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Tutorial II: Constitutive Models for Crystalline Solids

Tutorial II: Constitutive Models for Crystalline Solids. Alberto M. Cuitiño Mechanical and Aerospace Engineering Rutgers University Piscataway, New Jersey cuitino@jove.rutgers.edu. IHPC-IMS Program on Advances & Mathematical Issues in Large Scale Simulation

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Tutorial II: Constitutive Models for Crystalline Solids

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  1. Tutorial II:Constitutive Models for Crystalline Solids Alberto M. Cuitiño Mechanical and Aerospace Engineering Rutgers University Piscataway, New Jersey cuitino@jove.rutgers.edu IHPC-IMS Program on Advances & Mathematical Issues in Large Scale Simulation (Dec 2002 - Mar 2003 & Oct - Nov 2003) Institute of High Performance Computing Institute for Mathematical Sciences, NUS

  2. Collaborators • Bill Goddard • Marisol Koslowski • Stephen Kuchnicki • Michael Ortiz • Raul Radovitzky • Laurent Stainier • Alejandro Strachan • Zisu Zhao Singapore 2003cuitiño@rutgers

  3. Polycrystals Direct FE simulation Phase stability, elasticity Energy barriers, paths Phase-boundary mobility Hierarchy of Scales SCS test ms Grains Single crystals time µs Microstructures ns Force Field nm µm mm length Singapore 2003cuitiño@rutgers

  4. General Framework Incremental Field Equations Singapore 2003cuitiño@rutgers

  5. General Framework Additive Decomposition of the Free Energy Multiplicative Decomposition of Deformation Gradient ELASTIC RESPONSE HARDENING Internal Variables Absolute Temperature ELASTIC RESPONSE with and Temperature-dependent elastic constants for Ta Singapore 2003cuitiño@rutgers

  6. Multiplicative Decomposition F = FeFp Initial Configuration Deformed Configuration PLASTIC EFFECTS Dislocation Slip ELASTIC EFFECTS Lattice deformation and rotation Fp Fe Intermediate Configuration Singapore 2003cuitiño@rutgers

  7. General Framework HARDENING Microscopic Description LOCAL CONSTITUTIVE LAW Current flow Stress In rate form where Piola-Kirchoff Stress Tensor Applied Resolved Shear Stress Slip-plane Normal Slip Direction Flow Rule Micro/Macro Relations Singapore 2003cuitiño@rutgers

  8. Dislocation Mobility (double-kink formation and thermally activated motion of kinks) We consider the thermally activated motion of dislocations within an obstacle-free slip plane. Under these conditions, the motion of the dislocations is driven by an applied resolved shear stress t and is hindered by the lattice resistance, which is weak enough that it may be overcome by thermal activation. The lattice resistance is presumed to be well-described by a Peierls energy function. FORMATION ENTHAPY FOR DOUBLE KINK Energy formation of kink pair. Estimated by atomistic calculations of the order of 1 eV (Xu and Moriarty, 1998) Kink proliferation is expected at Then, Estimated by atomistic calculations of the order of few Gpa (Xu and Moriarty, 1998) Singapore 2003cuitiño@rutgers

  9. Dislocation Mobility (double-kink formation and thermally activated motion of kinks) From Orowan’s equation and transition state theory, with where, r = dislocation density b = Burgers vector l = mean free-path of kinks nD = Debye Frequency STRAIN-RATE AND TEMPERATURE DEPENDENT EFFECTIVE PEIERLS STRESS Singapore 2003cuitiño@rutgers

  10. Forest Hardening (close-range interactions between primary and forest) In the forest-dislocation theory of hardening, the motion of dislocations, which are the agents of plastic deformation in crystals, is impeded by secondary -or forest- dislocations crossing the slip plane. As the moving and forest dislocations intersect, they form jogs or junctions of varying strengths which, provided the junction is sufficiently short, may be idealized as point obstacles. Moving dislocations are pinned down by the forest dislocations and require a certain elevation of the applied resolved shear stress in order to bow out and bypass the pinning obstacles. The net effect of this mechanism is macroscopic hardening. STRENGTH OF OBSTACLE PAIR IN BCC CRYSTALS Energetic condition for bow-out process Retaining dominant terms, Bow-out mechanism in BCC crystals Singapore 2003cuitiño@rutgers

  11. Forest Hardening (distribution function of obstacle-pair strength ) We assume that the point obstacles are randomly distributed over the slip plane with a mean density na of obstacles per unit area. We also assume that the obstacle pairs spanned by dislocation segments are nearest-neighbors in the obstacle ensemble. PROBABILITY DENSITY FUNCTION ASSOCIATED DISTRIBUTION FUNCTION NOTE 1: The function just derived provides a complete description of the distribution of the obstacle-pair strengths when the point obstacles are of infinite strength and, consequently, impenetrable to the dislocations. NOTE 2:It is interesting to note that the probability density of obstacle-pair strengths for BCC differs markedly from FCC crystals. This difference owes to the different bow-out configurations for the two crystal classes and the comparatively larger values of the Peierls stress. PROBABILITY DENSITY FUNCTION for FCC Singapore 2003cuitiño@rutgers

