Lecture 3: Dislocation dynamics: from microscopic models ...€¦ · Equilibrium of the blue points...

Lecture 3:Dislocation dynamics:

from microscopic models to macroscopic crystal plasticity

Régis Monneau

Paris-Est University

Sapporo; July 30, 2010

R. Monneau Lecture 3

Plan of the talk : Hierarchy of scales

(1) Frenkel-Kontorova (atomic)

⇓ (ε1 → 0)

(2) Peierls-Nabarro (micro)

⇓ (ε2 → 0)

(3) Dynamics of dislocations (meso)

⇓ (ε3 → 0)

(4) Crystal plasticity (macro)


2D Frenkel-Kontorova model

(ε = ε1)


Motion of a dislocation


2D Frenkel-Kontorova

Points only move on the horizontal axis with I = (I1, I2) ∈ Z2

e1

e2

I =−1

2

2

I = 0



strong springs / weak springs, stress σ

I =−1

2

2

I = 0

e1

e2

−σ

σ



Assumption : antisymmetric horizontal displacement

U(I1,I2) = −U(I1,−I2−1)

I =−1

2

2

I = 0

e1

e2



I =−1

2

2

I = 0

e1

e2


Half plane

I =−1

2

2

I = 0

e1

e2


Equilibrium of the blue points

e2

e1

I = 02

σ

−σ

I

For I = (I1, I2) with I2 ≥ 1∑J∈Z2, |J−I|=1

(UJ − UI) = 0

(harmonic blue springs)


Dynamics of the red points

e2

e1

I = 02

σ

−σI

For I = (I1, I2) with I2 = 0

d

dtUI = εε2σ − εW ′(UI) +

∑J ∈ Z2, J2 ≥ 0,|J − I| = 1

(UJ − UI)


The full 2D Frenkel-Kontorova

∑J∈Z2, |J−I|=1

(UJ − UI) = 0 for I2 ≥ 1

d

dtUI = εε2σ − εW ′(UI) +

∑J ∈ Z2, J2 ≥ 0,|J − I| = 1

(UJ − UI) for I2 = 0

(1)


Potential W

smooth periodic potential W (a+ 1) = W (a)

W(a)

a0 1 2 3−1

W describes the mist between the two half planes.


Homogenization of 1D FK models

Fully overdamped FK[Forcadel, Imbert, M., (2009)]

FK with acceleration[Forcadel, Imbert, M., (preprint 2010)]

FK with parabolic rescaling[Alibaud, Briani, M. (2010)]


Convergence to the Peierls-Nabarro model

(ε = ε1 → 0)


Hyperbolic rescaling

uε(X, t) = UXε

(t

ε

)for X = (X1, X2) ∈ (εZ)× (εN)


Convergence ε→ 0

Baby Thm 1 (Formal convergence FK → PN) [Fino, Ibrahim, M.]

As ε→ 0, we have formallyuε → u0

with u0 solution of the PN model.


Peierls-Nabarro model

X

X

1

2

X =02

Ω

For u0(X1, X2, t) :0 = ∆u0 on Ω = X2 > 0

u0t = ε2σ −W ′(u0) +

∂u0

∂X2for ∂Ω = X2 = 0


Reformulation of the PN model

0 = ∆u0 on Ω

u0t = ε2σ −W ′(u0) +

∂u0

∂X2on ∂Ω

=⇒ V (x, t) = u0(x, 0, t) satises with ε = ε2

Vt = εσ −W ′(V ) + I[V (·, t)] for x ∈ R (2)

with the Lévy-Khintchine formula

I[w](x) =1π

∫R

dz

z2

(w(x+ z)− w(x)− zw′(x)1|z|≤1

)


Reformulation of the PN model

0 = ∆u0 on Ω

u0t = ε2σ −W ′(u0) +

∂u0

∂X2on ∂Ω

=⇒ V (x, t) = u0(x, 0, t) satises with ε = ε2

Vt = εσ −W ′(V ) + I[V (·, t)] for x ∈ R (2)

with the Lévy-Khintchine formula

I[w](x) =1π

∫R

dz

z2

(w(x+ z)− w(x)− zw′(x)1|z|≤1

)R. Monneau Lecture 3

Vt = εσ −W ′(V ) + I[V (·, t)] for x ∈ R

Rescaled to (anisotropic) mean curvature motion on RN

[Imbert, Souganidis (2009)]

Rescaled to a line tension model on RN (variational approach)[Garroni, Müller (2006)]

Homogenization of PN model on RN

[M., Patrizi (preprint 2010)]


