Extremal model for amorphous media plasticity

Thermally activated intermittent flow in amorphous solids

Using mean field theory and a mesoscale elastoplastic model, we analyze the steady state shear rheology of thermally activated amorphous solids. At sufficiently high temperature and driving rates, flow is continuous and described by well-established rheological flow laws such as Herschel-Bulkley and logarithmic rate dependence. However, we find that these flow laws change in the regime of intermittent flow, where collective events no longer overlap and serrated flow becomes pronounced. In this regime, we identify a thermal activation stress scale, xa(T,), that wholly captures the effect of driving rate  and temperature T on average flow stress, stress drop (avalanche) size and correlation lengths. Different rheological regimes are summarized in a dynamic phase diagram for the amorphous yielding transition. Theoretical predictions call for a need to re-examine the rheology of very slowly sheared amorphous matter much below the glass transition.

Loss of memory of an elastic line on its way to limit cycles

Oscillatory-driven amorphous materials forget their initial configuration and converge to limit cycles. Here we investigate this memory loss under a nonquasistatic drive in a minimal model system, with quenched disorder and memory encoded in a spatial pattern, where oscillating protocols are formally replaced by a positive-velocity drive. We consider an elastic line driven athermally in a quenched disorder with biperiodic boundary conditions and tunable system size, thus controlling the area swept by the line per cycle as would the oscillation amplitude. The convergence to disorder-dependent limit cycle is strongly coupled to the nature of its velocity dynamics depending on system size. Based on the corresponding phase diagram, we propose a generic scenario for memory formation in disordered systems under finite driving rate.

Mechanical excitation and marginal triggering during avalanches in sheared amorphous solids

We study plastic strain during individual avalanches in overdamped particle-scale molecular dynamics (MD) and mesoscale elastoplastic models (EPM) for amorphous solids sheared in the athermal quasistatic limit. We show that the spatial correlations in plastic activity exhibit a short length scale that grows as t^{3/4} in MD and ballistically in EPM, which is generated by mechanical excitation of nearby sites not necessarily close to their stability thresholds, and a longer lengthscale that grows diffusively for both models and is associated with remote marginally stable sites. These similarities in spatial correlations explain why simple EPMs accurately capture the size distribution of avalanches observed in MD, though the temporal profiles and dynamical critical exponents are quite different.

Finite-Disorder Critical Point in the Yielding Transition of Elastoplastic Models

Upon loading, amorphous solids can exhibit brittle yielding, with the abrupt formation of macroscopic shear bands leading to fracture, or ductile yielding, with a multitude of plastic events leading to homogeneous flow. It has been recently proposed, and subsequently questioned, that the two regimes are separated by a sharp critical point, as a function of some control parameter characterizing the intrinsic disorder strength and the degree of stability of the solid. In order to resolve this issue, we have performed extensive numerical simulations of athermally driven elastoplastic models with long-range and anisotropic realistic interaction kernels in two and three dimensions. Our results provide clear evidence for a finite-disorder critical point separating brittle and ductile yielding, and we provide an estimate of the critical exponents in 2D and 3D.

Yield precursor in primary creep of colloidal gels

Colloidal gels under constant moderate stress flow only after a prolonged solid-like deformation. Predicting the time-dependent yielding of the gels would facilitate control of their mechanical stability and transport, but early detectable signs of such delayed solid-to-fluid transition remain unknown. We show that the shear rate of colloidal gels under constant stress can forecast an eventual yielding during the earliest stage of deformation known as primary creep. The shear rate before failure exhibits a characteristic power-law decrease as a function of time, distinct from the linear viscoelastic response. We model this early-stage behavior as a series of uncorrelated local plastic events that are thermally activated, which illuminates the exponential dependence of the yield time on the applied stress. By revealing underlying viscoplasticity, this precursor to yield in the macroscopic shear rate provides a convenient tool to predict the yielding of a gel well in advance of its actual occurrence.

Clusters in an Epidemic Model with Long-Range Dispersal

In the presence of long-range dispersal, epidemics spread in spatially disconnected regions known as clusters. Here, we characterize exactly their statistical properties in a solvable model, in both the supercritical (outbreak) and critical regimes. We identify two diverging length scales, corresponding to the bulk and the outskirt of the epidemic. We reveal a nontrivial critical exponent that governs the cluster number and the distribution of their sizes and of the distances between them. We also discuss applications to depinning avalanches with long-range elasticity.

Residual stress in athermal soft disordered solids: insights from microscopic and mesoscale models

In soft amorphous materials, shear cessation after large shear deformation leads to configurations having residual shear stress. The origin of these states and the distribution of the local shear stresses within the material is not well understood, despite its importance for the change in material properties and consequent applications. In this work, we use molecular dynamics simulations of a model dense non-Brownian soft amorphous material to probe the non-trivial relaxation process towards a residual stress state. We find that, similar to thermal glasses, an increase in shear rate prior to the shear cessation leads to lower residual stress states. We rationalise our findings using a mesoscopic elasto-plastic description that explicitly includes a long range elastic response to local shear transformations. We find that after flow cessation the initial stress relaxation indeed depends on the pre-sheared stress state, but the final residual stress is majorly determined by newly activated plastic events occurring during the relaxation process, a scenario consistent with the phenomenology of avalanche dynamics in the low shear rate limit of steadily sheared amorphous solids. Our simplified coarse grained description not only allows capturing the phenomenology of residual stress states but also rationalising the altered material properties that are probed using small and large deformation protocols applied to the relaxed material.

