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Dynamic frictional slip along an interface between plastically compressible solids is analyzed. The plane strain, small deformation initial/boundary value problem formulation and the numerical method are identical to those in Shi et al. (Int J Fract 162:51, 2010) except that here the material constitutive relation allows for plastic compressibility. The interface is characterized by a rate and state dependent friction law. The specimens have an initial compressive stress and are subject to shear loading by edge impact near the interface. Two loading conditions are analyzed, one giving rise to a crack-like mode of slip propagation and the other to a pulse-like mode of slip propagation. In both cases, the initial compressive stress is taken to vary with plastic compressibility such that the associated initial effective stress is the same for all values of plastic compressibility. The volume change for the crack-like slip mode is mainly plastic while the elastic volume change plays a larger role for the pulse-like mode. For the crack-like slip mode, the proportion of plastic dissipation in the material increases with the increasing plastic compressibility, but the effect of plastic compressibility on the energy partitioning for the pulse-like slip mode is much smaller. The predicted propagation speeds approach a speed about the dilational wave speed for both the crack-like and pulse-like slip modes and this speed is not sensitive to the value of the plastic compressibility parameter. Plastic dissipation is found to be mainly associated with the deformation induced by the loading wave rather than with the deformation arising from slip propagation. The amplitude of the slip rate in the slip pulses is found to be largely governed by the value of the initial compressive stress regardless of the value of plastic compressibility.

The line crack models, including linear elastic fracture mechanics (LEFM), cohesive crack model (CCM), and extended finite element method (XFEM), rest on the century-old hypothesis of constancy of materials’ fracture energy. However, the type of fracture test presented here, named the gap test, reveals that, in concrete and probably all quasibrittle materials, including coarse-grained ceramics, rocks, stiff foams, fiber composites, wood, and sea ice, the effective mode I fracture energy depends strongly on the crack-parallel normal stress, in-plane or out-of-plane. This stress can double the fracture energy or reduce it to zero. Why hasn’t this been detected earlier? Because the crack-parallel stress in all standard fracture specimens is negligible, and is, anyway, unaccountable by line crack models. To simulate this phenomenon by finite elements (FE), the fracture process zone must have a finite width, and must be characterized by a realistic tensorial softening damage model whose vectorial constitutive law captures oriented mesoscale frictional slip, microcrack opening, and splitting with microbuckling. This is best accomplished by the FE crack band model which, when coupled with microplane model M7, fits the test results satisfactorily. The lattice discrete particle model also works. However, the scalar stress–displacement softening law of CCM and tensorial models with a single-parameter damage law are inadequate. The experiment is proposed as a standard. It represents a simple modification of the three-point-bend test in which both the bending and crack-parallel compression are statically determinate. Finally, a perspective of various far-reaching consequences and limitations of CCM, LEFM, and XFEM is discussed.

Pore fluids are ubiquitous throughout the lithosphere and are commonly invoked as the cause of induced seismicity and slow earthquakes. We perform lab experiments to address these questions for drained fault conditions and low pore pressure. We shear simulated faults at effective normal stress σn′$\\left({\\sigma }_{n}^{\\prime }\\right)$of 20 MPa and pore pressures Pp from 1 to 4 MPa. We document the full range of lab earthquake behaviors from slow slip to elasto‐dynamic rupture and show that slow slip can be explained by the slip rate dependence of the critical rheologic stiffness without dilatancy hardening or other fluid effects. Our fault permeabilities ranges from 10−18 to 10−17 m2 with an initial porosity of 0.1 and estimated fluid diffusion time ≈1 s. Slow slip and quasi‐dynamic fault motion may arise from high Pp at higher pressures but dilatancy strengthening is not a general requirement. Plain Language Summary Earthquakes begin and propagate within the fluid‐saturated rocks of Earth's crust. Many investigators have suggested that high pore fluid pressure (Pp) is essential for slow earthquakes and tremor. These studies rely on the idea that changes in Pp can impact rupture propagation speed by dilatant volume increase during faulting with concurrent increases in fault effective normal stress. Thus, understanding the processes that produce slow‐slip versus dynamically propagating rupture is integral to seismic hazard forecasting. Here, we describe experiments on granular faults that produce the full spectrum of slip observed in nature. We measure the mechanical and hydraulic behavior of the faults and determine that frictional and fluid‐driven processes occur in conjunction. Importantly, we demonstrate that frictional processes are sufficient to explain slow‐slip when fluid migration is not inhibited. We demonstrate that for low pore fluid pressures, the full transition from slow slip to dynamic rupture events can be explained as a frictional effect via the critical rheologic stiffness. Key Points The frictional stability transition does not require dilatant hardening for granular fault zones sheared at low pore pressures Slow earthquakes and quasi‐dynamic fault slip can be explained by the strain‐rate dependence of the critical fault stiffness (Kc) For the effective normal stresses studied, pore pressure has a negligible impact on frictional stability and the mode of fault slip

