Slip pulses at a sheared frictional viscoelastic/nondeformable interface

Fifty years of Schallamach waves: from rubber friction to nanoscale fracture

The question of how soft polymers slide against hard surfaces is of significant scientific interest, given its practical implications. Specifically, such systems commonly show interesting stick-slip dynamics, wherein the interface moves intermittently despite uniform remote loading. The year 2021 marked the 50th anniversary of the publication of a seminal paper by Adolf Schallamach (Wear, 1971), which first revealed an intimate link between stick-slip and moving detachment waves, now called Schallamach waves. We place Schallamach's results in a broader context and review subsequent investigations of stick-slip, before discussing recent observations of solitary Schallamach waves. This variant is not observable in standard contacts so that a special cylindrical contact must be used to quantify its properties. The latter configuration also reveals the occurrence of a dual wave-the so-called separation pulse-that propagates in a direction opposite to Schallamach waves. We show how the dual wave and other, more general, Schallamach-type waves can be described using continuum theory and provide pointers for future research. In the process, fundamental analogues of Schallamach-type waves emerge in nanoscale mechanics and interface fracture. The result is an ongoing application of lessons learnt from Schallamach-type waves to better understand these latter phenomena. This article is part of the theme issue 'Nanocracks in nature and industry'.

Propagating Schallamach-type waves resemble interface cracks

Intermittent motion, called stick-slip, is a friction instability that commonly occurs during relative sliding of two elastic solids. In adhesive polymer contacts, where elasticity and interface adhesion are strongly coupled, stick-slip arises due to the propagation of slow detachment waves at the interface. Here we analyze two distinct detachment waves moving parallel (Schallamach wave) and antiparallel (separation wave) to applied remote sliding. Both waves cause slip in the same direction, travel at speeds much lesser than any elastic wave speed, and are therefore describable using the same perturbative elastodynamic framework with identical boundary conditions. A numerical scheme is used to obtain interface stresses and velocities for arbitrary Poisson ratio, along with closed-form solutions for incompressible solids. Our calculations reveal a close correspondence between moving detachment waves and bimaterial interface cracks, including the nature of the singularity and the functional forms of the stresses. Based on this correspondence, and coupled with a fracture analogy for dynamic friction, we develop a phase diagram showing domains of possible occurrence of stick-slip via detachment waves vis-á-vis steady interface sliding. Our results have interesting implications for sliding and stick-slip phenomena at soft interfaces.

Slow wave propagation in soft adhesive interfaces

Stick-slip in sliding of soft adhesive surfaces has long been associated with the propagation of Schallamach waves, a type of slow surface wave. Recently it was demonstrated using in situ experiments that two other kinds of slow waves-separation pulses and slip pulses-also mediate stick-slip (Viswanathan et al., Soft Matter, 2016, 12, 5265-5275). While separation pulses, like Schallamach waves, involve local interface detachment, slip pulses are moving stress fronts with no detachment. Here, we present a theoretical analysis of the propagation of these three waves in a linear elastodynamics framework. Different boundary conditions apply depending on whether or not local interface detachment occurs. It is shown that the interface dynamics accompanying slow waves is governed by a system of integral equations. Closed-form analytical expressions are obtained for the interfacial pressure, shear stress, displacements and velocities. Separation pulses and Schallamach waves emerge naturally as wave solutions of the integral equations, with oppositely oriented directions of propagation. Wave propagation is found to be stable in the stress regime where linearized elasticity is a physically valid approximation. Interestingly, the analysis reveals that slow traveling wave solutions are not possible in a Coulomb friction framework for slip pulses. The theory provides a unified picture of stick-slip dynamics and slow wave propagation in adhesive contacts, consistent with experimental observations.

Nucleation and propagation of solitary Schallamach waves

We isolate single Schallamach waves--detachment fronts that mediate inhomogeneous sliding between an elastomer and a hard surface--to study their creation and dynamics. Based on measurements of surface displacement using high-speed in situ imaging, we establish a Burgers vector for the waves. The crystal dislocation analogs of nucleation stress, defect pinning, and configurational force are demonstrated. It is shown that many experimentally observed features can be quantitatively described using a conventional model of a dislocation line in an elastic medium. We also highlight the evolution of nucleation features, such as surface wrinkles, with consequences for interface delamination.

Slip dynamics at a patterned rubber/glass interface during stick-slip motions

We report on an experimental study of heterogeneous slip instabilities generated during stick-slip motions at a contact interface between a smooth rubber substrate and a patterned glass lens. Using a sol-gel process, the glass lens is patterned with a lattice of parallel ridges (wavelength, 1.6 μm, amplitude 0.35 μm). Friction experiments using this patterned surface result in the systematic occurrence of stick-slip motions over three orders of magnitude in the imposed driving velocity while stable friction is achieved with a smooth surface. Using a contact imaging method, real-time displacement fields are measured at the surface of the rubber substrate. Stick-slip motions are found to involve the localized propagation of transverse interface shear cracks whose velocity is observed to be remarkably independent on the driving velocity.

Fracture and friction: Stick-slip motion

We discuss the stick-slip motion of an elastic block sliding along a rigid substrate. We argue that for a given external shear stress this system shows a discontinuous nonequilibrium transition from a uniform stick state to uniform sliding at some critical stress which is nothing but the Griffith threshold for crack propagation. An inhomogeneous mode of sliding occurs when the driving velocity is prescribed instead of the external stress. A transition to homogeneous sliding occurs at a critical velocity, which is related to the critical stress. We solve the elastic problem for a steady-state motion of a periodic stick-slip pattern and derive equations of motion for the tip and resticking end of the slip pulses. In the slip regions we use the linear friction law and do not assume any intrinsic instabilities even at small sliding velocities. We find that, as in many other pattern forming system, the steady-state analysis itself does not select uniquely all the internal parameters of the pattern, especially the primary wavelength. Using some plausible analogy to first-order phase transitions we discuss a "soft" selection mechanism. This allows to estimate internal parameters such as crack velocities, primary wavelength and relative fraction of the slip phase as functions of the driving velocity. The relevance of our results to recent experiments is discussed.

Friction and fracture

Consider a block placed on a table and pushed sideways until it begins to slide. Amontons and Coulomb found that the force required to initiate sliding is proportional to the weight of the block (the constant of proportionality being the static coefficient of friction), but independent of the area of contact. This is commonly explained by asserting that, owing to the presence of asperities on the two surfaces, the actual area in physical contact is much smaller than it seems, and grows in proportion to the applied compressive force. Here we present an alternative picture of the static friction coefficient, which starts with an atomic description of surfaces in contact and then employs a multiscale analysis technique to describe how sliding occurs for large objects. We demonstrate the existence of self-healing cracks that have been postulated to solve geophysical paradoxes about heat generated by earthquakes, and we show that, when such cracks are present at the atomic scale, they result in solids that slip in accord with Coulomb's law of friction. We expect that this mechanism for friction will be found to operate at many length scales, and that our approach for connecting atomic and continuum descriptions will enable more realistic first-principles calculations of friction coefficients.

Jamming creep of a frictional interface

We measure the displacement response of a frictional multicontact interface between identical polymer glasses to a biased shear force oscillation. We evidence the existence, for maximum forces close below the nominal static threshold, of a jamming creep regime governed by an aging-rejuvenation competition acting within the micrometer-sized contacting asperities. The time dependence of the creep process deviates from the standard Rice-Ruina [J. R. Rice and A. L. Ruina, J. Appl. Mech. 50, 343 (1983)] phenomenology at early times; this suggests the possibility of an aging-rejuvenation competition at much smaller scales, within the nanometer-thick adhesive junctions.