Yield stress fluids display complex dynamics, in particular when driven into the transient regime between the solid and the flowing state. Inspired by creep experiments on dense amorphous materials, we implement mesocale elasto-plastic descriptions to analyze such transient dynamics in athermal systems. Both our mean-field and space-dependent approaches consistently reproduce the typical experimental strain rate responses to different applied steps in stress. Moreover, they allow us to understand basic processes involved in the strain rate slowing down (creep) and the strain rate acceleration (fluidization) phases. The fluidization time increases in a power-law fashion as the applied external stress approaches a static yield stress. This stress value is related to the stress over-shoot in shear start-up experiments, and it is known to depend on sample preparation and age. By calculating correlations of the accumulated plasticity in the spatially resolved model, we reveal different modes of cooperative motion during the creep dynamics.
We develop an elasto-plastic description for the transient dynamics prior to steady flow of athermally yielding materials. Our mean-field model not only reproduces the experimentally observed non-linear time dependence of the shear-rate response to an external shear-stress, but also allows for the determination of the different physical processes involved in the onset of the re-acceleration phase after the initial critical slowing down and a distinct well defined fluidization phase. The evidenced power-law dependence of the fluidization time on the distance of the applied to an age dependent s…
Yield stress materials fail when the imposed stress crosses a critical threshold. A well-known dynamical response to the applied stress is the phenomenon of creep where the cumulative deformation grows sublinearly with time, prior to failure or arrest. Using extensive molecular dynamics simulations, we study such response for a model amorphous system, in the athermal limit, and probe how the annealing history of the initial state determines the observed behaviour to an applied shear stress. Further, we analyze the microscopic dynamics in the vicinity of the yield threshold, using large systems…
We study the statistical properties of the yielding transition in model amorphous solids in the limit of slow, athermal deformation. Plastic flow occurs via alternating phases of elastic loading punctuated by rapid dissipative events in the form of collective avalanches. We investigate their characterization through energy vs. stress drops and at multiple stages of deformation, thus revealing a change of spatial extent of the avalanches and degree of stress correlations as deformation progresses. We show that the statistics of stress and energy drops only become comparable for large events in …
The distribution of local residual stresses (threshold to instability) that controls the statistical properties of plastic flow in athermal amorphous solids is examined with an atomistic simulation technique. For quiescent configurations, the distribution has a pseudogap (power-law) form with an exponent that agrees well with global yielding statistics. As soon as deformation sets in, the pseudogap region gives way to a system size dependent plateau at small residual stresses that can be understood from the statistics of local residual stress {\em differences} between plastic events. Results f…
We propose a method to predict the value of the external strain where a generic amorphous solid will fail by a plastic response (i.e. an irreversible deformation), solely on the basis of measurements of the nonlinear elastic moduli. While usually considered fundamentally different, with the elastic properties describing reversible phenomena and plastic failure epitomizing irreversible behavior, we show that the knowledge of some nonlinear elastic moduli is enough to predict where plasticity sets in.
