Recent years have seen substantial progress in elucidating the statistical physics of the vibrational properties of structural glasses. Although many real-world glasses relevant for science and technology are molecular, the majority of computational studies concerning the mechanical and vibrational properties of structural glasses employ simple atomistic glass-forming models. Thus, the effects of stiff molecular constraints on mechanical and vibrational glass physics remain largely unexplored. In this work, we directly compare the properties of a molecular computer glass with those of an atomistic model featuring the same inter-molecular interaction potential, and created using the same formation protocol. We find that the molecular glass features a higher degree of mechanical disorder, with larger mesoscopic correlation lengths, while at the same time it hosts a lower number of soft, quasilocalized vibrations per atom – compared to the atomistic glass. We rationalize these differences by accounting for the reduction in the effective number of degrees of freedom induced by the stiff molecular constraints. We additionally find that nonlinear plastic modes — that carry plastic deformation in driven glassy solids — couple much more strongly to volumetric strains in the molecular glass. Future research directions are discussed.

Non-active disordered solids feature quasilocalized excitations that control plasticity, similar to crystal lattice defects, and these excitations can be identified via harmonic or anharmonic analyses of the potential energy landscape. Here we explore whether such ideas can be extended to active matter, focusing on dense packings of self-propelled rods. We generalize the definition of nonlinear excitations to force landscapes that incorporate active, non-conservative forces and find that force-based cubic excitations robustly predict future plastic events, enabling control of active solids.

Materials deform elasto-plastically and fail under various loading conditions, typically quantified by the stress triaxiality, which is the ratio between the dilatational (hydrostatic) stress and the deviatoric (shear-like) one. We show that the elasto-plastic deformation of glasses approaching failure qualitatively differ for large and small stress triaxiality levels. Specifically, in the former limit, glasses reveal a strong hyperelastic (nonlinear elastic) response with minute plasticity, largely independently of the quenching rate across the glass transition. Yet, glassy disorder gives rise to significant elastic (reversible) nonaffine deformation, accompanied by the formation of micro-cavities. A small fraction of the latter is irreversible, i.e., survives unloading prior to the onset of failure, and may serve as nucleation sites for failure in the form of large-scale cavitation, upon which the glass loses a significant fraction of its load-bearing capacity. These results are contrasted with glass behavior in the limit of vanishing stress triaxiality and their universality across different glass formers is demonstrated. Finally, the implications of our findings for understanding glass deformation and failure under realistic stress conditions are discussed.

The dramatic slowdown of dynamics in supercooled liquids approaching the glass transition remains one of the central unresolved problems in condensed matter physics. We review approaches that attribute this slowdown to growing thermodynamic or structural length scales and discuss their difficulties in accounting for recent numerical results. These limitations motivate the present review, which critically examines alternative theories in which the glassy slowdown is instead controlled by localized excitations and their elastic interactions. After reviewing key phenomenology with a focus on the fragility of liquids, dynamical heterogeneities, thermodynamics-dynamics correlation, and the effect of kinetic rules and swap algorithms, we compare elastic descriptions based on homogeneous and local heterogeneous elasticity to excitation-based theories incorporating nonlinear responses. Results are compiled to relate global and local elastic moduli, the Debye-Waller factor, and the density of excitations, leading to a quantitative theory testable in experiments. The thermal evolution of the excitation spectrum provides a parameter-free account of the activation energy, while their elastic interactions quantitatively reproduce dynamical heterogeneities via thermal avalanche processes. Synthesized together, these results lead to a framework where the evolution of the excitation spectrum, rather than the growth of a thermodynamic length scale, governs fragility in simple glass-forming liquids -- yet mean-field concepts of dynamical transitions remain central to describing excitations and building a real-space picture of relaxation.

