Frictional state evolution laws and the non-linear nucleation of dynamic shear rupture

Abstract

We assess if a characteristic length for a non-linear interfacial slip instability follows from theoretical descriptions of sliding friction. We examine friction laws and their coupling with the elasticity of bodies in contact and show that such a length does not always exist. We consider a range of descriptions for frictional strength and show that the area needed to support a slip instability is negligibly small for laws that are more faithful to experimental data. This questions whether a minimum earthquake size exists and shows that the nucleation phase of dynamic rupture contains discriminatory information on the nature of frictional strength evolution.

Figures (5)

• Figure 1: (a) Illustration of one-parameter family of state evolution laws for sliding friction, (\ref{['eq:interm']}). (b) Definition of slip $\delta$ and slip rate $V$ for an example of in-plane distribution of relative interfacial displacement. (c,d) Results from numerical solutions for slip-rate evolution for rupture of a rate-weakening interface ($a<b$) at late stages of quasi-static instability. The spatial distribution of a diverging slip rate is shown at instants in time, with time progress corresponding to darkening greyscale. Slip rate is scaled by its instantaneous value at $x=0$, which is diverging and about which the distribution is symmetric. Distance $x$ is scaled by $\epsilon L_b$ where $L_b$ is an elasto-frictional length scale (Appendix A.3). Solutions are found for different values of parameter $\epsilon$ and the same value of parameter $a/b=0.6$. The slip rate in (d) diverges over distances ten-times smaller than that in (c). The solutions approach the same distribution (red-dashed) as the instability progresses.
• Figure 2: Results from linear stability analysis of diverging slip-rate solutions (\ref{['eq:div']}). Illustration of eigenvalue trajectories in complex plane (black curves) as (a) $\epsilon$ is fixed and $a/b$ is varied, and (b) $a/b$ is fixed and $\epsilon$ is varied. (a) At fixed $\epsilon$, loss of stability occurs via Hopf bifurcations $a/b$ is increased Viesca:2016bga. Open circles are eigenvalue positions for the six least stable modes at fixed values of $a/b$. (b) At fixed $a/b$, a stable blow-up solution (\ref{['eq:div']}) becomes unstable and the eigenvalues tend towards a purely real set (red circles) as the state evolution law transitions from aging to slip ($\epsilon\rightarrow0$).
• Figure 3: Numerical solutions (black) for eigenvalues $\lambda\geq0$. Pointwise divergence is a stable, attractive solution for ${a/b<0.3781...}$ Above this critical value, unstable modes appear sequentially. Eigenvalues for the first seven unstable modes are shown, with asymptotic behavior as $a/b\rightarrow 1$ (red-dashed). Two positive eigenvalues at $a/b=0.65$ (red circles) shown for comparison with Figure \ref{['figeig']}b.
• Figure 4: Snapshots of the distribution of slip velocity $V$ over positions along the interface $x$ at instants in time for a fixed value $a/b=0.65$ and two values of $\epsilon$ near a stability transition point. Time snapshots are at equal intervals of the peak slip rate. As the slip law is approached with decreasing $\epsilon$, an abrupt transition from localized to migrating slip occurs for such large values of $a/b$.
• Figure C.1: Snapshots of diverging slip velocity $V$ with distance along the interface $x$. The instability in slip rate occurs within a steady-state rate-weakening patch ($a/b<1$) confined to $|x|\leq 2\epsilon L_b$. For $|x|>2\epsilon L_b$, the frictional response is steady-state rate neutral ($a/b=1$), which alone cannot sustain an instability. The instability is self-sustaining within the rate-weakening patch despite the size of the rate-weakening patch being much smaller than critical wavelength $\lambda_{cr}$ predicted by linear stability analysis.