Notations
=========

We will note by:


.. csv-table::

    ":math:`\mathrm{Id}` ", "identity matrix"
    ":math:`\text{tr}A` ", "trace of tensor :math:`A`"
    ":math:`{A}^{T}` ", "transposed from the :math:`A` tensor"
    ":math:`\text{det}A` ", "determinant of :math:`A`"
    ":math:`\langle X\rangle` ", "positive part of :math:`X`"
    ":math:`\tilde{A}` ", "deviatory part of the :math:`A` tensor defined by :math:`\tilde{A}=A-(\frac{1}{3}\text{tr}A)\text{Id}`"
    ":math:`:` ", "doubly contracted product: :math:`A:B=\sum _{i,j}{A}_{\text{ij}}{B}_{\text{ij}}=\text{tr}({\text{AB}}^{T})`"
    ":math:`Ä` ", "tensor product: :math:`(\mathrm{AÄB}{)}_{\text{ijkl}}={A}_{\text{ij}}{B}_{\text{kl}}`"
    ":math:`{A}_{\text{eq}}` ", "equivalent von Mises value defined by :math:`{A}_{\text{eq}}=\sqrt{\frac{3}{2}\tilde{A}:\tilde{A}}`"
    ":math:`{Ñ}_{X}A` ", "gradient: :math:`{Ñ}_{X}A=\frac{\partial A}{\partial X}`"
    ":math:`{\text{div}}_{x}A` ", "discrepancy: :math:`({\text{div}}_{x}A{)}_{i}=\sum _{j}\frac{\partial {A}_{\text{ij}}}{\partial {x}_{j}}`"
    ":math:`\lambda`, :math:`\mu` ", "Lamé coefficients: :math:`\lambda =\frac{\mathrm{E\nu }}{(1+\nu )(1-\mathrm{2\nu })}`, :math:`m=\frac{E}{2(1+\nu )}`"
    ":math:`E` ", "Young's module"
    ":math:`\nu` ", "Poisson's ratio"
    ":math:`K` ", "compression stiffness modulus: :math:`\mathrm{3K}=\mathrm{3\lambda }+\mathrm{2m}=\frac{E}{(1-\mathrm{2\nu })}`"
    "*T*", "temperature"
    ":math:`{T}_{\text{ref}}` ", "reference temperature"
    ":math:`{Z}_{g}` ", "austenite proportion"
    ":math:`{Z}_{i}` ", "proportion of the four phases :math:`\alpha`: ferrite, pearlite, bainite and martensite"


Moreover, in the context of time discretization, all the quantities evaluated at the previous instant are indexed by :math:`{}^{-}`, the quantities evaluated at the instant :math:`t+\Delta t` are not indexed and the increments are designated by :math:`\Delta`. We thus have:

:math:`\Delta Q=Q-{Q}^{-}`