβ, β dx = lim

ε→0

1 | log ε| min

Curlβ=µ

• C1(γ)\Cε(γ)

Cβ, β dx = min

β= 1

r Γ(θ) Curlβ=b⊗tH1

γ

• S1CΓ, Γ ds =
• S1CΓ0, Γ0 ds

This is a variational characterization of what is called the pre-logarithmic factor.

Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 17 /21

LACK OF LOWER SEMICONTINUITY: MICROSTRUCTURE The line tension energy

• γ

ϕ0(b(x) ⊗ t(x)) dH1(x) is not lower semi-continuous w.r.t. the weak convergence of measures (weak convergence of 1-currents).

Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 18 /21

LACK OF LOWER SEMICONTINUITY: MICROSTRUCTURE The line tension energy

• γ

ϕ0(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 is given by

• γ

¯ ϕ0(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

− →

• γ

¯ ϕ0(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 the

linear elastic case) in presence of dislocations we obtain the same line tension limit.

• 2. We would like to remove the kinematic constraints (dilute

dislocations)

• 3. There might be a Ginzburg-Landau type formulation that enforces

concentration on lines with two difficulties:

◮ The energy density is anisotropic and depends only on the symmetric

part of the strain field

◮ 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) Etot(v) = Eelast.(v) + Emisfit(v)

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) Etot(v) = Eelast.(v)

• Long-range elastic

energy induced by the slip

+ Emisfit(v)

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) Etot(v) = Eelast.(v)

• Long-range elastic

energy induced by the slip

+ Emisfit(v)

• Interfacial energy that

penalizes slips not compatible with the lattice

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)

• Long-range elastic

energy induced by the slip

• +

Emisfit(v)

• Interfacial energy that

penalizes slips not compatible with the lattice

• Eε(v) =

Z

Q

Z

Q

(v(x) − v(y))tK(x − y)(v(x) − v(y)) dx dy + 1 ε Z

Q

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)

• Long-range elastic

energy induced by the slip

• +

Emisfit(v)

• Interfacial energy that

penalizes slips not compatible with the lattice

• Eε(v) =

Z

Q

Z

Q

(v(x) − v(y))tK(x − y)(v(x) − v(y)) dx dy + 1 ε Z

Q

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.-M¨

uller, preprint) (Compactness) If Eε(vε) ≤ C| log ε|, then (up to a subsequence) ∃ aε ∈ Z2 and v ∈ 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

Sv tv

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