variational models for dislocations lecture 1...

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

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

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

Screw dislocation

Edge dislocation

Dislocation loop

Continuum definition of dislocations:Lines on slip planes separating regions undergoing different slips (Volterra 1905).

MICROSCOPIC DESCRIPTION OF DISLOCATIONS

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

Dislocation coreBurgers circuit

Burgers vector

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)

= βe + βp

Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0

(D-D model)

(Peierls-Nabarro)

= βe + βp

Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0

(D-D model)

(Peierls-Nabarro)

= βe + βp

Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0

(D-D model)

(Peierls-Nabarro)

= βe + βp

Curl∇u = −(∇τ [u] ∧ n) dH2 Σ = µ 6= 0

I singularities of the Curl of the elastic strain (D-D model)

I regions where the slip is not uniform (Peierls-Nabarro)

These defects favor the slip =⇒ Plastic behaviour

(Caterpillar, Lloyd, Molina-Aldareguia 2003)

(Crease on a carpet, Cacace 2004)

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

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.

DIFFERENT SCALES

THE DISCRETE MODELFor simplicity we consider the cubic lattice.

Four-point interaction quadratic energy with interaction coefficientsBij(`− `′) with finite range.

E(u) =3∑

∑`, `′∈lattice bonds

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 ∫

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)

THE EIGEN-DEFORMATION (Ariza-Ortiz, ARMA 2005)

E(u, βp) =3∑

∑`, `′∈lattice bonds

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)

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)

|u(i)− u(j)− βp(< i , j >)|2

minβp

α = dβp

is the discrete dislocation densityGeneralization of Frenkel Kontorowa ’38 (for the analysis see e.g. Bonilla Carpio ’05, Fino-Ibrahim-Monneau ’11)

|u(i)− u(j)− βp(< i , j >)|2

minβp

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)

|u(i)− u(j)− βp(< i , j >)|2

minβp

α = dβp

is the discrete dislocation densityGeneralization of Frenkel Kontorowa ’38 (for the analysis see e.g. Bonilla Carpio ’05, Fino-Ibrahim-Monneau ’11)

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)

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.

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)

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

ε = core radius

Straight dislocation

The linear elastic energy outside the core behaves as the logarithm of the core radius

h∼ |b|2 log

Precisely

limε→0

| log ε|

∫CR (γ)\Cε(γ)

〈Cβ0, β0〉 dx = limε→0

| log ε|

∫CR (γ)\Cε(γ)

〈Cβ0, β0〉 dx =

〈CΓ0, Γ0〉 ds

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)

CONSEQUENCES

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

+ Emisfit(v)

qInterfacial energy that

penalizes slips not

compatible with the lattice

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

+ Emisfit(v)

qInterfacial energy that

penalizes slips not

slip plane Interfacial energy

Q = (0, 1)2 ⊆ R2

∼ lattice spacing

Long-range elastic

+ Emisfit(v)q

Interfacial energy that

penalizes slips not

Bulk elastic energy

Q = (0, 1)2 ⊆ R2

∼ lattice spacing

Long-range elastic

+ Emisfit(v)q

penalizes slips not

Eε(v) =

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

∫QW (v) dx

Bulk elastic energy

Q = (0, 1)2 ⊆ R2

∼ lattice spacing

Long-range elastic

+ Emisfit(v)q

penalizes slips not

Eε(v) =

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

∫QW (v) dx

The interfacial EnergyWe write v in a basis given by two normalized Burgers vectors v = v1b1 + v2b2

Emisfit(v) =1

∫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

b 2 b 2

The Long-Range Elastic Interaction

NOTE: In order to understand the properties of the singular kernel, consider thefollowing elementary example

π minu|x3=0

|∇u|2dx =

|v(x)− v(y)|2

|x − y |3dx dy =: [v ]2

Similarly the term Eelast.(v) is obtained minimizing the bulk elastic energygiven the slip on x3 = 0

Eelast.(v) =

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

Then: K(t) ∈M2×2

|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

The rescaled functional

Fε(v) =1

| log ε|

(v(x)−v(y))tK(x−y)(v(x)−v(y)) dx dy+1

ε| log ε|

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

A Simple Nonlocal 1D case(Alberti-Bouchitte-Seppecher ’94 - Model for fluid-fluid phase transition at the boundary)

Eε(v) =

|v(x)− v(y)|2

|x − y |2dx dy +

∫IW (v) dx

With W a double well potential (with zeros α and β).

Note: The nonlocal term is the H12 1 dimensional seminorm

“Cost” of a jump in the H12 norm.

Eε(vε) ∼∫Q

|vε(x)− vε(y)|2

|x − y |2 dx dy + l.o.t.

(α− β)2

|x − y |2 dx dy + l.o.t.

= 4(α− β)2| log ε| + l.o.t.

Eε(v) =

|v(x)− v(y)|2

|x − y |2dx dy +

∫IW (v) dx

Eε(vε) ∼∫Q

|vε(x)− vε(y)|2

|x − y |2 dx dy + l.o.t.

(α− β)2

|x − y |2 dx dy + l.o.t.

= 4(α− β)2| log ε| + l.o.t.

Eε(v) =

|v(x)− v(y)|2

|x − y |2dx dy +

∫IW (v) dx

Eε(vε) ∼∫Q

|vε(x)− vε(y)|2

|x − y |2 dx dy + l.o.t.

(α− β)2

|x − y |2 dx dy + l.o.t.

= 4(α− β)2| log ε| + l.o.t.

Eε(v) =

|v(x)− v(y)|2

|x − y |2dx dy +

∫IW (v) dx

Eε(vε) ∼∫Q

|vε(x)− vε(y)|2

|x − y |2 dx dy + l.o.t.

(α− β)2

|x − y |2 dx dy + l.o.t.

= 4(α− β)2| log ε| + l.o.t.

Eε(v) =

|v(x)− v(y)|2

|x − y |2dx dy +

∫IW (v) dx

Eε(vε) ∼∫Q

|vε(x)− vε(y)|2

|x − y |2 dx dy + l.o.t.

(α− β)2

|x − y |2 dx dy + l.o.t.

= 4(α− β)2| log ε| + l.o.t.

The rescaled functionalWe rescale by | log ε|

FABSε (v) =

| log ε|

|v(x)− v(y)|2

|x − y |2dx dy +

| 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.

FABSε (v) =

| log ε|

|v(x)− v(y)|2

|x − y |2dx dy +

| log ε|ε

∫IW (v) dx

FABSε (v)

FABSε (v) =

| log ε|

|v(x)− v(y)|2

|x − y |2dx dy +

| log ε|ε

∫IW (v) dx

FABSε (v)

FABSε (v) =

| log ε|

|v(x)− v(y)|2

|x − y |2dx dy +

| log ε|ε

∫IW (v) dx

FABSε (v)

FABSε (v) =

| log ε|

|v(x)− v(y)|2

|x − y |2dx dy +

| log ε|ε

∫IW (v) dx

FABSε (v)

COMPARISON WITH CAHN-HILLIARD MODEL

Gradient Traceregularization regularization

MAIN FEATURES ∫I|v′|2dx

|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

variational models for dislocations lecture 1...

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