Earthquake-like ruptures break the contacts that form the frictional interface separating contacting bodies and mediate the onset of frictional motion (stick-slip). The slip (motion) of the interface immediately resulting from the rupture that initiates each stick-slip event is generally much smaller than the total slip logged over the duration of the event. Slip after the onset of friction is generally attributed to continuous motion globally attributed to ‘dynamic friction’. Here we show, by means of direct measurements of real contact area and slip at the frictional interface, that sequences of myriad hitherto invisible, secondary ruptures are triggered immediately in the wake of each initial rupture. Each secondary rupture generates incremental slip that, when not resolved, may appear as steady sliding of the interface. Each slip increment is linked, via fracture mechanics, to corresponding variations of contact area and local strain. Only by accounting for the contributions of these secondary ruptures can the accumulated interface slip be described. These results have important ramifications both to our fundamental understanding of frictional motion as well as to the essential role of aftershocks within natural faults in generating earthquake-mediated slip. Conventionally, a continuous motion or “dynamic friction” is expected to take place after the initial rupture under friction. Here, the authors perform direct measurement of real contact and slip at the frictional interface and show that the secondary rupture takes place after each initial rupture.

When a frictional interface is subject to a localized shear load, it is often (experimentally) observed that local slip events propagate until they arrest naturally before reaching the edge of the interface. We develop a theoretical model based on linear elastic fracture mechanics to describe the propagation of such precursory slip. The model’s prediction of precursor lengths as a function of external load is in good quantitative agreement with laboratory experiments as well as with dynamic simulations, and provides thereby evidence to recognize frictional slip as a fracture phenomenon. We show that predicted precursor lengths depend, within given uncertainty ranges, mainly on the kinetic friction coefficient, and only weakly on other interface and material parameters. By simplifying the fracture mechanics model, we also reveal sources for the observed nonlinearity in the growth of precursor lengths as a function of the applied force. The discrete nature of precursors as well as the shear tractions caused by frustrated Poisson’s expansion is found to be the dominant factors. Finally, we apply our model to a different, symmetric setup and provide a prediction of the propagation distance of frictional slip for future experiments.

The transition from static to dynamic friction is often described as a fracture instability. However, studies on slow sliding processes aimed at understanding frictional instabilities and earthquakes report slow friction transients that are usually explained by empirical rate-and-state formulations. We perform very slow ( ∼ nm/s) macroscopic-scale sliding experiments and show that the onset of frictional slip is governed by continuous non-monotonic dynamics originating from a competition between contact aging and shear-induced rejuvenation. This allows to describe both our non-monotonic dynamics and the simpler rate-and-state transients with a single evolution equation.

In this paper we investigate a new class of elliptic variational–hemivariational inequalities without the relaxed monotonicity condition of the generalized subgradient. The inequality describes the mathematical model of the steady state flow of incompressible fluid of Bingham type in a bounded domain. The boundary condition represents a generalization of the no leak condition, and a multivalued and nonmonotone version of a nonlinear Navier–Fujita frictional slip condition. The analysis provides results on existence of solution to a variational–hemivariational inequality, continuous dependence of the solution on the data, existence of solutions to optimal control problems, and the dependence of the solution on the yield limit. The proofs profit from results of nonsmooth analysis and the theory of multivalued pseudomontone operators.

