dislocations dynamics: from microscopic models to ... seminars/stefaniapatrizi... · dislocations...

Dislocations dynamics: from microscopicmodels to macroscopic crystal plasticity

Stefania Patrizi

Post-Doctoral fellow, WIAS Berlin

K.A.U.S.TMarch 25th 2015

Stefania Patrizi

S. P. and E. Valdinoci, Relaxation times for atomdislocations in crystals, preprintS. P. and E. Valdinoci, Crystal dislocations with differentorientations and collisions, to appear in Arch. RationalMech. Anal.S. P. and E. Valdinoci, Homogenization and Orowan’slaw for anisotropic fractional operators of any order,Nonlinear Analysis: Theory, Methods and Applications,Available online 27 August 2014

Stefania Patrizi

Dislocations

Dislocations are defect lines in crystalline solids whose motionis directly responsible for the plastic deformation of thesematerials. Their typical length is of order of 10−6m withthickness of order of 10−9m.

Geometry of an edge dislocation

Stefania Patrizi

Motion of a dislocation line

Stefania Patrizi

Dislocations can be described at several scales by differentmodels:

1 atomic scale (Frenkel-Kontorova model)

2 microscopic scale (Peierls-Nabarro model)

3 mesoscopic scale (Discrete dislocation dynamics)

4 macroscopic scale (elasto-visco-plasticity with density ofdislocations)

Stefania Patrizi

From PN to DDD

Peierls-Nabarro modelwwww�(ε→ 0)

Discrete dislocation dynamics

Stefania Patrizi

The Peierls-Nabarro modelWe consider a straight dislocation line parallel to e3.

which are certain “line defects” in the crystal. To simplify the presentation, we will assume that the materialis invariant by integer translations in the direction e3. Because of this assumption, we can simply consider thecross section of the crystal in the plane (e1, e2) where each atom is now assumed to have a position I ∈ Z2 inthe perfect crystal. We also assume that each atom I can have a displacement UI ∈ R in the direction e1, suchthat the effective position of the atom I is I + UIe1.

On Figure 1 below is represented a view of the perfect crystal. On Figure 2 we can see a schematic viewof a edge dislocation in the crystal. On this picture, the upper part {I2 ≥ 0} of the crystal has been expandedto the right of a vector 1

2e1, while the lower part {I2 ≤ −1} of the crystal has been contracted to the left ofa vector − 1

2e1. The net difference between these two vectors is e1 and is called the Burgers vector of thisdislocation.

e1

e2

I =−122

I = 0

Figure 1: Perfect crystal

I =−122

I = 0

e1

e2

Figure 2: Schematic view of a edge dislocation inthe crystal

In order to describe a edge dislocation in our formalism, let us make a few assumptions. We will assume thatthe dislocation defects are essentially described by the mismatch between the two planes I2 = 0 and I2 = −1,like on Figure 2. For this reason, and also in order to simplify the analysis, we assume that the displacementof the crystal satisfies the following antisymmetry property

U(I1,−I2) = −U(I1,I2−1) for all I = (I1, I2) ∈ Z2. (2.1)

Let us also define the discrete gradient

(∇dU)I =

(UI+e1 − UI

UI+e2 − UI

).

Remark that defects in the crystal can be seen as regions where the discrete gradient is not small.

Formalism for a edge dislocation with Burgers vector e1

In our formalism, a edge dislocation like the one of Figure 2, can be represented by a displacement UI satisfying⎧⎪⎨⎪⎩

U(I1,0) = −U(I1,−1) → 0 as I1 → −∞

U(I1,0) = −U(I1,−1) → 1

2as I1 → +∞.

Because we assume that the dislocation core lies in the two planes I2 = 0 and I2 = −1, it is reasonable toassume that all the components of the discrete gradient are small, except components UI+e2 −UI for I = (I1, I2)with I2 = −1. More precisely, we assume that there exists a small δ > 0 such that

{|UI+e1 − UI | ≤ δ for all I = (I1, I2) ∈ Z2

|UI+e2 − UI | ≤ δ for all I = (I1, I2) ∈ Z2 with I2 = −1.(2.2)

2

Assumptionsthe dislocation defects are described by the mismatch between the twoplanes I2 = 0 and I2 = −1the displacement of the crystal is antysimmetric wrt the plane e1e3

any atoms move only in the direction e1

the displacement is independent of e3

Stefania Patrizi

The Peierls-Nabarro model

The P-N model is a continuous model where a dislocation isdescribed by means of a scalar phase field defined over the slipplane.

