diff --git a/dumux/geomechanics/elastic/model.hh b/dumux/geomechanics/elastic/model.hh index 1b7ca4fea0be4f9802f5915a1ac88c998a3a5427..bce39074e947616beab3cf9311ea9d424b43ebf2 100644 --- a/dumux/geomechanics/elastic/model.hh +++ b/dumux/geomechanics/elastic/model.hh @@ -21,27 +21,32 @@ * \ingroup Elastic * \brief A geomechanical model. * - * This model describe the deformation of the solid body, using theory of linear elasticity: + * This model describes the deformation of a solid body using the theory of linear elasticity. + * The momentum balance equation of the solid body can be expressed by \f[ - \boldsymbol{\nabla}\cdot\boldsymbol{\sigma_s} + \mathbf{F} + \rho_s \mathbf{g} = \rho\ddot{\mathbf{u}} + \nabla\cdot\boldsymbol{\sigma} + \rho \mathbf{g} + \mathbf{f} = \rho\ddot{\mathbf{u}}, \f] - * We assume the quasi-static conditions, so the acceleration term \f$\rho\ddot{\mathbf{u}} \approx 0\f$. + * where \f$\boldsymbol{\sigma} \f$ is the stress tensor, \f$\rho \f$ is the density of + * the solid, \f$\mathbf{f} \f$ in \f$\mathrm{N/m^3} \f$ is the external force acting on the body per unit volume (e.g. magnetism), + * and \f$\mathbf{u} = \mathbf{x} - \mathbf{x}_{\mathrm{initial}} \f$ is the displacement, + * defined as the difference in material points \f$\mathbf{x} \f$ and \f$\mathbf{x}_{\mathrm{initial}} \f$ + * in the deformed and undeformed (initial) state, respectively. The model assumes quasi-static conditions, + * that is, the above momentum balance equation is solved under the assumption that the acceleration term + * \f$\rho\ddot{\mathbf{u}} \approx 0\f$. * - * For isotropic materials, the stress tensor \f$\boldsymbol{\sigma_s} \f$ can be calculated after Hookes' Law + * Per default, Hookes' Law is used for expressing the stress tensor \f$\boldsymbol{\sigma} \f$ as a function of the + * displacement: \f[ - {\boldsymbol {\sigma_s}}= \lambda \mathrm{tr}(\varepsilon) \boldsymbol{\mathrm I}+ 2G \varepsilon, + \boldsymbol{\sigma} = \lambda\mathrm{tr}(\boldsymbol{\varepsilon}) \mathbf{I} + 2G \boldsymbol{\varepsilon}, \f] * with \f[ - {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left[{\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{\mathrm {T} }\right]. + \boldsymbol{\varepsilon} = \frac{1}{2} \left[ \nabla\mathbf{u} + (\nabla\mathbf{u})^{\mathrm{T}} \right]. \f] * - * Gravity can be enabled or disabled via the property system. + * Primary variables are the displacements in each direction \f$\mathbf{u} \f$. + * Gravity can be enabled or disabled via a runtime parameter. * - * The equations are discretized using a vertex-centered finite volume (box) scheme as spatial discretization. The time discretization is not needed due to the quasi-static conditions. - * PrimaryVariables are the displacements in each direction \f$\mathbf{u} \f$. - * - * Pay attention to the sign convention, where tension has a positive! */ #ifndef DUMUX_GEOMECHANICS_ELASTIC_MODEL_HH #define DUMUX_GEOMECHANICS_ELASTIC_MODEL_HH diff --git a/dumux/geomechanics/poroelastic/model.hh b/dumux/geomechanics/poroelastic/model.hh index 852ccd6bd5eabba078cb659cfce3e0df0120c423..98d590e4f4133e4b17e2c1d22a340a4e0a614e95 100644 --- a/dumux/geomechanics/poroelastic/model.hh +++ b/dumux/geomechanics/poroelastic/model.hh @@ -21,32 +21,39 @@ * \ingroup PoroElastic * \brief A poroelastic geomechanical model * - * The adapted geomechanical model describes the deformation