Search Results (196 total)

We propose that there exists a generic class of glass-forming systems that have competing states (of crystalline order or not) which are locally close in energy to the ground state (which is typically unique). Upon cooling, such systems exhibit patches (or clusters) of these competing states which become locally stable in the sense of having a relatively high local shear modulus. It is in between these clusters where aging, relaxation, and plasticity under strain can take place. We demonstrate explicitly that relaxation events that lead to aging occur where the local shear modulus is low (even negative) and result in an increase in the size of local patches of relative order. We examine the aging events closely from two points of view. On the one hand we show that they are very localized in real space, taking place outside the patches of relative order, and from the other point of view we show that they represent transitions from one local minimum in the potential surface to another. This picture offers a direct relation between structure and dynamics, ascribing the slowing down in glass-forming systems to the reduction in relative volume of the amorphous material which is liquidlike. While we agree with the well-known Adam-Gibbs proposition that the slowing down is due to an entropic squeeze (a dramatic decrease in the number of available configurations), we do not agree with the Adam-Gibbs (or the Volger-Fulcher) formulas that predict an infinite relaxation time at a finite temperature. Rather, we propose that generically there should be no singular crisis at any finite temperature: the relaxation time and the associated correlation length (average cluster size) increase at most superexponentially when the temperature is lowered.

Numerical simulations of turbulent channel flows, with or without additives, are limited in the extent of the Reynolds number (Re) and Deborah number (De). The comparison of such simulations to theories of drag reduction, which are usually derived for asymptotically high Re and De, calls for some care. In this paper we present a study of drag reduction by rodlike polymers in a turbulent channel flow using direct numerical simulation and illustrate how these numerical results should be related to the recently developed theory.

We address the relaxation dynamics in hydrogen-bonded supercooled liquids near (but above) the glass transition, measured via broadband dielectric spectroscopy (BDS). We propose a theory based on decomposing the relaxation of the macroscopic dipole moment into contributions from hydrogen-bonded clusters of s molecules, with s_min≤s≤s_max. The existence of s_max is translated into a sum rule on the concentrations of clusters of size s. We construct the statistical mechanics of the supercooled liquid subject to this sum rule as a constraint, to estimate the temperature-dependent density of clusters of size s. With a theoretical estimate of the relaxation time of each cluster, we provide predictions for the real and imaginary parts of the frequency-dependent dielectric response. The predicted spectra and their temperature dependence are in accord with measurements, explaining a host of phenomenological fits like the Vogel-Fulcher fit and the stretched exponential fit. Using glycerol as a particular example, we demonstrate quantitative correspondence between theory and experiments. The theory also demonstrates that the α peak and the “excess wing” stem from the same physics in this material. The theory also shows that in other hydrogen-bonded glass formers the excess wing can develop into a β peak, depending on the molecular material parameters (predominantly the surface energy of the clusters). We thus argue that α and β peaks can stem from the same physics. We address the BDS in constrained geometries (pores) and explain why recent experiments on glycerol did not show a deviation from bulk spectra. Finally, we discuss the dc part of the BDS spectrum and argue why it scales with the frequency of the α peak, providing an explanation for the remarkable data collapse observed in experiments.

A classical problem in elasticity theory involves an inhomogeneity embedded in a material of given stress and shear moduli. The inhomogeneity is a region of arbitrary shape whose stress and shear moduli differ from those of the surrounding medium. In this paper we present a semianalytic method for finding the stress tensor for an infinite plate with such an inhomogeneity. The solution involves two conformal maps, one from the inside and the second from the outside of the unit circle to the inside, and respectively outside, of the inhomogeneity. The method provides a solution by matching the conformal maps on the boundary between the inhomogeneity and the surrounding material. This matching converges well only for relatively mild distortions of the unit circle due to reasons which will be discussed in the article. We provide a comparison of the present result to known previous results.

In this Letter, we suggest a simple and physically transparent analytical model of pressure driven turbulent wall-bounded flows at high but finite Reynolds numbers Re. The model provides an accurate quantitative description of the profiles of the mean-velocity and Reynolds stresses (second order correlations of velocity fluctuations) throughout the entire channel or pipe, for a wide range of Re, using only three Re-independent parameters. The model sheds light on the long-standing controversy between supporters of the century-old log-law theory of von Kàrmàn and Prandtl and proposers of a newer theory promoting power laws to describe the intermediate region of the mean velocity profile.

The understanding of dynamic failure in amorphous materials via the propagation of free boundaries like cracks and voids must go beyond elasticity theory, since plasticity intervenes in a crucial and poorly understood manner near the moving free boundary. We focus on failure via a cavitation instability in a radially symmetric stressed material and set up the free boundary dynamics taking both elasticity and viscoplasticity into account using the recently proposed athermal shear transformation zone theory. We demonstrate that this theory predicts the existence (in amorphous systems) of fast cavitation modes accompanied by extensive plastic deformations and discuss the revealed physics.

