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OUTLINE

• 1 DDD simulations
• elastic properties
• 2 DDD simulations
• connection with atomic scale
• 3 Typical problems at mesoscale
• 4 Coupling with the continuum
• Full DDD
• Constitutive modelling
• Continuum theory of dislocations ?

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(2-D, 2.5-D) 3-D SIMULATIONS
interactions P-K force line tension junctions
dislocation flux (basic DDD)
stress equilibrium (full DDD)
mobility laws cross-slip ............
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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)
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DISLOCATIONS AND DEFECT CLUSTERS
(radiation damage fatigue)
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JUNCTIONS -gt FOREST HARDENING
The Lomer-Cottrell lock (Saada 1960 , Schoeck
Frydman 1972)
R. Madec et al. 2001
D. Rodney R. PHillips, 2000 (MS)
same critical stresses (t ? mb/l)
junctions elastic problem gt no free parameter
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STRENGTH OF THE FOREST
Up to large strains, insensitive to SFE,
Cross-slip, rotations, long range stresses,
GNDs, patterning
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Local rules
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LOCAL RULES
series connection with atomic scale needs models
rate equations, elastic models (Escaig, Schoeck
et al.,..) gt saddle points, equilibrium (MS
simulations) fast events (MD simulations)

• Dislocation theory is not finished ....
• FCCs cross-slip (Cu) scaling laws
• BCCs 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
  ..)

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MOBILITY / MICROSTRUCTURE
fast moving dislocations (Zbib et al.), climb
velocities ?
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CROSS-SLIP (FCCs) MESOSCOPIC VIEW
multiplication/annihilation dynamic
recovery pattern formation precipitate
bypassing textures
b
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MESOSCOPIC LOCAL RULES
..... mesoscale simulations are weak in chemistry
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• Typical issues for basic DDDs
• single crystal
• hardening patterning
• interactions between slip systems
• composition of mechanisms
• (ex Peierls stress forest hardening)

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MASS SIMULATIONS
Cu, 100 stress axis
.
Not necessarily the best way for understanding
hardening
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CELLS and CROSS-SLIP
"similitude principle" ? structure of internal
stress
(111) foil t 3 mm
10 mm
(110) foil t 3 mm
von Mises stress
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• Interactions between slip systems

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INTERACTION COEFFICIENTS (FCCs)
tc ambvr gt
(Franciosi et al., 1980)
12x12 144 gt 6
junctions a3 Lomer a2 glissile a1ortho
Hirth
a0 self a1copla coplanar
the collinear interaction(b SP, b CSP)
acoli measurement by model simulations acoli
15 a3
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THE COLLINEAR INTERACTION
P1 ? P2 same b
exhausts the mobile dislocations leaves small
stable debris
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COLLINEAR INTERACTIONS
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STRESS-STRAIN CURVES IN MULTISLIP
Cu, 300 K, 100, 8 active slip systems DD
simulation critical stresses reconstructed
from
B 4 slip systems are de-activated
A the cross-slip systems of B remain
active A', B' same but without collinear
interaction
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Coupling with the continuum full DDD
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NANO-GRAINS
s
d (nm-1/2)
Atomistic simulations d 30-50 nm but still no
Hall-Petch law ! DDD simulations ?
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HALL-PETCH
scaling k ? (atomic ? meso ?) dislocation-grain
boundary (local rule at mesoscale) continuum
modelling ?
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3D MMC COMPOSITE(full DDD)
There are size effects in s001and density/stress
gradients but how can we compose mechanisms
forest hardening (basic DDD) load
transfer (FE) size effects (full DDD) ?
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CONSTITUTIVE MODELLING
Basic DDD dislocation-based models
crystal plasticity codes
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
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HARDENING MATRIX
This constitutive formulation is parameter-free
for copper crystals It is inserted into a crystal
plasticity FE code (boundary conditions ..)
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Cu crystals
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STAGE I - STAGE II
Prediction of slip systems
No information needed about dislocation
mstructures as long as there is no change in
deformation path Next step Bauschinger test
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STRAIN LOCALIZATIONS JERKY FLOW
FE code for polycrystals (A. Beaudoin) no
gradient term, incompatibity stresses Constitutiv
e formulation
s qe ? F( )
Al-Mg alloy 2 10-4 s-1 Type B
All types of bands and dynamic behaviour
S. Kok et al. Acta Mater. 51 (2003) 3651
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CONTINUUM THEORY OF DISLOCATIONS ?
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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,almo
st everything small strains few achievements
till now (in 3-D) can we draw models from it ?