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

Abstract

Stability of undrained shearing in a classical Cauchy continuum has been first analyzed by Rice (J Geophys Res 80(11):1531–1536, 1975) 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 (Int J Numer Anal Methods Geomech 9:339–414, 1985; Int J Numer Anal Methods Geomech 10:177–190, 1986) 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 (Géotechnique 46(3):441–456, 1996; Géotechnique 46(3):457–472, 1996) 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 (Géotechnique 37:271–283, 1987) 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.