The medium will be R2, endowed with coordinates (x , y).

The disregistry of the upper half crystal {y > 0} relative to thelower half {y < 0} is given by v(x), which is a transitionbetween 0 and 1:

{v(−∞) = 0, v(+∞) = 1v ′ > 0.

Stefania Patrizi


The total energy is given by

E =12

∫

(R2)+|∇V (x , y)|2dxdy

︸︷︷︸elastic energy

+

∫

RW (V (x ,0))dx

︸︷︷︸misfit energy

where V : (R2)+ → R represents (twice) the (scalar)displacement and it is such that

V (x ,0) = v(x).

The potential W satisfiesW (u + 1) = W (u) ∀u ∈ R (periodicity)W (Z) = 0 < W (u) ∀u ∈ R \ Z (minimum property)

Stefania Patrizi


A critical point of the energy satisfies{

∆V (x , y) = 0 (x , y) ∈ (R2)+

∂yV (x ,0) = W ′(V (x ,0)) x ∈ R

The system can be rewritten for

v(x) = V (x ,0)

as follows−(−∆)

12 v = W ′(v) in R

where(−∆)

12 v = F−1(|ξ|v) for any v ∈ S(Rn)

and F is the Fourier transform.

Stefania Patrizi


The phase transition v therefore satisfies

−(−∆)12 v = W ′(v) in R

v ′ > 0v(−∞) = 0, v(+∞) = 1, v(0) = 1

2

In the original PN model:

W (v) =1

4π2 (1− cos(2πv))

andv(x) =

12

+1π

arctan(2x)

Stefania Patrizi


More in general, we consider the layer function v solution of, fors ∈ (0,1)

−(−∆)sv = W ′(v) in Rv ′ > 0v(−∞) = 0, v(+∞) = 1, v(0) = 1

2

(1)

where for ϕ ∈ C2(Rn) ∩ L∞(Rn)

Is[ϕ](x) := −(−∆)sϕ(x) = c(n, s) PV∫

Rn

ϕ(x + y)− ϕ(x)

|y |n+2s dy .

Under additional assumptions on the potential W , the existenceof a unique solution of (1) was proven by Cabré-Sire andPalatucci-Savin-Valdinoci

Asymptotic estimates have been proved by Dipierro, Figalli,Monneau, Palatucci, Savin, Valdinoci in different papers.

Stefania Patrizi

From PN to DDD

We consider the evolutive version of the PN model:

∂tv = Isv −W ′(v) + σε(t , x) in R× R+

whereσε(t , x) = ε2sσ(ε2s+1t , εx)

is an exterior stress acting on the material and ε is a smallpositive parameter. Then we consider the following rescaling:

vε(t , x) := v(

tε1+2s ,

xε

)

and we look at the equation satisfied by the rescaled function vε

Stefania Patrizi

From PN to DDD

(PN)ε

(vε)t =1ε

(Isvε −

1ε2s W ′(vε) + σ(t , x)

)in (0,+∞)× R

vε(0, ·) = v0ε on R,

On W and σ we assume:

W ∈ C3,α(R) for some 0 < α < 1W (v + 1) = W (v) for any v ∈ RW = 0 on Z, W > 0 on R \ ZW ′′(0) > 0.

σ ∈ BUC([0,+∞)× R)

‖σx‖L∞([0,+∞)×R) + ‖σt‖L∞([0,+∞)×R) ≤ M|σx (t , x + h)− σx (t , x)| ≤ M|h|α, x ,h ∈ R, t > 0α ∈ (s,1).

Stefania Patrizi

From PN to DDD

Let v be the layer solution.

DefinitionGiven x0 ∈ R, we say that the function

v(

x − x0

ε

)

is a transition layer centered at x0 and positively oriented and

v(

x0 − xε

)− 1

is a transition layer centered at x0 and negatively oriented.

Stefania Patrizi

From PN to DDD

We consider as initial condition the state obtained bysuperposing N copies of the transition layer, centered atx0

1 , ..., x0N , N − K of them positively oriented and the remaining

K negatively oriented, that is

v0ε (x) =

ε2s

βσ(0, x) +

N∑

i=1

v

(ζi

x − x0i

ε

)− K ,

where ζ1, ..., ζN ∈ {−1,1},N∑

i=1

(ζi)− = K , 0 ≤ K ≤ N and

β := W ′′(0) > 0.

