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 is written :math:`A_{eq}=\sqrt{\cfrac{3}{2}\boldsymbol{A}_d:\boldsymbol{A}_d}`.

The notations defined in :numref:`r7.01.17-table_notations_latines` and :numref:`r7.01.17-table_notations_grecques` will be progressively supplemented by other symbols during the presentation of the Barcelona model.

.. _r7.01.17-table_notations_latines:

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


 * - :math:`e`
   - Void index
 * - :math:`e^p`
   - Plastic void index
 * - :math:`f_1, f_2`
   - Plasticity criteria
 * - :math:`f_1, f_2`
   - Flow potentials respectively associated with plasticity criteria :math:`f_1, f_2`
 * - :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 of second order (: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`)
 * - :math:`p_c`
   - Capillary pressure, otherwise called suction (:math:`p_c=p_g-p_l`)
 * - :math:`p_g`
   - Pressure of the gas phase
 * - :math:`p_l`
   - Liquid phase pressure

.. _r7.01.17-table_notations_grecques:

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

 * - :math:`\boldsymbol{\varepsilon}`
   - Total deformation tensor
 * - :math:`\varepsilon_v`
   - Total volume deformation
 * - :math:`\dot{\lambda}_1,\dot{\lambda}_2`
   - Plastic multipliers respectively associated with the plasticity criteria :math:`f_1, f_2`
 * - :math:`\boldsymbol{\sigma}`
   - Stress tensor
 * - :math:`\boldsymbol{\sigma}''`
   - Net stress tensor (:math:`\boldsymbol{\sigma}''=\boldsymbol{\sigma}+p_g\boldsymbol{I}`)
 * - :math:`\sigma''_{eq}`
   - Net von Mises equivalent stress (:math:`\sigma''_{eq}=\sigma_{eq}`)
 * - :math:`\sigma''_m`
   - Net mean stress