F. Roters, M. Friák, J. Neugebauer, D. Raabe Multiscale modeling of metal forming considering microstructure and texture: micro to macro F. Roters, M. Friák, J. Neugebauer, D. Raabe Department of Microstructure Physics and Metal Forming 06. April 2009

Polycrystal theory and simulation Small scale crystal plasticity Overview Motivation Polycrystal theory and simulation Small scale crystal plasticity Large scale polycrystal mechanics Quantum mechanics and crystal mechanics

Motivation Overview Products Boundary conditions Complex engineering materials Performance in service Consider microstructure and texture Multiscale models

Polycrystal theory and simulation Small scale crystal plasticity Overview Motivation Polycrystal theory and simulation Small scale crystal plasticity Large scale polycrystal mechanics Quantum mechanics and crystal mechanics

Al Bicrystals, low angle g.b. [112] 7.4°, v Mises strain experiment von Mises strain [1] SSD viscoplastic phenomen. model dislocation- based model; g.b. model 10% 20% 30% 40% 50%

grain1 grain2 grain3 grain4…. ? Polycrystal mechanics: homogenization stress / strain in grain1 grain2 grain3 grain4…. ? ?

. . Single crystal yield surface, Taylor Bishop-Hill slip system 2 slip system 1 imposed stress internal stress total stress 1 crystal, 2 slip systems: 1 crystal, 1 slip system: . T different stresses same strain

. Single crystal yield surface, Taylor Bishop-Hill Many crystals, many slip systems: . imposed strain grain 1 grain 2 grain 3 grain 4 stress in grain 1 stress in grain 2 stress in grain 3 stress in grain 4

Homogeneity and boundary conditions – meso-scale 3% 8% 15% M. Sachtleber, Z. Zhao, D. Raabe: Mater. Sc. Engin. A 336 (2002) 81

Crystal Mechanics FEM (General): full field; direct CPFEM D. Raabe: Adv. Mater. 14 (2002) 639; Acta Mater. 49 (2001) 3433

Crystal mechanics FEM (General): CPFEM & homogenization D. Raabe: Adv. Mater. 14 (2002) 639; Acta Mater. 49 (2001) 3433

Polycrystal theory and simulation Small scale crystal plasticity Overview Motivation Polycrystal theory and simulation Small scale crystal plasticity Large scale polycrystal mechanics Quantum mechanics and crystal mechanics

Nanoindentation (smaller is stronger) Cu, 60° conical, tip radius 1μm, loading rate 1.82mN/s, loads: 4000μN, 6000μN, 8000μN, 10000μN [-110] [11-2] [111] Misorientation angle 0° 20° Hardness and GND* in one experiment Higher GND density at smaller scales responsible ? [11-2] rotations experiment 3D EBSD dislocation-based CPFEM simulation [-110] [111] [11-2] - + [-110] [111] [11-2] * GND: geometrically necessary dislocations (accomodate curvature) Zaafarani, Raabe, Singh, Roters, Zaefferer: Acta Mater. 54 (2006) 1707; Zaafarani, Raabe, Roters, Zaefferer: Acta Mater. 56 (2008) 31

Example: Micro-bending

Crystal Mechanics FEM, grain scale mechanics (2D) Experiment (DIC, EBSD) v Mises strain Simulation (CP-FEM) v Mises strain

exp., grain orientation, side A exp., grain orientation, side B Crystal plasticity FEM, grain scale mechanics (3D) 5mm 5mm exp., grain orientation, side A exp., grain orientation, side B equivalent strain 8mm 21mm 1mm FE mesh Zhao, Rameshwaran, Radovitzky, Cuitino, Roters, Raabe (IJP, 2008)

Crystal plasticity FEM, grain scale mechanics (3D) D. Kumar, T.R. Bieler, P. Eisenlohr, D.E. Mason, M.A. Crimp, F. Roters, D. Raabe: Journal of Engineering and Materials Technology (Transactions of ASME) 130 (2008) 021012-1 - 021012-12 and IJP 2009 in press

Polycrystal theory and simulation Small scale crystal plasticity Overview Motivation Polycrystal theory and simulation Small scale crystal plasticity Large scale polycrystal mechanics Quantum mechanics and crystal mechanics

10 billion grains in an auto part too many crystals

Homogenization and cluster models in CPFEM Raabe, Roters: Intern. J. Plast. 20 (2004) 339; Raabe et al.: Adv. Eng. Mater. 4 (2002) 169; Zhao, Mao, Roters, Raabe: Acta Mater. 52 (2004) 1003

Crystal Plasticity FEM: large scale

Example: crystal plasticity FEM for automotive Numerical Laboratory

Polycrystal theory and simulation Small scale crystal plasticity Overview Motivation Polycrystal theory and simulation Small scale crystal plasticity Large scale polycrystal mechanics Quantum mechanics and crystal mechanics

Ab initio alloy design: Ti alloys for medical application plane wave pseudopotential (VASP) cutoff energy: 170 eV 8×8×8 Monkhorst supercells of 2×2×2 cubic unit cells total of 16 atoms 48 bcc and 28 hcp configurations Hershey homogenization discrete FFT crystal elasticity FEM Approach: DFT*: design elastically soft BCC Ti; understand ground state; obtain single crystal elastic constants Polycrystal coarse graining including texture and anisotropy * DFT: density functional theory

Elastic properties: Ti-Nb system Ti-18.75at.%Nb Ti-25at.%Nb Ti-31.25at.%Nb Az=3.210 Az=2.418 Az=1.058 [001] [100] [010] Young‘s modulus surface plots Pure Nb Az=0.5027 Az= 2 C44/(C11 − C12) Hershey FFT FEM D. Ma, M. Friák, J. Neugebauer, D. Raabe, F. Roters: phys. stat. sol. B 245 (2008) 2642

stress strain Discrete FFTs, stress and strain; different anisotropy Hershey, FEM, FFT similar for random texture Ti-35wt.%Nb-7wt.%Zr-5wt.%Ta: 59.9 GPa (elastic isotropic) strain

a) Advanced characteriation of microstructure b) Multiscale models Summary Simulation of complex materials, products, and processes (boundary condtions) requires a) Advanced characteriation of microstructure b) Multiscale models c) Advanced mechanical testing d) Quantum mechanics for engineering applications