The layer solution

[Cabré, Solà-Morales, (2005)]

x

1

0

φ

0 = −W ′(φ) + I[φ] on R

φ′ > 0, φ(−∞) = 0, φ(+∞) = 1, φ(0) =12

assuming

W ′′(0) > 0


The layer solution

[Cabré, Solà-Morales, (2005)]

x

1

0

φ

0 = −W ′(φ) + I[φ] on R

φ′ > 0, φ(−∞) = 0, φ(+∞) = 1, φ(0) =12

assuming

W ′′(0) = 1


Convergence to Dislocation points dynamics

(ε = ε2 → 0)


Parabolic rescaling

vε(x, t) = V

(x

ε,t

ε2

)for x ∈ R

1

2

3

xx xx1 2 3

vε

Well-prepared initial datax0

1 < x02 < ... < x0

N

vε0(x) = εσ +N∑i=1

φ

(x− x0

i

ε

)


Convergence ε→ 0

Thm 2 (Convergence PN → DP) [Gonzalez, M. (preprint 2010)]

As ε→ 0, we have

vε → v0 in L1loc(R× [0,+∞))

with

v0(x, t) =N∑i=1

H(x− xi(t)), (H = Heaviside function)

1

2

3

xx xx1 2 3

vε


Convergence ε→ 0

Thm 2 (Convergence PN → DP) [Gonzalez, M. (preprint 2010)]

As ε→ 0, we have

vε → v0 in L1loc(R× [0,+∞))

with

v0(x, t) =N∑i=1

H(x− xi(t)), (H = Heaviside function)

1

2

3

xx xx1 2 3

v0


Dislocation points dynamics

dxidt

= −γ

σ +∑j 6=i

V ′(xi − xj)

xi(0) = x0

i

∣∣∣∣∣∣∣∣∣∣for i = 1, ..., N

with γ =

(∫R

(φ′)2)−1

V (x) = − 1π

ln |x|

φ(x)−H(x) ∼ − 1πx

as |x| → +∞


Ansatz for the proof

Ansatz

vε(x, t) = εσ +N∑i=1

φ

(x− xi(t)

ε

)−εxi(t)ψ

(x− xi(t)

ε

)

with the corrector ψ solving

Lψ −W ′′(φ)ψ = φ′ + γ−1(W ′′(φ)−W ′′(0))


Ansatz for the proof

Ansatz

vε(x, t) = εσ +N∑i=1

φ

(x− xi(t)

ε

)− εxi(t)ψ

(x− xi(t)

ε

)

with the corrector ψ solving

I[ψ]−W ′′(φ)ψ = φ′ + γ−1(W ′′(φ)−W ′′(0))


Straight dislocations lines

parallel dislocation lines


Points

Dislocations


Dislocation points dynamicswith periodic obstacles

(ε = ε3)


Precipitates = obstacles


Stress σ

smooth periodic stress σ(a+ 1) = σ(a)

a0 1 2−1

σ (a)

σ describes the obstacles to the motion of dislocations


Dynamics with two-body interactions

Velocity

Dislocations


Nε dislocations x01 < ... < x0

Nε

dxidt

= −γ

σ(xi) +∑j 6=i

V ′(xi − xj)

xi(0) = x0

i

(3)

v0(x, t) =Nε∑i=1

H(x− xi(t))

1

2

3

xx xx1 2 3

v0


Convergence to crystal plasticity

(ε = ε3 → 0)


Hyperbolic rescaling

wε(x, t) = εv0

(x

ε,t

ε

)


Convergence ε→ 0

Thm 3 (Convergence DP → CP) [Forcadel, Imbert, M. (2009)]

As ε→ 0, we have

wε → w0 in L∞loc(R× [0,+∞))

where w0 is a solution to the crystal plasticity model.

ε(x)

x

w


Convergence ε→ 0

Thm 3 (Convergence DP → CP) [Forcadel, Imbert, M. (2009)]

As ε→ 0, we have

wε → w0 in L∞loc(R× [0,+∞))

where w0 is a solution to the crystal plasticity model.

(x)

x

w0


Dynamics for densities of dislocations

Crystal (elasto-visco-) plasticity

w0t = H(w0

x, I[w0]) (4)

Mechanical interpretation

w0 = plastic strain

w0t = plastic strain velocity

w0x = density of dislocations

I[w0] = self-stress created by the density of dislocations

w0t = H(w0

x, I[w0]) is the visco-plastic law




w0t = H(w0

x, I[w0])

ρ

σ

σH( , )




w0t = H(w0

x, I[w0])

with the Orowan's law

H(ρ, σ) = γρσ if σ = 0 and ρ ≥ 0

Study of self-similar solutions of wt = (I[w])|wx|[Biler, Karch, M. (2010)]


Mechanical interpretation

I[w0] = σ stress

w0x = ρ dislocation density

1. Orowan's law (plastic strain velocity)

ep = σ|ρ|

2. Norton law with threshold (elasto-visco-plasticity)

ep = Csign(σ)((|σ| − σc)+)m

3. By homogenization, we nd

ep = H(ρ, σ)



w0t = H(w0

x, I[w0])

conservation law for ρ = w0x and σ = I[w0]

ρt = (H(ρ, σ))x


How to compute H ?

