Deformation and flow of amorphous solids: Insights from elastoplastic models

TL;DR: In this article, the authors review the physical insight provided by elastoplastic models into practical issues such as strain localization, creep and steady-state rheology, but also the fundamental questions that they address with respect to criticality at the yielding point and the statistics of avalanches of plastic events.

Abstract: The deformation and flow of disordered solids, such as metallic glasses and concentrated emulsions, involves swift localized rearrangements of particles that induce a long-range deformation field. To describe these heterogeneous processes, elastoplastic models handle the material as a collection of 'mesoscopic' blocks alternating between an elastic behavior and plastic relaxation, when they are too loaded. Plastic relaxation events redistribute stresses in the system in a very anisotropic way. We review not only the physical insight provided by these models into practical issues such as strain localization, creep and steady-state rheology, but also the fundamental questions that they address with respect to criticality at the yielding point and the statistics of avalanches of plastic events. Furthermore, we discuss connections with concurrent mean-field approaches and with related problems such as the plasticity of crystals and the depinning of an elastic line.

Summary

A. Short introduction to out-of-equilibrium transitions

• Statistical physics is largely concerned with phase transitions, whereby some properties of a system abruptly change upon the small variation of a control parameter.
• The paradigmatic example of an equilibrium phase transition is the Ising model, which consists of spins positioned on a lattice and interacting with their first neighbors.
• The latter are said to belong to the same universality class as the Ising model.

1. Avalanches in sandpile models

• As models featuring threshold dynamics and a toppling rule, EPM are also connected to the somewhat simpler sandpile models, introduced in Sec. III.A.
• Let us clarify some concepts using the latter class of systems.
• This means that small avalanches are more frequent than larger ones, but in such a fashion that no typical or characteristic size can be established, which has been called self-organized criticality.
• Let us note that the extremal dynamics used to trigger avalanches can be substituted by a very slow uniform loading of the columns of sand if some randomness is introduced in the stability thresholds.
• Different regimes of avalanches can be seen when the deposition rate is varied or, somewhat equivalently, when one inspects them at different frequencies ω.

2. Stress drops and avalanches in EPM

• Similarly to the instabilities in sandpile models, the plastic events occurring in EPM can trigger avalanches of successive ruptures.
• To facilitate the comparison with experiments or atomistic simulations, these avalanches are usually quantified by looking at the time series of the macroscopic stress σ(t) and, more specifically, at the stress drops ∆σ associated with plastic relaxation (in this chapter, the authors use a lowercase symbol σ for the stress to underscore that it is an intensive variable).
• Close to criticality, the duration T of these drops and their extensive size S ≡ ∆σLd in a system of volume Ld in d dimensions most often display statistics formally similar to Eq. (31), viz.
• T−τ ′ g(T/Tcut), (32) where the upper cut-offs Scut and Tcut entering the scaling functions f and g will typically depend on system size, e.g. Scut ∝ Ldf .
• In the following, the authors will pay particular attention to the possible impact of the peculiarities of the quadrupolar stress redistribution in EPM, notably its fluctuating sign, on the avalanche statistics.

B. Avalanches in mean-field models

• Shortly after the emergence of the first EPM, mean-field approximations were exploited to determine the statistics of avalanches.
• An exponent τ = 3/2 is then consistently found in the avalanche size scaling of Eq. (32).
• 43 For instance, Sornette (1992) proposed to map the Burridge-Knopoff model for earthquakes, introduced in Sec. III.A, onto a fiber bundle which carries a load equally shared among unbroken fibers (see Sec. IX.C).
• An avalanche of failures lasts as long as this extremal load remains below the initial load, so its size is given by the walker’s survival time close to an absorbing boundary, whence an exponent τ = 3/2.
• By contrast, stress fluctuations in disordered solids deplete local stresses close to σy, so much so that p(σ − y ) = 0 and p(σ) varies substantially close to σy.

1. Experiments

• Various experimental settings have been designed to characterize avalanche statistics in deformed amorphous solids in the last decade, even though experiments are still trailing behind the theoretical predictions and numerical computations in this area.
• A granular packing subject to the simultaneous application of pressure and shear was also shown to display stress drops with power-law statistics by Denisov et al. (2016).
• The power-law exponents, which seem to lie in a relatively broad range in Fig. 15c, were not fitted, but, upon rescaling, were reported to be in good agreement with the meanfield value τ = 3/2.
• From the gradient of the image intensity, they quantified the local pressure acting on each grain, hence the energy stored in it, and tracked the fluctuations of the global energy.
• This allowed them to define avalanches as spontaneous energy drops, with a dissipated power E related to granular rearrangements.

2. Atomistic simulations

• In parallel to experiments, stress drops have been analyzed in atomistic simulations of the deformation of glassy materials.
• In a 2D packing of soft spheres, Maloney and Lemâıtre (2004) measured power-law distributed energy drops with an exponent τ = 0.5 − 0.7 comparable to that obtained in Durian (1997)’s foam experiments.
• On the contrary, exponential distributions of stress drops and energy drops were then reported in athermal systems of particles interacting with three distinct potentials in 2D (Maloney and Lemâıtre, 2006), but also with a more realistic potential for a metallic glass in 3D (Bailey et al., 2007).
• All these studies were however limited to fairly small system sizes.
• The authors also mention that, opposing the rather consensual view of scale-free avalanches and non-trivial spatiotemporal correlations, Dubey et al. (2016) suggested that the characteristics of the stick-slip behavior stemmed from trivial finite-size effects.

D. Avalanche statistics in EPM

• The large amount of statistics afforded by EPM can enlighten the debate about the criticality of the yielding transition and the existence (or not) of a unique class of universality by overcoming the uncertainty and limitations of some experimental measurements.
• In the last years, EPM have tended to challenge the strict amalgamation of the yielding transition with the depinning one.

1. Avalanche sizes in the quasistatic limit

• Avalanches are most easily defined in the limit of quasistatic driving, in which the external load is kept fixed during avalanches (Sec. II.C).
• In a second series of simulations, apparently inspired by Talamali et al. (2011), the system was pulled by a spring of stiffness k ∈ [0.1, 1], moving adiabatically, hence an external stress Σ = k(γtot − γ), where γtot is the position of the spring and γ is the plastic strain.
• Still, they definitely deviate from the mean-field value, too.
• Σy, in which sites are randomly ‘kicked’ to trigger an avalanche, they found an exponent τ larger than that of the quasistatic simulations described above: Besides, power-law distributions were reported for the avalanche durations, with exponents τ ′ ' 1.6 in 2D and 1.9 in 3D.
• In parallel, extremal dynamics were implemented and yielded smaller exponents for the same models, τ ' 1.2 in 2D and τ ' 1.3 in 3D, closer to previous quasistatic approaches, even though not devoid of finite-size effects.

