  In this study, we analyse the famous Aw-Rascle system in which the difference
between the actual and the desired velocities (the offset function) is a
gradient of a singular function of the density. This leads to a dissipation in
the momentum equation which vanishes when the density is zero. The resulting
system of PDEs can be used to model traffic or suspension flows in one
dimension with the maximal packing constraint taken into account. After proving
the global existence of smooth solutions, we study the so-called "hard
congestion limit", and show the convergence of a subsequence of solutions
towards a weak solution of an hybrid freecongested system. In the context of
suspension flows, this limit can be seen as the transition from a suspension
regime, driven by lubrication forces, towards a granular regime, driven by the
contact between the grains.
