lecture 1: introduction to dislocation dynamics · lecture 1: introduction to dislocation dynamics...

Lecture 1:Introduction to dislocation dynamics

Régis Monneau

Paris-Est University

Sapporo; July 28, 2010

R. Monneau Lecture 1

Physical motivation


Experiment 1 : compression of a cylinder


Experiment 1 : compression of a cylinder

compression of a micro-pillar of metal


Experiment 2 : traction of a sample

F−F

l


Experiment 2 : stress-strain curve

Y

F

F

l

−l

0

0l


Experiment 2 : Persistent plastic strain

Y

F

F

l

−l

0

0l


Experiment 2 : Persistent plastic strain

l0

l > l 0

initial length

after unloading


Propagation of a defect


Concept of dislocation introduced independently

by Orowan, Polanyi and Taylor in 1934.


E. Orowan M. Polanyi G.I. Taylor(1902-1989) (1891-1976) (1886-1975)


Plan of the Lecture

I Description of dislocations

I Dynamics with normal velocity

I The stress created by a dislocation

I Regularization of the singular stress on the dislocation core

I Dislocation dynamics and Non-uniqueness of distribution solutions


Description of dislocations


Observation by electronic microscopy (1956)

Denition : a dislocation is a curve of crystal defects.

Length = 10−6m, Thickness = 10−9m

Velocity ' 10−6ms−1 to 102ms−1


Dislocations


3D continuous model

elastic medium

dislocation line


2D continuous model

defect

elastic medium


2D atomic model


A dislocation in cubic Boron Nitride

High Resolution Transmission Electronic Microscopy


Perfect crystal


The Burgers vector


J.M. Burgers (1895-1981)


Consequences

closed loopsdislocations goes to the surface of the crystal


Consequences

closed loops

dislocations goes to the surface of the crystal


Consequences

closed loops

dislocations goes to the surface of the crystal

triple junctions

b

b

b

1

3

2

2 3b + b + b = 0

1

bi ∈ Z3


Frank network


Singular deformation of the crystal


Motion of a dislocation


Dynamics with normal velocity


Planar dislocations

e3

dislocation curve

slip plane


Dislocation dynamics with normal velocity c

c ntΓ

t∆

Γt

Γt+∆t

dΓtdt

= c nΓt


Velocity c

c = resolved Peach-Koehler force

c = σ : (b⊗ e3) =3∑i=1

σi3bi

with

σ ∈ R3×3

sym, stress (σij = σji)

b ∈ R3, Burgers vector

e3 ∈ R3 unit normal to the slip plane


The stress created by a dislocation


V. Volterra (1860-1940)


Volterra 1907

3e

b

Burgers vector

Ω


Volterra 1907

3e

b

Burgers vector

Ω

ρ(x′) =

1 if x′ = (x1, x2) ∈ Ω0 otherwise

u(x) = (u1, u2, u3) displacement for x = (x′, x3) ∈ R3

σ(x) = (σij)i,j=1,2,3 stress (σij = σji)


Volterra 1907

3e

b

Burgers vector

Ω

3∑i=1

∂σij∂xi

= 0 on R3\ x3 = 0 (elasticity)

u(x′, 0+)− u(x′, 0−) = b ρ(x′) for x′ ∈ R2 (jump)

σi3(x′, 0+)− σi3(x′, 0−) = 0

σij =∑

k,l=1,2,3

Λijkl ekl(u), (Λijkl = elastic coef.)

ekl(u) =12

(∂uk∂xl

+∂ul∂xk

)R. Monneau Lecture 1

Isotropic homogeneous elasticity

Λijkl = λδijδkl + µ (δikδjl + δilδjk)

Lamé coef. λ, µ such that for m > 0

∑i,j,k,l=1,2,3

Λijkl eij ekl ≥ m∑

i,j=1,2,3

(eij)2, for all eij = eji

⇐⇒ elasticity stability

⇐⇒ coercivity to apply Lax-Milgram thm

⇐⇒

µ > 0 and 3λ+ 2µ > 0

ν =λ

2(λ+ µ)∈ (−1, 1/2) (Poisson ratio)

Example : Aluminium crystal (almost isotropic) with ν ∼ 0, 33


Explicit expression of the stress

for a simplied scalar model


Simplied scalar model

u(x) ∈ Rσ = ∇u ∈ R3 on R3\ x3 = 0c = σ · e3 = σ3

div σ = 0 on R3\ x3 = 0

u(x′, 0+)− u(x′, 0−) = 2ρ(x′) for x′ ∈ R2

∂u

∂x3(x′, 0+)− ∂u

∂x3(x′, 0−) = 0


Simplied scalar model

If u(x′,−x3) = −u(x′, x3), then∆u = 0 on R3\ x3 > 0

u = ρ on x3 = 0

and

c =∂u

∂x3=: I[ρ] on x3 = 0

I is a Dirichlet to Neumann operator.


