Alloy yield strength modeling with MatCalc · PDF filePage 2 Model overview •...

Alloy yield strength modeling with MatCalc(rel. 6.00.0200)

P. Warczok

Model overview

• Contributions to yield strength, σYS

• Intrinsic strength, σi

• Work hardening, σdisl

• Grain/subgrain boundary strengthening, σgb , σsgb

• Solid solution strengthening, σss

• Precipitation strenghtening, σprec

( )precsssgbgbdisliYS f σσσσσσσ ,,,,,=

Intrinsic strength, σi

Model overview








Work hardening, σdisl

• Taylor equation

ρασ GbAdisl =

- Taylor factor

- Shear stress

- Burger’s vector

- Dislocation density

αGbρ

( )υ−=12EG


• Taylor equation

ρασ GbAdisl =

- Taylor factor

- Shear stress

- Burger’s vector


αGbρ


• Taylor equation

ρασ GbAdisl =

- Taylor factor

- Shear stress

- Burger’s vector


αGbρ


• Taylor equation

• Two parameter model

2211 ραρασ GbAGbAdisl +=

Model overview








Grain/subgrain boundary strengthening, σgb , σsgb

• Hall-Petch equation

- Grain diameter

- Subgrain diameter

- Constant

Dδ

nk

Dkgb

gb =σδ

σ sgbsgb

k=

Model overview








Solid solution strengthening, σss

- Coefficient for element i

- Element i content in the prec. Domain

(mole fraction)

- Exponent for element i

- Exponent for substitutional elements

- Exponent for interstitial elements

- Global exponent

icik

( ) ( )totm

totm

submtotm

subi

m

i

mnii

subi

mniiss ckck

1

int

int

int

+

= ∑∑σ

insubmintmtotm


- Coefficient for element i

- Element i content in the prec. domain

- Exponent for element i

- Exponent for substitutional elements

- Exponent for interstitial elements

- Global exponent

icik

( ) ( )totm

totm

submtotm

subi

m

i

mnii

subi

mniiss ckck

1

int

int

int

+

= ∑∑σ

insubmintmtotm


( ) ( )totm

totm

submtotm

subi

m

i

mnii

subi

mniiss ckck

1

int

int

int

+

= ∑∑σ

Solid solution strengthening, σss• Solid solution strengthening, σss

( ) ( )totm

totm

submtotm

subi

m

i

mnii

subi

mniiss ckck

1

int

int

int

+

= ∑∑σ

Model overview








Precipitation strengthening, σprec• Some general parameters/settings

• Angle between dislocation line and Burger’s vector θ (edge:screw ratio, θ = 0

for pure screw, θ = π/2 for pure edge)


• Equivalent radius, req (describes precipitate-dislocation interference area)

- Precipitate mean radiusmr

meq rr4π

=


• Mean distance between the precipitate surfaces, LS

ss

classclassmclassV

S rrrN

LSS

242

3ln 2

,,

−+=∑π ∑

∑=

classclassmclassV

classclassmclassV

ss rN

rNr

,,

2,,

32

- Precipitate number density within the class

- Precipitate mean radius within the class

classVN ,

classmr ,

Precipitation strengthening, σprec• Derived from τprec

• Non-shearable particles (Orowan mechanism)

• Shearable particles (weak or strong)