  12. Forest Hardening (distribution function of obstacle-pair strength) We extend the preceding analysis to include point obstacle with finite strength. Heaviside Function PROBABILITY DENSITY FUNCTION STRENGTH OF OBSTACLE FORMED BY DISLOCATIONS OF SYSTEMS a ANDb ASSOCIATED DISTRIBUTION FUNCTION where Probability that the weakest of two obstacles forming a pair be of type b Number of obstacles of type b per unit area of the slip plane a, which is estimated as with Singapore 2003cuitiño@rutgers

  13. Forest Hardening (percolation motion of dislocations through a random array) From non-equilibrium statistical concepts, we obtain KINETIC EQUATION OF EVOLUTION Then, the incremental plastic strain driven by and increment in the resolved shear stress can be expressed by where AVERAGE DISTANCE BETWEEN OBSTACLES AVERAGE NUMBER OF JUMPS BEFORE DISLOCATION SEGMENTS ATTAIN STABLE POSITIONS Singapore 2003cuitiño@rutgers

  14. Forest Hardening (hardening relations in rate form) where, HARDENING MODULUS CHARACERISTIC STRAIN PARTICULAR CASE: OBSTACLES OF UNIFORM STRENGTH Singapore 2003cuitiño@rutgers

  15. Dislocation Intersection(jog formation energy) Unfavorable Junction After intersection JOG FORMATION ENERGY Before intersection Favorable Junction Reaction coordinate Details of Intersection Process Singapore 2003cuitiño@rutgers

  16. Dislocation Intersection(jog formation energy) Higher mobility of EDGE SEGMENTS Larger population of SCREW SEGMENTS INITIAL Screw Segment Edge Segment Slip Plane FINAL ENERGY FORMATION OF A EDGE SEGMENT INTERSECTING A SCREW SEGMENT Further assuming where r is the ratio between edge and screw energies Singapore 2003cuitiño@rutgers

  17. Dislocation Intersection(jog formation energy) Example of normalized jog-formation energy for r = 1.77 Computed value of r for Ta from atomistic calculations (Wang et al., 2000) Singapore 2003cuitiño@rutgers

  18. Dislocation Intersection(obstacle strength) By equating the energy expended in forming jogs with the potential energy released as a result of the motion of dislocation, we obtain that the forest obstacles become transparent to the motion of primary dislocations when Obstacle strength at zero temperature Invoking transition state theory concepts, OBSTACLE STRENGTH where Singapore 2003cuitiño@rutgers

  19. Dislocation Evolution Rate of generation of dynamic sources induced by cross-slip DISLOCATION PRODUCTION by breeding by double cross-slip Dislocation length emitted by source prior to saturation where Energy Barrier for cross slip Length of screw segment effecting cross slip Mean-free path between cross-slip events Assuming the the mean-free path is inversely proportional to the square root of the dislocation density, we obtain where where DISLOCATION ANNIHILATION by cross-slip Singapore 2003cuitiño@rutgers

  20. Dislocation Evolution DISLOCATION MULTIPLICATION RATE or This rate equation expresses a competition between the dislocation multiplication and annihilation mechanisms. For small slip strains, the multiplication term dominates and the dislocation density grows as a quadratic function of the slip rate. By contrast, for large strains, the rates of multiplication and annihilation balance out and saturation sets in. After saturation is attained, the dislocation density remains essentially unchanged. It should be carefully noted, that the saturation slip strain is a function of temperature and strain rate. Singapore 2003cuitiño@rutgers

  21. Dislocation Evolution PRODUCTION = DISLOCATION PRODUCTION by breeding by double cross-slip + DISLOCATION PRODUCTION by Frank-Reed sources TOTAL DISLOCATION MULTIPLICATION Thermally activated process with activation energy Ecross Singapore 2003cuitiño@rutgers

  22. INTERACTION FORCE Dislocation Evolution ANNIHILATION Maximum Annihilation Radius Imposed velocity Escaping Dislocation No Annihilation Trapped Dislocation Annihilation Annihilation Radius (T,Strain Rate) Minimum Annihilation Radius Singapore 2003cuitiño@rutgers

