Introducing Linear Marginal Stability Hypothesis for Obtaining Slip-Front-Propagation Velocity with the Slip- and Slip-Velocity-Dependent Friction Law

Abstract:

The slip-front-propagation velocity (SFPV) on the interface between two media has been attracted the interests of researchers, including seismological studies. Actually, we have also considered an infinitely long viscoelastic block on a rigid substrate, and if we assume the friction law depending on the slip velocity with the quadratic form, SFPV has been analytically obtained (Suzuki and Matsukawa, 2019). The purpose of the present presentation is to extend the result to the model with the friction law depending on both the slip and slip velocity.

We introduce here the Linear Marginal Stability Hypothesis (LMSH). This hypothesis requires linearizing the governing equation to obtain the SFPV. This assumes the plane wave solution near the front, u~exp(i (kx -ωt)), where u is the slip, and k and ω are the complex wave number and frequency, respectively. We should obtain the values of four quantities, the real part of k (k_r), the imaginary part of k (k_i), the real part of ω (ω_r), and the imaginary part of ω (ω_i) by using four equations, the real and imaginary parts of the dispersion relation, the growth and propagation stabilities. The SFPV is given by ω_i/k_i. Note that ω_i and k_i must be positive.

Based on the idea of LMSH, we linearize the governing equation (e.o.m. of the block) to obtain the terms describing the friction stress, -C₁ u + C₂ \dot{u}, where C₁ and C₂ are constants, and the overdot stands for the temporal differentiation. To treat both hardening and weakening behaviors, we allow the positive and negative C₁ and C₂.

We obtained the cubic equation for ω_i by employing LMSH. We have found that the C₁ -C₂ phase space can be divided into 12 regions in terms of the numbers of the real and positive solutions. The future purpose will be to categorize the numbers of the real and positive solutions for k_i, and systematically understand the SFPV.