Creep, stick-slip, and dry-friction dynamics: Experiments and a heuristic model

Statistical laws of stick-slip friction at mesoscale

Friction between two rough solid surfaces often involves local stick-slip events occurring at different locations of the contact interface. If the apparent contact area is large, multiple local slips may take place simultaneously and the total frictional force is a sum of the pinning forces imposed by many asperities on the interface. Here, we report a systematic study of stick-slip friction over a mesoscale contact area using a hanging-beam lateral atomic-force-microscope, which is capable of resolving frictional force fluctuations generated by individual slip events and measuring their statistical properties at the single-slip resolution. The measured probability density functions (PDFs) of the slip length δxs, the maximal force Fc needed to trigger the local slips, and the local force gradient [Formula: see text] of the asperity-induced pinning force field provide a comprehensive statistical description of stick-slip friction that is often associated with the avalanche dynamics at a critical state. In particular, the measured PDF of δxs obeys a power law distribution and the power-law exponent is explained by a new theoretical model for the under-damped spring-block motion under a Brownian-correlated pinning force field. This model provides a long-sought physical mechanism for the avalanche dynamics in stick-slip friction at mesoscale.

Transition between the stick and slip states in a simplified model of magnetic friction

We introduce a simplified model of magnetic friction and investigate its behavior using both numerical and analytical methods. When resistance coefficient γ is large, the movement of the system obeys the thermally activated process. In contrast, when γ is sufficiently small, the slip and stick states behave as separate metastable states, and the lattice velocity depends on the probability that the slip state appears. We evaluate the velocities in both cases using several approximations and compare the results with those of numerical simulations.

Predicting frictional aging from bulk relaxation measurements

The coefficient of static friction between solids normally increases with the time they have remained in static contact before the measurement. This phenomenon, known as frictional aging, is at the origin of the difference between static and dynamic friction coefficients but has remained difficult to understand. It is usually attributed to a slow expansion of the area of atomic contact as the interface changes under pressure. This is however challenging to quantify as surfaces have roughness at all length scales. In addition, friction is not always proportional to the contact area. Here we show that the normalized stress relaxation of the surface asperities during frictional contact with a hard substrate is the same as that of the bulk material, regardless of the asperities' size or degree of compression. This result enables us to predict the frictional aging of rough interfaces based on the bulk material properties of two typical polymers: polypropylene and polytetrafluoroethylene.

Scaling theory for the statistics of slip at frictional interfaces

Slip at a frictional interface occurs via intermittent events. Understanding how these events are nucleated, can propagate, or stop spontaneously remains a challenge, central to earthquake science and tribology. In the absence of disorder, rate-and-state approaches predict a diverging nucleation length at some stress σ^{*}, beyond which cracks can propagate. Here we argue for a flat interface that disorder is a relevant perturbation to this description. We justify why the distribution of slip contains two parts: a power law corresponding to "avalanches" and a "narrow" distribution of system-spanning "fracture" events. We derive novel scaling relations for avalanches, including a relation between the stress drop and the spatial extension of a slip event. We compute the cut-off length beyond which avalanches cannot be stopped by disorder, leading to a system-spanning fracture, and successfully test these predictions in a minimal model of frictional interfaces.

Optimizing Earthquake Nowcasting With Machine Learning: The Role of Strain Hardening in the Earthquake Cycle

Nowcasting is a term originating from economics, finance, and meteorology. It refers to the process of determining the uncertain state of the economy, markets or the weather at the current time by indirect means. In this paper, we describe a simple two-parameter data analysis that reveals hidden order in otherwise seemingly chaotic earthquake seismicity. One of these parameters relates to a mechanism of seismic quiescence arising from the physics of strain-hardening of the crust prior to major events. We observe an earthquake cycle associated with major earthquakes in California, similar to what has long been postulated. An estimate of the earthquake hazard revealed by this state variable time series can be optimized by the use of machine learning in the form of the Receiver Operating Characteristic skill score. The ROC skill is used here as a loss function in a supervised learning mode. Our analysis is conducted in the region of 5° × 5° in latitude-longitude centered on Los Angeles, a region which we used in previous papers to build similar time series using more involved methods (Rundle & Donnellan, 2020, https://doi.org/10.1029/2020EA001097; Rundle, Donnellan et al., 2021, https://doi.org/10.1029/2021EA001757; Rundle, Stein et al., 2021, https://doi.org/10.1088/1361-6633/abf893). Here we show that not only does the state variable time series have forecast skill, the associated spatial probability densities have skill as well. In addition, use of the standard ROC and Precision (PPV) metrics allow probabilities of current earthquake hazard to be defined in a simple, straightforward, and rigorous way.