of the solid matrix filled with fluids, using linear poro-elasticity: + * This model describes the deformation of a porous medium using the theory of linear poroelasticity. + * The momentum balance equation of a porous medium can be expressed by \f[ - \boldsymbol{\nabla}\cdot\boldsymbol{\sigma} + \mathbf{F} + \rho \mathbf{g} = \rho\ddot{\mathbf{u}} + \nabla\cdot\boldsymbol{\sigma_{\mathrm{eff}}} + \rho \mathbf{g} + \mathbf{f} = \rho\ddot{\mathbf{u}}, \f] - * where the density of the matrix \f$\rho = \phi \sum_{\beta} S_{\beta} \rho_{\beta} + (1-\phi) \rho_s\f$. + * where \f$\boldsymbol{\sigma_{\mathrm{eff}}} \f$ is the effective stress tensor, + * \f$\rho = (1 - \phi) \rho_s + \phi \rho_f \f$ is the average density of solids and fluids within the porous medium, + * \f$\mathbf{f} \f$ in \f$\mathrm{N/m^3} \f$ is the external force acting on the body per unit volume (e.g. magnetism), + * and \f$\mathbf{u} = \mathbf{x} - \mathbf{x}_{\mathrm{initial}} \f$ is the displacement, + * defined as the difference in material points \f$\mathbf{x} \f$ and \f$\mathbf{x}_{\mathrm{initial}} \f$ + * in the deformed and undeformed (initial) state, respectively. The model assumes quasi-static conditions, + * that is, the above momentum balance equation is solved under the assumption that the acceleration term + * \f$\rho\ddot{\mathbf{u}} \approx 0\f$. * - * We assume the quasi-static conditions, so the acceleration term \f$\rho\ddot{\mathbf{u}} \approx 0\f$. - * - * For isotropic materials, the stress tensor \f$\boldsymbol{\sigma} \f$ can be calculated after Hookes' Law, including the effective fluid pressure. + * Using the concept of the effective stress, the effective stress tensor \f$\boldsymbol{\sigma_{\mathrm{eff}}} \f$ is + * determined by the stress tensor \f$\boldsymbol{\sigma} \f$ , the effective pore pressure \f$p_{\mathrm{eff}} \f$ and the Biot's coefficient \f$\alpha \f$ : \f[ - {\boldsymbol {\sigma}}= \lambda \mathrm{tr}(\varepsilon) \boldsymbol{\mathrm I}+ 2G \varepsilon - \alpha p_{\mathrm{eff}} \boldsymbol{\mathrm I}, + \boldsymbol{\sigma_{\mathrm{eff}}} = \boldsymbol{\sigma} - \alpha p_{\mathrm{eff}} \mathbf{I} \f] - * with + * + * Per default, Hookes' Law is used for expressing the stress tensor \f$\boldsymbol{\sigma} \f$ as a function of the + * displacement: \f[ - {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left[{\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{\mathrm {T} }\right], + \boldsymbol{\sigma} = \lambda\mathrm{tr}(\boldsymbol{\varepsilon}) \mathbf{I} + 2G \boldsymbol{\varepsilon}, \f] - * and + * with \f[ - p_{\mathrm{eff}} = \sum_{\beta} S_{\beta} p_{\beta}. + \boldsymbol{\varepsilon} = \frac{1}{2} \left[ \nabla\mathbf{u} + (\nabla\mathbf{u})^{\mathrm{T}} \right]. \f] - * Gravity can be enabled or disabled via a runtime parameter. * - * The equations are discretized using a vertex-centered finite volume (box) scheme as spatial discretization. The time discretization is not needed due to the quasi-static conditions. - * PrimaryVariables are the displacements in each direction \f$\mathbf{u} \f$. + * Primary variables are the displacements in each direction \f$\mathbf{u} \f$. + * Gravity can be enabled or disabled via a runtime parameter. * - * Pay attention to the sign convention, where tension has a positive! */ #ifndef DUMUX_GEOMECHANICS_POROELASTIC_MODEL_HH #define DUMUX_GEOMECHANICS_POROELASTIC_MODEL_HH