Notations
=========

We will note by:


.. csv-table::

    ":math:`\mathrm{Id}` ", "second order identity tensor"
    ":math:`\mathrm{II}` ", "fourth-order identity tensor"
    ":math:`\text{tr}A` ", "trace of the second order tensor :math:`\mathrm{A}`"
    ":math:`\tilde{A}` ", "deviatory part of the :math:`\mathrm{A}` tensor defined by :math:`\tilde{A}=A-(\frac{1}{3}\text{tr}A)\text{Id}`"
    ":math:`{A}_{m}` ", "hydrostatic part of the :math:`\mathrm{A}` tensor defined by :math:`{A}_{m}\mathrm{=}\frac{\text{tr}\mathrm{A}}{3}`"
    ":math:`{A}_{\text{eq}}` ", "equivalent von Mises value defined by :math:`{A}_{\text{eq}}=\sqrt{\frac{3}{2}\tilde{A}:\tilde{A}}`"
    ":math:`\mathrm{:}` ", "doubly contracted product: :math:`\mathrm{A}\mathrm{:}\mathrm{B}\mathrm{=}\mathrm{\sum }_{i,j}{A}_{\text{ij}}{B}_{\text{ij}}\mathrm{=}\text{tr}({\text{AB}}^{\mathrm{T}})`"
    ":math:`\mathrm{\otimes }` ", "tensor product: :math:`(\mathrm{A}\mathrm{\otimes }\mathrm{B}{)}_{\text{ijkl}}\mathrm{=}{A}_{\text{ij}}{B}_{\text{kl}}`"
    ":math:`\lambda ,\mu ,E,\nu ,K` ", "isotropic elasticity coefficients"
    ":math:`\dot{p}` ", "equivalent plastic deformation rate :math:`\dot{p}=\sqrt{\frac{2}{3}{\tilde{\dot{\mathrm{\varepsilon }}}}^{p}:{\tilde{\dot{\mathrm{\varepsilon }}}}^{p}}`"


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

:math:`Q={Q}^{\text{-}}+\Deltaq`

The numerical resolution is done by a :math:`\theta` -method, with :math:`0\le \mathrm{\theta }\le 1`. For all quantities, we define:

:math:`{Q}^{\theta }={Q}^{\text{-}}+\theta \Deltaq`