Continuum Models Of Dislocation Dynamics And Dislocation .

Continuum Models of DislocationDynamics and Dislocation StructuresM. OrtizCalifornia Institute of TechnologyGordon Research Conference on PhysicalMetallurgyThe Holderness School, Plymouth, NHJuly 27, 2004Michael OrtizGRC 07/04

Outline The case for multiscale simulation The case for multiscale modeling The lengthscale hierarchy of polycrystallinemetals The quasicontinuum method Phase-field dislocation dynamics Subgrid models of martensite Subgrid models of dislocation structuresMichael OrtizGRC 07/04

Machining – Experimental ValidationChip Morphology Validation(Courtesy of Third Wave Systems Inc)(Courtesy of IWH, Switzerland)FE simulation(Marusich and Ortiz, IJNME 95)Michael OrtizGRC 07/04

Machining – Experimental Validation(Courtesy of Third Wave Systems Inc)Al 7050AL7010Courtesy of BAE SystemsCutting Force ValidationResidual Stress Validation General trends predicted, but discrepanciesMichael Ortizremain!GRC 07/04

Validation and Verification Fidelity of simulation codes is critically limitedby uncertainties in engineering (empirical)material models Main sources of error and uncertainty– Discretization errors (spatial temporal)– Uncertainties in data: Material properties Model geometry Loading and boundary conditions – Empiricism of constitutive models Need to reduce uncertainty in engineeringconstitutive models for codes to be predictive!Michael OrtizGRC 07/04

Limitations of empirical modelsEarsDeep-drawn cupGrain structure of polycrystalline W(Courtesy of Clyde Briant) Conventional engineering plasticity models fail topredict earing in deep drawing Prediction of earing requires consideration ofpolycrystalline structure, texture developmentMichael OrtizGRC 07/04

Limitations of empirical modelsHall-Petch scaling(NJ Petch,J. Iron and Steel Inst.,174, 1953, pp. 25-28.)Lamellar structure Dislocation pile-upin shocked Taat Ti grain boundary(MA Meyers et al 95)(I. Robertson) Conventional plasticity models fail to predictscaling, size effects.Michael OrtizGRC 07/04

The case for multiscale computing Empirical models fail because they do notproperly account for microstructure The empirical approach does not provide asystematic means of eliminating uncertaintyfrom material models Instead, concurrent multiscale computing:– Model physics at first-principles level, fine lengthscales– Compute on multiple lengthscales simultaneously– Fully resolve the fine scales Bypasses the need to model at coarselengthscalesMichael OrtizGRC 07/04

µsPolycrystalsnstimemsMetal plasticity - Multiscale islocationdynamicsLatticedefects, EoSnmµmlengthmmMichael OrtizGRC 07/04

Multiscale computing - FeasibilityASCI computing systems roadmap Computing power is growing rapidly, but Michael OrtizGRC 07/04

Multiscale computing – FeasibilityTa quadrupole(T. Arias 00)FCC ductile fracture (Courtesy F.F. Abraham) Au nanoindentation(F.F. Abraham 03)(Knap and Ortiz 03) Computing power is growing rapidly, butMichael Ortiz109 1023GRC 07/04

Multiscale computing – FeasibilityPolycrystalline W (Courtesy of C. Briant)Grain-boundarysliding modelSingle-crystalplasticity model(A.M. Cuitiño and R. Radovitzky 02)Michael OrtizGRC 07/04

Multiscale computing – FeasibilityCold-rolled @ 42%polycrystalline TaExperimentalcold-rolled texture(A.M. Cuitiño and R. Radovitzky 03)PolefigureMichael OrtizGRC 07/04

Multiscale computing – FeasibilityDNS of polycrystals: ConvergenceCoarse mesh192 elmts/grainIntermediate mesh1536 elmts/grainFine mesh12288 el/grain(A.M. Cuitiño and R. Radovitzky 03) Numerical convergence extremely slow!Michael OrtizGRC 07/04

Multiscale computing - Feasibility 109 elements at our disposal (106elements/processor x 1000 processors) 1000 elements/coordinate direction 20 elements/grain/direction (8000elements/grain) 50 grains/direction (125K grains) 2.5 mm specimen for 50 µm grains Not enough for complex engineeringsimulations! Subgrain scales still unresolved, needmodeling!Michael OrtizGRC 07/04

Metal plasticity - Multiscale modelingPolycrystalsµsnstimemsAccessibleto cationdynamicsEngineeringcalculationsAccessibleto directnumericalsimulationLatticedefects, EoSnmµmlengthmmMichael OrtizGRC 07/04

