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*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}`"
    "", ""
    ":", "doubly contracted product: :math:`A:B=\sum _{i,j}{A}_{\text{ij}}{B}_{\text{ij}}=\text{tr}({\text{AB}}^{T})`"
    "", ""
    ":math:`\otimes` ", "tensor product: :math:`(A\otimes B{)}_{\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:`{\nabla }_{X}A` ", "gradient: :math:`{\nabla }_{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 ,\mu ,E,\nu ,K` ", "isotropic elasticity coefficients"
    "", ""
    ":math:`{\sigma }_{y}` ", "elastic limit"
    "", ""
    ":math:`\alpha` ", "thermal expansion coefficient"
    "", ""
    ":math:`T` ", "temperature"
    "", ""
    ":math:`{T}_{\text{ref}}` ", "reference temperature"


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}^{-}`