• Coherency effect

• Modulus effect

• Anti-phase boundary effect

• Stacking fault effect

• Interfacial effect

Non-shearable particles

−

−−

=υ

θπυ

12

cos12

J

bri 2=

−=

i

eq

snsh r

rL

JGb 2ln

12 υπτ


−

−−

=υ

θπυ

12

cos12

J

meq rr4π

= bri 2=

If req < 2ri , then

( )2ln12 s

nsh LJGb

υπτ

−=

−=

i

eq

snsh r

rL

JGb 2ln

12 υπτ


−

−−

=υ

θπυ

12

cos12

J

meq rr4π

=

−=

i

eq

snsh r

rL

JGb 2ln

12 υπτ

bri 2=





• Modulus effect




Shearable particles• Some general parameters/settings

• Dislocation bending angle, ψ - weak and strong particles

0° ≤ ψ ≤ 120° “strong” particles

120° ≤ ψ ≤ 180° “weak” particles


• Mean distance between the precipitate surfaces, L

ss

classclassmclassV

S rrrN

LSS

242

3ln 2

,,

−+=∑π

∑∑

=

classclassmclassV

classclassmclassV

ss rN

rNr

,,

2,,

32

Strong particles


• Mean distance between the precipitate surfaces, L

Weak particles

21

2cos

−

=ψ

Seff LL


• Dislocation line tension, T

• Simple model

• Advanced model

2

2GbT =

−

−+=

i

Sstrong r

LGbT ln1

sin314

22

υθυυ

π

−

−+=

i

effweak r

LGbT ln1

sin314

22

υθυυ

π


• Dislocation line tension, T

• Simple

• Advanced


• Inner cut-off radius, ri (multiplication of Burger’s vector)

−

−+=

i

Sstrong r

LGbT ln1

sin314

22

υθυυ

π

−

−+=

i

effweak r

LGbT ln1

sin314

22

υθυυ

π





• Modulus effect




Coherency effect• Strain field due to precipitation/matrix misfit

• Strong particles

• Weak particles

( ) 4/1

3

322

,sin1352.2cos2

+=

brGT

Lmstrong

Sstrongcoh

εθθτ

( ) 2/133322

,sin1127.4cos3416.1

+=

weak

eq

Sweakcoh T

brGL

εθθτ

vollin ∆=∆=92

32ε

- Linear misfit

- Volumetric misfit

lin∆

vol∆

Coherency effect• Shearable particles – Coherency effect

• Strain field due to precipitation/matrix misfit


• Weak particles

( ) 4/1

3

322

,sin1352.2cos2

+=

brGT

Lmstrong

Sstrongcoh

εθθτ

( ) 2/133322

,sin1127.4cos3416.1

+=

weak

m

Sweakcoh T

brGL

εθθτ

vollin ∆=∆=92

32ε

- Linear misfit

- Volumetric misfit

lin∆

vol∆



• Weak particles

( ) 4/1

3

322

,sin1352.2cos2

+=

brGT

Lmstrong

Sstrongcoh

εθθτ

( ) 2/133322

,sin1127.4cos3416.1

+=

weak

m

Sweakcoh T

brGL

εθθτ

vollin ∆=∆=92

32ε

- Linear misfit

- Volumetric misfit

lin∆

vol∆



• Weak particles

( ) 4/1

3

322

,sin1352.2cos2

+=

brGT

Lmstrong

Sstrongcoh

εθθτ

( ) 2/133322

,sin1127.4cos3416.1

+=

weak

eq

Sweakcoh T

brGL

εθθτ





• Modulus effect




Modulus effect• Elastic properties of precipitate and matrix differ dislocation

energy inside and outside the precipitate differ

• 2 models

• Nembach

• Siems

• Nembach model


• Weak particles

Modulus effect

Sstrong bL

Fmodmod, =τ

2/3

modmod, 2

2

=

weakS

weakweak T

FbLTτ

- Particle shear moduluspG

85.02

mod 05.0

−=

br

bGGF eqP

Modulus effect• Siems model

2/12

mod, 12

8.0

−=

EE

bLT p

S

strongstrongτ

- Particle Poisson ratiopυ

( ) ( )

( )i

Sp

eq

Sp

i

eqP

p

rLG

rLG

rr

G

EE

log1

log1log1

υ

υυ

−

−+−=

4/32

mod, 12

−=

EE

bLT p

S

weakweakτ

( ) ( )

( )i

effp

eq

effp

i

eqP

p

rL

G

rL

Grr

G

EE

log1

log1log1

υ

υυ

−

−+−=

Strong Weak

• Nembach model


• Weak particles

Modulus effect

Sstrong bL

Fmodmod, =τ

2/3

modmod, 2

2

=

weakS

weakweak T

FbLTτ

- Particle shear moduluspG

85.02

mod 05.0

−=

br

bGGF eqP

- Particle shear moduluspG- Particle Poisson ratiopυ

Modulus effect• Elastic properties of precipitate and matrix differ dislocation

energy inside and outside the precipitate differ

• 2 models

• Nembach

• Siems

strongmod,τ

weakmod,τ





• Modulus effect




Anti-phase boundary (APB) effect• Dislocation passing through ordered precipitate increases the energy

