Ratings
=========




Generalities
-----------

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

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

We note:


.. csv-table::

    ":math:`{I}_{1}\mathrm{=}\text{tr}(\sigma )` ", "first constraint invariant"
    ":math:`s\mathrm{=}\sigma \mathrm{-}\frac{{I}_{1}}{3}I` ", "deviatory stress tensor"
    ":math:`{s}_{\mathit{II}}\mathrm{=}\sqrt{s\mathrm{.}s}` ", "second invariant of the deviatory stress tensor"
    ":math:`{\sigma }_{\text{max}}` ", "major main constraint"
    ":math:`{\sigma }_{\text{min}}` ", "minor main constraint"
    ":math:`\tilde{\varepsilon }\mathrm{=}\varepsilon \mathrm{-}\frac{\text{Tr}(\varepsilon )}{3}I` ", "deformation deviator"
    ":math:`{\varepsilon }_{V}\mathrm{=}\text{tr}(\varepsilon )` ", "volume deformation"
    ":math:`\text{cos}(3\theta )\mathrm{=}{2}^{1\mathrm{/}2}{3}^{3\mathrm{/}2}\frac{\text{det}(s)}{{s}_{\text{II}}^{3}}` "," :math:`\theta` being the Lode angle"
    ":math:`{\dot{\gamma }}_{p}\mathrm{=}\sqrt{\frac{2}{3}{\tilde{\dot{\varepsilon }}}_{\text{ij}}^{p}{\tilde{\dot{\varepsilon }}}_{\text{ij}}^{p}}` ", "cumulative plastic deviatory deformations"
    ":math:`{\dot{\gamma }}_{\text{vp}}\mathrm{=}\sqrt{\frac{2}{3}{\tilde{\dot{\varepsilon }}}_{\text{ij}}^{\text{vp}}{\tilde{\dot{\varepsilon }}}_{\text{ij}}^{\text{vp}}}` ", "cumulative viscoplastic deviatory deformations"
    ":math:`{\xi }_{p}` ", "plastic work hardening parameter"
    ":math:`{\xi }_{\text{vp}}` ", "viscoplastic work hardening parameter"
    ":math:`{G}^{\text{visc}}` ", "function controlling the evolution of viscous deformations and describing the flow direction"
    ":math:`\tilde{G}\mathrm{=}G\mathrm{-}\frac{\text{Tr}(G)}{3}I` ", ":math:`G` deviator"
    ":math:`G\mathrm{=}\text{Tr}(G)` ", "trace of :math:`G`"
    ":math:`{\tilde{G}}_{\mathit{II}}\mathrm{=}\sqrt{\tilde{G}\mathrm{\cdot }\tilde{G}}` ", ":math:`\tilde{G}` standard"
    ":math:`\psi` ", "angle of dilatance"
    ":math:`{f}^{d}` ", "elastoplastic load surface"
    ":math:`{f}^{\text{vp}}` ", "viscoplastic load surface"





Sign convention
--------------------


* In Code_Aster, the sign convention is that of the mechanics of continuous media:

In compression: :math:`\sigma <0`; :math:`\varepsilon \mathrm{=}\frac{\mathrm{\partial }u}{\mathrm{\partial }x}<0`

In traction: :math:`\sigma >0`; :math:`\varepsilon \mathrm{=}\frac{\mathrm{\partial }u}{\mathrm{\partial }x}>0`


* In model LETK, the sign convention is that of soil mechanics:

In compression: :math:`\sigma >0`

Contract: :math:`{\varepsilon }_{v}>0`

In traction: :math:`\sigma <0`

Distance: :math:`{\varepsilon }_{v}<0`

**Note:**

To integrate this law in*Code_Aster* as it stands, you must change the sign of all fields at the input of the routine corresponding to the law of behavior and at its output.

At the start of the routine:

:math:`\begin{array}{c}{\sigma }_{\text{L\&K}}^{\mathrm{-}}\mathrm{=}\mathrm{-}{\sigma }^{\mathrm{-}}\\ {\varepsilon }_{\text{L\&K}}^{\mathrm{-}}\mathrm{=}\mathrm{-}{\varepsilon }^{\mathrm{-}}\\ \Delta {\varepsilon }_{\text{L\&K}}\mathrm{=}\mathrm{-}\Delta \varepsilon \end{array}`

At the end of the routine:

:math:`\begin{array}{c}\sigma \mathrm{=}\mathrm{-}{\sigma }_{\text{L\&K}}\\ \varepsilon \mathrm{=}\mathrm{-}{\varepsilon }_{\text{L\&K}}\\ \Delta \varepsilon \mathrm{=}\mathrm{-}\Delta {\varepsilon }_{\text{L\&K}}\end{array}`




Model parameters
--------------------


.. csv-table::

    "**Notion**", "**Description**"
    ":math:`{P}_{a}` ", "atmospheric pressure"
    ":math:`{\sigma }_{c}` ", "resistance in simple compression, intervening in the expression of the criteria"
    ":math:`{H}_{0}^{\text{ext}}` ", "parameter controlling the resistance in extension, intervening in the expression of the criteria"
    ":math:`{\sigma }_{\text{point1}}` "," :math:`{\sigma }_{\text{min}}` of the intersection between peak and intermediate thresholds"
    ":math:`{x}_{\text{ams}}` ", "non-zero parameter involved in pre-peak work hardening laws"
    ":math:`\eta` ", "non-zero parameter involved in post-peak work hardening laws"
    ":math:`{a}_{0}` ", "value of :math:`a` on the damage threshold"
    ":math:`{m}_{0}` ", "value of :math:`m` on the damage threshold"
    ":math:`{s}_{0}` ", "value of s on the damage threshold"
    ":math:`{a}_{\text{pic}}` ", "value of :math:`a` on the peak threshold"
    ":math:`{m}_{\text{pic}}` ", "value of m on the peak threshold"
    ":math:`{\xi }_{\text{pic}}` ", "level of work hardening required at :math:`{\xi }_{p}` to reach the peak threshold"
    ":math:`{a}_{e}` ", "value of :math:`a` on the cleavage threshold"
    ":math:`{m}_{e}` ", "value of :math:`m` on the cleavage threshold"
    ":math:`{\xi }_{e}` ", "level of work hardening required at :math:`{\xi }_{p}` to reach the cleavage threshold"
    ":math:`{m}_{\text{ult}}` ", "value of :math:`m` on the residual threshold"
    ":math:`{\xi }_{\text{ult}}` ", "level of work hardening required at :math:`{\xi }_{p}` to reach the residual threshold"
    ":math:`{m}_{v\mathrm{-}\text{max}}` ", "value of :math:`m` on the maximum viscoplastic threshold"
    ":math:`{\xi }_{v\mathrm{-}\text{max}}` ", "value of :math:`{\xi }_{v}` for which the maximum viscoplastic criterion is reached"
    ":math:`{A}_{v}` ", "parameter characterizing the amplitude of the creep speed"



.. csv-table::

    ":math:`{n}_{v}` ", "exponent involved in the formula controlling the creep kinetics"
    ":math:`{\mu }_{\mathrm{0,}v}` ", "parameter relating to pre-peak dilatance"
    ":math:`{\xi }_{\mathrm{0,}v}` ", "parameter relating to pre-peak dilatance"
    ":math:`{\mu }_{1}` ", "parameter relating to the dilatance in post peak"
    ":math:`{\xi }_{1}` ", "parameter relating to post-peak dilatance"