Revisiting mean-field elasto-plastic models at the mesoscopic scale

Encompassing very dissimilar systems (such as foams, emulsions, or granular materials), amorphous materials are composed of constituents of different shapes and sizes, such as bubbles in a soap foam or sand grains in a sandpile, so that they exhibit a structural disorder that plays a determinant role in their mechanical properties, while challenging their very description. Several elasto-plastic models have been developed at the mesoscopic scale, in order to account for the plasticity in sheared amorphous materials, such as the Hébraud- Lequeux (HL) model [P. Hébraud & F. Lequeux, Phys.Rev.Lett. 81, 2934 (1998)]. They have proven to be rather successful in reproducing certain features observed in amorphous systems, but not all at once. Moreover, a consistent picture connecting them is still missing. Here we discuss the physical ingredients that are put in such mean-field models, distinguishing between thermal and mechanical noises in the mean-field dynamics of amorphous materials. We focus in particular on the role of structural disorder, implemented by means of a distribution of energy barriers for the system to overcome locally when an external constant shear rate is applied to the material, and discuss specifically its implications for a generalization of the HL model [E. Agoritsas et al., Eur.Phys.J.E 38, 71 (2015)].