  To study the microscopic origins of friction, we build a framework to
describe the collective behaviour of a large number of individual
micro-junctions forming a macroscopic frictional interface. Each micro-junction
can switch in time between two states: A pinned state characterized by a
displacement-dependent force, and a slipping state characterized by a
time-dependent force. Instead of tracking each micro-junction individually, the
state of the interface is described by two coupled distributions for (i) the
stretching of pinned junctions and (ii) the time spent in the slipping state.
We show how this framework represents an overarching structure for important
models existing in the friction literature. We then use it to study
systematically the effect of the time-scale that controls the duration of the
slipping state. We first find the steady-state friction force as a function of
the sliding velocity. As the framework allows for a whole family of
micro-junction behaviour laws, we show how these laws can be chosen to obtain
monotonic (strengthening or weakening) or non-monotonic velocity dependence at
the macroscale. By then considering transient situations, we predict that the
macroscopic static friction coefficient is strongly influenced by the way the
interface was prepared, in particular by the slip dynamics of the previous
sliding event. We also show that slow slip spontaneously occurs in the
framework for a wide range of behaviour laws.
