1. Meso-Scale Simulation and Measurement of Dislocation/Grain Boundary Interactions AFOSR Grant Number: FA9550-05-0068, 0088 Robert H. Wagoner, PI, Myoung-Gyu Lee, Hojun Lim Department of Materials Science and Engineering Ohio State University B. L. Adams, PI, Colin Landon, Josh Kacher Department of Mechanical Engineering Brigham Young University

2. (Lee, Wagoner, OSU) Fundamental Role of Grain Boundaries: Meso-scale Simulation and Measurement Tensile Specimen Grain Orientations Single crystal properties Two-scale Simulation Grain Scale Dislocation Scale Predictions (Lee, Wagoner, OSU) Verification (Lee, Wagoner, OSU) (Homer, Adams, BYU) Slip Activity Stress Choice of Materials Stress/strain, Hall-Petch relations (Lim, Wagoner, OSU)

3. Fundamental Role of Grain Boundaries: Meso-scale Simulation and Measurement 3-D Curvature Recovery via Oblique Double Sectioning (ODS) Opacity Limitations on Curvature Recovery Verification of ODS Recovery True (Left), Recovered (Right) Verification of Experimental Resolution x1x1 x2x2 x3x3 gcgc gbgb gaga 3-D only (Adams, Homer, Lemmon, and Landon, BYU)

4. Summary of Results as of AFOSR Review, Nov. 3, 2006

5. Simulation

6. 6 Procedure Two-Scale ModelInput (OIM) Tensile Specimen Grain Orientations Slip Activity Lattice Curvature BYU AFOSR FA9550-05-1-0088 Predictions Single-Crystal Properties OSU AFOSR FA9550-05-0068 Superdislocations at the center of elements Generalized pileup configuration Grain Scale Dislocation Scale

7. 7 Numerical Tests of Simulation Procedure 1-D Pileup 2-D Pileup CPU: < 2min. (2.8 GHz PC) Mesh independent Reproduce analytical solutions Numerically stable Dislocation pileup with Superdislocation concept

8. 8 Constitutive Equations: SCCE-T Slip activities (Asaro & Needleman, 1985) Hardening of slip systems (Peirce et al., 1982) SCCE-T: Single Crystal Constitutive Equations - Texture Arbitrary parameters: ≥ 6 (m, h ii, h ij, h 1 , h 2 , h 3  

9. 9 Constitutive Equations: SCCE-D Slip activities (Asaro & Needleman, 1985) Hardening of slip systems SCCE-D: Single Crystal Constitutive Equations - Dislocation Arbitrary parameters: 4 (m,  0, k a, k b  

10. 10 SCCE-D: Orowan hardening model Forest dislocation Active (moving) dislocation Slip plane  l Effective forest dislocation density H  Orowan model [ E. Orowan, 1948]    n()n()

11. 11 Results: Constitutive Equations Iron (BCC) Copper (FCC) SCCE-T vs. SCCE-D

12. Measurement

13. 13 Characterization of complete curvature tensor  κ - 6 of 9 lattice curvatures  1 of 2 boundary inclination parameters, Full orientation characterization, g 3-D Curvature (Under Development) 2-D Curvature (Currently Available) gcgc gbgb Lattice Curvature gaga 3-D only gaga gbgb gcgc x1x1 x2x2 x3x3 x1x1 x2x2 x3x3  κ - All 9 lattice curvatures  Full boundary inclination description,  Full orientation characterization, g

14. 14 Experimental resolution limits for lattice curvature Exp. data.5/dx

15. 15 Oblique Double-Sectioning –Combination of serial sectioning and stereology –2 parallel section-cuts for direct measurement of grain boundary character

16. 16 Registry and interpolation Alignment of layers –Reference marks –Grains –Triple-Junction Distribution Interpolation of boundaries to obtain GBCD –Meshing Algorithm

17. 17 Application: Fe-3%Si Multi-crystal Input (OIM) Verification Lattice Curvatures 0 0.02 Simulated (CPU=7h) Measured (rad/  m) (BYU)

18. Results since AFOSR Review, Nov. 3, 2006

19. Simulations

20. 20 Parametric Tests: Bi-crystal BA Simple bi-crystal structure Iron single crystal properties Dislocation=mobile + immobile Only mobile density can be piled up near the grain boundary Apply grain boundary strength  A B  45 o Mat.+Mech. (  *=5  y )  total =  app +  defect F super =  total · (b x  ) Force on Superdislocation where Obstacle force F obs =  = n  Y ·A Slip transmission F super ≥ F obs Slip transmission

21. 21 Parametric Tests: Dislocation Density Dislocation density (1/mm 2 )  = 1%  = 5%  = 10% Von Mises Stress at 10% strain Dislocation density on various slip systems Total dislocation density at different strain levels

22. 22 Parametric Tests: Size dependence Stress vs. grain size (d)Stress vs. grain size -1/2 (d -1/2 ) Constant grain boundary strength: 5*300 MPa Different grain sizes with same grain configuration  *=5  y =1,500 MPa

23. 23 Parametric Tests Crystal orientation (grain boundary  = 45°) Crystal Orientation Simulation Method Mech.Mech.+Mat. Infinite τ* Mech.+Mat. Finite τ* 0°0°377.92409.76403.33 15°377.58410.14404.40 30°378.82413.77408.12 45°379.62415.91409.90 Δσ2.046.155.57 Grain boundary orientation (crystal  = 45°) Grain boundary orientation Simulation Method Mech.Mech.+Mat. Infinite τ* Mech.+Mat. Finite τ* 0°0°380.23408.63399.69 15°380.01414.42406.80 30°379.20414.47407.25 45°379.62415.91409.90 Δσ1.037.2810.21 Eng.Stress at 10% strain (MPa)

24. Experiments

25. 25 Measuring shifts to 1/20 of a pixel increase resolution of rotation by at least a factor of ten The correlation based method is also sensitive to lattice strains Cross Correlation Technique: Promising New Method Ref: Angus Wilkinson (Oxford University)

26. 26 Cross Correlation Technique Reference Image Comparison image at adjacent scan point A region in the reference image is placed over the comparison image and progressively shifted. The correlation intensity is recorded and forms the correlation image.

