  The rupture of dry frictional interfaces occurs through the propagation of
fronts breaking the contacts at the interface. Recent experiments have shown
that the velocities of these rupture fronts range from quasi-static velocities
proportional to the external loading rate to velocities larger than the shear
wave speed. The way system parameters influence front speed is still poorly
understood. Here we study steady-state rupture propagation in a one-dimensional
(1D) spring-block model of an extended frictional interface, for various
friction laws. With the classical Amontons--Coulomb friction law, we derive a
closed-form expression for the steady-state rupture velocity as a function of
the interfacial shear stress just prior to rupture. We then consider an
additional shear stiffness of the interface and show that the softer the
interface, the slower the rupture fronts. We provide an approximate closed form
expression for this effect. We finally show that adding a bulk viscosity on the
relative motion of blocks accelerates steady-state rupture fronts and we give
an approximate expression for this effect. We demonstrate that the 1D results
are qualitatively valid in 2D. Our results provide insights into the
qualitative role of various key parameters of a frictional interface on its
rupture dynamics. They will be useful to better understand the many systems in
which spring-block models have proved adequate, from friction to granular
matter and earthquake dynamics.
