Our current understanding of the microscopic origin of gradient terms · 2004. 7. 26. · Outline Motivations Linking micro to meso-scale Properties of the correlation functions Gradient

Our current understanding of themicroscopic origin of gradient terms

I. Groma

Department of General Physics, Eötvös University Budapest

FF. CsikorM. Zaiser

E. Van der Giessen

http://metal.elte.hu/.̃groma

. – p.1/14

Outline

• Motivations• Linking micro to meso-scale• Properties of the correlation functions• Gradient term• Numerical results• Extension for multiple slip• Conclusions

. – p.2/14

Motivations

l

τ

γ

Local plasticity

τclass(γ, γ̇, ...)

. – p.3/14

Motivations

l

τ

γ

Local plasticity

τclass(γ, γ̇, ...)

Phenomenological nonlocal plasticity

τ(γ, γ̇, ...) = τclass(γ, γ̇, ...) + l2µ

d2

d~r2γ

. – p.3/14

Earlier examples

Fluid dynamics, Boltzmann equation

∂∂tf + ~v

∂∂~v

f(t, ~v, ~r) + ~F (~r) ∂∂~r

f(t, ~v, ~r) = δfcδt⇓

Navier − Stokes Equations

Plasma physics

BBGKY hierarhy

⇓

V lasov′sEquation

. – p.4/14

2D dislocation dynamics.

Equation of motion

~vi = B~b

0

@

NX

j 6=i

τind(~ri − ~rj) + τext

1

A , τind(~r) =bµ

2π(1 − ν)x(x2 − y2)(x2 + y2)2

For dislocations with ± Burgers vectors

∂ρ+(~r1, t)

∂t+~b

d

d~r1

»

ρ+(~r1, t)τext +

Z

{ρ++(~r1, ~r2, t) − ρ+−(~r1, ~r2, t)} τind(~r1 − ~r2)d~r2–

= 0

∂ρ−(~r1, t)

∂t−~b d

d~r1

»

ρ−(~r1, t)τext −Z

{ρ−−(~r1, ~r2, t) − ρ−+(~r1, ~r2, t)} τind(~r1 − ~r2)d~r2–

= 0

. – p.5/14


Equation of motion

~vi = B~b

0

@

NX

j 6=i


1

A , τind(~r) =bµ

2π(1 − ν)x(x2 − y2)(x2 + y2)2

N particle density function

fN (t, ~r1, ~r2...~rN )d~r1d~r2...d~rN = fN (t + ∆t, ~r′1, ~r

′2...~r

′N

)d~r′1d~r′2...d~r

′N

~r′i = ~ri + ~vi∆t d~r′i = d~ri(1 + dv

ix/dxi∆t)


∂ρ+(~r1, t)

∂t+~b

d

d~r1

»

ρ+(~r1, t)τext +

Z


= 0

∂ρ−(~r1, t)

∂t−~b d

d~r1

»



= 0

. – p.5/14


Equation of motion

~vi = B~b

0

@

NX

j 6=i


1

A , τind(~r) =bµ

2π(1 − ν)x(x2 − y2)(x2 + y2)2



′2...~r

′N

)d~r′1d~r′2...d~r

′N


ix/dxi∆t)

From this

∂fN

∂t+

NX

i=1

∂

∂~ri{fN (~r1, ~r2, ..., ~rN )~vi} = 0.


∂ρ+(~r1, t)

∂t+~b

d

d~r1

»

ρ+(~r1, t)τext +

Z


= 0

∂ρ−(~r1, t)

∂t−~b d

d~r1

»



= 0

. – p.5/14


Equation of motion

~vi = B~b

0

@

NX

j 6=i


1

A , τind(~r) =bµ

2π(1 − ν)x(x2 − y2)(x2 + y2)2



′2...~r

′N

)d~r′1d~r′2...d~r

′N


ix/dxi∆t)

With integration

∂ρ1(~r1, t)

∂t+~b

d

d~r1

»

ρ1(~r1, t)τext +

Z

ρ2(~r1, ~r2, t)τind(~r1 − ~r2)d~r2–

= 0


∂ρ+(~r1, t)

∂t+~b

d

d~r1

»

ρ+(~r1, t)τext +

Z


= 0

∂ρ−(~r1, t)

∂t−~b d

d~r1

»



= 0

. – p.5/14


Equation of motion

~vi = B~b

0

@

NX

j 6=i


1

A , τind(~r) =bµ

2π(1 − ν)x(x2 − y2)(x2 + y2)2



′2...~r

′N

)d~r′1d~r′2...d~r

′N


ix/dxi∆t)


∂ρ+(~r1, t)

∂t+~b

d

d~r1

»

ρ+(~r1, t)τext +

Z


= 0

∂ρ−(~r1, t)

∂t−~b d

d~r1

»



= 0

. – p.5/14

Long range ⇔ Short range

Let

ρss′ (~r1, ~r2, t) = ρs(~r1)ρs′ (~r2)(1 + dss′ (~r1, ~r2)) s, s′ ∈ {+,−}

-2 -1 0 1 2-2

-1

0

1

2

(x-x') ρ 1/2

( y-y

') ρ

1/2

X ray line profile analysis

0.0001

0.001

0.01

0.1

1

-4 -2 0 2 4

I(q)

q 108 m-1

a.