2. Connection with other critical exponents

• A discussion on the density of zones close to yielding and its connection with the critical exponents was opened up by Lin et al. (2014a).
• Denoting x ≡ σy−σ the distance to threshold of local stresses, a stark contrast was emphasized between depinning-like models, with only positive stress increments and p(x) ∼ x0 for small x, and EPM, where a pseudo-gap emerges at small x, viz., p(x) ∼ xθ with θ >.
• Shortly afterwards, Lin et al. (2014b) proposed to link p(x) with P (S), in a scaling description of the yielding transition.
• Their scaling argument can be summarized as follows.
• Now, in a stationary situation, on average this stress drop must balance the stress increase that is applied 47 to trigger an avalanche.

3. At finite strain rates

• Seeking to narrow the gap between experiments and EPM, Liu et al. (2016) analyzed the EPM stress signal with methods mimicking the experimental ones and studied the effect of varying the applied shear rate γ̇.
• These results coincide very well with MD simulations in the quasistatic limit and support the nascent convergence towards an avalanche size exponent τ ' 1.25 in 2D or 3D EPM, deviating from the mean-field value 3/2.
• Much more tentatively, there might be a downward trend of τ with increasing dimensions, which would be compatible with Jagla (2015)’s suggestion τ ' 1.1−1.2 above the upper critical dimension.
• Interestingly, Liu et al. (2016) observe a systematic crossover towards higher values of τ when the shear rate is increased, so that τ reaches τ ' 1.5 at intermediate γ̇, before entering the high-γ̇ regime of pure viscous flow.
• At the same time, the external driving starts to dominate over the signed stress fluctuations originating from mechanical noise; this nudges the system into a depinning-like scenario, with an exponent θ in p(x) ∼ xθ decreasing towards zero as γ̇ reaches finite values both in 2D and 3D.

4. Insensitivity to EPM simplifications and settings

• At present, technical difficulties still hamper a clear discrimination between theoretical predictions through experiments.
• The simplifications used in the models thus need to be carefully examined.
• Budrikis et al. (2017) investigated the effect of the scalar approximation of the stress (see Sec. II.C.2) by comparing the results of a scalar model to those of a finite-element-based fully tensorial model, under different deformation protocols (uniaxial tension, biaxial deformation, pure shear, simple shear) and in both 2D and 3D.
• Irrespective of the dimension, and (most of the) 48 loading and boundary conditions, a universal scaling function is observed for the avalanche distribution, shown in Fig. 18 and coinciding with Le Doussal and Wiese (2012)’s proposal P (S) = A 2 √ π Heterogeneous deformations, such as bending and indentation, were also considered and yielded similar values for τ .
• An independent length scale enters the problem and the yield stress.

5. Effects of inertia

• Without the assumption of instantaneous stress redistribution, stress waves are expected to propagate throughout the system (see Sec. III.D and Fig. 9), in a ballistic way or a diffusive one depending on the damping.
• Karimi et al. (2017) exploited this type of model to study Salerno and Robbins (2013)’s claim, based on extensive atomistic simulations in the quasistatic regime, that inertial effects drive the system into a new class of universality.
• A characteristic hump (or secondary peak) of large events emerges in the avalanche size distribution P (S), similarly to Fisher et al. (1997)’s findings.
• In Karimi et al. (2017)’s work, both the relative weight and the scaling with the system size of this peak are controlled by the damping coefficient Γ, a dimensionless parameter that quantifies the relative impact of dissipation.

6. Avalanche shapes

• In addition to their duration and size, further insight has been gained into the avalanche dynamics by considering their average temporal signal, i.e., the ‘shape’ of the bursts.
• In the latter example, the magnetization of a film changes mostly changes via the motion of domain walls4; its rate of change is recorded as a time series V (t).
• 4 Rigorously speaking, this is true in the central part of the hysteresis loop near the coercive field.
• This would explain the gradually more symmetric shapes observed for increasing T (see the evolution of the asymmetry parameter in the inset of Fig. 19b).
• In addition, the shapes obtained by sorting the avalanches according to their sizes (see Fig.19c) collapsed well with the scaling form proposed by Dobrinevski et al. (2015), with a shape exponent γ ' 1.8 (note the difference with respect to the mean-field value γ = 2).

VII. STEADY-STATE BULK RHEOLOGY

• In this Chapter the authors redirect the focus to materials that flow rather than fail.
• This is the relevant framework for foams, dense emulsions, colloidal suspensions, and various other soft glassy materials exhibiting a yield stress.
• Such a nontrivial dependence on the shear rate γ̇ proves that the rheology of these materials cannot be understood as a mere sequence of γ̇-independent elastic loading phases interspersed with γ̇-independent plastic events (Puglisi and Truskinovsky, 2005).
• In particular, the authors will see that the local yielding and healing dynamics (notably via rules R2 and R4 in Sec. II.A) play a crucial role in determining the flow properties at finite γ̇.
• More generally, different flow regimes will be delineated, depending on the material time scales at play.

A. Activation-based (glassy) rheology v. dissipation-based (jammed) rheology

• The rheology of glasses was long thought to be tightly connected with that of jammed systems such as foams (Liu and Nagel, 1998).
• In SGR, where the Arrhenius law is controlled by a fixed effective temperature x, as γ̇ increases, blocks can accumulate more elastic strain before a plastic event is activated.
• Most other EPM dedicated to the study of steady-state rheology consider systems close to the athermal regime, in particular foams and dense emulsions of large droplets, which in practice undergo negligible thermal fluctuations.
• This will be the focus of the rest of this chapter.
• These avalanches are gradually perturbed and cut off as γ̇ is increased, while higher shear rates add further more local corrections.

B. Athermal rheology in the limit of low shear rates

• At vanishingly low shear rates γ̇ the nonlinear response of athermal materials is anchored in the critical dynamics discussed in Sec. VI.
• So far, the authors have seen that, within mean-field models, the dynamics of shear wave propagation and the (heavy-tailed) statistics of mechanical noise fluctuations may affect the low-shear-rate rheology, and that the finite dimension of space introduces deviations from mean-field predictions due to correlations in the noise.
• The scaling relation (40) seems to involve fewer parameters of the problem than Eq. (38); one should nonetheless bear in mind that in depinning problems the relevant effective potential V entering mean-field reasoning nontrivially depends on different properties of the system.
• No firm theoretical consensus has been reached yet regarding the flow exponent β and how universal it is.
• VI, the mechanical noise fluctuations induced by the alternate sign of the elastic propagator most probably play a prominent role in these deviations close to criticality.