The existence of high-density bedding planes is a typical characteristic of shale oil reservoirs. Understanding the behavior of hydraulic fracturing in high-density laminated rocks is significant for promoting shale oil production. In this study, a hydraulic fracturing model considering tensile failure and frictional slip of the bedding planes is established within the framework of the unified pipe-interface element method (UP-IEM). The model developed for simulating the interaction between the hydraulic fracture and the bedding plane is validated by comparison with experimental results. The hydraulic fracturing patterns in sealed and unsealed bedding planes are compared. Additionally, the effects of differential stress, bedding plane permeability, spacing, and the friction coefficient of the bedding plane are investigated. The results showed that a single main fracture crossing the bedding planes is more likely to form in sealed bedding planes under high differential stress. The decrease in bedding plane permeability and the increase in the friction coefficient also promote the fracture propagating perpendicular to the bedding planes. Shale with high-density bedding planes has a poorer fracturing effect than that with low-density bedding planes, as the hydraulic fracture is prone to initiate and propagate along the bedding planes. Moreover, higher injection pressure is needed to maintain fracture propagation along the bedding. An increase in bedding density will lead to a smaller fracturing area. Fracturing fluid seepage into the bedding planes slows shale fracturing. It is recommended that increasing the injection flow rate, selecting alternative fracturing fluids, and employing multi-well/multi-cluster fracturing may be efficient methods to improve energy production in shale oil reservoirs.

Fluid-induced seismicity in tectonically inactive regions has been attributed to fluid overpressurization of rock fractures during natural resource extraction and storage. We conducted a series of triaxial shear-flow experiments on sawcut fractures in granite and showed that the fracture responses can be dissimilar under various fluid pressurization conditions. For pressure-controlled fluid pressurization, a uniform fluid pressure distribution can be promoted by lowering pressurization rate and enhancing fracture permeability. However, during volume-controlled fluid pressurization, a high pressurization rate causes a drastic increase in fluid pressure before fracture failure. In this case, our analytical model reveals that the fracture area and normal stiffness also influence fluid pressure variations. The maximum seismic moment predicted by this model is well validated by the experimental data for the cases with low pressurization rates. The discrepancy between the analytical and experimental data increases with higher fluid overpressure ratio owing to the assumption of uniform fluid pressure distribution in the model. The sensitivity analysis demonstrates the importance of fracture size estimation in the maximum seismic moment prediction. Our model can potentially be applied to control the fluid overpressurization of rock fractures and to mitigate the risks of fluid-induced seismicity.

Fluid injection-induced fracture slip during hydraulic stimulation of shales may be seismic or aseismic with the slip mode potentially influencing the evolution of permeability and subsequent shale gas production. We report a series of friction-permeability tests with constant and stepped velocities on planar saw-cut fractures of Longmaxi shale, Green River shale and Marcellus shale. In particular we explore the additive effect of stepped velocity on fracture permeability evolution relative to the background permeability driven at constant velocity. Fracture permeability decreases at larger slip displacement at constant velocity presumably due to asperity degradation and clay swelling. Sudden up-steps in slip velocity temporarily enhance fracture permeability as a result of shear dilation on hard minerals, but permeability net decreases with increasing slip displacement as wear products fill the pore space. Fracture surface roughness is the link between the fracture permeability and friction coefficient, which are both influenced by mineralogical composition. The fractures and sheared-off particles in the tectosilicate-rich and carbonate-rich shales dilate to increase fracture permeability, whereas asperity comminution readily occurs in the phyllosilicate-rich shale to reduce fracture permeability. The results potentially improve our ability to facilitate shale gas extraction and to mitigate the associated seismic risks.