Yielding transitions in athermal amorphous materials resemble critical phenomena. Historically, they have been described by the Herschel-Bulkley rheological formula, which implies singular behaviors at yield points. In this paper, I examine this class of phenomena using an elementary version of the thermodynamic shear-transformation-zone (STZ) theory, focusing on the role of the effective disorder temperature, and paying special attention to scaling and dimensional arguments. I find a wide variety of Herschel-Bulkley-like rheologies but, for fundamental reasons not specific to the STZ theory, …
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 H{é}braud-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,…
We present the results of numerical simulations of an atomistic system undergoing plastic shear flow in the athermal, quasistatic limit. The system is shown to undergo cascades of local rearrangements, associated with quadrupolar energy fluctuations, which induce system-spanning events organized into lines of slip oriented along the Bravais axes of the simulation cell. A finite size scaling analysis reveals subextensive scaling of the energy drops and participation numbers, linear in the length of the simulation cell, in good agreement with the observed real-space structure of the plastic even…
We study the linear response to strain in a mean field elastoplastic model for athermal amorphous solids, incorporating the power-law mechanical noise spectrum arising from plastic events. In the "jammed" regime of the model, where the plastic activity exhibits a non-trivial slow relaxation referred to as aging, we find that the stress relaxes incompletely to an age-dependent plateau, on a timescale which grows with material age. We determine the scaling behaviour of this aging linear response analytically, finding that key scaling exponents are universal and independent of the noise exponent …
We study the elastic theory of amorphous solids made of particles with finite range interactions in the thermodynamic limit. For the elastic theory to exist one requires all the elastic coefficients, linear and nonlinear, to attain a finite thermodynamic limit. We show that for such systems the existence of non-affine mechanical responses results in anomalous fluctuations of all the nonlinear coefficients of the elastic theory. While the shear modulus exists, the first nonlinear coefficient B_2 has anomalous fluctuations and the second nonlinear coefficient B_3 and all the higher order coeffic…
When submitted to a constant mechanical load, many amorphous solids display power law creep followed by fluidization. A fundamental understanding of these processes is still far from being achieved. Here, we characterize creep and fluidization on the basis of a mesoscopic viscoplastic model that includes thermally activated yielding events and a broad distribution of energy barriers, which may be lowered under the effect of a local deformation. We relate the creep exponent observed before fluidization to the width of barrier distribution and to the specific form of stress redistribution follow…
Amorphous solids such as coffee foam, toothpaste or mayonnaise display a transient creep flow when a stress $Σ$ is suddenly imposed. The associated strain rate is commonly found to decay in time as $\dotγ \sim t^{-ν}$, followed either by arrest or by a sudden fluidisation. Various empirical laws have been suggested for the creep exponent $ν$ and fluidisation time $τ_f$ in experimental and numerical studies. Here, we postulate that plastic flow is governed by the difference between $Σ$ and the transient yield stress $Σ_t(γ)$ that characterises the stability of configurations visited by the syst…
We apply principal component analysis, a method frequently used in image processing and unsupervised machine learning, to characterize particle displacements observed in the steady shear flow of amorphous solids. PCA produces a low-dimensional representation of the data and clearly reveals the dominant features of elastic (i.e. reversible) and plastic deformation. We show that the principal directions of PCA in the plastic regime correspond to the soft (i.e. zero energy) modes of the elastic propagator that governs the redistribution of shear stress due to localized plastic events. Projections…
We study theoretically the yielding of sheared amorphous materials as a function of increasing levels of initial sample annealing prior to shear, in three widely used constitutive models and three widely studied annealing protocols. In thermal systems we find a gradual progression, with increasing annealing, from smoothly "ductile" yielding, in which the sample remains homogeneous, to abruptly "brittle" yielding, in which it becomes strongly shear banded. This progression arises from an increase with annealing in the size of an overshoot in the underlying stress-strain curve for homogeneous sh…
We report computer simulations of oscillatory athermal quasi-static shear deformation of dense amorphous samples of a three dimensional model glass former. A dynamical transition is observed as the amplitude of the deformation is varied: for large values of the amplitude the system exhibits diffusive behavior and loss of memory of the initial conditions, whereas localization is observed for small amplitudes. Our results suggest that the same kind of transition found in driven colloidal systems is present in the case of amorphous solids (e.g. metallic glasses). The onset of the transition is sh…