C. Athermal rheology at finite shear rates

• The regime of finite driving rates was already targeted by early EPM, including that of Picard et al. (2005) (see Sec. IV.A).
• This is due to the postulated elastic stress accumulation above the threshold σy for a fixed duration τ on average.
• Thus, EPM and experiments highlight the sensitivity of the finite-shear-rate rheology to the specific microscopic interactions between particles or dynamical rules at play.
• The mechanical noise felt in a given region, then, results from the superposition of a large number of events and its distribution acquires a Gaussian shape (Liu, 2016).
• Most EPM works consider strain-controlled protocols (defined in Sec. II.C).

VIII. RELAXATION, AGING AND CREEP PHENOMENA

• So far EPM have mostly been exploited to investigate the macroscopic flow behavior and flow profiles (Sec. V), characterize stationary flow (Sec. VII), or study fluctuations and correlations in the steady flow close to criticality, where one finds scale-free avalanches (Sec. VI).
• Still, some works, however few, are concerned with relaxation, aging, and creep phenomena.
• This section is dedicated to both the dynamics in the temperature assisted relaxation of disordered systems and to the transient dynamics under loading , prior to yielding or complete arrest.
• The latter phenomenon can be either an athermal process, provided that the stress load is above, but close to, the yielding point, or thermally assisted creep, in response to a load below the dynamical yield stress.

A. Relaxation and aging

• A striking feature in the theory of viscous liquids is their response to an external perturbation, close to the glass transition:.
• The main obstacle had been to probe the right parameter range, notably with respect to temperature and also length scales.
• From the outset, Cipelletti et al. (2000) suggested that the faster than exponential relaxation stems from the elastic deformation fields generated by local relaxation events.
• T) (Cipelletti et al., 2003) cannot be captured in EPM at present, but could be included by implementing hybrid models that consider smaller-scale dynamics as well.

IX. RELATED TOPICS

• Amorphous solids seem to form a specific class of materials.
• The phenomenology exposed in the previous chapters suggests underlying theoretical connections with other problems.
• And, indeed, EPM are related to a spectrum of other models, notwithstanding physical differences, in particular in the interaction kernels.
• This section reviews, and attempts to compare to EPM, some of these related approaches, from mesoscale models for crystalline plasticity and elastic line depinning to fiber bundles, fuse networks and random spring models.
• The ample connections with seismology, hinted at in Sec. III, and tribology (Jagla, 2018; Lastakowski et al., 2015; Persson, 1999) – the latter being plausibly mediated by fracture mechanics (Svetlizky and Fineberg, 2014; Svetlizky et al., 2017) – will not be discussed here.

1. Crystal plasticity

• Like amorphous solids, driven crystalline materials respond elastically to infinitesimal deformations, via an affine deformation of their structure, but undergo plastic deformation under higher loading.
• To be energetically favorable, plastic deformation increments must somehow preserve the regular stacking of atoms.
• For a perfect crystal, such a criterion would predict an elastic limit of around 5%.
• Dislocations are line defects obtained by making a half-plane cut in a perfect crystal and mismatching the cut surfaces before stitching them back together.
• The stress field around a dislocation is well known (it decays inversely proportionally to the distance to the line) and the attractive or repulsive interactions between dislocations can also be rigorously computed.

2. Models and results

• Mesoscale dislocation models, which exist in several variants (Field Dislocation Model, Continuum Dislocation Dynamics), bear formal similarities with EPM.
• In EPM, such effects would belong to the rules that govern the onset of a plastic event.
• Also, large avalanches are cut off due to strain hardening, which is one possible explanation for the macroscopic smoothness of the deformation.
• These fluctuations may be “mild”, with bursts superimposed on a relatively constant, seemingly uncorrelated fluctuation background, which is the case for many bulk samples, especially those with an fcc (face-centered cubic) structure.
• A meanfield rationalization of these phenomena considers the density ρm of mobile dislocations and expresses its evolution with the strain γ as dρm dγ = A− Cρm + √ 2Dρmξ (γ) , where A is a nucleation rate, C is the rate of annihilation of dislocation pairs, and D controls the intensity of the white noise ξ.

3. Relation to EPM

• The macroscopic phenomenology and, to some extent, the mesoscopic one share many similarities: Microscopic defects interact via long-range interactions and their activity is, in some conditions, controlled by temperature.
• Globally, the dynamics are highly intermittent at low shear rates and involve scale-invariant avalanches, as indicated, inter alia, by acoustic emission measurements on stressed ice crystals (Miguel et al., 2001).
• The phenomenological similarity is paralleled by a proximity in the models.
• Conversely, quadrupolar interactions may be directly implemented in mesoscale models of crystal plasticity, for instance in Eq. (2) of (Papanikolaou et al., 2012).
• Rottler et al. (2014) numerically investigated the transition between the dislocation-mediated plasticity of crystals and the shear-transformation-based deformation of amorphous solids.

1. The classical depinning problem

• In several systems, an interface is driven through a disordered medium by a uniform external force.
• This interface can be a magnetic or ferroelectric domain wall, the water front (contact line) in a wetting problem, the fracture front, or even charge density waves and arrays of vortices in superconductors.
• In all these cases, the interplay between the quenched disorder (e.g., due to impurities) and the elastic interactions along the interface is at the root of a common phenomenology and a universal dynamical response.
• If the external force is weak, the interface will advance and soon get pinned and unable to advance any further.
• This is the well documented dynamical phase transition known as depinning.

2. Models

• The most celebrated model to describe the depinning problem is the quenched Edwards-Wilkinson (QEW) equation.
• Of course, the QEW model just mentioned is minimal.
• Some of its variants take into account additional ingredients.
• Charge density waves and vortices involve a periodic elastic structure, in fracture and wetting the elastic interactions are long-ranged, and anharmonic corrections to elasticity or anisotropies could also be relevant.

3. Phenomenology

• Around these points, at vanishing temperature, the steady-state interface h(x) displays a self-affine geometry (in the sense that it is invariant under dimensional rescaling, viz., h(ax) ∼ aζh(x)) above a microscopic length scale, with characteristic roughness exponents: (i) ζeq, (ii) ζdep, and (iii) ζff .
• When the applied force approaches zero, macroscopic movement can be observed only at finite temperatures and at very long times.
• Tψ and the size `av is finite at the transition.
• One of the remarkable lessons learned from this simple model is the possibility to relate transport and geometry.

