Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Coupling Density Functional Theory with Continuum Mechanics for<br...
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Multi-scale Modeling<br />mm<br />μm<br />nm<br />Å<br />
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Multi-scale Modeling<br />mm<br />mm<br />Å<br />μm<br />nm<br />Å...
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Two Examples:<br />(1) β-Ti Alloys for implants<br />(2) Mg-Li All...
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloy design<br />1. Motivation<br />2. Phase analysis<br />3...
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloys design<br />1. Motivation<br />2. Phase analysis<br />...
1. Motivation<br />
1. Motivation<br />Main challenges in designing the bone replacement:<br />(1) Bio-compatibility<br />(2) Reduce the elast...
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloys design<br />1. Motivation<br />2. Phase analysis<br />...
2. Phase Analysis<br />DFT<br />
Nb<br />Ti<br />unwanted hcp-based phase<br />that is stiffer and stable<br />2. Phase Analysis<br />wanted bcc-based phas...
2. Phase Analysis<br />
2. Phase Analysis<br />
2. Phase Analysis<br />XRD<br />DFT<br />
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloys design<br />1. Motivation<br />2. Phase analysis<br />...
3. Elastic Properties<br />Ab-initio calculation:<br />Equilibrium lattice constants<br />Lattice constants<br />Minimum e...
3. Elastic Properties<br />Ab-initio calculation: Equilibrium elastic constants<br />ε, strain tensor<br />δ, strain<br />...
3. Elastic Properties<br />Ab initio calculation results of the elastic constants:<br />C11, C12, C44: elastic stiffness c...
3. Elastic Properties<br />Young‘s modulus surface plots<br />Pure Nb<br />Ti-25at.%Nb<br />Ti-31.25at.%Nb<br />Ti-18.75at...
3. Elastic Properties<br />
single-crystalline<br />C11, C12, C44, B0<br />micro-scale<br />macro-scale<br />polycrystalline<br />Young modulus<br />3...
3. Elastic Properties<br />theory: bcc <br />polycrystals<br />MECHANICAL<br />INSTABILITY!!<br />
3. Elastic Properties<br />theory: bcc <br />polycrystals<br />MECHANICAL<br />INSTABILITY!!<br />
3. Elastic Properties<br />Ti-hcp: 117 GPa<br />theory: bcc <br />polycrystals<br />MECHANICAL<br />INSTABILITY!!<br />
Ultra-sonic measurement<br />exp. polycrystals !<br />bcc+hcp phases<br />3. Elastic Properties<br />Ti-hcp: 117 GPa<br />...
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloys design<br />1. Motivation<br />2. Phase analysis<br />...
4. Elastic Constants as Input of CPFEM<br />Required input data of the materials properties in crystal plasticity finite e...
4. Elastic Constants as Input of CPFEM<br /> Plane strain compression:<br /> (1) Influence of the elastic anistropy <br />...
4. Elastic Constants as Input of CPFEM<br />      Elastic constants of a single crystal<br />      flow curve from the com...
4. Elastic Constants as Input of CPFEM<br />0°<br />90°<br />0°<br />εh=0<br />α-fiber<br />εh=30%<br />γ-fiber<br />εh=60...
4. Elastic Constants as Input of CPFEM<br />      Elastic constants of a single crystal<br />     Textured and non texture...
4. Elastic Constants as Input of CPFEM<br />
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloys design<br />1. Motivation<br />2. Phase analysis<br />...
5. Summary<br />Thermodynamic stability of hcp- and bcc-Ti was studied<br />Configurational entropy at finite temperature ...
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Mg-Li alloy design<br />1. Motivation<br />2. Elastic properties<b...
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Mg-Li alloy design<br />1. Motivation<br />2. Elastic properties<b...
1. Motivation<br />Magnesium Bad<br />Magnesium Good<br /><ul><li> Magnesium (and its alloys) are generally hcp
 Not ductile, textures
 Problematic for industrial applications (anisotropy)
 Magnesium (and its alloys) are light weight and relatively strong
 Ideal lightweight structural material</li></ul>How can hcp magnesium be transformed into bcc/fcc magnesium?<br />
1. Motivation<br />hcp<br />+<br />bcc<br />hcp<br />bcc<br />Ultra light-weight structural material<br /><ul><li>rLi = 0....
Homogenize to get isotropic polycrystal elastic constants
Analyze engineering ratio’s</li></ul>Physical Limitations<br /><ul><li>Ordered alloys (periodic structures)
Ground state calculations (0 K)</li></li></ul><li>Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Mg-Li alloy desi...