Thermally activated flow in models of amorphous solids

Amorphous solids yield at a critical value Σ_{c} of the imposed stress Σ through a dynamical phase transition. While sharp in athermal systems, the presence of thermal fluctuations leads to the rounding of the transition and thermally activated flow even below Σ_{c}. Here we study the steady-state thermal flow of amorphous solids using a mesoscopic elastoplastic model. In the Hébraud-Lequex (HL) model we provide an analytical solution of the thermally activated flow at low temperature. We then propose a general scaling law that also describes the transition rounding. Finally, we find that the scaling law holds in numerical simulations of the HL model, a two-dimensional (2D) elastoplastic model, and previously published molecular dynamics simulations of 2D Lennard-Jones glass.

Quasistatic kinetic avalanches and self-organized criticality in deviatorically loaded granular media

The behavior of granular media under quasistatic loading has recently been shown to attain a stable evolution state corresponding to a manifold in the space of micromechanical variables. This state is characterized by sudden transitions between metastable jammed states, involving the partial micromechanical rearrangement of the granular medium. Using numerical simulations of two-dimensional granular media under quasistatic biaxial compression, we show that the dynamics in the stable evolution state is characterized by scale-free avalanches well before the macromechanical stationary flow regime traditionally linked to a self-organized critical state. This, together with the nonuniqueness and the nonmonotony of macroscopic deformation curves, suggests that the statistical avalanche properties and the susceptibilities of the system cannot be reduced to a function of the macromechanical state. The associated scaling exponents are nonuniversal and depend on the interactions between particles. For stiffer particles (or samples at low confining pressure) we find distributions of avalanche properties compatible with the predictions of mean-field theory. The scaling exponents decrease below the mean-field values for softer interactions between particles. These lower exponents are consistent with observations for amorphous solids at their critical point. We specifically discuss the relationship between microscopic and macroscopic variables, including the relation between the external stress drop and the internal potential energy released during kinetic avalanches.

Yielding in an Integer Automaton Model for Amorphous Solids under Cyclic Shear

We present results on an automaton model of an amorphous solid under cyclic shear. After a transient, the steady state falls into one of three cases in order of increasing strain amplitude: (i) pure elastic behavior with no plastic activity, (ii) limit cycles where the state recurs after an integer period of strain cycles, and (iii) irreversible plasticity with longtime diffusion. The number of cycles N required for the system to reach a periodic orbit diverges as the amplitude approaches the yielding transition between regimes (ii) and (iii) from below, while the effective diffusivity D of the plastic strain field vanishes on approach from above. Both of these divergences can be described by a power law. We further show that the average period T of the limit cycles increases on approach to yielding.

Elastoplastic Approach Based on Microscopic Insights for the Steady State and Transient Dynamics of Sheared Disordered Solids

We develop a framework to study the mechanical response of athermal amorphous solids via a coupling of mesoscale and microscopic models. Using measurements of coarse-grained quantities from simulations of dense disordered particulate systems, we present a coherent elastoplastic model approach for deformation and flow of yield stress materials. For a given set of parameters, this model allows us to match consistently transient and steady state features of driven disordered systems with diverse preparation histories under both applied shear-rate and creep protocols.

Spatial Clustering of Depinning Avalanches in Presence of Long-Range Interactions

Disordered elastic interfaces display avalanche dynamics at the depinning transition. For short-range interactions, avalanches correspond to compact reorganizations of the interface well described by the depinning theory. For long-range elasticity, an avalanche is a collection of spatially disconnected clusters. In this Letter we determine the scaling properties of the clusters and relate them to the roughness exponent of the interface. The key observation of our analysis is the identification of a Bienaymé-Galton-Watson process describing the statistics of the number of clusters. Our work has concrete importance for experimental applications where the cluster statistics is a key probe of avalanche dynamics.

Simple and Broadly Applicable Definition of Shear Transformation Zones

Plastic deformation in amorphous solids is known to be carried by stress-induced localized rearrangements of a few tens of particles, accompanied by the conversion of elastic energy to heat. Despite their central role in determining how glasses yield and break, the search for a simple and generally applicable definition of the precursors of those plastic rearrangements-the so-called shear transformation zones (STZs)-is still ongoing. Here we present a simple definition of STZs-based solely on the harmonic approximation of a glass's energy. We explain why and demonstrate directly that our proposed definition of plasticity carriers in amorphous solids is more broadly applicable compared to anharmonic definitions put forward previously. Finally, we offer an open-source library that analyzes low-lying STZs in computer glasses and in laboratory materials such as dense colloidal suspensions for which the harmonic approximation is accessible. Our results constitute a physically motivated methodological advancement towards characterizing mechanical disorder in glasses, and understanding how they yield.

