Description of document versions
====================================


.. csv-table::

    "**Version** **Aster**", "**Author (s)**
    **Organization (s)**", "**Description of changes**"
    "7.4", "R.Fernandes, C.Chavant EDF -R&D/ AMA ", "Initial text"





Recalibration of the criterion on triaxial compression

Taking the general expression of the criterion under the conditions of a triaxial in compression, we find:

:math:`f={(\frac{g(s)}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{a({\gamma }^{p})}}-(-\frac{m({\gamma }^{p})k({\gamma }^{p})}{\sqrt{6}{\sigma }_{c}}\frac{g(s)}{{h}_{c}^{0}}-\frac{m({\gamma }^{p})k({\gamma }^{p})}{3{\sigma }_{c}}{I}_{1}+s({\gamma }_{p})\mathrm{.}k({\gamma }^{p}))` :math:`={(\frac{\sqrt{\frac{2}{3}}∣{\sigma }_{1}-{\sigma }_{3}∣\mathrm{.}h}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{a({\gamma }^{p})}}+\frac{1}{{\sigma }_{c}}(\frac{m({\gamma }^{p})k({\gamma }^{p})}{{\sigma }_{c}\sqrt{6}}\frac{\sqrt{\frac{2}{3}}∣{\sigma }_{1}-{\sigma }_{3}∣\mathrm{.}h}{{\sigma }_{c}{h}_{c}^{0}}+\frac{m({\gamma }^{p})k({\gamma }^{p})}{3{\sigma }_{c}}({\sigma }_{1}+2{\sigma }_{3})-s({\gamma }^{p})\mathrm{.}k({\gamma }^{p}))`

:math:`={(\frac{\sqrt{\frac{2}{3}}∣{\sigma }_{1}-{\sigma }_{3}∣}{{\sigma }_{c}{h}_{c}^{0}})}^{\frac{1}{a({\gamma }^{p})}}+(\frac{m({\gamma }^{p})k({\gamma }^{p})}{{\sigma }_{c}\sqrt{6}}\sqrt{\frac{2}{3}}∣{\sigma }_{1}-{\sigma }_{3}∣+\frac{m({\gamma }^{p})k({\gamma }^{p})}{3{\sigma }_{c}}({\sigma }_{1}+2{\sigma }_{3})-s({\gamma }^{p})\mathrm{.}k({\gamma }^{p}))`

:math:`={(\frac{\sqrt{\frac{2}{3}}}{{\sigma }_{c}})}^{\frac{1}{a({\gamma }^{p})}}{(∣{\sigma }_{1}-{\sigma }_{3}∣)}^{\frac{1}{a({\gamma }^{p})}}+(\frac{m({\gamma }^{p})\mathrm{.}k({\gamma }^{p})}{3{\sigma }_{c}}∣{\sigma }_{1}-{\sigma }_{3}∣+\frac{m({\gamma }^{p})\mathrm{.}k({\gamma }^{p})}{3{\sigma }_{c}}({\sigma }_{1}+2{\sigma }_{3})-s({\gamma }^{p})\mathrm{.}k({\gamma }^{p}))`

:math:`={(\frac{\sqrt{\frac{2}{3}}}{{\sigma }_{c}})}^{\frac{1}{a({\gamma }^{p})}}{(∣{\sigma }_{1}-{\sigma }_{3}∣)}^{\frac{1}{a({\gamma }^{p})}}+(\frac{m({\gamma }^{p})\mathrm{.}k({\gamma }^{p})}{3{\sigma }_{c}}({\sigma }_{3}-{\sigma }_{1})+\frac{m({\gamma }^{p})\mathrm{.}k({\gamma }^{p})}{3{\sigma }_{c}}({\sigma }_{1}+2{\sigma }_{3})-s({\gamma }^{p})\mathrm{.}k({\gamma }^{p}))`

:math:`={(\frac{\sqrt{\frac{2}{3}}}{{\sigma }_{c}})}^{\frac{1}{a({\gamma }^{p})}}{(∣{\sigma }_{1}-{\sigma }_{3}∣)}^{\frac{1}{a({\gamma }^{p})}}+(\frac{m({\gamma }^{p})\mathrm{.}k({\gamma }^{p})}{{\sigma }_{c}}({\sigma }_{3})-s({\gamma }^{p})\mathrm{.}k({\gamma }^{p}))`

:math:`={(\frac{\sqrt{\frac{2}{3}}}{{\sigma }_{c}})}^{\frac{1}{a({\gamma }^{p})}}{(∣{\sigma }_{1}-{\sigma }_{3}∣)}^{\frac{1}{a({\gamma }^{p})}}-{\sqrt{\frac{2}{3}}}^{\frac{1}{a({\gamma }^{p})}}(\frac{m({\gamma }^{p})}{{\sigma }_{c}}(-{\sigma }_{3})+s({\gamma }^{p}))`