We study the rheology of amorphous solids in the limit of negligible thermal fluctuations. On the basis of general arguments, the flow curve is shown to result from an interplay between the time scales of the macroscopic driving and the (cascades of) local particle rearrangements. Such rearrangements are known to induce a redistribution of the elastic stress in the system. Although mechanical noise, i.e., the local stress fluctuations arising from this redistribution, is widely believed to activate new particle rearrangements, we provide evidence that casts severe doubt on the analogy with the…
Using numerical simulations, we study the failure of an amorphous solid under quasi-static expansion starting from a homogeneous high-density state. During the volume expansion, we demonstrate the existence of instabilities manifesting via saddle-node bifurcation in which a minimum meets a saddle. During all such events, the smallest eigenvalue of the Hessian matrix vanishes as a square-root singularity. The plastic instabilities are manifested via sudden jumps in pressure and energy, with the largest event happening when a cavity appears, leading to the yielding of the material. We show that …
When an amorphous solid is deformed homogeneously, the response exhibits heterogeneous plastic instabilities with localized cooperative rearrangement of cluster of particles. The heterogeneous behavior plays an important role in deciding the mechanical properties of amorphous solids. In this paper, we employ computer simulation to study the characteristics and the spatial correlations of these clusters characterized by the non-affine displacements in amorphous solids under simple shear deformation in the athermal quasistatic limit. The clusters with large displacements are found to be homogene…
We numerically study the local stress distribution within athermal, isotropically stressed, mechanically stable, packings of bidisperse frictionless disks above the jamming transition in two dimensions. Considering the Fourier transform of the local stress, we find evidence for algebraically increasing fluctuations in both isotropic and anisotropic components of the stress tensor at small wavenumbers, contrary to recent theoretical predictions. Such increasing fluctuations imply a lack of self-averaging of the stress on large length scales. The crossover to these increasing fluctuations define…
Applying constant tensile stress to a piece of amorphous solid results in a slow extension, followed by an eventual rapid mechanical collapse. This "creep" process is of paramount engineering concern, and as such was the subject of study in a variety of materials, for more than a century. Predictive theories for $τ_w$, the expected time of collapse, are lacking, mainly due to its dependence on a bewildering variety of parameters, including temperature, system size, tensile force, but also the detailed microscopic interactions between constituents. The complex dependence of the collapse time on…
The slow flow of amorphous solids exhibits striking heterogeneities: swift localised particle rearrangements take place in the midst of a more or less homogeneously deforming medium. Recently, experimental as well as numerical work has revealed spatial correlations between these flow heterogeneities. Here, we use molecular dynamics (MD) simulations to characterise the rearrangements and systematically probe their correlations both in time and in space. In particular, these correlations display a four-fold azimuthal symmetry characteristic of shear stress redistribution in an elastic medium and…
We present an extensive numerical study of dynamical heterogeneities and their influence on diffusion in an athermal mesoscopic model for actively deformed amorphous solids. At low strain rates the stress dynamics are governed by cooperative regions of plastic events. On the basis of scaling arguments as well as an extensive numerical study of an athermal elasto-plastic model, we show that there is a direct link between the self-diffusion coefficient and the size of cooperative regions at low strain rates. Both depend strongly on rate and on system size. A measure of the mean square displaceme…
We present results on a series of 2D atomistic computer simulations of amorphous systems subjected to simple shear in the athermal, quasistatic limit. The athermal quasistatic trajectories are shown to separate into smooth, reversible elastic branches which are intermittently broken by discrete catastrophic plastic events. The onset of a typical plastic event is studied with precision, and it is shown that the mode of the system which is responsible for the loss of stability has structure in real space which is consistent with a quadrupolar source acting on an elastic matrix. The plastic event…
Granular materials such as sand, powders, and grains are omnipresent in daily life, industrial applications, and earth-science [1]. When unperturbed, they form stable structures that resemble the ones of other amorphous solids like metallic and colloidal glasses [2]. It is commonly conjectured that all these amorphous materials show a universal mechanical response when sheared slowly, i.e., to have an elastic regime, followed by yielding [3]. Here we use X-ray tomography to determine the microscopic dynamics of a cyclically sheared granular system in three dimensions. Independent of the shear …
From soft polymeric gels to hardened cement paste, amorphous solids under constant load exhibit a pronounced time-dependent deformation called creep. The microscopic mechanism of such a phenomenon is poorly understood in amorphous materials and constitutes an even greater challenge in densely packed and chemically reactive granular systems. Both features are prominently present in hydrating cement pastes composed of calcium silicate hydrate (C-S-H) nanoparticles, whose packing density increases as a function of time, while cement hydration is taking place. Performing nano-indentation tests and…
