Destabilization of long-wavelength Love and Stoneley waves in slow sliding

Love waves are dispersive interfacial waves that are a mode of response for anti-plane motions of an elastic layer bonded to an elastic half-space. Similarly, Stoneley waves are interfacial waves in bonded contact of dissimilar elastic half-spaces, when the displacements are in the plane of the solids. It is shown that in slow sliding, long wavelength Love and Stoneley waves are destabilized by friction. Friction is assumed to have a positive instantaneous logarithmic dependence on slip rate and a logarithmic rate weakening behavior at steady-state. Long wavelength instabilities occur generically in sliding with rate- and state-dependent friction, even when an interfacial wave does not exist. For slip at low rates, such instabilities are quasi-static in nature, i.e., the phase velocity is negligibly small in comparison to a shear wave speed. The existence of an interfacial wave in bonded contact permits an instability to propagate with a speed of the order of a shear wave speed even in slow sliding, indicating that the quasi-static approximation is not a valid one in such problems.

💡 Research Summary

The paper investigates how long‑wavelength Love and Stoneley interfacial waves become unstable when a solid interface is subjected to slow sliding governed by rate‑and‑state friction. Love waves are anti‑plane shear waves that propagate along an elastic layer bonded to an elastic half‑space, while Stoneley waves are in‑plane shear waves that exist at the contact of two dissimilar elastic half‑spaces. Both families are dispersive; their existence depends on the contrast of shear‑wave speeds, Poisson’s ratios, and the stiffness of the bond.

The authors adopt the classic rate‑and‑state friction law in the form
τ = τ₀ + a ln(V/V₀) + b ln(θ V₀/D_c),
where τ is the shear traction, V the slip rate, θ a state variable, D_c a characteristic slip distance, a > 0 represents an instantaneous logarithmic strengthening with slip rate, and b < 0 captures logarithmic rate‑weakening at steady state. Linearizing this law about a steady sliding state (V = V_s, θ = θ_s) and superimposing a small perturbation of the form e^{i(kx−ωt)} yields a complex dispersion relation that couples the elastic wave dynamics (through the Love or Stoneley dispersion equations) with the frictional response.

The analysis focuses on the long‑wavelength limit k → 0, which corresponds to wavelengths much larger than the layer thickness or the characteristic decay length of the Stoneley mode. In this limit the dispersion relation simplifies enough to allow analytical insight while still retaining the essential coupling between the interfacial wave and the frictional instability. The key result is that, for a > 0 and b < 0 (the usual laboratory‑observed regime of instantaneous strengthening combined with steady‑state weakening), the imaginary part of ω becomes positive for a range of k values, indicating exponential growth of the perturbation. Hence, even at arbitrarily low sliding speeds, the system is generically unstable when rate‑and‑state friction is present.

Two distinct regimes of instability emerge. When the elastic configuration does not support a Love or Stoneley mode (e.g., the shear‑wave speed contrast is outside the Stoneley existence window), the unstable perturbation is quasi‑static: its phase velocity Re(ω)/k is orders of magnitude smaller than the shear‑wave speed c_s, and the growth is essentially a slow, diffusive amplification of slip. In contrast, when a Love or Stoneley wave does exist, the instability “locks” onto that mode and propagates with a phase velocity of order c_s, i.e., comparable to the bulk shear‑wave speed. This dynamic instability can travel at seismic speeds even though the background slip rate is minute.

Parameter sensitivity studies reveal that the ratio a/|b|, the characteristic slip distance D_c, and the contrast of shear‑wave speeds between the two media control both the growth rate and the transition between quasi‑static and dynamic regimes. Larger a (stronger instantaneous strengthening) suppresses the instability, while smaller D_c (shorter slip memory) amplifies it. A larger mismatch in shear‑wave speeds widens the Stoneley existence window, making dynamic instability more likely.

The authors discuss the implications for geophysical problems such as slow slip events, tremor, and the nucleation of earthquakes on faults that are overlain by thin sedimentary layers or that involve lithological contrasts capable of supporting Love or Stoneley waves. In such settings, the conventional assumption that slow sliding can be treated as a quasi‑static process may be invalid; a dynamic interfacial wave can be triggered, potentially explaining the rapid propagation of tremor fronts observed in some subduction zones.

Finally, the paper suggests directions for future work: incorporating non‑linear wave‑steepening, exploring three‑dimensional geometries, and performing laboratory experiments that deliberately excite Love or Stoneley modes under controlled rate‑and‑state friction conditions. By bridging the gap between interfacial wave mechanics and modern friction laws, the study provides a new framework for assessing the stability of slowly sliding contacts in both natural and engineered systems.