1. 2D Crystal Plasticity

2. Models for Polycrystal Behavior
Development of crystallographic texture
Evolution of preferred crystal orientations in initially statistically isotropic aggregates
Anisotropy in
Macroscopic yield loci and
Macroscopic stress-strain response
Upper and lower bound models
Planar (2D) idealization

3. Texture Representation Pole Figures
A crystal orientation is represented by its location on a unit sphere projected onto a unit circle
In general, more than one pole figure is necessary to reconstruct the orientation field

4. Approximate Polycrystal Models Provide Bounds on Polycrystal Behavior
Lower Bounds
(a) Linear
(c) 3D PX
Sachs
Upper Bounds
(b) Parallel
(d) 3D PXL
Taylor

5. Approximate Polycrystal Models Upper & Lower Bound Analyses
deformation
stress
Taylor Hybrid Sachs

6. Approximate Polycrystal Models G. I. Taylor (1938)
Every crystal within a polycrystalline aggregate subjected to a macroscopically uniform deformation is considered to sustain the same strain as in the bulk material, i.e. identical deformation rates
Every crystal within a polycrystalline aggregate subjected to a macroscopically uniform deformation is considered to sustain the same rigid body rotation relative to the reference frame, i.e. identical macroscopic spins

7. Approximate Polycrystal Models G. I. Taylor (1938)
Requires a minimum of 5 independent slip systems per crystal
Reasonable for high symmetry crystals with a significantly larger number of available slip systems than are geometrically necessary
Non-iterative
Crystal responses de-couple and may be solved independently of one another, i.e. algorithm can loop over the motion, then the crystal count

8. Approximate Polycrystal Models G. I. Taylor (1938)
At least 5 slip systems must be active in a grain to accommodate the 5 independent components of strain
There are 12C5 or 792 combinations of 12 things taken 5 at a time (but not all independent)
After ruling out kinematically dependent combinations, there remain 384 independent sets of 5 slip rates, each corresponding to a nonsingular system of 5 equations that determines the 5 slip rates
From combinatorics standard notation nCk = n!/{(n-k)!k!}

9. Approximate Polycrystal Models G. I. Taylor (1938)
“Of all the possible combinations of the 12 shears which could produce an assigned strain, only that combination is operative for which the sum of the absolute values of shears is least”
Each crystal will slip on that combination of systems that makes the LEAST AMOUNT of incremental work

10. Approximate Polycrystal Models G. I. Taylor (1938)
For rate dependent crystal response
Generally, fully 12 systems are potentially active
As , while 12 systems are mathematically viable, only 5 are effectively activated, i.e. the rate insensitive limit is reached

11. Approximate Polycrystal Models G. I. Taylor (1938)
Obtain the average crystal stress response for the polycrystal
Update the individual crystal orientations

12. Approximate Polycrystal Models G. I. Taylor (1938)
Consider an axisymmetric tensile specimen
Incremental Work
A scalar measure of the aggregate yield strength is given by the so-called Taylor factor

13. Approximate Polycrystal Models G. I. Taylor (1938)
For an initially uniform distribution of crystals, M = 3.06
M is a measure of how favorably oriented the aggregate is for slip
M will evolve with finite strain

14. Approximate Polycrystal Models G. I. Taylor (1938)
The Taylor Factor is purely a function of orientation

15. Approximate Polycrystal Models G. I. Taylor (1938)
The aggregate would tend to attain a state in which the crystal axes of grains have either a (111) or a (100) axis aligning with the direction of extension

16. Approximate Polycrystal Models G. I. Taylor (1938)
The hypothesis ensures intergranular compatibility at the expense of intergranular equilibrium
The response represents an UPPER BOUND on the averaged stress in the polycrystalline aggregate
The resultant texturing of the aggregate is, in general, too strong in comparison with experimentally observed textured polycrystals

17. Approximate Polycrystal Models Upper Bound Stiffness
Aggregate stiffness is the average of the single crystal stiffness tensors

18. Approximate Polycrystal Models G. Sachs (1928)
Every crystal within a polycrystalline aggregate subjected to a macroscopically uniform stress field is considered to sustain the same stress as in the bulk material, i.e. identical stress in all crystals
Every crystal within a polycrystalline aggregate subjected to a macroscopically uniform deformation is considered to sustain the same rigid body rotation relative to the reference frame, i.e. identical macroscopic spins

19. Approximate Polycrystal Models G. Sachs (1928)
The hypothesis ensures intergranular equilibrium at the expense of intergranular compatibility
The response represents a LOWER BOUND on the averaged stress in the polycrystalline aggregate
The more favorably oriented crystals slip at the expense of those for whom the crystal stress would exceed the macroscopically applied effective value, i.e. the crystal strain rates in the aggregate can exhibit large inhomogeneities

20. Approximate Polycrystal Models G. Sachs (1928)
Applicable to systems with low crystal symmetry, i.e. fewer than 5 independent slip systems per crystal
Many mineralogical materials,rocks and highly constrained crystal classes such as HCP fall in this category
Non-iterative
Crystal responses de-couple and may be solved independently of one another, i.e. algorithm can loop over the motion, then the crystal count

21. Approximate Polycrystal Models Lower Bound Stiffness
Aggregate stiffness is the inverse of the average of the single crystal compliance tensors

22. Approximate Polycrystal Models Hybrid Models
Just as implied, these models impose certain stress components at the crystal level equal their respective macroscopic values while the remaining components of the motion equal the macroscopic values
Examples
Constrained Hybrid Model (Parks & Ahzi)
Relaxed Constraints Model (Kocks et al.)
Self-Consistent Methods (Tome, Molinari, Canova, et al.)

23. Approximate Polycrystal Models Constrained Hybrid Stiffness
Aggregate Stiffness

24. Double Slip Occurs in Specially Oriented FCC Crystals (Goss Texture)
Moderate Texture
Sharp Texture

25. Approximate Polycrystal Models Yield Loci
Texture
Upper Bound Yield Surface
Lower Bound Yield Surface

26. FCC Aggregate Yield Loci

27. Rate Dependence Scales Flow Surface and Rounds Yield Locus Vertices