Low frequency quasi-localized modes of amorphous glasses appear to exhibit universal density of states, depending on the frequencies as $D(ω) \sim ω^4$. To date various models of glass formers with short range binary interaction, and network glasses with both binary and ternary interactions, were shown to conform with this law. In this paper we examine granular amorphous solids with long-range electrostatic interactions, and find that they exhibit the same law. To rationalize this wide universality class we return to a model proposed by Gurevich, Parshin and Schober (GPS) and analyze its predictions for interaction laws with varying spatial decay, exploring this wider than expected universality class. Numerical and analytic results are provided for both the actual system with long range interaction and for the GPS model.
The question of the local stability of the (replica-symmetric) amorphous solid state is addressed for a class of systems undergoing a continuous liquid to amorphous-solid phase transition driven by the effect of random constraints. The Hessian matrix, associated with infinitesimal fluctuations around the stationary point corresponding to the amorphous solid state, is obtained. The eigenvalues of this Hessian matrix are all shown to be strictly positive near the transition, except for one--the zero mode associated with the spontaneously broken continuous translational symmetry of the system. Th…
Goldstone modes in the amorphous solid state, resulting from the spontaneous breaking of translational symmetry due to random localisation of particles, are discussed. Starting from a microscopic model with quenched disorder, the broken symmetry is identified to be that of relative translations of the replicas. Goldstone excitations, corresponding to pure shear deformations, are constructed from long wavelength distortions of the order parameter. The elastic free energy is computed, and it is shown that Goldstone fluctuations destroy localisation in two spatial dimensions, yielding a two-dimen…
In sufficiently high spatial dimensions, the formation of the amorphous (i.e. random) solid state of matter, e.g., upon sufficent crosslinking of a macromolecular fluid, involves particle localization and, concommitantly, the spontaneous breakdown of the (global, continuous) symmetry of translations. Correspondingly, the state supports Goldstone-type low energy, long wave-length fluctuations, the structure and implications of which are identified and explored from the perspective of an appropriate replica field theory. In terms of this replica perspective, the lost symmetry is that of relative…
Prestress in amorphous solids bears the memory of their formation, and plays a profound role in their mechanical properties, from stiffening or softening elastic moduli to shifting frequencies of vibrational modes, as well as directing yielding and solidification in the nonlinear regime. Here we develop a set of mathematical tools to investigate elasticity of prestressed discrete networks, which disentangles the effects from disorder in configuration and disorder in prestress. Applying these methods to prestressed triangular lattices and a computational model of amorphous solids, we demonstrat…
Plasticity in amorphous solids is mediated by localized quadrupolar instabilities, but the mechanism by which an amorphous solid eventually fails or melts is debated. In this work we argue that these phenomena can be investigated in the model problem of an elastic continuum with quadrupolar defects, at finite temperature. This problem is posed and the collective behavior of the defects is analytically investigated. Using both renormalization group and field-theoretic techniques, it is found that the model has a yielding/melting transition of spinodal type.
Under sufficient permanent random covalent bonding, a fluid of atoms or small molecules is transformed into an amorphous solid network. Being amorphous, local structural properties in such networks vary across the sample. A natural order parameter, resulting from a statistical-mechanical approach, captures information concerning this heterogeneity via a certain joint probability distribution. This joint probability distribution describes the variations in the positional and orientational localization of the particles, reflecting the random environments experienced by them, as well as further i…
We derive a phase diagram for amorphous solids and liquid supercooled water and explain why the amorphous solids of water exist in several different forms. Application of large-deviation theory allows us to prepare such phases in computer simulations. Along with nonequilibrium transitions between the ergodic liquid and two distinct amorphous solids, we establish coexistence between these two amorphous solids. The phase diagram we predict includes a nonequilibrium triple point where two amorphous phases and the liquid coexist. While the amorphous solids are long-lived and slowly-aging glasses, …
The damping or attenuation coefficient of sound waves in solids due to impurities scales with the wavevector to the fourth power, also known as Rayleigh scattering. In amorphous solids, Rayleigh scattering may be enhanced by a logarithmic factor although computer simulations offer conflicting conclusions regarding this enhancement and its microscopic origin. We present a tensorial replica field-theoretic derivation based on heterogeneous or fluctuating elasticity (HE), which shows that long-range (power-law) spatial correlations of the elastic constants, is the origin of the logarithmic enhanc…
Using numerical simulations, we have studied the yielding response, in the athermal quasi static 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 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 vi…
An approximation scheme for model disordered solids is proposed that leads to the fully analytical evaluation of the elastic constants under explicit account of the inhomogeneity (nonaffinity) of the atomic displacements. The theory is in quantitative agreement with simulations for central-force systems and predicts the vanishing of the shear modulus at the isostatic point with the linear law μ ~ (z - 2d), where z is the coordination number. The vanishing of rigidity at the isostatic point is shown to be a consequence of the canceling out of positive affine and negative nonaffine terms.
