On the seismic cycle seen as a relaxation oscillation

Abstract

An earthquake is commonly described as a stick-slip frictional instability occurring along preexisting crustal faults. The seismic cycle of earthquake recurrence is characterized by long periods of quasi-static evolution, which precede sudden slip events accompanied by elastic wave radiation: the earthquake. This succession of processes over two well-distinguished time-scales recalls the behavior of nonlinear relaxation oscillations. We explore this connection by studying, in the framework of rate-and-state friction, the sliding of two identical slabs of elastic solid driven in opposite directions with a constant relative velocity. Our first innovation is to establish that the motion of a spring–block system is an asymptotic mechanical analogue of the frictional sliding of a single interface from which elastic waves radiate. Due to wave reflection at the boundaries, the equivalent mass of the block M = k(h/c _s )²/12 is not independent of the equivalent spring stiffness k, where h/2 denotes the slab thickness and c _s is the shear wave speed. Considering a non-monotonic friction law, we show that the relaxation oscillation regime is reached when the characteristic time-scale of frictionless oscillations is much greater than the characteristic time of frictional memory effects: (M/k)^1/2 ≫ L/V _*. We combine a composite approximation of the stick-slip cycle and numerical studies to show that the interfacial relaxation oscillations result from the subtle interplay of the non-monotonic properties of the friction law driving the long stress build-up of the quasi-static phase, and the inertial control of the fast slip phase originating from the wave propagation. We discuss the geophysical consequences for earthquake mechanics, and connections between the rate-and-state and Coulomb models of friction.

Keywords:

• earthquake mechanics
• relaxation oscillation
• stick-slip
• rate-and-state friction
• non-monotonic
• continuation
• dynamical system
• stress drop control

Acknowledgements

This study has been partially supported by the European Research and Training Network DIGA (Degradation and Instabilities in Geomaterials with Application to Hazard & Mitigation; RTN/DIGA–HPRN-CT-2002-00220, Oct. 2002-Sep. 2006). TP thanks E.C. Aifantis for the funding opportunity and his interest in the subject. JHPD is supported by Newnham College, Cambridge, and by the Royal Society through a University Research Fellowship.

Notes

Notes

1. More accurately, the first term that is neglected should be small in comparison with the last term retained.

2. It could be realized in our model if some impedance boundary condition were applied at y = ±h/2.

3. Note the hypotheses vφ ≪ 1 or vφ ≫ 1 break down at the point I.

4. This is not quite right actually. Looking at Figure (3) shows that there is a fast transition phase where μ(t) is attracted or repelled from μ _ss when v > V, the inertial phase having not started or ended yet.