General ratings
===================

All quantities evaluated at the previous moment are indexed by :math:`{}^{-}`. The quantities evaluated at moment :math:`t+\Delta t` are not indicated. Increments are designated by :math:`\Delta`. We thus have:

.. math::
 :label: eq-1

    Q\ mathrm {=} Q (t+\ Delta t)\ mathrm {=} Q (t) +\ Delta Q\ mathrm {=} {Q} ^ {\ mathrm {-}} ^ {\ mathrm {-}}} +\ Delta Q

For the calculation of derivatives, we will note :math:`\dot{Q}` as a derivative of :math:`Q` with respect to time. Tensors and vectors will be noted in **bold**.


.. csv-table::

    ":math:`{Z}_{{k}_{f}\mathrm{=}\mathrm{1,4}}` ", "Proportion of ferritic phases (ferrite, pearlite, bainite and martensite)"
    ":math:`{Z}_{\gamma }` ", "Proportion of the austenitic phase"
    ":math:`{Z}_{f}` ", "Sum of all ferritic phases"
    ":math:`{\varepsilon }_{\gamma }^{\text{th}}` ", "Thermal deformation of the austenitic phase"
    ":math:`{\varepsilon }_{f}^{\text{th}}` ", "Thermal deformation of ferritic phases"
    ":math:`\Delta {\varepsilon }_{f\gamma }^{{T}_{\mathit{ref}}}` ", "Compactness difference between hot and cold phase"
    ":math:`{\varepsilon }^{\mathit{th}}` ", "Thermal deformation"
    ":math:`{T}^{\text{ref}}` ", "Reference temperature"
    ":math:`{\alpha }_{\gamma }` ", "Average expansion coefficient of the austenitic phase"
    ":math:`{\alpha }_{f}` ", "Average expansion coefficient of ferritic phases"
    ":math:`{Z}_{\gamma }^{R}` ", "Indicator of the reference metallurgical phase (value :math:`1` when the reference phase is the austenitic phase and :math:`0` when the reference phase is the ferritic phase)"
    ":math:`\sigma` ", "Stress tensor"
    ":math:`\text{~}` ", "Deviatoric operator"
    ":math:`\langle \mathrm{\cdot }\rangle` ", "Positive part"
    ":math:`{()}_{\mathit{eq}}` ", "Equivalent value of a tensor in the Von Mises sense"
    ":math:`\mathit{Id}` ", "Identity tensor"
    ":math:`A` ", "Hooke elasticity tensor"
    ":math:`\lambda`, :math:`\mu` ", "Lamé coefficients"
    ":math:`E` ", "Young's module"
    ":math:`\nu` ", "Poisson's ratio"
    ":math:`K` ", "Compressibility module :math:`\mathrm{3K}=3\lambda +2\mu`"
    ":math:`T` ", "Temperature"
    ":math:`t` ", "Time"
    ":math:`{g}_{Z}^{\mathit{pt}}` ", "Function for transformation plasticity"
    ":math:`{R}_{\mathrm{0,}k}` ", "Phase :math:`k` linear work hardening coefficient"
    ":math:`{\theta }_{\gamma ,{k}_{f}}` ", "Proportion of work-hardening restoration during austenite to ferrite transformation"
    ":math:`{\theta }_{{k}_{f},\gamma }` ", "Proportion of work-hardening restoration during ferrite-to-austenite transformation"
    ":math:`{C}_{k}`, :math:`{m}_{k}` ", "Phase :math:`k` viscous restoration coefficients"
    ":math:`{g}^{\text{re},v}` ", "Function for restoring viscous isotropic work hardening"
    ":math:`{h}^{\text{re},v}` ", "Function for restoring viscous kinematic work hardening"
    ":math:`{g}_{{k}_{f}}^{\text{re},m}` ", "Function for the restoration of metallurgical isotropic work hardening (cold phases)"
    ":math:`{g}_{\gamma }^{\text{re},m}` ", "Function for the restoration of metallurgical isotropic work hardening (hot phase)"
    ":math:`{h}_{{k}_{f}}^{\text{re},m}` ", "Function for the restoration of metallurgical kinematic work hardening (cold phases)"
    ":math:`{h}_{\gamma }^{\text{re},m}` ", "Function for the restoration of metallurgical kinematic work hardening (hot phase)"
    ":math:`\overline{R}` ", "Homogenized linear work hardening coefficient (multiphase)"
    ":math:`{\sigma }_{c,k}` ", "Phase :math:`k` elasticity limit"
    ":math:`{\overline{\sigma }}_{c}` ", "Elasticity limit of homogenized material (multiphase)"
    ":math:`{X}_{k}` ", "Recall tensor for kinematic work hardening and for phase :math:`k`"
    ":math:`\overline{R}` ", "Recall tensor for homogenized (multiphase) kinematic work hardening"
    ":math:`{\eta }_{k}`, :math:`{n}_{k}` ", "Material coefficients for the viscosity of phase :math:`k`"
    ":math:`\overline{\eta }`, :math:`\overline{n}` ", "Material coefficients for homogenized (multiphase) viscosity"