Rejuvenation and shear banding in model amorphous solids

We measure the local yield stress, at the scale of small atomic regions, in a deeply quenched two-dimensional glass model undergoing shear banding in response to athermal quasistatic deformation. We find that the occurrence of essentially a single plastic event suffices to bring the local yield stress distribution to a well-defined value for all strain orientations, thus essentially erasing the memory of the initial structure. It follows that in a well-relaxed sample, plastic events cause the abrupt (nucleation-like) emergence of a local softness contrast and thus precipitate the formation of a band, which, in its early stages, is measurably softer than the steady-state flow. Moreover, this postevent yield stress ensemble presents a mean value comparable to that of the inherent states of a supercooled liquid around the mode-coupling temperature T_{MCT}. This, we argue, explains that the transition between brittle and ductile yielding in amorphous materials occurs around a comparable parent temperature. Our data also permit to capture quantitatively the contributions of pressure and density changes and demonstrate unambiguously that they are negligible compared with the changes of softness caused by structural rejuvenation.

Elementary plastic events in amorphous silica

Plastic instabilities in amorphous materials are often studied using idealized models of binary mixtures that do not capture accurately molecular interactions and bonding present in real glasses. Here we study atomic-scale plastic instabilities in a three-dimensional molecular dynamics model of silica glass under quasistatic shear. We identify two distinct types of elementary plastic events, one is a standard quasilocalized atomic rearrangement while the second is a bond-breaking event that is absent in simplified models of fragile glass formers. Our results show that both plastic events can be predicted by a drop of the lowest nonzero eigenvalue of the Hessian matrix that vanishes at a critical strain. Remarkably, we find very high correlation between the associated eigenvectors and the nonaffine displacement fields accompanying the bond-breaking event, predicting the locus of structural failure. Both eigenvectors and nonaffine displacement fields display an Eshelby-like quadrupolar structure for both failure modes, rearrangement, and bond breaking. Our results thus clarify the nature of atomic-scale plastic instabilities in silica glasses, providing useful information for the development of mesoscale models of amorphous plasticity.

Theory for the density of interacting quasilocalized modes in amorphous solids

Quasilocalized modes appear in the vibrational spectrum of amorphous solids at low frequency. Though never formalized, these modes are believed to have a close relationship with other important local excitations, including shear transformations and two-level systems. We provide a theory for their frequency density, D_{L}(ω)∼ω^{α}, that establishes this link for systems at zero temperature under quasistatic loading. It predicts two regimes depending on the density of shear transformations P(x)∼x^{θ} (with x the additional stress needed to trigger a shear transformation). If θ>1/4, then α=4 and a finite fraction of quasilocalized modes form shear transformations, whose amplitudes vanish at low frequencies. If θ<1/4, then α=3+4θ and all quasilocalized modes form shear transformations with a finite amplitude at vanishing frequencies. We confirm our predictions numerically.

Diffusion in Mesoscopic Lattice Models of Amorphous Plasticity

We present results on tagged particle diffusion in a mesoscale lattice model for sheared amorphous material in athermal quasistatic conditions. We find a short time diffusive regime and a long time diffusive regime whose diffusion coefficients depend on system size in dramatically different ways. At short time, we find that the diffusion coefficient, D, scales roughly linearly with system length, D∼L^{1.05}. This short time behavior is consistent with particle-based simulations. The long-time diffusion coefficient scales like D∼L^{1.6}, close to previous studies which found D∼L^{1.5}. Furthermore, we show that the near-field details of the interaction kernel do not affect the short time behavior but qualitatively and dramatically affect the long time behavior, potentially causing a saturation of the mean-squared displacement at long times. Our finding of a D∼L^{1.05} short time scaling resolves a long standing puzzle about the disagreement between the diffusion coefficient measured in particle-based models and mesoscale lattice models of amorphous plasticity.

Avalanche Behavior in Creep Failure of Disordered Materials

We present a mesoscale elastoplastic model of creep in disordered materials, which considers temperature-dependent stochastic activation of localized deformation events that are coupled by internal stresses, leading to collective avalanche dynamics. We generalize this stochastic plasticity model by introducing damage in terms of a local strength that decreases, on statistical average, with increasing local plastic strain. The model captures failure in terms of strain localization in a catastrophic shear band concomitant with a finite-time singularity of the creep rate. The statistics of avalanches in the run-up to failure is characterized by a decreasing avalanche exponent τ that, at failure, approaches the value τ=1.5 typical of a critical branching process. The average avalanche rate exhibits an inverse Omori law as a function of time to failure. The distribution of interavalanche times turns out to be consistent with the epidemic-type aftershock sequences (ETAS) model of earthquake statistics.

Crackling Dynamics in the Mechanical Response of Knitted Fabrics

Crackling noise, which occurs in a wide range of situations, is characterized by discrete events of various sizes, often correlated in the form of avalanches. We report experimental evidence that the mechanical response of a knitted fabric displays such broadly distributed events both in the force signal and in the deformation field, with statistics analogous to that of earthquakes or soft amorphous materials. A knit consists of a regular network of frictional contacts, linked by the elasticity of the yarn. When deformed, the fabric displays spatially extended avalanchelike yielding events resulting from collective interyarn contact slips. We measure the size distribution of these avalanches, at the stitch level from the analysis of nonelastic displacement fields and externally from force fluctuations. The two measurements yield consistent power law distributions reminiscent of those found in other avalanching systems. Our study shows that a knitted fabric is not only a thread-based metamaterial with highly sought after mechanical properties, but also an original, model system, with topologically protected structural order, where an intermittent, scale-invariant response emerges from minimal ingredients, and thus a significant landmark in the study of out-of-equilibrium universality.

