  The origin of the abrupt shear thickening observed in some dense suspensions
has been recently argued to be a transition from frictionless (lubricated) to
frictional interactions between immersed particles. The Wyart-Cates rheological
model, built on this scenario, introduced the concept of fraction of frictional
contacts $f$ as the relevant order parameter for the shear thickening
transition. Central to the model is the "equation-of-state" relating $f$ to the
applied stress $\sigma$, which is directly linked to the distribution of the
normal components of non-hydrodynamics interparticle forces. Here, we develop a
model for this force distribution, based on the so-called $q$-model that we
borrow from granular physics. This model explains the known $f(\sigma)$ in the
simple case of sphere contacts displaying only sliding friction, but also
predicts strong deviation from this "usual" form when stronger kinds of
constraints are applied on relative motion. We verify these predictions in the
case of contacts with rolling friction, in particular a broadening of the
stress range over which shear thickening occurs. We finally discuss how a
similar approach can be followed to predict $f(\sigma)$ in systems with other
variations from the canonical system of monodisperse spheres with sliding
friction, in particular the case of large bidispersity.
