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Communication dans un congrès Année : 2012

Friction laws versus material interfaces for earthquake rupture dynamics: seismological implications

Résumé

Summary: A material interface condition based on the RCCM law is used as a boundary condition for seismic rupture to mimic damage in the fault zone. This constitutive law may provide quasi-stable solutions and variable nucleation lengths as a function of the loading and the thickness of the damaged layer. Introduction Earthquake rupture is classically modeled as a friction-dominated process, where friction arises from the roughness of the contact between the two sides of a fault. During the metastable nucleation and the dynamic rupture phases, the traction on the fault decreases reflecting a weakening of the interface. Seismological observations show that this traction evolution can be represented as a simple slip weakening process, where the traction drops to its final value over a critical slip weakening distance, usually referred to as Dc. The simplest phenomenological model is a linear slip weakening friction law (LSW)[1], with a fixed slip weakening rate and fracture energy release between the static and the dynamic friction states. The nucleation length associated to this law is universal [2] and can hardly account for the occurrence of both small and large earthquakes. The friction dissipation does not depend on the dynamics but is rather a mechanical property, which appears to be strongly scale-dependent as suggested when comparing seismological and laboratory observations. Laboratory experiments on rocks sliding in steady-state conditions have shown a more complex evolution where the traction on the interface depends on the slip and slip velocity, as well as other interface state parameters that govern both the weakening and the healing phases. A classical example of these rate-and-state friction laws is the Dietrich-Ruina law [3] Where is the friction coefficient, v is the slip rate, a and b are constant parameters, and the * points to a reference state. The state variable  obeys an ageing law, associated with of the time evolution of the population of contacts along the sliding interface, including a healing process. Depending on the sign of a-b, the friction law can exhibit either a weakening (a-b <0) or a strengthening (a-b >0) behavior [4]. Rate-and-state friction laws allow modeling the evolution of the fault interface during seismic cycles, i.e., loading-and-rupture cycles. A fault interface is not however a zero-order surface discontinuity. Even though slip during an earthquake is localized along a thin layer, and the rupture propagates along very large distance, fault interfaces appear as a highly heterogeneous damaged shear zone produced by the repeated occurrence of earthquakes along that interface. Surface (friction) and volume (damage) dissipation are therefore contributing to the energy release during a seismic rupture. Since the wavelength associated with seismic observations is significantly larger than the width of the fault zone, it is generally believed that an interface constitutive law remains a valid phenomenological description as a result of an implicit homogenization of the dissipation processes within the interface. In this framework, we explore here seismological implications of the RCCM (Raous-Cangemi-Cocu-Monerie) interface law [5] and the differences with classical linear slip weakening, for both the rupture nucleation and dynamics phases. A material interface We assume a fault zone with thickness h in a 2D in-plane geometry and homogeneous properties in the direction parallel to the fault. The fault zone is described as a material interface. The interface dissipation results both from the interface damage weakening and the friction mobilization. We define a state variable  that describes the damage evolution. ranges between 0 and 1:  indicating a cohesive interface and  a completely damage interface respectively. We assume a partition of the traction into a reversible and an irreversible contribution as a linear function of . The former corresponds to the traction elastically stored by the interface which can be given back during a quasi-static discharge. The latter contribution is associated to the irreversible friction. Separating also the energy into a reversible elastic part and an irreversible contribution related to the dissipation and written in a Dupré form, we come out with a particular slip weakening relationship, hereinafter indicated as RCCM: In this relation,  is the tangential traction, n is the normal traction, d the dynamic friction coefficient, u is the slip, 1/2w is the energy stored by the layer and u is the yield level which can be written as , where Ct is the interface stiffness. The traction evolution exhibits a non linear behavior with slip, with a power law decay at the beginning, a minimum which depends on both the energy stored by the interface and the stiffness Ct, and a final asymptotic value related to the dynamic friction (Figure 1). Equating the elastic energy stored by the interface to the energy associated to a deformation of the layer we retrieve that the stiffness of the interface is Where is the harmonic mean of the shear modulus. The smaller the layer thickness is, the larger the stiffness of the interface is, and the larger the stored elastic energy is before dynamic rupture instability. When comparing the RCCM traction-slip evolution with the Traction-slip evolutions inferred from earthquake kinematic rupture inversion models [6], we retrieve a yield traction of the order of 10-15 MPa and a thickness of the fault zone of about one hundred meters, which is comparable with the size of the damaged zone observed on exhumed faults. Figure 1: Traction evolution with slip for the RCCM law Nucleation During the nucleation phase, the fault interface evolves quasi-statically under slow time-increasing remote loading condition. The quasi-static equilibrium equation relates the stress and the space derivative of the slip by the Hilbert transform: 0 is the initial stress, G* the reduced shear modulus, i.e., G/(1-), with  the Poisson coefficient. Finally a(t) is half the length of the evolving slip zone. We assume a peaked-shape of the traction, with the maximum value at the yield level. As the remote load continuously increases with time, the slip zone is extending with time. To solve the quasi-static equation together with