We present a linearized shear-transformation-zone (STZ) theory of glassy dynamics in which the internal STZ transition rates are characterized by a broad distribution of activation barriers. For slowly aging or fully aged systems, the main features of the barrier-height distribution are determined by the effective temperature and other near-equilibrium properties of the configurational degrees of freedom. Our theory accounts for the wide range of relaxation rates observed in both structural glasses and soft glassy materials such as colloidal suspensions. We find that the frequency dependent loss modulus is not just a superposition of Maxwell modes. Rather, it exhibits an $α$ peak that rises near the viscous relaxation rate and, for nearly jammed, glassy systems, extends to much higher frequencies in accord with experimental observations. We also use this theory to compute strain recovery following a period of large, persistent deformation and then abrupt unloading. We find that strain recovery is determined in part by the initial barrier-height distribution, but that true structural aging also occurs during this process and determines the system's response to subsequent perturbations. In particular, we find by comparison with experimental data that the initial deformation produces a highly disordered state with a large population of low activation barriers, and that this state relaxes quickly toward one in which the distribution is dominated by the high barriers predicted by the near-equilibrium analysis. The nonequilibrium dynamics of the barrier-height distribution is the most important of the issues raised and left unresolved in this paper.
We study shear yielding and steady state flow of glassy materials with molecular dynamics simulations of two standard models: amorphous polymers and bidisperse Lennard-Jones glasses. For a fixed strain rate, the maximum shear yield stress and the steady state flow stress in simple shear both drop linearly with increasing temperature. The dependence on strain rate can be described by a either a logarithm or a power-law added to a constant. In marked contrast to predictions of traditional thermal activation models, the rate dependence is nearly independent of temperature. The relation to more re…
The `soft glassy rheology' (SGR) model gives an appealing account of the flow of nonergodic soft materials in terms of the local yield dynamics of mesoscopic elements. Newtonian, power-law, and yield-stress fluid regimes arise on varying a `noise temperature', x. Here we extend the model, to capture the idea that the noise is largely caused by yield itself. The extended model can account for the viscosity-bifurcation and shear-banding effects reported recently in a wide range of soft materials. A variant model may shed light on shear banding and strain-rate hysteresis seen in glassy star polym…
We study the transient start-up dynamics of a fluidity model which captures the rheological behavior of aging soft glassy materials, in a rectilinear shear flow upon application of step shear rate. We observe that when the steady state flow curve is non-monotonic the system shows transient and/or apparent steady state shear banding in close qualitative agreement with experimental observations. Due to a competetion between aging and rejuvenation during the start up, we show that there is an apparent steady state banding at large times even for shear rates such that the steady state flow curve a…
The nonlinear rheology of a soft glassy material is captured by its constitutive relation, shear stress vs shear rate, which is most generally obtained by sweeping up or down the shear rate over a finite temporal window. For a huge amount of complex fluids, the up and down sweeps do not superimpose and define a rheological hysteresis loop. By means of extensive rheometry coupled to time-resolved velocimetry, we unravel the local scenario involved in rheological hysteresis for various types of well-studied soft materials. We introduce two observables that quantify the hysteresis in macroscopic …
In this work, we study the deformation behavior of thin films of various soft glassy materials that are simultaneously subjected to two creep flow fields, rotational shear flow by applying torque and elongational flow by applying normal force. The generic behavior under the combined fields is investigated in different soft glassy materials with diverse microstructure such as: hair gel, emulsion paint, shaving foam and clay suspension. Increase in strength of one stress component while keeping the other constant, not only leads to an expected enhanced deformation in its own direction, but also …
The response of glasses to mechanical loading often leads to the formation of inhomogeneous flow patterns that strongly affect materials properties. Among them, shear bands are ubiquitous in a wide variety of materials, ranging from soft matter systems to metallic alloys. Shear banding is associated with strain localization, i.e. the deformation of the sheared glassy solid is localized in space in form of band-like structures. These structures are often precursors to catastrophic failure, implying that a proper understanding of the underlying mechanisms could lead to the design of smarter mate…
Continuum modeling of Soft Glassy Materials under shear
Università di Roma “Tor Vergata”（罗马托尔维尔加塔大学）；ENSL, CNRS, Laboratoire de physique（里昂高等师范学校，法国国家科学研究中心，物理实验室）；Univ. de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière（里昂大学，克莱贝尔·伯纳德里昂第一大学，法国国家科学研究中心，光物质研究所）；Institut Universitaire de France (IUF)（法兰西学院）；Eindhoven University of Technology（埃因霍温理工大学）；CNR-IAC（意大利国家研究委员会应用化学研究所）
We use considerations of energy balance and dissipation to derive a self-consistent version of the shear-transformation-zone (STZ) theory of plastic deformation in amorphous solids. The theory is generalized to include arbitrary spatial orientations of STZs. Continuum equations for elasto-plastic material and their energy balance properties are discussed.
