Mathematical modelingof dislocation dynamics

Régis Monneau

Thursday 13th, December 2007 CERMICS-ENPC

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Plan of the talkIntroduction to dislocationsSharp interface modelingMathematical results for the dynamicsLink with MCM

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Introduction to dislocations

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Traction of a sample

F−F

l

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Plasticity

Y

F

F

l−l0

0l

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Persistent plastic strain

Y

F

F

l−l0

0l

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Persistent plastic strain

l0

l > l 0

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Scenario 1

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Scenario 1

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Scenario 1

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Scenario 1

IMPOSSIBLE !!

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Scenario 2

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Scenario 2

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Scenario 2

Concept of dislocation (1934): Orowan; Polanyi; Taylor

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Scenario 2

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Scenario 2

POSSIBLE !!

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Description of dislocations

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Observation of dislocationsDislocations in metallic alloys Al-Mg

Definition: a dislocation is a line of crystal defects.Length = 10−6m, Thickness = 10−9m.

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Observation of dislocationsDislocations in metallic alloys Al-Mg

Definition: a dislocation is a line of crystal defects.

Length = 10−6m, Thickness = 10−9m.

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Observation of dislocationsDislocations in metallic alloys Al-Mg

Definition: a dislocation is a line of crystal defects.Length = 10−6m, Thickness = 10−9m.

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A very brief summary of the history

1934: introdution of the concept of dislocationto explain plasticity [Orowan / Polanyi / Taylor]

1956: first observation of dislocations

∼ 1970: treatises on dislocations equilibrium[Nabarro / Hirth &Lothe, ...]

Since 1990: dislocations dynamicsexplored by computer simulations

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A very brief summary of the history

1934: introdution of the concept of dislocationto explain plasticity [Orowan / Polanyi / Taylor]

1956: first observation of dislocations

∼ 1970: treatises on dislocations equilibrium[Nabarro / Hirth &Lothe, ...]

Since 1990: dislocations dynamicsexplored by computer simulations

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A very brief summary of the history

1934: introdution of the concept of dislocationto explain plasticity [Orowan / Polanyi / Taylor]

1956: first observation of dislocations

∼ 1970: treatises on dislocations equilibrium[Nabarro / Hirth &Lothe, ...]

Since 1990: dislocations dynamicsexplored by computer simulations

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A very brief summary of the history

1934: introdution of the concept of dislocationto explain plasticity [Orowan / Polanyi / Taylor]

1956: first observation of dislocations

∼ 1970: treatises on dislocations equilibrium[Nabarro / Hirth &Lothe, ...]

Since 1990: dislocations dynamicsexplored by computer simulations

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3D continuous model

elastic medium

dislocation line

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2D continuous model

defect

elastic medium

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2D atomic model

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Perfect crystal

=⇒ elasticity at large scale

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Perfect crystal

=⇒ elasticity at large scale. – p.24/62

Singular deformation of the crystal

dislocation = topological defect

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Singular deformation of the crystal

dislocation = topological defect

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How do dislocations move ?

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A sharp interface modelingof dislocation dynamics

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3D continuous model

elastic medium

dislocation line

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2D-3D coupled system

R \3

3D elastic field2D dislocation

Γ Γ

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Computation of the stress

Σ

div σ = 0 with [u] = b on Σ

b : Burgers vectorσ : stress[Volterra 1905]

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Computation of the stress

Σ

div σ = 0 with [u] = b on Σ

σ = Λ : e with e = (e(u) − δΣ · (b ⊗ e3)sym)

e(u) = (∇u)sym =1

2

(

∇u + t∇u)

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Dislocation dynamics

We define E(Γ) =∫

R3

12e : Λ : e with e = e(Γ)

c ∆t n Γt

ΓtΓt+ ∆t

dΓt

dt= c nΓt

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Dislocation dynamics

We define E(Γ) =∫

R3

12e : Λ : e with e = e(Γ)

c ∆t nΓt

ΓtΓt+ ∆t

dΓt

dt= c nΓt

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Dislocation dynamics

We define E(Γ) =∫

R3

12e : Λ : e with e = e(Γ)

c ∆t nΓt

ΓtΓt+ ∆t

dΓt

dt= c nΓt

with c = “ −∇ΓE(Γt)′′ = σ : (b ⊗ e3)

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A sharp interface modeling

Ωt

ρ(t, x1, x2) =

1 if (x1, x2) ∈ Ωt

0 otherwise

andΓt = ∂Ωt . – p.34/62

A sharp interface modeling

dΓt

dt= c nΓt

with c = c(ρ) = −(−∆)1

2ρ

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A difficulty of the classical theory

σ ∼ 1r

r

dislocation line

stress

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Regularization

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Peierls-Nabarro modelReplace the energy by

E(ρ) +

∫

R2

W (ρ)

with

W

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Notion of core function

x

xχ( )

We consider

regularized stress = χ ? (classical stress)

c = c0 ? ρ with c0 = −(−∆)1

2χ

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Notion of core function

x

xχ( )

