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

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Singapore 2003 cuiti ñ o@rutgers Bill Goddard Marisol Koslowski Stephen Kuchnicki Michael Ortiz Raul Radovitzky Laurent Stainier Alejandro Strachan Zisu Zhao Collaborators

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Singapore 2003 cuiti ñ o@rutgers Hierarchy of Scales length time mm nmµmµm ms µs ns Phase stability, elasticity Energy barriers, paths Phase-boundary mobility Microstructures Grains Direct FE simulation Polycrystals Single crystals SCS test Force Field

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Singapore 2003 cuiti ñ o@rutgers Motivation The aim of this work is concerned with the development of a micromechanical model of the hardening, rate-sensitivity and thermal softening of BCC crystals. We place primary emphasis on the derivation closed-form analytical expressions describing the macroscopic behavior of the crystals amenable to implementation as constitutive relations within a standard finite-element code. In developing the model, we follow the well-established paradigm of micromechanical modeling, consisting of: –the identification of the dominant or rate-limiting `unit' processes operating at the microscale; –the identification of the macroscopic forces driving the unit processes; –the analysis of the response of the unit processes to the macroscopic driving forces; and –the determination of the average or macrocospic effect of the combined operation of all the micromechanical unit processes. W e specifically consider the following unit processes: –double-kink formation and thermally activated motion of kinks; – the close-range interactions between primary and forest dislocation, and the subsequent formation of jogs; –the percolation motion of dislocations through a random array of forest dislocations introducing short-range obstacles of different strengths; –dislocation multiplication due to breeding by double cross-slip; and –dislocation pair-annihilation.

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Singapore 2003 cuiti ñ o@rutgers General Framework Incremental Field Equations

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Singapore 2003 cuiti ñ o@rutgers General Framework Additive Decomposition of the Free Energy Multiplicative Decomposition of Deformation Gradient HARDENINGELASTIC RESPONSE with and Temperature-dependent elastic constants for Ta Absolute Temperature Internal Variables

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Singapore 2003 cuiti ñ o@rutgers Multiplicative Decomposition Initial Configuration F = F e F p Deformed Configuration Intermediate Configuration ELASTIC EFFECTS Lattice deformation and rotation PLASTIC EFFECTS Dislocation Slip FpFp FeFe

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Singapore 2003 cuiti ñ o@rutgers General Framework HARDENING Applied Resolved Shear Stress Flow Rule In rate formwhere Microscopic Description Piola-Kirchoff Stress Tensor Slip Direction Slip-plane Normal Micro/Macro Relations Current flow Stress LOCAL CONSTITUTIVE LAW

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Singapore 2003 cuiti ñ o@rutgers Dislocation Mobility ( double-kink formation and thermally activated motion of kinks) FORMATION ENTHAPY FOR DOUBLE KINK 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 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. 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)

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

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Singapore 2003 cuiti ñ o@rutgers 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 Bow-out mechanism in BCC crystals Energetic condition for bow-out process Retaining dominant terms,

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Singapore 2003 cuiti ñ o@rutgers 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. 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 n 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. PROBABILITY DENSITY FUNCTION for FCC

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

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

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

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

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

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Singapore 2003 cuiti ñ o@rutgers 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)

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Singapore 2003 cuiti ñ o@rutgers 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

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

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Singapore 2003 cuiti ñ o@rutgers 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.

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Singapore 2003 cuiti ñ o@rutgers Dislocation Evolution TOTAL DISLOCATION MULTIPLICATION DISLOCATION PRODUCTION by Frank-Reed sources PRODUCTION DISLOCATION PRODUCTION by breeding by double cross-slip =+ Thermally activated process with activation energy E cross

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

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Singapore 2003 cuiti ñ o@rutgers Comparison with Experiment ORIENTATION DEPENDENCE Sensitivity to misalignment

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Singapore 2003 cuiti ñ o@rutgers Comparison with Experiment (Theory and Experiment) TEMPERATURE DEPENDENCE