The dispersion of elastic waves in disordered solids plays a key role in determining the vibrational density of states and harmonic wave attenuation rates. As such, the availability of robust computational approaches to the precise extraction of the dispersion is of key importance. Here we present a simple method – the imposed wave method (IWM) – , which provides direct access to the dispersion of elastic waves in computer models of solids, without any fitting involved. We directly benchmark the method against the `ground-truth' obtained from direct diagonalization of solids' hessian matrices, to find good agreement. We discuss limitations of and finite-size effects in the method, and show that exploiting the method's finite-size scaling provides access to a fundamental quantifier of mechanical disorder that determines wave attenuation rates and spectral widths.

All solids, whether crystalline or disordered, support elastic wave propagation with a linear dispersion relation in the long-wavelength limit. These waves, corresponding to low-frequency phonons, feature a vibrational density of states that follows Debye's classical model. Deviations from Debye's predictions with increasing frequency can emerge from phonon dispersion nonlinearity and from non-phononic vibrational modes, which exist in non-crystalline solids due to structural disorder. Both nonlinear phonon dispersion in disordered solids and its relative contribution to non-Debye anomalies, most notably manifested by the controversial boson peak, remain poorly understood. Here we show that nonlinear phonon dispersion in a broad range of disordered solids, including elastic networks and various glasses, emerge from a mesoscopic, disorder-induced lengthscale, which also controls wave attenuation. We subsequently use analysis and large-scale computer simulations to quantitatively determine the relative contributions of nonlinear phonon softening and non-phononic vibrations to the onset of non-Debye anomalies and to the boson peak. We show that the relative magnitude of the two contributions strongly depends on the strength of disorder of the solid, e.g., controlled by the thermal history upon glass formation, and that for realistic laboratory glasses both pieces of physics significantly contribute to the boson peak. These findings constitute basic progress in understanding disordered solids.

A recently formulated mean-field model of glasses features an equilibrium, zero-temperature Replica-Symmetry-Breaking (RSB) transition in some parameter range. In this range, the model's solution in the Replica-Symmetric phase is expressed in terms of an effective, self-consistent random potential for uncoupled degree of freedoms, where the transition to the RSB phase is characterized by the emergence of spectral-edge localized modes and a pseudogapped quartic vibrational spectrum, resulting in a finite spin-glass susceptibility. These properties are preserved in numerical solutions of the model under non-equilibrium conditions, i.e., upon an instantaneous quench. Inspired by recent advances in computer glasses, we define a micromechanical response function --- the linear response to local force monopoles --- in the framework of the mean-field model. We establish exact relations between the force monopole stiffness and global susceptibilities, which suggest a close correspondence between the non-equilibrium RSB phase of the model and the above-mentioned effective random potential description. As such, the obtained micromechanical observables constitute a concrete realization of the collective degrees of freedom of the model, offering a bridge between a glassy mean-field model and finite-dimensional glasses. We show that the model's vibrational spectrum can be computed solely from the monopole response statistics and, by employing a marginal stability criterion, we extract a characteristic frequency/stiffness scale of soft glassy modes, which is related to the boson peak in finite-dimensional, laboratory glasses.

Understanding the statistical mechanics of low-energy excitations in structural glasses has been the focus of extensive research efforts in the past decades due to their key roles in determining the low-temperature mechanical and transport properties of these intrinsically nonequilibrium materials. While it is established that glasses feature low-energy nonphononic excitations that follow a non-Debye vibrational density of states, we currently lack a well-founded theory of these fundamental objects and their vibrational spectra. A recent theory-that builds on the so-called heterogeneous-elasticity theory (HET) and its extensions-provides explicit predictions for the scaling of the low-frequency tail of the nonphononic spectrum of glasses, the localization properties of the vibrational modes that populate this tail, and its connections to glass formation history and to the form of the distribution of weak microscopic (interatomic) stresses. Here, we employ computer models of structural glasses to quantitatively test these predictions. Our findings do not support the HET's predictions regarding the nature and statistics of low-energy excitations in glasses, highlighting the need for additional theoretical developments.

Glassy systems reveal a wide variety of generic behaviors, which lack a unified theoretical description. Here, we study a mean-field model, recently shown to reproduce the universal non-phononic vibrational spectra of glasses, under oscillatory driving forces. The driven mean-field model, featuring a disordered Hamiltonian structure, naturally predicts the salient dynamical phenomena in cyclically deformed glasses. Specifically, it features an oscillatory yielding transition, characterized by an absorbing-to-diffusive transition in the system's microscopic trajectories and large-scale hysteresis. The model also reveals dynamic slowing-down from both sides of the transition, as well as mechanical and thermal annealing effects that mirror their glass counterparts. Finally, we demonstrate a non-equilibrium ensemble equivalence between the driven post-yielding dynamics at fixed quenched disorder and quenched disorder averages of the non-driven system, along with memory formation.

Disordered athermal biopolymer materials, such as collagen networks that constitute a major component in extracellular matrices and various connective tissues, are initially soft and compliant but stiffen dramatically under strain. Such network materials are topologically sub-isostatic and feature strong rigidity scale separation between the bending and stretching response of the constituent polymer fibers. Recently, a comprehensive scaling theory of the athermal strain-stiffening phase transition has been developed, providing predictions for all mean-field critical exponents characterizing the transition in terms of the distance to the critical strain and of the small rigidity scales ratio. Here, we employ large-scale computer simulations, at and away from criticality, to test the analytic predictions. We find that all numerical critical exponents are in quantitative agreement with the analytically predicted ones. Moreover, we find that all predicted mean-field exponents remain valid whether the driving strain is shear, i.e., volume-preserving, or dilation, and independent of the degree of the network's sub-isostaticity, thus establishing the universality of the strain-stiffening phase transition with respect to the symmetry of the driving strain and the network's topology.

Numerical investigations of disordered solids often involve employing a fixed simulation-box geometry and periodic boundary conditions, the latter are intended to minimize surface effects. One consequence of this methodological approach is that the resulting solids typically feature residual macroscopic stresses and could feature negative shear moduli, in which case they are deemed unstable to shear. In this work, we review and validate via numerical simulations the statistical theory that explains how the fraction of shear-unstable solids within an ensemble of solids created via the same protocol depends on the system size. We further demonstrate and rationalize how the fraction of disordered soft-sphere packings that are unstable to shear depends on their proximity to the unjamming point at which solidity is lost. Finally, we show that residual-stress-free structural glasses created in a fixed-box geometry have the same vibrational and elastic properties as glasses that do feature residual stresses. This result seemingly contradicts previous work that shows that residual-stress-free samples created using variable box shape feature different vibrational properties. These observations raise questions about the equivalence of constant-stress vs constant box-geometry ensembles in the thermodynamic limit, which are discussed together with future research directions.

A hallmark of glasses is an excess of low-frequency, nonphononic vibrations. It is manifested as a terahertz peak-the boson peak-in the ratio of the vibrational density of state (VDoS) and Debye's VDoS of phonons. Here, using experimental data, extensive computer simulations, and a mean-field model, we show that the nonphononic part of the VDoS itself features both a universal power-law tail and a peak at higher frequencies, entirely accounted for by quasilocalized nonphononic vibrations, whose existence and spectra power-law tail were recently established in computer glasses. We rationalize the variation of the peak's frequency and magnitude with glasses' thermal history, which is much weaker than the variation of the tail and may follow an opposite trend, and show that the peak's modes are composed of many spatially coupled quasilocalized nonphononic vibrations. Our results shed light on the origin, nature, and properties of the boson peak in glasses.

Under decompression, disordered solids undergo an unjamming transition where they become under-coordinated and lose their structural rigidity. The mechanical and vibrational properties of these materials have been an object of theoretical, numerical, and experimental research for decades. In the study of low-coordination solids, understanding the behavior and physical interpretation of observables that diverge near the transition is of particular importance. Several such quantities are length scales (ξ or l) that characterize the size of excitations, the decay of spatial correlations, the response to perturbations, or the effect of physical constraints in the boundary or bulk of the material. Additionally, the spatial and sample-to-sample fluctuations of macroscopic observables such as contact statistics or elastic moduli diverge approaching unjamming. Here, we discuss important connections between all of these quantities and present numerical results that characterize the scaling properties of sample-to-sample contact and shear modulus fluctuations in ensembles of low-coordination disordered sphere packings and spring networks. Overall, we highlight three distinct scaling regimes and two crossovers in the disorder quantifiers χz and χμ as functions of system size N and proximity to unjamming δz. As we discuss, χX relates to the standard deviation σX of the sample-to-sample distribution of the quantity X (e.g., excess coordination δz or shear modulus μ) for an ensemble of systems. Importantly, χμ has been linked to experimentally accessible quantities that pertain to sound attenuation and the density of vibrational states in glasses. We investigate similarities and differences in the behaviors of χz and χμ near the transition and discuss the implications of our findings on current literature, unifying findings in previous studies.

Many fibrous materials are modeled as elastic networks featuring a substantial separation between the stiffness scales that characterize different microscopic deformation modes of the network's constituents. This scale separation has been shown to give rise to emergent complexity in these systems' linear and nonlinear mechanical response. Here we study numerically a simple model featuring said stiffness scale separation in two-dimensions and show that its mechanical response is governed by the competition between the characteristic stiffness of collective nonphononic soft modes of the stiff subsystem, and the characteristic stiffness of the soft interactions. We present and rationalize the behavior of the shear modulus of our complex networks across the unjamming transition at which the stiff subsystem alone loses its macroscopic mechanical rigidity. We further establish a relation in the soft-interaction-dominated regime between the shear modulus, the characteristic frequency of nonphononic vibrational modes, and the mesoscopic correlation length that marks the crossover from a disorder-dominated response to local mechanical perturbations in the near field, to a linear, continuumlike response in the far field. The effects of spatial dimension on the observed scaling behavior are discussed, in addition to the interplay between stiffness scales in strain-stiffened networks, which is relevant to understanding the nonlinear mechanics of non-Brownian fibrous biomatter.

The flowing, jamming, and avalanche behavior of granular materials is satisfyingly universal and vexingly hard to tune: A granular flow is typically intermittent and will irremediably jam if too confined. Here, we show that granular metamaterials made from particles with a negative Poisson’s ratio yield more easily and flow more smoothly than ordinary granular materials. We first create a collection of auxetic grains based on a re-entrant mechanism and show that each grain exhibits a negative Poisson’s ratio regardless of the direction of compression. Interestingly, we find that the elastic and yielding properties are governed by the high compressibility of granular metamaterials: At a given confinement, they exhibit lower shear modulus, lower yield stress, and more frequent, smaller avalanches than materials made from ordinary grains. We further demonstrate that granular metamaterials promote flow in more complex confined geometries, such as intruder and hopper geometries, even when the packing contains only a fraction of auxetic grains. Moreover, auxetic granular metamaterials exhibit enhanced impact absorption. Our findings blur the boundary between complex fluids and metamaterials and could help in scenarios that involve process, transport, and reconfiguration of granular materials.

There have been some interesting recent advances in understanding the notion of mechanical disorder in structural glasses and the statistical mechanics of these systems' low-energy excitations. Here we contribute to these advances by studying a minimal model for structural glasses' elasticity in which the degree of mechanical disorder-as characterized by recently introduced dimensionless quantifiers-is readily tunable over a very large range. We comprehensively investigate a number of scaling laws observed for various macro, meso and microscopic elastic properties, and rationalize them using scaling arguments. Interestingly, we demonstrate that the model features the universal quartic glassy vibrational density of states as seen in many atomistic and molecular models of structural glasses formed by cooling a melt. The emergence of this universal glassy spectrum highlights the role of self-organization (toward mechanical equilibrium) in its formation, and elucidates why models featuring structural frustration alone do not feature the same universal glassy spectrum. Finally, we discuss relations to existing work in the context of strain stiffening of elastic networks and of low-energy excitations in structural glasses, in addition to future research directions.

Materials typically fail under complex stress states, essentially involving dilatational (volumetric) components that eventually lead to material decohesion/separation. It is therefore important to understand dilatational irreversible deformation -- i.e., dilatational plasticity -- en route to failure. In the context of glasses, much focus has been given to shear (volume-preserving) plasticity, both in terms of the stress states considered and the corresponding material response. Here, using a recently-developed methodology and extensive computer simulations, we shed basic light on the elementary processes mediating dilatational plasticity in glasses. We show that plastic instabilities, corresponding to singularities of the glass Hessian, generically feature both dilatational and shear irreversible strain components. The relative magnitude and statistics of the strain components depend both on the symmetry of the driving stress (e.g., shear vs.~hydrostatic tension) and on the cohesive (attractive) part of the interatomic interaction. We further show that the tensorial shear component of the plastic strain is generally non-planar and also extract the characteristic volume of plastic instabilities. Elucidating the fundamental properties of the elementary micro-mechanical building blocks of plasticity in glasses sets the stage for addressing larger-scale, collective phenomena in dilatational plasticity such as topological changes in the form of cavitation and ductile-to-brittle transitions. As a first step in this direction, we show that the elastic moduli markedly soften during dilatational plastic deformation approaching cavitation.

In addition to Goldstone phonons that generically emerge in the low-frequency vibrational spectrum of any solid, crystalline or glassy, structural glasses also feature other low-frequency vibrational modes. The nature and statistical properties of these modes-often termed "excess modes"-have been the subject of decades-long investigation. Studying them, even using well-controlled computer glasses, has proven challenging due to strong spatial hybridization effects between phononic and nonphononic excitations, which hinder quantitative analyses of the nonphononic contribution DG(ω) to the total spectrum D(ω), per frequency ω. Here, using recent advances indicating that DG(ω)=D(ω)-DD(ω), where DD(ω) is Debye's spectrum of phonons, we present a simple and straightforward scheme to enumerate nonphononic modes in computer glasses. Our analysis establishes that nonphononic modes in computer glasses indeed make an additive contribution to the total spectrum, including in the presence of strong hybridizations. Moreover, it cleanly reveals the universal DG(ω)∼ω4 tail of the nonphononic spectrum, and opens the way for related analyses of experimental spectra of glasses.

It is now established that glasses feature low-frequency, nonphononic excitations, in addition to phonons that follow Debye's vibrational density of state (VDoS). Extensive computer studies demonstrated that these nonphononic, glassy excitations follow a universal non-Debye VDoS ${\cal D}_{\rm G}(\omega)\!\sim\!\omega^4$, at low frequencies $\omega$. Yet, due to intrinsic difficulties in disentangling ${\cal D}_{\rm G}(\omega)$ from the total VDoS ${\cal D}(\omega)$, which is experimentally accessible through various scattering techniques, the $\omega^4$ tail of ${\cal D}_{\rm G}(\omega)$ lacked direct experimental support. We develop a procedure to extract ${\cal D}_{\rm G}(\omega)$ from the measured ${\cal D}(\omega)$, based on recent advances in understanding low-frequency excitations in glasses, and apply it to available datasets for diverse glasses. The resulting analysis shows that the $\omega^4$ tail of the nonphononic vibrational spectra of glasses is nontrivially consistent with a broad range of experimental observations. It also further supports that ${\cal D}_{\rm G}(\omega)$ makes an additive contribution to ${\cal D}(\omega)$.

A hallmark of structural glasses and other disordered solids is the emergence of excess low-frequency vibrations on top of the Debye spectrum DDebye(ω) of phonons (ω denotes the vibrational frequency), which exist in any solid whose Hamiltonian is translationally invariant. These excess vibrations-a signature of which is a THz peak in the reduced density of states D(ω)/DDebye(ω), known as the boson peak-have resisted a complete theoretical understanding for decades. Here, we provide direct numerical evidence that vibrations near the boson peak consist of hybridizations of phonons with many quasilocalized excitations; the latter have recently been shown to generically populate the low-frequency tail of the vibrational spectra of structural glasses quenched from a melt and of disordered crystals. Our results suggest that quasilocalized excitations exist up to and in the vicinity of the boson-peak frequency and, hence, constitute the fundamental building blocks of the excess vibrational modes in glasses.