The flow of fluids in channels, pipes, or ducts, as in any other wall-bounded flow (like water along the hulls of ships or air on airplanes) is hindered by a drag, which increases manyfold when the fluid flow turns from laminar to turbulent. A major technological problem is how to reduce this drag in order to minimize the expense of transporting fluids like oil in pipelines, or to move ships in the ocean. It was discovered that minute concentrations of polymers can reduce the drag in turbulent flows by up to 80%. While experimental knowledge had accumulated over the years, the fundamental theory of drag reduction by polymers remained elusive for a long time, with arguments raging whether this is a “skin” or a “bulk” effect. In this Colloquium the phenomenology of drag reduction by polymers is summarized, stressing both its universal and nonuniversal aspects, and a recent theory is reviewed that provides a quantitative explanation of all the known phenomenology. Both flexible and rodlike polymers are treated, explaining the existence of universal properties like the maximum drag reduction asymptote, as well as nonuniversal crossover phenomena that depend on the Reynolds number, on the nature of the polymer and on its concentration. Finally other agents for drag reduction are discussed with a stress on the important example of bubbles.

We address the interesting temperature range of a glass forming system where the mechanical properties are intermediate between those of a liquid and a solid. We employ an efficient Monte Carlo method to calculate the elastic moduli, and show that in this range of temperatures the moduli are finite for short times and vanish for long times, where short and long depend on the temperature. By invoking some exact results from statistical mechanics we offer an alternative method to compute shear moduli using molecular dynamics simulations, and compare those to the Monte Carlo method. The final conclusion is that these systems are not “viscous fluids” in the usual sense, as their actual time-dependence concatenates solidlike materials with varying local shear moduli.

We study a recently introduced model of one-component glass-forming liquids whose constituents interact with an anisotropic potential. This system is interesting per se and as a model of liquids such as glycerol (interacting via hydrogen bonds) which are excellent glass formers. We work out the statistical mechanics of this system, encoding the liquid and glass disorder using appropriate quasiparticles (36 of them). The theory provides a full explanation of the glass transition phenomenology, including the identification of a diverging length scale and a relation between the structural changes and the diverging relaxation times.

We develop an athermal shear-transformation-zone (STZ) theory of plastic deformation in spatially inhomogeneous, amorphous solids. Our ultimate goal is to describe the dynamics of the boundaries of voids or cracks in such systems when they are subjected to remote, time-dependent tractions. The theory is illustrated here for the case of a circular hole in an infinite two-dimensional plate, a highly symmetric situation that allows us to solve much of the problem analytically. In spite of its special symmetry, this example contains many general features of systems in which stress is concentrated near free boundaries and deforms them irreversibly. We depart from conventional treatments of such problems in two ways. First, the STZ analysis allows us to keep track of spatially heterogeneous, internal state variables such as the effective disorder temperature, which determines plastic response to subsequent loading. Second, we subject the system to stress pulses of finite duration, and therefore are able to observe elastoplastic response during both loading and unloading. We compute the final deformations and residual stresses produced by these stress pulses. Looking toward more general applications of these results, we examine the possibility of constructing a boundary-layer theory that might be useful in less symmetric situations.

The long-range elastic model, which is believed to describe the evolution of a self-affine rough crack front, is analyzed to linear and nonlinear orders. It is shown that the nonlinear terms, while important in changing the front dynamics, do not change the scaling exponent which characterizes the roughness of the front. The scaling exponent thus predicted by the model is much smaller than the one observed experimentally. The inevitable conclusion is that the gap between the results of experiments and the model that is supposed to describe them is too large and some new physics has to be invoked for another model.

The statistical mechanics of simple glass forming systems in two dimensions is worked out. The glass disorder is encoded via a Voronoi tesselation, and the statistical mechanics is performed directly in this encoding. The theory provides, without free parameters, an explanation of the glass transition phenomenology, including the identification of two different temperatures, T_g and T_c, the first associated with jamming and the second associated with crystallization at very low temperatures.

The stability of a rapid dynamic crack in a two-dimensional infinite strip is studied in the framework of linear elastic fracture mechanics supplemented with a modified principle of local symmetry. It is predicted that a single crack becomes unstable by a finite wavelength oscillatory mode at a velocity v_c, 0.8c_R<v_c<0.85c_R, where c_R is the Rayleigh wave speed. The relevance of this theoretical calculation to the oscillatory instability reported in the companion experimental Letter is discussed.

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 of other state variables that fall in and out of equilibrium with it. Our analysis points to a limitation of either the step-strain procedure used by DA in their simulations, or the STZ theory in its ability to describe rapid transients in stress-strain curves, or perhaps to both. Once we allow for this limitation, we are able to bring our theoretical predictions into accurate agreement with the simulations.

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 distribution of STZ transition thresholds. Although this theory contains no conventional thermal fluctuations, the concept of an effective temperature is essential for understanding how the STZ density is related to the state of disorder of the system.