Beyond quality and quantity: Spatial distribution of contact encodes frictional strength

Classically, the quantity of contact area A_{R} between two bodies is considered a proxy for the force of friction. However, bond density across the interface-quality of contact-is also relevant, and contemporary debate often centers around the relative importance of these two factors. In this work, we demonstrate that a third factor, often overlooked, plays a significant role in static frictional strength: The spatial distribution of contact. We perform static friction measurements, μ, on three pairs of solid blocks while imaging the contact plane. By using linear regression on hundreds of image-μ pairs, we are able to predict future friction measurements with three to seven times better accuracy than existing benchmarks, including total quantity of contact area. Our model has no access to quality of contact, and we therefore conclude that a large portion of the interfacial state is encoded in the spatial distribution of contact, rather than its quality or quantity.

The evolution of rock friction is more sensitive to slip than elapsed time, even at near-zero slip rates

Nearly all frictional interfaces strengthen as the logarithm of time when sliding at ultra-low speeds. Observations of also logarithmic-in-time growth of interfacial contact area under such conditions have led to constitutive models that assume that this frictional strengthening results from purely time-dependent, and slip-insensitive, contact-area growth. The main laboratory support for such strengthening has traditionally been derived from increases in friction during "load-point hold" experiments, wherein a sliding interface is allowed to gradually self-relax down to subnanometric slip rates. In contrast, following step decreases in the shear loading rate, friction is widely reported to increase over a characteristic slip scale, independent of the magnitude of the slip-rate decrease-a signature of slip-dependent strengthening. To investigate this apparent contradiction, we subjected granite samples to a series of step decreases in shear rate of up to 3.5 orders of magnitude and load-point holds of up to 10,000 s, such that both protocols accessed the phenomenological regime traditionally inferred to demonstrate time-dependent frictional strengthening. When modeling the resultant data, which probe interfacial slip rates ranging from 3 .[Formula: see text]. to less than [Formula: see text], we found that constitutive models where low slip-rate friction evolution mimics log-time contact-area growth require parameters that differ by orders of magnitude across the different experiments. In contrast, an alternative constitutive model, in which friction evolves only with interfacial slip, fits most of the data well with nearly identical parameters. This leads to the surprising conclusion that frictional strengthening is dominantly slip-dependent, even at subnanometric slip rates.

Shear adhesive strength between epoxy resin and copper surfaces: a density functional theory study

Shear adhesive strengths of epoxy resin for copper and copper oxide surfaces are estimated based on quantum chemical calculations. Shear adhesion has periodicity, and its origin is revealed.

Memory effects in friction: the role of sliding heterogeneities

We report on memory effects involved in the unsteady-state frictional response of a contact interface between a silicone rubber and a spherical glass probe when it is perturbed by changes in the orientation of the driving motion or by velocity steps. From measurements of the displacement fields at the interface, we show that observed memory effects can be accounted for by the non-uniform distribution of the sliding velocity within the contact interface. As a consequence of these memory effects, the friction force may no longer be aligned with respect to the sliding trajectory. In addition, stick–slip motions with a purely geometrical origin are also evidenced. These observations are adequately accounted for by a friction model that takes into account heterogeneous displacements within the contact area. When a velocity dependence of the frictional stress is incorporated in this model, unsteady-state regimes induced by velocity steps are also adequately described. The good agreement between the model and experiments outlines the role of space heterogeneities in memory effects involved in soft matter friction.

