2D Crystal Plasticity. Models for Polycrystal Behavior Development of crystallographic texture...

2D 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 (2D) idealization

Texture RepresentationPole 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 ModelsProvide Bounds on Polycrystal Behavior

• Lower Bounds– (a) Linear– (c) 3D PX– Sachs

• Upper Bounds– (b) Parallel– (d) 3D PXL– Taylor

Approximate Polycrystal ModelsUpper & Lower Bound Analyses

Taylor Hybrid Sachs

deformation

stress

Approximate Polycrystal ModelsG. 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


• 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


• 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


•“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


• 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

g

g g

o

N

g

N 0


• Obtain the average crystal stress response for the polycrystal

• Update the individual crystal orientations

T T g T g

g

ggA

gggTgg RPskwWRRRW

1


• Consider an axisymmetric tensile specimen

• Incremental Work

• A scalar measure of the aggregate yield strength is given by the so-called Taylor factor

dW Td gd min d

M T

min d d

g g,


• 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


• The Taylor Factor is purely a function of orientation


• 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


• 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 ModelsUpper Bound Stiffness

• Aggregate stiffness is the average of the single crystal stiffness tensors

T g Mg 1Dg

T g Mg 1Dg

T Mg 1D

Approximate Polycrystal ModelsG. 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


• 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


• 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 ModelsLower Bound Stiffness

• Aggregate stiffness is the inverse of the average of the single crystal compliance tensors

Dg MgT g

Dg MgT g

D Mg T

T Mg 1D

Approximate Polycrystal ModelsHybrid 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 ModelsConstrained Hybrid Stiffness

g I 32 (c)(c)

(c)c c 13 I

Dg g g 1D

T g 1Mg 1 g g 1

D

Aggregate Stiffness

Double Slip Occurs in Specially OrientedFCC Crystals (Goss Texture)

Moderate Texture Sharp Texture

Approximate Polycrystal ModelsYield Loci

• Texture

• Upper Bound Yield Surface

• Lower BoundYield Surface

FCC Aggregate Yield Loci

Rate Dependence Scales Flow Surface and Rounds Yield Locus Vertices

2D Crystal Plasticity. Models for Polycrystal Behavior Development of crystallographic texture...

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