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

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

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**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**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**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**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**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**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 )**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)**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**Results: Constitutive Equations**SCCE-T vs. SCCE-D Copper (FCC) Iron (BCC)**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**Exp. data**.5/dx Experimental resolution limits for lattice curvature**Oblique Double-Sectioning**Combination of serial sectioning and stereology 2 parallel section-cuts for direct measurement of grain boundary character Oblique Double-Sectioning**Alignment of layers**Reference marks Grains Triple-Junction Distribution Interpolation of boundaries to obtain GBCD Meshing Algorithm Registry and interpolation**0.02**0 Application: Fe-3%Si Multi-crystal Input (OIM) Lattice Curvatures Verification Measured (BYU) Simulated (CPU=7h) (rad/mm)**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**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**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)**Parametric Tests**Eng.Stress at 10% strain (MPa)**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)**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.**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**Cross Correlation Technique: Algorithm**• This results in a system of 2 independent equations for each region of interest with 8 unknowns**Cross Correlation Technique: Algorithm**• Using the deformation gradient tensor you can find the strain and rotation gradients**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**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)**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**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**2007 work: New specimen/OIM***Grain Boundary at 5° Total dislocation density (simulated)**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**2007 work : Grain boundary transmission**Curvature plot with infinite GB Exp. curvature Obstacle strength with slip transmissivity**2007~ work : Adaptive or Other Mesh refinement (FEA, OIM)**5mm 1.4mm 1mm 10mm**Results: Constitutive Equations**Standard Deviation (Average % errors) Units: MPa (%)**Results: Constitutive Equations**Uni-axial Compression of Polycrystals Iron (BCC) Copper (FCC)**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**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**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**Summary III: Verification**• Preliminary result with reasonable distribution: • Meso model is CPU efficient (7hr/10 grains)**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