  23. Comparison with Experiment ORIENTATION DEPENDENCE Sensitivity to misalignment Singapore 2003cuitiño@rutgers

  24. Comparison with Experiment(Theory and Experiment) TEMPERATURE DEPENDENCE Singapore 2003cuitiño@rutgers

  25. Comparison with Experiment(Theory and Experiment) STRAIN-RATE DEPENDENCE Singapore 2003cuitiño@rutgers

  26. Microscopic PredictionsSlip Strains TEMPERATURE DEPENDENCE Singapore 2003cuitiño@rutgers

  27. Microscopic PredictionsSlip Strains STRAIN-RATE DEPENDENCE Singapore 2003cuitiño@rutgers

  28. Microscopic PredictionsDislocation Densities TEMPERATURE DEPENDENCE Singapore 2003cuitiño@rutgers

  29. Microscopic PredictionsDislocation Densities STRAIN-RATE DEPENDENCE Singapore 2003cuitiño@rutgers

  30. Pressure Dependency 540 GPa Application of EoS 1/2a<111> edge dislocation in (110) plane Find Jacobian of F (=det(F)) Core energy (eV/b) 1/2a<111> screw dislocation Use EoS to find hydrostatic pressure Volume (1/V0) MD Data Evaluate core energies and elastic moduli Data from MP group Goddard, Strachan, Cagin, Wu Singapore 2003cuitiño@rutgers

  31. Pressure and Plasticity • [213] Ta single crystal; • Strain speed: 10-3 /s; With core energy data Without core energy data Singapore 2003cuitiño@rutgers

  32. Pressure in Core Energy + EoS Closer examination of core energy effects Elastic range shows expected behavior Plastic behavior converges with increasing pressure Singapore 2003cuitiño@rutgers

  33. Material Parameters b = 2.86e-10 m ; T_Debye = 340 Singapore 2003cuitiño@rutgers

  34. Comparison with Experiment TEMPERATURE DEPENDENCE Mitchell and Spitzig, 1965 EXPERIMENT THEORY Singapore 2003cuitiño@rutgers

  35. Comparison with Experiment STRAIN-RATE DEPENDENCE EXPERIMENT THEORY Mitchell and Spitzig, 1965 Singapore 2003cuitiño@rutgers

  36. Material Parameters ATOMISTICS from WANG, STRACHAN, CAGIN and GODDARD Singapore 2003cuitiño@rutgers

  37. Multiscale Modeling Final Remarks • Multiscale modeling leads to material parameters which quantify well-defined physical entities • The material parameters for Ta have been determined independently in two ways: • Both approaches have yielded ostensibly identical material parameters! • Same agreement with experiment would have been obtained if the parameters had been determined directly by simulation in the absence of data. • This provides validation of modeling and simulation paradigm (as a complement to experimental science). Fitting Atomistic calculations Singapore 2003cuitiño@rutgers

  38. Another Study Case Meso- Macro-scale Nanostructure-properties relationships Constitutive Laws Inverse problem Direct problem Force Fields and MD Elastic, dielectric constants Nucleation Barrier Domain wall and interface mobility Phase transitions Anisotropic Viscosity ab initio QM EoS of various phases Torsional barriers Vibrational frequencies Singapore 2003cuitiño@rutgers

  39. Full-Field Coupled Electromechanical • Features: • - Utilizes multiplicative decomposition of the deformation gradient into elastic-piezoelectric and phase transitional parts, • Accounts for amorphous + several orientation of crystalline phases tracking mass concentration of each phase • -Phase transformation is thermodynamically driven • Electric Gibbs free energy for one component, • -Stress and electric displacement are that of the volume average of components , • -Weak (integral) formulation based on generaized principle of virtual work Singapore 2003cuitiño@rutgers

  40. Macroscale simulation: b-version ALLOWS FOR ARBITRARY SHAPES AND GENERAL ELECTROMECHANICAL BC IN 2D and 3D Mechanically driven non-polar (T3G) to polar (all-trans) transformation Load Initial condition Non-polar (T3G) Deformed Undeformed (all-trans) Complex nucleation of polar phase Singapore 2003cuitiño@rutgers

  41. Macroscale simulation: Stress Red (1) = All Trans Blue (0) = T3G Transformed Region (T3G) (all-trans) Normal stress along chains Singapore 2003cuitiño@rutgers

  42. Macroscale simulation: Stress Red (1) = All Trans Blue (0) = T3G Transformed Region (T3G) (all-trans) Normal stress perpendicular to chains Singapore 2003cuitiño@rutgers

  43. Macroscale simulation: Stress Red (1) = All Trans Blue (0) = T3G Transformed Region (T3G) (all-trans) Shear Singapore 2003cuitiño@rutgers

  44. Macroscale simulation:Electric Displacement Red (1) = All Trans Blue (0) = T3G Transformed Region (T3G) (all-trans) Along the Chains Singapore 2003cuitiño@rutgers

  45. Macroscale simulation:Electric Displacement Red (1) = All Trans Blue (0) = T3G Transformed Region (T3G) (all-trans) Normal to the Chains Singapore 2003cuitiño@rutgers

  46. Macroscale simulation: Electric Potential Red (1) = All Trans Blue (0) = T3G Transformed Region (T3G) (all-trans) Singapore 2003cuitiño@rutgers

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