2. General ratings#
All quantities evaluated at the previous moment are indexed by \({}^{-}\). The quantities evaluated at moment \(t+\Delta t\) are not indicated. Increments are designated by \(\Delta\). We thus have:
For the calculation of derivatives, we will note \(\dot{Q}\) as a derivative of \(Q\) with respect to time. Tensors and vectors will be noted in bold.
\({Z}_{{k}_{f}\mathrm{=}\mathrm{1,4}}\) |
Proportion of ferritic phases (ferrite, pearlite, bainite and martensite) |
\({Z}_{\gamma }\) |
Proportion of the austenitic phase |
\({Z}_{f}\) |
Sum of all ferritic phases |
\({\varepsilon }_{\gamma }^{\text{th}}\) |
Thermal deformation of the austenitic phase |
\({\varepsilon }_{f}^{\text{th}}\) |
Thermal deformation of ferritic phases |
\(\Delta {\varepsilon }_{f\gamma }^{{T}_{\mathit{ref}}}\) |
Compactness difference between hot and cold phase |
\({\varepsilon }^{\mathit{th}}\) |
Thermal deformation |
\({T}^{\text{ref}}\) |
Reference temperature |
\({\alpha }_{\gamma }\) |
Average expansion coefficient of the austenitic phase |
\({\alpha }_{f}\) |
Average expansion coefficient of ferritic phases |
\({Z}_{\gamma }^{R}\) |
Indicator of the reference metallurgical phase (value \(1\) when the reference phase is the austenitic phase and \(0\) when the reference phase is the ferritic phase) |
\(\sigma\) |
Stress tensor |
\(\text{~}\) |
Deviatoric operator |
\(\langle \mathrm{\cdot }\rangle\) |
Positive part |
\({()}_{\mathit{eq}}\) |
Equivalent value of a tensor in the Von Mises sense |
\(\mathit{Id}\) |
Identity tensor |
\(A\) |
Hooke elasticity tensor |
\(\lambda\), \(\mu\) |
Lamé coefficients |
\(E\) |
Young’s module |
\(\nu\) |
Poisson’s ratio |
\(K\) |
Compressibility module \(\mathrm{3K}=3\lambda +2\mu\) |
\(T\) |
Temperature |
\(t\) |
Time |
\({g}_{Z}^{\mathit{pt}}\) |
Function for transformation plasticity |
\({R}_{\mathrm{0,}k}\) |
Phase \(k\) linear work hardening coefficient |
\({\theta }_{\gamma ,{k}_{f}}\) |
Proportion of work-hardening restoration during austenite to ferrite transformation |
\({\theta }_{{k}_{f},\gamma }\) |
Proportion of work-hardening restoration during ferrite-to-austenite transformation |
\({C}_{k}\), \({m}_{k}\) |
Phase \(k\) viscous restoration coefficients |
\({g}^{\text{re},v}\) |
Function for restoring viscous isotropic work hardening |
\({h}^{\text{re},v}\) |
Function for restoring viscous kinematic work hardening |
\({g}_{{k}_{f}}^{\text{re},m}\) |
Function for the restoration of metallurgical isotropic work hardening (cold phases) |
\({g}_{\gamma }^{\text{re},m}\) |
Function for the restoration of metallurgical isotropic work hardening (hot phase) |
\({h}_{{k}_{f}}^{\text{re},m}\) |
Function for the restoration of metallurgical kinematic work hardening (cold phases) |
\({h}_{\gamma }^{\text{re},m}\) |
Function for the restoration of metallurgical kinematic work hardening (hot phase) |
\(\overline{R}\) |
Homogenized linear work hardening coefficient (multiphase) |
\({\sigma }_{c,k}\) |
Phase \(k\) elasticity limit |
\({\overline{\sigma }}_{c}\) |
Elasticity limit of homogenized material (multiphase) |
\({X}_{k}\) |
Recall tensor for kinematic work hardening and for phase \(k\) |
\(\overline{R}\) |
Recall tensor for homogenized (multiphase) kinematic work hardening |
\({\eta }_{k}\), \({n}_{k}\) |
Material coefficients for the viscosity of phase \(k\) |
\(\overline{\eta }\), \(\overline{n}\) |
Material coefficients for homogenized (multiphase) viscosity |