Grain-size effect in 3D polycrystalline microstructure including texture evolution

1. Grain-size effect in 3D polycrystalline microstructure including texture evolution Bin Wen and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering101 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/

2. Motivations Grain/crystal Macro Meso Twinning Inter-grain slip Grain boundary Mechanical properties of material are extremely essential to the quality of products. Preference on material properties requires efficient modeling and designing in virtual environment.

3. Motivations Adequate description of material properties using appropriate mathematical and physical models Use appropriate model to capture the plastic slip in polycrystals and simulate the mechanical properties of the material. Couple the macro-scale finite element simulation with underlying meso-scale constitutive model.

4. Outlines Constitutive Model based on crystal slip theory Texture evolution of polycrystalline material Grain size effect model Geometric processing techniques Multiscale simulation with homogenization method Conclusions

5. Modeling of realistic 3D polycrystalline microstructure Mesh Mechanical response Load Deformed microstructure Virtual interrogation of microstructure

6. Microstructure constitutive model The meso-scale microstructure is represented with a polycrystal aggregate. Each microstructure contains several grains having different orientations. The initial orientation is assigned randomly and the constitutive model of the crystal is using the rate independent continuum slip theory developed by Anand. Total Lagrangian algorithm is adopted.

7. Microstructure constitutive model Active slip systems is determined by comparing trial shear stress with slip resistance. Where the hardening matrix is The update of the slip resistance is based on shear strain increment

8. Microstructure constitutive model The plastic and elastic deformation gradient can then be updated. Cauchy stress and PK-I stress are also ready to be calculated as well as homogenized equivalent value.

9. Verification of constitutive model Example: a uniaxial compression virtual test of a cubic microstructure.

10. Texture Evolution in a discrete manner The texture is represented by various orientations of grains in microstruture. Properties of polycrystals (such as strength, heat conductivity, etc ) are highly dependent on the texture. Crystals with random texture generally demonstrate isotropic characters while those having preferable texture distribution show anisotropic. The evolution of texture along with microstructure deformation can be tracked by the elastic twist while assuming plastic deformation causes glide on slip planes only. The change of slip system is evaluated as

11. Rodrigues representation and ODF The orientation of a grain is described by a rotation around a specific axis in the real space. This rotation can be conveniently expressed using vector (or point) in 3D Rodrigues space. Considering the symmetries of a crystal (cubic structure for FCC), the Rodrigues space is able to be contracted into a finite fundamental zone. The texture is represented using discrete Orientation Distribution Function (ODF) in that fundamental zone.

12. Texture evolution Initiated with random texture, a microstructure subjected to different deformation form (boundary condition) gives distinct evolution of textures. {110} {111} {110} {111} Initial random texture simple compression, szz=1.0

13. Texture evolution Plane strain in y-z plane, szz=1.0 {110} {111} {110} {111} Simple shear, syz=0.6

14. Grain size effect Resistance to plastic flow in crystals is dominated by dislocation density. The presence and motion of dislocations lead to permanent deformation and strain-hardening and can cause incompatibility in crystals. As the lattice incompatibility can be measured by elastic deformation gradient, it is reasonable to quantifies the incompatibility in Fe (Acharya and Bassani, 2000) A evaluation form of dislocation density is considered as Where slip system lattice incompatiblity is the unique skew symmetric tensor defined by slip normal.

15. Grain size effect Bailey-Hirsch relationship: Differentiate both sides with respect to time, a isotropic single crystal hardening law is obtained Substitute k1 and k2 with initial strain-hardening rate The first term considers hardening by strain, and the second term considers the effect from strain gradient, which is affected majorly by grain size.

16. Examples with different grain size 12x12x12 grid with cubic grain 24x24x24 grid with cubic grain 24x24x24 grid with phase field grain

17. Examples of different grain size 24x24x24 elements 12x12x12 elements 24x24x24 grains 12x12x12 grains 8x8x8 grains 6x6x6 grains 64 grains 12x12x12 grains 6x6x6 grains 4x4x4 grains 64 realistic Stress-strain curves using grain size effect model. All of the microstructures are subjected to compression in z direction and stretch in the other two.

18. Realistic grain 64 grains Domain decomposition Stress field

19. Voronoi Tessellation method and microstructures (a) (b) (c) • Steps: • Sample a set of points; • Calculate the grain boundaries with V.T.; • Generate the grains.

20. Conforming mesh generation Advantages: No restriction on grains Fully adaptive to microstructure geometries Element numbers manageable Simulate the “real” microstructures without assuming unrealistic grain boundaries

21. Conforming mesh generation Conforming grids with 4097 elements Pixel grids with 20×20×20 elements

22. D A C D D C C G I I O O G K G G L M M H N N G M J N J Mesh Generation and Domain Decomposition Mesh the grains Domain decomposition Split into brick elements

23. Conforming grids example Microstructure deformation Mechanical response Equivalent Stress field

24. Multi-scale simulation In material processing, the mechanical response of a work piece is highly interested. To get an reliable prediction of material property in processing, accurate micro-scale constitutive model is needed. Given the previous microstructure constitutive models, a multi-scale simulation can be naturally implemented. The microstructure is coupled with macrostructure through homogenization assumption.

25. Multi-scale homogenization The macro-scale values are obtained from the microstructure in the form of volume average. In this work, Cauchy stress and its derivative with respect to deformation gradient are returned, and the PK stress in macro-continuum is calculated using deformation gradient at that point. where

26. Some examples A cubic macrostructure consisting of 6x6x6 elements is compressed along z direction and stretched in the other two. The orientations of grains in all the microstructures are randomly assigned.

27. Examples Strain field Stress field

28. Examples All microstructures have the same texture. Strain field Stress field