Stefania Patrizi

From PN to DDD

Let us introduce the solution (xi(t))i=1,...,N to the system

(DDD)

xi = γ

∑

j 6=i

ζiζjxi − xj

2s|xi − xj |1+2s − ζiσ(t , xi)

in (0,Tc)

xi(0) = x0i ,

where γ := 1/∫R(v ′(x))2dx ,

0 < Tc ≤ +∞ is the first time when a collision between twoparticles occurs:

xi(t) < xi+i(t), for any t ∈ [0,Tc) and any i

∃ i0 such that xi0(Tc) = xi0+1(Tc)

Stefania Patrizi

From PN to DDD

Theorem (S. P., E. Valdinoci)Let

v0(t , x) =N∑

i=1

H(ζi(x − xi(t)))− K ,

where H is the Heaviside function and (xi(t))i=1,...,N is thesolution to (DDD).Then, for every ε > 0 there exists a unique solution vε to (PN)ε.Furthermore, as ε→ 0+

lim sup(t′,x′)→(t,x)ε→0+

vε(t ′, x ′) ≤ (v0)∗(t , x)

lim inf(t′,x′)→(t,x)ε→0+

vε(t ′, x ′) ≥ (v0)∗(t , x),

for any (t , x) ∈ [0,Tc)× R.

Stefania Patrizi

From PN to DDD

The case K = 0, i.e., the case in which the dislocations have allthe same orientation and therefore

Tc = +∞,

was already studied by:

Gonzales, Monneau: s = 12

Dipierro, Palatucci, Valdinoci: s ∈(1

2 ,1)

Dipierro, Figalli, Valdinoci: s ∈(0, 1

2

)

Stefania Patrizi

From PN to DDD

We have (optimal) estimates for Tc in the following situations:

2 particles (+-):

Tc < +∞ if either σ ≤ 0 or θ0 <

(1

2s‖σ‖∞

) 12s

Tc ≤sθ1+2s

0

(2s + 1)γ(1− 2sθ2s0 ‖σ‖∞)

if θ0 ≥(

12s‖σ‖∞

) 12s

it may happen that Tc = +∞.

3 particles (+-+)N alternates particlesN particles, two of which sufficiently close each other

Stefania Patrizi

From PN to DDD

What happens for t ≥ Tc?

Suppose that there are two particles with different orientationand that a collision occurs at a time 0 < Tc < +∞, and

x1(Tc) = x2(Tc) = xc

Theorem

Assume N = 2 and K = 1. Let vε be the solution to (PN)ε, then

limt→T−c

limε→0+

vε(t , x) = 0 for any x 6= xc

lim supt→T−cε→0+

vε(t , xc) ≥ 1.

Stefania Patrizi

From PN to DDD

Theorem (S. P., E. Valdinoci)

Assume N = 2 and K = 1 and let vε the solution of (PN)ε.Then there exists ε0 > 0 such that for any ε < ε0 there existTε, %ε > 0 such that

Tε = Tc + o(1), %ε = o(1) as ε→ 0,

vε(Tε, x) ≤ %ε for any x ∈ R.

Theorem (S. P., E. Valdinoci)Assume in addition σ ≡ 0. Then there exist ε0 > 0 and c > 0such that for any ε < ε0 we have

|vε(t , x)| ≤ %εec Tε−tε2s+1 , for any x ∈ R and t ≥ Tε.

Stefania Patrizi

From PN to DDD

t=0

x1

x2

0 0

εt=T

ρε

εt>T

Figure 1: Evolution of the dislocation function in case of two particles.

Stefania Patrizi

From PN to DDD

x x x1 2 3

000

t=Tε

ρε

t=0

t>T2ε

zxεε

2

Figure 2: Evolution of the dislocation function in case of three particles.

Stefania Patrizi

From PN to DD

Peierls-Nabarro modelwwww�(ε→ 0)

Dislocation density model

Stefania Patrizi

From PN to DDD

Let us go back to the evolutive PN model in dimension n:

∂tu = I1[u(t , ·)]−W ′ (u) + σ (t , x) in R+ × Rn

where for ϕ ∈ C2(Rn) ∩ L∞(Rn)

I1[ϕ](x) := PV∫

RN(ϕ(x + z)− ϕ(x))

1|z|n+1 g

(z|z|

)dz.

We assume:

g ∈ C(Sn−1), g > 0, g even;

W ∈ C1,1(R) and W (v + 1) = W (v) for any v ∈ R;

σ ∈ C0,1(R+ × Rn) and σ(t + 1, x) = σ(t , x), σ(t , x + k) = σ(t , x)for any k ∈ Zn and (t , x) ∈ R+ × Rn.