4. Similarities and differences with EPM

• The manifest qualitative similarity between the yielding transition and the depinning one has enticed many researchers to look for a unification of these theories.
• The analogy has promoted the vision of yielding as a critical phenomenon and has given rise to interesting advances, but, in their opinion, the belief in a strict equivalence of the problems has been deceptive in some regards.
• This formal similarity between the two classes of phenomena seems to buttress the application of results from the depinning problem (hence mean field, owing to the long range of the elastic propagator) to the question of, e.g., avalanche statistics in disordered solids (see Sec. VI).
• Let us now mention a subclass of problems that may be more closely related to EPM: the so called “plastic depinning”.

1. Brief introduction to cracks and fracture

• In partial overlap with the scope of EPM, the question of the failure of hard solids under loading, e.g. in tension, has attracted much attention over the last centuries.
• 2γa︸︷︷︸ surface energy , where E is the Young modulus of the material, γ is the interfacial energy, and Σ is the applied stress.
• While the material is being fractured, the crack propagates along a rough, scale-invariant frontline (see Fig. 25a), characterized by the in-plane roughness exponent ζ‖.
• The crack produced when tearing apart two sandblasted Plexiglas sheets stuck together through annealing undergoes a stick-slip motion at small scales that is reminiscent of dry solid friction (Måløy and Schmittbuhl, 2001), which in turn may tell us about earthquake dynamics (Svetlizky and Fineberg, 2014).
• In (rock) fracture, the microruptures very generally do not have time to heal on the time scale of the deformation; without recovery process, the material is thus permanently damaged.

2. Fiber bundles

• Arguably, the simplest way to model fracture is to consider two blocks bound by N aligned fibers.
• These fibers share the global load and break irreversibly when their elongation x exceeds a randomly distributed threshold; this is the basis of fiber-bundle models (Herrmann and Roux, 2014).
• Analytical progress is possible in this intrinsically mean-field model.
• The gradual shift to an exponent τ = 3/2 then signals imminent failure.

3. Fuse networks

• Unfortunately, the picture promoted by mean-field or 1D fiber bundles is incapable of describing the heterogeneous and anisotropic propagation of cracks.
• The voltages Vi are imposed at two opposite edges of the system, as depicted in Fig. 25c.
• It can then be understood that failure occurs along a line of burnt fuses, the “crack” line, provided that there is finite disorder (θ > 0) and the network is large (Shekhawat et al., 2013).
• These differences are not negligible in any way.
• Nevertheless, the process of fracture can be mimicked in the random Ising models by imposing spin +1 (-1) on the left edges of the sample and monitoring the interface line between the +1 and -1 domains.

4. Spring models

• From a mechanical perspective, should one replace the voltage Vi in Eq. (53) with the displacement ui at node i, viz., H′nc = 1 2 ∑ 〈i,j〉 Kij (ui − uj)2 , (54) the interpretation of the Hamiltonian as the energy of a network of random springs of stiffness Kij will become apparent.
• The x, y, and z components of the dispacements in H′nc decouple, so that model is actually scalar (De Gennes, 1976).
• As bonds are gradually removed in a random fashion, the initially rigid system transitions to a non-solid state with vanishing elastic moduli at a critical bond fraction pc.
• Instead of gradually destroying bonds, he cranked up the fraction p of bonds by randomly connecting pairs of neighbours until bonds percolated throughout the system; this occurred at a critical fraction pc, supposedly corresponding to gel formation.

5. Beyond random spring models

• Refinements have been suggested to bring random fuse (or spring) networks closer to models of material deformation and fracture.
• At the opposite end, perfect plasticity is mirrored by the saturation of the fuse intensity past a threshold.
• Another limitation of the models stems directly from the description of the bonds on a regular lattice: let alone the presence of soft modes in several cases, the (Hc-based) central-force model, discretized on a triangular lattice, displays an anisotropic tensile failure surface (despite an isotropic linear response), with an anisotropy ratio of 50% (Monette and Anderson, 1994).
• As with EPM, the following step in the endeavor to refine the description led to the introduction of a finiteelement approach, which relies on a continuum description down to the scale of one mesh element.
• In the case of large φ and brittle failure, a description of compressive failure under uniaxial stress as a critical phenomenon analogous to depinning was proposed by Girard et al. (2010) and elaborated by Weiss et al. (2014).

X. OUTLOOK

• In the last ten years, EPM have become an essential theoretical tool to understand the flow of solids.
• A more unexpected emerging avenue is the study of systems with internal activity, such as living tissues or dense cell assemblies.
• Scaling relations between critical exponents have been proposed (Aguirre and Jagla, 2018; Lin et al., 2014b) and tested in diverse EPM, but analytical calculations beyond mean field are scant.
• EPM and other theoretical studies have proposed possible mechanisms that may influence the continuous or discontinuous character of the transition (see Sec. V).
• Wortel et al. (2016)’s work 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.

ACKNOWLEDGMENTS

• The authors acknowledge financial support from ERC grant ADG20110209 .
• E.E.F acknowledges financial support from ERC Grant No. ADG291002 .
• The authors thank M. V. Duprez for professional help with the graphics.

Figures

FIG. 3 Schematic macroscopic response of amorphous solids to deformation. (a) Evolution of the shear stress Σ with the imposed shear strain γ, with a stress overshoot Σmax. In the event of material failure, which is generally preceded by strain localization, the stress dramatically drops down. (b) Steady-state flow curve, i.e., dependence of the steady-state shear stress Σss on the shear rate γ̇, represented with semi-logarithmic axes. If the flow is split into macroscopic shear bands, a stress plateau is generally observed.

FIG. 5 Comparison between different indicators based on the local structure to predict future plastic rearrangements. (a) Contour maps of the participation ratio in the 1% softest vibrational modes for two numerical samples of a binary metallic glass (Ding et al., 2014). (b) Correlation between the locations of future rearrangements and diverse local properties in an instantaneously quenched binary glass model. The following properties have been considered: local yield stress (τy), participation in the soft vibrational modes (PF), lowest shear modulus (2µI), local potential energy (PE), short-range order (SRO), local density (ρ). From (Patinet et al., 2016). (c) Identification of soft particles (thick black contours) by a machinelearning algorithm (SVM), in a compressed granular pillar. Particles are colored from gray to red according to their actual nonaffine motion (D2min value). From (Cubuk et al., 2015).

FIG. 23 EPM characterization of creep. (a) Strain rate ̇ as a function of time t for different applied stresses Σ in the EPM of Bouttes and Vandembroucq (2013). (b) Non-linear compliance J ≡ Σ0 as a function of time for different applied stresses Σ0, obtained with Merabia and Detcheverry (2016)’s mesoscopic model. (c) Dependence of the fluidization time tf on Σ0. Panels (b) and (c) are extracted from (Merabia and Detcheverry, 2016).