A(n) = exp[ρn2 ln(n/R)]√

ρR ≈ 1

. – p.6/14

Structure of equations

With ρ = ρ+ + ρ− and κ = ρ+ − ρ−

∂ρ(~r, t)

∂t+~b

d

d~r[κ(~r, t) {τsc(~r) + τext − τf(~r) + τb(~r)}] = f(ρ, τext, ...)

∂κ(~r, t)

∂t+~b

d

d~r[ρ(~r, t) {τsc(~r) + τext − τf(~r) + τb(~r)}] = 0

τf(~r) =µb

2π(1 − ν)C(τ)p

ρ(~r)

τb(~r) = −µb

2π(1 − ν)D1

ρ(~r)

∂κ(~r)

∂x

. – p.7/14



∂ρ(~r, t)

∂t+~b

d


∂κ(~r, t)

∂t+~b

d


τsc(~r) =

Z

κ(~r1, t)τind(~r − ~r1)d~r1, τind =x(x2 − y2)(x2 + y2)2

τf(~r) =µb

2π(1 − ν)C(τ)p

ρ(~r)

τb(~r) = −µb

2π(1 − ν)D1

ρ(~r)

∂κ(~r)

∂x

. – p.7/14



∂ρ(~r, t)

∂t+~b

d


∂κ(~r, t)

∂t+~b

d


Field equations (Kröner)

∆2χ = −Ab∂κ∂y

, τsc =∂2

∂x∂yχ

τf(~r) =µb

2π(1 − ν)C(τ)p

ρ(~r)

τb(~r) = −µb

2π(1 − ν)D1

ρ(~r)

∂κ(~r)

∂x

. – p.7/14



∂ρ(~r, t)

∂t+~b

d


∂κ(~r, t)

∂t+~b

d



∆2χ = −Ab∂κ∂y

, τsc =∂2

∂x∂yχ

τf(~r) =

Z

ρ(~r1)da(~r − ~r1)τind(~r − ~r1)d ~r1 da = d+− − d−+

τf(~r) =µb

2π(1 − ν)C(τ)p

ρ(~r)

τb(~r) = −µb

2π(1 − ν)D1

ρ(~r)

∂κ(~r)

∂x

. – p.7/14



∂ρ(~r, t)

∂t+~b

d


∂κ(~r, t)

∂t+~b

d



∆2χ = −Ab∂κ∂y

, τsc =∂2

∂x∂yχ

τf(~r) =

Z

ρ(~r1)da(~r − ~r1)τind(~r − ~r1)d ~r1 da = d+− − d−+

τb(~r) =

Z

κ(~r1)d(~r − ~r1)τind(~r − ~r1)d ~r1 d = d+− + d−+ + d++ + d−−

τf(~r) =µb

2π(1 − ν)C(τ)p

ρ(~r)

τb(~r) = −µb

2π(1 − ν)D1

ρ(~r)

∂κ(~r)

∂x

. – p.7/14



∂ρ(~r, t)

∂t+~b

d


∂κ(~r, t)

∂t+~b

d



∆2χ = −Ab∂κ∂y

, τsc =∂2

∂x∂yχ

τf(~r) = ρ(~r)

Z

da(~r)τind(~r)d~r,

τb(~r) = −dκ(~r)

d~r

Z

~rd(~r)τind(~r)d~r

τf(~r) =µb

2π(1 − ν)C(τ)p

ρ(~r)

τb(~r) = −µb

2π(1 − ν)D

1

ρ(~r)

∂κ(~r)

∂x

. – p.7/14



∂ρ(~r, t)

∂t+~b

d


∂κ(~r, t)

∂t+~b

d



∆2χ = −Ab∂κ∂y

, τsc =∂2

∂x∂yχ

τf(~r) =µb

2π(1 − ν)C(τ)p

ρ(~r)

τb(~r) = −µb

2π(1 − ν)D1

ρ(~r)

∂κ(~r)

∂x

. – p.7/14

Plasticity

By definition

bκ = −∂γ∂x

b∂κ

∂t= −∂γ̇

∂x

Rate of deformationγ̇ = κ

»

τ − τf −µb

2π(1 − ν)D

1

ρ(~r)

∂κ(~r)

∂x

–

Gradient term

τeff = τ +µ

2π(1 − ν)D1

ρ(~r)

∂2γ(~r)

∂x2

. – p.8/14

Finite size effect

y x

L 0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

2.5

Effe

ctiv

e st

ress

[ µbρ

01/2/(

2 π(1

- ν))

]

Strain[bρ01/2D]

-2 -1 0 1 2

-4

-2

0

2

4

-4

-2

0

2

4

γ [ D

bρ01

/2]

κ [ ρ

0]

x [ρ0-1/2]