Recently introduced constitutive equations for the rheology of dense, disordered materials are investigated in the context of stick-slip experiments in boundary lubrication. The model is based on a generalization of the shear transformation zone (STZ) theory, in which plastic deformation is represented by a population of mesoscopic regions which may undergo non affine deformations in response to stress. The generalization we study phenomenologically incorporates the effects of aging and glassy relaxation. Under experimental conditions associated with typical transitions from stick-slip to stea…
Despite qualitative differences in their underlying physics, both hard and soft glassy materials exhibit almost identical linear rheological behaviors. We show that these nearly universal properties emerge naturally in a shear-transformation-zone (STZ) theory of amorphous plasticity, extended to include a broad distribution of internal thermal-activation barriers. The principal features of this barrier distribution are predicted by nonequilibrium, effective-temperature thermodynamics. Our theoretical loss modulus $G"(ω)$ has a peak at the $α$ relaxation rate, and a power law decay of the form …
We present an STZ-based analysis of numerical simulations by Haxton and Liu (HL). The extensive HL data sharply test the basic assumptions of the STZ theory, especially the central role played by the effective disorder temperature as a dynamical state variable. We find that the theory survives these tests, and that the HL data provide important and interesting constraints on some of its specific ingredients. Our most surprising conclusion is that, when driven at various constant shear rates in the low-temperature glassy state, the HL system exhibits a classic glass transition, including super-…
We model a sheared disordered solid using the theory of Shear Transformation Zones (STZs). In this mean-field continuum model the density of zones is governed by an effective temperature that approaches a steady state value as energy is dissipated. We compare the STZ model to simulations by Shi, et al.(Phys. Rev. Lett. 98 185505 2007), finding that the model generates solutions that fit the data,exhibit strain localization, and capture important features of the localization process. We show that perturbations to the effective temperature grow due to an instability in the transient dynamics, bu…
Glassy solids may undergo a fluidization (yielding) transition upon deformation whereby the material starts to flow plastically. It has been a matter of debate whether this process is controlled by a specific time scale, from among different competing relaxation/kinetic processes. Here, two constitutive models of cage relaxation are examined within the microscopic model of nonaffine elasto-plasticity. One (widely used) constitutive model implies that the overall relaxation rate is dominated by the fastest between the structural ($α$) relaxation rate and the shear-induced relaxation rate. A dif…
Yielding transitions in athermal amorphous materials resemble critical phenomena. Historically, they have been described by the Herschel-Bulkley rheological formula, which implies singular behaviors at yield points. In this paper, I examine this class of phenomena using an elementary version of the thermodynamic shear-transformation-zone (STZ) theory, focusing on the role of the effective disorder temperature, and paying special attention to scaling and dimensional arguments. I find a wide variety of Herschel-Bulkley-like rheologies but, for fundamental reasons not specific to the STZ theory, …
We use Shear Transformation Zone (STZ) theory to develop a deformation map for amorphous solids as a function of the imposed shear rate and initial material preparation. The STZ formulation incorporates recent simulation results [Haxton and Liu, PRL 99 195701 (2007)] showing that the steady state effective temperature is rate dependent. The resulting model predicts a wide range of deformation behavior as a function of the initial conditions, including homogeneous deformation, broad shear bands, extremely thin shear bands, and the onset of material failure. In particular, the STZ model predicts…
We use energetic considerations to deduce the form of a previously uncertain coupling term in the shear-transformation-zone (STZ) theory of plastic deformation in amorphous solids. As in the earlier versions of the STZ theory, the onset of steady deformation at a yield stress appears here as an exchange of dynamic stability between jammed and plastically deforming states. We show how an especially simple ``quasilinear'' version of this theory accounts qualitatively for many features of plasticity such as yielding, strain softening, and strain recovery. We also show that this minimal version of…
Yield stress fluids (YSFs) display a dual nature highlighted by the existence of a yield stress such that YSFs are solid below the yield stress, whereas they flow like liquids above it. Under an applied shear rate $\dotγ$, the solid-to-liquid transition is associated with a complex spatiotemporal scenario. Still, the general phenomenology reported in the literature boils down to a simple sequence that can be divided into a short-time response characterized by the so-called "stress overshoot", followed by stress relaxation towards a steady state. Such relaxation can be either long-lasting, whic…
The competing effects of slow structural relaxations (aging) and deformation at constant strain rate on the shear yield stress $τ^y$ of simple model glasses are examined using molecular simulations. At long times, aging leads to a logarithmic increase in density and $τ^y$. The yield stress also rises logarithmically with rate, but shows a sharp transition in slope at a rate that decreases with increasing age. We present a simple phenomenological model that includes both intrinsic rate dependence and the change in properties with the total age of the system at yield. As predicted by the model, …