Weakly non-linear creep of amorphous polymers near their glass transition, comparisons between models and experiment
Ecole Supérieure de Physique et Chimie Industrielles (ESPCI), Université Pierre et Marie Curie, Paris (UPMC)（巴黎高等物理化工学院，皮埃尔和玛丽·居里大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
Creep is defined as time-dependent deformation and rupture processes taking place within a material subjected to a constant applied stress smaller than its athermal, time-independent strength. This time-dependence is classically attributed to thermal activation of local deformation events. The phenomenology of creep is characterized by several ubiquitous but empirical rheological and scaling laws. We focus here on primary creep following the onset of loading, for which a power law decay of the strain-rate is observed, with the exponent p varying between '0.4 and 1, this upper bound defining th…
Although the notion of mechanical noise is expected to play a key role in the non-linear rheology of athermally sheared amorphous systems, its characterization has so far remained elusive. Here, we show using molecular dynamic simulations that in spite of the presence of strong spatio-temporal correlations in the system, the local stress exhibits normal diffusion under the effect of the mechanical noise in the finite driving regime. The diffusion constant appears to be proportional to the mean plastic activity. Our data suggests that the corresponding proportionality constant is density indepe…
We investigate the down-hill creep of a layer of granular material on a slope caused by an oscillatory variation of the size of the particles. The material is modeled as an athermal two dimensional polydisperse system of soft disks under the action of gravity. The slope angle is below the critical rest angle and therefore the system reaches an equilibrium configuration under static external conditions. However, under a protocol in which particles slowly change size in a quasistatic oscillatory way the system is observed to creep down in a synchronized way with the oscillation. We measure the c…
Granular packings display a wealth of mechanical features which are of widespread significance. One of these features is creep: the slow deformation under applied stress. Creep is common for many other amorphous materials such as many metals and polymers. The slow motion of creep is challenging to understand, probe and control. We probe the creep properties of packings of soft spheres with a sinking ball viscometer. We find that in our granular packings, creep persists up to large strains and has a power law form, with diffusive dynamics. The creep amplitude is exponentially dependent on both …
We consider the slow and athermal deformations of amorphous solids and show how the ensuing sequence of discrete plastic rearrangements can be mapped onto a directed network. The network topology reveals a set of highly connected regions joined by occasional one-way transitions. The highly connected regions include hierarchically organized hysteresis cycles and sub-cycles. At small to moderate strains this organization leads to near-perfect return point memory. The transitions in the network can be traced back to localized particle rearrangements (soft-spots) that interact via Eshelby-type def…
The yielding transition that occurs in amorphous solids under athermal quasistatic deformation has been the subject of many theoretical and computational studies. Here, we extend this analysis to include thermal effects at finite shear rate, focusing on how temperature alters avalanches. We derive a nonequilibrium phase diagram capturing how temperature and strain rate effects compete, when avalanches overlap, and whether finite-size effects dominate over temperature effects. The predictions are tested through simulations of an elastoplastic model in two dimensions and in a mean-field approxim…
Since the 1970's, theories of deformation and failure of amorphous, solidlike materials have started with models in which stress-driven, molecular rearrangements occur at localized flow defects via "shear transformations". This picture is the basis for the modern theory of "shear transformation zones" (STZ's), which is the focus of this review. We begin by describing the structure of the theory in general terms and by showing several applications, specifically: interpretation of stress-strain measurements for a bulk metallic glass, analysis of numerical simulations of shear banding, and the us…
We couple the recently developed self-learning metabasin escape algorithm, which enables efficient exploration of the potential energy surface (PES), with shear deformation to elucidate strain-rate and temperature effects on the shear transformation zone (STZ) characteristics in two-dimensional amorphous solids. In doing so, we report a transition in the STZ characteristics that can be obtained through either increasing the temperature or decreasing the strain rate. The transition separates regions having two distinct STZ characteristics. Specifically, at high temperatures and high strain rate…
The behavior of shear-oscillated amorphous materials is studied using a coarse-grained model. Samples are prepared at different degrees of annealing and then subject to athermal and quasistatic oscillatory deformations at various fixed amplitudes. The steady-state reached after several oscillations is fully determined by the initial preparation and the oscillation amplitude, as seen from stroboscopic stress and energy measurements. Under small oscillations, poorly annealed materials display shear-annealing, while ultra-stabilized materials are insensitive to them. Yet, beyond a critical oscill…
