Continuous model
==============

Here we describe the model regardless of whether it is used in mechanics alone, or in hydro-mechanical calculations under total or effective stress. Thus the notation :math:`\mathrm{\sigma }` used in the following paragraphs should be interpreted according to use.




Elastic behavior
----------------------

The elastic behavior is given by a linear law. The two parameters characterizing this behavior are the elasticity module :math:`E` and the Poisson's ratio :math:`\mathrm{\nu  }`.




Plastic behavior
----------------------

The wording adopted comes from the document [:ref:`1 <1>`].




Charging surface
~~~~~~~~~~~~~~~~~~

:math:`F(\sigma ,\gamma )\text{=}({\sigma }_{3}\text{-}{\sigma }_{1})\text{-}\sqrt{\text{-}{\sigma }_{3}\text{.}m{\sigma }_{c}\text{+}S(\gamma ){\sigma }_{c}^{2}}\text{-}b(\gamma )\text{.}(1\text{-}\frac{{\sigma }_{3}}{{\sigma }_{3}^{b\text{-}d}})`

where:


* :math:`\mathrm{\gamma }` is the work hardening parameter (defined in paragraph :ref:`3.2.3 <Ref100116782>`)


* :math:`S` characterizes the state of damage and fracturing of the rock


* :math:`m` is a parameter for smoothing the model


* :math:`{\mathrm{\sigma }}_{c}` is the strength of healthy rock without any damage :math:`{\mathrm{\sigma }}_{c}>0`


* :math:`b` is a function of the work-hardening variable with parabolic evolution that characterizes post-rupture behavior


* :math:`{\sigma }_{3}^{b\text{-}d}` is the intersection between the line :math:`{\sigma }_{1}\text{=}\alpha {\sigma }_{3}` (:math:`\mathrm{\alpha }` being a parameter of the model) and the criterion at the time of the break (:math:`\gamma \text{=}{\gamma }^{\text{rup}}`). :math:`{\sigma }_{3}^{b\text{-}d}>0`


* :math:`{\mathrm{\sigma }}_{1}` and :math:`{\mathrm{\sigma }}_{3}` are the main major and minor constraints: :math:`{\mathrm{\sigma }}_{1}<{\mathrm{\sigma }}_{2}<{\mathrm{\sigma }}_{3}`




Plastic flow potential
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The plastic flow potential is given by a function derived from the Drucker-Prager criterion:

:math:`G(\sigma ,\gamma )\text{=}\eta (\gamma ){I}_{1}\text{+}\sqrt{\frac{3}{2}}{s}_{\text{II}}\text{=}\eta (\gamma ){I}_{1}\text{+}{\sigma }_{\text{eq}}` with :math:`\eta (\gamma )\text{=}\frac{2\text{sin}\phi (\gamma )}{3\text{+}\text{sin}\phi (\gamma )}`, :math:`{\sigma }_{\text{eq}}\text{=}\sqrt{\frac{3}{2}}{s}_{\text{II}}`, and :math:`\phi (\gamma )` the equivalent angle of friction.


.. _Ref100116782:




Work hardening parameter :math:`\mathrm{\gamma }`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The work hardening parameter :math:`\mathrm{\gamma }` that we are considering takes the following values:

:math:`\gamma \text{=}0` at the initiation of damage

:math:`\gamma \text{=}{\gamma }^{\text{rup}}` at breakup

:math:`\gamma \text{=}{\gamma }^{\text{res}}` at the beginning of residual resistance

It is defined by placing itself in triaxial compression by: :math:`\gamma \text{=}{\varepsilon }_{1}^{p}`, and we can then show that :math:`{\gamma }^{p}\text{=}\mid \frac{{\varepsilon }_{\nu }^{p}}{3}\text{-}\gamma \mid`. So we have :math:`\gamma \text{=}\frac{{\varepsilon }_{\nu }^{p}}{3}\pm {\gamma }^{p}` which must be positive.




Other settings
~~~~~~~~~~~~~~~~~~

Parameters :math:`S{\sigma }_{c}^{2},m{\sigma }_{c},\phi ,b` vary as follows depending on the work hardening parameter :math:`\gamma`:


