Notations
=========

The notations and sign conventions used are those of the mechanics of continuous media. A bold character represents a tensor of order two and an openwork character represents a tensor of order four.

For a tensor of order two :math:`\boldsymbol{a}`, its decomposition between deviatory and volume parts, respectively noted :math:`\boldsymbol{a}_d` and :math:`a_v`, is written as follows:
 
.. math::
\ boldsymbol {a} =\ boldsymbol {a} _d +\ frac {a_v} {3}\ boldsymbol {I},\ quad\ text {with}\ quad a_v =\ boldsymbol {I} =\ boldsymbol {I}:\ boldsymbol {I}:\ boldsymbol {a} _d=0

where we note :math:`\boldsymbol{I}=\boldsymbol{\delta}` the second-order identity tensor. We also write :math:`a_m=\cfrac{a_v}{3}`. 

For a tensor of order two :math:`\boldsymbol{a}` of the same nature as a deformation, we denote its equivalent von Mises norm by 
:math:`a_{eq}=\sqrt{\cfrac{2}{3}\boldsymbol{a}_d:\boldsymbol{a}_d}`. For a tensor of order two :math:`\boldsymbol{A}` of the same nature as a constraint, 
It says :math:`A_{eq}=\sqrt{\cfrac{3}{2}\boldsymbol{A}_d:\boldsymbol{A}_d}`.

The notations defined in :numref:`r7.01.44-table_notations_latines` and :numref:`r7.01.44-table_notations_grecques` will be progressively supplemented by other symbols during the presentation of the CSSM model.

.. _r7.01.44-table_notations_latines:

.. list-table:: Notations (lettres latines).

 * - :math:`\mathbb{C}`
   - Isotropic linear elasticity tensor
 * - :math:`f, F_i`
   - Plasticity criteria
 * - :math:`\boldsymbol{I}`
   - Second order identity operating on vectors
 * - :math:`\mathbb{I}`
   - Fourth-order identity operating on symmetric second-order tensors
 * - :math:`\mathbb{J}`
   - Spotlight on the space of hydrostatic tensors (:math:`\mathbb{J}:\boldsymbol{a}=\cfrac{a_v}{3}\boldsymbol{I}`)
 * - :math:`\mathbb{K}`
   - Projector on the space of symmetric tensors with zero trace (:math:`\mathbb{K}:\boldsymbol{a}=(\mathbb{I}-\mathbb{J}):\boldsymbol{a}=\boldsymbol{a}_d`)

.. _r7.01.44-table_notations_grecques:

.. list-table:: Notations (alphabet grec).

 * - :math:`\boldsymbol{\alpha}_i,\boldsymbol{\varepsilon}^p,\xi,\gamma`
   - Internal variables
 * - :math:`\boldsymbol{\varepsilon}`
   - Total deformation tensor
 * - :math:`\varepsilon_v`
   - Total volume deformation
 * - :math:`\dot{\lambda},\dot{\lambda}_i`
   - Plastic multipliers respectively associated with the plasticity criteria :math:`f, F_i`
 * - :math:`\boldsymbol{\sigma}`
   - Stress tensor
 * - :math:`\sigma_{eq}`
   - Equivalent von Mises stress
 * - :math:`\sigma_m`
   - Average stress
 * - :math:`\psi`
   - Energy potential (volume density)