Random critical point separates brittle and ductile yielding transitions in amorphous materials

We combine an analytically solvable mean-field elasto-plastic model with molecular dynamics simulations of a generic glass former to demonstrate that, depending on their preparation protocol, amorphous materials can yield in two qualitatively distinct ways. We show that well-annealed systems yield in a discontinuous brittle way, as metallic and molecular glasses do. Yielding corresponds in this case to a first-order nonequilibrium phase transition. As the degree of annealing decreases, the first-order character becomes weaker and the transition terminates in a second-order critical point in the universality class of an Ising model in a random field. For even more poorly annealed systems, yielding becomes a smooth crossover, representative of the ductile rheological behavior generically observed in foams, emulsions, and colloidal glasses. Our results show that the variety of yielding behaviors found in amorphous materials does not necessarily result from the diversity of particle interactions or microscopic dynamics but is instead unified by carefully considering the role of the initial stability of the system.

Orientation of plastic rearrangements in two-dimensional model glasses under shear

The plastic deformation of amorphous solids is mediated by localized shear transformations involving small groups of particles rearranging irreversibly in an elastic background. We introduce and compare three different computational methods to extract the size and orientation of these shear transformations in simulations of a two-dimensional athermal model glass under simple shear. We find that the shear angles are broadly distributed around the macroscopic shear direction, with a more or less Gaussian distribution with a standard deviation of around 20^{∘}. The distributions of sizes and orientations of shear transformations display no substantial sensitivity to the shear rate. These results can notably be used to refine the description of rearrangements in elastoplastic models.

Soft modes and strain redistribution in continuous models of amorphous plasticity: the Eshelby paradigm, and beyond?

The deformation of disordered solids relies on swift and localised rearrangements of particles. The inspection of soft vibrational modes can help predict the locations of these rearrangements, while the strain that they actually redistribute mediates collective effects. Here, we study soft modes and strain redistribution in a two-dimensional continuous mesoscopic model based on a Ginzburg-Landau free energy for perfect solids, supplemented with a plastic disorder potential that accounts for shear softening and rearrangements. Regardless of the disorder strength, our numerical simulations show soft modes that are always sharply peaked at the softest point of the material (unlike what happens for the depinning of an elastic interface). Contrary to widespread views, the deformation halo around this peak does not always have a quadrupolar (Eshelby-like) shape. Instead, for finite and narrowly-distributed disorder, it looks like a fracture, with a strain field that concentrates along some easy directions. These findings are rationalised with analytical calculations in the case where the plastic disorder is confined to a point-like 'impurity'. In this case, we unveil a continuous family of elastic propagators, which are identical for the soft modes and for the equilibrium configurations. This family interpolates between the standard quadrupolar propagator and the fracture-like one as the anisotropy of the elastic medium is increased. Therefore, we expect to see a fracture-like propagator when extended regions on the brink of failure have already softened along the shear direction and thus rendered the material anisotropic, but not failed yet. We speculate that this might be the case in carefully aged glasses just before macroscopic failure.

Local yield stress statistics in model amorphous solids

We develop and extend a method presented by Patinet, Vandembroucq, and Falk [Phys. Rev. Lett. 117, 045501 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.045501] to compute the local yield stresses at the atomic scale in model two-dimensional Lennard-Jones glasses produced via differing quench protocols. This technique allows us to sample the plastic rearrangements in a nonperturbative manner for different loading directions on a well-controlled length scale. Plastic activity upon shearing correlates strongly with the locations of low yield stresses in the quenched states. This correlation is higher in more structurally relaxed systems. The distribution of local yield stresses is also shown to strongly depend on the quench protocol: the more relaxed the glass, the higher the local plastic thresholds. Analysis of the magnitude of local plastic relaxations reveals that stress drops follow exponential distributions, justifying the hypothesis of an average characteristic amplitude often conjectured in mesoscopic or continuum models. The amplitude of the local plastic rearrangements increases on average with the yield stress, regardless of the system preparation. The local yield stress varies with the shear orientation tested and strongly correlates with the plastic rearrangement locations when the system is sheared correspondingly. It is thus argued that plastic rearrangements are the consequence of shear transformation zones encoded in the glass structure that possess weak slip planes along different orientations. Finally, we justify the length scale employed in this work and extract the yield threshold statistics as a function of the size of the probing zones. This method makes it possible to derive physically grounded models of plasticity for amorphous materials by directly revealing the relevant details of the shear transformation zones that mediate this process.

Local and global avalanches in a two-dimensional sheared granular medium

We present the experimental and numerical studies of a two-dimensional sheared amorphous material composed of bidisperse photoelastic disks. We analyze the statistics of avalanches during shear including the local and global fluctuations in energy and changes in particle positions and orientations. We find scale-free distributions for these global and local avalanches denoted by power laws whose cutoffs vary with interparticle friction and packing fraction. Different exponents are found for these power laws depending on the quantity from which variations are extracted. An asymmetry in time of the avalanche shapes is evidenced along with the fact that avalanches are mainly triggered by the shear bands. A simple relation independent of the intensity is found between the number of local avalanches and the global avalanches they form. We also compare these experimental and numerical results for both local and global fluctuations to predictions from mean-field and depinning theories.