:math:`={(\frac{\sqrt{\frac{2}{3}}}{{\sigma }_{c}})}^{\frac{1}{a({\gamma }^{p})}}\left[{(∣{\sigma }_{1}-{\sigma }_{3}∣)}^{\frac{1}{a({\gamma }^{p})}}-{({\sigma }_{c})}^{\frac{1}{a({\gamma }^{p})}}(\frac{m({\gamma }^{p})}{{\sigma }_{c}}(-{\sigma }_{3})+s({\gamma }^{p}))\right]`




Normalization of Q

:math:`Q=\frac{1}{{h(\theta )}^{5}}\left[(1+\frac{{\gamma }_{\text{cjs}}}{2}\mathrm{cos}(3\theta ))\frac{s}{{s}_{\text{II}}}+\frac{{\gamma }_{\text{cjs}}\sqrt{54}}{{\mathrm{6.s}}_{\text{II}}^{2}}\text{dev}(\frac{\partial \text{det}(\underline{\underline{s}})}{\partial s})\right]`

We put :math:`t=\frac{\partial \text{det}(\underline{\underline{s}})}{\partial s}` and :math:`t=\text{dev}(\frac{\partial \text{det}(\underline{\underline{s}})}{\partial s})` (see Reference document CJS R7.01.13)

:math:`{Q}_{\text{II}}^{2}=\mathrm{Q.}Q=\frac{1}{{h(\theta )}^{10}}\left[{1+\frac{{\gamma }_{\text{cjs}}}{2}\mathrm{cos}(3\theta )}^{2}+\frac{3}{2}\mathrm{.}\frac{{{\gamma }_{\text{cjs}}}^{2}}{{{s}_{\text{II}}}_{4}}{t}^{d}\mathrm{.}{t}^{d}+\frac{{\gamma }_{\text{cjs}}\sqrt{54}}{{\mathrm{3.s}}_{\text{II}}^{3}}(1+\frac{{\gamma }_{\text{cjs}}}{2}\mathrm{cos}(3\theta ))s\mathrm{.}{t}^{d}\right]`

To evaluate this expression, we place ourselves in the case where

.. image:: images/Object_344.svg
    :width: 10
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.. _RefImage_Object_344.svg:

is diagonal for the sake of simplifying the calculations.

So: :math:`\begin{array}{cc}s=& \left[\begin{array}{}{s}_{1}\\ {s}_{2}\\ {s}_{3}\\ 0\\ 0\\ 0\end{array}\right]\end{array}` and :math:`\begin{array}{cc}{t}^{d}=\frac{1}{3}& \left[\begin{array}{}{\mathrm{2s}}_{2}{s}_{3}-{s}_{1}{s}_{2}-{s}_{1}{s}_{3}\\ {\mathrm{2s}}_{1}{s}_{3}-{s}_{1}{s}_{2}-{s}_{2}{s}_{3}\\ 2{s}_{1}{s}_{2}-{s}_{1}{s}_{3}-{s}_{2}{s}_{3}\\ 0\\ 0\\ 0\end{array}\right]\end{array}`

Using the property of

.. image:: images/Object_347.svg
    :width: 10
    :height: 14


.. _RefImage_Object_347.svg:

:, we :math:`s={s}_{1}+{s}_{2}+{s}_{3}` shows that :math:`{s}_{\text{II}}^{4}=({s}_{1}^{2}{s}_{2}^{2}+{s}_{1}^{2}{s}_{3}^{2}+{s}_{2}^{2}{s}_{3}^{2})` and therefore:

:math:`\begin{array}{cccc}{t}^{\mathrm{d.}}{t}^{d}=\frac{1}{9}& \mid \begin{array}{}{\mathrm{2s}}_{2}{s}_{3}-{s}_{1}{s}_{2}-{s}_{1}{s}_{3}\\ {\mathrm{2s}}_{1}{s}_{3}-{s}_{1}{s}_{2}-{s}_{2}{s}_{3}\\ 2{s}_{1}{s}_{2}-{s}_{1}{s}_{3}-{s}_{2}{s}_{3}\\ 0\\ 0\\ 0\end{array}& \mid \begin{array}{}{\mathrm{2s}}_{2}{s}_{3}-{s}_{1}{s}_{2}-{s}_{1}{s}_{3}\\ {\mathrm{2s}}_{1}{s}_{3}-{s}_{1}{s}_{2}-{s}_{2}{s}_{3}\\ 2{s}_{1}{s}_{2}-{s}_{1}{s}_{3}-{s}_{2}{s}_{3}\\ 0\\ 0\\ 0\end{array}& =\frac{{{s}_{\text{II}}}^{4}}{6}\end{array}`

We also show from property :math:`{s}_{1}+{s}_{2}+{s}_{3}=0` that :math:`{s}_{1}^{3}{s}_{2}^{3}{s}_{3}^{3}={\mathrm{3s}}_{1}{s}_{2}{s}_{3}=3.\text{det}(s)` and therefore:

