3. 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 \(\mathrm{\sigma }\) used in the following paragraphs should be interpreted according to use.

3.1. Elastic behavior#

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

3.2. Plastic behavior#

The wording adopted comes from the document [1].

3.2.1. Charging surface#

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

where:

  • \(\mathrm{\gamma }\) is the work hardening parameter (defined in paragraph 3.2.3)

  • \(S\) characterizes the state of damage and fracturing of the rock

  • \(m\) is a parameter for smoothing the model

  • \({\mathrm{\sigma }}_{c}\) is the strength of healthy rock without any damage \({\mathrm{\sigma }}_{c}>0\)

  • \(b\) is a function of the work-hardening variable with parabolic evolution that characterizes post-rupture behavior

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

  • \({\mathrm{\sigma }}_{1}\) and \({\mathrm{\sigma }}_{3}\) are the main major and minor constraints: \({\mathrm{\sigma }}_{1}<{\mathrm{\sigma }}_{2}<{\mathrm{\sigma }}_{3}\)

3.2.2. Plastic flow potential#

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

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

3.2.3. Work hardening parameter \(\mathrm{\gamma }\)#

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

\(\gamma \text{=}0\) at the initiation of damage

\(\gamma \text{=}{\gamma }^{\text{rup}}\) at breakup

\(\gamma \text{=}{\gamma }^{\text{res}}\) at the beginning of residual resistance

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

3.2.4. Other settings#

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

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

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

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

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

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

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