We consider

regularized stress = χ ? (classical stress)

c = c0 ? ρ with c0 = −(−∆)1

2χ. – p.39/62

Core functionFor the Peierls-Nabarro model we have

W (ρ) =1

ζ(1 − cos(2πρ))

withχ(ξ1, ξ2) = e−ζ

√ξ2

1+ξ2

2

ζ > 0 : Peierls-Nabarro parameter

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Dynamics of a single dislocation

dΓt

dt= c nΓt

with c = c(ρ) = c0 ? ρ

where c0(x1, x2) is a fixed kernel

⇐⇒ ∂ρ

∂t= (c0 ? ρ) |∇ρ| on R

2

Under an exterior stress field c1, we get

∂ρ

∂t= (c1 + c0 ? ρ) |∇ρ| on R

2

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Dynamics of a single dislocation

dΓt

dt= c nΓt

with c = c(ρ) = c0 ? ρ

where c0(x1, x2) is a fixed kernel

⇐⇒ ∂ρ

∂t= (c0 ? ρ) |∇ρ| on R

2

Under an exterior stress field c1, we get

∂ρ

∂t= (c1 + c0 ? ρ) |∇ρ| on R

2

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Dynamics of a single dislocation

dΓt

dt= c nΓt

with c = c(ρ) = c0 ? ρ

where c0(x1, x2) is a fixed kernel

⇐⇒ ∂ρ

∂t= (c0 ? ρ) |∇ρ| on R

2

Under an exterior stress field c1, we get

∂ρ

∂t= (c1 + c0 ? ρ) |∇ρ| on R

2

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Mathematical studiesof the dynamics

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Mathematical difficulty

How to define the evolutionwith the change of topology ?

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Mathematical difficulty !

Change of topology = you can be affraid !

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Level Sets method

Front in the plan xyFront = intersection of the surface with the plan xy

Level Sets equation:∂f

∂t= c(x, t)|∇f |

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Level Sets method

Front in the plan xyFront = intersection of the surface with the plan xy

Level Sets equation:∂f

∂t= c(x, t)|∇f |

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Notion of solution

∂ρ

∂t= (c1 + c0 ? ρ) |∇ρ| on R

2

Viscosity solutions (introduced by Crandall and Lions)for Hamilton-Jacobi equations.

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Notion of solution

∂ρ

∂t= (c1 + c0 ? ρ) |∇ρ| on R

2

Viscosity solutions (introduced by Crandall and Lions)for Hamilton-Jacobi equations.

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Known results (general c0)

Short time existence, uniqueness[Alvarez, Hoch, Le Bouar, M.], [Forcadel]Convergent schemes[Alvarez, Carlini, M., Rouy], [Ghorbel, M.]Fast Marching schemes[Carlini, Cristiani, Forcadel], [Carlini, Falcone,Forcadel, M.]

Long time (c ≥ 0)[Alvarez, Cardaliaguet, M.], [Barles, Ley],[Cardaliaguet, Marchi], [Cannarsa, Cardaliaguet]Long time (general c)[Barles, Cardaliaguet, Ley, Monneau]

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Known results (general c0)

Short time existence, uniqueness[Alvarez, Hoch, Le Bouar, M.], [Forcadel]Convergent schemes[Alvarez, Carlini, M., Rouy], [Ghorbel, M.]Fast Marching schemes[Carlini, Cristiani, Forcadel], [Carlini, Falcone,Forcadel, M.]Long time (c ≥ 0)[Alvarez, Cardaliaguet, M.], [Barles, Ley],[Cardaliaguet, Marchi], [Cannarsa, Cardaliaguet]

Long time (general c)[Barles, Cardaliaguet, Ley, Monneau]

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Known results (general c0)

Short time existence, uniqueness[Alvarez, Hoch, Le Bouar, M.], [Forcadel]Convergent schemes[Alvarez, Carlini, M., Rouy], [Ghorbel, M.]Fast Marching schemes[Carlini, Cristiani, Forcadel], [Carlini, Falcone,Forcadel, M.]Long time (c ≥ 0)[Alvarez, Cardaliaguet, M.], [Barles, Ley],[Cardaliaguet, Marchi], [Cannarsa, Cardaliaguet]Long time (general c)[Barles, Cardaliaguet, Ley, Monneau]

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Main difficulty

Physics =⇒∫

R2 c0 = 0 and c0(−x) = c0(x).

=⇒ no inclusion principle

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Main difficulty

Physics =⇒∫

R2 c0 = 0 and c0(−x) = c0(x).

