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_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⁾(τd)−c⁽⁰⁾_p(τd)line, obtained from a family of steady-state pulse solutions parameterized by the driving shear stress τd, is unstable. Nevertheless, and remarkably, the c⁽⁰⁾_p[L⁽⁰⁾]line is a dynamic attractor such that the unsteady dynamics of slip pulses (when they exist)—whether growing ( ̇L(t)>0) or decaying ( ̇L(t)<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 ̇L(t)/cp(t)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.