FIG. 15 Distributions of stress drops in the deformation of amorphous materials. (a) Distribution of stress drops ∆σ in a foam that is strained in a Couette cell, for three different strain rates. From (Lauridsen et al., 2002) with APS permission. The solid line in this logarithmic plot has a slope of −0.8. (b) Distribution D(s) of stress drops of normalized magnitude s in a metallic glass (Cu47.5Zr47.5Al5). Adapted from (Sun et al., 2010). The red line represents a power law with exponent τ ' 1.49. (c) Distribution D(s) of force fluctuation sizes s in a sheared granular system, for different shear rates and at constant confining pressure P = 9.6 kPa. Adapted from (Denisov et al., 2016) with permission. The data suggest truncated power laws D(s) ∼ s−τ exp(−s/γ̇µ), with τ = 1.5 and µ = 0.5.

FIG. 17 Probability densities P (x) of the distances x to the yield threshold in EPM : (a) In 2D and (b) in 3D systems, whose size L is varied. From (Lin et al., 2014b). (c) In a 3D system, as the shear rate is varied. The inset shows the rescaled distribution of avalanche sizes. From (Liu et al., 2016).

FIG. 10 Sketches illustrating the difference between thermal fluctuations ξT and mechanical noise ξpl; the variable l represents the local strain configuration. ξT are thermal kicks within a fixed potential energy landscape (PEL), whereas σ(t) and its fluctuations ξpl tilt the local PEL up and down From (Agoritsas et al., 2015).

FIG. 24 The depinning picture. (a) Connection between transport and geometry in depinning. From (Ferrero et al., 2013). (i) Snapshot of a domain wall in a 2D ferromagnet. (ii) Typical velocity-force characteristics. (iii) Crossover lengths `opt and `av representing the optimal excitation and the deterministic avalanches, respectively. (iv) Geometric crossover diagram. (b) Steady-state structure factor S(q) of the line in the limit of vanishing temperature for different forces (curves are shifted for clarity). Adapted from (Kolton et al., 2006). (c) Comparison of the depinning and yielding critical transitions in a correspondence (v ↔ γ̇), (f ↔ Σ).

FIG. 21 Dependence of steady-state EPM flow curves on the shear rate γ̇. (a) Difference ∆σ0 ≡ Σ − Σy (circles) between the steady-state stress Σ and Σy as a function of γ̇, in Nicolas et al. (2014a)’s model. From (Liu et al., 2016). The data suggest two different scaling regimes: Close to criticality the Herschel-Bulkley exponent is n = 0.65, whereas at high γ̇ n tends towards 1/2. (b) Flow curves for the same EPM at relatively high γ̇ with a shear-rate-dependent local shear modulus G0(γ̇) ∼ γ̇ψ1 and plastic event ‘duration’ γc(γ̇) ∼ γ̇−ψ2 . These dependences introduce corrections to the Herschel-Bulkley exponent, n = (1 + ψ1 − ψ2)/2 (pay attention to the horizontal axis on the plot). From (Agoritsas and Martens, 2017). (c) Flow-curves obtained from stress-imposed EPM simulations in 2D. The dashed line is a fit to a Herschel-Bulkley law with n ≈ 0.66. From (Lin et al., 2014b).

FIG. 12 Experimental observations of shear bands and material failure. (a) In a granular packing of 90µm glass beads under biaxial compression. (b, right) In a bulk metallic glass under uniaxial tension. The composite glass shown in (b, left), reinforced with dendrites, displays a more ductile response to tension. From (Le Bouil et al., 2014) and (Hofmann et al., 2008), respectively.

TABLE III List of values measured for the critical exponents characterizing avalanches in EPM. Only values measured in EPM with extremal dynamics (or akin) and a quadrupolar propagator are reported. Mean field values are added for comparison.

FIG. 25 Observation and modeling of crack propagation. (a) Raw image of the front of an in-plane crack propagating between Plexiglas plates. The intact region appears in black and the image-processed front line is shown in (b). The roughness along the propagation (z ) direction has a power-law spectrum characterized by the roughness exponent ζ‖. From (Schmittbuhl and Måløy, 1997). (c) Sketch of the random fuse network and (d-e) failure process for distinct probability density functions for the thresholds, p(x) ∼ xθ. The crack has been colored in red. From (Shekhawat et al., 2013).

FIG. 7 Spatial variations of the mechanical and configurational properties of glasses. (a) Maps of the weaker local shear modulus in a 2D Lennard-Jones glass. Black (white) represents values larger (smaller) than the mean value. Distances are in particle size units. From (Tsamados et al., 2009). (b-c) Maps of the local contact-resonance frequency, which is related to the indentation modulus, measured by atomic force acoustic microscopy in (b) a bulk metallic glass (PdCuSi) and (c) a crystal (SrTiO3). The latter clearly appears to be mechanically more homogeneous. The radius of contact is of order 10 nm. From (Wagner et al., 2011).

FIG. 16 Distributions P (S) of avalanche sizes obtained with 2D EPM in the quasistatic limit, in diverse settings: (a) Under extremal dynamics, where the system (of linear size L = 256) is driven by spring of variable stiffnesses k, indicated in the legend. The fitted exponent is τ ' 1.25. From (Talamali et al., 2011). (b) In strain-controlled simulations with the ‘image sum’ implementation of the elastic propagator kernel (see Sec. III.C.2), with fits to Eq. (33). Notice that the driving springs are much stiffer than in panel (a). From (Budrikis and Zapperi, 2013). (c) Under extremal dynamics, for systems of different sizes L. The exponent reported for the unscaled curves (inset) is τ ' 1.2, while the rescaled curve shown in the main plot was fitted with τ ' 1.36. From (Lin et al., 2014b).

FIG. 6 Average stress redistribution around a shear transformation. (a) Experimental measurement in very dense emulsions. Adapted from (Desmond and Weeks, 2015). (b) Average response to an imposed ST obtained in atomistic simulations with the binary Lennard-Jones glass used by Puosi et al. (2014). (c) Simplified theoretical form, given by Eq. (4). From (Martens et al., 2012). Note that the absolute values are not directly comparable between the graphs and that in (b-c) the central blocks are artificially colored.