Different universality classes at the yielding transition of amorphous systems

We study the yielding transition of a two-dimensional amorphous system under shear by using a mesoscopic elasto-plastic model. The model combines a full (tensorial) description of the elastic interactions in the system and the possibility of structural reaccommodations that are responsible for the plastic behavior. The possible structural reaccommodations are encoded in the form of a "plastic disorder" potential, which is chosen independently at each position of the sample to account for local heterogeneities. We observe that the stress must exceed a critical value σ_{c} in order for the system to yield. In addition, when the system yields a flow curve (relating stress σ and strain rate γ[over ̇]) of the form γ[over ̇]∼(σ-σ_{c})^{β} is obtained. Remarkably, we observe the value of β to depend on some details of the plastic disorder potential. For smooth potentials a value of β≃2.0 is obtained, whereas for potentials obtained as a concatenation of smooth pieces a value β≃1.5 is observed in the simulations. This indicates a dependence of critical behavior on details of the plastic behavior. In addition, by integrating out nonessential, harmonic degrees of freedom, we derive a simplified scalar version of the model that represents a collection of interacting Prandtl-Tomlinson particles. A mean-field treatment of this interaction reproduces the difference of β exponents for the two classes of plastic disorder potentials and provides values of β that compare favorably with those found in the full simulations.

Universal features of amorphous plasticity

Plastic yielding of amorphous solids occurs by power-law distributed deformation avalanches whose universality is still debated. Experiments and molecular dynamics simulations are hampered by limited statistical samples, and although existing stochastic models give precise exponents, they require strong assumptions about fixed deformation directions, at odds with the statistical isotropy of amorphous materials. Here, we introduce a fully tensorial, stochastic mesoscale model for amorphous plasticity that links the statistical physics of plastic yielding to engineering mechanics. It captures the complex shear patterning observed for a wide variety of deformation modes, as well as the avalanche dynamics of plastic flow. Avalanches are described by universal size exponents and scaling functions, avalanche shapes, and local stability distributions, independent of system dimensionality, boundary and loading conditions, and stress state. Our predictions consistently differ from those of mean-field depinning models, providing evidence that plastic yielding is a distinct type of critical phenomenon.

The universality class for plastic yield in amorphous materials remains controversial. Here authors present a tensorial mesoscale model that captures both complex shear patterns and avalanche scaling behaviour, which differs from mean-field models and suggests a distinct type of critical phenomenon.

Scaling description of non-local rheology

The plastic flow of amorphous materials displays non-local effects, characterized by a cooperativity length scale ξ. We argue that these effects enter in the more general description of surface phenomena near critical points. Using this approach, we obtain a scaling relation between exponents that describe the strain rate profiles in shear driven and pressure driven flow, which we confirm both in numerical models and experimental data. We find empirically that the cooperative length follows closely the characteristic length previously extracted in homogenous bulk flows. This analysis shows that the often used mean field exponents fail to capture quantitatively the non-local effects. Our analysis also explains the unusually large finite size effects previously observed in pressure driven flows.

Avalanches dynamics in reaction fronts in disordered flows

We report on numerical studies of avalanches of an autocatalytic reaction front in a porous medium. The front propagation is controlled by an adverse flow resulting in upstream, static, or downstream regimes. In an earlier study focusing on front shape, we identified three different universality classes associated with this system by following the front dynamics experimentally and numerically. Here, using numerical simulations in the vicinity of the second-order transition, we identify an avalanche dynamics characterized by power-law distributions of avalanche sizes, durations, and lateral extensions. The related exponents agree well with the quenched-Kardar-Parisi-Zhang theory, which describes the front dynamics. However, the geometry of the propagating front differs slightly from that of the theoretical one. We show that this discrepancy can be understood in terms of the nonquasistatic correction induced by the finite front velocity.

Coarse graining atomistic simulations of plastically deforming amorphous solids

The primary mode of failure in disordered solids results from the formation and persistence of highly localized regions of large plastic strains known as shear bands. Continuum-level field theories capable of predicting this mechanical response rely upon an accurate representation of the initial and evolving states of the amorphous structure. We perform molecular dynamics simulations of a metallic glass and propose a methodology for coarse graining discrete, atomistic quantities, such as the potential energies of the elemental constituents. A strain criterion is established and used to distinguish the coarse-grained degrees-of-freedom inside the emerging shear band from those of the surrounding material. A signal-to-noise ratio provides a means of evaluating the strength of the signal of the shear band as a function of the coarse graining. Finally, we investigate the effect of different coarse graining length scales by comparing a two-dimensional, numerical implementation of the effective-temperature description in the shear transformation zone (STZ) theory with direct molecular dynamics simulations. These comparisons indicate the coarse graining length scale has a lower bound, above which there is a high level of agreement between the atomistics and the STZ theory, and below which the concept of effective temperature breaks down.

