1 Introduction
2 Outline of Namo-Version 2
2.1 Framework for Modeling Cluster Formation
2.1.1 Problem description
2.1.2 Modeling strategy
2.1.3 Constitutive laws and equations embedded in the numerical code
2.1.4 Nucleation laws
Model | Expression for j
0
| Expression for ΔG* | Adjustable Parameters |
---|---|---|---|
NA-model (cluster formation) |
\( j_{0}^{{c_{1} }} \,\left( {1 - \;\rho /\rho_{c} } \right) + \;j_{0}^{{c_{2} }} \)
|
\( \frac{{\left( {A_{0}^{c} } \right)^{3} }}{{\left( {R\,T\,\ln \left( {\overline{C} /C_{\text{e}}^{\text{Mg}} } \right) - \;\Delta G_{\text{s}} } \right)^{2} }} \)
|
\( j_{0}^{{c_{1} }} ,\;j_{0}^{{c_{2} }} \) and A
0
c
(since the other parameters are fixed) |
AA-model (nucleation of β″ in Al matrix) |
j
0
m
(1 − ρ/ρ
c
) |
\( \frac{{\left( {A_{0}^{m} } \right)^{3} }}{{\left( {R\,T\ln \left( {\overline{C} /C_{\text{e}}^{\text{Mg}} } \right)} \right)^{2} }} \)
| none (since j
0
m
and A
0
m
are the same as in NaMo-version 1) |
AA-model (nucleation of β′ at dislocations) |
j
0
d
|
\( \frac{{\left( {A_{0}^{d} } \right)^{3} }}{{\left( {R\,T\ln \left( {\overline{C} /C_{\text{e}}^{\text{Mg}} } \right)} \right)^{2} }} \)
|
j
0
d
and A
0
d
|
2.2 Growth Laws
Model | Expression for the Growth Rate | Adjustable Parameters |
---|---|---|
NA-model (cluster formation) |
\( \frac{{{\text{d}}r}}{{{\text{d}}t}} = \;M_{0} \exp \left( { - \frac{{Q_{d} }}{RT}} \right)\left( {\overline{C} - \;C_{\text{i}}^{\text{Mg}} } \right) \)
| none (since M
0 and Q
d
are fixed) |
AA-model (nucleation of β″ in Al matrix and nucleation of β′ at dislocations) |
\( \frac{{{\text{d}}r}}{{{\text{d}}t}} = \;\left( {\frac{{\overline{C} - \;C_{i}^{\text{Mg}} }}{{C_{\text{p}}^{\text{Mg}} - \;C_{i}^{\text{Mg}} }}} \right)\;\left( {\frac{{D_{\text{eff}}^{\text{Mg}} }}{r}} \right) \)
where
\( D_{\text{eff}}^{\text{Mg}} = \;D^{\text{Mg}} \,\left[ {1 + \,\left( {\rho \,a_{c} \,\frac{{D_{c}^{\text{Mg}} }}{{D^{\text{Mg}} }}\,} \right)} \right] \)
| none (since a
c
and D
c
Mg
are fixed) |
2.2.1 Continuity equation
2.2.2 Correction term for the reduced thermodynamic stability of clusters
2.2.3 Evolution equations for the dislocation density
2.2.4 Framework for calculating the macroscopic yield strength
Parameter | Sub-Equations | Adjustable Parameters | |
---|---|---|---|
Clusters | Hardening precipitates | ||
σ
p
|
\( \sigma_{p\,1} = \;\frac{{M_{r} \,\overline{{F_{1} }} \,}}{{b\,\ell_{1} }} \)
|
\( \sigma_{p\,2} = \;\frac{{M_{r} \,\overline{{F_{2} }} \,}}{{b\,\ell_{2} }} \)
| None (since M
r
and b are fixed) |
\( \overline{F} \)
|
\( \overline{{F_{1} }} = \;\frac{{\sum\limits_{i} {N_{i} \,F_{i} } }}{{\sum\limits_{i} {N_{i} } }} \)
where
\( F_{i} = \;2\,\beta_{1} \,G\,b^{2} \,\left( {\frac{{r_{i} }}{{r_{c\,1} }}} \right)\;^{{^{{^{{^{\dag } }} }} }} \)
|
\( \overline{{F_{2} }} = \;\frac{{\sum\limits_{j} {N_{j} \,F_{j} } }}{{\sum\limits_{j} {N_{j} } }} \)
where
\( F_{j} = \;2\,\beta_{1} \,G\,b^{2} \,\left( {\frac{{r_{j} }}{{r_{c\,2} }}} \right)\;^{{^{{^{{^{\dag \dag } }} }} }} \)
|
r
c1 and β
1 (since the other parameters are the same as in NaMo-Version 1) |
ℓ |
\( \ell_{1} = \;\left( {\frac{{N_{v\,1} \,\,\overline{{r_{1} }} \,\;\overline{{F_{1} }} }}{{\beta_{1} \,G\,b^{2} }}} \right)^{\, - 1/2} \)
|
\( \ell_{2} = \;\left( {\frac{{N_{v\,2} \,\,\overline{{r_{2} }} \,\,\overline{{F_{2} }} }}{{\beta_{1} \,G\,b^{2} }}} \right)^{\, - 1/2} \)
| None (when r
c1 and β
1 are fixed) |
3 Experimental
Alloy | Si | Mg | Cu | Mn | Fe | Others | Al |
---|---|---|---|---|---|---|---|
AA6005 | 0.60 | 0.59 | 0.13 | 0.09 | 0.22 | — | balance |
3.1 Heat Treatment and Cold Deformation Schedules