Amorphous solids are ubiquitous among natural and man-made materials. Often used as structural materials for their attractive mechanical properties, their utility depends critically on their response to applied stresses. Processes underlying such mechanical response, and in particular the yielding behaviour of amorphous solids, are not satisfactorily understood. Although studied extensively[1-14], observed yielding behaviour can be gradual and depend significantly on conditions of study, making it difficult to convincingly validate existing theoretical descriptions of a sharp yielding transiti…
Amorphous solids have excess soft modes in addition to the phonon modes described by the Debye theory. Recent numerical results show that if the phonon modes are carefully removed, the density of state of the excess soft modes exhibit universal quartic scaling, independent of the interaction potential, preparation protocol, and spatial dimensions. We hereby provide a theoretical framework to describe this universal scaling behavior. For this purpose, we extend the mean-field theory to include the effects of finite dimensional fluctuation. Based on a semi-phenomenological argument, we show that…
By means of a finite elements technique we solve numerically the dynamics of an amorphous solid under deformation in the quasistatic driving limit. We study the noise statistics of the stress-strain signal in the steady state plastic flow, focusing on systems with low internal dissipation. We analyze the distributions of avalanche sizes and durations and the density of shear transformations when varying the damping strength. In contrast to avalanches in the overdamped case, dominated by the yielding point universal exponents, inertial avalanches are controlled by a non-universal damping depend…
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…
Systems undergoing an equilibrium phase transition from a liquid state to an amorphous solid state exhibit certain universal characteristics. Chief among these are the fraction of particles that are randomly localized and the scaling functions that describe the order parameter and (equivalently) the statistical distribution of localization lengths for these localized particles. The purpose of this Paper is to discuss the origins and consequences of this universality, and in doing so, three themes are explored. First, a replica-Landau-type approach is formulated for the universality class of sy…
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…
Sound attenuation in low temperature amorphous solids originates from their disordered structure. However, its detailed mechanism is still being debated. Here we analyze sound attenuation starting directly from the microscopic equations of motion. We derive an exact expression for the zero-temperature sound damping coefficient. We verify that the sound damping coefficients calculated from our expression agree very well with results from independent simulations of sound attenuation. The small wavevector analysis of our expression shows that sound attenuation is primarily determined by the non-a…
The holographic duality has proven successful in linking seemingly unrelated problems in physics.Recently, intriguing correspondences between the physics of soft matter and gravity are emerging,including strong similarities between the rheology of amorphous solids, effective field theories for elasticity and the physics of black holes. However, direct comparisons between theoretical predictions and experimental/simulation observations remain limited. Here, we study the effects of non-linear elasticity on the mechanical and thermodynamic properties of amorphous materials responding to shear, us…
We present a theoretical derivation of acoustic phonon damping in amorphous solids based on the nonaffine response formalism for the viscoelasticity of amorphous solids. The analytical theory takes into account the nonaffine displacements in transverse waves and is able to predict both the ubiquitous low-energy diffusive damping $\sim k^{2}$, as well as a novel contribution to the Rayleigh damping $\sim k^{4}$ at higher wavevectors and the crossover between the two regimes observed experimentally. The coefficient of the diffusive term is proportional to the microscopic viscous (Langevin-type) …
We present results of atomistic simulations of a new model of a magnetic amorphous solid subjected to external mechanical strains and magnetic fields. The model employed offers new perspectives on important effects like Barkhausen noise and magnetostriction. It is shown that the plastic response in such systems exhibit singularities characterized by unexpected exponents requiring careful theoretical reasoning. The spatial structure of the plastic events requires a new coarse grained elasto-magnetic theory which is provided here.
