Tutorial I: Mechanics of Ductile Crystalline Solids

1. Tutorial I:Mechanics of Ductile 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 • Michael Ortiz • Alejandro Strachan • Habin Su • Shanfu Zheng Singapore 2003 cuitiño@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 2003 cuitiño@rutgers

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

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

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

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

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

9. Effective Energy Elastic interaction Core energy External field where with m Displacementjump across S SlipSurface S Singapore 2003 cuitiño@rutgers

10. Elastic interaction • Displacement field: • Elastic distortion: • Elastic interaction: • Green function for an isotropic crystal: Singapore 2003 cuitiño@rutgers

11. b Elastic Interaction with A1 R A2 (Hirth and Lothe,1969) Singapore 2003 cuitiño@rutgers

12. External Field with: applied shear stress forestdislocations Singapore 2003 cuitiño@rutgers

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

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

15. Phase-Field Energy Elastic energy Core regularization Core regularization factor Singapore 2003 cuitiño@rutgers

16. PX d Energy minimizing phase-field Unconstrained minimization problem: if with Singapore 2003 cuitiño@rutgers

17. 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: Singapore 2003 cuitiño@rutgers

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

19. Solution Procedure and compute the reactions: • Stick predictor. Set • Reaction projection • Phase-field evaluation Singapore 2003 cuitiño@rutgers

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

21. Macroscopic averages • Slip • Dislocation density • Obstacle concentration • Shear stress Singapore 2003 cuitiño@rutgers

22. Forest Hardening Parameters Obstacle distribution BOUNDARY CONDITIONS Periodic OBSTACLE STRENGTH Uniform, f = 10 G b2 PEIERLS STRESS tp= 0 Singapore 2003 cuitiño@rutgers

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

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

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

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

27. Cyclic loading a b c Stress-strain curve. d e f Evolution of dislocation density with strain. i g h Singapore 2003 cuitiño@rutgers

28. Irreversibility/Cyclic Loading Singapore 2003 cuitiño@rutgers

29. Poisson ratio effects Evolution of dislocation density with strain. Stress-strain curve. Stress-strain curve Dislocation density vs.strain b Singapore 2003 cuitiño@rutgers

30. Obstacle density Singapore 2003 cuitiño@rutgers

31. Multiple Glide Singapore 2003 cuitiño@rutgers

32. Physics Capturing Capabilities • 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. Ortiz,1999 Singapore 2003 cuitiño@rutgers

33. Summary • Phase-field model. • Closed form solution at zero temperature. • Temperature effects. • Strain rate effects. • Dislocation structures in grain boundaries. Singapore 2003 cuitiño@rutgers

34. Another Study Case: FerroElectrics 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 2003 cuitiño@rutgers

35. Mesoscale Ti = 0 En = 0 u Mechanical and Dielectric constants from atomistics G0 assumed equal to Gm Singapore 2003 cuitiño@rutgers

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

37. Mesoscale Simulations: Stress Evolution of stress Singapore 2003 cuitiño@rutgers

38. Polarization Evolution of polarization Singapore 2003 cuitiño@rutgers

39. Hysteresis Loop Increasing G0 Hysteresis Loop Energy Singapore 2003 cuitiño@rutgers

40. 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 Singapore 2003 cuitiño@rutgers