FIG. 22 Structural relaxation in quiescent systems. (a) Diffusivity D(t) (i.e., mean-square displacement divided by time) of tracer particles measured in an EPM, at four temperatures, increasing from bottom (blue) to top (black). (b) Rescaled self-part of the intermediate scattering function S(q, t) for t in the first ascending regime of D(t) in panel (a). The motion is close to ballistic (linear in time), with τ ≈ q−1 in Eq. (41), and the form factor β ' 2, defined in the same equation, implies a compressed exponential relaxation. From (Ferrero et al., 2014). (c) Relaxation of thin Zr67Ni33 metallic glass ribbons with time, measured by the the decay of the X-ray photon intensity autocorrelation g2 at T = 373K, for different waiting times tw and wavectors Q (shown in the inset). The characteristic relaxation time τ(Q, tw) was determined by fitting R = g2 − 1 to the KWW form of Eq. (41), which yielded a shape parameter b ' 1.8± 0.08. From (Ruta et al., 2013).

FIG. 20 Steady-state flow curves obtained in variants of Picard’s EPM. (a) Rescaled flow curves in Picard’s original EPM, for different system sizes, in logarithmic representation. The inset shows a typical stress-strain curve at low shear rate, starting from a stress-free configuration. From (Picard et al., 2005). (b) Non-monotonic flow curve obtained in Picard’s model with a long local restructuring time τres, plotted with semi-logarithmic axes. Inset: average local shear rate in the flowing regions. For γ̇ < γ̇c a mechanical instability leads to shear-banding, with a coexistence of a flowing band and a static region. From (Martens et al., 2012). (c) Flow curve obtained in a variant of Picard’s model. The solid line is a fit to a Herschel-Bulkley equation, with exponent n = 0.56. Inset: same data, in linear representation. From (Nicolas et al., 2014a).

FIG. 14 Strain localization in an EPM. Maps of the cumulated plastic strain p at different rescaled ‘times’ 〈 p〉; δ quantifies the weakening of the local yield stresses when they are renewed, as explained in the main text (the bottom row is strain weakening, while the top row is not). Darker colors represent larger values of p. The principal strain directions are the horizontal and vertical directions. Adapted from (Vandembroucq and Roux, 2011).

FIG. 1 Overview of amorphous solids. From left to right, top row : cellular phone case made of metallic glass (1); toothpaste (2); mayonnaise (3); coffee foam (4); soya beans (5). Second row : a transmission electron microscopy (TEM) image of a fractured bulk metallic glass (Cu50Zr45Ti5) by X. Tong et. al (Shanghai University, China); TEM image of blend (PLLA/PS) nanoparticles obtained by miniemulsion polymerization, from L. Becker Peres et al. (UFSC, Brazil); emulsion of water droplets in silicon oil observed with an optical microscope by N. Bremond (ESPCI Paris); a soap foam filmed in the lab by M. van Hecke (Leiden University, Netherlands); thin nylon cylinders of different diameters pictured with a camera, from T. Miller et al. (University of Sydney, Australia). The white scale bars are approximate. Just below, a chart of different amorphous materials, classified by the size and the damping regime of their elementary particles. At the bottom: some popular modeling approaches, arranged according to the length scales of the materials for which they were originally developed. STZ stands for the shear transformation zone theory of Langer (2008), and SGR for the soft glassy rheology theory of Sollich et al. (1997).

FIG. 19 Avalanche shapes in experiments and EPM. (a) Experimental avalanche shapes, for avalanches of fixed duration (top) and fixed sizes (bottom) in a bulk metallic glasses (BMG) and a granular system. From (Denisov et al., 2017). (b) Avalanche shapes at different fixed durations in a strain-controlled EPM simulation at fixed γ̇. From (Liu et al., 2016). (c) Avalanche shapes at fixed sizes. From (Budrikis et al., 2017).

TABLE I Classification of some of the main EPM in the literature.

FIG. 9 Average displacement field induced by an ST in an underdamped elastic medium, computed with a basic Finite Element routine. The plotted snapshots correspond to different delays after the transformation was (artificially) triggered at the origin: (a) ∆t = 2, (b) ∆t = 10, (c) ∆t = 1000. Red hues indicate larger displacements. Adapted from (Nicolas et al., 2015)

Full text

Deformation and flow of amorphous solids: An updated review of mesoscale
elastoplastic models
Alexandre Nicolas
LPTMS, CNRS, Univ. Paris-Sud,
Universit´e Paris-Saclay, 91405 Orsay,
France.
Ezequiel E. Ferrero
Centro At´omico Bariloche,
8400 San Carlos de Bariloche, R´ıo Negro,
Argentina
Kirsten Martens and Jean-Louis Barrat
Univ. Grenoble Alpes, CNRS,
LIPhy, 38000 Grenoble,
France
The deformation and flow of disordered solids, such as metallic glasses and concentrated
emulsions, involves swift localized rearrangements of particles that induce a long-range
deformation field. To describe these heterogeneous processes, elastoplastic models han-
dle the material as a collection of ‘mesoscopic’ blocks alternating between an elastic
behavior and plastic relaxation, when they are loaded above a threshold. Plastic relax-
ation events redistribute stresses in the system in a very anisotropic way. We review not
only the physical insight provided by these models into practical issues such as strain lo-
calization, creep and steady-state rheology, but also the fundamental questions that they
address with respect to criticality at the yielding point and the statistics of avalanches
of plastic events. Furthermore, we discuss connections with concurrent mean-field ap-
proaches and with related problems such as the plasticity of crystals and the depinning
of an elastic line.
arXiv:1708.09194v4 [cond-mat.dis-nn] 6 Jul 2018

