Alloy yield strength modeling with MatCalc · PDF filePage 2 Model overview •...
Transcript of Alloy yield strength modeling with MatCalc · PDF filePage 2 Model overview •...
Alloy yield strength modeling with MatCalc(rel. 6.00.0200)
P. Warczok
Page 2
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 ,,,,,=
Page 3
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 ,,,,,=
Page 4
Intrinsic strength, σi
Page 5
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 ,,,,,=
Page 6
Work hardening, σdisl
• Taylor equation
ρασ GbAdisl =
- Taylor factor
- Shear stress
- Burger’s vector
- Dislocation density
αGbρ
Page 7( )υ−=12EG
Work hardening, σdisl
• Taylor equation
ρασ GbAdisl =
- Taylor factor
- Shear stress
- Burger’s vector
- Dislocation density
αGbρ
Page 8
Work hardening, σdisl
• Taylor equation
ρασ GbAdisl =
- Taylor factor
- Shear stress
- Burger’s vector
- Dislocation density
αGbρ
Page 9
Work hardening, σdisl
• Taylor equation
• Two parameter model
2211 ραρασ GbAGbAdisl +=
Page 10
Work hardening, σdisl
• Taylor equation
• Two parameter model
2211 ραρασ GbAGbAdisl +=
Page 11
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 ,,,,,=
Page 12
Grain/subgrain boundary strengthening, σgb , σsgb
• Hall-Petch equation
- Grain diameter
- Subgrain diameter
- Constant
Dδ
nk
Dkgb
gb =σδ
σ sgbsgb
k=
Page 13
Grain/subgrain boundary strengthening, σgb , σsgb
• Hall-Petch equation
- Grain diameter
- Subgrain diameter
- Constant
Dδ
nk
Dkgb
gb =σδ
σ sgbsgb
k=
Page 14
Grain/subgrain boundary strengthening, σgb , σsgb
• Hall-Petch equation
- Grain diameter
- Subgrain diameter
- Constant
Dδ
nk
Dkgb
gb =σδ
σ sgbsgb
k=
Page 15
Grain/subgrain boundary strengthening, σgb , σsgb
• Hall-Petch equation
- Grain diameter
- Subgrain diameter
- Constant
Dδ
nk
Dkgb
gb =σδ
σ sgbsgb
k=
Page 16
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 ,,,,,=
Page 17
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
Page 18
Solid solution strengthening, σss
- 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
Page 19
Solid solution strengthening, σss
( ) ( )totm
totm
submtotm
subi
m
i
mnii
subi
mniiss ckck
1
int
int
int
+
= ∑∑σ
Page 20
Solid solution strengthening, σss• Solid solution strengthening, σss
( ) ( )totm
totm
submtotm
subi
m
i
mnii
subi
mniiss ckck
1
int
int
int
+
= ∑∑σ
Page 21
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 ,,,,,=
Page 22
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)
Page 23
Precipitation strengthening, σprec• Some general parameters/settings
• Equivalent radius, req (describes precipitate-dislocation interference area)
- Precipitate mean radiusmr
meq rr4π
=
Page 24
Precipitation strengthening, σprec• Some general parameters/settings
• 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 ,
Page 25
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
Page 26
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
Page 27
Non-shearable particles
−
−−
=υ
θπυ
12
cos12
J
bri 2=
−=
i
eq
snsh r
rL
JGb 2ln
12 υπτ
Page 28
Non-shearable particles
−
−−
=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
Page 29
Non-shearable particles
−
−−
=υ
θπυ
12
cos12
J
meq rr4π
=
−=
i
eq
snsh r
rL
JGb 2ln
12 υπτ
bri 2=
Page 30
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
Page 31
Shearable particles• Some general parameters/settings
• Dislocation bending angle, ψ - weak and strong particles
0° ≤ ψ ≤ 120° “strong” particles
120° ≤ ψ ≤ 180° “weak” particles
Page 32
Shearable particles• Some general parameters/settings
• 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
Page 33
Shearable particles• Some general parameters/settings
• Mean distance between the precipitate surfaces, L
Weak particles
21
2cos
−
=ψ
Seff LL
Page 34
Shearable particles• Some general parameters/settings
• Mean distance between the precipitate surfaces, L
Weak particles
21
2cos
−
=ψ
Seff LL
Page 35
Shearable particles• Some general parameters/settings
• 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
Page 36
Shearable particles• Some general parameters/settings
• 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
Page 37
Shearable particles• Some general parameters/settings
• Dislocation line tension, T
• Simple
• Advanced
Page 38
Shearable particles• Some general parameters/settings
• 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
Page 39
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
Page 40
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∆
Page 41
Coherency effect• Shearable particles – 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
m
Sweakcoh T
brGL
εθθτ
vollin ∆=∆=92
32ε
- Linear misfit
- Volumetric misfit
lin∆
vol∆
Page 42
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
m
Sweakcoh T
brGL
εθθτ
vollin ∆=∆=92
32ε
- Linear misfit
- Volumetric misfit
lin∆
vol∆
Page 43
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
Page 44
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
Page 45
Modulus effect• Elastic properties of precipitate and matrix differ dislocation
energy inside and outside the precipitate differ
• 2 models
• Nembach
• Siems
Page 46
Modulus effect• Elastic properties of precipitate and matrix differ dislocation
energy inside and outside the precipitate differ
• 2 models
• Nembach
• Siems
Page 47
• Nembach model
• Strong particles
• 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
Page 48
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
Page 49
• Nembach model
• Strong particles