The case for multiscale modeling It is not possible to fully resolve material anddeformation microstructures in complexengineering applications directly by brute force Instead, multiscale modeling:– Identify relevant structures and mechanisms at alllengthscales– Bridge lengthscales by: Building models of effective behavior (coarsegraining) Computing material parameters from first principles(parameter passing) Approaches?Michael OrtizGRC 07/04

LaminationµsPolycrystalsnstimemsMultiscale modeling - ticedefects, EoSnmEngineeringcalculationsPhase-field DDQuasi-continuumµmlengthmmmmMichael OrtizGRC 07/04

µsPolycrystalsnstimemsMultiscale modeling - sDislocationdynamicsLatticedefects, EoSnmQuasi-continuumµmlengthmmmmMichael OrtizGRC 07/04

Quasicontinuum - ReductionTadmor, Ortiz and Phillips,Phil. Mag. A, 76 (1996) 1529.Knap and Ortiz, J. Mech. Phys. Solids, 49 (2001) 1899.Michael OrtizGRC 07/04

Quasicontinuum – Cluster sumsMerging of clusters near atomistic limitMichael OrtizGRC 07/04

Quasicontinuum - AdaptivityLongest-edge bisectionof tetrahedron (1,4,a,b)along longest edge (a,b)and of ring of tetrahedraincident on (a,b)Michael OrtizGRC 07/04

QC - Nanoindentation of [001] Au(Movie) Nanoindentation of[001] Au, 2x2x1micrometers Spherical indenter, R 7and 70 nm Johnson EAM potential Total number of atoms 0.25 10 12 Initial number of nodes 10,000 Final number of nodes 100,000Detail of initial computational mesh(Knap and Ortiz, PRL 90 2002-226102)Michael OrtizGRC 07/04

QC - Nanoindentation of [001] Au7 nm indenter, depth 0.92 nmMichael OrtizGRC 07/04

QC - Nanoindentation of [001] Au7 nm indenter, depth 0.92 nmMichael OrtizGRC 07/04

QC - Nanoindentation of [001] Au70 nm indenter, depth 0.75 nmMichael OrtizGRC 07/04

QC - Nanoindentation of [001] Au70 nm indenter, depth 0.75 nm Michael OrtizGRC 07/04

QC - Nanovoid cavitation in Al 72x72x72 cell sample Initial radius R 2a Ercolessi and Adams(Europhys. Lett. 26,583, 1994) EAMpotential. Total number of atoms 16x106 Initial number of nodes 34,000Close-up of internal void(Marian, Knap and Ortiz 04)Michael OrtizGRC 07/04

QC - Nanovoid cavitation in rialmaterialfailure(underinvestigation)failure (under slocationlockingdislocation me,followingfollowingtheinteratomicthe interatomicpotential’spotential’sshapeshape20p 540ε (%)vMichael OrtizGRC 07/04

QC - Nanovoid cavitation in Al353025p (GPa)2015105005101520253035εv (%)αβBCδαABDislocation structures, first yield pointCDβγγ δ DAMichael OrtizGRC 07/0440

QC - Nanovoid cavitation in Al353025p (GPa)201510500510152025303540εv (%)Dislocation types:A - Conventional½〈110〉{111}B - Anomalous½〈110〉{001}Dislocation structures, hardening stageMichael OrtizGRC 07/04

QC - Nanovoid cavitation in Al353025p (GPa)201510500510152025303540εv (%)Unconfined plasticflow carried ion structures, second yield pointMichael OrtizGRC 07/04

Quasicontinuum The Quasicontinuum method is an example of amultiscale method based on:– Kinematic constraints (coarse-graining)– Clusters (sampling)– Adaptivity (spatially adapted resolution) The Quasicontinuum method is an example of aconcurrent multiscale computing: it resolvescontinuum and atomistic lengthscalesconcurrently during same calculation Challenges:– Dynamics (internal reflections)– Finite temperature (heat conduction)– Transition to dislocation dynamicsMichael OrtizGRC 07/04

µsPolycrystalsnstimemsMultiscale modeling - ineeringcalculationsPhase-field DDLatticedefects, EoSnmµmlengthmmmmMichael OrtizGRC 07/04

Phase-field dislocation dynamicsdislocation line(slip area)(Burgerscircuit)Michael OrtizGRC 07/04

Phase-field dislocation dynamicsImpenetrable obstacles(pinning)(Humphreys and Hirsch ’70)Obstacles of finite strength(dissipative interaction)Michael OrtizGRC 07/04

