The asymptotic stress-strain field at a stationary crack tip in elastic-ideally plastic f.c.c. and b.c.c. single crystals, as determined by Rice et al. [1], turns out to be locally constant within angular sectors. It involves shear displacement discontinuities at sector boundaries, that can be interpreted as strain localization bands. The numerical analysis of the same problem using finite strain crystal elastoplasticity in [2] reveals that the condition of constant stress state in each sector must be relieved because of possible local unloading, but also that the strain localization patterns pertain. Discrete models based on dislocation dynamics also lead to strongly localized dislocation distributions near the crack tip and to the progressive formation of the predicted sectors [3]. Experimental observations in a b.c.c. single crystal in [4] confirms the existence of such intense deformation bands radiating from the crack tip. The fact that the strong strain gradients developing in the vicinity of the crack tip may affect the local mechanical response of a crystalline solid suggests that generalized continuum theories, including Cosserat, strain gradient and non local models, could be helpful for computing more realistic local stress-strain fields. The strain gradient model used in [5] results in a substantial increase of the tractions ahead of the tip of a mode I crack within a domain of characteristic size related to the constitutive intrinsic length. This monitoring of the local strain field enables one to improve the prediction of subsequent crack growth. The use of a generalized continuum model also strongly affects the localized deformation modes as demonstrated in [6] in the case of single crystals. In particular, classical crystal plasticity theory predicts two types of deformation bands in single crystals undergoing single slip:slip bands lying in the slip plane of the locally activated slip system, or kink bands lying in a plane normal to the slip direction of the slip system [7]. The formation of a kink band is associated with the development of strong lattice rotation gradients at its boundary and may therefore be precluded if the model incorporates additional hardening due to lattice curvature [6]. The present work investigates the effect of a generalized continuum theory on the localized deformation patterns arising at the crack tip in elastoplastic f.c.c. single crystals. Important consequences are expected regarding crack branching after stable crack growth. Scripta mater. 44 (2001) 953–958

Earthquake faults are the result of severe strain localization in rocks deep down in the earth's crust. This localized deformation is controlled by the size of the microstructure and various Thermo-Hydro-Mechanical (THM) couplings, whose modeling is central for understanding earthquake nucleation and seismic energy release. We model this challenging system using the Cosserat theory and by considering large shear deformations during seismic slip. Our numerical results justify the rate and state friction law (Dieterich, 1992; Ruina, 1983a), which describes empirically the fault tribology. This shows the importance of the underlying physics of our model for fault friction. In our analyses traveling shear bands along the thickness of the fault are present, leading to oscillations in the fault's frictional response. Existing numerical analyses presented in Rattez et al. (2018a, 2018b, 2018c) do not capture this behavior, which goes beyond the established models of uniform shear (Lachenbruch, 1980) and shear on a mathematical plane (Rice, 2006a). Recent experimental results, that insulate thermal pressurization from other weakening mechanisms (Badt et al., 2020), corroborate our numerical results. Our results motivate us to extend the classical model of thermal pressurization in Mase & Smith (1987) and Rice (2006b) to incorporate different strain localization modes, temperature and pore fluid pressure boundary conditions.In particular, we start our analysis by using the normalized coupled system of partial differential equations that include the THM couplings for the case of a Cosserat continuum. We then perform a bifurcation analysis, which indicates that traveling shear bands are possible inside the fault gouge. Next, we derive our non linear mesh independent numerical results accounting for the influence of large displacements by using and Adaptive Lagrangian Eulerian (ALE) procedure. We introduce viscosity in our numerical analyses for the rate and state phenomenology to emerge. We corroborate our numerical results comparing them to similar laboratory experiments. Furthermore, we modify the Volterra integral equation of the classical model of thermal pressurization in Mase & Smith (1987) and Rice (2006b), which we solve by a semi analytical procedure, in order to capture the effect of isothermal drained boundary conditions and traveling shear bands. Our results reappraise and extend the established models of frictional weakening due to thermal pressurization during coseismic slip.