The equilibrium amorphous solid state -- formed, e.g., by adequately randomly crosslinking the constituents of a macromolecular fluid -- is a heterogeneous state characterized by a universal distribution of particle localization lengths. Near to the crosslink-density-controlled continuous amorphous-solidification transition, this distribution obeys a scaling form: it has a single peak at a lengthscale that diverges (along with the width of the distribution) as the transition is approached. The modulus controlling macroscale elastic shear deformations of the amorphous solid does not depend on t…
It has been suggested that high-density amorphous (HDA) ice is a structurally arrested form of high-density liquid (HDL) water, while low-density amorphous (LDA) ice is a structurally arrested form of low-density liquid (LDL) water. Recent experiments and simulations have been interpreted to support the possibility of a second "distinct" high-density structural state, named very high-density amorphous (VHDA) ice, questioning the LDL-HDL hypothesis. We test this interpretation using extensive computer simulations, and find that VHDA is a more stable form of HDA and that in fact VHDA should be c…
We consider the yielding behavior of amorphous solids under cyclic shear deformation and show that it can be mapped into a random walk in a confining potential with an absorbing boundary. The resulting dynamics is governed by the first passage time into the absorbing state and suffices to capture the essential qualitative features recently observed in atomistic simulations of amorphous solids. Our results provide insight into the mechanism underlying yielding and its robustness. When the possibility of activated escape from absorbing states is added, it leads to a unique determination of a thr…
We describe the phase diagram of amorphous solid water by performing molecular dynamics simulations. Our simulations follow different paths in the phase diagram: isothermal compression/decompression, isochoric cooling/heating and isobaric cooling/heating. We are able to identify low-density amorphous (LDA), high-density amorphous (HDA), and very-high density amorphous (VHDA) ices. The density $ρ$ of these glasses at different pressure $P$ and temperature $T$ agree well with experimental values. We also study the radial distribution functions of glassy water. We obtain VHDA by isobaric heating …
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…
The yield stress is a defining feature of amorphous materials which is difficult to analyze theoretically, because it stems from the strongly non-linear response of an arrested solid to an applied deformation. Mode-coupling theory predicts the flow curves of materials undergoing a glass transition, and thus offers predictions for the yield stress of amorphous solids. We use this approach to analyse several classes of disordered solids, using simple models of hard sphere glasses, soft glasses, and metallic glasses for which the mode-coupling predictions can be directly compared to the outcome o…
Fatigue caused by cyclic bending of a piece of material, resulting in its mechanical failure, is a phenomenon that had been studied for ages by engineers and physicists alike. In this Letter we study such fatigue in a strip of athermal amorphous solid. On the basis of atomistic simulations we conclude that the crucial quantity to focus on is the {\em accumulated damage}. Although this quantity exhibits large sample-to-sample fluctuations, its dependence on the loading determines the statistics of the number of cycles to failure. Thus we can provide a scaling theory for the Wöhler plots of mean…
The high frequency dynamics of low- (LDA) and high-density amorphous-ice (HDA) and of cubic ice (I_c) has been measured by inelastic X-ray Scattering (IXS) in the 1-15 nm^{-1} momentum transfer (Q) range. Sharp phonon-like excitations are observed, and the longitudinal acoustic branch is identified up to Q = 8nm^{-1} in LDA and I_c and up to 5nm^{-1} in HDA. The narrow width of these excitations is in sharp contrast with the broad features observed in all amorphous systems studied so far. The "crystal-like" behavior of amorphous ices, therefore, implies a considerable reduction in the number o…
We discuss the strain dependence of the acoustic properties of amorphous metals in both normal and superconducting states, in the temperature range 0.1 mK $ \le T \le 1 $K. A crossover is found when the strain energy is of the order of the effective interaction energy between tunneling systems at the corresponding temperature. Our results provide clear evidence for the interaction between tunneling systems, whose energy is in quantitative agreement with theoretical expectations, and reveal that without the knowledge of the corresponding strain dependences, the measured temperature dependences …