Proposition (Fourier expression of the operator I)

We have

I[ρ] = − (−∆)12 ρ

and for ξ′ = (ξ1, ξ2)I[ρ](ξ′) = −|ξ′| ρ(ξ′)


Formal proof

The function v =∂u

∂x3satises

∆v = 0 on R3\ x3 > 0

v = I[ρ] on x3 = 0

Then

I[I[ρ]] =∂v

∂x3=∂2u

∂x23

= ∆u−∆x′u = 0−∆x′ρ

and I is a square root of −∆x′ .


Sketch of the proof

Step 1 : Show thatσ = ∇u− 2ρe3δ0(x3)

solvesdiv σ = 0 on R3

Step 2 : Compute I[ρ](ξ′) = σ3(ξ′)


Theorem (Lévy-Khintchine formula for the operator I)

We have (for ρ smooth enough)

I[ρ](x′) =1

2π

∫R2

dz

|z|3ρ(x′ + z)− ρ(x′)− z · ∇ρ(x′) · 1|z|<1


Formal proof

We have for some distribution L

I[ρ] = L ? ρ and − |ξ′|ρ(ξ′) = I[ρ](ξ′) = L(ξ′)ρ(ξ′)

We simply have to compute L. Because we haveL is homogeneous of degree − 3L is invariant by rotations,

we formally get

L(ξ′) =∫

R2

dx′ L(x′) eix′·ξ′ ∼ (x′)2 · (x′)−3 ∼ (x′)−1 ∼ (ξ′) ∼ K|ξ′|

It is more dicult to see that K = −1.


Proposition (Maximum principle for the operator I)

At the maximum of ρ (smooth), we have

I[ρ] ≤ 0

ProofMaximum at x′0 =⇒ ∇ρ(x′0) = 0 and ρ(x′) ≤ ρ(x′0).Then

I[ρ](x′0) =1

2π

∫R2

dz

|z|3ρ(x′0 + z)− ρ(x′0)

≤ 0


Proposition (Singular stress on the dislocation)

For ρ = 1x1<0, we have

c = I[ρ] =1πx1

and σ =1π

eθr

with

x1 = r cos θx3 = r sin θ

x

x

1

3

x2

x

θe

θ

Sketch of the proofSet σ = (−∂3ψ, 0, ∂1ψ) with ψ(x1, x3) and compute the curl of σ.


Explicit expression of the stress

for linear elasticity

(results admitted)


Theorem (Lévy operator for linear elasticity)

For general elasticity coecients satisfying the stability condition, we have

c = L ? ρ =1

2π

∫R2

dzg(z/|z|)|z|3

ρ(x′ + z)− ρ(x′)− z · ∇ρ(x′) · 1|z|<1

with

g(−z) = g(z)

and

L(λξ′) = |λ|L(ξ′) ≤ 0, for all λ ∈ R

Remark : in physical applications, we usually have g ≥ 0, but there are alsophysical examples of unstable dislocations where g is not non-negative.


Unstable dislocations in brass (= zinc + copper)

[Head (1967)]


Proposition (Lévy operator for isotropic elasticity)

In the case of isotropic elasticity with a dislocation in the slip plane (x1, x2)and with Burgers vector b = e1, we have

L(ξ1, ξ2) = −µ2

(ξ2

2 + 11−ν ξ

21√

ξ21 + ξ2

2

)≤ 0 with ν ∈ (−1, 1/2)

and

g(z1, z2) = (2γ − 1)z21 + (2− γ)z2

2 ≥ 0 with γ =1

1− ν∈ (1/2, 2)

Even for isotropic elasticity, the Lévy operator is anisotropic (if ν 6= 0) :the dislocation loop wants to be elongated in the direction of the Burgersvector b.

b


Anisotropic evolution of a circle


Regularization of the singular stress

on the dislocation core


Regularization of the singular stress

For ρ = 1x1<0, we have

x

c

1

physical stress

core thickness = ε


Regularizations of the singular stress

Phase eld regularization

Convolution by the core function

Monotone regularization


Regularizations of the singular stress

Phase eld regularization (ex : Peierls-Nabarro model) :

Total energy = elastic energy + Emist(ρ)

with

Emist(ρ) =∫W (ρ) with W double-well potential

Convolution by the core function χ :

creg = χ ? (L ? ρ) = c0 ? ρ

andχ(ξ′) = e−ε|ξ

′| for the classical Peierls-Nabarro model


Main diculty

Physics =⇒∫

R2

c0 = 0 and c0(−x) = c0(x).

