Mechanics near a jamming transition: a minimalist model

Concentrated suspensions of soft particles exhibit a pasty-like mechanical behaviour due to the jamming of the particles. In particular these solutions display a yield stress and typically exhibit very non-linear mechanical responses. Here, we introduce a very naive mode-coupling model that describes in a mean-field way the mechanics of such systems close to the jamming transition. Despite its simplicity this model correctly reproduces many features characteristic of real systems, in particular, non-linear features of the apparent viscoelastic modulus. It is also shown that, as linear response theory can not be applied, different mechanical tests provide different information. Moreover our model is a helpful tool to understand, in terms of trajectories in phase space, the origin of the observed non-linearities, which are essentially of two types.