It has been established that the low frequency quasi-localized modes of amorphous solids at zero temperature exhibit universal density of states, depending on the frequencies as $D(ω) \sim ω^4$. It remains an open question whether this universal law extends to finite temperatures. In this Letter we show that well quenched model glasses at temperatures as high as $T_g/3$ possess the same universal density of states. The only condition required is that {\em average} particle positions stabilize before thermal diffusion destroys the cage structure of the material. The universal density of quasi-l…
Failure of amorphous solids is fundamental to various phenomena, including landslides and earthquakes. Recent experiments indicate that highly plastic regions form elongated structures that are especially apparent near the maximal shear stress $Σ_{\max}$ where failure occurs. This observation suggested that $Σ_{\max}$ acts as a critical point where the length scale of those structures diverges, possibly causing macroscopic transient shear bands. Here we argue instead that the entire solid phase ($Σ<Σ_{\max}$) is critical, that plasticity always involves system-spanning events, and that their m…
We perform an extensive numerical study of avalanche behavior in a 2D LJ glass at T=0, sheared at finite strain rates $\dotγ$. From the finite size analysis of stress fluctuations and of transverse diffusion we show that flip-flip correlations remain relevant at all realistic strain rates. We predict that the avalanche size scales as $\dotγ^{-1/d}$, with $d$ the space dimension.
An important aspect of the physics of amorphous solids is the onset of irreversible behavior usually associated with yield. Here we study amorphous solids under periodic shear using quasi-static molecular dynamics simulations and observe a transition from reversible to irreversible deformation at a critical strain amplitude. We find that for small strain amplitudes the system exhibits a noisy but repetitive limit-cycle, similar to return point memory \cite{sethna1993hysteresis}. However, for large strain amplitudes the behavior becomes chaotic (shows sensitivity to initial conditions) and thus…
We build a minimal, mean-field, model of plasticity of amorphous solids, based upon a phenomenology of dissipative events derived, in a preceding paper [A. Lemaitre, C. Caroli, arXiv:0705.0823] from extensive molecular simulations. It reduces to the dynamics of an ensemble of identical shear transformation zones interacting via the dynamic noise due to the long ranged elastic fields induced by zone flips themselves. We find that these ingredients are sufficient to generate flip avalanches with a power-law scaling with system size, analogous to that observed in molecular simulations. We further…
It has recently been shown that yield in amorphous solids under oscillatory shear is a dynamical transition from asymptotically periodic to asymptotically chaotic, diffusive dynamics. However, the type and universality class of this transition are still undecided. Here we show that the diffusive behavior of the vector of coordinates of the particles comprising an amorphous solid when subject to oscillatory shear, is analogous to that of a particle diffusing in a percolating lattice, the so-called "ant in the labyrinth" problem, and that yield corresponds to a percolation transition in the latt…
Shanghai Jiao Tong University（上海交通大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of Montpellier and CNRS（蒙彼利埃大学与法国国家科学研究中心）；Chengdu University of Technology（成都理工大学）
Amorphous solids that underwent a strain in one direction such that they responded in a plastic manner `remember' that direction also when relaxed back to a state with zero mean stress. We address the question `what is the order parameter that is responsible for this memory?' and is therefore the reason for the different subsequent responses of the material to strains in different directions. We identify such an order parameter which is readily measurable, we discuss its trajectory along the stress-strain curve, and propose that it and its probability distribution function must form a necessar…
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…
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…
We demonstrate using the classical density functional theory (DFT) model that an intermediate degree of mass localization in the amorphous state is essential for producing the boson peak. The localization length $\ell$ is identified from the width of the gaussian density profile in terms of which the inhomogeneous density n(x) of the solid is expressed in DFT. At a fixed average density, there exists a limiting value $\ell_0$ of $\ell$ signifying a minimum mass localization in the amorphous state. For more delocalized states ($\ell>\ell_0$) occurrence of boson peak is unfeasible.
