Notations
=========

We will note by:


+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`\mathit{Id}`       |identity matrix                                                                                                                                                |
+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`\mathit{Tr}A`      |Tensor trace :math:`A`                                                                                                                                         |
+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`\tilde{A}`         |deviatoric part of tensor :math:`A` defined by                                                                                                                 |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+                          + .. image:: images/Object_11.svg                                                                                                                               +
|                          |    :width: 128                                                                                                                                                |
+                          +    :height: 41                                                                                                                                                +
|                          |                                                                                                                                                               |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`\text{:}`          |doubly contracted product:                                                                                                                                     |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+                          + .. image:: images/Object_12.svg                                                                                                                               +
|                          |    :width: 128                                                                                                                                                |
+                          +    :height: 41                                                                                                                                                +
|                          |                                                                                                                                                               |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                          |tensor product:                                                                                                                                                |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+                          + .. image:: images/Object_14.svg                                                                                                                               +
|                          |    :width: 181                                                                                                                                                |
+                          +    :height: 43                                                                                                                                                +
|                          |                                                                                                                                                               |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{A}_{\mathit{eq}}` |equivalent Von Mises value in the Hill sense defined by                                                                                                        |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+                          + .. image:: images/Object_16.svg                                                                                                                               +
|                          |    :width: 128                                                                                                                                                |
+                          +    :height: 41                                                                                                                                                +
|                          |                                                                                                                                                               |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+                          +                                                                                                                                                               +
|                          |                                                                                                                                                               |
+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`M`                 |Hill anisotropy matrix                                                                                                                                         |
+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|,, E,, *K*                |isotropic elasticity coefficients                                                                                                                              |
+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`\alpha`            |coefficient of thermal expansion                                                                                                                               |
+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`T`                 |temperature                                                                                                                                                    |
+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{T}_{\mathit{ref}}`|reference temperature                                                                                                                                          |
+--------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+


Moreover, in the context of time discretization, all the quantities evaluated at the previous instant are indexed by :math:`\text{-}`, 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::
 :label: eq-1

    \ mathrm {\ Delta} Q=Q- {Q} ^ {\ text {-}}