Non-local and non-linear problems in the mechanics of disordered systems: application to granular media and rigidity problems

Many studies originally made in the field of statistical physics have led to results which are directly related to different features of random packings of granular materials. The authors outline some of these developments: geometrical properties of packings, electrical transport in the pore or the grain space, mechanics of granular arrays. They mainly focus on the following points: (i) The geometrical properties of random close packings of grains which can be studied using the tools of stereology. (ii) The distribution of contacts which controls the transport characteristics in the grain phase. (iii) The rigidity of granular structures which can be mapped onto a lattice model and which can be approached by a central-force percolation model. (iv) The non-linear properties due to the asymmetry of the contact behaviour between two grains, which can be studied on randomly depleted lattice models. Such properties, which come from the existence of non-linear local laws, are extended to the strong irreversible behaviour of fractured structures. (v) Finally, they consider some minimal structure approaches which model the distribution of stresses along a granular system.