FROM MESOSCALE SIMULATIONS

TO MULTISCALE MODELLING

Discrete Dislocation PlasticityCambridge, 1-2 July 2004

L. Kubin &, B. Devincre LEM, CNRS-ONERA, F-ChâtillonT. Hoc Ecole Centrale Paris, F-Châtenay MalabryR. Madec DPTA, CEA, F-Bruyères-Le-Châtel

polycrystalcontinuum framework

electronic scale

dislocationcore properties

microstructuresingle crystal

dislocation core

dislocation:elastic properties

atomic scale

OUTLINE

1 DDD simulationselastic properties

2 DDD simulationsconnection with atomic scale

3 Typical problems at mesoscale

4 Coupling with the continuum

Full DDD

Constitutive modelling

Continuum theory of dislocations ?

interactionsP-K force

line tensionjunctions

mobility lawscross-slip............

(2-D, 2.5-D) 3-D SIMULATIONS

dislocation flux (basic DDD) +stress equilibrium (full DDD)

3 = (Periodic) boundary conditions

1 = Elastic properties

2 = Local rules

Discretisationtime (10-9 - 10-10 s) & space

line & character

10 -15 m

FCC, BCC, HCP, DC ..

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Elastic properties of dislocations

can be very complicated

not an issue for DDDs

down to a few Burgers vectors

dislocations vs. small loops, other dislocations, obstacles (planar glide)

DISLOCATIONS AND DEFECT CLUSTERS (radiation damage & fatigue)

Drift mechanismKratochvil 1986

Sweeping mechanismSharp-Makin 1964 ....Ghoniem et al.

. .

b

Atomistic vision: the cluster is absorbed(Rodney & Martin 1999)

JUNCTIONS -> FOREST HARDENING

D. Rodney & R. PHillips, 2000 (MS)

R. Madec et al. 2001

The Lomer-Cottrell lock (Saada 1960 , Schoeck & Frydman 1972)

≈ same critical stresses ( b/l)

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cette image.

junctions ≈ elastic problem => no free parameter

STRENGTH OF THE FOREST

Cu Al Cu & Ag (Basinski, 1979)DD

Up to large strains,≈ insensitive to:

SFE, Cross-slip, rotations, long range stresses, GNDs,& patterning

€

τ=αμb ρf

fccs: 0.35 ±0.15

Local rules

series connection with atomic scaleneeds models : rate equations, elastic models (Escaig, Schoeck et al.,..)

=> saddle points, equilibrium (MS simulations)fast events (MD simulations)

Dislocation theory is not finished ....

FCCs : cross-slip (Cu) & scaling lawsBCCs : ab initio core structure (C. Woodward)

kink-pair mechanism (J. Moriarty)

*solute atoms and screw dislocation cores (BCCs, Ti)

* dislocation generation in defect-free volumes- homogeneous (S. Yip)- heterogeneous

(crack tips, surfaces, nanoindentation, grain boundaries, interphases, epitaxial layers ..)

LOCAL RULES

MOBILITY / MICROSTRUCTURE

b = Bv; v << v

Obstacle : d-d interactions(athermal) FCCs :

Patterning

√

Thermally activated obstacle: the lattice No pattern

v = voExp[-G()]/kT .. up to medium-high temperature or large strains

≠ √

Nb 50K

fast moving dislocations (Zbib et al.), climb velocities ?

CROSS-SLIP (FCCs): MESOSCOPIC VIEW

multiplication/annihilation

dynamic recovery

pattern formation

precipitate bypassing

textures

Local rule :

P exp[-G(*, )/kT)

Friedel-Escaig elastic model

atomistic models -> Jacobsen et al(1997 ..)Rao et al. (1999)

seen in literature: P = 0 (planar slip), P = 1(perfect screws)

b

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b

MESOSCOPIC LOCAL RULES

dissociation : attractive stress between the partials

€

=γ/bp

precipitate: glide resistance inside the precipitate (shearing)

grain boundary/interface: dislocations are blocked, absorbed (re-emitted), cross ?

..... mesoscale simulations are weak in chemistry

Typical issues for basic DDDs

single crystal:

hardening & patterning

interactions between slip systems

composition of mechanisms(ex: Peierls stress + forest

hardening)

MASS SIMULATIONS

Cu, [100] stress axis

.

