J. S. Langer

Dynamics of viscoplastic deformation in amorphous solids

We propose a dynamical theory of low-temperature shear deformation in amorphous solids. Our analysis is based on molecular-dynamics simulations of a two-dimensional, two-component noncrystalline system. These numerical simulations reveal behavior typical of metallic glasses and other viscoplastic materials, specifically, reversible elastic deformation at small applied stresses, irreversible plastic deformation at larger stresses, a stress threshold above which unbounded plastic flow occurs, and a strong dependence of the state of the system on the history of past deformations. Microscopic observations suggest that a dynamically complete description of the macroscopic state of this deforming body requires specifying, in addition to stress and strain, certain average features of a population of two-state shear transformation zones. Our introduction of these state variables into the constitutive equations for this system is an extension of earlier models of creep in metallic glasses. In the treatment presented here, we specialize to temperatures far below the glass transition and postulate that irreversible motions are governed by local entropic fluctuations in the volumes of the transformation zones. In most respects, our theory is in good quantitative agreement with the rich variety of phenomena seen in the simulations. {copyright} {ital 1998} {ital The American Physical Society}

Statistical theory of the decay of metastable states

A procedure is outlined for the first-principles calculation of the rate of decay of a metastable phase. The model on which the calculation is based is sufficiently general to describe a wide variety of thermally activated nucleation and growth processes, hopefully even including decay of superflow in liquid helium. The theory provides a definite prescription for calculating both the activation free-energies and the fundamental fluctuation rates which appear in the conventional formulae describing such processes.

Theory of the condensation point

Abstract This paper is a report of some studies leading to a new mathematical description of the condensation point for a simple class of models of first-order phase transitions. The paper consists of three main parts. In the first part it is pointed out that, although the conventional droplet model of condensation predicts that the free energy has an essential singularity at the condensation point, this singularity is so weak as to be experimentally unobservable. Furthermore the analytic continuation of the free energy beyond the singularity describes a metastable phase according to the assumptions of the model. The second part of the paper is devoted to the study of a soluble functional integral that exhibits an essential singularity similar to that found in the droplet model. A method is developed for computing the singular properties of such integrals in cases where it is not possible to evaluate the integrals exactly. In the third part of the paper this method is applied to a simple model of a ferromagnet at temperatures well below the Curie point. Most of the really characteristic features of the droplet model are recovered in this calculation. The detailed results have a bearing on problems of phase coexistence, surface energies, and possibly even condensation rates.

Theory of spinodal decomposition in alloys

Abstract A statistical theory of the thermally driven composition fluctuations in a binary alloy is developed for the purpose of studying the phenomenon of spinodal decomposition. The theory can be stated in the form of a Fokker-Planck equation, which reduces, upon taking a suitable moment, to the nonlinear generalized diffusion equation which has been the basis of recent work in this field. Using the full Fokker-Planck equation, it is possible to compute the lifetime of the stationary solutions of the diffusion equation, and thus to study the rate at which the structure of the alloy coarsens during aging.

Deformation and Failure of Amorphous, Solidlike Materials

Since the 1970s, theories of deformation and failure of amorphous, solidlike materials have started with models in which stress-driven, molecular rearrangements occur at localized flow defects via shear transformations. This picture is the basis for the modern theory of shear transformation zones (STZs), which is the focus of this review. We begin by describing the structure of the theory in general terms and by showing several applications, specifically, interpretation of stress-strain measurements for a bulk metallic glass, analysis of numerical simulations of shear banding, and the use of the STZ equations of motion in free-boundary calculations. In the second half of this review, we focus for simplicity on what we call an athermal model of amorphous plasticity, and use that model to illustrate how the STZ theory emerges within a systematic formulation of nonequilibrium thermodynamics.

Pattern propagation in nonlinear dissipative systems

Abstract We discuss the problem of pattern selection in situations where a stable, nonuniform state of a nonlinear dissipative system propagates into an initially unstable, homogeneous region. Our strategy is to consider this process as a generalization of front propagation in a nonlinear diffusion problem for which rigorous results are known; and we point out that these known properties are consistent with a marginal-stability hypothesis that has been suggested in the theory of dendritic crystal growth. We then describe a more general interpretation of the marginal-stability hypothesis and, finally, present numerical evidence for its validity from three different pattern-forming models.

Patterns of seismic activity preceding large earthquakes

We analyze the patterns of seismic activity which precede large events in a mechanical model of a fault. The model generates a statistical distribution of events similar to that observed for a single fault, with a scaling region consistent with the Gutenberg-Richter law at small and moderate magnitudes, and an excess of events at large magnitudes. We find only slight variation in the scaling behavior during a loading cycle. However, we do observe systematic variations in space and time of the overall rate of activity. In the model, the activity accelerates dramatically preceding a large event and is usually a maximum hi the neighborhood of the future epicenter. These results are compared to California seismicity data, where we find that activity patterns vary regionally. Looking at patterns of activity in the San Francisco Bay Area since 1948, we find an increase of activity on the Calaveras fault near San Jose beginning in the 1980s which, if our model is relevant, would forecast a large earthquake in that region. The 1989 Loma Prieta earthquake occurred on the San Andreas fault within 30 km of the section of the Calaveras fault showing increased activity.

Shear-transformation-zone theory of plastic deformation near the glass transition

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 by 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 by-product, a fundamental reinterpretation of the dynamic yield stress in amorphous materials. In order to put all these concepts together in a realistic context, I conclude with a reexamination of the experimentally observed rheological behavior of a bulk metallic glass. That reexamination serves as a test of the STZ dynamics, confirming that system parameters obtained from steady-state properties such as the viscosity can be used to predict transient behaviors.

Strain localization in a shear transformation zone model for amorphous solids

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 [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, but unstable systems do not always develop shear bands. Nonlinear energy dissipation processes interact with perturbation growth to determine whether a material exhibits strain localization. By estimating the effects of these interactions, we derive a criterion that determines which materials exhibit shear bands based on the initial conditions alone. We also show that the shear band width is not set by an inherent diffusion length scale but instead by a dynamical scale that depends on the imposed strain rate.

Dynamics of shear-transformation zones in amorphous plasticity: Formulation in terms of an effective disorder temperature.

This investigation extends earlier studies of a shear-transformation-zone (STZ) theory of plastic deformation in amorphous solids. The main purpose 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 STZs equilibrates with the effective temperature allows the theory to be compared directly with experimentally measured properties of metallic glasses, including their calorimetric behavior. The coupling between the effective temperature and mechanical deformation suggests an explanation of shear-banding instabilities.