Robert H. Wagoner, PI, Myoung-Gyu Lee, Hojun Lim Department of Materials Science and Engineering

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

3. x3 x2 x1 3-D only Fundamental Role of Grain Boundaries: Meso-scale Simulation and Measurement (Adams, Homer, Lemmon, and Landon, BYU) ga gb gc Verification of Experimental Resolution 3-D Curvature Recovery via Oblique Double Sectioning (ODS) Verification of ODS Recovery True (Left), Recovered (Right) Opacity Limitations on Curvature Recovery

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

5. Simulation

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

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

8. Constitutive Equations: SCCE-T SCCE-T: Single Crystal Constitutive Equations - Texture Slip activities (Asaro & Needleman, 1985) Hardening of slip systems (Peirce et al., 1982) • Arbitrary parameters: ≥ 6 (m, hii, hij, h1b, h2b, h3b )

9. Constitutive Equations: SCCE-D SCCE-D: Single Crystal Constitutive Equations - Dislocation Slip activities (Asaro & Needleman, 1985) Hardening of slip systems • Arbitrary parameters: 4 (m, r0, ka, kb)

10. l t SCCE-D: Orowan hardening model Forest dislocation n(a) a q Slip plane a Active (moving) dislocation Orowan model [ E. Orowan, 1948] Effective forest dislocation density Hab

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

12. Measurement

13. x3 x2 x1 3-D only Characterization of complete curvature tensor 2-D Curvature (Currently Available) 3-D Curvature (Under Development) ga ga gc x3 gc x2 gb gb x1 Lattice Curvature • κ - 6 of 9 lattice curvatures • 1 of 2 boundary inclination parameters, Full orientation characterization, g • κ - All 9 lattice curvatures • Full boundary inclination description, • Full orientation characterization, g

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

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

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

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

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

19. Simulations

20. B A f Mat.+Mech. (t*=5sy) B A Parametric Tests: Bi-crystal Force on Superdislocation • 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 Slip transmission Fsuper= Stotal · (b x x) Stotal = sapp+sdefect where Fobs=t*A= nsY·A Obstacle force Slip transmission Fsuper ≥ Fobs f= 45o

21. Parametric Tests: Dislocation Density Von Mises Stress at 10% strain e = 10% e = 5% e = 1% Dislocation density (1/mm2) Total dislocation density at different strain levels Dislocation density on various slip systems

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

23. Parametric Tests Eng.Stress at 10% strain (MPa)

24. Experiments

25. Cross Correlation Technique: Promising New Method • 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 Ref: Angus Wilkinson (Oxford University)

26. Reference Image Cross Correlation Technique • 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. 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. y x

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

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

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

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

32. Plans: 2007 (or 2008?)

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

34. 2007 work : New material-Minimum Alloy Steel Desirable Material Characteristics Stress- Strain Curves Hall-Petch Slopes • High Hall-Petch Slopes • Good Ductility / Hardening • Grain Size • Good OIM imaging/polishing Choice: Minimum Alloy Steel Composition K11 Grain size attained (OSU) : 80mm ~ 1500mm Initial Grain Orientations Measured Lattice Curvature

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

36. Plans: 2008~2011

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 t*=f (slip transmission), current model: t*=constant. • Compare H-P slope: simulation, measurement (Use range of real grain size) • Extend to HCP materials

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

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

40. Extra Slides

41. Results: Constitutive Equations Standard Deviation (Average % errors) Units: MPa (%)

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

43. If the region of interest were centered here then r would be a unit vector in the (-1,1,-3) direction High Resolution Strain and Rotation Measurement • 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

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

47. Summary VI: High Resolution Strain Measurement • 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