  Friction in static and sliding contact of rough surfaces is important in
numerous physical phenomena. We seek to understand macroscopically observed
static and sliding contact behavior as the collective response of a large
number of microscopic asperities. We develop an efficient numerical framework
that can be used to investigate how the macroscopic response of multiple
frictional contacts depends on long-range elastic interactions, different
constitutive assumptions about the deforming contacts and their local shear
resistance, and surface roughness. We approximate the contact between two rough
surfaces as that between a regular array of discrete deformable elements
attached to a elastic block and a rigid rough surface. The deformable elements
are viscoelastic or elasto/viscoplastic with a range of relaxation times, and
the elastic interaction between contacts is long-range. We find that the model
reproduces main macroscopic features of evolution of contact and friction for a
range of constitutive models of the elements, suggesting that macroscopic
frictional response is robust with respect to the microscopic behavior.
Viscoelasticity/viscoplasticity contributes to the increase of friction with
contact time and leads to a subtle history dependence. Interestingly,
long-range elastic interactions only change the results quantitatively compared
to the meanfield response. We find that sustained increase in the static
friction coefficient during long hold times suggests viscoelastic response of
the underlying material with multiple relaxation time scales. We also find that
the experimentally observed proportionality of the direct effect in velocity
jump experiments to the logarithm of the velocity jump points to a complex
material-dependent shear resistance at the microscale.