This paper examines the stability of a previously proposed version of the shear-transformation-zone (STZ) theory of plasticity where the total STZ population is determined by an effective temperature and compares it to experimental results for a metallic glass. In particular, the addition of effective temperature dynamics to the shear transformation zone theory leads to the existence of a range of strain rates for which the strain localizes into shear bands. Yet while the steady-state results qualitatively agree, the instability of the dynamics of the system while loading begins better describ…
Experimental measurements of the onset of granular flow are directly compared to predictions of the "shear transformation zone" (STZ) theory of amorphous plasticity. The STZ equations make it possible, on a coarse grained level, to incorporate the anisotropy of the particle contact network and its dynamic evolution including changes in the contact network direction under shear reversal. The STZ theory and experiment show qualitative agreement in the transient and steady state shear forces and velocity profiles during the start of shear and shear reversal.
A two-dimensional version of the shear-transformation-zone (STZ) theory by Falk and Langer is explored numerically. Two different geometries are used to simulate uniaxial tension experiments where materials are subjected to constant strain rates. In the first setup, a rectangular specimen is given an imperceptible indentation, allowing it to neck at the center. The dynamics are explored systematically by varying both the straining capability of the STZs (epsilon_0) and the external strain rate. Higher values of epsilon_0 increase the plastic flow and result in sharper necks. Decreased values o…
Simulations of the structure and dynamics of fluid films confined to a thickness of a few molecular diameters are described. Confining walls introduce layering and in-plane order in the adjacent fluid. The latter is essential to transfer of shear stress. As the film thickness is decreased, by increasing pressure or decreasing the number of molecular layers, the entire film may undergo a phase transition. Spherical molecules tend to crystallize, while short chain molecules enter a glassy state with strong local orientational and translational order. These phase transitions lead to dramatic chan…
We study the stability of amorphous solids, focusing on the distribution P(x) of the local stress increase x that would lead to an instability. We argue that this distribution is singular P(x)x^θ, where the exponent θ is non-zero if the elastic interaction between rearranging regions is non-monotonic, and increases with the interaction range. For a class of finite dimensional models we show that stability implies a lower bound on θ, which is found to lie near saturation. For quadrupolar interactions these models yield θ ~ 0.6 for d=2 and θ~ 0.4 in d=3 where d is the spatial dimension, accurate…
Atomistic Framework for Glassy Polymer Viscoelasticity Across Twenty Frequency Decades
University of Milan（米兰大学）；University of Göttingen（哥廷根大学）；Graz University of Technology（格拉茨工业大学）；Polymer Competence Center Leoben GmbH（莱奥本聚合物能力中心有限公司）；US Army DEVCOM Army Research Laboratory（美国陆军DEVCOM陆军研究实验室）
Using molecular simulations and theory, we develop an explicit mapping of the contribution of molecular relaxation modes in glassy thermosets to the shear modulus, where the relaxations were tuned by altering the polarity of side groups. Specifically, motions at the domain, segmental, monomer, and atomic levels are taken from molecular dynamics snapshots and directly linked with the viscoelasticity through a framework based in the lattice dynamics of amorphous solids. This unique approach provides direct insight into the roles of chemical groups in the stress response, including the timescale …
Sheared concentrated colloids with short range polymer coats are examined via simulations. Distributions of force are found to be sums of exponentials. The 'hydrodynamic clustering' underlying the shear thickening effect is shown, in this system, to be a network of percolating coat contacts, with coats both in compression and extension. The geometry and kinetics of this network are explored along with its relation to the bulk stress tensor. Particles experience strong fluctuations in force over epochs $\it{circa}$ 10% strain. Density fluctuations up to glassy volume fractions give rise to scat…
Soft glassy materials such as mayonnaise, wet clays, or dense microgels display under external shear a solid-to-liquid transition. Such a shear-induced transition is often associated with a non-monotonic stress response, in the form of a stress maximum referred to as "stress overshoot". This ubiquitous phenomenon is characterized by the coordinates of the maximum in terms of stress $σ_\text{M}$ and strain $γ_\text{M}$ that both increase as weak power laws of the applied shear rate. Here we rationalize such power-law scalings using a continuum model that predicts two different regimes in the li…
This investigation extends earlier studies of a shear-transformation-zone (STZ) theory of plastic deformation in amorphous solids. My main purpose here is to explore the possibility that the configurational degrees of freedom of such systems fall out of thermodynamic equilibrium with the heat bath during persistent mechanical deformation, and that the resulting state of configurational disorder may be characterized by an effective temperature. The further assumption that the population of STZ's equilibrates with the effective temperature allows the theory to be compared directly with experimen…
We report results of non-equilibrium molecular dynamics simulations of a one-component glassy system under the influence of a shear flow, with the aim of investigating shear induced ordering of this system. In spite of the very low temperature, the system transforms into a strained crystalline state through well defined nucleation events. Various characteristics of the observed ordering at different strain rates and temperatures are discussed. We also define and discuss the transition rates.