2. Elastic Properties: Bulk Modulus<br />Li<br />
2. Elastic Properties: Shear Modulus<br />Optimal G (17 GPa)<br />around bcc phase <br />boundary (70 at % Mg)<br />bcc <b...
2. Elastic Properties:Young‘s Modulus<br />Optimal E (45 GPa)<br />around bcc phase <br />boundary (70 at % Mg)<br />bcc <...
2. Elastic Properties: Poisson‘s Ratio<br />Softer alloys have a higher n<br />Softer alloys have a <br />lower n<br />Li<...
Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Mg-Li alloy design<br />1. Motivation<br />2. Elastic properties<b...
3. Analysis of the Elastic Properties<br /><ul><li>Defined as
Based on experimental observations
Measure of ductile vs. brittle behavior</li></ul>G         Resisting Plastic Flow<br />B        Bond Strength<br />Opposit...
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Dierk Raabe Ab Initio Simulations In Metallurgy

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Dierk Raabe Ab Initio Simulations In Metallurgy

  1. 1. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Coupling Density Functional Theory with Continuum Mechanics for<br />Alloy Design<br />D. Ma*, M. Friák, W. Counts, D. Raabe, J. Neugebauer<br />Max Planck Institute for Iron Research, Düsseldorf, Germany<br />
  2. 2. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Multi-scale Modeling<br />mm<br />μm<br />nm<br />Å<br />
  3. 3. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Multi-scale Modeling<br />mm<br />mm<br />Å<br />μm<br />nm<br />Å<br />
  4. 4. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Two Examples:<br />(1) β-Ti Alloys for implants<br />(2) Mg-Li Alloys for lightweight structures<br />
  5. 5. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloy design<br />1. Motivation<br />2. Phase analysis<br />3. Elastic properties<br />4. Elastic constants as input for CPFEM<br />5. Summary<br />
  6. 6. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloys design<br />1. Motivation<br />2. Phase analysis<br />3. Elastic properties<br />4. Elastic constants as Input for CPFEM<br />5. Summary<br />
  7. 7. 1. Motivation<br />
  8. 8. 1. Motivation<br />Main challenges in designing the bone replacement:<br />(1) Bio-compatibility<br />(2) Reduce the elastic stiffness<br />(3) Stabilize the β-phase<br />Ti-Nb binary system<br />~20GPa<br />~70GPa<br />>100GPa<br />M. Niinomi, Sci. Tech. Adv. Mater. 2003<br />M. Niinomi, Mater. Sci. Eng. 1998<br />
  9. 9. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloys design<br />1. Motivation<br />2. Phase analysis<br />3. Elastic properties<br />4. Elastic constants as Input for CPFEM<br />5. Summary<br />
  10. 10. 2. Phase Analysis<br />DFT<br />
  11. 11. Nb<br />Ti<br />unwanted hcp-based phase<br />that is stiffer and stable<br />2. Phase Analysis<br />wanted bcc-based phase <br />that is softer but metastable<br />BCC structure of Ti-Nb alloy<br />HCP structure of Ti-Nb alloy<br />
  12. 12. 2. Phase Analysis<br />
  13. 13. 2. Phase Analysis<br />
  14. 14. 2. Phase Analysis<br />XRD<br />DFT<br />
  15. 15. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloys design<br />1. Motivation<br />2. Phase analysis<br />3. Elastic properties<br />4. Elastic constants as Input for CPFEM<br />5. Summary<br />
  16. 16. 3. Elastic Properties<br />Ab-initio calculation:<br />Equilibrium lattice constants<br />Lattice constants<br />Minimum energy<br />Bulk modulus<br />
  17. 17. 3. Elastic Properties<br />Ab-initio calculation: Equilibrium elastic constants<br />ε, strain tensor<br />δ, strain<br />U, elastic energy density<br />B, bulk modulus<br />
  18. 18. 3. Elastic Properties<br />Ab initio calculation results of the elastic constants:<br />C11, C12, C44: elastic stiffness constants<br />AZ, Zener‘s ratio<br />EH: homogenized Young‘s modulus by Hershey‘s model<br />
