Theory ====== 1. Mechanical models ------------------- 1.1 Mechanical problem ^^^^^^ Let :math:`\Omega_t \subset R^3` be the spatial domain at time :math:`t`. The mechanical problem in the quasi-static regime consists in finding the velocity field :math:`\dot u$($t$,$x$)` and the stress field :math:`$\sigma$($t$,$x$)` solutions of: :math:`\mathbf{\sigma} + \rho\mathbf{g} = 0 \in \mathbf{\Omega}_t`, :math:`\frac{D\mathbf{\sigma}}{Dt}= \mathcal{M}(\mathbf{\sigma} (t),\mathbf{d},\dots )` \in :math:`\mathbf{\Omega}_t`, :math:`\dot u = \dot u_0 \text{ on } \Gamma_{u,t} ,` and :math:`\mathbf{\sigma}\cdot n = F_0 \text{ on } \Gamma_{\sigma t}`. .. image:: figures/patata.jpg In this set of equations, :math:`$\rho$` is the density, :math:`$g$` the vector of gravity acceleration, :math:`$\mathbf{d}=\frac{1}{2}(\nabla \dot u + \nabla \dot u^T)$` the strain rate tensor, also written as :math:`$\mathbf{\dot \epsilon}(t)$, $\Gamma_{u,t}$` and :math:`$\Gamma_{\sigma , t }$` are parts of the boundary with a given velocity :math:`($\dot u_0$)` and the traction vector (:math:`$F_0$`), and :math:`$\frac{D\sigma}{Dt}$` is an objective time derivative (Jaumann rate) of :math:`\sigma` defined as: :math:`\frac{D\sigma}{Dt}=\dot \sigma - \omega \sigma + \sigma \omega`, with :math:`\omega = (\nabla v − \nabla v^T)` the corotational rate tensor. :math:`$\mathcal{M}$` represents a functional constitutive law that corresponds to an elastic, elasto-plastic or elasto-visco-plastic rheology defined as: :math:`\mathcal{M}(\mathbf{\sigma}, \mathbf{d}, \mathbf{d_p}) = 2G ( \mathbf{d}-\mathbf{d_p}) + \lambda \text{tr}( \mathbf{d}-\mathbf{d_p}) \mathbf{I} -\frac{G}{\eta} \mathbf{dev \sigma}`, where :math:`{$\mathbf{d_p}$}` is the plastic part of the strain rate tensor :math:`\mathbf{d}`, :math:`$\mathbf{I}$` is the identity tensor, :math:`$tr$` the trace operator, :math:`$G$` and :math:`$\lambda$` are the Lamé parameters and :math:`\eta` is the viscosity. 1.2 Elastic, viscous and plastic rheologies ^^^^^ Various elasto-visco-plastic rheologies can be accounted for numerically. The Drucker-Prager shear failure criterion and the tensile failure criterion are commonly used to define plastic yielding, and are defined with a friction angle `φ`, a cohesion `C`, and a tensile strength :math:`T` leading to the following yield envelopes: :math:`{f}_{DP} =J(\mathbf{s}) + \alpha {I}_{1} - {p}_0` and :math:`{f}_{T} ={I}_{1} - T`, where :math:`{I}_{1}=tr(\mathbf{\sigma})` is the first invariant of the stress tensor and :math:`J(\mathbf{s})` is the second invariant of the deviatoric stress tensor, with :math:`\mathbf{s}=\mathbf{\sigma}-p` and :math:`p` is the mean stress (or pressure). The same invariants characterize the strain tensor :math:`\mathbf{\varepsilon}`. Conventionnally, negative stress and strain values correspond to compression and "shortening". Parameters :math:`\alpha=\frac{6sin \phi}{3-sin \phi}` and :math:`p_0=\alpha.C.tan \phi`. The plastic part of the strain rate tensor :math:`\mathbf{d_p}=\mathbf{d}-\mathbf{d_e}` is given by the non-associative flow rule: :math:`\mathbf{d_p} = \lambda_p \cdot \frac{\partial G(\sigma)}{\partial \sigma}`, and :math:`G(\sigma) = J(\sigma) + \frac{6sin\psi}{3-sin\psi} \cdot I_1(\sigma)`, where :math:`G` is the plastic potential, :math:`\psi` is the dilatancy angle, and :math:`\lambda_p` is the plastic multiplier. Maxwell visco-elasticity in turn relates the deviatoric stress :math:`s` and the strain rate :math:`d`. The effective viscosity obeys a power law rheology, where the viscous deviatoric strain rate corresponds to : :math:`d_v = \gamma_0 \cdot J(s)^{n-1} \cdot e^{\frac{Q-LV}{RT}} \cdot s`, where :math:`\gamma_0` is the initial fluidity, the inverse of a non-linear viscosity. :math:`T` is the absolute temperature at the onset of the model, :math:`R` is the ideal gas constant, and :math:`Q`, :math:`L` and :math:`n` are material activation energy, activation volume and power-law exponent. 