2
CONTENTS
Frequently used notations 3
Introduction 4
I. General phenomenology 5
A. What are amorphous solids? 5
B. What controls the dynamics of amorphous solids? 6
1. Athermal systems 6
2. Thermal systems 7
3. Potential Energy Landscape 7
C. Jagged stress-strain curves and localized rearrangements 8
Evidence 9
Quantitative description 9
Variations 10
Structural origins of rearrangements 11
D. Nonlocal effects 12
Idealized elastic propagator 12
Exact induced field and variations 13
II. The building blocks of elastoplastic models (EPM) 14
A. General philosophy of the models 14
B. Thermal fluctuations 15
C. Driving 16
1. Driving protocol 16
2. Symmetry of the driving 16
D. Driving rate and material time scales 17
E. Spatial disorder in the mechanical properties 17
F. Spatial resolution of the model 18
G. Bird’s eye view of the various models 19
III. Elastic couplings and the interaction kernel 19
A. Sandpile models and first-neighbor stress redistribution 19
B. Networks of springs 21
C. Elastic propagators 22
1. Pointwise transformation in a uniform medium 22
2. Technical issues with pointwise transformations and possible remediations 23
3. Variations: Soft modes and lattice symmetries; tensoriality; convection 23
D. Approaches resorting to Finite-Element methods 24
IV. Mean-field treatments of mechanical noise 25
A. Uniform redistribution of stress 25
B. Random stress redistribution 26
1. Deviations from uniform mean field 26
2. The H´ebraud-Lequeux model 27
3. Fraction of sites close to yielding 27
C. Validity of the above ‘mean-field’ approximations 27
1. Uniform mean field 27
2. White-noise fluctuations 27
3. Heavy-tailed fluctuations 28
4. Structure of the elastic propagator and soft modes 29
D. A mechanical noise activation temperature? 29
1. The Soft Glassy Rheology model (SGR) 29
2. Mechanical noise v. thermal fluctuations 30
E. Connection with the diffusion of tracers 31
F. Continuous approaches based on plastic disorder potentials 32
V. Strain localization: From transient heterogeneities to permanent shear bands 33
A. Two opposite standpoints 33
1. The shear-banding instability from the standpoint of rheology 34
2. The mechanics of bands in a solid 34
B. Spatial correlations in driven amorphous solids 36
1. Spatial correlations 36
2. Cooperative effects under inhomogeneous driving 37
3. Cooperative effects due to boundaries 38
C. Alleged causes of permanent shear localization or fracture 39
1. Long breakdowns (rearrangements), slow recovery 39
2. Influence of initial stability (aging) and shear rejuvenation (softening): 39
3. Shear bands like it hot 41

3
VI. Critical Behavior and Avalanches at the Yielding Transition 41
A. Short introduction to out-of-equilibrium transitions 41
1. Avalanches in sandpile models 42
2. Stress drops and avalanches in EPM 42
B. Avalanches in mean-field models 42
C. Experimental observations and atomistic simulations of avalanches 44
1. Experiments 44
2. Atomistic simulations 44
D. Avalanche statistics in EPM 44
1. Avalanche sizes in the quasistatic limit 45
2. Connection with other critical exponents 46
3. At finite strain rates 47
4. Insensitivity to EPM simplifications and settings 47
5. Effects of inertia 48
6. Avalanche shapes 49
VII. Steady-state bulk rheology 50
A. Activation-based (glassy) rheology v. dissipation-based (jammed) rheology 51
B. Athermal rheology in the limit of low shear rates 51
C. Athermal rheology at finite shear rates 53
D. Strain-driven vs. stress driven protocols 53
VIII. Relaxation, Aging and Creep phenomena 54
A. Relaxation and aging 54
B. Creep 56
IX. Related topics 58
A. Mesoscale models of crystalline plasticity 58
1. Crystal plasticity 58
2. Models and results 58
3. Relation to EPM 59
B. Depinning transition 60
1. The classical depinning problem 60
2. Models 60
3. Phenomenology 60
4. Similarities and differences with EPM 61
C. Fiber bundle, fuse networks and continuum models for the study of cracks and fracture 63
1. Brief introduction to cracks and fracture 63
2. Fiber bundles 63
3. Fuse networks 64
4. Spring models 65
5. Beyond random spring models 66
X. Outlook 67
Acknowledgments 68
References 69
FREQUENTLY USED NOTATIONS
Σ Macroscopic shear stress
Σ
y
Macroscopic yield stress
σ Local shear stress
σ
y
Local yield stress
µ Shear modulus
γ Shear strain
˙γ Shear rate
EPM Elastoplastic model
MD Molecular dynamics
rhs (lhs) right-hand side (left-hand side)
ST Shear transformation

4
FIG. 1 Overview of amorphous solids. From left to right, top row: cellular phone case made of metallic glass (1); toothpaste
(2); mayonnaise (3); coffee foam (4); soya beans (5). Second row: a transmission electron microscopy (TEM) image of a
fractured bulk metallic glass (Cu
50
Zr
45
Ti
5
) by X. Tong et. al (Shanghai University, China); TEM image of blend (PLLA/PS)
nanoparticles obtained by miniemulsion polymerization, from L. Becker Peres et al. (UFSC, Brazil); emulsion of water droplets
in silicon oil observed with an optical microscope by N. Bremond (ESPCI Paris); a soap foam filmed in the lab by M. van
Hecke (Leiden University, Netherlands); thin nylon cylinders of different diameters pictured with a camera, from T. Miller et al.
(University of Sydney, Australia). The white scale bars are approximate. Just below, a chart of different amorphous materials,
classified by the size and the damping regime of their elementary particles. At the bottom: some popular modeling approaches,
arranged according to the length scales of the materials for which they were originally developed. STZ stands for the shear
transformation zone theory of Langer (2008), and SGR for the soft glassy rheology theory of Sollich et al. (1997).
INTRODUCTION
19th-century French Chef Marie-Antoine Carˆeme (1842) claims that ‘mayonnaise’ comes from the French verb
‘manier’ (‘to handle’), because of the continuous whipping that is required to make the mixture of egg yolk, oil, and
vinegar thicken. This etymology may be erroneous, but what is certain is that the vigorous whipping of these liquid
ingredients can produce a viscous substance, an emulsion consisting of oil droplets dispersed in a water-based phase.
At high volume fraction of oil, mayonnaise even acquires some resistance to changes of shape, like a solid; it no longer
yields to small forces, such as its own weight. Similar materials, sharing solid and liquid properties, pervade our
kitchens and fridges: Chantilly cream, heaps of soya grains or rice are but a couple of examples. They also abound on
our bathroom shelves (shaving foam, tooth paste, hair gel), and in the outside world (sand heaps, clay, wet concrete),
see Fig. 1 for further examples. All these materials will deform, and may flow, if they are pushed hard enough, but will
preserve their shape otherwise. Generically known as amorphous (or disordered) solids, they seem to have no more in
common than what the etymology implies: their structure is disordered, that is to say, deprived of regular pattern at
“any” scale, as liquids, but they are nonetheless solid. So heterogeneous a categorization may make one frown, but
has proven useful in framing a unified theoretical description (Barrat and de Pablo, 2007). In fact, the absence of long
range order or of a perceptible microstructure makes the steady-state flow of amorphous solids simpler, and much less
dependent on the preparation and previous deformation history, than that of their crystalline counterparts. A flowing
amorphous material is therefore a relatively simple realization of a state of matter driven far from equilibrium by an
external action, a topic of current interest in statistical physics.
A matter of clear industrial interest, the prediction of the mechanical response of such materials under loading is a

