Introduction on Crystal plasticity강의명 : 금속가공학특론 ( A M B 2 0 0 4 )

정영웅

창원대학교 신소재공학부

YJ E O N G @ C H A N G W O N . A C . K R

연구실 : # 5 2 - 2 1 2 전화 : 0 5 5 - 2 1 3 - 3 6 9 4

H O M E PA G E : H T T P : / / YO U N G U N G .G I T H U B . I O

Mathematical treatment on polycrystal behavior

!" =0 0 00 0 00 0 %&''

(#* = !( + ,(#*

#1#2

#3#4

(#- = !( + ,(#-...

Local inhomogeneity should be influenced

by orientation

%. =? 0 00 ? 00 0 1''

.#* = %. + 2.#*

.#- = %. + 2.#-...

Local inhomogeneity should be influenced

by orientation

Some classical models§Sachs (or more correctly static model; often is assumed for Schmid factor estimations)§ !" = 0 regardless of orientation§ Which leads to " = %"§ Then, one could obtain & from constitutive law (Linear Hooke’s law ' = (:*)§ Summing up ∑,-./0

0# ,-./0 ',-./0 = &§ But it turns out & ≠ 3& (compatibility is not satisfied; ⟨"⟩ = %" is naturally satisfied.

§Taylor model§ 6& = 0 regardless of orientation§ Which leads to & = 3&§ Then, one could obtain " from constitutive law (* = ℂ: &)§ Summing up ∑,-./0

0 # ,-./0 ',-./0 = &§ But it turns out & ≠ 3& (compatibility is not satisfied; ⟨"⟩ = %" is naturally satisfied.

Self-consistent approach§Self-consistent approach§ !" ≠ 0 and %& ≠ 0, both of which are usually determined by following

§ the Eshelby approach (!" = (): %&)

§ () = + − - ./: -: 0) where - is the Eshelby tensor

§ Both strain compatibility and force equilibrium are simultaneously satisfied

§Important contributions

oG. I. Taylor (1938)o Upper bound

oSachs (1928) – Static modelo Lower bound

oKröner (1958) – self-consistent scheme for elasticity

oR. Hill’s self-consistent scheme (1965) – elastoplastic

oMolinari et al. (1987) – Anisotropic inclusion embedded in isotropic HEM

oCarlos Tomé and Ricardo Lebensohn (1993) – anisotropic inclusion in anisotropy HEM

Self-consistent estimation of macroscopic propertiesSelf-consistent estimation

!" ≠ 0 and %& ≠ 0, which is usually determined following the Eshelby approach (!" = (): %&)() = + − - ./: -: 0) where - is the Eshelby tensor

At the same time, compatibility and force equilibrium are simultaneously satisfied

Elastic self-consistent estimation& = ℂ: " for various grains in various orientations0& = 2ℂ: 2" for polycrystal consisting of such grains.

Forpolycrystalsconsistingofsinglecrystalinvariousorientations?CanwerelateH of each individual grains to 0H?Can we obtain 0H that satisfies

2" = 0H: 0&which can be a function of H of grains in various orientation while satisfying" = H: & and " = 0H: & , and " = 2", & = 0&?That’s corresponding to finding self-consistent 0H that represents the polycrystal.

Anisotropic elasticity (Hooke-Cauchy law)

ℂ = )./

2ℂ = 0)./

Self-consistent estimation of macroscopic properties

Visco-Plastic self-consistent estimation! = ℂ$%: ( for various grains in various orientations)! = *ℂ$%: *( for polycrystal consisting of such grains.

Self-consistent estimation+, ≠ 0 and /0 ≠ 0, which is usually determined following

the Eshelby approach (+, = 12: /0)12 = 3 − 5 67: 5: )2 where 5 is the Eshelby tensor

At the same time, compatibility and force equilibrium are simultaneously satisfied

anisotropic viscous fluid (Newtonian fluid’s law)

Forpolycrystalsconsistingofsinglecrystalinvariousorientations?

CanwerelatedℂNO of each individual grains to *ℂNO?Can we obtain *ℂ that satisfies

*, = *ℂNO: )0which can be a function of P of grains in various orientation while satisfying, = *ℂNO: 0 and , = *ℂNO: 0 , and , = *,, 0 = )0?That’s corresponding to finding self-consistent *ℂNO that represents the polycrystal.

Viscous plastic deformation should be accommodated by disl. slips

bn

aRSTU←WU

[100]

[010] Crystal axes(CA)

b

nX

→½ X

½ X

+spin

Slip sys. axes (SA)

b

n

ε"#$% = a"(

$%←*%a#+$%←*%ε(+

*% + a"+$%←*%a#(

$%←*%ε+(*%

Crystal deformation by disl. slipViscous plastic deformation should

be accommodated by disl. slips

bn

b

n

Slip sys. axes (SA)

[100]

[010]

a"#$%←*%

Crystal axes (CA)

=- ½ -

½ -+

spin

a"#$%←*% =

b( n( t(b+ n+ t+b1 n1 t1

where 6 = 7×9

ε"#$% = a":

$%←*%a#;$%←*%ε:;

*% 2nd order tensor transformation rule

Among 9 components of ε:;*%, only ε(+*% and ε+(*% non-zero: (k=1,l=2) or (k=2,k=1)

ε"#*% =

012- 0

12- 0 0

0 0 0

ε"#$% = b"n#

12- + n"b#

12- ε"#

$% =12b"n# + b#n" -

ε"#$% = ?"#γ where ?"# =

(+b"n# + b#n"

This is for a single slip system.

ε"#$% = A

B

C DE F;"GFHFIJKF.

?"#B γF ε"#

$% = AB

C DE F;"GFHFIJKF.

?"#B γF

Time derivative form.

Rate-sensitive formula

ε#$%& = (

)

* +, -.#/

-0-123-.

5#$) γ-

γ- 3 ∝ τ9::-

→γ-

γ∘=

τ9::-

τ∘-

=/3

m: SRSτ9::- : Resolved shear stress on slip system s

=@ ⋅ B- ⋅ C-

D∘)

=/3

=@:5)

D∘)

=/3

5-: Schmid tensor

ε#$%& = (

)

* +, -.#/-0-123-.

5#$)@:5)

D∘)

=/3

F = G + I =(

)

5-γ- +(

)

J-γ-

Grain spin rate

J#$- =

1

2b#-n$

- − b$-n#

-

Recall

5#$- =

1

2b#-n$

- + b$-n#

-

@ ⋅ B- ⋅ C- = σ#$n#b$ =1

2σ#$ b#n$ + b$n# =

1

2σ$# b#n$ + b$n#

If @ is symmetric; meaning σ#$ = σ$#

Slip and Lattice rotations

Dislocation slip is a simple shear deformation; which involves a) pure shear, b) spin (lattice rotation)

!" =0 &" 00 0 00 0 0

'( = 12 +( + +( -

.( = 12 +( − +( -

+RBR (BC)+Inclusion spin

Polycrystal aggregate§The polycrystalline aggregate is represented by a statistical population of discrete orientations.

Intragranular fluctuation (inhomogeneity) is discarded in VPSC

- See full-field crystal plasticity models (FFT, FEM …)

(Often) Dummy lines

VPSC input texture file

Orientation notation (B: Bunge), # of Grains

Discrete orientation represents each grain (inclusion): 3 Euler angles followed by weight

Macroscopic properties in VPSC follow from ‘weighted’ average of individual grains

!ε#$ = &'

()))f 'ε#$

' = ⟨ε#$⟩