Amplitude equations for polycrystalline materials with interaction between composition and stress

We investigate the ability of frame-invariant amplitude equations [G. H. Gunaratne, Q. Ouyang, and H. Swinney, Phys. Rev. E {\bf 50}, 2802 (1994)] to describe quantitatively the evolution of polycrystalline microstructures and we extend this approach to include the interaction between composition and stress. Validations for elemental materials include studies of the Asaro-Tiller-Grinfeld morphological instability of a stressed crystal surface, polycrystalline growth from the melt, grain boundary energies over a wide range of misorientation, and grain boundary motion coupled to shear deformation. Amplitude equations with accelerated strain relaxation in the solid are shown to model accurately the Asaro-Tiller-Grinfeld instability. Polycrystalline growth is also well described. However, the survey of grain boundary energies shows that the approach is only valid for a restricted range of misorientations as a direct consequence of an amplitude expansion. This range covers approximately half the complete range allowed by crystal symmetry for some fixed reference set of density waves used in the expansion. Over this range, coupled motion to shear is well described by known geometrical rules and a transition from coupling to sliding motion is also reproduced. Amplitude equations for alloys are derived phenomenologically in a Ginzburg-Landau spirit. Vegard's law is shown to be naturally described by seeking a gauge invariant form of those equations under a transformation that corresponds to a lattice expansion and deviations from Vegard's law can be easily incorporated. Those equations realistically describe the dilute alloy limit and have the same flexibility as conventional phase-field models for incorporating arbitrary free-energy/composition curves...Comment: 28 page

Effect of shear-coupled grain boundary motion on coherent precipitation

We examine the interaction between precipitates and grain boundaries, which undergo shear-coupled motion. The elastic problem, emerging from grain boundary perturbations and an elastic mismatch strain induced by the precipitates, is analysed. The resulting free elastic energy contains interaction terms, which are derived numerically via the integration of the elastic energy density. The interaction of the shear-coupled grain boundary and the coherent precipitates leads to potential elastic energy reductions. Such a decrease of the elastic energy has implications on the grain boundary shape and also on the solubility limit near the grain boundary. By energy minimisation we are able to derive the grain boundary shape change analytically. We apply the results to the Fe-C system to predict the solubility limit change of cementite near an $\alpha$-iron grain boundary.Comment: 8 page

On the velocity-strengthening behavior of dry friction

The onset of frictional instabilities, e.g. earthquakes nucleation, is intimately related to velocity-weakening friction, in which the frictional resistance of interfaces decreases with increasing slip velocity. While this frictional response has been studied extensively, less attention has been given to steady-state velocity-strengthening friction, in spite of its potential importance for various aspects of frictional phenomena such as the propagation speed of interfacial rupture fronts and the amount of stored energy released by them. In this note we suggest that a crossover from steady-state velocity-weakening friction at small slip velocities to steady-state velocity-strengthening friction at higher velocities might be a generic feature of dry friction. We further argue that while thermally activated rheology naturally gives rise to logarithmic steady-state velocity-strengthening friction, a crossover to stronger-than-logarithmic strengthening might take place at higher slip velocities, possibly accompanied by a change in the dominant dissipation mechanism. We sketch a few physical mechanisms that may account for the crossover to stronger-than-logarithmic steady-state velocity-strengthening and compile a rather extensive set of experimental data available in the literature, lending support to these ideas.Comment: Updated to published version: 2 Figures and a section adde

Velocity-strengthening friction significantly affects interfacial dynamics, strength and dissipation

Frictional interfaces are abundant in natural and manmade systems and their dynamics still pose challenges of fundamental and technological importance. A recent extensive compilation of multiple-source experimental data has revealed that velocity-strengthening friction, where the steady-state frictional resistance increases with sliding velocity over some range, is a generic feature of such interfaces. Moreover, velocity-strengthening friction has very recently been linked to slow laboratory earthquakes and stick-slip motion. Here we elucidate the importance of velocity-strengthening friction by theoretically studying three variants of a realistic rate-and-state friction model. All variants feature identical logarithmic velocity-weakening friction at small sliding velocities, but differ in their higher velocity behaviors. By quantifying energy partition (e.g. radiation and dissipation), the selection of interfacial rupture fronts and rupture arrest, we show that the presence or absence of velocity-strengthening friction can significantly affect the global interfacial resistance and the total energy released during frictional instabilities ("event magnitude"). Furthermore, we show that different forms of velocity-strengthening friction (e.g. logarithmic vs. linear) may result in events of similar magnitude, yet with dramatically different dissipation and radiation rates. This happens because the events are mediated by interfacial rupture fronts with vastly different propagation velocities, where stronger velocity-strengthening friction promotes slower rupture. These theoretical results may have significant implications on our understanding of frictional dynamics.Comment: 9 pages, 6 figure

Modeling of grain boundary dynamics using amplitude equations

We discuss the modelling of grain boundary dynamics within an amplitude equations description, which is derived from classical density functional theory or the phase field crystal model. The relation between the conditions for periodicity of the system and coincidence site lattices at grain boundaries is investigated. Within the amplitude equations framework we recover predictions of the geometrical model by Cahn and Taylor for coupled grain boundary motion, and find both $\langle100\rangle$ and $\langle110\rangle$ coupling. No spontaneous transition between these modes occurs due to restrictions related to the rotational invariance of the amplitude equations. Grain rotation due to coupled motion is also in agreement with theoretical predictions. Whereas linear elasticity is correctly captured by the amplitude equations model, open questions remain for the case of nonlinear deformations.Comment: 21 pages. We extended the discussion on the geometrical nonlinearities in Section

Instabilities at Frictional Interfaces: Creep Patches, Nucleation and Rupture Fronts

The strength and stability of frictional interfaces, ranging from tribological systems to earthquake faults, are intimately related to the underlying spatially-extended dynamics. Here we provide a comprehensive theoretical account, both analytic and numeric, of spatiotemporal interfacial dynamics in a realistic rate-and-state friction model, featuring both velocity-weakening and strengthening behaviors. Slowly extending, loading-rate dependent, creep patches undergo a linear instability at a critical nucleation size, which is nearly independent of interfacial history, initial stress conditions and velocity-strengthening friction. Nonlinear propagating rupture fronts -- the outcome of instability -- depend sensitively on the stress state and velocity-strengthening friction. Rupture fronts span a wide range of propagation velocities and are related to steady state fronts solutions.Comment: Typos and figures corrected. Supplementary information at: http://www.weizmann.ac.il/chemphys/bouchbinder/frictional_instabilities.htm

Effective Elastic Moduli in Solids with High Crack Density

We investigate the weakening of elastic materials through randomly distributed circles and cracks numerically and compare the results to predictions from homogenization theories. We find a good agreement for the case of randomly oriented cracks of equal length in an isotropic plane-strain medium for lower crack densities; for higher densities the material is weaker than predicted due to precursors of percolation. For a parallel alignment of cracks, where percolation does not occur, we analytically predict a power law decay of the effective elastic constants for high crack densities, and confirm this result numerically.Comment: 8 page