Theory
1. Mechanical models
1.1 Mechanical problem
Let \(\Omega_t \subset R^3\) be the spatial domain at time \(t\). The mechanical problem in the quasi-static regime consists in finding the velocity field \(\dot u$($t$,$x$)\) and the stress field \($\sigma$($t$,$x$)\) solutions of:
\(\mathbf{\sigma} + \rho\mathbf{g} = 0 \in \mathbf{\Omega}_t\),
\(\frac{D\mathbf{\sigma}}{Dt}= \mathcal{M}(\mathbf{\sigma} (t),\mathbf{d},\dots )\) in \(\mathbf{\Omega}_t\),
\(\dot u = \dot u_0 \text{ on } \Gamma_{u,t} ,\) and \(\mathbf{\sigma}\cdot n = F_0 \text{ on } \Gamma_{\sigma t}\).
In this set of equations, \($\rho$\) is the density, \($g$\) the vector of gravity acceleration, \($\mathbf{d}=\frac{1}{2}(\nabla \dot u + \nabla \dot u^T)$\) the strain rate tensor, also written as \($\mathbf{\dot \epsilon}(t)$, $\Gamma_{u,t}$\) and \($\Gamma_{\sigma , t }$\) are parts of the boundary with a given velocity \(($\dot u_0$)\) and the traction vector (\($F_0$\)), and \($\frac{D\sigma}{Dt}$\) is an objective time derivative (Jaumann rate) of \(\sigma\) defined as:
\(\frac{D\sigma}{Dt}=\dot \sigma - \omega \sigma + \sigma \omega\), with \(\omega = (\nabla v − \nabla v^T)\) the corotational rate tensor.
\($\mathcal{M}$\) represents a functional constitutive law that corresponds to an elastic, elasto-plastic or elasto-visco-plastic rheology defined as:
\(\mathcal{M}(\mathbf{\sigma}, \mathbf{d}, \mathbf{d_p}) = 2G ( \mathbf{d}-\mathbf{d_p}) + \lambda \text{tr}( \mathbf{d}-\mathbf{d_p}) \mathbf{I} -\frac{G}{\eta} \mathbf{dev \sigma}\),
where \({$\mathbf{d_p}$}\) is the plastic part of the strain rate tensor \(\mathbf{d}\), \($\mathbf{I}$\) is the identity tensor, \($tr$\) the trace operator, \($G$\) and \($\lambda$\) are the Lamé parameters and \(\eta\) 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 \(T\) leading to the following yield envelopes:
\({f}_{DP} =J(\mathbf{s}) + \alpha {I}_{1} - {p}_0\) and \({f}_{T} ={I}_{1} - T\),
where \({I}_{1}=tr(\mathbf{\sigma})\) is the first invariant of the stress tensor and \(J(\mathbf{s})\) is the second invariant of the deviatoric stress tensor, with \(\mathbf{s}=\mathbf{\sigma}-p\) and \(p\) is the mean stress (or pressure). The same invariants characterize the strain tensor \(\mathbf{\varepsilon}\). Conventionnally, negative stress and strain values correspond to compression and “shortening”. Parameters \(\alpha=\frac{6sin \phi}{3-sin \phi}\) and \(p_0=\alpha.C.tan \phi\).
The plastic part of the strain rate tensor \(\mathbf{d_p}=\mathbf{d}-\mathbf{d_e}\) is given by the non-associative flow rule:
\(\mathbf{d_p} = \lambda_p \cdot \frac{\partial G(\sigma)}{\partial \sigma}\), and \(G(\sigma) = J(\sigma) + \frac{6sin\psi}{3-sin\psi} \cdot I_1(\sigma)\),
where \(G\) is the plastic potential, \(\psi\) is the dilatancy angle, and \(\lambda_p\) is the plastic multiplier.
Maxwell visco-elasticity in turn relates the deviatoric stress \(s\) and the strain rate \(d\). The effective viscosity obeys a power law rheology, where the viscous deviatoric strain rate corresponds to :
\(d_v = \gamma_0 \cdot J(s)^{n-1} \cdot e^{\frac{Q-LV}{RT}} \cdot s\),
where \(\gamma_0\) is the initial fluidity, the inverse of a non-linear viscosity. \(T\) is the absolute temperature at the onset of the model, \(R\) is the ideal gas constant, and \(Q\), \(L\) and \(n\) 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<www.gmsh.info>`_ 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.
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