Stress propagation in locally loaded packings of disks and pentagons

The mechanical strength and flow of granular materials can depend strongly on the shapes of individual grains. We report quantitative results obtained from photoelasticimetry experiments on locally loaded, quasi-two-dimensional granular packings of either disks or pentagons exhibiting stick-slip dynamics. Packings of pentagons resist the intruder at significantly lower packing fractions than packings of disks, transmitting stresses from the intruder to the boundaries over a smaller spatial extent. Moreover, packings of pentagons feature significantly fewer back-bending force chains than packings of disks. Data obtained on the forward spatial extent of stresses and back-bending force chains collapse when the packing fraction is rescaled according to the packing fraction of steady state open channel formation, though data on intruder forces and dynamics do not collapse. We comment on the influence of system size on these findings and highlight connections with the dynamics of the disks and pentagons during slip events.

Relaxation process of magnetic friction under sudden changes in velocity

Although there have been many studies of statistical mechanical models of magnetic friction, most of these have focused on the behavior in the steady state. In this study, we prepare a system composed of a chain and a lattice of Ising spins that interact with each other, and we investigate the relaxation of the system when the relative velocity v changes suddenly. The situation where v is given is realized by attaching the chain to a spring, the other end of which moves with a constant velocity v. Numerical simulation finds that, when the spring constant has a moderate value, the relaxation of the frictional force is divided into two processes, which are a sudden change and a slow relaxation. This behavior is also observed on regular solid surfaces, although caused by different factors than our model. More specifically, the slow relaxation process is caused by relaxation of the magnetic structure in our model but is caused by creep deformation in regular solid surfaces.

Model of magnetic friction with infinite-range interaction

We investigate a model of magnetic friction with the infinite-range interaction by mean-field analysis and a numerical simulation, and compare its behavior with that of the short-range model that we considered previously [Komatsu, Phys. Rev. E 100, 052130 (2019)2470-004510.1103/PhysRevE.100.052130]. This infinite-range model always obeys the Stokes law when the temperature is higher than the critical value, T_{c}, whereas it shows a crossover or transition from the Dieterich-Ruina law to the Stokes law when the temperature is lower than T_{c}. Considering that the short-range model in our previous study shows a crossover or transition irrespective of whether the temperature is above or below the equilibrium transition temperature, the behavior in the high-temperature state is the major difference between these two models.

Millisecond dynamics of deformation bands during discontinuous creep in an AlMg polycrystal

Formation of bands of macroscopic strain localization during staircase creep in an AlMg polycrystal is studied by the acoustic emission (AE) technique and high-speed video recording with an image acquisition rate up to 50 000 frames per second. The simultaneous measurements by two methods allow us to distinguish different types of embryo deformation bands and concomitant AE signals, and to establish correlations between the band evolution and the acoustic response. It is found that the fastest stages of band formation, associated with its emergence to the surface and subsequent accelerated expansion, generate complex AE bursts in the frequency band ∼0.05-1 MHz. The correlations hidden in the complex structure of an individual acoustic burst are investigated by methods of statistical and fractal analysis. On the other hand, relationships between average parameters of various physical responses to discontinuous creep are assessed. Particularly, a close correspondence is found between the envelope of the acoustic burst and the rate of stress change during formation of a single deformation band. Evolution of dynamical behavior of embryo bands with increasing creep stress is discussed. Notably, a qualitative change in the AE waveform observed on approaching the ultimate stress is considered from the viewpoint of anticipation of the oncoming fracture.

Length Scale Effect in Frictional Aging of Silica Contacts

Friction between two solid surfaces often exhibits strong rate and slip-history dependence, which critically determines the dynamic stability of frictional sliding. Empirically, such an evolutional effect has been captured by the rate-and-state friction (RSF) law based on laboratory-scale experiments; but its applicability for generic sliding interfaces under different length scales remains unclear. In this Letter, frictional aging, the key manifestation of the evolutional behavior, of silica-silica contacts is studied via slide-hold-slide tests with apparent contact size spanning across 3 orders of magnitude. The experimental results demonstrate a clear and strong length scale dependency in frictional aging characteristics. Assisted by a multiasperity RSF model, we attribute the length scale effect to roughness-dependent true contact area evolution as well as scale-dependent friction stress due to nonconcurrent slip.