4 10-5

5 10-5

6 10-5

7 10-5

8 10-5

9 10-5

0,0001

0,00011

0,00012

0 0,0001 0,0002 0,0003 0,0004 0,0005 0,0006 0,0007 0,0008

/

γ /P sysNot necessarily the best wayfor understanding hardening

CELLS and CROSS-SLIP

(111) foilt = 3 m

10 m

von Misesstress

(110) foilt = 3 m

"similitude principle" ?

structure of internal stress

Interactions between slip systems

INTERACTION COEFFICIENTS (FCCs)

€

bb1

bj

b2

a0:: self

a1copla: coplanar

junctions

a3 : Lomer

a2 : glissile

a1ortho : Hirth

+ the collinear interaction(b SP, b CSP): acoli

measurement by model simulations:

acoli ≈ 15 a3

c = b√

€

τcs =μb asu

u∑ ρu

(Franciosi et al., 1980)12x12 = 144 => 6

THE COLLINEAR INTERACTION

P1 ≠ P2same bb

exhausts the mobile dislocations

leaves small stable debris

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COLLINEAR INTERACTIONS

SiGe/Si: (after Stach et al., 2000) Al-6Mg in situ(Mills)

STRESS-STRAIN CURVES IN MULTISLIP

Cu, 300 K, [100], 8 active slip systems DD simulation

critical stresses reconstructed from

A : the cross-slip systems of B remain active

A', B' same but without collinear interaction

€

cs = μb as

uρu

u

∑

B: 4 slip systems are de-activated

Coupling with the continuum : full DDD

NANO-GRAINS

d (nm-1/2)

Cu

J. Weertman 1997

Ni

d (nm)

d-1/2

Atomistic simulations: d ≈ 30-50 nm but still no Hall-Petch law !

DDD simulations ?

HALL-PETCH

scaling k ? (atomic ? meso ?)dislocation-grain boundary(local rule at mesoscale)

continuum modelling ?

1 - pile-ups

d

Ndisl d joint Ndisl d

k/√d(yield)

+kγ/d (flow)

€

s =kγ

d==>

€

dρ s

dγ= +

k

d

2 - storage

3 - GNDs

€

δγδx

∝γd

3D MMC COMPOSITE(full DDD)

(001 view)

S. Groh 2003

7 mThere are size effects in 001and density/stress gradients but:how can we compose mechanisms :

forest hardening (basic DDD)+ load transfer (FE)

+ size effects (full DDD) ?

CONSTITUTIVE MODELLING

Basic DDD simulations are used to feed (tensorial) dislocation-based constitutive models

Avoids/limits parameter fitting

Many possible applications up to large strains

= atomic + meso + continuum

Basic DDD + dislocation-based models

+ crystal plasticity codes

HARDENING MATRIX

cf. Kocks-Mecking Teodosiu et al.

€

cs = μb asuρ

u

u

∑ dρ s

dγs=

1

b

1

K1,2asuρ

u

u

∑ − 2yρ s ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

(Franciosi, 1980) storage recovery

Forest densities only, no space variable

€

hsu =μ

2

asu

asuρu

u

∑

1

K1,2asuρ

u

u

∑ − 2 y ρu ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

interaction coefficients (measured by DDD)critical annihilation distance (cross-slip models)mean-free path (from experiment)

This constitutive formulation is parameter-free for copper crystals

It is inserted into a crystal plasticity FE code (boundary conditions ..)

Cu crystals(Diehl, 1956)

[531]

(M

Pa )

γ

"Al": ys = 500 nm

Cu: ys = 50 nm

"Ag": ys = 12 nm

(T. Takeuchi, 1974)l/lo

F/So

(MPa)[001]4 activeslip systemsinstead of 8

STAGE I - STAGE II

(MP

a)

γ

γ

No information needed

about dislocation structures

as long as there is no

change in deformation path

Next step: Bauschinger test

Prediction of slip systems

STRAIN LOCALIZATIONS: JERKY FLOW

FE code for polycrystals (A. Beaudoin): no gradient term, incompatibity stresses

Constitutive formulation: ˙ ε a

Al-Mg alloy

2 10-4 s-1

Type B

€

˙ ε a =

All types of bands and dynamic behaviour

= F( )

S. Kok et al. Acta Mater. 51 (2003) 3651

CONTINUUM THEORY OF DISLOCATIONS ?

A. El Azab (2000 ..)

finite crystal distortionselastic fieldsdislocation structure

statistical dislocation dynamics framework, 3D, accounts for all reactions

≈ analytical version of a full DDD simulationcomplex (≈ 60 equations)goes to large strains

SUMMARY

All DDDs:

local rules (need atomistic input)

small strains

Basic DDD :

tool for modelling/understanding

interactions, microstructures, strain hardening

more powerful, modelling & connection with crystal plasticity codes

Fulll DDD :

applications: ...potentially,almost everything

small strains

few achievements till now (in 3-D)

can we draw models from it ?