Series No. | Plastic Strain (Pct) | Storage Time t
1 (At RT) | Storage Time t
2(At RT) |
---|---|---|---|
A1 | 0 | from 1 min to 1 week | 0 |
A2 | 2 | 1 min | 1 hour, 1 week, 2 months |
A3 | 2 | 6 hours | 1 hour, 1 week, 2 months |
A4 | 2 | 1 week | 1 hour, 1 week, 2 months |
A5 | 10 | 1 min | 1 hour, 1 week, 2 months |
A6 | 10 | 6 hours | 1 hour, 1 week, 2 months |
A7 | 10 | 1 week | 1 hour, 1 week, 2 months |
Series No. | Plastic Strain (Pct) | Holding Times | ||
---|---|---|---|---|
t
1 (at RT) |
t
2 (at RT) |
t
3 (at 458 K) (185 °C) | ||
B1 | 0 | 1 min | 0 | from 10 min to 1 week |
B2 | 0 | 1 hour | 0 | from 10 min to 1 week |
B3 | 0 | 1 week | 0 | from 1 min to 24 hours |
B4 | 0 | 4 weeks | 0 | from 10 min to 1 week |
B5 | 0 | 10 weeks | 0 | from 10 min to 5 hours |
B6 | 0 | 18 weeks | 0 | from 10 min to 5 hours |
B7 | 2 | 1 min | 1 hour | from 1 min to 5 hours |
B8 | 2 | 1 min | 2 months | from 10 min to 24 hours |
B9 | 2 | 10 weeks | 1 hour | from 10 min to 5 hours |
B10 | 2 | 10 weeks | 2 months | from 10 min to 5 hours |
B11 | 10 | 1 min | 1 hour | from 1 min to 5 hours |
B12 | 10 | 1 min | 2 months | from 10 min to 24 hours |
3.2 Tensile Testing
3.3 Supplementary DSC and TEM Investigations
4 Calibration procedure
4.1 Default Parameter Values
Parameter | Value | Ref†
|
---|---|---|
a
c
(m2) | 1.0 × 10−18
| [53] |
b (m) | 2.84 × 10−10
| [67] |
A
0
m
(kJ/mol) | 18 | [21] |
C
p
Mg
(wt pct) | 59 | [21] |
D
0
Mg
(m2/s) | 2.2 × 10−4
| |
\( D_{C_{0}}^{{\text{Mg}}}\,({\text{m}}^{2}/{\text{s}}) \)
| 1.0 × 10−4
| |
G (GPa) | 27 | [67] |
ΔG
s
(kJ/mol) | 2.7 | [47] |
j
0
m
(#/m3s) | 3.07 × 1036
| [21] |
k
1 (m−1) | 4.0 × 108
| [22] |
k
2
|
C-dependent | [22] |
k
3(m/s) | 5.68 | [56] |
k
4(Jm/mol) | 0.0048 | [56] |
k
Cu
(MPa/wt pct2/3) | 46.4 | [21] |
k
Mg
(MPa/wt pct2/3) | 15.0 | [21] |
k
Mn
(MPa/wt pct2/3) | 80.0 | [21] |
k
Si
(MPa/wt pct2/3) | 33.0 | [21] |
M
r
| 3.1 | |
M
0 (m/s wt pct) | 6000 | [32] |
Q
c
(kJ/mol) | 90 | [53] |
Q
d
(kJ/mol) | 130 | |
r
c2 (m) | 5.0 × 10−9
| |
U
s
(kJ/mol) | 160 | [35] |
V
m
(m3/mol) | 7.62 × 10−5
| [21] |
α
| 0.30 | [57] |
γ
c
(J/m2) | 0.02 | [70] |
Ω
c
(kJ/mol) | 18.3 | [55] |
ρ
c
(#/m2) (value at ɛ
p
= 0.025) | 2.0 × 1013
| [22] |
σ
i
(MPa) | 10 | [52] |
4.2 Determination of Adjustable Parameters in Nucleation Laws
4.3 Determination of Adjustable Parameters in Modified Yield Strength Model
Parameter | Value | Comments |
---|---|---|
A
0
c
(kJ/mol) | 3 | calculated from Eq. [15] |
A
0
d
(kJ/mol) | 11 | calculated from Eq. [16] |
\( j_{0}^{{c_{1} }} \) (#/m3s) | 5.5 × 1042
| calculated using data from Ref. [66] |
\( j_{0}^{{c_{2} }} \)(#/m3s) | 5.5 × 1039
| equilibrium value of \( j_{0}^{{c_{1} }} \) at ρ = ρ
c
|
j
0
d
(#/m3s) | 5.0 × 1033
| calculated using data from Ref. [41] |
r
c1 (m) | 7.0 × 10−9
| calculated using data from Series A1 |
β
1
| 0.44 | calculated using data from Series A1 |
5 Experimental Validation of Namo-Version 2
5.1 Validation of Different Sub-Models
Nano-Constituent | Experimental and Simulation Conditions | Source | |||
---|---|---|---|---|---|
Alloy Designation | Type of Treatment | Type of Data Available | Assumed Taylor Factor | ||
Clusters†
| Al-0.62 Mg-0.93Si | NA |
N
v and \( \overline{r} \)
| — | Ref. [66] |
β″
††
| AA6005 | AA |
N
v, \( \overline{r} \) and HV | 2.7 | Ref. [21] |
Pre-β″ and β″
††
| AA6082 | AA |
N
v, \( \overline{r} \) and HV | 2.7 | Ref. [17] |
β″ and β′††
| AA6060 | AA and CD + AA |
N
v
, \( \overline{r} \) and HV | 2.7 | Ref. [5] |
β″ and β′††
| AA6005 | CD + AA |
N
v, \( \overline{r} \) and YS | 3.1 | Ref. [41] |
Dislocations††
| AA6005 | CD + AA |
ρ
| — | Ref. [56] |