In the case I[w0] = 0, dislocation density ρ = w0x

Look for particular solutions

xj(t+T ) = xj+1(t)

=⇒

xj(t) = h

(vt+

j

ρ

)with

h = hull functionh(a+ 1) = 1 + h(a)

=⇒ uniqueness of the velocity v

SetH(ρ, 0) = −vρ


Eective Hamiltonian H(ρ, σ)

H=0

H<0

H >0

σ

ρ


Pinning eect if∫

[0,1) σ = 0

H=0

H<0

H >0σ

ρ

pinning effect


Barriers for smooth enough interactions

H=0

H<0

H >0σ

ρ

barriers as 1/ ρ

pinning effect


obstacleσ

dislocationsρ

σ =σobst

ρ

[Hirth, Lothe (1992), p. 766]


Eective Hamiltonian H(ρ, σ) (Numerics)

[Cacace, Chambolle, M. (preprint 2010)]

DISLOCATION DENSITY p

EX

TE

RN

AL

ST

RE

SS

L


Eective Hamiltonian σ 7→ H(ρ, σ)

0 0.5 1 1.5 2 2.5 3 3.5 4−1

0

1

2

3

4

5

6

7

8

External stress L

Effe

ctiv

e H

amilt

onia

n

p=0.1

p=0.5

p=1.0

p=1.1

p=2.0

p=2.1


Homogenization of dislocation curves


motion

Periodic obstacles

Limit for density of dislocations ?


Homogenization for curves

motion

A single plane and a single Burgers vector


Plastic displacement

0

4

3

2

1


Odd integer part

E(w) = l +12

for l ≤ w < l + 1, l ∈ Z

x

(x)E

−1−2−30.5

−0.5

−1.5

1.5

2.5

1 2 3 4


We consider solutions w(x, t) ofwt = |∇w| J ? E w(·, t)− w(x, t) (x)+c(x) on RN × (0,+∞)

w(·, 0) = 1εw0(εx) at t = 0

Here c(x) represents the stress eld created by the obstacles (precipitates,other xed dsilocations, ...) that are assumed to be periodic :

c(x+ k) = c(x) for all k ∈ ZN

and

0 ≤ J(z) =g(z/|z|)|z|N+1

· 1|z|≥δ with δ xed

withg(−z) = g(z) ≥ 0 and g ∈ C(SN−1; R)


Let

wε(x, t) = εw

(x

ε,t

ε

)

Thm 3' (Convergence DD → CP) [Forcadel, Imbert, M. (2009)]

Assume w0 ∈W 2,∞(RN ) and the technical condition :

g > 0 if N ≥ 2

Then we havewε → w0 in L∞loc((0,+∞)× RN )

where w0 is the solution of the limit equation for some suitable function H


The limit equation

w0t = H(∇w0, I[w0(·, t)]) on RN × (0,+∞)

w0(·, 0) = w0 at t = 0

with

I[w](x) =∫

RN

dzg(z/|z|)|z|N+1

w(x+ z)− w(x)− z · ∇w(x) · 1|z|<1


Formal proposition (the cell problem)

If wε → w0 and

wε(t, x) ' w0(t, x) + εv(xε

)locally

then v(y) is called the corrector and is assymptotically a solution of the

following cell problem :λ = |p+∇v(y)| · c(y) + σ +Mp[v](y) on RN

v(y + k) = v(y) for all k ∈ ZN

with

p = ∇w0(x, t)σ = I[w0(·, t)](x) (Long range)

λ = w0t (x, t) = H(p, σ)

Mp[v](y) =∫

RN

dz J(z) E v(y + z)− v(y) + p · z − p · zR. Monneau Lecture 3

Main idea = splitting of interactions

Interactions = Short range + Long range


Conclusion


Hierarchy of scales

(1) Frenkel-Kontorova (atomic)

⇓ (ε1 → 0)

(2) Peierls-Nabarro (micro)

⇓ (ε2 → 0)

(3) Dynamics of dislocations (meso)

⇓ (ε3 → 0)

(4) Crystal plasticity (macro)


Lecture 3: Dislocation dynamics: from microscopic models ...€¦ · Equilibrium of the blue points...

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