1. Max-Planck-Institut für Eisenforschung, Düsseldorf Coupling Density Functional Theory with Continuum Mechanics for Alloy Design D. Ma*, M. Friák, W. Counts, D. Raabe, J. Neugebauer Max Planck Institute for Iron Research, Düsseldorf, Germany

2. Max-Planck-Institut für Eisenforschung, Düsseldorf Multi-scale Modeling mm μm nm Å

3. Max-Planck-Institut für Eisenforschung, Düsseldorf Multi-scale Modeling mm mm Å μm nm Å

4. Max-Planck-Institut für Eisenforschung, Düsseldorf Two Examples: (1) β-Ti Alloys for implants (2) Mg-Li Alloys for lightweight structures

5. Max-Planck-Institut für Eisenforschung, Düsseldorf β-Ti alloy design 1. Motivation 2. Phase analysis 3. Elastic properties 4. Elastic constants as input for CPFEM 5. Summary

6. Max-Planck-Institut für Eisenforschung, Düsseldorf β-Ti alloys design 1. Motivation 2. Phase analysis 3. Elastic properties 4. Elastic constants as Input for CPFEM 5. Summary

7. 1. Motivation

8. 1. Motivation Main challenges in designing the bone replacement: (1) Bio-compatibility (2) Reduce the elastic stiffness (3) Stabilize the β-phase Ti-Nb binary system ~20GPa ~70GPa >100GPa M. Niinomi, Sci. Tech. Adv. Mater. 2003 M. Niinomi, Mater. Sci. Eng. 1998

9. Max-Planck-Institut für Eisenforschung, Düsseldorf β-Ti alloys design 1. Motivation 2. Phase analysis 3. Elastic properties 4. Elastic constants as Input for CPFEM 5. Summary

10. 2. Phase Analysis DFT

11. Nb Ti unwanted hcp-based phase that is stiffer and stable 2. Phase Analysis wanted bcc-based phase that is softer but metastable BCC structure of Ti-Nb alloy HCP structure of Ti-Nb alloy

12. 2. Phase Analysis

13. 2. Phase Analysis

14. 2. Phase Analysis XRD DFT

15. Max-Planck-Institut für Eisenforschung, Düsseldorf β-Ti alloys design 1. Motivation 2. Phase analysis 3. Elastic properties 4. Elastic constants as Input for CPFEM 5. Summary

16. 3. Elastic Properties Ab-initio calculation: Equilibrium lattice constants Lattice constants Minimum energy Bulk modulus

17. 3. Elastic Properties Ab-initio calculation: Equilibrium elastic constants ε, strain tensor δ, strain U, elastic energy density B, bulk modulus

18. 3. Elastic Properties Ab initio calculation results of the elastic constants: C11, C12, C44: elastic stiffness constants AZ, Zener‘s ratio EH: homogenized Young‘s modulus by Hershey‘s model

19. 3. Elastic Properties Young‘s modulus surface plots Pure Nb Ti-25at.%Nb Ti-31.25at.%Nb Ti-18.75at.%Nb [001] [100] [010] Az=3.210 Az=1.058 Az=0.5027 Az=2.418 The elastic properties of the Ti-Nb binary alloys become isotropic as the Nb content increases

20. 3. Elastic Properties

21. single-crystalline C11, C12, C44, B0 micro-scale macro-scale polycrystalline Young modulus 3. Elastic Properties 64μ H4 + 16(4C11 + 5C12)μ H3 + [3(C11+ 2C12) × (5C11+ 4C12) -8(7C11 – 4C12)C44]μ H2 -(29C11 – 20C12)(C11+2C12)C44 μ H –3(C11 + 2C12)2(C11 – C12)C44 = 0 “scale-jumping” (across the meso-scale)

22. 3. Elastic Properties theory: bcc polycrystals MECHANICAL INSTABILITY!!

23. 3. Elastic Properties theory: bcc polycrystals MECHANICAL INSTABILITY!!

24. 3. Elastic Properties Ti-hcp: 117 GPa theory: bcc polycrystals MECHANICAL INSTABILITY!!

25. Ultra-sonic measurement exp. polycrystals ! bcc+hcp phases 3. Elastic Properties Ti-hcp: 117 GPa theory: bcc polycrystals MECHANICAL INSTABILITY!!

26. Max-Planck-Institut für Eisenforschung, Düsseldorf β-Ti alloys design 1. Motivation 2. Phase analysis 3. Elastic properties 4. Elastic constants as input for CPFEM 5. Summary

27. 4. Elastic Constants as Input of CPFEM Required input data of the materials properties in crystal plasticity finite element method

28. 4. Elastic Constants as Input of CPFEM Plane strain compression: (1) Influence of the elastic anistropy (2) predict the texture evolution Bending test: Homogenized elastic properties of textured and non-texture materials

29. 4. Elastic Constants as Input of CPFEM Elastic constants of a single crystal flow curve from the compression test on solution annealed Ti30at.%Nb Random texture The plastic property is kept, and only the elastic property is varied!!!