:math:`\begin{array}{ccccc}\frac{{\gamma }_{\text{cjs}}\mathrm{.}\sqrt{54}}{{\mathrm{3.s}}_{\text{II}}^{3}}\mathrm{s.}{t}^{d}=\frac{{\gamma }_{\text{cjs}}\mathrm{.}\sqrt{54}}{{\mathrm{9.s}}_{\text{II}}^{3}}& & \mid \begin{array}{}{s}_{1}\\ {s}_{2}\\ {s}_{3}\\ 0\\ 0\\ 0\end{array}& \text{.}\mid \begin{array}{}{\mathrm{2s}}_{2}{s}_{3}-{s}_{1}{s}_{2}-{s}_{1}{s}_{3}\\ {\mathrm{2s}}_{1}{s}_{3}-{s}_{1}{s}_{2}-{s}_{2}{s}_{3}\\ 2{s}_{1}{s}_{2}-{s}_{1}{s}_{3}-{s}_{2}{s}_{3}\\ 0\\ 0\\ 0\end{array}& =\frac{{\gamma }_{\text{cjs}}\mathrm{.}\sqrt{54}}{{s}_{\text{II}}^{3}}\text{det}(s)={\gamma }_{s}\mathrm{.}\mathrm{cos}(3\theta )\end{array}`

We deduce from this as follows:

:math:`{Q}_{\text{II}}^{2}=\frac{1}{{h(\theta )}^{10}}\left[{(1+\frac{{\gamma }_{\text{cjs}}}{2}\text{cos}(3\theta ))}^{2}+\frac{{{\gamma }_{\text{cjs}}}^{2}}{4}+{\gamma }_{\text{cjs}}\text{cos}(3\theta )(1+\frac{{\gamma }_{\text{cjs}}}{2}\text{cos}(3\theta ))\right]`



Projection angle framing

We remind you that :math:`\mathrm{cos}{\phi }_{s}\underset{s\to 0}{\to }\frac{3}{(\beta  {\text{'}}^{2}+3)\sqrt{{(\frac{3}{\beta  {\text{'}}^{2}+3})}^{2}-\frac{1}{4}+\frac{1}{2(1+{\gamma }_{\text{cjs}}\mathrm{cos}(3\theta ))}+\frac{{\gamma }_{\text{cjs}}^{2}-1}{{4(1+{\gamma }_{\text{cjs}}\mathrm{cos}(3\theta ))}^{2}}}}`

We pose: :math:`X(\psi )=\frac{1}{2(1+{\gamma }_{\text{cjs}}\mathrm{cos}(\psi ))}+\frac{{\gamma }_{\text{cjs}}^{2}-1}{{4(1+{\gamma }_{\text{cjs}}\mathrm{cos}(\psi ))}^{2}}` where :math:`\psi \in [\mathrm{0,2}\pi [`

Note that: :math:`X(-\psi )=X(\psi )`, the function

.. image:: images/Object_359.svg
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    :height: 14


.. _RefImage_Object_359.svg:

Being even we restricted the study interval to :math:`\psi \in [\mathrm{0,2}\pi [`.

The resolution of :math:`\frac{\mathrm{dX}}{d\psi }=0` results in :math:`\frac{{\gamma }_{\text{cjs}}\text{sin}(\psi )}{{2(1+{\gamma }_{\text{cjs}}\text{cos}(\psi ))}^{3}}\mathrm{.}{\gamma }_{\text{cjs.}}({\gamma }_{\text{cjs}}+\mathrm{cos}(3\psi ))=0`

From this we deduce that the lower and upper limits of function :math:`X` are:

:math:`\{\begin{array}{}X(\psi =0)=\frac{1}{4}\\ X({\psi }_{\text{cjs}})=\frac{1}{4(1-{\gamma }_{\text{cjs}}^{2})}\text{où}{\psi }_{\text{cjs}}\text{est tel que cos}({\psi }_{\text{cjs}})=-{\gamma }_{\text{cjs}}\end{array}`

We can thus give the following framework for :math:`\mathrm{cos}{\phi }_{s}`: :math:`\mathrm{cos}{\phi }_{s}^{\text{min}}\le \mathrm{co}{\phi }_{s}\le \mathrm{cos}{\phi }_{s}^{\text{max}}` with:

:math:`\{\begin{array}{}\mathrm{cos}{\phi }_{s}^{\text{min}}=\frac{3}{(\beta  {\text{'}}^{2}+3)\sqrt{{(\frac{3}{\beta  {\text{'}}^{2}+3})}^{2}+\frac{{{\gamma }_{\text{cjs}}}^{2}}{4(1-{\gamma }_{\text{cjs}}^{2})}}}\\ \mathrm{cos}{\phi }_{s}^{\text{max}}=1\end{array}`