by creating the APB


• Weak particles

2/1

, 3869.0

= APBeqstrong

SstrongAPB

rwTbL

γτ

−

=

S

eqAPB

weak

APBeqweak

SweakAPB L

rT

rT

sbL πβγγ

τ3

1622 22/3

,

- APB energy

- Interaction parameter between

the leading and trailing dislocation

- Number of pair dislocations

APBγ

,w β

s


by creating the APB


• Weak particles

2/1

, 3869.0

= APBeqstrong

SstrongAPB

rwTbL

γτ

−

=

S

eqAPB

weak

APBeqweak

SweakAPB L

rT

rT

sbL πβγγ

τ3

1622 22/3

,

- APB energy

- Interaction parameter between

the leading and trailing dislocation

- Number of pair dislocations

APBγ

,w β

s

2/1

, 3869.0

= APBeqstrong

SstrongAPB

rwTbL

γτ

−

=

S

eqAPB

weak

APBeqweak

SweakAPB L

rT

rT

sbL πβγγ

τ3

1622 22/3

,


by creating the APB


• Weak particles

2/1

, 3869.0

= APBeqstrong

SstrongAPB

rwTbL

γτ

−

=

S

eqAPB

weak

APBeqweak

SweakAPB L

rT

rT

sbL πβγγ

τ3

1622 22/3

,





• Modulus effect




Stacking fault (SF) effect• Passing dislocation creates a stacking fault – energy difference

between the SF in the precipitate and matrix

( ) 42 2effeqeffSFPSFMSF WrWF −−= γγ

SFPSFM

SFeff

KWγγ +

=2

( )( )( )υπ

θυυ−

−−=

182cos222

pSF

GbK

- Burger’s vector of particle

- Stacking fault energy of particle

- Stacking fault energy of matrix

SFPγ

SFMγ

pb




• Weak particles

S

SFstrongSF bL

F=,τ

2/3

, 22

=

weak

SF

S

weakweakSF T

FbLTτ ( ) 42 2

effeqeffSFPSFMSF WrWF −−= γγ

SFPSFM

SFeff

KWγγ +

=2

( )( )( )υπ

θυυ−

−−=

182cos222

pSF

GbK



( ) 42 2effeqeffSFPSFMSF WrWF −−= γγ

SFPSFM

SFeff

KWγγ +

=2

( )( )( )υπ

θυυ−

−−=

182cos222

pSF

GbK

- Burger’s vector of particle

- Stacking fault energy of particle

- Stacking fault energy of matrix

SFPγ

SFMγ

pb




• Weak particles

S

SFstrongSF bL

F=,τ

2/3

, 22

=

weak

SF

S

weakweakSF T

FbLTτ





• Modulus effect




Interfacial effect• Passing dislocation increases the area of precipitate/matrix interface


• Weak particles

bF PMγ2int =

- Precipitate/matrix interface energyPMγ

Sstrong bL

Fintint, =τ

2/3

intint, 2

2

=

weakS

weakweak T

FbLTτ



• Weak particles

bF PMγ2int =Sstrong bL

Fintint, =τ

2/3

intint, 2

2

=

weakS

weakweak T

FbLTτ

- Precipitate/matrix interface energyPMγ



• Weak particles

Sstrong bL

Fintint, =τ

2/3

intint, 2

2

=

weakS

weakweak T

FbLTτ

Precipitation strengthening, σprec• Evaluation

• Identifying the regime for each precipitate separately

• Summation of the calculated τregime of each precipitate

• Conversion of τprec to σprec



• Summation of the calculated τi,regime of each precipitate


Regime of each precipitate• For every precipitate phase i: Values of τ evaluated for each of three

regimes (Non-shearable, shearable weak, shearable strong)