We focus on the probability distribution function (pdf) $P(Δγ; γ)$ where $Δγ$ are the {\em measured} strain intervals between plastic events in an athermal strained amorphous solids, and $γ$ measures the accumulated strain. The tail of this distribution as $Δγ\to 0$ (in the thermodynamic limit) scales like $Δγ^η$. The exponent $η$ is related via scaling relations to the tail of the pdf of the eigenvalues of the {\em plastic modes} of the Hessian matrix $P(λ)$ which scales like $λ^θ$, $η=(θ-1)/2$. The numerical values of $η$ or $θ$ can be determined easily in the unstrained material and in the …
A number of current theories of plasticity in amorphous solids assume at their basis that plastic deformations are spatially localized. We present in this paper a series of numerical experiments to test the degree of locality of plastic deformation. These experiments increase in terms of the stringency of the removal of elastic contributions to the observed elasto-plastic deformations. It is concluded that for all our simulational protocols the plastic deformations are not localized, and their scaling is sub-extensive. We offer a number of measures of the magnitude of the plastic deformation, …
We demonstrate that irreversible structural reorganization is not necessary for the observation of yield behaviour in an amorphous solid. While the majority of solids strained to their yield point do indeed undergo an irreversible reorganization, we find a significant fraction of solids exhibit 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 non-affine motions driven by the applied strain. The empirical similarity of the dependence of the ratio of stress over strain…
A continuum model for growth of solids is developed, considering adatom deposition, surface diffusion, and configuration dependent incorporation rate. For amorphous solids it is related to surface energy densities. The high adatom density leads to growth enhanced dynamics of (a) Mullins' classical equation [J. Appl. Phys. {\bf 28}, 333 (1957)] without, and (b) of the Asaro-Tiller-Grinfeld-Srolovitz instability with lateral stress in the growing film. The latter mechanism is attributed to morphologies found in recent experiments.
The shear-modulus and yield-stress of amorphous solids are important material parameters, with the former determining the rate of increase of stress under external strain and the latter being the stress value at which the material flows in a plastic manner. It is therefore important to understand how these parameters can be related to the inter-particle potential. Here a scaling theory is presented such that given the inter-particle potential, the dependence of the yield stress and the shear modulus on the density of the solid can be predicted in the athermal limit. It is explained when such p…
Glasses have a large excess of low-frequency vibrational modes in comparison with continuous elastic body, the so-called Boson Peak, which appears to correlate with several crucial properties of glasses, such as transport or fragility. I review recent results showing that the Boson Peak is a necessary consequence of the weak connectivity of the solid. I explain why in assemblies repulsive spheres the boson peak shifts up to zero frequency as the pressure is lowered toward the jamming threshold, and derive the corresponding exponent. I show how these ideas capture the main low-frequency feature…
Glasses are known to exhibit quantitative universalities at low temperatures, the most striking of which is the ultrasonic attenuation coefficient 1/Q. In this work we develop a theory of coupled generic blocks with a certain randomness property to show that universality emerges essentially due to the interactions between elastic blocks, regardless of their microscopic nature.
We derive expressions for the lowest nonlinear elastic constants of amorphous solids in athermal conditions (up to third order), in terms of the interaction potential between the constituent particles. The effect of these constants cannot be disregarded when amorphous solids undergo instabilities like plastic flow or fracture in the athermal limit; in such situations the elastic response increases enormously, bringing the system much beyond the linear regime. We demonstrate that the existing theory of thermal nonlinear elastic constants converges to our expressions in the limit of zero tempera…
