2 D Crystal Plasticity Models for Polycrystal Behavior

2 D Crystal Plasticity

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 (2 D) idealization

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

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

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

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

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

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 12 C 5 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

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

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

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

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

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 texture @ finite strain

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

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

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

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

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

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

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

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

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. )

Approximate Polycrystal Models Constrained Hybrid Stiffness Aggregate Stiffness

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

Approximate Polycrystal Models Yield Loci • Texture • Upper Bound Yield Surface • Lower Bound Yield Surface

FCC Aggregate Yield Loci

Rate Dependence Scales Flow Surface and Rounds Yield Locus Vertices