  19. 19. 3. Elastic Properties<br />Young‘s modulus surface plots<br />Pure Nb<br />Ti-25at.%Nb<br />Ti-31.25at.%Nb<br />Ti-18.75at.%Nb<br />[001]<br />[100]<br />[010]<br />Az=3.210<br />Az=1.058<br />Az=0.5027<br />Az=2.418<br /> The elastic properties of the Ti-Nb binary alloys become isotropic as the Nb content increases<br />
  20. 20. 3. Elastic Properties<br />
  21. 21. single-crystalline<br />C11, C12, C44, B0<br />micro-scale<br />macro-scale<br />polycrystalline<br />Young modulus<br />3. Elastic Properties<br />64μ H4 + 16(4C11 + 5C12)μ H3 + [3(C11+ 2C12) <br />× (5C11+ 4C12) -8(7C11 – 4C12)C44]μ H2<br />-(29C11 – 20C12)(C11+2C12)C44 μ H<br />–3(C11 + 2C12)2(C11 – C12)C44 = 0 <br />“scale-jumping”<br />(across the meso-scale)<br />
  22. 22. 3. Elastic Properties<br />theory: bcc <br />polycrystals<br />MECHANICAL<br />INSTABILITY!!<br />
  23. 23. 3. Elastic Properties<br />theory: bcc <br />polycrystals<br />MECHANICAL<br />INSTABILITY!!<br />
  24. 24. 3. Elastic Properties<br />Ti-hcp: 117 GPa<br />theory: bcc <br />polycrystals<br />MECHANICAL<br />INSTABILITY!!<br />
  25. 25. Ultra-sonic measurement<br />exp. polycrystals !<br />bcc+hcp phases<br />3. Elastic Properties<br />Ti-hcp: 117 GPa<br />theory: bcc <br />polycrystals<br />MECHANICAL<br />INSTABILITY!!<br />
  26. 26. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloys design<br />1. Motivation<br />2. Phase analysis<br />3. Elastic properties<br />4. Elastic constants as input for CPFEM<br />5. Summary<br />
  27. 27. 4. Elastic Constants as Input of CPFEM<br />Required input data of the materials properties in crystal plasticity finite element method<br />
  28. 28. 4. Elastic Constants as Input of CPFEM<br /> Plane strain compression:<br /> (1) Influence of the elastic anistropy <br /> (2) predict the texture evolution<br /> Bending test:<br /> Homogenized elastic properties of textured and non-texture materials<br />
  29. 29. 4. Elastic Constants as Input of CPFEM<br /> Elastic constants of a single crystal<br /> flow curve from the compression test on solution annealed Ti30at.%Nb<br />Random texture<br />The plastic property is kept, and only the elastic property is varied!!!<br />
  30. 30. 4. Elastic Constants as Input of CPFEM<br />0°<br />90°<br />0°<br />εh=0<br />α-fiber<br />εh=30%<br />γ-fiber<br />εh=60%<br />90°<br />φ1 (0°~90°)<br />εh=90%<br />Φ(0°~90°)<br />φ2=45°<br />
  31. 31. 4. Elastic Constants as Input of CPFEM<br /> Elastic constants of a single crystal<br /> Textured and non texture<br />
  32. 32. 4. Elastic Constants as Input of CPFEM<br />
  33. 33. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />β-Ti alloys design<br />1. Motivation<br />2. Phase analysis<br />3. Elastic properties<br />4. Elastic Constants as Input of CPFEM<br />5. Summary<br />
  34. 34. 5. Summary<br />Thermodynamic stability of hcp- and bcc-Ti was studied<br />Configurational entropy at finite temperature stabilizes bcc Ti-Nbphase<br />Volume fractions have been calculated using the Gibbs construction <br />Polycrystalline two-phase Young’s modulus has been theoretically predicted employing the Hershey and CPFEM homogenization methods <br />Very good agreement between theoretical prediction and experiment<br />The calculated elastic constants (DFT) can be used as input for CPFEM <br />Nb SHOULD BE THE PRIMARY ALLOYING ELEMENTS IN Ti FOR HUMAN IMPLANT MATERIALS<br />
  35. 35. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Mg-Li alloy design<br />1. Motivation<br />2. Elastic properties<br />3. Analysis of the Elastic Properties<br />4. Summary<br />
  36. 36. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Mg-Li alloy design<br />1. Motivation<br />2. Elastic properties<br />3. Analysis of the Elastic Properties<br />4. Summary<br />
  37. 37. 1. Motivation<br />Magnesium Bad<br />Magnesium Good<br /><ul><li> Magnesium (and its alloys) are generally hcp
  38. 38. Not ductile, textures
  39. 39. Problematic for industrial applications (anisotropy)
  40. 40. Magnesium (and its alloys) are light weight and relatively strong
