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The last few years have seen an increasing interest of the statistical physics community in the study of the yielding phenomenon of driven amorphous materials. Foams, gels, colloidal glasses, granular materials or bulk metallic glasses respond elastically when a small external strain is applied, but they yield and flow due to internal plastic rearrangements if the drive is large enough. The dynamical crossover between these two regimes can be understood as a dynamical phase transition and it has been shown to be accompanied by critical-like phenomena like growing correlation lengths and avalanches. Nevertheless, specificities of this transition are still to be understood in extent.

Considerable efforts have been devoted for more than 30 years to the study of a related problem, the depinning of elastic manifolds moving in random media*. In the latter case, (mostly) exact exponents, scaling functions, and avalanche shapes were derived using scaling analysis and renormalization techniques. For the yielding transition, the (slow) process of consensus building has not converged yet, but there are reasons to believe that the results on avalanche statistics obtained in the depinning problem cannot be directly transposed to this field, because the propagator controlling stress redistribution is partly negative, which affects the density of sites close to yielding. Whether this features only induces an effective dimensional reduction, or whether it exhibits a completely distinct set of critical exponents, still needs to be clarified. Scaling relations between critical exponents have been proposed and tested in diverse elasto-plastic models (EPM), yet analytical calculations beyond mean field are scant. Recent efforts to relate EPM to problems of motion through a disordered landscape open new vistas for the understanding of yielding and transport properties under slow driving, but there is still no consensual theory explaining the flow exponents (the low-shear-rate rheology).

The situation is somewhat similar on the experimental side: The depinning phenomenon has benefited from a very detailed experimental characterization in various systems (magnetic domain walls, contact lines, vortices), including avalanche statistics and shapes, which has permitted comparison to the theory. Amorphous plasticity is not on quite so good a footing, with only a few attempts to characterize the distribution of stress drops in deformed systems. The situation is however improving, thanks to several recent efforts, e.g. those combining mechanical deformation and confocal microscopy in colloidal glasses.

The foregoing discussion is related to the critical aspects of the yielding phenomenon. In a number of real systems, the onset of flow is in fact discontinuous and implies a coexistence between flowing and immobile states. EPM and other theoretical studies have proposed

possible mechanisms that may influence the continuous or discontinuous character of the transition. Nevertheless, it turns out to be experimentally difficult to control the transition in a systematic way by changing some experimental parameter. Recent works on weakly vibrated granular media represents a notable exception, insofar as the intensity of external shaking could be used to continuously tweak the flow curve towards nonmonotonicity. Similar systems of vibrated grains have also permitted the experimental realization of a Gardner transition, a transition which may be important for the theory of glasses and which has been associated with shear yielding. On a related note, cutting-edge atomistic simulations suggest that the ductility or brittleness of the yielding phenomenon hinges on the initial preparation of the glass, rather than the microscopic interactions between particles or the dynamics.

In  rheology, considerable attention has recently been devoted to the study of transient regimes. For instance, one can mention the study of the load curves at fixed shear rate, that can exhibit stress overshoots depending on the initial preparation and non-trivial scalings of the time to reach the stationary state. Other examples include creep under imposed stress, the dynamics of relaxation and the residual stresses after sudden cessation of the driving, and oscillatory regimes. In the latter category, the Large Amplitude Oscillatory Strain (LAOS) protocol probes the nonlinear behavior and the frequency dependent one at the same time, and therefore involves a complex interplay between plastic deformation and internal relaxation.

A more unexpected emerging avenue is the study of systems with internal activity, such as living tissues or dense cell assemblies. The general ideas exploited for the description of amorphous systems can indeed be expanded to incorporate new types of events, such as cell division (assimilated to a local anisotropic dilation) and cell death (local isotropic contraction). At present, new experimental tools are providing information on the statistical fluctuations in such systems.


*Systems such as magnetic and ferroelectric domain walls, contact lines in wetting, fracture fronts, and arrays of vortices in type-II superconductors, present a common phenomenology when we consider them as elastic objects embedded in a disordered medium and driven by an external force. If the external force is weak, the elastic object eventually gets pinned in the disordered landscape and its steady velocity is zero. If the force is strong enough, instead, the manifold will overcome even the largest pinning centers, reaching a steady state of mean finite velocity. This dynamical phase transition, known as the depinning transition,  is well-documented in a literature that nowadays goes well beyond the depinning itself, describing also the equilibrium problem of the elastic line, thermally activated dynamical regimes, different effective elasticities and disorder types, the fast flow regime at large driving and, in all these cases, the relation between geometry and transport properties.
Note: Some text find here has been extracted from "Deformation and flow of amorphous solids: Insights from elastoplastic models", A. Nicolas et al. Rev. Mod. Phys. 90, 045006 (2018)
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