The dynamics of unsteady frictional slip pulses

Self-healing slip pulses are major spatiotemporal failure modes of frictional systems, featuring a characteristic size $L(t)$ and a propagation velocity $c_{\mathrm{p}}(t)$ ($t$ is time). Here, we develop a theory of slip pulses in realistic rate-and-state dependent frictional systems. We show that slip pulses are intrinsically unsteady objects – in agreement with previous findings – yet their dynamical evolution is closely related to their unstable steady-state counterparts. In particular, we show that each point along the time-independent $L^{(0)}({\tau }_{\mathrm{d}})\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}{c}_{\mathrm{p}}^{(0)}({\tau }_{\mathrm{d}})$ line, obtained from a family of steady-state pulse solutions parameterized by the driving shear stress ${\tau }_{\mathrm{d}}$, is unstable. Nevertheless, and remarkably, the $c_{\mathrm{p}}^{(0)}[L^{(0)}]$ line is a dynamic attractor such that the unsteady dynamics of slip pulses (when they exist) – whether growing ($\dot{L}(t)\text{\hspace{0.17em}⁣}>\text{\hspace{0.17em}⁣}0$) or decaying ($\dot{L}(t)\text{\hspace{0.17em}⁣}<\text{\hspace{0.17em}⁣}0$) – reside on the steady-state line. The unsteady dynamics along the line are controlled by a single slow unstable mode. The slow dynamics of growing pulses, manifested by $\dot{L}(t)\mathrm{/}{c}_{\mathrm{p}}(t)\text{\hspace{0.17em}⁣}\ll \text{\hspace{0.17em}⁣}1$, explain the existence of sustained pulses, i.e.~pulses that propagate many times their characteristic size without appreciably changing their properties. Our theoretical picture of unsteady frictional slip pulses is quantitatively supported by large-scale, dynamic boundary-integral method simulations.