Phase-field dislocation dynamics012Michael OrtizGRC 07/04

Phase-field dislocation dynamicsMichael OrtizGRC 07/04

Phase-field dislocation dynamics012Michael OrtizGRC 07/04

Phase-field dislocation dynamicsabcdefghiStress-strain curveDislocation density(Movie)Michael OrtizGRC 07/04

Fractal dimensionPhase-field dislocation dynamicssimulationProbabilityAvalanchesMiguel (2001)EnergyMichael OrtizGRC 07/04

Phase-field dislocation dynamics Dislocation dynamics approaches rely onanalytical solutions of linear elasticity to reducethe dimensionality of the problem from 3(crystal) to 1 (dislocation lines): semi-inverseapproach Phase-field dislocation dynamics with pairwisePeierls potential reduces dimensionality further,from 3 (crystal) to 0 (point obstacles) Challenges:– Large three-dimensional ensembles– Atomistic dislocation cores– Dislocation reactions, junctionsMichael OrtizGRC 07/04

LaminationµsPolycrystalsnstimemsMultiscale modeling - sDislocationdynamicsLatticedefects, EoSnmµmlengthmmmmMichael OrtizGRC 07/04

Twinning - Microstructures(Cu-Al-Ni, C. Chu and R. D. James)Michael OrtizGRC 07/04

Crystal plasticity - MicrostructuresDipolar dislocation wallsLabyrinth structure in fatiguedcopper single crystal(Jin and Winter 84)Nested bands in copper single crystalfatigued to saturation(Ramussen and Pedersen 80)Michael OrtizGRC 07/04

Crystal plasticity - MicrostructuresDislocation wallsLamellar dislocation structurein 90% cold-rolled Ta(Hughes and Hansen 97)Dislocation wallsLamellar structurein shocked Ta(Meyers et al 95) Lamellar structures are universally found on themicron scale in highly-deformed crystals These microstructures are responsible for thesoft behavior of crystals and for size effectsMichael OrtizGRC 07/04

Microstructures – Sequential ial laminateaveragedeformation43averagestressMichael OrtizGRC 07/04

Nematic elastomers - Lamination(Courtesy of de Simone and Dolzmann)Central region ofsample atmoderate stretch(Courtesy of Kunderand Finkelmann)Blandon et al. 93De Simone and Dolzmann 00De Simone and Dolzmann 02Michael OrtizGRC 07/04

Solid/solid transitions in ironTemperature ( C) Commonly observed solid/solid transitions inFe:– α(bcc) ε(hcp) at p 13 GPa, coexisting phases p 20 GPa– ε(hcp) α(bcc)(Bundy,1964) at p 16 GPa, coexisting phases p 5 GPaγ(fcc)α(bcc)ε(hcp)Pressure (Kbar)Phase diagram for Feε platelets in 0.1%C steelshocked to 20 GPa(Bowden and Kelly, 1967)Michael OrtizGRC 07/04

Phase transitions in Fe – Effect of shearHydrostatic CompressionShear CompressionInitial model with 7 total variants (1 bcc/6 hcp)Michael OrtizGRC 07/04

Phase transitions in Fe – Effect of shear V 1 / 3 F ε12 ε 21 V1/ 3 1/ 3 V ApproximateExperimentalValueMichael OrtizGRC 07/04

Phase transitions in Fe – Effect of shearrank-3laminatesmixed statesrank-1laminatesApproximateExperimentalValue Shear lowers bcc to hcp transition pressure. bcc to hcp transition path involves mixed statesin the form of rank-1 and rank-3 laminates Michael OrtizGRC 07/04

ECAP – Lamination(Beyerlein et al ‘03)X2Die Entryx2x1Shear PlaneϕX1Die ExitEvolution of microstructure(sequential lamination)(Sivakumar and Ortiz 03)Michael OrtizGRC 07/04

Crystal plasticity – size effectsShocked Ta(Meyers et al 95)LaminateHall-Petch effect!BranchingLiF impact(Meir and Clifton 86)Michael OrtizGRC 07/04

Subgrid microstructures - Lamination Sequential lamination supplies microstructures‘on demand’ and is another example ofconcurrent multiscale computing Sub-grid microstructural information isrecovered locally at the Gauss-point level But: Effective response is known explicitly invery few cases (e.g., nematic elastomers) Instead: Consider easy-to-generate specialmicrostructures, such as sequential laminates– Off-line (Dolzmann 99; Dolzmann & Walkington 00)– Concurrently with the calculations (Aubry et al. 03)Michael OrtizGRC 07/04