Stefania Patrizi

From PN to DD

∂tu = I1[u(t , ·)]−W ′ (u) + σ (t , x) in R+ × Rn

We consider the following initial condition: for ε > 0

u(0, x) =1ε

u0(εx) on Rn,

where u0 ∈W 2,∞(Rn).

The parameter ε takes into account the fact that the number ofdislocations is increasing of order 1/ε.

We want to identify at large scale an evolution model for thedynamics of a density of dislocations (DD)

Stefania Patrizi

From PN to DD

{∂tu = I1[u(t , ·)]−W ′ (u) + σ (t , x) in R+ × Rn

u(0, x) = 1εu0(εx) on Rn

ww�

uε(t , x) := εu(

tε,xε

)

ww�

Homogenization problem:{∂tuε = I1[uε(t , ·)]−W ′ (uε

ε

)+ σ

( tε ,

xε

)in R+ × Rn

uε(0, x) = u0(x) on Rn

Stefania Patrizi

From PN to DD

Theorem (R. Monneau, S. P.)

The solution uε converges towards the solution u0 of thefollowing homogenized problem:

{∂tu = H(∇u, I1[u(t , ·)]) in R+ × Rn

u(0, x) = u0(x) on Rn,

locally uniformly in (t , x), for some effective Hamiltonian H suchthat

for any (p,L) ∈ Rn × R, H(p,L) is defined through the socalled cell-problem;H : Rn × R→ R is continuous;H(p, ·) is non-decreasing for any p ∈ Rn.

Stefania Patrizi

From PN to DD

Mechanical interpretation of the homogenization result:the homogenized equation

{∂tu = H(∇u, I1[u(t , ·)]) in R+ × RN

u(0, x) = u0(x) on RN ,

can be interpreted as the plastic flow rule in a model formacroscopic crystal plasticity. In the equation:

u0 is the plastic strain;∂tu0 is the plastic strain velocity;∇u0 is the dislocation density;I1[u0] is the internal stress created by the density ofdislocations contained in a slip plane.

Stefania Patrizi

From PN to DD

Assume:n = 1σ ≡ 0I1 = −(−∆)

12

W ∈ C4,β(R), 0 < β < 1; W (v + 1) = W (v), ∀v ∈ R;W > 0 on R \ Z; W ′′(0) > 0.

Orowan’s law:H(p,L) ∼ c0|p|L

for small p and L, where c0 > 0

Theorem (R. Monneau, S. P.)Let p0, L0 ∈ R. Then there exists c0 > 0 such that

H(δp0, δL0)

δ2 → c0|p0|L0 as δ → 0+.

Stefania Patrizi

From PN to DD

For s > 12 :

{∂tuε = ε2s−1Is[uε(t , ·)]−W ′ (uε

ε

)+ σ

( tε ,

xε

)in R+ × Rn

uε(0, x) = u0(x) on Rn.

For s < 12 :

{∂tuε = Is[uε(t , ·)]−W ′ ( uε

ε2s

)+ σ

( tε2s ,

xε

)in R+ × Rn

uε(0, x) = u0(x) on Rn.

Stefania Patrizi

From PN to DD

Theorem (S.P., E. Valdinoci)

For s > 12 , uε → u0 converges locally uniformly in (t , x)

towards the solution u0 of{∂tu = H1(∇xu) in R+ × RN

u(0, x) = u0(x) on RN ,

For s < 12 , the solution uε converges locally uniformly in

(t , x) towards the solution u0 of{∂tu = H2(I1[u]) in R+ × RN

u(0, x) = u0(x) on RN ,

where the effective Hamiltonians H1(p)and H2(L) are definedthrough a cell-problem.

Stefania Patrizi

From PN to DD

Assume:n = 1σ ≡ 0Is = −(−∆)s

W ∈ C4,β(R), 0 < β < 1; W (v + 1) = W (v), ∀v ∈ R;W > 0 on R \ Z; W ′′(0) > 0.

Theorem (S.P., E. Valdinoci)Let p0, L0 ∈ R with p0 6= 0. Then there exists c0 > 0 such that

H(δp0, δ2sL0)

δ1+2s → c0|p0|L0 as δ → 0+.

Stefania Patrizi

dislocations dynamics: from microscopic models to ... seminars/stefaniapatrizi... · dislocations...

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