In soft amorphous materials, shear cessation after large shear deformation leads to structures 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 s…
We address the cross effects between mechanical strains and magnetic fields on the plastic response of magnetoelastic amorphous solids. It is well known that plasticity in non-magnetic amorphous solids under external strain $γ$ is dominated by the co-dimension 1 saddle-node bifurcation in which an eigenvalue of the Hessian matrix vanishes at $γ_P$ like $\sqrt{γ_P-γ}$. This square-root singularity determines much of the statistical physics of elasto-plasticity, and in particular that of the stress-strain curves under athermal-quasistatic conditions. In this Letter we discuss the much richer phy…
Amorphous magnetic solids, like metallic glasses, exhibit a novel effect: the growth of magnetic order as a function of mechanical strain under athermal conditions in the presence of a magnetic field. The magnetic moment increases in steps whenever there is a plastic event. Thus plasticity induces the magnetic ordering, acting as the effective noise driving the system towards equilibrium. We present results of atomistic simulations of this effect in a model of a magnetic amorphous solid subjected to pure shear and a magnetic field. To elucidate the dependence on external strain and magnetic fi…
Stress vs. strain fluctuations in athermal amorphous solids are an example of `crackling noise' of the type studied extensively in the context of elastic membranes moving through random potentials. Contrary to the latter, we do not have a stochastic equation whose statistics agree with the measured ones. On the other hand we show in this Letter that the statistics of the former exhibit 'density scaling' when the interparticle potential can be well approximated by a power law. The distributions of sizes of dissipative events for a wide range of densities and system sizes follow a single scaling…
Microstructural dynamics in amorphous particle packings is commonly probed by quantifying particle displacements. While rigidity in particle packings emerges when displacement of particles are hindered, it is not obvious how the typically disordered displacement metrics connect to mechanical response. Particle rotations, in contrast, are much less sensitive to confinement effects, while still sensitive to the mechanics of the packing. So far, little attention has been paid to connect microscopic rotational motion to mechanics of athermal amorphous packings. We demonstrate through experimental …
We simulate a densely jammed, athermal assembly of repulsive soft particles immersed in a solvent. Starting from an initial condition corresponding to a quench from a high temperature, we find non-trivial slow dynamics driven by a gradual release of stored elastic energy, with the root mean squared particle speed decaying as a power law in time with a fractional exponent. This decay is accompanied by the presence within the assembly of spatially localised and temporally intermittent `hot-spots' of non-affine deformation, connected by long-ranged swirls in the velocity field, reminiscent of the…
Under applied shear strain, granular and amorphous materials deform via particle rearrangements, which can be small and localized or organized into system-spanning avalanches. While the statistical properties of avalanches under quasi-static shear are well-studied, the dynamics during avalanches is not. In numerical simulations of sheared soft spheres, we find that avalanches can be decomposed into bursts of localized deformations, which we identify using an extension of persistent homology methods. We also study the linear response of unstable systems during an avalanche, demonstrating that e…
By comparing the response to external strains in metallic glasses and in Lenard-Jones glasses we find a quantitative universality of the fundamental plastic instabilities in the athermal, quasistatic limit. Microscopically these two types of glasses are as different as one can imagine, the latter being determined by binary interactions, whereas the former by multiple interactions due to the effect of the electron gas that cannot be disregarded. In spite of this enormous difference the plastic instability is the same saddle-node bifurcation. As a result the statistics of stress and energy drops…
We present a mathematical description of amorphous solid deformation and plasticity by extending the concept of instantaneous normal modes (INMs) to deformed systems, which allows us to retain the effect of strain on the vibrational density of states (VDOS). Starting from the nonaffine lattice dynamics (NALD) description of elasticity and viscoelasticity of glasses, we formulate the linear response theory up to large deformations by considering the strain-dependent tangent modulus at finite values of shear strain. The (nonaffine) tangent shear modulus is computed from the vibrational density o…
Mechanical properties are of central importance to materials sciences, in particular if they depend on external stimuli. Here we investigate the rheological response of amorphous solids, namely col- loidal glasses, to external forces. Using confocal microscopy and computer simulations, we establish a quantitative link between the macroscopic creep response and the microscopic single-particle dy- namics. We observe dynamical heterogeneities, namely regions of enhanced mobility, which remain localized in the creep regime, but grow for applied stresses leading to steady flow. These different beha…