We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence. We construct, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem. The statistics of the auxiliary linear model are dominated by “statistically preserved structures“ which are associated with exact conservation laws. The latter can be used, for example, to determine the value of the anomalous scaling exponent of the second order structure function. The approach is equally applicable to shell models and to the Navier-Stokes equations.

We address the role of the nature of material disorder in determining the roughness of cracks, which grow by damage nucleation and coalescence ahead of the crack tip. We highlight the role of quenched and annealed disorders in relation to the length scales d and ξ_c associated with the disorder and the damage nucleation, respectively. In two related models, one with quenched disorder in which d≃ξ_c, the other with annealed disorder in which d⪡ξ_c, we find qualitatively different roughening properties for the resulting cracks in two dimensions. The first model results in random cracks with an asymptotic roughening exponent ζ≈0.5. The second model shows correlated roughening with ζ≈0.66. The reasons for the qualitative difference are rationalized and explained.

Dynamic fracture in a wide class of materials reveals a “fracture energy” Γ much larger than the expected nominal surface energy due to the formation of two fresh surfaces. Moreover, the fracture energy depends on the crack velocity, Γ=Γ(v). We show that a simple dynamical theory of viscoplasticity coupled to asymptotic pure linear elasticity provides a possible explanation to the above phenomena. The theory predicts tip blunting characterized by a dynamically determined crack tip radius of curvature. In addition, we demonstrate velocity selection for cracks in fixed-grip strip geometry accompanied by the identification of Γ and its velocity dependence.

Drag reduction by polymers is bounded between two universal asymptotes, the von Kármán log law of the law and the maximum drag reduction (MDR) asymptote. It is theoretically understood why the MDR asymptote is universal, independent of whether the polymers are flexible or rodlike. The crossover behavior from the Newtonian von Kármán log law to the MDR is, however, not universal, showing different characteristics for flexible and rodlike polymers. In this paper we provide a theory for this crossover phenomenology.

Drag reduction by bubbles in stationary turbulent flows is sensitive to the compressibility of the bubbles. Without this dynamical effect the bubbles only renormalize the fluid density and viscosity, an effect that by itself can only lead to a small percentage of drag reduction. We show in this paper that the dynamics of bubbles and their effect on the compressibility of the mixture can lead to a much higher drag reduction.

Fracture paths in quasi-two-dimensional (2D) media (e.g., thin layers of materials or paper) are analyzed as self-affine graphs h(x) of height h as a function of length x. We show that these are multiscaling, in the sense that nth order moments of the height fluctuations across any distance ℓ scale with a characteristic exponent that depends nonlinearly on the order of the moment. Having demonstrated this, one rules out a widely held conjecture that fracture in 2D belongs to the universality class of directed polymers in random media. In fact, 2D fracture does not belong to any of the known kinetic roughening models. The presence of multiscaling offers a stringent test for any theoretical model; we show that a recently introduced model of quasistatic fracture passes this test.

We construct a simple analytic model for wall-bounded turbulence, containing only four adjustable parameters. Two of these parameters are responsible for the viscous dissipation of the components of the Reynolds stress tensor. The other two parameters control the nonlinear relaxation of these objects. The model offers an analytic description of the profiles of the mean velocity and the correlation functions of velocity fluctuations in the entire boundary region, from the viscous sublayer, through the buffer layer, and further into the log-law turbulent region. In particular, the model predicts a very simple distribution of the turbulent kinetic energy in the log-law region between the velocity components: the streamwise component contains a half of the total energy whereas the wall-normal and cross-stream components contain a quarter each. In addition, the model predicts a very simple relation between the von Kármán slope κ and the turbulent velocity in the log-law region v⁺ (in wall units): v⁺=6κ. These predictions are in excellent agreement with direct numerical simulation data and with recent laboratory experiments.

Structure functions of rough fracture surfaces in isotropic materials exhibit complicated scaling properties due to the broken isotropy in the fracture plane generated by a preferred propagation direction. Decomposing the structure functions into the even order irreducible representations of the SO(2) symmetry group indexed by (m=0,2,4,…) results in a lucid and quickly convergent description. The scaling exponent of the isotropic sector (m=0) dominates at small length scales. One can reconstruct the anisotropic structure functions using only the isotropic and the first nonvanishing anisotropic sector (m=2) [or at most the next one (m=4)]. The scaling exponent of the isotropic sector should be observed in a proposed, yet unperformed, experiment.

The apparent similarity of microbranching instabilities in different brittle materials gave rise to a widely held belief that many aspects of the postinstability physics were universal. We propose that the physics determining the typical length and time scales characterizing the postinstability patterns differ greatly from material to material. We offer a scaling theory connecting the pattern characteristics to material properties (like molecular weight) in brittle plastics like PMMA, and stress the fundamental differences with patterns in glass which are crucially influenced by three-dimensional dynamics. In both cases the present ab initio theoretical models are still too far from reality, disregarding some fundamental physics of the phenomena.