. – p.9/14

Finite size effect

-2 -1 0 1 2

-4

-2

0

2

4

-4

-2

0

2

4

γ [ D

bρ01

/2]

κ [ ρ

0]

x [ρ0-1/2]

. – p.9/14

Finite size effect

-2 -1 0 1 2

-4

-2

0

2

4

-4

-2

0

2

4

γ [ D

bρ01

/2]

κ [ ρ

0]

x [ρ0-1/2]

if ρ = 1/l2

τ =µb

2π(1 − ν)D1

ρ(~r)

∂κ(~r)

∂x

κ(~r) = τ1

l22π(1 − ν)

µbDx

. – p.9/14

Finite size effect

-2 -1 0 1 2

-4

-2

0

2

4

-4

-2

0

2

4

γ [ D

bρ01

/2]

κ [ ρ

0]

x [ρ0-1/2]

if ρ = 1/l2

τ =µb

2π(1 − ν)D1

ρ(~r)

∂κ(~r)

∂x

κ(~r) = τ1

l22π(1 − ν)

µbDx

at the boundary −L/2 ρ = −κτ = − µb

2π(1 − ν)D1

κ(~r)

∂κ(~r)

∂x

κ(~r) = κ0 exp

−τ 2π(1 − ν)µbD

x

ff

. – p.9/14

Boundary layer

-2 -1 0 1 2

-4

-2

0

2

4

-4

-2

0

2

4

γ [ D

bρ01

/2]

κ [ ρ

0]x [ρ0

-1/2]

. – p.10/14

Nontrivial problems (Erik, Serge)

2w 2 3h=

2h 2hf

2wf

U· hΓ·=

U· hΓ·=

x1

x2

. – p.11/14


Γ(%)

τ/µ

x10

3

0 0.25 0.5 0.75 10

0.5

1

1.5

2

2.5

3 ∆τn=0

(i)

(iii)discretecontinuum

|

. – p.11/14


-1 0 1-1

0

1400350300250200150100500

ρL2

-1 0 1-1

0

14003002001000

-100-200-300-400

κL2

. – p.11/14

Multiple slip?

∂ρi(~r, t)

∂t+~bi

d

d~r[κi(~r, t)

˘

τsc(~r) + τiext − τ if (~r) + τ ib(~r)

¯

] = fi(ρi, τiext, ...)

∂κi(~r, t)

∂t+~bi

d

d~r[ρi(~r, t)

˘

τ isc(~r) + τiext − τ if (~r) + τ ib(~r)

¯

] = 0,

τ ib(~r) = −N

X

j=1

Dij(ρk)~bjdκj(~r)

d~r

Dij(ρk, γ)

. – p.12/14

Multiple slip?

∂ρi(~r, t)

∂t+~bi

d

d~r[κi(~r, t)

˘

τsc(~r) + τiext − τ if (~r) + τ ib(~r)

¯

] = fi(ρi, τiext, ...)

∂κi(~r, t)

∂t+~bi

d

d~r[ρi(~r, t)

˘

τ isc(~r) + τiext − τ if (~r) + τ ib(~r)

¯

] = 0,

τ ib(~r) = −N

X

j=1

Dij(ρk)~bjdκj(~r)

d~r

Dij(ρk, γ). – p.12/14

Temperature?, Climb?

Temperature

vix = Bg(T )bτ(xi, yi) +p

kTBgζix < ζ

ix(t)ζ

ix(0) >= δ(t)

. – p.13/14


Temperature


kTBgζix < ζ

ix(t)ζ

ix(0) >= δ(t)

Climb

viy = −Bc(T )bσ11(xi, yi) +p

kTBcζiy < ζ

iy(t)ζ

iy(0) >= δ(t)

. – p.13/14


Temperature


kTBgζix < ζ

ix(t)ζ

ix(0) >= δ(t)

Climb

viy = −Bc(T )bσ11(xi, yi) +p

kTBcζiy < ζ

iy(t)ζ

iy(0) >= δ(t)

Fokker-Planck type equation

∂

∂tρ± ∓

d

d~rρ±B̂

d

d~rV (~r) ∓ d

d~rD̂

d

d~r(ρ+ − ρ−) + kT

d

d~rB̂

d

d~rρ± = f(ρ±, ...)

. – p.13/14

Summary

Continuum theory of dislocation dynamics• Balance equation for ρ and κ• Self-consistent (long range) field• Flow stress• Back stress, gradient term

. – p.14/14

includegraphics [angle=0,width=1.4cm]{eltecimer.ps} Outline $ $Motivations $ $Earlier examples $ $2D dislocation dynamics. $ $Long range $Leftrightarrow $ Short range $ $Structure of equations $ $Plasticity $ $ Finite size effect $ $Boundary layer $ $Nontrivial problems (Erik, Serge)$ $Multiple slip? $ $Temperature?, Climb? $ $Summary $ $

Documents

Our current understanding of the microscopic origin of gradient terms · 2004. 7. 26. · Outline Motivations Linking micro to meso-scale Properties of the correlation functions Gradient