1. :math:`(m{\sigma }_{c})(\gamma )\text{=}\{\begin{array}{cc}\gamma \frac{(m{\sigma }_{c}{)}^{\text{rup}}\text{-}(m{\sigma }_{c}{)}^{\text{end}}}{{\gamma }^{\text{rup}}}\text{+}(m{\sigma }_{c}{)}^{\text{end}}\text{=}{p}_{m\sigma }\gamma \text{+}(m{\sigma }_{c}{)}^{\text{end}}& \begin{array}{cc}\text{si}& \gamma \le {\gamma }^{\text{rup}}\end{array}\\ (m{\sigma }_{c}{)}^{\text{rup}}& \begin{array}{cc}\text{si}& \gamma \ge {\gamma }^{\text{rup}}\end{array}\end{array}`


2. :math:`(S{\sigma }_{c}^{2})(\gamma )\text{=}\{\begin{array}{cc}\gamma \frac{(S{\sigma }_{c}^{2}{)}^{\text{rup}}\text{-}(S{\sigma }_{c}^{2}{)}^{\text{end}}}{{\gamma }^{\text{rup}}}\text{+}(S{\sigma }_{c}^{2}{)}^{\text{end}}\text{=}{p}_{S{\sigma }^{2}}\gamma \text{+}(S{\sigma }_{c}^{2}{)}^{\text{end}}& \begin{array}{cc}\text{si}& \gamma \le {\gamma }^{\text{rup}}\end{array}\\ (S{\sigma }_{c}^{2}{)}^{\text{rup}}& \begin{array}{cc}\text{si}& \gamma \ge {\gamma }^{\text{rup}}\end{array}\end{array}`


3. :math:`\phi (\gamma )\text{=}\{\begin{array}{cc}\frac{{\phi }^{\text{rup}}\text{-}{\varphi }^{\text{end}}}{{\gamma }^{\text{rup}}}\gamma \text{+}{\phi }^{\text{end}}& \text{si}\gamma \le {\gamma }^{\text{rup}}\\ \frac{{\phi }^{\text{res}}\text{-}{\phi }^{\text{rup}}}{{\gamma }^{\text{res}}\text{-}{\gamma }^{\text{rup}}}\gamma \text{+}\frac{{\phi }^{\text{rup}}{\gamma }^{\text{res}}\text{-}{\phi }^{\text{res}}{\gamma }^{\text{rup}}}{{\gamma }^{\text{res}}\text{-}{\gamma }^{\text{rup}}}& \text{si}{\gamma }^{\text{rup}}\le \gamma \le {\gamma }^{\text{res}}\\ {\phi }^{\text{res}}& \text{sinon}\end{array}`


1. :math:`b(\gamma )\text{=}\{\begin{array}{cc}0& \text{si}\gamma \le {\gamma }^{\text{rup}}\\ a{\gamma }^{2}+d\gamma +c& \text{si}{\gamma }^{\text{rup}}\le \gamma \le {\gamma }^{\text{res}}\\ {b}^{\text{res}}& \text{si}\gamma \ge {\gamma }^{\text{res}}\end{array}` where :math:`a\text{=}\text{-}\frac{{b}^{\text{res}}}{({\gamma }^{\text{rup}}\text{-}{\gamma }^{\text{res}}{)}^{2}}`, :math:`d\text{=}\frac{{\mathrm{2b}}^{\text{res}}{\gamma }^{\text{res}}}{({\gamma }^{\text{rup}}\text{-}{\gamma }^{\text{res}}{)}^{2}}`, :math:`c\text{=}\frac{{b}^{\text{res}}{\gamma }^{\text{rup}}({\gamma }^{\text{rup}}\text{-}2{\gamma }^{\text{res}})}{({\gamma }^{\text{rup}}\text{-}{\gamma }^{\text{res}}{)}^{2}}`, and :math:`{b}^{\text{res}}\text{=}\beta \text{-}\sqrt{(S{\sigma }_{c}^{2}{)}^{\text{rup}}}`


1. :math:`{\sigma }_{3}^{b\text{-}d}\text{=}\frac{\text{-}(m{\sigma }_{c}{)}^{\text{rup}}\text{-}\sqrt{((m{\sigma }_{c}{)}^{\text{rup}}{)}^{2}\text{+}4(1\text{-}\alpha {)}^{2}(S{\sigma }_{c}^{2}{)}^{\text{rup}}}}{2(1\text{-}\alpha {)}^{2}}`


1. the coefficients :math:`\alpha ,\beta ,(S{\sigma }_{c}^{2}{)}^{\text{rup}},(S{\sigma }_{c}^{2}{)}^{\text{end}},(m{\sigma }_{c}{)}^{\text{rup}},(m{\sigma }_{c}{)}^{\text{end}},{\phi }^{\text{end}},{\phi }^{\text{rup}},{\phi }^{\text{res}}` are given.