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Singapore 2003 cuiti ñ o@rutgers Comparison with Experiment (Theory and Experiment) STRAIN-RATE DEPENDENCE

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Singapore 2003 cuiti ñ o@rutgers Microscopic Predictions Slip Strains TEMPERATURE DEPENDENCE

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Singapore 2003 cuiti ñ o@rutgers Microscopic Predictions Slip Strains STRAIN-RATE DEPENDENCE

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Singapore 2003 cuiti ñ o@rutgers Microscopic Predictions Dislocation Densities TEMPERATURE DEPENDENCE

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Singapore 2003 cuiti ñ o@rutgers Microscopic Predictions Dislocation Densities STRAIN-RATE DEPENDENCE

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Singapore 2003 cuiti ñ o@rutgers Core energy (eV/b) 1/2a edge dislocation in (110) plane 1/2a screw dislocation Volume (1/V 0 ) 540 GPa MD Data Application of EoS Find Jacobian of F (=det(F)) Use EoS to find hydrostatic pressure Evaluate core energies and elastic moduli Data from MP group Goddard, Strachan, Cagin, Wu Pressure Dependency

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Singapore 2003 cuiti ñ o@rutgers [213] Ta single crystal; Strain speed: 10 -3 /s; Without core energy data With core energy data Pressure and Plasticity

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

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Singapore 2003 cuiti ñ o@rutgers Material Parameters MATERIAL PROPERTY FITTED FROM EXPERIMENT L kink /b13 E kink [eV] 0.70 U edge / b 2 0.200 U edge / U screw 1.77 E cross [eV] 0.65 b = 2.86e-10 m ; T_Debye = 340

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Singapore 2003 cuiti ñ o@rutgers Comparison with Experiment TEMPERATURE DEPENDENCE Mitchell and Spitzig, 1965 EXPERIMENT THEORY

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Singapore 2003 cuiti ñ o@rutgers Comparison with Experiment Mitchell and Spitzig, 1965 EXPERIMENT THEORY STRAIN-RATE DEPENDENCE

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Singapore 2003 cuiti ñ o@rutgers Material Parameters MATERIAL PROPERTY FITTED FROM EXPERIMENT COMPUTED BY ATOMISTICS L kink /b1317 E kink [eV] 0.700.73 U edge / b 2 0.2000.216 U edge / U screw 1.77 E cross [eV] 0.65- ATOMISTICS from WANG, STRACHAN, CAGIN and GODDARD MATERIAL PROPERTY FITTED FROM EXPERIMENT L kink /b13 E kink [eV] 0.70 U edge / b 2 0.200 U edge / U screw 1.77 E cross [eV] 0.65

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Singapore 2003 cuiti ñ o@rutgers 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

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

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Singapore 2003 cuiti ñ o@rutgers 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 Full-Field Coupled Electromechanical

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

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Singapore 2003 cuiti ñ o@rutgers Macroscale simulation: Stress Blue (0) = T 3 G Red (1) = All Trans (T 3 G) (all-trans) Transformed Region Normal stress along chains

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Singapore 2003 cuiti ñ o@rutgers Macroscale simulation: Stress Blue (0) = T 3 G Red (1) = All Trans (T 3 G) (all-trans) Transformed Region Normal stress perpendicular to chains

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Singapore 2003 cuiti ñ o@rutgers Macroscale simulation: Stress Blue (0) = T 3 G Red (1) = All Trans (T 3 G) (all-trans) Transformed Region Shear

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Singapore 2003 cuiti ñ o@rutgers Macroscale simulation: Electric Displacement Blue (0) = T 3 G Red (1) = All Trans (T 3 G) (all-trans) Transformed Region Along the Chains

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Singapore 2003 cuiti ñ o@rutgers Blue (0) = T 3 G Red (1) = All Trans (T 3 G) (all-trans) Transformed Region Normal to the Chains Macroscale simulation: Electric Displacement

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Singapore 2003 cuiti ñ o@rutgers Macroscale simulation: Electric Potential Blue (0) = T 3 G Red (1) = All Trans (T 3 G) (all-trans) Transformed Region

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