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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 ?