Direct Observation of Percolation in the Yielding Transition of Colloidal Glasses

When strained beyond the linear regime, soft colloidal glasses yield to steady-state plastic flow in a way that is similar to the deformation of conventional amorphous solids. Because of the much larger size of the colloidal particles with respect to the atoms comprising an amorphous solid, colloidal glasses allow us to obtain microscopic insight into the nature of the yielding transition, as we illustrate here combining experiments, atomistic simulations, and mesoscopic modeling. Our results unanimously show growing clusters of nonaffine deformation percolating at yielding. In agreement with percolation theory, the spanning cluster is fractal with a fractal dimension d_{f}≃2, and the correlation length diverges upon approaching the critical yield strain. These results indicate that percolation of highly nonaffine particles is the hallmark of the yielding transition in disordered glassy systems.

Connecting Local Yield Stresses with Plastic Activity in Amorphous Solids

In model amorphous solids produced via differing quench protocols, a strong correlation is established between local yield stress measured by direct local probing of shear stress thresholds and the plastic rearrangements observed during remote loading in shear. This purely local measure shows a higher predictive power for identifying sites of plastic activity when compared with more conventional structural properties. Most importantly, the sites of low local yield stress, thus defined, are shown to be persistent, remaining predictive of deformation events even after fifty or more such plastic rearrangements. This direct and nonperturbative approach gives access to relevant transition pathways that control the stability of amorphous solids. Our results reinforce the relevance of modeling plasticity in amorphous solids based on a gradually evolving population of discrete and local zones preexisting in the structure.

From depinning transition to plastic yielding of amorphous media: A soft-modes perspective

A mesoscopic model of amorphous plasticity is discussed in the context of depinning models. After embedding in a d+1-dimensional space, where the accumulated plastic strain lives along the additional dimension, the gradual plastic deformation of amorphous media can be regarded as the motion of an elastic manifold in a disordered landscape. While the associated depinning transition leads to scaling properties, the quadrupolar Eshelby interactions at play in amorphous plasticity induce specific additional features like shear-banding and weak ergodicity breakdown. The latters are shown to be controlled by the existence of soft modes of the elastic interaction, the consequence of which is discussed in the context of depinning.

Finite-size effects in a model for plasticity of amorphous composites

We discuss the plastic behavior of an amorphous matrix reinforced by hard particles. A mesoscopic depinning-like model accounting for Eshelby elastic interactions is implemented. Only the effect of a plastic disorder is considered. Numerical results show a complex size dependence of the effective flow stress of the amorphous composite. In particular, the departure from the mixing law shows opposite trends associated to the competing effects of the matrix and the reinforcing particles, respectively. The reinforcing mechanisms and their effects on localization are discussed. Plastic strain is shown to gradually concentrate on the weakest band of the system. This correlation of the plastic behavior with the material structure is used to design a simple analytical model. The latter nicely captures reinforcement size effects in (logN/N)(1/2), where N is the linear size of the system, observed numerically. Predictions of the effective flow stress accounting for further logarithmic corrections show a very good agreement with numerical results.

Spontaneous instabilities and stick-slip motion in a generalized Hébraud-Lequeux model

We revisit the Hébraud-Lequeux (HL) model for the rheology of jammed materials and argue that a possibly important time scale is missing from HL's initial specification. We show that our generalization of the HL model undergoes interesting oscillating instabilities for a wide range of parameters, which lead to intermittent, stick-slip flows under constant shear rate. The instability we find is akin to the synchronization transition of coupled elements that arises in many different contexts (neurons, fireflies, financial bankruptcies, etc.). We hope that our scenario could shed light on the commonly observed intermittent, serrated flows of glassy materials under shear.

Nonlinear plastic modes in disordered solids

We propose a theoretical framework within which a robust micromechanical definition of precursors to plastic instabilities, often termed soft spots, naturally emerges. They are shown to be collective displacements (modes) z[over ̂] that correspond to local minima of a barrier function b(z[over ̂]), which depends solely on inherent structure information. We demonstrate how some heuristic searches for local minima of b(z[over ̂]) can a priori detect the locus and geometry of imminent plastic instabilities with remarkable accuracy, at strains as large as γ_{c}-γ∼10^{-2} away from the instability strain γ_{c}. Our findings suggest that the a priori detection of the entire field of soft spots can be effectively carried out by a systematic investigation of the landscape of b(z[over ̂]).

Criticality in the Approach to Failure in Amorphous Solids

Failure of amorphous solids is fundamental to various phenomena, including landslides and earthquakes. Recent experiments indicate that highly plastic regions form elongated structures that are especially apparent near the maximal shear stress Σmax where failure occurs. This observation suggested that Σmax acts as a critical point where the length scale of those structures diverges, possibly causing macroscopic transient shear bands. Here, we argue instead that the entire solid phase (Σ<Σmax) is critical, that plasticity always involves system-spanning events, and that their magnitude diverges at Σmax independently of the presence of shear bands. We relate the statistics and fractal properties of these rearrangements to an exponent θ that captures the stability of the material, which is observed to vary continuously with stress, and we confirm our predictions in elastoplastic models.

