1. IHPC-IMS Program on Advances & Mathematical Issues in Large Scale Simulation (Dec 2002 - Mar 2003 & Oct - Nov 2003) Tutorial I: Mechanics of Ductile 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

2. Singapore 2003 cuiti ñ o@rutgers Collaborators Bill Goddard Marisol Koslowski Michael Ortiz Alejandro Strachan Habin Su Shanfu Zheng

3. 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

4. Singapore 2003 cuiti ñ o@rutgers The Role of Dislocations Initial ELASTICITY PLASTICITY Dislocation Motion Dislocation Generation area swept by dislocation

5. Singapore 2003 cuiti ñ o@rutgers Anatomy of a Dislocation Loop Animations from: http://uet.edu.pk/dmems/ b : burgers vector Screw Segment Edge Segment Mixed Segment Area swept by dislocation loop Glide Plane Mixed Screw Edge

6. Singapore 2003 cuiti ñ o@rutgers Dislocation Motion and Arrest OBSTACLES DISLOCATION FOREST Animation from: zig.onera.fr/DisGallery/ junction.html

7. Singapore 2003 cuiti ñ o@rutgers Tracking Dislocation in ONE Plane (Ortiz, 1999)

8. Singapore 2003 cuiti ñ o@rutgers Overview Effective Dislocation Energy Core Energy Dislocation Interaction Irreversible Obstacle Interaction Macroscopic Averages Equilibrium configurations Closed form solution at zero temperature. Metropolis Monte Carlo algorithm and mean field approximation at finite temperatures.

9. Singapore 2003 cuiti ñ o@rutgers Effective Energy wherewith Elastic interactionCore energyExternal field Slip Surface S Displacement jump across S m

10. Singapore 2003 cuiti ñ o@rutgers Elastic interaction  Displacement field:  Elastic distortion:  Elastic interaction:  Green function for an isotropic crystal:

11. Singapore 2003 cuiti ñ o@rutgers Elastic Interaction with b R A2A2 A1A1 (Hirth and Lothe,1969)

12. Singapore 2003 cuiti ñ o@rutgers External Field applied shear stress forest dislocations with:

13. Singapore 2003 cuiti ñ o@rutgers Core Energy Ortiz and Phillips, 1999 d: inter-planar distance

14. Singapore 2003 cuiti ñ o@rutgers Phase-Field Energy Minimization with respect to  gives: core regularization factor elastic energy

15. Singapore 2003 cuiti ñ o@rutgers Phase-Field Energy Core regularization Elastic energy Core regularization factor

16. Singapore 2003 cuiti ñ o@rutgers Energy minimizing phase-field Unconstrained minimization problem: if with PXPX dd

17. Singapore 2003 cuiti ñ o@rutgers Irreversible Process and Kinetics  Irreversible dislocation-obstacle interaction may be built into a variational framework, we introduce the incremental work function: incremental work dissipated at the obstacles  Primary and forest dislocations react to form a jog:  Updated phase-field follows from:  Short range obstacles:

18. Singapore 2003 cuiti ñ o@rutgers Irreversible Process and Kinetics   g g Kuhn-Tucker optimality conditions: Equilibrium condition:

19. Singapore 2003 cuiti ñ o@rutgers Solution Procedure 1.Stick predictor. Set 2.Reaction projection 3.Phase-field evaluation and compute the reactions:

20. Singapore 2003 cuiti ñ o@rutgers Closed-form solution Dislocation loops Calculations are gridless and scale with the number of obstacles

21. Singapore 2003 cuiti ñ o@rutgers Macroscopic averages  Slip  Dislocation density  Obstacle concentration  Shear stress

22. Singapore 2003 cuiti ñ o@rutgers Forest Hardening Obstacle distribution BOUNDARY CONDITIONS Periodic OBSTACLE STRENGTH Uniform, f = 10 G b 2 PEIERLS STRESS  p = 0 Parameters

23. Singapore 2003 cuiti ñ o@rutgers Monotonic loading Stress-strain curve. Evolution of dislocation density with strain.

24. Singapore 2003 cuiti ñ o@rutgers Dislocation Patterns Evolution of dislocation pattern as a function of slip strain

25. Singapore 2003 cuiti ñ o@rutgers Interaction with Obstacles Detail of the evolution of the dislocation pattern showing dislocations bypassing a pair of obstacles

26. Singapore 2003 cuiti ñ o@rutgers Fading memory a c d b e f Stress-strain curve. Three dimensional view of the evolution of the phase-field, showing the the switching of the cusps.

27. Singapore 2003 cuiti ñ o@rutgers Stress-strain curve. a Cyclic loading b c d ef gh i Evolution of dislocation density with strain.

28. Singapore 2003 cuiti ñ o@rutgers Irreversibility/Cyclic Loading

29. Singapore 2003 cuiti ñ o@rutgers Poisson ratio effects b Stress-strain curveDislocation density vs.strain Stress-strain curve. Evolution of dislocation density with strain.

30. Singapore 2003 cuiti ñ o@rutgers Obstacle density

31. Singapore 2003 cuiti ñ o@rutgers Multiple Glide

32. Singapore 2003 cuiti ñ o@rutgers Physics Capturing Capabilities Ortiz,1999  The aim of this study is to develop a phase- field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals.  This representation enables to identify individual dislocation lines and arbitrary dislocation geometries, including tracking intricate topological transitions such as loop nucleation, pinching and the formation of Orowan loops.  This theory permits the coupling between slip systems, consideration of obstacles of varying strength, anisotropy, thermal and strain rate effects.

33. Singapore 2003 cuiti ñ o@rutgers Summary 1. Phase-field model. 2.Closed form solution at zero temperature. 3.Temperature effects. 4.Strain rate effects. 5.Dislocation structures in grain boundaries.

34. Singapore 2003 cuiti ñ o@rutgers Another Study Case: FerroElectrics 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

35. Singapore 2003 cuiti ñ o@rutgers Mesoscale T i = 0 E n = 0 u Mechanical and Dielectric constants from atomistics G 0 assumed equal to G m

36. Singapore 2003 cuiti ñ o@rutgers Evolution of Polarized Region Nucleate at end boundary Propagate along chain Loop movie Evolution of interface Nucleate at side boundary Propagate along chain

37. Singapore 2003 cuiti ñ o@rutgers Mesoscale Simulations: Stress Evolution of stress

38. Singapore 2003 cuiti ñ o@rutgers Polarization Evolution of polarization

39. Singapore 2003 cuiti ñ o@rutgers Increasing G 0 Hysteresis Loop Energy Hysteresis Loop

40. Singapore 2003 cuiti ñ o@rutgers Towards material design: sensitivity analysis and inverse problem Sensitivity analysis: “taking derivatives across the scales” Derivative of the area in the hysteresis loop (loss) with respect to percentage of CTFE Nucleation energy From atomistics: Nucleation energy decreases 3% per 1 mol % CTFE From mesoscale: Hysteresis decreases 2% per 1% reduction in nucleation energy Multi-scale: Hysteresis decreases XX % per 1 mol % CTFE