Shear Controls Frictional Aging by Erasing Memory

We simultaneously measure the static friction and the real area of contact between two solid bodies. These quantities are traditionally considered equivalent, and under static conditions both increase logarithmically in time, a phenomenon coined aging. Here we show that the frictional aging rate is determined by the combination of the aging rate of the real area of contact and two memory-erasure effects that occur when shear is changed (e.g., to measure static friction.) The application of a static shear load accelerates frictional aging while the aging rate of the real area of contact is unaffected. Moreover, a negative static shear-pulling instead of pushing-slows frictional aging, but similarly does not affect the aging of contacts. The origin of this shear effect on aging is geometrical. When shear load is increased, minute relative tilts between the two blocks prematurely erase interfacial memory prior to sliding, negating the effect of aging. Modifying the loading point of the interface eliminates these tilts and as a result frictional aging rate becomes insensitive to shear. We also identify a secondary memory-erasure effect that remains even when all tilts are eliminated and show that this effect can be leveraged to accelerate aging by cycling between two static shear loads.

Ageing of Polymer Frictional Interfaces: The Role of Quantity and Quality of Contact

When two objects are in contact, the force necessary for one to start sliding over the other is larger than the force necessary to keep the sliding motion going. This difference between static and dynamic friction is thought to result from a reduction in the area of real contact upon the onset of slip. Here, we resolve the structure in the area of contact on the molecular scale by means of environment-sensitive molecular rotors using (super-resolution) fluorescence microscopy and fluorescence lifetime imaging. We demonstrate that the macroscopic friction force is not only controlled by the area of real contact but also controlled by the "quality" of that area of real contact, which determines the friction per unit contact area. We show that the latter is affected by the local density of the contacting surfaces, a parameter that can be expected to change in time at any interface that involves glassy, amorphous materials.

Linear Aging Behavior at Short Timescales in Nanoscale Contacts

Nanoscale silica-silica contacts were recently found to exhibit logarithmic aging for times ranging from 0.1 to 100 s, consistent with the macroscopic rate and state friction laws and several other aging processes. Nanoscale aging in this system is attributed to progressive formation of interfacial siloxane bonds between surface silanol groups. However, understanding or even data for contact behavior for aging times <0.1  s, before the onset of logarithmic aging, is limited. Using a combination of atomic force microscopy experiments and kinetic Monte Carlo simulations, we find that aging is nearly linear with aging time at short timescales between ∼ 5 and 90 ms. We demonstrate that aging at these timescales requires the existence of a particular range of reaction energy barriers for interfacial bonding. Specifically, linear aging behavior consistent with experiments requires a narrow peak close to the upper bound of this range of barriers. These new insights into the reaction kinetics of interfacial bonding in nanoscale aging advance the development of physically based rate and state friction laws for nanoscale contacts.

Model of magnetic friction obeying the Dieterich-Ruina law in the steady state

We propose a model of magnetic friction and investigate the relation between the frictional force and the relative velocity of surfaces in the steady state. The model comprises two square lattices adjacent to each other, the upper of which is subjected to an external force, and the magnetic interaction acts as a kind of potential barrier that prevents the upper lattice from moving. We consider two surface types for the upper lattice: smooth and rough. The behavior of this model is classified into two domains, which we refer to as domains I and II. In domain II, the external force is dominant compared with other forces, whereas in domain I, the velocity of the lattice is suppressed by the magnetic interaction and obeys the Dieterich-Ruina law. This characteristic property can be observed regardless of whether the surface is smooth or rough.

How collective asperity detachments nucleate slip at frictional interfaces

Sliding at a quasi-statically loaded frictional interface can occur via macroscopic slip events, which nucleate locally before propagating as rupture fronts very similar to fracture. We introduce a microscopic model of a frictional interface that includes asperity-level disorder, elastic interaction between local slip events, and inertia. For a perfectly flat and homogeneously loaded interface, we find that slip is nucleated by avalanches of asperity detachments of extension larger than a critical radius [Formula: see text] governed by a Griffith criterion. We find that after slip, the density of asperities at a local distance to yielding [Formula: see text] presents a pseudogap [Formula: see text], where θ is a nonuniversal exponent that depends on the statistics of the disorder. This result makes a link between friction and the plasticity of amorphous materials where a pseudogap is also present. For friction, we find that a consequence is that stick-slip is an extremely slowly decaying finite-size effect, while the slip nucleation radius [Formula: see text] diverges as a θ-dependent power law of the system size. We discuss how these predictions can be tested experimentally.