5
challenge for Mechanical Engineering, too. This problem naturally brings in its wake many questions of fundamental
physics. Obviously, it is not exactly solvable, since it involves the coupled mechanics equations of the N  1
elementary constituents of the macroscopic material; this is a many-body problem with intrinsic disorder and very
few symmetries. Two paths can be considered as alternatives: (i) searching for empirical laws in the laboratory,
and/or (ii) proposing approximate, coarse-grained mathematical models for the materials. The present review is a
pedagogical journey along the second path.
FIG. 2 Scientific position of elastoplastic modeling.
Along this route, substantial assumptions are made to sim-
plify the problem. The prediction capability of models hinges
on the accuracy of these assumptions. Following their dis-
tinct interests and objectives, different scientific communities
have adopted different modeling approaches. Material scien-
tists tend to include a large number of parameters, equations
and rules, in order to reproduce different aspects of the mate-
rial behavior simultaneously . Statistical physicists aspire for
generality and favor minimal models, or even toy models, in
which the parameter space is narrowed down to a few vari-
ables. At the interface between these approaches, “elasto-
plastic” models (EPM) consider an assembly of mesoscopic
material volumes that alternate between an elastic regime
and plastic relaxation, and interact among themselves. As
simple models, they aim to describe a general phenomenol-
ogy for all amorphous materials, but they may also include
enough physical parameters to address material particulari-
ties, in view of potential applications. They rely on simple
assumptions to connect the microsopic phenomenology to the
macroscopic behavior and therefore have a central position
in the endeavor to bridge scales in the field (Rodney et al.,
2011). To some extent, EPM can be compared to classical
lattice models of magnetic systems, which permit the explo-
ration of a number of fundamental and practical issues, by
retaining a few key features such as local exchange and long range dipolar interactions, spin dynamics, local symme-
tries, etc., without explicit incorporation of the more microscopic ingredients about the electronic structure.
This review aims to articulate a coherent overview of the state of the art of these EPM, starting in Sec. I with
the microscopic observations that guided the coarse-graining efforts. We will discuss several possible practical imple-
mentations of coarse-grained systems of interacting elastoplastic elements, considering the possible attributes of the
building blocks (Sec. II) and the more technical description of their mutual interactions (Sec. III). Section IV is then
concerned with the widespread approximations of the effect of the stress fluctuations resulting from these interactions.
In Sec. V we describe the current understanding of strain localization based on the study of EPM. Section VI focuses
on the statistical marks of criticality encountered when the system is driven extremely slowly, especially in terms of
the temporal and spatial organization of stress fluctuations in ‘avalanches’, while Sec. VII describes the bulk rheology
of amorphous solids in response to a shear deformation. Section VIII gives a short perspective on the much less
studied phenomena of creep and aging. The review ends on a discussion of the relation between EPM and several
other descriptions of mechanical response in disordered systems, in Sec. IX, and some final outlooks.
These sections are largely self-contained and can thus be read separately. Sections I and II are both particularly
well suited as entry points for newcomers in the field, while Sections III and IV might be more technical and of greater
relevance for the experts interested in the implementation of EPM. Finally, Sections V to VIII focus on applications
of the models to specific physical phenomena and are largely independent from each other.
I. GENERAL PHENOMENOLOGY
A. What are amorphous solids?
From a mechanical perspective, amorphous solids are neither perfect solids nor simple liquids. Albeit solid, some
of these materials are made of liquid to a large extent and appear soft. Nevertheless, at rest they preserve a solid
structure, and will flow only if a sufficient load is applied to them. Accordingly, in the rheology of complex fluids

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Deformation and flow of amorphous solids: an updated review of mesoscale elastoplastic models" ?

The deformation and flow of disordered solids, such as metallic glasses and concentrated emulsions, involves swift localized rearrangements of particles that induce a long-range deformation field. The authors review not only the physical insight provided by these models into practical issues such as strain localization, creep and steady-state rheology, but also the fundamental questions that they address with respect to criticality at the yielding point and the statistics of avalanches of plastic events. Furthermore, the authors discuss connections with concurrent mean-field approaches and with related problems such as the plasticity of crystals and the depinning of an elastic line. 

Q2. What are the likely candidates for microscopic origins of long rearrangements?

Attractive interactions in adhesive colloidal systems (Irani et al., 2014) and directional bonds in molecular systems are tentative candidates for possible microscopic origins of long rearrangements, i.e., long time delays before the destabilized region reaches another stable configuration. 

Q3. What is the definition of shear banding in complex fluids?

Shear-banding in complex fluids is interpreted as the consequence of the presence of an instability in the constitutive curve, i.e., the flow curve Σ0 = f(γ̇) that would be obtained if the flow were macroscopically homogeneous. 

Q4. What is the reason why the yielding transition is not converged?

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. 

Q5. How can the authors measure the spatial extent of correlations in the flow?

The spatial extent of correlations in the flow can be quantified by cooperativity or correlation lengths ξ in bulk flows, brought within reach by the computational efficiency of EPM. 

Q6. How did the authors bring the thermal activation of STs into play?

Dynamics were brought into play via the implementation of an event-driven (Kinetic Monte Carlo) scheme determining the thermal activation of STs, in the wake of the pioneering works of Bulatov and Argon (1994a). 

Q7. What were the first methods used to determine the statistics of avalanches?

Shortly after the emergence of the first EPM, mean-field approximations were exploited to determine the statistics of avalanches. 

Q8. What is the thickness of the shear bands?

These bands are typically 10 to 50nm or even 100 nm-thin (Bokeloh et al., 2011; Schuh et al., 2007), i.e., much thinner than the adiabatic shear bands encountered in crystalline metals and alloys, which are about 10− 100µm-thick. 

Q9. How does it show that homogeneous flow is unstable to perturbations?

it is easy to show that homogeneous flow in decreasing portions of the constitutive curve is unstable to perturbations and gives in to co-existing bands. 

Q10. What can be done to capture the q-dependence of the experimental intermediate scattering functions?

For instance the q-dependence of the experimental intermediate scattering functions S(q, t) (Cipelletti et al., 2003) cannot be captured in EPM at present, but could be included by implementing hybrid models that consider smaller-scale dynamics as well. 

Q11. Why is the EPM equation of motion akin to Eq. (47)?

the EPM equation of motion [Eq. (5)] cannot always be reduced to an expression akin to Eq. (47), because of the memory effects contained in the plastic activity variable n. 

Q12. What is the dominant view of the shear banding observed at low strain rates?

The dominant view is that it is however not the initial cause of the shear banding observed at low strain rates, as ∆T is small in this case. 

Q13. What is the effect of small oscillatory stress modulations on a granular packing?

Studying a related effect, Pons et al. (2015) have shown that applying small oscillatory stress modulations to a granular packing subjected to a small loading can dramatically fluidize it: