Plastic response of a 2D Lennard-Jones amorphous solid: detailed analysis of the local rearrangements at very slow strain rate

Ductile-to-brittle transition and yielding in soft amorphous materials: perspectives and open questions

Soft amorphous materials are viscoelastic solids ubiquitously found around us, from clays and cementitious pastes to emulsions and physical gels encountered in food or biomedical engineering. Under an external deformation, these materials undergo a noteworthy transition from a solid to a liquid state that reshapes the material microstructure. This yielding transition was the main theme of a workshop held from January 9 to 13, 2023 at the Lorentz Center in Leiden. The manuscript presented here offers a critical perspective on the subject, synthesizing insights from the various brainstorming sessions and informal discussions that unfolded during this week of vibrant exchange of ideas. The result of these exchanges takes the form of a series of open questions that represent outstanding experimental, numerical, and theoretical challenges to be tackled in the near future.

Yielding Is an Absorbing Phase Transition with Vanishing Critical Fluctuations

The yielding transition in athermal complex fluids can be interpreted as an absorbing phase transition between an elastic, absorbing state with high mesoscopic degeneracy and a flowing, active state. We characterize quantitatively this phase transition in an elastoplastic model under fixed applied shear stress, using a finite-size scaling analysis. We find vanishing critical fluctuations of the order parameter (i.e., the shear rate), and relate this property to the convex character of the phase transition (β>1). We locate yielding within a family of models akin to fixed-energy sandpile (FES) models, only with long-range redistribution kernels with zero modes that result from mechanical equilibrium. For redistribution kernels with sufficiently fast decay, this family of models belongs to a short-range universality class distinct from the conserved directed percolation class of usual FES, which is induced by zero modes.

Topological characterization of rearrangements in amorphous solids

In amorphous materials, plasticity is localized and occurs as shear transformations. It was recently shown by Wu et al. that these shear transformations can be predicted by applying topological defect concepts developed for liquid crystals to an analysis of vibrational eigenmodes [Z. W. Wu et al., Nat. Commun. 14, 2955 (2023)10.1038/s41467-023-38547-w]. This study relates the -1 topological defects to the displacement fields expected of an Eshelby inclusion, which are characterized by an orientation and the magnitude of the eigenstrain. A corresponding orientation and magnitude can be defined for each defect using the local displacement field around each defect. These parameters characterize the plastic stress relaxation associated with the local structural rearrangement and can be extracted using the fit to either the global displacement field or the local field. Both methods provide a reasonable estimation of the molecular-dynamics-measured stress drop, confirming the localized nature of the displacements that control both long-range deformation and stress relaxation.

Solid-that-Flows Picture of Glass-Forming Liquids

This perspective article reviews arguments that glass-forming liquids are different from those of standard liquid-state theory, which typically have a viscosity in the mPa·s range and relaxation times on the order of picoseconds. These numbers grow dramatically and become 1012 - 1015 times larger for liquids cooled toward the glass transition. This translates into a qualitative difference, and below the "solidity length" which is roughly one micron at the glass transition, a glass-forming liquid behaves much like a solid. Recent numerical evidence for the solidity of ultraviscous liquids is reviewed, and experimental consequences are discussed in relation to dynamic heterogeneity, frequency-dependent linear-response functions, and the temperature dependence of the average relaxation time.

Instantaneous normal modes of glass-forming liquids during the athermal relaxation process of the steepest descent algorithm

Understanding glass formation by quenching remains a challenge in soft condensed matter physics. Recent numerical studies on steepest descent dynamics, which is one of the simplest models of quenching, revealed that quenched liquids undergo slow relaxation with a power law towards mechanical equilibrium and that the late stage of this process is governed by local rearrangements of particles. These advances motivate the detailed study of instantaneous normal modes during the relaxation process because the glassy dynamics is considered to be governed by stationary points of the potential energy landscape. Here, we performed a normal mode analysis of configurations during the steepest descent dynamics and found that the dynamics is driven by almost flat directions of the potential energy landscape at long times. These directions correspond to localized modes and we characterized them in terms of their statistics and structure using methods developed in the study of local minima of the potential energy landscape.

Characteristics and correlations of nonaffine particle displacements in the plastic deformation of athermal amorphous materials

When an amorphous solid is deformed homogeneously, it exhibits heterogeneous plastic instabilities with a localized cooperative rearrangement of a cluster of particles in response. The heterogeneous behavior plays an important role in deciding the mechanical properties of amorphous solids. In this paper, we employ computer simulations to study the characteristics and the spatial correlations of these clusters characterized by their non-affine displacements in amorphous solids under simple shear deformation in the athermal quasistatic limit. The clusters with large displacements are found to be homogeneously distributed in space in the elastic regime, followed by a localization within a system-spanning shear band after yielding. The distributions of the displacement field exhibit a power-law nature in the elastic regime with an exponential cutoff post yielding. The non-affine displacements show strong spatial correlations, which become long-ranged with increasing strain. From our results, it is evident that the decay of the correlation functions is exponential in nature in the elastic regime. The yielding transition is marked by an abrupt change in the decay after which it is well described by a power-law with an exponential cutoff. These results demonstrate a scale-free character of non-affine correlations in the steady flow regime. These results are found to be robust and independent of the strain window over which the total non-affine displacement is calculated.

Relevance of Shear Transformations in the Relaxation of Supercooled Liquids

While deeply supercooled liquids exhibit divergent viscosity and increasingly heterogeneous dynamics as the temperature drops, their structure shows only seemingly marginal changes. Understanding the nature of relaxation processes in this dramatic slowdown is key for understanding the glass transition. Here, we show by atomistic simulations that the heterogeneous dynamics of glass-forming liquids strongly correlate with the local residual plastic strengths along soft directions computed in the initial inherent structures. The correlation increases with decreasing temperature and is maximum in the vicinity of the relaxation time. For the lowest temperature investigated, this maximum is comparable with the best values from the literature dealing with the structure-property relationship. However, the nonlinear probe of the local shear resistance in soft directions provides here a real-space picture of relaxation processes. Our detection method of thermal rearrangements allows us to investigate the first passage time statistics and to study the scaling between the activation energy barriers and the residual plastic strengths. These results shed new light on the nature of relaxations of glassy systems by emphasizing the analogy between the thermal relaxations in viscous liquids and the plastic shear transformation in amorphous solids.

Power fluctuations in sheared amorphous materials: A minimal model

The importance of mesoscale fluctuations in flowing amorphous materials is widely accepted, without a clear understanding of their role. We propose a mean-field elastoplastic model that admits both stress and strain-rate fluctuations, and investigate the character of its power distribution under steady shear flow. The model predicts the suppression of negative power fluctuations near the liquid-solid transition; the existence of a fluctuation relation in limiting regimes but its replacement in general by stretched-exponential power-distribution tails; and a crossover between two distinct mechanisms for negative power fluctuations in the liquid and the yielding solid phases. We connect these predictions with recent results from particle-based, numerical microrheological experiments.

Using delaunay triangularization to characterize non-affine displacement fields during athermal, quasistatic deformation of amorphous solids

We investigate the non-affine displacement fields that occur in two-dimensional Lennard-Jones models of metallic glasses subjected to athermal, quasistatic simple shear (AQS). During AQS, the shear stress versus strain displays continuous quasi-elastic segments punctuated by rapid drops in shear stress, which correspond to atomic rearrangement events. We capture all information concerning the atomic motion during the quasi-elastic segments and shear stress drops by performing Delaunay triangularizations and tracking the deformation gradient tensor F_α associated with each triangle α. To understand the spatio-temporal evolution of the displacement fields during shear stress drops, we calculate F_α along minimal energy paths from the mechanically stable configuration immediately before to that after the stress drop. We find that quadrupolar displacement fields form and dissipate both during the quasi-elastic segments and shear stress drops. We then perform local perturbations (rotation, dilation, simple and pure shear) to single triangles and measure the resulting displacement fields. We find that local pure shear deformations of single triangles give rise to mostly quadrupolar displacement fields, and thus pure shear strain is the primary type of local strain that is activated by bulk, athermal quasistatic simple shear. Other local perturbations, e.g. rotations, dilations, and simple shear of single triangles, give rise to vortex-like and dipolar displacement fields that are not frequently activated by bulk AQS. These results provide fundamental insights into the non-affine atomic motion that occurs in driven, glassy materials.

On the yielding of a point-defect-rich model crystal under shear: insights from molecular dynamics simulations

In real crystals and at finite temperatures point defects are inevitable. Under shear their dynamics severely influence the mechanical properties of these crystals, giving rise to non-linear effects, such as ductility. In an effort to elucidate the complex behavior of crystals under plastic deformation it is crucial to explore and to understand the interplay between the timescale related to the equilibrium point-defect diffusion and the shear-induced timescale. Based on extensive non-equilibrium molecular dynamics simulations we present a detailed investigation on the yielding behavior of cluster crystals, an archetypical model for a defect-rich crystal: in such a system clusters of overlapping particles occupy the lattice sites of a regular (FCC) structure. In equilibrium particles diffuse via site-to-site hopping while maintaining the crystalline structure intact. We investigate these cluster crystals at a fixed density and at different temperatures where the system remains in the FCC structure: temperature allows us to vary the diffusion timescale appropriately. We then expose the crystal to shear, thereby choosing shear rates which cover timescales that are both higher and lower than the equilibrium diffusion timescales. We investigate the macroscopic and microscopic response of our cluster crystal to shear and find that the yielding scenario of such a system does not rely on the diffusion of the particles - it is rather related to the plastic deformation of the underlying crystalline structure. The local bond order parameters and the measurement of local angles between neighboring clusters confirm the cooperative movement of the clusters close to the yield point. Performing complementary, related simulations for an FCC crystal formed by harshly repulsive particles reveals similarities in the yielding behavior between both systems. Still we find that the diffusion of particles does influence characteristic features in the cluster crystal, such as a less prominent increase of order parameters close to the yield point. Our simulations provide for the first time an insight into the role of the diffusion of defects in the yielding behavior of a defect-rich crystal under shear. These observations will thus be helpful in the development of theories for the plastic deformation of defect-rich crystals.

Unified view of avalanche criticality in sheared glasses

Plastic events in sheared glasses are considered an example of so-called avalanches, whose sizes obey a power-law probability distribution with the avalanche critical exponent τ. Although the so-called mean-field depinning (MFD) theory predicts a universal value of this exponent, τ_{MFD}=1.5, such a simplification is now known to connote qualitative disagreement with realistic systems. Numerically and experimentally, different values of τ have been reported depending on the literature. Moreover, in the elastic regime, it has been noted that the critical exponent can be different from that in the steady state, and even criticality itself is a matter of debate. Because these confusingly varying results have been reported under different setups, our knowledge of avalanche criticality in sheared glasses is greatly limited. To gain a unified understanding, in this work, we conduct a comprehensive numerical investigation of avalanches in Lennard-Jones glasses under athermal quasistatic shear. In particular, by excluding the ambiguity and arbitrariness that has crept into the conventional measurement schemes, we achieve high-precision measurement and demonstrate that the exponent τ in the steady state follows the prediction of MFD theory, τ_{MFD}=1.5. Our results also suggest that there are two qualitatively different avalanche events. This binariness leads to the nonuniversal behavior of the avalanche size distribution and is likely to be the cause of the varying values of τ reported thus far. To investigate the dependence of criticality and universality on applied shear, we further study the statistics of avalanches in the elastic regime and the ensemble of the first avalanche event in different samples, which provide information about the unperturbed system. We show that while the unperturbed system is indeed off-critical, criticality gradually develops as shear is applied. The degree of criticality is encoded in the fractal dimension of the avalanches, which starts from zero in the off-critical unperturbed state and saturates in the steady state. Moreover, the critical exponent τ is consistent with the prediction of the MFD τ_{MFD} universally, regardless of the amount of applied shear, once the system becomes critical.

Universality of plastic instability and mechanical yield in metallic glasses

The generic response of a wide range of amorphous solids is the average increase of stress upon external loading until the yielding transition point, after which elasto-plastic steady state sets in. The stress-strain response comprises of a series of elastic branches interspersed with plastic drops. The ubiquitousness of these phenomena indicates universality, independent of material property, but the literature predominantly deals with specific materials. In pursuit of generality among different amorphous systems, we undertake a careful investigation in the mechanical response of metallic glasses using computer simulation. By comparing our results of multi-body metallic glass potentials to those obtained from pairwise Lennard-Jones glasses, we show that the mechanism of plastic instabilities is universal and independent of the details of the underlying potential. We also investigate the yielding transition in terms of the overlap parameterQ₁₂, which has been successfully used Lennard-Jones glasses. The yielding is unambiguously identified as a first-order phase transition. These observations conform the nature of plastic instabilities and mechanical yield as universal and independent of microscopic interactions.

Origin of reversible and irreversible atomic-scale rearrangements in a model two-dimensional network glass

In this contribution, we investigate the fundamental mechanism of plasticity in a model two-dimensional network glass. The glass is generated by using a Monte Carlo bond-switching algorithm and subjected to athermal simple shear deformation, followed by subsequent unloading at selected deformation states. This enables us to investigate the topological origin of reversible and irreversible atomic-scale rearrangements. It is shown that some events that are triggered during loading recover during unloading, while some do not. Thus, two kinds of elementary plastic events are observed, which can be linked to the network topology of the model glass.

Connecting shear localization with the long-range correlated polarized stress fields in granular materials

One long-lasting puzzle in amorphous solids is shear localization, where local plastic deformation involves cooperative particle rearrangements in small regions of a few inter-particle distances, self-organizing into shear bands and eventually leading to the material failure. Understanding the connection between the structure and dynamics of amorphous solids is essential in physics, material sciences, geotechnical and civil engineering, and geophysics. Here we show a deep connection between shear localization and the intrinsic structures of internal stresses in an isotropically jammed granular material subject to shear. Specifically, we find strong (anti)correlations between the micro shear bands and two polarized stress fields along two directions of maximal shear. By exploring the tensorial characteristics and the rotational symmetry of force network, we reveal that such profound connection is a result of symmetry breaking by shear. Finally, we provide the solid experimental evidence of long-range correlated inherent shear stress in an isotropically jammed granular system.

Precursors to plastic failure in a numerical simulation of CuZr metallic glass

We deform, in pure shear, a thin sample of Cu₅₀Zr₅₀ metallic glass using a molecular dynamics simulation up to, and including, failure. The experiment is repeated ten times in order to have average values and standard deviations. Although failure occurs at the same value of the externally imposed strain for the ten samples, there is significant sample-to-sample variation in the specific microscopic material behavior. Failure can occur along one, two, or three planes, located at the boundaries of previously formed shear bands (SBs). These SBs form shortly before failure. However, well before their formation and at external strains where plastic deformation just begins to be significant, non-affine displacement organizes itself along localized bands. The SBs subsequently form at the edges of these non-affine-displacement-bands, and present an alternating rotation-quadrupole structure, as found previously by Şopu et al (2017 Phys. Rev. Lett. 119 195503) in the case of a notched sample loaded in tension. The thickness of SBs is roughly determined by the available plastic energy. The onset of shear banding is accompanied by a sharp increase in the rate of change of the rotation angle localization, the strain localization, and the non-affine square displacement.

Rheological similarities between dense self-propelled and sheared particulate systems

Different from previous modeling of self-propelled particles, we develop a method to propel particles with a constant average velocity instead of a constant force. This constant propulsion velocity (CPV) approach is validated by its agreement with the conventional constant propulsion force (CPF) approach in the flowing regime. However, the CPV approach shows its advantage of accessing quasistatic flows of yield stress fluids with a vanishing propulsion velocity, while the CPF approach is usually unable to because of finite system size. Taking this advantage, we realize cyclic self-propulsion and study the evolution of the propulsion force with the propelled particle displacement, both in the quasistatic flow regime. By mapping the shear stress and shear rate to the propulsion force and propulsion velocity, we find similar rheological behaviors of self-propelled systems to sheared systems, including the yield force gap between the CPF and CPV approaches, propulsion force overshoot, reversible-irreversible transition under cyclic propulsion, and propulsion bands in plastic flows. These similarities suggest underlying connections between self-propulsion and shear, although they act on systems in different ways.

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.

Compaction of elastic granular materials: inter-particles friction effects and plastic events

The uni-axial compaction of granular materials made of elastic neo-Hookean particles is investigated in the quasi-static regime. Two-dimensional disk assemblies are simulated using the Finite Element model coupled with Contact Dynamics method for dealing both with finite deformations of the particles and contact interactions. Due to large deformations of the particles, the packing fraction of the system increases continuously during the compaction process, reaching values close to 1. The influence of the coefficient of friction between the particles on the macroscopic and micro-structural behaviors of the system is thoroughly discussed.

Microscopic precursors of failure in soft matter

The mechanical properties of soft matter are of great importance in countless applications, in addition of being an active field of academic research. Given the relative ease with which soft materials can be deformed, their non-linear behavior is of particular relevance. Large loads eventually result in material failure. In this Perspective article, we discuss recent work aiming at detecting precursors of failure by scrutinizing the microscopic structure and dynamics of soft systems under various conditions of loading. In particular, we show that the microscopic dynamics is a powerful indicator of the ultimate fate of soft materials, capable of unveiling precursors of failure up to thousands of seconds before any macroscopic sign of weakening.

Particle pinning suppresses spinodal criticality in the shear-banding instability

Strained amorphous solids often fail mechanically by creating a shear band. It had been understood that the shear-banding instability is usefully described as crossing a spinodal point (with disorder) in an appropriate thermodynamic description. It remained contested, however, whether the spinodal is critical (with divergent correlation length) or not. Here we offer evidence for critical spinodal by using particle pinning. For a finite concentration of pinned particles the correlation length is bounded by the average distance between pinned particles, but without pinning it is bounded by the system size.

Rheology in dense assemblies of spherocylinders: Frictional vs. frictionless

Using molecular dynamics simulations, we study the steady shear flow of dense assemblies of anisotropic spherocylindrical particles of varying aspect ratios. Comparing frictionless and frictional particles we discuss the specific role of frictional inter-particle forces for the rheological properties of the system. In the frictional system we evidence a shear-thickening regime, similar to that for spherical particles. Furthermore, friction suppresses the alignment of the spherocylinders along the flow direction. Finally, the jamming density in frictional systems is rather insensitive to variations in aspect ratio, quite contrary to what is known from frictionless systems.

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.

Effect of Pinning on the Yielding Transition of Amorphous Solids

Using numerical simulations, we have studied the yielding response, in the athermal quasistatic limit, of a model amorphous material having inclusions in the form of randomly pinned particles. We show that, with increasing pinning concentration, the plastic activity becomes more spatially localized, resulting in smaller stress drops, and a corresponding increase in the magnitude of strain where yielding occurs. We demonstrate that, unlike the spatially heterogeneous and avalanche led yielding in the case of the unpinned glass, for the case of large pinning concentration, yielding takes place via a spatially homogeneous proliferation of localized events.

Avalanches, thresholds, and diffusion in mesoscale amorphous plasticity

We present results on a mesoscale model for amorphous matter in athermal, quasistatic (a-AQS), steady-state shear flow. In particular, we perform a careful analysis of the scaling with the lateral system size L of (i) statistics of individual relaxation events in terms of stress relaxation S, and individual event mean-squared displacement M, and the subsequent load increments Δγ, required to initiate the next event; (ii) static properties of the system encoded by x=σ_{y}-σ, the distance of local stress values from threshold; and (iii) long-time correlations and the emergence of diffusive behavior. For the event statistics, we find that the distribution of S is similar to, but distinct from, the distribution of M. We find a strong correlation between S and M for any particular event, with S∼M^{q} with q≈0.65. The exponent q completely determines the scaling exponents for P(M) given those for P(S). For the distribution of local thresholds, we find P(x) is analytic at x=0, and has a value P(x)|_{x=0}=p_{0} which scales with lateral system length as p_{0}∝L^{-0.6}. The size dependence of the average load increment 〈Δγ〉 appears to be asymptotically controlled by the plateau behavior of P(x) rather than by a subsequent apparent power-law behavior. Extreme value statistics arguments lead thus to a scaling relation between the exponents governing P(x) and those governing P(S). Finally, we study the long-time correlations via single-particle tracer statistics. The value of the diffusion coefficient is completely determined by 〈Δγ〉 and the scaling properties of P(M) (in particular from 〈M〉) rather than directly from P(S) as one might have naively guessed. Our results (i) further define the a-AQS universality class, (ii) clarify the relation between avalanches of stress relaxation and diffusive behavior, and (iii) clarify the relation between local threshold distributions and event statistics.

Slow Coarsening in Jammed Athermal Soft Particle Suspensions

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 nontrivial 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 localized and temporally intermittent "hot spots" of nonaffine deformation, connected by long-ranged swirls in the velocity field, reminiscent of the local plastic events and long-ranged elastic propagation that have been intensively studied in sheared amorphous materials. The pattern of hot spots progressively coarsens, with the hot-spot size and separation slowly growing over time, and the associated correlation length in particle speed increasing as a sublinear power law. Each individual spot, however, exists only transiently within an overall picture of strongly intermittent dynamics.

Potential energy landscape activations governing plastic flows in glass rheology

While glasses are ubiquitous in natural and manufactured materials, the atomic-level mechanisms governing their deformation and how these mechanisms relate to rheological behavior are still open questions for fundamental understanding. Using atomistic simulations spanning nearly 10 orders of magnitude in the applied strain rate we probe the atomic rearrangements associated with 3 characteristic regimes of homogeneous and heterogeneous shear flow. In the low and high strain-rate limits, simulation results together with theoretical models reveal distinct scaling behavior in flow stress variation with strain rate, signifying a nonlinear coupling between thermally activated diffusion and stress-driven motion. Moreover, we find the emergence of flow heterogeneity is closely correlated with extreme values of local strain bursts that are not readily accommodated by immediate surroundings, acting as origins of shear localization. The atomistic mechanisms underlying the flow regimes are interpreted by analyzing a distance matrix of nonaffine particle displacements, yielding evidence of various barrier-hopping processes on a fractal potential energy landscape (PEL) in which shear transformations and liquid-like regions are triggered by the interplay of thermal and stress activations.

Distinguishing failure modes in oligomeric polymer nanopillars

Brittle failure is ubiquitous in amorphous materials that are sufficiently cooled below their glass transition temperature, T_g. This catastrophic failure mode is limiting for amorphous materials in many applications, and many fundamental questions surrounding it remain poorly understood. Two challenges that prevent a more fundamental understanding of the transition between a ductile response at temperatures near T_g to brittle failure at lower temperatures are (i) a lack of computationally inexpensive molecular models that capture the failure modes observed in experiments and (ii) the lack of quantitative metrics that can distinguish various failure mechanisms. In this work, we use molecular dynamics simulations to capture ductile-to-brittle transition in glass-forming oligomeric polymer systems where we systematically vary both the temperature relative to T_g and the form of the interaction potential to induce a variety of failure modes. We characterized the effects of this new potential on macroscopic mechanical properties as well as microscopic structural and dynamical response during deformation. Finally, we develop several quantitative metrics to distinguish between different failure modes, and we find that the transition between catastrophic brittle failure, necking, and homogeneous plastic flow is gradual as the temperature is increased.

Effective forces in thermal amorphous solids with generic interactions

In thermal glasses at temperatures sufficiently lower than the glass transition, the constituent particles are trapped in their cages for a sufficiently long time such that their time-averaged positions can be determined before diffusion and structural relaxation takes place. The effective forces are those that hold these average positions in place. In numerical simulations the effective forces F_{ij} between any pair of particles can be measured as a time average of the bare forces f_{ij}(r_{ij}(t)). In general, even if the bare forces come from two-body interactions, thermal dynamics dress the effective forces to contain many-body interactions. Here, we develop the effective theory for systems with generic interactions, where the effective forces are derivable from an effective potential and in turn they give rise to an effective Hessian whose eigenvalues are all positive when the system is stable. In this Rapid Communication, we offer analytic expressions for the effective theory, and demonstrate the usefulness and the predictive power of the approach.

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.

Energy Fluctuations in Slowly Sheared Granular Materials

Here we show the first experimental measurement of the particle-scale energy fluctuations ΔE in a slowly sheared layer of photoelastic disks. Starting from an isotropically jammed state, applying shear causes the shear-induced stochastic strengthening and weakening of particle-scale energies, whose statistics and dynamics govern the evolution of the macroscopic stress-strain curve. We find that the ΔE behave as a temperaturelike noise field, showing a novel, Boltzmann-type, double-exponential distribution at any given shear strain γ. Following the framework of the soft glassy rheology theory, we extract an effective temperature χ from the statistics of the energy fluctuations to interpret the slow startup shear (shear starts from an isotropically jammed state) of granular materials as an "aging" process: Starting below one, χ gradually approaches one as γ increases, similar to those of spin glasses, thermal glasses, and bulk metallic glasses.

Plastic flow and localization in an amorphous material: Experimental interpretation of the fluidity

We present a thorough study of the plastic response of a granular material progressively loaded. We study experimentally the evolution of the plastic field from a homogeneous one to a heterogeneous one and its fluctuations in terms of incremental strain. We show that the plastic field can be decomposed in two components evolving on two decoupled strain increment scales. We argue that the slowly varying part of the field can be identified with the so-called fluidity field introduced recently to interpret the rheological behavior of amorphous materials. This fluidity field progressively concentrates along a macroscopic direction corresponding to the Mohr-Coulomb angle.

Disordered Nanoparticle Packings under Local Stress Exhibit Avalanche-Like, Environmentally Dependent Plastic Deformation

Nanoindentation experiments on disordered nanoparticle packings performed both in an atomic force microscope and in situ in a transmission electron microscope are used to investigate the mechanics of plastic deformation. Under an applied load, these highly porous films exhibit load drops, the magnitudes of which are consistent with an exponential population distribution. These load drops are attributed to local rearrangements of a small number of particles, which bear similarities to shear transformation zones and to the T1 process, both of which have been previously predicted for disordered packings. An increase in the relative humidity results in an increase in the number of observed load drops, indicating that the strength of the particle interactions has a significant effect on the modes of plastic deformation. These results suggest how disordered nanoparticle packings may be expected to behave in devices operating under varying environments.

Long-range stress correlations in viscoelastic and glass-forming fluids

A simple and rigorous approach to obtain stress correlations in viscoelastic liquids (including supercooled liquid and equilibrium amorphous systems) is proposed. The long-range dynamical correlations of local shear stress are calculated and analyzed in 2-dimensional space. It is established how the long-range character of the stress correlations gradually emerges as the relevant dynamical correlation length l grows in time. The correlation range l is defined by momentum propagation due to acoustic waves and vorticity diffusion which are the basic mechanisms for transmission of shear stress perturbations. We obtain the general expression defining the time- and distance-dependent stress correlation tensor in terms of material functions (generalized relaxation moduli). The effect of liquid compressibility is quantitatively analyzed; it is shown to be important at large distances and/or short times. The revealed long-range stress correlation effect is shown to be dynamical in nature and unconnected with static structural correlations in liquids (correlation length ξ_s). Our approach is based on the assumption that ξ_s is small enough as reflected in weak wave-number dependencies of the generalized relaxation moduli. We provide a simple physical picture connecting the elucidated long-range fluctuation effect with anisotropic correlations of the (transient) inherent stress field, and discuss its implications.

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.

Critical exponents of the yielding transition of amorphous solids

We investigate numerically the yielding transition of a two-dimensional model amorphous solid under external shear. We use a scalar model in terms of values of the total local strain, derived from the full (tensorial) description of the elastic interactions in the system, in which plastic deformations are accounted for by introducing a stochastic "plastic disorder" potential. This scalar model is seen to be equivalent to a collection of Prandtl-Tomlinson particles, which are coupled through an Eshelby quadrupolar kernel. Numerical simulations of this scalar model reveal that the strain rate versus stress curve, close to the critical stress, is of the form γ[over ̇]∼(σ-σ_{c})^{β}. Remarkably, we find that the value of β depends on details of the microscopic plastic potential used, confirming and giving additional support to results previously obtained with the full tensorial model. To rationalize this result, we argue that the Eshelby interaction in the scalar model can be treated to a good approximation in a sort of "dynamical" mean field, which corresponds to a Prandtl-Tomlinson particle that is driven by the applied strain rate in the presence of a stochastic noise generated by all other particles. The dynamics of this Prandtl-Tomlinson particle displays different values of the β exponent depending on the analytical properties of the microscopic potential, thus giving support to the results of the numerical simulations. Moreover, we find that other critical exponents that depend on details of the dynamics show also a dependence with the form of the disorder, while static exponents are independent of the details of the disorder. Finally, we show how our scalar model relates to other elastoplastic models and to the widely used mean-field version known as the Hébraud-Lequeux model.

Yielding of a model glass former: An interpretation with an effective system of icosahedra

We consider the yielding under simple shear of a binary Lennard-Jones glass former whose super-Arrhenius dynamics are correlated with the formation of icosahedral structures. We recast this glass former as an effective system of icosahedra [Pinney et al., J. Chem. Phys. 143, 244507 (2015)JCPSA60021-960610.1063/1.4938424]. Looking at the small-strain region of sheared simulations, we observe that shear rates affect the shear localization behavior particularly at temperatures below the glass transition as defined with a fit to the Vogel-Fulcher-Tamman equation. At higher temperature, shear localization starts immediately on shearing for all shear rates. At lower temperatures, faster shear rates can result in a delayed start in shear localization, which begins close to the yield stress. Building from a previous work which considered steady-state shear [Pinney et al., J. Chem. Phys. 143, 244507 (2015)JCPSA60021-960610.1063/1.4938424], we interpret the response to shear and the shear localization in terms of a local effective temperature with our system of icosahedra. We find that the effective temperatures of the regions undergoing shear localization increase significantly with increasing strain (before reaching a steady-state plateau).

Strain Distribution Across an Individual Shear Band in Real and Simulated Metallic Glasses

Because of the fast dynamics of shear band formation and propagation along with the small size and transient character of the shear transformation zones (STZs), the elementary units of plasticity in metallic glasses, the description of the nanoscale mechanism of shear banding often relies on molecular dynamics (MD) simulations. However, the unrealistic parameters used in the simulations related to time constraints may raise questions about whether quantitative comparison between results from experimental and computational analyses is possible. Here, we have experimentally analyzed the strain field arising across an individual shear band by nanobeam X-ray diffraction and compared the results with the strain characterizing a shear band generated by MD simulations. Despite their largely different spatiotemporal scales, the characteristic features of real and simulated shear bands are strikingly similar: the magnitude of the strain across the shear band is discontinuous in both cases and the direction of the principal strain axes exhibits the same antisymmetric profile. This behavior can be explained by considering the mechanism of STZ activation and percolation at the nanoscale, indicating that the nanoscale effects of shear banding are not limited to the area within the band but they extend well into the surrounding elastic matrix. These findings not only demonstrate the reliability of MD simulations for explaining (also quantitatively) experimental observations of shear banding but also suggest that designed experiments can be used the other way around to verify numerical predictions of the atomic rearrangements occurring within a band.

Understanding the mechanisms of amorphous creep through molecular simulation

Molecular processes of creep in metallic glass thin films are simulated at experimental timescales using a metadynamics-based atomistic method. Space-time evolutions of the atomic strains and nonaffine atom displacements are analyzed to reveal details of the atomic-level deformation and flow processes of amorphous creep in response to stress and thermal activations. From the simulation results, resolved spatially on the nanoscale and temporally over time increments of fractions of a second, we derive a mechanistic explanation of the well-known variation of creep rate with stress. We also construct a deformation map delineating the predominant regimes of diffusional creep at low stress and high temperature and deformational creep at high stress. Our findings validate the relevance of two original models of the mechanisms of amorphous plasticity: one focusing on atomic diffusion via free volume and the other focusing on stress-induced shear deformation. These processes are found to be nonlinearly coupled through dynamically heterogeneous fluctuations that characterize the slow dynamics of systems out of equilibrium.

Universal scaling of the stress-strain curve in amorphous solids

The yielding transition of amorphous solids is a phase transition with a special type of universality. Critical exponents and scaling relations have been defined and proposed near the yield stress. We show here that, even in the initial stage of shear far below the yield stress, the stress-strain curve of amorphous solids also shows critical scaling with universal exponents. The key point is to remove the elastic part of the strain, and the shear stress exhibits a sublinear scaling with the plastic strain. We show how this critical scaling is related to the finite size effect of the minimum strain to trigger the first plastic avalanche after a quench. We point out that this sublinear scaling between the stress and the plastic strain implies the divergence of a high-order shear modulus. A scaling relation is derived between two exponents characterizing the stress-strain curve and the density distribution of the local stabilities, respectively. We test the critical scaling of the stress-strain curve using both mesoscopic and atomistic simulations and get satisfying agreement in two and three dimensions.

Mechanical failure in amorphous solids: Scale-free spinodal criticality

The mechanical failure of amorphous media is a ubiquitous phenomenon from material engineering to geology. It has been noticed for a long time that the phenomenon is "scale-free," indicating some type of criticality. In spite of attempts to invoke "Self-Organized Criticality," the physical origin of this criticality, and also its universal nature, being quite insensitive to the nature of microscopic interactions, remained elusive. Recently we proposed that the precise nature of this critical behavior is manifested by a spinodal point of a thermodynamic phase transition. Demonstrating this requires the introduction of an "order parameter" that is suitable for distinguishing between disordered amorphous systems. At the spinodal point there exists a divergent correlation length which is associated with the system-spanning instabilities (known also as shear bands) which are typical to the mechanical yield. The theory, the order parameter used and the correlation functions which exhibit the divergent correlation length are universal in nature and can be applied to any amorphous solid that undergoes mechanical yield. The phenomenon is seen at its sharpest in athermal systems, as is explained below; in this paper we extend the discussion also to thermal systems, showing that at sufficiently high temperatures the spinodal phenomenon is destroyed by thermal fluctuations.