We review the anomalous behavior of solid He-4 at low temperatures with particular attention to the role of structural defects present in solid. The discussion centers around the possible role of two level systems and structural glassy components for inducing the observed anomalies. We propose that the origin of glassy behavior is due to the dynamics of defects like dislocations formed in He-4. Within the developed framework of glassy components in a solid, we give a summary of the results and predictions for the effects that cover the mechanical, thermodynamic, viscoelastic, and electro-elast…
We use the internal-variable, effective-temperature thermodynamics developed in two preceding papers to reformulate the shear-transformation-zone (STZ) theory of amorphous plasticity. As required by the preceding analysis, we make explicit approximations for the energy and entropy of the STZ internal degrees of freedom. We then show that the second law of thermodynamics constrains the STZ transition rates to have an Eyring form as a function of the effective temperature. Finally, we derive an equation of motion for the effective temperature for the case of STZ dynamics.
Notched bars of bulk metallic glasses, Pd$_{40}$Ni$_{40}$P$_{20}$ and Zr$_{52.5}$Cu$_{17.9}$Ni$_{14.6}$Al$_{10}$Ti$_{5}$, were deformed under 3-point bending conditions, resulting in the formation of shear bands before failure. The immediate environment of shear bands was investigated in detail using fluctuation electron microscopy to extract information on the strain-induced modifications of the medium-range order (MRO) and its lateral extension. Characteristic material-independent gradients were observed for the tensile and compressive sides of the samples indicating the impact of the local …
We develop an athermal version of the shear-transformation-zone (STZ) theory of amorphous plasticity in materials where thermal activation of irreversible molecular rearrangements is negligible or nonexistent. In many respects, this theory has broader applicability and yet is simpler than its thermal predecessors. For example, it needs no special effort to assure consistency with the laws of thermodynamics, and the interpretation of yielding as an exchange of dynamic stability between jammed and flowing states is clearer than before. The athermal theory presented here incorporates an explicit …
The nucleation of cavities in a homogenous polymer under tensile strain is investigated in a coarse-grained molecular dynamics simulation. In order to establish a causal relation between local microstructure and the onset of cavitation, a detailed analysis of some local properties is presented. In contrast to common assumptions, the nucleation of a cavity is neither correlated to a local loss of density nor, to the stress at the atomic scale and nor to the chain ends density in the undeformed state. Instead, a cavity in glassy polymers nucleates in regions that display a low bulk elastic modul…
We extend our earlier shear-transformation-zone (STZ) theory of amorphous plasticity to include the effects of thermally assisted molecular rearrangements. This version of our theory is a substantial revision and generalization of conventional theories of flow in noncrystalline solids. As in our earlier work, it predicts a dynamic transition between jammed and flowing states at a yield stress. Below that yield stress, it now describes thermally assisted creep. We show that this theory accounts for the experimentally observed strain-rate dependence of the viscosity of metallic glasses, and that…
The shear-transformation-zone (STZ) theory of plastic deformation in glass-forming materials is reformulated in light of recent progress in understanding the roles played the effective disorder temperature and entropy flow in nonequilibrium situations. A distinction between fast and slow internal state variables reduces the theory to just two coupled equations of motion, one describing the plastic response to applied stresses, and the other the dynamics of the effective temperature. The analysis leading to these equations contains, as a byproduct, a fundamental reinterpretation of the dynamic …
We propose that the super-Arrhenius relaxation rates observed in glassy materials are determined by thermally nucleated rearrangements of increasing numbers of molecules at decreasing temperatures. In our model of this mechanism, string-like fluctuations in the neighborhood of shear transformation zones (STZ's) provide routes along which rearrangements can propagate, and the entropy associated with long enough strings allows the rearrangement energy to be distributed stably in the surrounding material. We further postulate that, at low enough temperatures, these fluctuations are localized on t…
We solve exactly and describe in detail a simplified scalar model for the low frequency shear rheology of foams, emulsions, slurries, etc. [P. Sollich, F. Lequeux, P. Hebraud, M.E. Cates, Phys. Rev. Lett. 78, 2020 (1997)]. The model attributes similarities in the rheology of such ``soft glassy materials'' to the shared features of structural disorder and metastability. By focusing on the dynamics of mesoscopic elements, it retains a generic character. Interactions are represented by a mean-field noise temperature x, with a glass transition occurring at x=1 (in appropriate units). The exact sol…
In the preceding paper, we developed an athermal shear-transformation-zone (STZ) theory of amorphous plasticity. Here we use this theory in an analysis of numerical simulations of plasticity in amorphous silicon by Demkowicz and Argon (DA). In addition to bulk mechanical properties, those authors observed internal features of their deforming system that challenge our theory in important ways. We propose a quasithermodynamic interpretation of their observations in which the effective disorder temperature, generated by mechanical deformation well below the glass temperature, governs the behavior…
In this work we study morphological characteristics of the critically-sized crystalline nuclei at initial stage of the shear-induced crystallization of a model single-component amorphous (glassy) system. These characteristics are estimated quantitatively through statistical treatment of the non-equilibrium molecular dynamics simulation results for the system under steady shear at various (fixed) values of the shear rate $\dotγ$ and at different temperatures. It is found that the sheared glassy system is crystallized through nucleation mechanism. From analysis of a time-dependent trajectories o…
Shear banding is a fundamental non-equilibrium phenomenon in amorphous solids. Due to the intrinsic entangling of three local atomic motions: shear, dilatation and rotation, the precise physical process of shear band emergence is still an enigma. To unveil this mystery, we formulate for the first time a theoretical protocol covering both affine and non-affine components of deformation, to decode these highly entangled local atomic-scale events. In contrast to the broad concept of shear transformation zone, plastic behavior can be demonstrated comprehensively as the operative manipulation of mo…
Rapidly cooling a liquid may result in a glass transition, creating an amorphous solid whose shear and bulk moduli are finite. Even when done with constant density, these resulting moduli depend strongly on the rate of cooling. Understanding this phenomenon calls for analyzing separately the "Born term" that exists also in perfectly ordered materials and the contributions of the "excess modes" that result from glassy disorder. We show that the Born term is very insensitive to the cooling rate, and all the variation in the shear modulus is due to the excess modes. We argue that this approach pr…
The shear-transformation-zone (STZ) theory of plastic deformation predicts that sufficiently soft, non-crystalline solids are linearly unstable against forming periodic arrays of microstructural shear bands. A limited nonlinear analysis indicates that this instability may be the mechanism responsible for strain softening in both constant-stress and constant-strain-rate experiments. The analysis presented here pertains only to one-dimensional banding patterns in two-dimensional systems, and only to very low temperatures. It uses the rudimentary form of the STZ theory in which there is only a si…
We perform a systematic study of the characteristics of shear transformation zones (STZs) that nucleate at free surfaces of two-dimensional amorphous solids subject to tensile loading using two different atomistic simulation methods, the standard athermal, quasistatic (AQ) approach and our recently developed self-learning metabasin escape (SLME) method to account for the finite temperature and strain-rate effects. In the AQ, or strain-driven limit, the nonaffine displacement fields of surface STZs decay exponentially away from their centers at similar decay rates as their bulk counterparts, th…
The shear-banding instability in quasi-statically driven bulk metallic glasses emerges from collective dynamics, mediated by shear transformation zones and associated non-local elastic interactions. It is also phenomenologically known that sharp structural features of shear bands are typically correlated to the sharpness of the plastic yielding transition, being predominant in commonly studied alloys composed of multiple different elements, that have very different atomic radii. However, in the opposite limit \remove{of high-entropy multicomponent alloys,} where elements' radii are relatively …