Avalanche-size distributions in mean-field plastic yielding models

We discuss the size distribution N(S) of avalanches occurring at the yielding transition of mean-field (i.e., Hebraud-Lequeux) models of amorphous solids. The size distribution follows a power law dependence of the form N(S)∼S(-τ). However (contrary to what is found in its depinning counterpart), the value of τ depends on details of the dynamic protocol used. For random triggering of avalanches we recover the τ=3/2 exponent typical of mean-field models, which, in particular, is valid for the depinning case. However, for the physically relevant case of external loading through a quasistatic increase of applied strain, a smaller exponent (close to 1) is obtained. This result is rationalized by mapping the problem to an effective random walk in the presence of a moving absorbing boundary.

On the relevance of disorder in athermal amorphous materials under shear

We show that, at least at a mean-field level, the effect of structural disorder in sheared amorphous media is very dissimilar depending on the thermal or athermal nature of their underlying dynamics. We first introduce a toy model, including explicitly two types of noise (thermal versus athermal). Within this interpretation framework, we argue that mean-field athermal dynamics can be accounted for by the so-called Hébraud-Lequeux (HL) model, in which the mechanical noise stems explicitly from the plastic activity in the sheared medium. Then, we show that the inclusion of structural disorder, by means of a distribution of yield energy barriers, has no qualitative effect in the HL model, while such a disorder is known to be one of the key ingredients leading kinematically to a finite macroscopic yield stress in other mean-field descriptions, such as the Soft-Glassy-Rheology model. We conclude that the statistical mechanisms at play in the emergence of a macroscopic yield stress, and a complex stationary dynamics at low shear rate, are different in thermal and athermal amorphous systems.

Scaling description of the yielding transition in soft amorphous solids at zero temperature

Yield stress materials flow if a sufficiently large shear stress is applied. Although such materials are ubiquitous and relevant for industry, there is no accepted microscopic description of how they yield, even in the simplest situations in which temperature is negligible and in which flow inhomogeneities such as shear bands or fractures are absent. Here we propose a scaling description of the yielding transition in amorphous solids made of soft particles at zero temperature. Our description makes a connection between the Herschel-Bulkley exponent characterizing the singularity of the flow curve near the yield stress Σc, the extension and duration of the avalanches of plasticity observed at threshold, and the density P(x) of soft spots, or shear transformation zones, as a function of the stress increment x beyond which they yield. We argue that the critical exponents of the yielding transition may be expressed in terms of three independent exponents, θ, df, and z, characterizing, respectively, the density of soft spots, the fractal dimension of the avalanches, and their duration. Our description shares some similarity with the depinning transition that occurs when an elastic manifold is driven through a random potential, but also presents some striking differences. We test our arguments in an elasto-plastic model, an automaton model similar to those used in depinning, but with a different interaction kernel, and find satisfying agreement with our predictions in both two and three dimensions.

Plasticity of a colloidal polycrystal under cyclic shear

We use confocal microscopy and time-resolved light scattering to investigate plasticity in a colloidal polycrystal, following the evolution of the network of grain boundaries as the sample is submitted to thousands of shear deformation cycles. The grain boundary motion is found to be ballistic, with a velocity distribution function exhibiting nontrivial power law tails. The shear-induced dynamics initially slow down, similarly to the aging of the spontaneous dynamics in glassy materials, but eventually reach a steady state. Surprisingly, the crossover time between the initial aging regime and the steady state decreases with increasing probed length scale, hinting at a hierarchical organization of the grain boundary dynamics.

Universal and non-universal features in coarse-grained models of flow in disordered solids

We study the two-dimensional (2D) shear flow of amorphous solids within variants of an elastoplastic model, paying particular attention to spatial correlations and time fluctuations of, e.g., local stresses. The model is based on the local alternation between an elastic regime and plastic events during which the local stress is redistributed. The importance of a fully tensorial description of the stress and of the inclusion of (coarse-grained) convection in the model is investigated; scalar and tensorial models yield similar results, while convection enhances fluctuations and breaks the spurious symmetry between the flow and velocity gradient directions, for instance when shear localisation is observed. Besides, correlation lengths measured with diverse protocols are discussed. One class of such correlation lengths simply scale with the spacing between homogeneously distributed, simultaneous plastic events. This leads to a scaling of the correlation length with the shear rate as γ̇(-1/2) in 2D in the athermal regime, regardless of the details of the model. The radius of the cooperative disk, defined as the near-field region in which plastic events induce a stress redistribution that is not amenable to a mean-field treatment, notably follows this scaling. On the other hand, the cooperative volume measured from the four-point stress susceptibility and its dependence on the system size and the shear rate are model-dependent.

Time-dependent elastic response to a local shear transformation in amorphous solids

The elastic response of a two-dimensional amorphous solid to induced local shear transformations, which mimic the elementary plastic events occurring in deformed glasses, is investigated via molecular-dynamics simulations. We show that for different spatial realizations of the transformation, despite relative fluctuations of order one, the long-time equilibrium response averages out to the prediction of the Eshelby inclusion problem for a continuum elastic medium. We characterize the effects of the underlying dynamics on the propagation of the elastic signal. A crossover from a propagative transmission in the case of weakly damped dynamics to a diffusive transmission for strong damping is evidenced. In the latter case, the full time-dependent elastic response is in agreement with the theoretical prediction, obtained by solving the diffusion equation for the displacement field in an elastic medium.

