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Dierk Raabe Ab Initio Simulations In Metallurgy

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  • 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

    90°

    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
  • 37. 1. Motivation
    Magnesium Bad
    Magnesium Good
    • Magnesium (and its alloys) are generally hcp
    • 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 material
    How can hcp magnesium be transformed into bcc/fcc magnesium?
  • 42. 1. Motivation
    hcp
    +
    bcc
    hcp
    bcc
    Ultra light-weight structural material
    • rLi = 0.58 g/cm3rMg = 1.74 g/cm3
  • 1. Motivation
    Use DFT to find the bcc MgLi alloy
    composition with optimal elastic properties
    Goal:
    11 different bcc alloys
    • Calculate single crystal Cij’s
    • 43. Homogenize to get isotropic polycrystal elastic constants
    • 44. Analyze engineering ratio’s
    Physical Limitations
    • Ordered alloys (periodic structures)
    • 45. Ground state calculations (0 K)
  • Max-Planck-Institut für Eisenforschung, Düsseldorf
    Mg-Li alloy design
    1. Motivation
    2. Elastic properties
    3. Analysis of the elastic Properties
    4. Summary
  • 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
  • 51. 3. Analysis of the Elastic Properties
    • Defined as
    • 52. Based on experimental observations
    • 53. Measure of ductile vs. brittle behavior
    G Resisting Plastic Flow
    B Bond Strength
    Opposition
    To
    Fracture
    • 1.75 critical value
    • 54. B/G > 1.75 DUCTILE
    • 55. B/G < 1.75 BRITTLE
    1.75 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
  • 57. 3. Analysis of the Elastic Properties
    • Defined as
    • 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
    • 66. Lead 1.41
  • 3. Analysis of the Elastic Properties
    Better than Al-Mg. Comparable to Al-Li.
  • 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