1 Introduction
2 The New Constitutive Model1
2.1 Kinetic Equation
Symbols | Variables | Unit |
---|---|---|
σ
| stress | MPa |
σ
⊥
| dislocation stress | MPa |
σ
s
| solute contribution to the precipitate stress | MPa |
σ
p
| precipitate contribution to the precipitate stress | MPa |
\( \hat{\sigma } \)
| hardening stress | MPa |
σ*
| saturation stress | MPa |
\( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon } \)
| total strain rate | s−1
|
\( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }^{p} \)
| viscoplastic strain rate | s−1
|
\( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }^{e} \)
| elastic strain rate | s−1
|
\( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }^{T} \)
| thermal strain rate | s−1
|
ρ
| dislocation density | m−2
|
ρ* | saturation dislocation density | m−2
|
N
v
| precipitate density | #/m3
|
\( \overline{F} \)
| mean obstacle strength | MPa/m |
T
| temperature | K |
\( \overline{C} \)
| mean solute concentration | wt pct |
C
e
| equilibrium solute concentration | wt pct |
C
p
| particle solute concentration | wt pct |
C
i
| interface solute concentration | wt pct |
α
T
| thermal dilatation coefficient | K−1
|
r
| particle radius | M |
ZH | Zener–Hollomon variable | — |
Parameters | Value | |
---|---|---|
M
| average Taylor factor | 3.06 |
E
| Young’s modulus | 70,000 MPa (room temperature) |
G
| shear modulus | 27,000 MPa (room temperature) |
b
| burger’s vector | 2.84·10−10 m |
α
| numerical constant | 0.3 |
\( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }_{0} \)
| reference strain rate | 60,000 s−1
|
m
| strain rate sensitivity | temperature dependency relation (Table IV) |
χ
| hardening factor | temperature dependency relation (Table IV) |
ρ
0
| initial dislocation density | 1012 m−2
|
k
1
| dislocation storage coefficient | 2.7·108 m−1
|
k
2
| dynamic recovery coefficient | from Zener–Hollomon relation (Eq. [5]) |
D
| solute diffusivity | 2.2·10−4 m2/s |
j
0
| reference nucleation rate | 3.07·1036 #/m3/s |
A
0
| energy barrier for nucleation | 18 kJ/mol |
Q
d
| diffusion activation energy | 130 kJ/mol |
β
| numerical constant | 0.53 |
r
c
| critical radius | 5.7·10−9 m |
γ | particle-matrix interface energy | 0.26 J/m2
|
V
m
| molar volume of the particle | 7.62·10−5 m3/mol |
T
c
| critical temperature | 423 K |
a
| reference stress in ZH relation | 30 MPa |
Q
| activation energy in ZH relation | 161 kJ/mol |
A
| reference strain rate in ZH relation | 2.35·1010 s−1
|
n
| exponent in ZH relation | 4.1 |
Name | Temperature | Initial Material State |
---|---|---|
A-ini | 20 °C | T6 heat treatment |
A-SSS | 20 °C | solute solution |
B-ini | 150 °C | T6 heat treatment |
B-SSS | 150 °C | solute solution |
C-ini | 250 °C | T6 heat treatment |
C-SSS | 250 °C | solute solution |
D-ini | 340 °C | T6 heat treatment |
D-SSS | 340 °C | solute solution |
2.2 Dislocation Hardening and Recovery
2.3 Precipitation Model
2.4 Implementation in FE Models
3 Parameter Determination
3.1 The Gleeble Tests of Reference 15
3.2 Parameter Fit to the Gleeble Tests
Strain Rate | Measured Stress | Predicted Stress |
---|---|---|
6.25·10−4 s−1
| 6.3 MPa | 6.5 MPa |
1.7·10−3 s−1
| 7.5 MPa | 8.3 MPa |
4.26·10−3 s−1
| 9.0 MPa | 10.3 MPa |
\( m = 320\;\exp \;{\left[ { - 0.0121 \cdot {\left( {T - 298} \right)}} \right]} + 4 \)
| |
χ \( = 4.6 \cdot {\left[ {\tanh {\left( {\frac{{T - 614}} {{51}}} \right)} + 1} \right]}\; + 1 \)
| if T ≤ 623 K |
χ = 0.0313 T–13.1 | if T > 623 K |