Elastic-plastic-brittle transitions and avalanches in disordered media

A spring lattice model with the ability to simulate elastic-plastic-brittle transitions in a disordered medium is presented. The model is based on bilinear constitutive law defined at the spring level and power-law-type disorder introduced in the yield and failure limits of the springs. The key parameters of the proposed model effectively control the disorder distribution, significantly affecting the stress-strain response, the damage accumulation process, and the fracture surfaces. The model demonstrates a plastic strain avalanche behavior for perfectly plastic as well as hardening materials with a power-law distribution, in agreement with the experiments and related models. The strength of the model is in its generality and ability to interpolate between elastic-plastic hardening and elastic-brittle transitions.

Avalanche localization and crossover scaling in amorphous plasticity

We perform large-scale simulations of a two-dimensional lattice model for amorphous plasticity with random local yield stresses and long-range quadrupolar elastic interactions. We show that as the external stress increases towards the yielding phase transition, the scaling behavior of the avalanches crosses over from mean-field theory to a different universality class. This behavior is associated with strain localization, which significantly depends on the short-range properties of the interaction kernel.

Yield stress in amorphous solids: a mode-coupling-theory analysis

The yield stress is a defining feature of amorphous materials which is difficult to analyze theoretically, because it stems from the strongly nonlinear response of an arrested solid to an applied deformation. Mode-coupling theory predicts the flow curves of materials undergoing a glass transition and thus offers predictions for the yield stress of amorphous solids. We use this approach to analyze several classes of disordered solids, using simple models of hard-sphere glasses, soft glasses, and metallic glasses for which the mode-coupling predictions can be directly compared to the outcome of numerical measurements. The theory correctly describes the emergence of a yield stress of entropic nature in hard-sphere glasses, and its rapid growth as density approaches random close packing at qualitative level. By contrast, the emergence of solid behavior in soft and metallic glasses, which originates from direct particle interactions is not well described by the theory. We show that similar shortcomings arise in the description of the caging dynamics of the glass phase at rest. We discuss the range of applicability of mode-coupling theory to understand the yield stress and nonlinear rheology of amorphous materials.

Spatial cooperativity in microchannel flows of soft jammed materials: a mesoscopic approach

The flow of amorphous solids results from a combination of elastic deformation and local structural rearrangements, which induce nonlocal elastic deformations. These elements are incorporated into a mechanically consistent mesoscopic model of interacting elastoplastic blocks. We investigate the specific case of channel flow with numerical simulations, paying particular attention to situations of strong confinement. We find that the simple picture of plastic events embedded in an elastic matrix successfully accounts for manifestations of spatial cooperativity. Shear rate fluctuations are observed in seemingly quiescent regions, and the velocity profiles in confined flows at high applied pressure deviate from those expected in the absence of nonlocal effects, in agreement with experimental data. However, we suggest a different physical origin for the large deviations observed when walls have rough surfaces, associated with "bumps" of the particles against the asperities of the walls.

A mesoscopic model for the rheology of soft amorphous solids, with application to microchannel flows

We have studied a mesoscopic model for the flow of amorphous solids. The model is based on key features identified at the microscopic level, namely periods of elastic deformation interspersed with localised rearrangements of particles that induce long-range elastic deformations. These long-range deformations are derived following a continuum mechanics approach, in the presence of solid boundaries, and are included in full in the model. Indeed, they mediate spatial cooperativity in the flow, whereby a localised rearrangement may lead a distant region to yield. In particular, we have simulated a channel flow and found manifestations of spatial cooperativity that are consistent with published experimental observations for concentrated emulsions in microchannels. Two categories of effects are distinguished. On the one hand, the coupling of regions subject to different shear rates, for instance, leads to finite shear rate fluctuations in the seemingly unsheared "plug" in the centre of the channel. On the other hand, there is convincing experimental evidence of a specific rheology near rough walls. We discuss the diverse possible physical origins for this effect, and we suggest that it may be associated with the bumps of particles into surface asperities as they slide along the wall.

Inhomogeneous shear flows in soft jammed materials with tunable attractive forces

We perform molecular dynamics simulations to characterize the occurrence of inhomogeneous shear flows in soft jammed materials. We use rough walls to impose a simple shear flow and study the athermal motion of jammed assemblies of soft particles in two spatial dimensions, both for purely repulsive interactions and in the presence of an additional short-range attraction of varying strength. In steady state, pronounced flow inhomogeneities emerge for all systems when the shear rate becomes small. Deviations from linear flow are stronger in magnitude and become very long lived when the strength of the attraction increases, but differ from permanent shear bands. Flow inhomogeneities occur in a stress window bounded by the dynamic and static yield stress values. Attractive forces enhance the flow heterogeneities because they accelerate stress relaxation, thus effectively moving the system closer to the yield stress regime where inhomogeneities are most pronounced. The present scenario for understanding the effect of particle adhesion on shear localization, which is based on detailed molecular dynamics simulations with realistic particle interactions, differs qualitatively from previous qualitative explanations and ad hoc theoretical modeling.