Shear bands as manifestation of a criticality in yielding amorphous solids

Amorphous solids increase their stress as a function of an applied strain until a mechanical yield point whereupon the stress cannot increase anymore, afterward exhibiting a steady state with a constant mean stress. In stress-controlled experiments, the system simply breaks when pushed beyond this mean stress. The ubiquity of this phenomenon over a huge variety of amorphous solids calls for a generic theory that is free of microscopic details. Here, we offer such a theory: The mechanical yield is a thermodynamic phase transition, where yield occurs as a spinodal phenomenon. At the spinodal point, there exists a divergent correlation length that is associated with the system-spanning instabilities (also known as shear bands), which are typical to the mechanical yield. The theory, the order parameter used, and the correlation functions that exhibit the divergent correlation length are universal in nature and can be applied to any amorphous solids that undergo mechanical yield.

Effect of composition and pressure on the shear strength of sodium silicate glasses: An atomic scale simulation study

The elastoplastic behavior of sodium silicate glasses is studied at different scales as a function of composition and pressure, with the help of quasistatic atomistic simulations. The samples are first compressed and then sheared at constant pressure to calculate yield strength and permanent plastic deformations. Changes occurring in the global response are then compared to the analysis of local plastic rearrangements and strain heterogeneities. It is shown that the plastic response results from the succession of well-identified localized irreversible deformations occurring in a nanometer-size area. The size and the number of these local rearrangements, as well as the amount of internal deviatoric and volumetric plastic deformation, are sensitive to the composition and to the pressure. In the early stages of the deformation, plastic rearrangements are driven by sodium mobility. Consequently, the elastic yield strength decreases when the sodium content increases, and the same when pressure increases. Finally, good correlation was found between global and local stress-strain relationships, reinforcing again the role of sodium ions as local initiators of the plastic behavior observed at larger scales.

Shear transformation distribution and activation in glasses at the atomic scale

We characterize shear transformations (STs) at the atomic scale in a model of amorphous silicon using a mapping on Eshelby inclusions. We investigate the effect of pressure, glass relaxation, as well as damage on the ST characteristics. We show that the characteristic ST effective volume, γ_{0}V_{0}, product of the ST plastic shear strain γ_{0} and volume V_{0}, does not depend significantly on an applied pressure but increases with accumulated plastic deformation from about 10Å^{3} in the pseudoelastic regime to about 60Å^{3} once plastic flow sets in. Furthermore, by using nudged elastic band calculations, we measure the energy barrier against ST activation. Analyzing different paths leading to either an isolated ST or an avalanche, we show that the barrier is systematically controlled by the first ST with an activation volume equal to the effective volume of the ST at the activated state, which represents only a fraction of the complete ST volume. The activation volume is also found smaller for avalanches, presumably because of accumulated local damage. This work provides essential information to build reliable mesoscale models of plasticity.

Yielding of glass under shear: A directed percolation transition precedes shear-band formation

Under external mechanical loading, glassy materials, ranging from soft matter systems to metallic alloys, often respond via formation of inhomogeneous flow patterns, during yielding. These inhomogeneities can be precursors to catastrophic failure, implying that a better understanding of their underlying mechanisms could lead to the design of smarter materials. Here, extensive molecular dynamics simulations are used to reveal the emergence of heterogeneous dynamics in a binary Lennard-Jones glass, subjected to a constant strain rate. At a critical strain, this system exhibits for all considered strain rates a transition towards the formation of a percolating cluster of mobile regions. We give evidence that this transition belongs to the universality class of directed percolation. Only at low shear rates, the percolating cluster subsequently evolves into a transient (but long-lived) shear band with a diffusive growth of its width. Finally, the steady state with a homogeneous flow pattern is reached. In the steady state, percolation transitions also do occur constantly, albeit over smaller strain intervals, to maintain the stationary plastic flow in the system.

Eshelby inclusions in granular matter: Theory and simulations

We present a numerical implementation of an active inclusion in a granular material submitted to a biaxial test. We discuss the dependence of the response to this perturbation on two parameters: the intragranular friction coefficient on one hand, and the degree of the loading on the other hand. We compare the numerical results to theoretical predictions taking into account the change of volume of the inclusion as well as the anisotropy of the elastic matrix.

Nonaffine displacements and the nonlinear response of a strained amorphous solid

We demonstrate that irreversible structural reorganization is not necessary for the observation of yield behavior in an amorphous solid. While the majority of solids strained to their yield point do indeed undergo an irreversible reorganization, we find that a significant fraction of solids exhibits yield via a reversible strain. We also demonstrate that large instantaneous strains in excess of the yield stress can result in complete stress relaxation, a result of the large nonaffine motions driven by the applied strain. The empirical similarity of the dependence of the ratio of stress over strain on the nonaffine mean-square displacement to that for the shear modulus obtained from quiescent liquid at nonzero temperature supports the proposition that rigidity depends on the size of the sampled configurational space only and is insensitive to how this space is sampled.

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.