A second scale of description of the material is the mesoscopic scale. Indeed many phenomenological models of the rheology of glassy materials are located at this intermediate scale. Peter Sollich, with whom I collaborated at King’s College in London, is precisely a specialist of the stochastic approach for this type of theoretical models and introduced what is considered to be a breakthrough in the field, the Soft Glassy Rheology (SGR). In this model, inspired from the so called trap model the system is divided conceptually in mesoscopic subcells to which one associates a local strain l (figure 2). This model is analytically soluble and provides a theoretical justification for the macroscopic stress-strain response as well as for the stress and energy drop distributions.
During my collaboration with Peter Sollich I started a systematic comparison of my atomistic results with the prediction/assumption of this mesoscopic model. Our study consists in extracting regularly snapshots of the glass at different macroscopical strains, then to partition the system in mesoscopic subcells (typically a cell contains about 30 atoms) and to apply a virtual defor- mation analysis on these cells by a PEM technique. For each snapshot this method permits to obtain mesoscopic quantities on each subcells, local strain l, minimum of the potential energy Emin, elastic constant k, yield strain lY    and stress σY    (figure 2), and to compare these mesoscopic quantities with the quantities introduced in SGR. Using this method we studied at a mesoscopic level the spatial propagation of the perturbation generated by localized plastic rearrangement and compared the results of the quasistatic and finite shear rate simulations. This approach allows for exciting new possibilities in order to estimate in a self consistent manner the effective temperature used in SGR.
Also, at the mesoscopic scale, phe- nomenological models called yield stress models were introduced in the last decade [1]. In collaboration with Lyd ́eric Bocquet (one of the authors of [1]) I compared the results of the yield models with the results obtained from the simulation of the polydisperse glass. Very promisingly the two approaches (atomic vs mesoscopic scale) showed quantitatively similar results at finite shear rates. By careful analysis of the spatiotemporal het- erogeneity of the flow (using sophisticated statistical tools such as the 4-point correla- tion function χ4) it appears that the flow of the glass can be classified in three regimes. For sufficiently large shear rates the flow appears homogeneous independently of the shear rate. At intermediate shear rates an heterogeneous cooperative behaviour builds up on a length scale ξ that scales as a power law of the shear rate, γ ̇−ν, where ν ≈ 0.9. Finally at low shear rates (and in the qua- sistatic simulations) a saturated regime is reached where finite size effects are observed [3]. These results open important avenues for the understanding, for example, of the flow of confined complex fluids or granulars.