=⇒ no inclusion principle

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Loosing the graph in finite time

−2 −1 0 1 2−2

0

10

0

X−Axis

Y−A

xis

−2 −1 0 1 2−2

0

10

−2 −1 0 1 2−2

0

10

0.000974359

X−Axis

Y−A

xis

−2 −1 0 1 2−2

0

10

−2 −1 0 1 2−2

0

10

0.00194872

X−Axis

Y−A

xis

−2 −1 0 1 2−2

0

10

−2 −1 0 1 2−2

0

10

0.00292308

X−Axis

Y−A

xis

−2 −1 0 1 2−2

0

10

−2 −1 0 1 2−2

0

10

0.00389744

X−Axis

Y−A

xis

−2 −1 0 1 2−2

0

10

−2 −1 0 1 2−2

0

10

0.00487179

X−Axis

Y−A

xis

−2 −1 0 1 2−2

0

10

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Link with MCM

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particular kernels

Let x ∈ Rn, and

J(−x) = J(x) =1

|x|n+1g

(

x

|x|

)

1|x|≥1 ≥ 0

We define

c0 = J −(∫

Rn

J

)

δ0

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particular kernels

Let x ∈ Rn, and

J(−x) = J(x) =1

|x|n+1g

(

x

|x|

)

1|x|≥1 ≥ 0

We define

c0 = J −(∫

Rn

J

)

δ0

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Rescaling

∂ρ

∂t= (c0 ? ρ) |∇ρ| on R

n

For ε > 0, we define

ρε(x, t) = ρ

(

x

ε,

t

ε2| ln ε|

)

=⇒

∂ρε

∂t= (cε

0 ? ρε) |∇ρε| on Rn

with

cε0(x) =

1

εn+1| ln ε|c0

(x

ε

)

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Rescaling

∂ρ

∂t= (c0 ? ρ) |∇ρ| on R

n

For ε > 0, we define

ρε(x, t) = ρ

(

x

ε,

t

ε2| ln ε|

)

=⇒

∂ρε

∂t= (cε

0 ? ρε) |∇ρε| on Rn

with

cε0(x) =

1

εn+1| ln ε|c0

(x

ε

)

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At large scale

ε

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Slepcev level sets formulation

∂ρ

∂t= (c0 ? ρ) |∇ρ| on R

n with ρ ∈ 0, 1

is replaced for ρ continuous by

∂ρ

∂t=

(

c0 ? 1ρ(·,t)≥ρ(x,t))

(x)

|∇ρ| on Rn

with≥ for subsolutions> for supersolutions

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Convergence to anisotropic MCM

Theorem 1 [Da Lio, Forcadel, M.]In the Slepcev formulation, and under certain regularityassumptions, as ε goes to zero, ρε converges to ρ0

solution of∂ρ0

∂t= Fg(D

2ρ0,∇ρ0)

where

Fg(M, p) = trace(

M · Ag

(

p

|p|

))

Ag

(

p

|p|

)

=

∫

Sn−1∩p⊥

(

1

2g(θ)θ ⊗ θ

)

dθ

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Similar results[Garroni, Muller] (Gamma limit, stationnary pb)[Merriman, Bence, Osher] algorithm[Evans], [Barles, Georgelin], [Ishii], [Ishii, Pires,Souganidis], ...

[Forcadel] : error estimate for a scheme for MCM

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Similar results[Garroni, Muller] (Gamma limit, stationnary pb)[Merriman, Bence, Osher] algorithm[Evans], [Barles, Georgelin], [Ishii], [Ishii, Pires,Souganidis], ...[Forcadel] : error estimate for a scheme for MCM

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Identification of the limit MCMTheorem 2 [Da Lio, Forcadel, M.]If u is smooth, then

Ag(p) = D2G(p), Fg(D2u,∇u) = |∇u| div ((∇G) (∇u))

derives from the energy∫

G(∇u) with

G = − 1

2πLg

Lg = pv

g(

x|x|

)

|x|n+1

. – p.57/62

Identification of the limit MCMTheorem 2 [Da Lio, Forcadel, M.]If u is smooth, then

Ag(p) = D2G(p), Fg(D2u,∇u) = |∇u| div ((∇G) (∇u))

derives from the energy∫

G(∇u) with

G = − 1

2πLg

Lg = pv

g(

x|x|

)

|x|n+1

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More properties

Theorem 3 [Da Lio, Forcadel, M.]

n = 2 : g ≥ 0 ⇐⇒ G convexn ≥ 3 : ∃G convex and g 6≥ 0

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An example in 2D

G(x1, x2) = γx21 + x2

2 with

γ = 11−ν

∈(

12 , 2

)

ν = Poisson ratio

g(x1, x2) = (2γ − 1)x21 + (2 − γ)x2

2

(for x21 + x2

2 = 1).

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Anisotropic evolution of a circle

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O. Alvarez (Univ. Rouen)G. Barles (Univ. Tours)A. Briani (Univ. Pise)P. Cardaliaguet (Univ. Brest)E. Carlini (post-doc Univ. Roma))F. Da Lio (Univ. Padoue)A. El Hajj (Post-doc Univ. orleans)M. Falcone (Univ. Roma)

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A. Finel (ONERA)N. Forcadel (Post-doc INRIA)A. Ghorbel (Univ. Sfax)P. Hoch (CEA)H. Ibrahim (PhD student CERMICS)C. Imbert (Univ. Dauphine)O. Ley (Univ. Tours)Y. Le Bouar (ONERA)R. Monneau (CERMICS)E. Rouy (Centrale Lyon)

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