Yohai Bar-Sinai

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.

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 velocity-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-front solutions.

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

Frictional interfaces abound in natural and man-made systems, yet their dynamics are not well-understood. Recent extensive experimental data have 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. This physical behavior has very recently been linked to slow stick-slip motion. Here we elucidate the importance of velocity-strengthening friction by theoretically studying three variants of a realistic friction model, all featuring 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 strengthening significantly affects the global interfacial resistance and the energy release during frictional instabilities. Furthermore, we show that different forms of strengthening may result in events of similar magnitude, yet with dramatically different dissipation and radiation rates. This happens because the events are mediated by 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.

Dynamic instabilities of frictional sliding at a bimaterial interface

Abstract Understanding the dynamic stability of bodies in frictional contact steadily sliding one over the other is of basic interest in various disciplines such as physics, solid mechanics, materials science and geophysics. Here we report on a two-dimensional linear stability analysis of a deformable solid of a finite height H, steadily sliding on top of a rigid solid within a generic rate-and-state friction type constitutive framework, fully accounting for elastodynamic effects. We derive the linear stability spectrum, quantifying the interplay between stabilization related to the frictional constitutive law and destabilization related both to the elastodynamic bi-material coupling between normal stress variations and interfacial slip, and to finite size effects. The stabilizing effects related to the frictional constitutive law include velocity-strengthening friction (i.e. an increase in frictional resistance with increasing slip velocity, both instantaneous and under steady-state conditions) and a regularized response to normal stress variations. We first consider the small wave-number k limit and demonstrate that homogeneous sliding in this case is universally unstable, independent of the details of the friction law. This universal instability is mediated by propagating waveguide-like modes, whose fastest growing mode is characterized by a wave-number satisfying kH ∼ O ( 1 ) and by a growth rate that scales with H−1. We then consider the limit kH → ∞ and derive the stability phase diagram in this case. We show that the dominant instability mode travels at nearly the dilatational wave-speed in the opposite direction to the sliding direction. In a certain parameter range this instability is manifested through unstable modes at all wave-numbers, yet the frictional response is shown to be mathematically well-posed. Instability modes which travel at nearly the shear wave-speed in the sliding direction also exist in some range of physical parameters. Previous results obtained in the quasi-static regime appear relevant only within a narrow region of the parameter space. Finally, we show that a finite-time regularized response to normal stress variations, within the framework of generalized rate-and-state friction models, tends to promote stability. The relevance of our results to the rupture of bi-material interfaces is briefly discussed.

Frictional Sliding without Geometrical Reflection Symmetry

The dynamics of frictional interfaces play an important role in many physical systems spanning a broad range of scales. It is well-known that frictional interfaces separating two dissimilar materials couple interfacial slip and normal stress variations, a coupling that has major implications on their stability, failure mechanism and rupture directionality. In contrast, interfaces separating identical materials are traditionally assumed not to feature such a coupling due to symmetry considerations. We show, combining theory and experiments, that interfaces which separate bodies made of macroscopically identical materials, but lack geometrical reflection symmetry, generically feature such a coupling. We discuss two applications of this novel feature. First, we show that it accounts for a distinct, and previously unexplained, experimentally observed weakening effect in frictional cracks. Second, we demonstrate that it can destabilize frictional sliding which is otherwise stable. The emerging framework is expected to find applications in a broad range of systems.

Spatial distribution of thermal energy in equilibrium.

The equipartition theorem states that in equilibrium, thermal energy is equally distributed among uncoupled degrees of freedom that appear quadratically in the systems Hamiltonian. However, for spatially coupled degrees of freedom, such as interacting particles, one may speculate that the spatial distribution of thermal energy may differ from the value predicted by equipartition, possibly quite substantially in strongly inhomogeneous or disordered systems. Here we show that for systems undergoing simple Gaussian fluctuations around an equilibrium state, the spatial distribution is universally bounded from above by 1/2k(B)T. We further show that in one-dimensional systems with short-range interactions, the thermal energy is equally partitioned even for coupled degrees of freedom in the thermodynamic limit and that in higher dimensions nontrivial spatial distributions emerge. Some implications are discussed.

Gaussian fluctuations of spatially inhomogeneous polymers

Inhomogeneous polymers, such as partially cofilin-bound actin filaments, play an important role in various natural and biotechnological systems. At finite temperatures, inhomogeneous polymers exhibit non-trivial thermal fluctuations. More broadly, these are relatively simple examples of fluctuations in spatially inhomogeneous systems, which are less understood compared to their homogeneous counterparts. Here we develop a statistical theory of torsional, extensional and bending Gaussian fluctuations of inhomogeneous polymers (chains), where the inhomogeneity is an inclusion of variable size and stiffness, using both continuum and discrete approaches. First, we analytically calculate the complete eigenvalue and eigenmode spectra within a continuum field theory. In particular, we show that the wavenumber inside and outside of the inclusion is nearly linear in the eigenvalue index, with a nontrivial coefficient. Second, we solve the corresponding discrete problem and highlight fundamental differences between the continuum and discrete spectra. In particular, we demonstrate that above a certain wavenumber the discrete spectrum changes qualitatively and discrete evanescent eigenmodes, which do not have continuum counterparts, emerge. The implications of these differences are explored by calculating fluctuation-induced forces associated with free-energy variations with either the inclusion properties (e.g. inhomogeneity formed by adsorbing molecules) or with an external geometric constraint. The former, which is the fluctuation-induced contribution to the adsorbing molecule binding force, is shown to be affected by short wavelengths and thus cannot be calculated using the continuum approach. The latter, on the other hand, is shown to be dominated by long wavelength shape fluctuations and hence is properly described by the continuum theory.