Stability analysis of undrained adiabatic shearing of a rock layer with Cosserat microstructure

Abstract

International audienceStability of undrained shearing in a classical Cauchy continuum has been first analyzed by Rice [1] who showed that instability occurs when the underlying drained deformation becomes unstable (i.e. in the softening regime of the corresponding drained stress-strain curve). However Vardoulakis [2, 3] has shown that Rice's linear stability analysis, if performed at the state of maximum deviator, leads to a sharp transition from infinitely stable to infinitely unstable behaviour, which indicates that the solution of the considered initial-value problem does not exist and consequently that the corresponding problem is mathematically ill-posed. Vardoulakis [4,5] proposed a regularization of the ill-posed problem in the softening regime by resorting to a second grade extension of plasticity theory. In this paper, the kinetics of a granular material is described by a Cosserat continuum as first suggested by Mühlhaus and Vardoulakis [6] and we incorporate the effect of shear heating due to the dissipation of the frictional work. The undrained adiabatic limit is applicable as soon as the slip event is sufficiently rapid and the shear zone broad enough to effectively preclude heat or fluid transfer as it is the case during an earthquake or a landslide. It is shown that shear heating has a destabilizing effect and that instability can occur in the hardening regime if the amount of dilatant strengthening is not sufficient as compared to the effect of thermal pressurization of the pore fluid. It is shown that the linear stability analysis with macro and micro inertia terms leads to the selection of a preferred wave length of the instability mode corresponding to the instability mode with fastest (but finite) growth coefficient

The importance of Thermo-Hydro-Mechanical couplings and microstructure to strain localization in 3D continua with application to seismic faults. Part II: Numerical implementation and post-bifurcation analysis

“…The values of the different parameters are chosen to represent a fault gouge at a seismogenic 165 depth of 7km, which is a typical centroidal depth for crustal faults. Furthermore, as this set of parameters was used in previous studies, we can compare our results to foregoing works on the subject and complement them (Rice, 2006, Sulem et al, 2011. The rotations are imposed at the upper and lower boundaries (ω The Drucker-Prager yield criterion is generalized for Cosserat continua, as in Rattez et al (2017a) with a friction coefficient which evolves with plastic shear deformation:…”

Failure in shear bands for granular materials: thermo-hydro-chemo-mechanical effects

“…Following Sulem et al (2011), we assume a Terzaghi decomposition of the stress tensor in effective stress s' ij and pore pressure p (s ij~s ' ij {pd ij ) and assume a Coulomb yield stress and plastic potential for the effective stresses…”

“…If the temperature is increased in undrained conditions, so that thermal pressurisation is active, Sulem et al (2011) showed that, even in the absence of dilatancy, the shear band will be formed in the hardening regime H . 0.…”

“…The details of the method are comprehensively presented by Sulem et al (2011) and Veveakis et al (2012), allowing application to a 7 km deep fault with the parameters listed in Table 1 The corresponding characteristic equations are solved numerically for each temperature regime, allowing calculation of the critical values of the normalised hardening modulus h cr 5 H cr /(s + p) below which instability sets in. The results are summarised in Table 2, where different values of the dilatancy b are used, from zero to the maximum value b 5 m (associative plasticity).…”

“…Defined as the critical wavelength number at which instability propagates with maximum finite speed, the analyses performed by Sulem et al (2011) and Veveakis et al (2012) revealed that the shear band thickness decreases with increasing temperature, having a lower threshold at D 5 3d 50 at high temperature areas where the chemical reaction is active.…”