Crystallographic Texture and Plastic Anisotropy

  • H. J. Bunge
Part of the Engineering Materials book series (ENG.MAT.)

Abstract

The structure of crystalline materials can be characterized by four structure levels:
  1. 1.

    Crystal Structure specifies the kind and position of atoms in the unit cell of the ideal crystal lattice.

     
  2. 2.

    Phase Structure specifies the sizes, shapes and mutual arrangement of single-phase volumes (volumes with constant crystal structure).

     
  3. 3.

    Grain Structure specifies the sizes, shapes, crystal lattice orientation, and mutual arrangement of monocrystal volumes (within the single-phase volumes).

     
  4. 4.

    Substructure specifies the kind, amount, arrangement, crystallographic orientation of all lattice defects, i.e. all deviations from the ideal crystal lattice such as point defects, dislocations, stacking faults, grain and phase boundaries, the surface, elastic strain, magnetization, electric polarization.

     

Keywords

Crystallization Anisotropy Chromium Torque Martensite 

List of Special Symbols

an

Relative shear stress of the glide system n

ant

Antisymmetric part of a tensor

bcc

Body centered cubic structure

Cλμν

Series expansion coefficients of the orientation distribution function f(g) (texture coefficients)

D(x)

Substructure function

d, di

Glide direction

dg

Orientation element (volume element of orientation space)

du

Displacement vector (after the deformation step dη)

dVg

Volume of the sample in which the crystal orientation is g

dW

Dissipated deformation work (monocrystal)

\(d\tilde W\)

Dissipated deformation work (polycrystal)

Deformation step

Solid angular element

f

Orientation density

f(g)

Orientation distribution function ODF

f′(g)

Orientation distribution function after the average lattice rotation \(\widetilde {\Delta g}\) (after texture spin)

fcc

Face centered cubic structure

Gijn

Glide system tensor

G(x)

Microstructure function

g = {φ1 ø, φ2}

Crystal orientation (Euler angles)

[gij]

Orientation matrix g

g(x)

Orientation-location function (orientation stereology)

gε

Orientation of the principle strain axes with respect to the sample coordinate system KA

gσ

Orientation of the principle stress axes with repect to the sample coordinate system KA

h

Normal direction to the lattice plane (hkl)

(hkl)

Miller indices of a crystal lattice plane

i

Number of the phase

KA

Sample coordinate system

KB

Crystal coordinate system

L, L0

Series expansion degree (series truncation)

M

Taylor factor (monocrystal)

\(\tilde M\)

Polycrystal average of the Taylor factor

M(λ)

Number of independent spherical harmonics of the degree λ (crystal symmetry)

m

Strain-rate sensitivity factor

mλμν

Series expansion coefficients of monocrystal Taylor factor M(g)

N

Number of glide systems

N(λ)

Number of independent spherical harmoncis of the degree λ (sample symmetry)

n,nj

Normal direction to the glide plane

P

Pole density

P(hk1)(αβ)

Pole density distribution function (pole figure)

q

Contraction ratio

qmin

Contraction ratio requiring minimum deformation work

R, Rij

Lattice rotation rate, rotation rate tensor

\(\tilde R\)

Polycrystal average of the rotation rate

r

Rotation axis (parameter of g)

r

Lankford parameter

ri

Numerical coefficients

sym

Symmetrical part of a tensor

Tλμν(g)

Generalized spherical harmonics

V

Total volume

VX

Volume of the polycrystalline subsample at the location X

Vx

Monocrystalline volume element at the location x

X={X1 X2, X3}

Location of a polycrystalline volume element

x={x1, x2, x3}

Location of a monocrystalline volume element

{αβ}

Spherical polar coordinates of a sample direction

αm

Numerical factors characterizing linear independent solutions

β

Angle in the sheet plane towards the rolling direction

γn

Glide rate in the glide system n

Δg

Lattice rotation (after the deformation step dη) (lattice spin)

\(\widetilde {\Delta g}\)

Average lattice rotation after the deformation step dη (texture spin)

ε,εij

Strain, strain tensor

\(\tilde \varepsilon \)

Averaged strain

η

Total deformation degree

σ, σij

Stress, stress tensor

σ1, σ2

Principle stresses

τn

Stress component falling into the glide system n

τ0n

Critical resolved shear stress in the glide system n

τ0

Reference shear stress (“hardness”)

ø

Texture changing rate

ø

Euler angle (parameter of g)

φ1

Euler angle (parameter of g)

φ2

Euler angle (parameter of g)

ω

Rotation angle (parameter of g)

~

Symbol characterizing polycrystal average

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© Springer-Verlag Berlin Heidelberg 2000

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  • H. J. Bunge

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