the interface law, we seek for a finite development of the slip function onto a Chebyshev polynomial basis. A unique solution is possible if we additionally impose the finiteness condition for the traction at the boundaries of the crack [2]. For linear slip-weakening friction law, the development leads to a linear system, which can be iteratively solved for different values of the crack length. For non-linear interface law, a perturbation approach can be used leading to solve both for the slip function and the crack size at each time t, when the solution is known at time t-t. When the time step is small, we can linearize the interface law around the solution found at the previous step, and iterate using a Newton-Raphson method, to obtain the solution at the following time step. In Figure 2, the slip and the traction space-time evolutions during the nucleation phase, in the case of the RCCM interface law, are illustrated up to the onset of the dynamic instability. The slip and traction evolutions for both the RCCM and the LSW laws exhibit similar behavior beside some important differences. For the RCCM law, the nucleation depends on the shape of the remote loading, in particular the width of the loading function. The larger the loading is, the smaller the nucleation length becomes. The nucleation length depends on the slope of the interface law at zero slip. The LSW universal nucleation length is found to be a lower asymptotic limit for the RCCM nucleation length, when comparing both LSW and RCCM laws with same weakening rates at yield, as a result of possible smaller RCCM stress drop when the slip increases. Since the weakening rate scales with the interface stiffness in the case of the RCCM law, the larger the stiffness is the smaller the nucleation length becomes. The RCCM law allows therefore the occurrence of a wide range of earthquake sizes, depending on the thickness of the damaged layer. Small earthquakes could be associated with less matured fault interfaces of smaller damaged width, while large earthquakes would be related with more matured fault interfaces of larger width. Finally, for well-peaked remote loading function, stable quasi-static solutions can be extended even for large external loading, indicating that in some cases slipping zones can accommodate remote loading without evolving into a dynamic rupture. Figure 2: Slip (left) and traction (right) as functions of space and time up to the triggering of the dynamic rupture. Dynamic rupture propagation We finally investigate the propagation of a seismic rupture with the RCCM law and compare it to the results obtained with a LSW law. Comparison is done assuming either the same slip critical distance Dc, or the same breakdown energy. We numerically solve the elastodynamic radiation within an elastic unbounded medium using a non-smooth spectral element method [7]. The dynamic rupture is solved using the non-smooth contact condition method. Specifically, we solve iteratively a Signorini criterion for the normal traction and a modified Coulomb criterion for the tangential traction. The initial condition is prescribed consistently as a perturbation of the traction profile obtained at the onset of the dynamic instability using the nucleation analysis described previously. We assume a peaked function for the tectonic loading and we investigate the supershear generation as a function of the strength parameter on the fault and the width of the loading function. The strength parameter is the ratio between the stress required to reach the yield level u, from the initial condition 0, and the stress drop, that is the difference between the initial and final stress d level on the fault. This parameter controls the asymptotic rupture speed. As in the LSW case, two modes of supershear rupture are found: a crack mode and a pulse mode, with no major differences for the same selection of parameters. Since the cohesive zone shrinks when the rupture progresses, both the LSW and RCCM ruptures tend to have the same singular behavior. However, due to the higher energy dissipation of the RCCM law, related to the damage processes of the interface, the distance from the nucleation at which supershear occurs is significantly increased compared to the LSW case. Damage processes tend to delay the onset of supershear, and make it less unlikely than in the case of a LSW rupture. Finally, when looking at the radiation field, high frequencies are strongly attenuated in the case of the RCCM law compared to the LSW law, with differences in amplitude which can be as high as one order of magnitude in the directive quadrant. This result still remains to be confronted with seismological observations. References [1]Y. Ida. Cohesive force across the tip of a longitudinal shear crack and Griffith’s specific surface energy. J. Geophys. Res., 77, 3796–3805, doi:10.1029/JB077i020p03796, 1972. [2]K. Uenishi and J. R. Rice. Universal nucleation length for slip-weakening rupture instability under nonuniform fault loading. J. Geophys. Res., 108 (B1), 2042, doi:10.1029/2001JB001681, 2003. [3]A. L. Ruina. Slip instability and state variable friction laws. J. Geophys. Res., 88, 10,359–10,370, doi:10.1029/ JB088iB12p10359, 1983. [4]A. Bizzarri and M. Cocco. Slip‐weakening behavior during the propagation of dynamic ruptures obeying rate‐ and state dependent friction laws, J. Geophys. Res., 108(B8), 2373, doi:10.1029/2002JB002198, 2003. [5]M. Raous, L. Cangémi and M. Cocou. A consistent model coupling adhesion, friction and unilateral contact, Computer Methods in Applied Mechanics and Engineering, 177, n° 3-4, 1999, 383-399, doi:10.1016/S0045-7825(98)00389-2. [6]E. Tinti, P. Spudich and M. Cocco. Earthquake fracture energy inferred from kinematic rupture models on extended faults, J. Geophys. Res., 110, B12303, doi:10.1029/2005JB003644, 2005- [7]G. Festa and J.-P. Vilotte. Influence of rupture initiation on the intersonic transition: crack-like versus pulse like modes. Geophys. Res. Lett., 33(15), L15320, 2006.
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hal-01251926, version 1 (07-01-2016)

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Gaetano Festa, Jean-Pierre Vilotte, Michel Raous. Friction laws versus material interfaces for earthquake rupture dynamics: seismological implications. Euromech 514 : New trends in Contact Mechanics, Michel Raous-Peter Wriggers, Mar 2012, Cargese - Corsica, France. ⟨hal-01251926⟩
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