  41. 41. Ideal lightweight structural material</li></ul>How can hcp magnesium be transformed into bcc/fcc magnesium?<br />
  42. 42. 1. Motivation<br />hcp<br />+<br />bcc<br />hcp<br />bcc<br />Ultra light-weight structural material<br /><ul><li>rLi = 0.58 g/cm3rMg = 1.74 g/cm3</li></li></ul><li>1. Motivation<br />Use DFT to find the bcc MgLi alloy <br />composition with optimal elastic properties<br />Goal:<br />11 different bcc alloys<br /><ul><li>Calculate single crystal Cij’s
  43. 43. Homogenize to get isotropic polycrystal elastic constants
  44. 44. Analyze engineering ratio’s</li></ul>Physical Limitations<br /><ul><li>Ordered alloys (periodic structures)
  45. 45. Ground state calculations (0 K)</li></li></ul><li>Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Mg-Li alloy design<br />1. Motivation<br />2. Elastic properties<br />3. Analysis of the elastic Properties<br />4. Summary<br />
  46. 46. 2. Elastic Properties: Bulk Modulus<br />Li<br />
  47. 47. 2. Elastic Properties: Shear Modulus<br />Optimal G (17 GPa)<br />around bcc phase <br />boundary (70 at % Mg)<br />bcc <br />Mg is <br />unstable<br />Li dominate alloys <br />are very soft<br />Li<br />Experiment is reasonably well reproduced<br />
  48. 48. 2. Elastic Properties:Young‘s Modulus<br />Optimal E (45 GPa)<br />around bcc phase <br />boundary (70 at % Mg)<br />bcc <br />Mg is <br />unstable<br />Li dominate alloys <br />are very soft<br />Li<br />Experiment is reasonably well reproduced<br />
  49. 49. 2. Elastic Properties: Poisson‘s Ratio<br />Softer alloys have a higher n<br />Softer alloys have a <br />lower n<br />Li<br />Experiment is reasonably well reproduced<br />
  50. 50. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Mg-Li alloy design<br />1. Motivation<br />2. Elastic properties<br />3. Analysis of the elastic Properties<br />4. Summary<br />
  51. 51. 3. Analysis of the Elastic Properties<br /><ul><li>Defined as
  52. 52. Based on experimental observations
  53. 53. Measure of ductile vs. brittle behavior</li></ul>G Resisting Plastic Flow<br />B Bond Strength<br />Opposition<br />To <br />Fracture<br /><ul><li>1.75 critical value
  54. 54. B/G > 1.75 DUCTILE
  55. 55. B/G < 1.75 BRITTLE</li></ul>1.75 is more a transition zone<br />
  56. 56. 3. Analysis of the Elastic Properties<br />Ductile Region<br />Brittle Region<br />Stiffer bcc Mg-Li alloys Ductile/brittle transition region<br />
  57. 57. 3. Analysis of the Elastic Properties<br /><ul><li>Defined as
  58. 58. Design Criteria
  59. 59. Maximum stiffness for minimum weight
  60. 60. Typical Values (MPa m3/kg)
  61. 61. Graphite Fiber 127.78
  62. 62. Graphite Fiber/epoxy 43.53
  63. 63. Steel 26.41
  64. 64. Aluminum 25.93
  65. 65. PET (polymer) 2.15
  66. 66. Lead 1.41</li></li></ul><li>3. Analysis of the Elastic Properties<br />Better than Al-Mg. Comparable to Al-Li.<br />
  67. 67. 3. Analysis of the Elastic Properties<br />
  68. 68. Max-Planck-Institut für Eisenforschung, Düsseldorf<br />Mg-Li alloy design<br />1. Motivation<br />2. Elastic properties<br />3. Analysis of the elastic Properties<br />4. Summary<br />
  69. 69. 4. Summary<br />DFT and homogenization schemes can be used to predict with reasonable accuracy elastic properties of polycrystalline metals<br />Optimal elastic properties of bcc MgLi alloys are observed around 70 at. % Mg<br />B/G for the optimal bcc Mg-Li alloys is in the brittle/ductile transition region<br />BCC MgLi has a better E/r than AlMg and a comparable E/r to Al-Li<br />BCC MgLi HAS POTENTIAL AS AN ULTRA-LIGHT <br />WEIGHT STRUCTURAL ALLOY<br />
  70. 70. Conclusions<br />+ Understanding trends (thermodynamics, mechanics)<br />+ Direct use of homogenization theory (elastic)<br />+ Extract engineering quantities for a rough but quick estimation<br />+ Get quantities that you cannot get elsewhere<br />- 0 K<br />- supercell size<br />- long calculation times<br />

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