Ratings ========= Generalities ----------- :math:`\sigma` refers to the effective stress tensor in small disturbances, noted in the form of the following vector: :math:`(\begin{array}{}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ \sqrt{2}{\sigma }_{12}\\ \sqrt{2}{\sigma }_{13}\\ \sqrt{2}{\sigma }_{23}\end{array})` We note: +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | :math:`{I}_{1}=\text{tr}(\sigma )` | premier invariant des contraintes | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ |:math:`s\mathrm{=}\sigma \mathrm{-}\frac{{I}_{1}}{3}I` | | | | tenseur des contraintes déviatoires | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ |:math:`{s}_{11}=\sqrt{\mathrm{s.s}}` | | | | deuxième invariant du tenseur des contraintes déviatoires | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | :math:`{\sigma }_{1}` | contrainte principale majeure | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | :math:`{\sigma }_{3}` | contrainte principale mineure | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ |:math:`e\mathrm{=}\varepsilon \mathrm{-}\frac{\mathit{Tr}(\varepsilon )}{3}I` | | | | déviateur des déformations | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | :math:`{\varepsilon }_{\nu }=\text{Tr}(\varepsilon )` | déformation volumique | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ |:math:`\text{cos}(3\theta )={2}^{1/2}{3}^{3/2}\frac{\mathrm{det}(s)}{{s}_{\mathrm{II}}^{3}}` | | | | | | | .. image:: images/Object_12.svg | | | :width: 13 | | | :height: 18 | | | | | | | | | .. _RefImage_Object_12.svg: | | | | | | étant l’angle de Lode | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ |:math:`{\gamma }^{p}\mathrm{=}\sqrt{\frac{2}{3}{e}_{\mathit{ij}}^{p}{e}_{\mathit{ij}}^{p}}` | | | | déformations déviatoriques plastiques cumulées | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | |normale de l’hypersurface de déformation | | .. image:: images/Object_14.svg | | | :width: 13 | | | :height: 18 | | | | | | | | | .. _RefImage_Object_14.svg: | | | | | | | | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | |fonction pilotant l’évolution des déformations plastiques et décrivant la direction d’écoulement | | .. image:: images/Object_15.svg | | | :width: 13 | | | :height: 18 | | | | | | | | | .. _RefImage_Object_15.svg: | | | | | | | | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ |:math:`\tilde{G}=G-\frac{\text{Tr}(G)}{3}I` | | | | déviateur de | | | | | | .. image:: images/Object_17.svg | | | :width: 13 | | | :height: 18 | | | | | | | | | .. _RefImage_Object_17.svg: | | | | | | | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ |:math:`G=\mathrm{Tr}(G)` |trace de | | | | | | .. image:: images/Object_19.svg | | | :width: 13 | | | :height: 18 | | | | | | | | | .. _RefImage_Object_19.svg: | | | | | | | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ |:math:`\tilde{{G}_{\mathrm{II}}}=\sqrt{\tilde{G}\mathrm{.}\tilde{G}}` |norme de | | | | | | .. image:: images/Object_21.svg | | | :width: 13 | | | :height: 18 | | | | | | | | | .. _RefImage_Object_21.svg: | | | | | | | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | :math:`\psi` | angle de dilatance | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | :math:`\phi` | angle de frottement | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | |surface de charge | | .. image:: images/Object_24.svg | | | :width: 13 | | | :height: 18 | | | | | | | | | .. _RefImage_Object_24.svg: | | | | | | | | +---------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ Model parameters -------------------- +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | **Notation** | **Description** | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |Pente du critère dans le plan | | .. image:: images/Object_25.svg | | | :width: 13 | .. image:: images/Object_26.svg | | :width: 13 | :width: 13 | | | :height: 13 | | | | | .. _RefImage_Object_25.svg: | | | | .. _RefImage_Object_26.svg: | | | | | | pour les très fortes contraintes (fonction de la nature minéralogique de la roche) | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |Cohésion du milieu. Représentatif de l’endommagement de la roche. | | .. image:: images/Object_27.svg | | | :width: 13 | | | :height: 18 | | | | | | | | | .. _RefImage_Object_27.svg: | | | | | | | | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |Caractérisation de la concavité du critère, fonction du niveau d’altération de la roche. Il définit l’influence de la composante de dilatance dans le comportement aux grandes déformations. | | .. image:: images/Object_28.svg | | | :width: 13 | | | :height: 18 | | | | | | | | | .. _RefImage_Object_28.svg: | | | | | | | | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | :math:`{\gamma }_{\text{ult}}` | Déformation déviatoire plastique correspondant au critère ultime | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | :math:`{\gamma }_{e}` | Déformation déviatoire plastique correspondant à la disparition complète de cohésion | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |Valeur de | | .. image:: images/Object_31.svg | | | :width: 13 | .. image:: images/Object_32.svg | | :width: 13 | :width: 13 | | | :height: 13 | | | | | .. _RefImage_Object_31.svg: | | | | .. _RefImage_Object_32.svg: | | | | | | du critère ultime atteinte en :math:`{\gamma }_{\text{ult}}` | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |Valeur de | | .. image:: images/Object_34.svg | | | :width: 13 | .. image:: images/Object_35.svg | | :width: 13 | :width: 13 | | | :height: 13 | | | | | .. _RefImage_Object_34.svg: | | | | .. _RefImage_Object_35.svg: | | | | | | du critère intermédiaire atteinte en :math:`{\gamma }_{e}` | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |Valeur de | | .. image:: images/Object_37.svg | | | :width: 13 | .. image:: images/Object_38.svg | | :width: 13 | :width: 13 | | | :height: 13 | | | | | .. _RefImage_Object_37.svg: | | | | .. _RefImage_Object_38.svg: | | | | | | du critère intermédiaire atteinte en :math:`{\gamma }_{e}` | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |Valeur de | | .. image:: images/Object_40.svg | | | :width: 13 | .. image:: images/Object_41.svg | | :width: 13 | :width: 13 | | | :height: 13 | | | | | .. _RefImage_Object_40.svg: | | | | .. _RefImage_Object_41.svg: | | | | | | du critère de pic atteinte au pic de contrainte | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |Valeur de | | .. image:: images/Object_42.svg | | | :width: 13 | .. image:: images/Object_43.svg | | :width: 13 | :width: 13 | | | :height: 13 | | | | | .. _RefImage_Object_42.svg: | | | | .. _RefImage_Object_43.svg: | | | | | | du critère de pic atteinte au pic de contrainte | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | :math:`\eta` | Exposant régulant l’écrouissage | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | :math:`{\sigma }_{c}` | Résistance à la compression simple | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | :math:`\gamma` | Premier paramètre réglant la dilatance | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | :math:`\zeta` | Second paramètre réglant la dilatance | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | :math:`{\gamma }_{\text{cjs}}` | Paramètre de forme du critère de plasticité dans le plan déviatoire | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |Module d’Young | | .. image:: images/Object_49.svg | | | :width: 13 | | | :height: 18 | | | | | | | | | .. _RefImage_Object_49.svg: | | | | | | | | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | :math:`\nu` | Coefficient de Poisson | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | :math:`{\sigma }_{\text{p1}}` | Intersection du critère intermédiaire et du critère de pic | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | :math:`{\sigma }_{\text{p2}}` | Intersection du critère intermédiaire et du critère ultime | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |Pression atmosphérique | | .. image:: images/Object_53.svg | | | :width: 13 | | | :height: 18 | | | | | | | | | .. _RefImage_Object_53.svg: | | | | | | | | +---------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+