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[SCG50-01] Homoclinic bifurcation in behavior of an asperity in a viscoelastic medium and effect of strength contrast
Keywords:Viscoelastic Relaxation, Earthquake sequence simulation, Seismic-aseismic transition
Interaction between slip on a fault and deformation of the surrounding medium is of great importance in determining the mode of the fault behavior such as repeating earthquakes, aseismic transients, steady slip, and pervasive bulk flow. Miyake and Noda (2019) developed a dynamic earthquake sequence simulation method for a fault embedded in a Maxwell linear viscoelastic medium and investigated the behavior of a rate-weakening patch. They found that there are two different seismic-aseismic transitions: seismic cycle (EQ)- steady slip (SS) transition and seismic cycle (EQ) - permanent stuck (ST) transition. In ST, the patch does not generate a seismic event because efficient viscoelastic relaxation dominates elastic loading to the patch and the slip rate there decreases towards zero. There are many previous studies on EQ-SS transition, and it is understood to be associated with Hopf bifurcation. On the other hand, the EQ-ST transition is not well understood. Miyake and Noda (2019) assumed uniform distribution of the frictional strength for simplicity. Ikari et al. (2011) reported that rate-weakening minerals showed frictional strength comparable to the Byerlee’s law, while rate-weakening minerals might have lower frictional strength. Because stress heterogeneity caused by the strength heterogeneity relaxes in viscoelastic medium, the effect of the strength contrast between the rate-weakening and rate-strengthening regions may be important unlike in the elastic medium.
In this study, investigated the EQ-ST transition in detail to understand it from the viewpoint of nonlinear dynamics. First, we performed a parametric study for relaxation time and frictional strength contrast using the simulation developed by Miyake and Noda (2019) for two-dimensional antiplane problems of a fault model with a velocity-weakening patche placed inside velocity-strengthening region. A rate- and state-dependent friction law (aging law) is assumed on the fault plane. For the frictional strength, the reference frictional coefficient f0 is changed between inside the rate-weakening patch (f0(rw)=0.6) and outside it (0.3<f0(rs)<0.6). The relaxation time tc is varied to see under when the EQ-ST transition takes place. As the initial conditions for the numerical simulations, we calculated a near-steady-state condition by simulations of increased state-evolution distance. As a result of a parametric study, it was found that the frictional strength contrast enhances permanent stuck of the patch, helping viscoelastic relaxation. At f0(rw)-f0(rs)=0.3, where the strength contrast is the largest in this study, the maximum relaxation time required to the permanent stuck was increased by more than one order of magnitude. Because the continuum model has large degree of freedom, we simplified it based on spatial averages over the rate-weakening and rate- strengthening regions. A parametric study on the simplified system with only two degrees of freedom showed remarkably similar results to the continuum model in terms of the phase diagram of the fault behavior and trajectories. The simplified model has two fixed points, one is a unstable spiral corresponding to the steady-state solution with the elastic medium, and the other is a saddle point. In EQ, the saddle point is outside the limit (seismic) cycle. As the relaxation time decreases or the strength contrast increases, the saddle point approaches to the limit cycle. The EQ-ST transition occurs when the saddle point touches the limit cycle and it disappears. Such bifurcation is referred to as a homoclinic bifurcation. This is a new type of the seismic-aseismic transition caused by efficient off-fault deformation. The condition for the EQ-ST transition revealed by the parametric study was explained by a new nondimensionalization of the relaxation time by a time scale of loading to the rate-weakening patch accounting for elastic loading and relaxation of the stress heterogeneity caused by the strength contrast.
In this study, investigated the EQ-ST transition in detail to understand it from the viewpoint of nonlinear dynamics. First, we performed a parametric study for relaxation time and frictional strength contrast using the simulation developed by Miyake and Noda (2019) for two-dimensional antiplane problems of a fault model with a velocity-weakening patche placed inside velocity-strengthening region. A rate- and state-dependent friction law (aging law) is assumed on the fault plane. For the frictional strength, the reference frictional coefficient f0 is changed between inside the rate-weakening patch (f0(rw)=0.6) and outside it (0.3<f0(rs)<0.6). The relaxation time tc is varied to see under when the EQ-ST transition takes place. As the initial conditions for the numerical simulations, we calculated a near-steady-state condition by simulations of increased state-evolution distance. As a result of a parametric study, it was found that the frictional strength contrast enhances permanent stuck of the patch, helping viscoelastic relaxation. At f0(rw)-f0(rs)=0.3, where the strength contrast is the largest in this study, the maximum relaxation time required to the permanent stuck was increased by more than one order of magnitude. Because the continuum model has large degree of freedom, we simplified it based on spatial averages over the rate-weakening and rate- strengthening regions. A parametric study on the simplified system with only two degrees of freedom showed remarkably similar results to the continuum model in terms of the phase diagram of the fault behavior and trajectories. The simplified model has two fixed points, one is a unstable spiral corresponding to the steady-state solution with the elastic medium, and the other is a saddle point. In EQ, the saddle point is outside the limit (seismic) cycle. As the relaxation time decreases or the strength contrast increases, the saddle point approaches to the limit cycle. The EQ-ST transition occurs when the saddle point touches the limit cycle and it disappears. Such bifurcation is referred to as a homoclinic bifurcation. This is a new type of the seismic-aseismic transition caused by efficient off-fault deformation. The condition for the EQ-ST transition revealed by the parametric study was explained by a new nondimensionalization of the relaxation time by a time scale of loading to the rate-weakening patch accounting for elastic loading and relaxation of the stress heterogeneity caused by the strength contrast.