FROM MESOSCALE SIMULATIONS TO MULTISCALE MODELLING Discrete Dislocation Plasticity Cambridge, 1-2 July 2004 L. Kubin &, B. Devincre LEM, CNRS-ONERA, F-Châtillon T. Hoc Ecole Centrale Paris, F-Châtenay Malabry R. Madec DPTA, CEA, F-Bruyères-Le-Châtel polycrystal continuum framework electronic scale dislocation core properties microstructure single crystal dislocation core dislocation: elastic properties atomic scale

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

interactions P-K force line tension junctions mobility laws cross-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 Discretisation time ( s) & space line & character  m FCC, BCC, HCP, DC..

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 mechanism Kratochvil 1986 Sweeping mechanism Sharp-Makin 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 The Lomer-Cottrell lock (Saada 1960, Schoeck & Frydman 1972) ≈ same critical stresses (    b/ ) junctions ≈ elastic problem => no free parameter

STRENGTH OF THE FOREST CuAlCu & Ag (Basinski, 1979) DD Up to large strains, ≈ insensitive to: SFE, Cross-slip, rotations, long range stresses, GNDs, & patterning fccs:  0.35 ±0.15

Local rules

series connection with atomic scale needs 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 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..) LOCAL RULES

MOBILITY / MICROSTRUCTURE  b = Bv; v << v Obstacle : d-d interactions (athermal) FCCs : Patterning   √  Thermally activated obstacle: the lattice No pattern v = v o Exp[-  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 b

MESOSCOPIC LOCAL RULES dissociation : attractive stress between the partials 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. Not necessarily the best way for understanding hardening

CELLS and CROSS-SLIP (111) foil t = 3  m 10  m von Mises stress (110) foil t = 3  m "similitude principle" ? structure of internal stress

Interactions between slip systems

INTERACTION COEFFICIENTS (FCCs) a 0: : self a 1copla : coplanar junctions a 3 : Lomer a 2 : glissile a 1ortho : Hirth + the collinear interaction(b SP, b CSP): a coli measurement by model simulations: a coli ≈ 15 a 3  c =  b√  (Franciosi et al., 1980) 12x12 = 144 => 6

THE COLLINEAR INTERACTION P1 ≠ P2 same b exhausts the mobile dislocations leaves small stable debris

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 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 ≈ 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 N disl  d    joint  N disl  d     k/√d  (yield )    +k  /d  (flow) ==> 2 - storage 3 - GNDs

3D MMC COMPOSITE (full DDD) (001 view) S. Groh  m There are size effects in  001 and 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. (Franciosi, 1980) storage recovery Forest densities only, no space variable 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]  (MPa)  "Al": y s = 500 nm Cu: y s = 50 nm "Ag": y s = 12 nm (T. Takeuchi, 1974)  l/l o F/S o (MPa) [001] 4 active slip systems instead of 8

STAGE I - STAGE II  (MPa)   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: Al-Mg alloy s -1 Type B 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 distortions elastic fields dislocation structure statistical dislocation dynamics framework, 3D, accounts for all reactions ≈ analytical version of a full DDD simulation complex (≈ 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 ?