1.3 Numerical implementation - ADELI ^^^^ ADELI (Hassani et al., 1997,Chery et al., 2001) is a 3D Finite Element algorithm developed to solve the differential equations of quasi-static equilibrium and mass conservation using a time-explicit dynamic Relaxation Method (Cundall & Board, 1988). This method is well known for being able to track the onset and the development of localized elasto-plastic deformation. ADELI has been widely used to simulate a variety of tectono-volcanic and geodynamic settings (e.g.,Chery2001, Cerpa2015, Gerbault2018, RuzGinouves2021, Novoa2019, Novoa2022). The three-dimensional space is discretized with tetrahedra, forming an unstructured mesh generated using the `GMSH`_ software (Geuzaine and Lemacle, 2009). More details regarding the equations and method can be found in eg. Chery et al. (2001) or Cerpa et al. (2015). 1.4 Other thermomechanical models: eg. FENICS (Felipe) ^^^^ 2. Geological framework (José) ---------------------------- 3. Slip and Dilation Tendencies (Cécile) ---------------------------- Plotting slip and dilation tendencies out from a domain in which the stress tensor is available is a tool to evaluate how measured fractures and faults locally on a field area are consistent with that a priori stress field (e.g. Ritz et al, 1995). It has been widely used in the Hazards and Geothermal communities for decades (eg. Jolie et al., 2016). 2.1 Definitions ^^^^^ 2.2 Combined slip and dilation tendencies and other stress ratio representations ^^^^^ 3. Krostov representations -------------------- 4. other probabilistic analyses tools -------------------------------- References ----------- { Cerpa, N. G., R. Araya, M. Gerbault, and R. Hassani, 2015: Relationship between slab dip and topography segmentation in an oblique subduction zone: Insights from numerical modeling. Geophysical Research Letters, 42 (14), 5786–5795, doi:10.1002/2015GL064047. Chery, J., M. D. Zoback, and R. Hassani, 2001: An integrated mechanical model of the San Andreas fault in central and northern California. Journal of Geophysical Research: Solid Earth, 106 (B10), 22 051–22 066, doi:10.1029/2001jb000382 Cundall, P., and M. Board, 1988: A microcomputer program for modeling large-strain plasticity problems. Prepared for the 6th International Congress on Numerical Methods in Geomechanics, 1988. Gerbault, M., R. Hassani, C. Novoa Lizama, and A. Souche, 2018: Three-Dimensional Failure Pat-653 terns Around an Inflating Magmatic Chamber. Geochemistry, Geophysics, Geosystems, 19 (3),654 749–771, doi:10.1002/2017GC007174. Geuzaine, C., and J.-F. Remacle, 2009: Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering,79 (11), 1309–1331, doi:https://doi.org/10.1002/nme.2579. Hassani, R., D. Jongmands, and J. Chery, 1997: Study of plate deformation and stress in subduction processes using two-dimensional numerical modes. Journal of Geophysical Research, 102. Novoa, C., and Coauthors, 2019: Viscoelastic relaxation : A mechanism to explain the decennial large surface displacements at the Laguna del Maule silicic volcanic complex. Earth and Planetary Science Letters, 521, 46–59, doi:10.1016/j.epsl.2019.06.005. Novoa, C., and Coauthors, 2022: The 2011 Cord´on Caulle eruption triggered by slip on the Liqui˜ne-Ofqui fault system. Earth and Planetary Science Letters, 583, doi:10.1016/j.epsl.2022. 117386. Ruz Ginouves, J., M. Gerbault, J. Cembrano, P. Iturrieta, F. Saez Leiva, C. Novoa, and R. Hassani, 2021: The interplay of a fault zone and a volcanic reservoir from 3D elasto-plastic models: Rheological conditions for mutual trigger based on a field case from the Andean Southern Volcanic Zone. Journal of Volcanology and Geothermal Research, 418, doi:10.1016/j.jvolgeores.2021.107317. }