( ) shshshshshshmm

strongim

strongSFim

strongAPBim

strongim

strongcoheristrongi/1

int,,,,,,mod,,,,, ττττττ ++++=

( ) shshshshshshmm

weakim

weakSFim

weakAPBim

weakim

weakcoheriweaki/1


nshi,τ



( ) shshshshshshmm

strongim

strongSFim

strongAPBim

strongim



( ) shshshshshshmm

weakim

weakSFim

weakAPBim

weakim

weakcoheriweaki/1


Orowani,τ



( ) shshshshshshmm

strongim

strongSFim

strongAPBim

strongim



( ) shshshshshshmm

weakim

weakSFim

weakAPBim

weakim

weakcoheriweaki/1


nshi,τ

Regime of each precipitate• τi,regime with the lowest value for the precipitate phase i is taken for

the further operations

( ) shshshshshshmm

strongim

strongSFim

strongAPBim

strongim



( ) shshshshshshmm

weakim

weakSFim

weakAPBim

weakim

weakcoheriweaki/1


nshi,τ

If coherency radius ≠ 0 and the mean radius of precipitate is greater than

the coherency radius the non-shearable regime is taken for further

calculation

Summation of τi,regime

sumsum

nshnsh

sum

shsh

mm

m

i

mnshi

m

m

i

mshiprec

11

,

1

,

+

= ∑∑ τττ

-Shearable particles contribution (weak and strong regime)

-Non-shearable particles contribution

shi,τ

nshi,τ


sumnsh

sum

nshsh

sum

sh

mmm

i

mnshi

mm

i

mshiprec

1

,,

+

= ∑∑ τττ

sumsum

nshnsh

sum

shsh

mm

m

i

mnshi

m

m

i

mshiprec

11

,

1

,

+

= ∑∑ τττ

• Precipitation strengthening, σprec (derived from τprec)

• Evaluation of τprec






precprec ατσ =

- Taylor factorα

Total yield strength, σYS• Summation of contributions

( )[ ] sigsigsigmm

precsssgbgbimdislYS

/1σσσσσσσ +++++=

Thank you for your attention !

Shape factor influence on LS

Precipitation hardening

4/126/1

32

−

+=

hhK

- Shape factorh

Sonderegger B., Kozeschnik E., Scripta Mater., 66 (2012) 52-55

−+=

∑ ss

classclassmclassV

S rrrN

KLSS

242

3ln 2

,,π

Precipitation hardening• Non-shearable particles (Orowan mechanism)

−=

i

eq

sOrowan r

rL

JGb 2ln

12 υπτ ( )nsscreweqscrewnsedgeeqedgeeq rPrPr ,,,, +=

mnsedgeeq rhhh

hr42

332

33

2222

32

,,π

+++

+=

mnsscreweq rhhh

hr42

9113

222

32

,,π

+++=

- Equivalent radius for edge disl.

- Equivalent radius for screw disl.

- Fraction of edge disl.

- Fraction of screw disl.

nsedgeeqr ,,

nsscreweqr ,,

edgeP

screwP

Precipitation hardening• Shearable particles – (e.g. coherency effect)

( )shscreweqscrewshedgeeqedgeeq rPrPr ,,,, +=

- Equivalent radius for edge disl.

- Equivalent radius for screw disl.

shedgeeqr ,,

shscreweqr ,,

hT

brGL

f

weak

eq

Sweakcoh

2/1333

, 27)(

=

εθτ

mshedgeeq rhh

hr451

622

33 22

32

,,π

++

+=

mshscreweq rhh

hr41

2213 2

32

,,π

++=

Precipitation hardening• Shearable particles – Coherency effect for non-spherical particles


• Weak particles

( ) Kb

rGTL

mstrong

Sstrongcoh

4/1

3

322

,sin1488.2cos1101.1

+=

εθθτ

( ) hT

brGL weak

eq

Sweakcoh

2/133322

, 27sin4736.3cos7310.2

+=

εθθτ ,if

4/126/1

32

−

+=

hhK

( )( )θθ

θθ22

22

sin4736.3cos7310.2sin1127.4cos3416.1

++

≤h

Summing up…• Total yield strength, σYS

• Summation of contributions

• Influence of strain rate,

( )[ ] sigsigsigmm

precsssgbgbimdislYS

/1σσσσσσσ +++++=

( )[ ]2

1

/11

Cmm

precsssgbgbimdislYS C

sigsigsig

++++++=ϕσσσσσσσ

ϕ

Alloy yield strength modeling with MatCalc · PDF filePage 2 Model overview •...

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Transcript of Alloy yield strength modeling with MatCalc · PDF filePage 2 Model overview •...