• 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
Page 50
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
Page 51
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
Page 52
Anti-phase boundary (APB) effect• Dislocation passing through ordered precipitate increases the energy
by creating the APB
• Strong particles
• 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
Page 53
Anti-phase boundary (APB) effect• Dislocation passing through ordered precipitate increases the energy
by creating the APB
• Strong particles
• 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
,
Page 54
Anti-phase boundary (APB) effect• Dislocation passing through ordered precipitate increases the energy
by creating the APB
• Strong particles
• Weak particles
2/1
, 3869.0
= APBeqstrong
SstrongAPB
rwTbL
γτ
−
=
S
eqAPB
weak
APBeqweak
SweakAPB L
rT
rT
sbL πβγγ
τ3
1622 22/3
,
Page 55
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
Page 56
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
Page 57
Stacking fault (SF) effect• Passing dislocation creates a stacking fault – energy difference
between the SF in the precipitate and matrix
• Strong particles
• 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
Page 58
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
Page 59
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
Page 60
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
Page 61
Stacking fault (SF) effect• Passing dislocation creates a stacking fault – energy difference
between the SF in the precipitate and matrix
• Strong particles
• Weak particles
S
SFstrongSF bL
F=,τ
2/3
, 22
=
weak
SF
S
weakweakSF T
FbLTτ
Page 62
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
Page 63
Interfacial effect• Passing dislocation increases the area of precipitate/matrix interface
• Strong particles
• Weak particles
bF PMγ2int =
- Precipitate/matrix interface energyPMγ
Sstrong bL
Fintint, =τ
2/3
intint, 2
2
=
weakS
weakweak T
FbLTτ
Page 64
Interfacial effect• Passing dislocation increases the area of precipitate/matrix interface
• Strong particles
• Weak particles
bF PMγ2int =Sstrong bL
Fintint, =τ
2/3
intint, 2
2
=
weakS
weakweak T
FbLTτ
- Precipitate/matrix interface energyPMγ
Page 65
Interfacial effect• Passing dislocation increases the area of precipitate/matrix interface
• Strong particles
• Weak particles
Sstrong bL
Fintint, =τ
2/3
intint, 2
2
=
weakS
weakweak T
FbLTτ
Page 66
Precipitation strengthening, σprec• Evaluation
• Identifying the regime for each precipitate separately
• Summation of the calculated τregime of each precipitate
• Conversion of τprec to σprec
Page 67
Precipitation strengthening, σprec• Evaluation
• Identifying the regime for each precipitate separately
• Summation of the calculated τi,regime of each precipitate
• Conversion of τprec to σprec
Page 68
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
int,,,,,,mod,,,,, ττττττ ++++=
nshi,τ
Page 69
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
int,,,,,,mod,,,,, ττττττ ++++=
Orowani,τ
Page 70
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
int,,,,,,mod,,,,, ττττττ ++++=
nshi,τ
Page 71
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
strongcoheristrongi/1
int,,,,,,mod,,,,, ττττττ ++++=
( ) shshshshshshmm
weakim
weakSFim
weakAPBim
weakim
weakcoheriweaki/1
int,,,,,,mod,,,,, ττττττ ++++=
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
Page 72
Precipitation strengthening, σprec• Evaluation
• Identifying the regime for each precipitate separately
• Summation of the calculated τi,regime of each precipitate
• Conversion of τprec to σprec
Page 73
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
Page 74
Summation of τi,regime
sumnsh
sum
nshsh
sum
sh
mmm
i
mnshi
mm
i
mshiprec
1
,,
+
= ∑∑ τττ
Page 75
sumsum
nshnsh
sum
shsh
mm
m
i
mnshi
m
m
i
mshiprec
11
,
1
,
+
= ∑∑ τττ
• Precipitation strengthening, σprec (derived from τprec)
• Evaluation of τprec
Summation of τi,regime
Page 76
Precipitation strengthening, σprec• Evaluation
• Identifying the regime for each precipitate separately
• Summation of the calculated τi,regime of each precipitate
• Conversion of τprec to σprec
Page 77
Precipitation strengthening, σprec• Evaluation
• Identifying the regime for each precipitate separately
• Summation of the calculated τi,regime of each precipitate
• Conversion of τprec to σprec
precprec ατσ =
- Taylor factorα
Page 78
Total yield strength, σYS• Summation of contributions
( )[ ] sigsigsigmm
precsssgbgbimdislYS
/1σσσσσσσ +++++=
Page 79
Total yield strength, σYS• Summation of contributions
( )[ ] sigsigsigmm
precsssgbgbimdislYS
/1σσσσσσσ +++++=
Page 80
Total yield strength, σYS• Summation of contributions
( )[ ] sigsigsigmm
precsssgbgbimdislYS
/1σσσσσσσ +++++=
Page 81
Thank you for your attention !
Page 82
Page 83
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
Page 84
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
Page 85
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
Page 86
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
Page 87
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
++=
Page 88
Precipitation hardening• Shearable particles – Coherency effect for non-spherical particles
• Strong 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
Page 89
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
Page 90
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
Page 91
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