27. 27 Cross Correlation Technique: Correlation image The peak intensity in the correlation image shows the x and y shift of the image to the pixel level. The center of the image correlates to a zero shift. Shifts can be measured to 1/20 of a pixel using a surface fitting scheme and the intensities. xx yy

28. 28 Cross Correlation Technique: Algorithm This results in a system of 2 independent equations for each region of interest with 8 unknowns

29. 29 Using the deformation gradient tensor you can find the strain and rotation gradients Cross Correlation Technique: Algorithm

30. 30 After analyzing a line scan any component of the strain or rotation gradient tensors can be displayed Components of Rotation Point Number Rotation (Rad) Cross Correlation Technique: Line scan

31. 31 An area scan can be analyzed to show the variation of any component of the strain or rotation tensor. The x and y axis indicate the position in the scan (This example was a 4 point x 4 point grid) Strain in the 1 1 direction Cross Correlation Technique: Area Scan

32. Plans: 2007 (or 2008?)

33. 33 2007 Plans Incorporate slip transmission criteria, determine physical  * (many more specimens) Ratio (  ) between mobile/immobile dislocation density  f (dislocation density), current model:  constant Improving cross section technique

34. 34 2007 work : New material-Minimum Alloy Steel C0.001Cr0.014 Mn0.13Mo0.003 P0.006Sn0.002 S0.005Al0.038 Si0.004Ti0.001 Cu0.023N0.003 Ni0.007Nb0.001 K 11 Measured Lattice Curvature Initial Grain Orientations Composition 1)High Hall-Petch Slopes 2)Good Ductility / Hardening 3)Grain Size 4)Good OIM imaging/polishing Desirable Material Characteristics Choice: Minimum Alloy Steel Stress- Strain CurvesHall-Petch Slopes Grain size attained (OSU) : 80  m ~ 1500  m

35. 35 2007 work: New specimen/OIM Total dislocation density (simulated) *Grain Boundary at 5°

36. Plans: 2008~2011

37. 37 2008-2011 Plans Recover elastic strain gradient by cross correlation (+ adaptive OIM) Develop high resolution OIM technique, couple with new adaptive OIM Parallel mesh refinement at grain boundaries and triple junctions (FEA, OIM) Parallelize Mech.+Mat. Simulation (Suitable for many grains) Grain boundary transmission criteria and Hall-Petch slopes for wide range of grain sizes  *=f (slip transmission), current model:  *=constant. Compare H-P slope: simulation, measurement (Use range of real grain size) Extend to HCP materials

38. 38 2007 work : Grain boundary transmission Curvature plot with infinite GB Exp. curvature Obstacle strength with slip transmissivity

39. 39 2007~ work : Adaptive or Other Mesh refinement (FEA, OIM) 10  m 5m5m 1m1m 1.4  m

40. Extra Slides

41. 41 Results: Constitutive Equations TCCEDCCE Cu [-112] 59 (177)10 (30) [-123] 64 (410)6 (46) Fe [011] 5 (12)3 (7) [-348] 8 (35)2 (3) Average 34 (159)5 (22) Units: MPa (%) Standard Deviation (Average % errors)

42. 42 Results: Constitutive Equations Iron (BCC) Copper (FCC) Uni-axial Compression of Polycrystals

43. 43 The shift in the EBSD pattern q at a region of interest (ROI) centered at point x with crystallographic direction r is related to the displacement gradient tensor a High Resolution Strain and Rotation Measurement If the region of interest were centered here then r would be a unit vector in the (-1,1,-3) direction

44. 44 Summary I: single Xl constitutive equations Novel two-scale simulation model based on Finite Element Method was developed Superdislocation concept is well validated with analytical pileup solution SCCE-D (4 parameters) fits real single Xl (no gb effects) SCCE-T (≥6 parameters) does not match single Xl (gb effects) Stress-strain response and texture evolution are similar with different single Xl models

45. 45 Summary II: Parametric tests Flow stress –Mech. Simul. < Mech+Mat. Simul. w/ finite boundary strength < Mech.+Mat. Simul. w/ infinite boundary strength High dislocation density is observed near the grain boundary at low strain level High dislocation density increases crystal hardness Dislocation density of grain interior becomes higher as the deformation proceeds due to the high slip activity Hall-Petch relation is observed with two-scale simulation model with finite grain boundary strength Bi-crystal analysis showed that grain boundary orientation is more sensitive to the dislocation pileup than crystal orientation

46. 46 Preliminary result with reasonable distribution: Meso model is CPU efficient (7hr/10 grains) Summary III: Verification

47. 47 Measuring shifts to 1/20 of a pixel increase resolution of rotation by at least a factor of ten The correlation based method is also sensitive to lattice strains Summary VI: High Resolution Strain Measurement