Multiscale analysis of dislocationsgkarali/anogia09/slides/garroni.pdf · Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21. TOPOLOGICAL SINGULARITIES

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


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


DISLOCATIONS

NOTE: The slip in general is not uniform ⇐⇒ DEFECTS (dislocations)

Dislocations are line defects in crystals (topological defects)At the microscopic level:


DISLOCATIONS



Dislocation coreBurgers circuit

Burgers vector


DISLOCATIONS




Burgers vector


DISLOCATIONS




Burgers vector


DISLOCATIONS




Burgers vector


DISLOCATIONS




Burgers vector


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




Du = ∇uL3 + ([u]⊗ n) dH2 Σ = βe + βp



Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0








Du = ∇uL3 + ([u]⊗ n) dH2 Σ = βe + βp



Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0






Why dislocations are importantDislocations in crystals favor the slip =⇒ Plastic behaviour

(Caterpillar, Lloyd, Molina-Aldareguia 2003)

(Crease on a carpet, Cacace 2004)


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.






































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.


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.


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()|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





|u(i)− u(j)− βp()|2



minβp





α = dβp






|u(i)− u(j)− βp()|2



minβp





α = dβp






|u(i)− u(j)− βp()|2



minβp





α = dβp



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




Ω

xi

• xi = cross section of a dislocation

• ε = core radius ∼ lattice spacing


µ =X

i

ξiδxi

with ξi ∈ Z.

• Consider a strain field β satisfyingZ∂Bε(xi )

β · t ds = ξi



Eε(µ, β) =

ZΩ

|β|2 dx




Ω

Bε(xi )



µ =X

i

ξiδxi


∂Bε(xi )

β · t ds = ξi



Eε(µ, β) =

ZΩ

|β|2 dx




Ω

Bε(xi )



µ =X

i

ξiδxi


∂Bε(xi )

β · t ds = ξi



Eε(µ, β) =

ZΩ

|β|2 dx


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|∑

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


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


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


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)




µ = b ⊗ tH1 γ

with


- Divµ = 0

loops

b

t

b1+ b2

b1 b2



The Elastic Energy

Eε(β, µ) =

ZΩ






µ = b ⊗ tH1 γ

with


- Divµ = 0

loops

b

t

b1+ b2

b1 b2



The Elastic Energy

Eε(β, µ) =

ZΩ




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.


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




β0 =1

rΓ0(θ)



such that


Σ

ε

R

CR

Cε

b

γ

2)

limε→0

1

| log ε|

ZR3\Cε(γ)


1

| log ε|

ZCR (γ)\Cε(γ)

〈Cβ0, β0〉 dx




β0 =1

rΓ0(θ)



such that


Σ

ε

R

CR

Cε

b

γ

2)

limε→0

1

| log ε|

ZR3\Cε(γ)


1

| log ε|

ZCR (γ)\Cε(γ)

〈Cβ0, β0〉 dx


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)




1

| log ε|

∫Ω


| log ε|∑

i

∫Cηρε (γ i

ε)\Cε(γ iε)


≥ 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)


i bi ⊗ tiH1 γi





1

| log ε|

∫Ω


| log ε|∑

i

∫Cηρε (γ i

ε)\Cε(γ iε)


≥ 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)


i bi ⊗ tiH1 γi





1

| log ε|

∫Ω


| log ε|∑

i

∫Cηρε (γ i

ε)\Cε(γ iε)


≥ 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)


i bi ⊗ tiH1 γi



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.




ϕ0(b ⊗ t) := limε→0

1

| log ε|min

Curlβ=µ


〈Cβ, β〉 dx

= limε→0

1

| log ε|min

Curlβ=µ

∫C1(γ)\Cε(γ)

〈Cβ, β〉 dx

= minβ= 1


∫S1

〈CΓ, Γ〉 ds =

∫S1

〈CΓ0, Γ0〉 ds





ϕ0(b ⊗ t) := limε→0

1

| log ε|min

Curlβ=µ


〈Cβ, β〉 dx

= limε→0

1

| log ε|min

Curlβ=µ

∫C1(γ)\Cε(γ)

〈Cβ, β〉 dx

= minβ= 1


∫S1

〈CΓ, Γ〉 ds =

∫S1

〈CΓ0, Γ0〉 ds





ϕ0(b ⊗ t) := limε→0

1

| log ε|min

Curlβ=µ


〈Cβ, β〉 dx

= limε→0

1

| log ε|min

Curlβ=µ

∫C1(γ)\Cε(γ)

〈Cβ, β〉 dx

= minβ= 1


∫S1

〈CΓ, Γ〉 ds =

∫S1

〈CΓ0, Γ0〉 ds



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


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


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.


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)


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


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



slip plane

Bulk elastic energy Q = (0, 1)2 ⊆ R2




∼ lattice spacing

Etot(v) = Eelast.(v)q

Long-range elastic


q

+ Emisfit(v)

qInterfacial energy that

penalizes slips not


q




slip plane Interfacial energy

Q = (0, 1)2 ⊆ R2




∼ lattice spacing


Long-range elastic


q

+ Emisfit(v)q

Interfacial energy that

penalizes slips not


q





Bulk elastic energy

Q = (0, 1)2 ⊆ R2




∼ lattice spacing


Long-range elastic


q

+ Emisfit(v)q


penalizes slips not


qEε(v) =

ZQ

ZQ

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

ε

ZQ

W (v) dx





Bulk elastic energy

Q = (0, 1)2 ⊆ R2




∼ lattice spacing


Long-range elastic


q

+ Emisfit(v)q


penalizes slips not


qEε(v) =

ZQ

ZQ

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

ε

ZQ

W (v) dx



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

Documents

Multiscale analysis of dislocationsgkarali/anogia09/slides/garroni.pdf · Adriana Garroni - Sapienza, Roma Multiscale analysis of dislocations Anogia 2009 3 /21. TOPOLOGICAL SINGULARITIES