30. 4. Elastic Constants as Input of CPFEM 0° 90° 0° εh=0 α-fiber εh=30% γ-fiber εh=60% 90° φ1 (0°~90°) εh=90% Φ(0°~90°) φ2=45°

31. 4. Elastic Constants as Input of CPFEM Elastic constants of a single crystal Textured and non texture

32. 4. Elastic Constants as Input of CPFEM

33. Max-Planck-Institut für Eisenforschung, Düsseldorf β-Ti alloys design 1. Motivation 2. Phase analysis 3. Elastic properties 4. Elastic Constants as Input of CPFEM 5. Summary

34. 5. Summary Thermodynamic stability of hcp- and bcc-Ti was studied Configurational entropy at finite temperature stabilizes bcc Ti-Nbphase Volume fractions have been calculated using the Gibbs construction Polycrystalline two-phase Young’s modulus has been theoretically predicted employing the Hershey and CPFEM homogenization methods Very good agreement between theoretical prediction and experiment The calculated elastic constants (DFT) can be used as input for CPFEM Nb SHOULD BE THE PRIMARY ALLOYING ELEMENTS IN Ti FOR HUMAN IMPLANT MATERIALS

35. Max-Planck-Institut für Eisenforschung, Düsseldorf Mg-Li alloy design 1. Motivation 2. Elastic properties 3. Analysis of the Elastic Properties 4. Summary

36. Max-Planck-Institut für Eisenforschung, Düsseldorf Mg-Li alloy design 1. Motivation 2. Elastic properties 3. Analysis of the Elastic Properties 4. Summary

38. Not ductile, textures

39. Problematic for industrial applications (anisotropy)

40. Magnesium (and its alloys) are light weight and relatively strong

41. Ideal lightweight structural materialHow can hcp magnesium be transformed into bcc/fcc magnesium?

43. Homogenize to get isotropic polycrystal elastic constants

46. 2. Elastic Properties: Bulk Modulus Li

47. 2. Elastic Properties: Shear Modulus Optimal G (17 GPa) around bcc phase boundary (70 at % Mg) bcc Mg is unstable Li dominate alloys are very soft Li Experiment is reasonably well reproduced

48. 2. Elastic Properties:Young‘s Modulus Optimal E (45 GPa) around bcc phase boundary (70 at % Mg) bcc Mg is unstable Li dominate alloys are very soft Li Experiment is reasonably well reproduced

49. 2. Elastic Properties: Poisson‘s Ratio Softer alloys have a higher n Softer alloys have a lower n Li Experiment is reasonably well reproduced

50. Max-Planck-Institut für Eisenforschung, Düsseldorf Mg-Li alloy design 1. Motivation 2. Elastic properties 3. Analysis of the elastic Properties 4. Summary

52. Based on experimental observations

54. B/G > 1.75 DUCTILE

55. B/G < 1.75 BRITTLE1.75 is more a transition zone

56. 3. Analysis of the Elastic Properties Ductile Region Brittle Region Stiffer bcc Mg-Li alloys Ductile/brittle transition region

58. Design Criteria

59. Maximum stiffness for minimum weight

60. Typical Values (MPa m3/kg)

61. Graphite Fiber 127.78

62. Graphite Fiber/epoxy 43.53

63. Steel 26.41

64. Aluminum 25.93

65. PET (polymer) 2.15

67. 3. Analysis of the Elastic Properties

68. Max-Planck-Institut für Eisenforschung, Düsseldorf Mg-Li alloy design 1. Motivation 2. Elastic properties 3. Analysis of the elastic Properties 4. Summary

69. 4. Summary DFT and homogenization schemes can be used to predict with reasonable accuracy elastic properties of polycrystalline metals Optimal elastic properties of bcc MgLi alloys are observed around 70 at. % Mg B/G for the optimal bcc Mg-Li alloys is in the brittle/ductile transition region BCC MgLi has a better E/r than AlMg and a comparable E/r to Al-Li BCC MgLi HAS POTENTIAL AS AN ULTRA-LIGHT WEIGHT STRUCTURAL ALLOY

70. Conclusions + Understanding trends (thermodynamics, mechanics) + Direct use of homogenization theory (elastic) + Extract engineering quantities for a rough but quick estimation + Get quantities that you cannot get elsewhere - 0 K - supercell size - long calculation times