Models for Polycrystalline Aggregate Behavior by pptfiles

VIEWS: 5 PAGES: 27

									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 Representation
Pole Figures

• A crystal              • In general, more
  orientation is           than one pole
  represented by its       figure is necessary
  location on a unit       to reconstruct the
  sphere projected         orientation field
  onto a unit circle
Approximate Polycrystal Models
Provide Bounds on Polycrystal Behavior


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

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

								
To top