=⇒ no inclusion principle


Graph can be lost in nite time

−2 −1 0 1 2−2

0

10

0

X−AxisY

−A

xis

−2

0

10

−2 −1 0 1 2−2

0

10

0.000974359

X−Axis

Y−

Axi

s

−2

0

10

−2 −1 0 1 2−2

0

10

0.00194872

X−Axis

Y−

Axi

s

−2

0

10

−2 −1 0 1 2−2

0

10

0.00292308

X−Axis

Y−

Axi

s

−2

0

10

−2 −1 0 1 2−2

0

10

0.00389744

X−Axis

Y−

Axi

s

−2

0

10

−2 −1 0 1 2−2

0

10

0.00487179

X−Axis

Y−

Axi

s

−2

0

10



ρ1

x’0

1/2

We have

creg(x′) =∫

R2\Bε

dz1

2πg(z/|z|)|z|3

ρ(x′ + z)− ρ(x′)−z · ∇ρ(x′) · 1|z|<1

= c0 ? ρ with

c0 = J −

(∫R2

J

)δ0

J(z) =1

2πg(z/|z|)|z|3

· 1|z|≥ε

i.e.

creg(x′) = J ? ρ− 12

∫R2

J



ρ1

x’0

1/2

We set

creg(x′) =∫

R2\Bε

dz1

2πg(z/|z|)|z|3

ρ(x′ + z)− ρ(x′)−z · ∇ρ(x′) · 1|z|<1

= c0 ? ρ with

c0 = J −

(∫R2

J

)δ0

J(z) =1

2πg(z/|z|)|z|3

· 1|z|≥ε

i.e.creg(x′) = J ? ρ− 1

2

∫R2 J


Monotone regularization for ρ = 1x1<0

x1

c = stress


Dislocation dynamics


n

ρ=1

tΩ

ρ=0

For ρ = 1Ωt , the normal velocity

c = c0 ? ρ + c1 with c1 = prescribed exterior stress eld


Proposition

The function ρ solves

ρt = c|∇ρ|

Formal proofIf ρ is smooth, then

ρ(x′ + cn∆t, t+ ∆t) ' ρ(x′, t)

i.e. by Taylor explansion

∆tρt + ∆tcn · ∇ρ ' 0

i.e.

ρt + cn · ∇ρ = 0 with n = − ∇ρ|∇ρ|

which implies the result.


Dislocation dynamics

n

ρ=1

tΩ

ρ=0

For ρ = 1Ωt , we have

ρt = (c0 ? ρ+ c1) |∇ρ|


Non-uniqueness of distribution solutions


First solution

A+A− A− A+00

t=0 t=1

We setΩ1t = B2−t(A+) ∪B2−t(A−)

andρ1 = 1Ω1

t

is a distribution solution of

ρt = c|∇ρ| with c = −1


Second solution

A+A− A− A+0

t=0 t=1

0

C+

C−

C+

C−

L = (A+)(C+)(A−)(C−)

0

We set with lozenge L = A+C+A−C− :

Ω2t = Ω1

t ∪ (L\ (Bt(C+) ∪Bt(C−)))

andρ2 = 1Ω2

t

is a distribution solution of

ρt = c|∇ρ| with c = −1R. Monneau Lecture 1

Necessity of a good notion of solution :

notion of viscosity solution (included in Lecture 2)


A few references

Some books about physics of dislocations[Read (1953)],[Nabarro (1969)],[Lardner (1974)],[Hull, Bacon (1984)],[Hirth, Lothe (1992)],[Bulatov, Cai (2006)]

Article[Alvarez, Hoch, Le Bouar, M. (2006)]


A few references

Non local, non monotone dynamics

ut =(c0 ? 1u≥0 + c1

)(x, t) |Du| on RN × (0, T )

Dynamics of sets (with interior ball)[Alvarez, Cardaliaguet, M. (2005)], [Cardaliaguet, Marchi (2006)],Level sets[Barles, Ley (2006)],Notion of weak solutions[Barles, Cardaliaguet, Ley, M. (2008)],[Barles, Cardaliaguet, Ley, Monteillet (2009)],Numerical method[Carlini, Forcadel, M., (preprint HAL)]For Fitzhugh-Nagumo equations[Giga, Goto, Ishii (1992)],[Soravia, Souganidis (1996)],[Barles, Cardaliaguet, Ley, Monteillet (2009)] (Notion of interior cones)[Barles, Ley, Mitake (2010)]


lecture 1: introduction to dislocation dynamics · lecture 1: introduction to dislocation dynamics...

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