Shear-Transformation-Zone Theory of Yielding in Athermal Amorphous Materials

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, conclude that the yielding transition is not truly critical. In particular, there is a correlation length that grows rapidly, but ultimately saturates near the yield point.Comment: 7 pages, 5 figure

Thermal Effects in Dislocation Theory

The mechanical behaviors of polycrystalline solids are determined by the interplay between phenomena governed by two different thermodynamic temperatures: the configurational effective temperature that controls the density of dislocations, and the ordinary kinetic-vibrational temperature that controls activated depinning mechanisms and thus deformation rates. This paper contains a review of the effective-temperature theory and its relation to conventional dislocation theories. It includes a simple illustration of how these two thermal effects can combine to produce a predictive theory of spatial heterogeneities such as shear-banding instabilities. Its main message is a plea that conventional dislocation theories be reformulated in a thermodynamically consistent way so that the vast array of observed behaviors can be understood systematically.Comment: 8 pages, 5 figure

Statistical Thermodynamics of Strain Hardening in Polycrystalline Solids

This paper starts with a systematic rederivation of the statistical thermodynamic equations of motion for dislocation-mediated plasticity proposed in 2010 by Langer, Bouchbinder and Lookman. It then uses that theory to explain the anomalous rate-hardening behavior reported in 1988 by Follansbee and Kocks, and to explore the relation between hardening rate and grain size reported in 1995 by Meyers et al. A central theme is the need for physics-based, nonequilibrium analyses in developing predictive theories of the strength of polycrystalline materials.Comment: 8 pages, 4 figure

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 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 yield stress in amorphous materials. In order to put all these concepts together in a realistic context, the paper concludes 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.Comment: 15 pages, four figure

Local Geometric Invariants of Integrable Evolution Equations

The integrable hierarchy of commuting vector fields for the localized induction equation of 3D hydrodynamics, and its associated recursion operator, are used to generate families of integrable evolution equations which preserve local geometric invariants of the evolving curve or swept-out surface.Comment: 15 pages, AMSTeX file (to appear in Journal of Mathematical Physics

Unified derivation of phase-field models for alloy solidification from a grand-potential functional

In the literature, two quite different phase-field formulations for the problem of alloy solidification can be found. In the first, the material in the diffuse interfaces is assumed to be in an intermediate state between solid and liquid, with a unique local composition. In the second, the interface is seen as a mixture of two phases that each retain their macroscopic properties, and a separate concentration field for each phase is introduced. It is shown here that both types of models can be obtained by the standard variational procedure if a grand-potential functional is used as a starting point instead of a free-energy functional. The dynamical variable is then the chemical potential instead of the composition. In this framework, a complete analogy with phase-field models for the solidification of a pure substance can be established. This analogy is then exploited to formulate quantitative phase-field models for alloys with arbitrary phase diagrams. The precision of the method is illustrated by numerical simulations with varying interface thickness.Comment: 36 pages, 1 figur