Ratings
=========

:math:`\sigma` refers to the stress tensor, arranged as a vector according to the convention:

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

We note:

:math:`{I}_{1}=\text{Trace}(\sigma )`

.. csv-table::

    ":math:`{\sigma }_{H}=\frac{1}{3}\text{tr}(\sigma )` ", "hydrostatic stress"
    ":math:`s=\sigma -\frac{1}{3}\text{tr}(\sigma )I` ", "the constraint deviator"
    ":math:`{\varepsilon }_{H}=\frac{1}{3}\text{tr}(\varepsilon )` ", "volume deformation"
    ":math:`\tilde{\varepsilon }=\varepsilon -\frac{1}{3}\text{tr}(\varepsilon )I` ", "the deflection deflector"
    ":math:`{\dot{\tilde{\varepsilon }}}_{\mathrm{eq}}=\sqrt{\frac{3}{2}\text{trace}(\dot{\tilde{\varepsilon }}\mathrm{.}\dot{\tilde{\varepsilon }})}` ", "the equivalent deformation rate"
    ":math:`{J}_{2}=\frac{1}{2}\text{trace}({s}^{2})` ", "the second invariant of constraints"
    ":math:`{\sigma }^{\text{eq}}=\sqrt{{\mathrm{3J}}_{2}}=\sqrt{\frac{3}{2}\text{trace}({s}^{2})}` ", "the equivalent stress"
    ":math:`{\tau }_{\text{oct}}=\sqrt{\frac{2}{3}{J}_{2}}=\sqrt{\frac{\text{trace}({s}^{2})}{3}}` ", ""
    ":math:`{\sigma }_{\text{oct}}={\sigma }_{H}=\frac{{I}_{1}}{3}=\frac{\text{trace}(\sigma )}{3}` ", ""



.. csv-table::

    ":math:`{f}_{c}^{\text{'}}` ", "initial break limit in simple compression"
    ":math:`{f}_{\text{cc}}^{\text{'}}` ", "initial break limit in bi compression"
    ":math:`\varphi {f}_{c}^{\text{'}}` ", "elastic limit in compression"
    ":math:`{f}_{t}^{\text{'}}` ", "initial tensile failure limit"
    ":math:`\alpha =\frac{{f}_{t}^{\text{'}}}{{f}_{c}^{\text{'}}}` ", "relationship between tensile and compression failure limit"
    ":math:`\beta  =\frac{{f}_{\text{cc}}^{\text{'}}}{{f}_{c}^{\text{'}}}` ", "relationship between rupture limit in bi-compression and simple compression"
    ":math:`{\kappa }_{{}_{t}}^{p}` ", "plastic deformation under tension"
    ":math:`{\lambda }_{t}` ", "plastic multiplier in traction"
    ":math:`{\kappa }_{{}_{c}}^{p}` ", "plastic deformation under compression"
    ":math:`{\lambda }_{c}` ", "plastic multiplier in compression"
    ":math:`{f}_{c}({\kappa }_{{}_{c}}^{p})` ", "compression work hardening curve"
    ":math:`{f}_{t}({\kappa }_{{}_{t}}^{p})` ", "tensile work hardening curve"
    ":math:`{\kappa }_{t}^{u}` ", "ultimate plastic deformation under traction"
    ":math:`{k}_{c}^{u}` ", "ultimate plastic deformation under compression"
    ":math:`{G}_{c}^{f}` ", "compression rupture energy (material characteristic)"
    ":math:`{G}_{t}^{f}` ", "tensile failure energy (material characteristic)"
    ":math:`\theta` ", "the maximum temperature during the load history"