Notations
==========

:math:`\sigma` refers to the effective stress tensor in small disturbances, noted in the form of the following vector:

:math:`\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ \sqrt{2}{\sigma }_{12}\\ \sqrt{2}{\sigma }_{13}\\ \sqrt{2}{\sigma }_{23}\end{array}`

We note:

.. csv-table::

    ":math:`{D}^{e}` ", "elasticity tensor"
    ":math:`{I}_{1}=\text{tr}(\sigma )` ", "first constraint invariant"
    ":math:`s=\sigma -\frac{{I}_{1}}{3}\mathrm{Id}` ", "deviatory stress tensor"
    ":math:`{s}_{\mathrm{II}}=\sqrt{s\mathrm{.}s}` ", "second invariant of the deviatory stress tensor"
    ":math:`{\sigma }_{\text{eq}}\mathrm{=}\sqrt{\frac{3}{2}{s}_{\mathit{ij}}{s}_{\mathit{ij}}}` ", "equivalent stress"
    ":math:`{I}_{1}^{\mathrm{el}}` ", "elastic stress prediction trace"
    ":math:`{s}^{\mathrm{el}}={\sigma }^{\mathrm{el}}-\frac{{I}_{1}^{\mathrm{el}}}{3}\mathrm{Id}` ", "deviatory stress tensor for elastic stress prediction"
    ":math:`{\sigma }_{\text{eq}}^{\mathit{el}}\mathrm{=}\sqrt{\frac{3}{2}{s}_{\mathit{ij}}^{\mathit{el}}{s}_{\mathit{ij}}^{\mathit{el}}}` ", "equivalent stress of elastic stress prediction"
    ":math:`\tilde{\varepsilon }=\varepsilon -\frac{\text{tr}(\varepsilon )}{3}\mathrm{Id}` ", "deformation deviator"
    ":math:`{\varepsilon }_{v}=\text{tr}(\varepsilon )` ", "volume deformation"
    ":math:`\dot{p}=\sqrt{\frac{2}{3}{\dot{\tilde{\varepsilon }}}_{\mathrm{ij}}^{\text{vp}}{\dot{\tilde{\varepsilon }}}_{\mathrm{ij}}^{\text{vp}}}` ", "cumulative viscoplastic deviatory deformations"
    ":math:`f` ", "viscoplastic load surface"
    ":math:`G` ", "viscoplastic flow potential"
    ":math:`{\alpha }_{0}`, :math:`{R}_{0}` and :math:`{\beta }_{0}` ", "work-hardening parameters corresponding to the elasticity threshold (:math:`p\mathrm{=}0`)"
    ":math:`{\alpha }_{\text{pic}}`, :math:`{R}_{\text{pic}}` and :math:`{\beta }_{\text{pic}}` ", "work-hardening parameters corresponding to peak :math:`(p\mathrm{=}{p}_{\mathit{pic}})`"
    ":math:`{\alpha }_{\text{ult}}`, :math:`{R}_{\text{ult}}` and :math:`{\beta }_{\text{ult}}` ", "work-hardening parameters corresponding to the ultimate threshold :math:`(p\mathrm{=}{p}_{\mathit{ult}})`"
    ":math:`\Phi` ", "magnitude of the speed of irreversible deformations"
    ":math:`A` ", "creep parameter"
    ":math:`n` ", "power of the law of creep"
    ":math:`{P}_{\mathit{ref}}` ", "reference pressure"





.. _Ref390748768: