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Lecture Micromechanics texture SFB 761

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  • 1. Why does a crystal rotate ?Dierk RaabeDsseldorf, Germany WWW.MPIE.DE [email protected] SFB Class 2012

2. Overview Roters et al. Acta Materi.58 (2010) 1 3. 2 4. Some dislocation kinematics and kinetics: phenomenatrue strainstress3 5. Plastic deformation of a single crystal by dislocation slip4 6. Plastic deformation of a single crystal by dislocation slip5 7. Plastic deformation of a single crystal by dislocation slip6 8. Plastic deformation of a single crystal by dislocation slip ddx b 1 n m bv dtX Z dt 7 9. Plastic deformation of a single crystal by dislocation slipSchmid factorS h id f t (orientation factor for that slip system) 8 10. Plastic deformation of a single crystal by dislocation slip 11. Plastic deformation of a single crystal by dislocation slip 12. Boundary condition: determines lab frame constraints 11 13. Single crystal plasticity: crystal shear and crystal rotation12 14. Plastic deformation of a single crystal by dislocation slip ddx b 1 n m bv dtX Z dt 13 15. Kinematics, displacementu u(x,y,z)u=u(x y z)(x(1),y,z) (x(2),y,z)12 u(1)(x,y,z)u(2)(x,y,z)u(1)(x y z)=u(2)(x y z)(x,y,z)=u (x,y,z)121 214 16. Kinematics, displacementu u(x,y,z)u=u(x y z)(x(1),y,z) (x(2),y,z)12 u(1)(x,y,z)u(2)(x,y,z)u(1)(x y z)u(2)(x y z)(x,y,z)u (x,y,z)12 1215 17. Kinematics, displacement, displacement gradient: general 16 18. Kinematics: Micro-to-macro-transition17 19. Geometrical interpretation ? 18 20. Geometrical interpretation 19 21. Complex boundary conditions mesoscopic boundary conditions one dislocation (grain / orientation neighborhood)parallel loops reactionsorientation change 20 22. Simplify boundary conditionsBoundary conditions:1) Upper bound treatment: iso-stress2) Lower bound treatment: iso-strainiso strain 21 23. Iso-stress: single slip system a ma sym b mbsym d md sym c mcsym 1 c b d a krit krit krit krit D 33TT D a D bcD krit aktiv d 11 22 24. Single crystal plasticity: multiple slip (or twinning) system/mit/mit23 25. Single crystal yield surface33 1 crystal, 1 slip system: aik nk a jlbl ijj critjslip system 1.. crit s 1 ( active) same strain 33 different 11 stresses slip system 2 crit s 2 ( active) critactive1 crystal, 2 slip systems:aik nks a jlbls ij crit 11 26. Iso-stress: multiple slip (or twinning) system 33Ds=1Ds=2 Vers.. krit (+)s1 ( aktiv) krit,(+) TBH S 11 krit,(-) s2 krit,(+) s2 ( aktiv) . krit,(-) s1 ( aktiv) VersDs=2 27. Single crystal plasticitybcc, fcc,bcc fccBcc: 24 systemsSection in stress spacekrit krit krit krit BCC, 48 systemskritkrit 26 28. Simplify boundary conditionsBoundary conditions:1) Upper bound treatment: iso-stress2) Lower bound treatment: iso-strainiso strain 27 29. The Taylor Model 28 30. Crystal yield surface, Taylor Bishop-Hill29 31. Crystal yield surface, Taylor Bishop-HillMany crystals, many slip systems: crystals grain 1 33grain 2ggrain 3grain 4imposed strain11 32. Homogeneity and boundary conditions meso-scale 3%8%15% 33. Simplify boundary conditions Boundary conditions: 1) Upper bound treatment: iso-stress (strain not compatible) 2) Lower bound treatment: iso-strain (forces not in equilibrium)32 34. Multiscale crystal plasticity FEM or FFT Raabe, Zhao, Park, Roters: Acta Mater. 50 (2002) 421 33 35. Crystal Mechanics FEM, grain scale mechanics (2D)Experiment (DIC, EBSD) v Mises strain Simulation(CP-FEM)(C )v Mises strainSachtleber,Sachtleber, Zhao, Raabe: Mater. Sc. Engin. A 336 (2002) 81 Mater. Engin. 34