Integrated QCM-Microtribometry: Friction of Single-Crystal MoS2 and Gold from μm/s to m/s

Two opposing microtribometry approaches have been developed over the past decade to help connect the dots between fundamental and practical tribology measurements: spring-based (e.g., AFM) approaches use low speed, low stiffness, and long relative slip length to quantify friction, while quartz crystal microbalance (QCM)-based approaches use high speed, high stiffness, and short relative slip length. Because the friction forces generated in these experiments are attributed to entirely different phenomena, it is unclear if or how the resulting friction forces are related. This study aims to resolve this uncertainty by integrating these distinct techniques into a single apparatus that allows two independent measurements of friction at a single interface. Alumina microspheres were tested against single-crystal MoS₂, a model nominally wear-free solid lubricant, and gold, a model metal control, at loads between 0.01 and 1 mN. The combined results from both measurement approaches gave friction coefficients (mean ± standard error) of 0.087 ± 0.007 and 0.27 ± 0.02 for alumina-MoS₂ and alumina-gold, respectively. The observed agreement between these methods for two different material systems suggests that friction in microscale contacts can be far less sensitive to external effects from compliance and slip speed than currently thought. Perhaps more importantly, this Article describes and validates a novel approach to closing the "tribology gap" while demonstrating how integration creates new opportunities for fundamental studies of practical friction.

Nonlinear denoising for characterization of solid friction under low confinement pressure

The present work investigates paper-paper friction dynamics by pulling a slider over a substrate. It focuses on the transition between stick-slip and inertial regimes. Although the device is classical, probing solid friction with the fewest contact damage requires that the applied load should be small. This induces noise, mostly impulsive in nature, on the recorded slider motion and force signals. To address the challenging issue of describing the physics of such systems, we promote here the use of nonlinear filtering techniques relying on recent nonsmooth optimization schemes. In contrast to linear filtering, nonlinear filtering captures the slider velocity asymmetry and, thus, the creep motion before sliding. Precise estimates of the stick and slip phase durations can thus be obtained. The transition between the stick-slip and inertial regimes is continuous. Here we propose a criterion based on the probability of the system to be in the stick-slip regime to quantify this transition. A phase diagram is obtained that characterizes the dynamics of this frictional system under low confinement pressure.

Spatiotemporal Dynamics of Frictional Systems: The Interplay of Interfacial Friction and Bulk Elasticity

Frictional interfaces are abundant in natural and engineering systems, and predicting their behavior still poses challenges of prime scientific and technological importance. At the heart of these challenges lies the inherent coupling between the interfacial constitutive relation—the macroscopic friction law—and the bulk elasticity of the bodies that form the frictional interface. In this feature paper, we discuss the generic properties of a minimal macroscopic friction law and the many ways in which its coupling to bulk elasticity gives rise to rich spatiotemporal frictional dynamics. We first present the widely used rate-and-state friction constitutive framework, discuss its power and limitations, and propose extensions that are supported by experimental data. We then discuss how bulk elasticity couples different parts of the interface, and how the range and nature of this interaction are affected by the system’s geometry. Finally, in light of the coupling between interfacial and bulk physics, we discuss basic phenomena in spatially extended frictional systems, including the stability of homogeneous sliding, the onset of sliding motion and a wide variety of propagating frictional modes (e.g., rupture fronts, healing fronts and slip pulses). Overall, the results presented and discussed in this feature paper highlight the inseparable roles played by interfacial and bulk physics in spatially extended frictional systems.

Effective drag of a rod in fluid-saturated granular beds

We measure the drag encountered by a vertically oriented rod moving across a sedimented granular bed immersed in a fluid under steady-state conditions. At low rod speeds, the presence of the fluid leads to a lower drag because of buoyancy, whereas a significantly higher drag is observed with increasing speeds. The drag as a function of the depth is observed to decrease from being quadratic at low speeds to appearing more linear at higher speeds. By scaling the drag with the average weight of the grains acting on the rod, we obtain the effective friction μ_{e} encountered over six orders of magnitude of speeds. While a constant μ_{e} is found when the grain size, rod depth, and fluid viscosity are varied at low speeds, a systematic increase is observed as the speed is increased. We analyze μ_{e} in terms of the inertial number I and viscous number J to understand the relative importance of inertia and viscous forces, respectively. For sufficiently high fluid viscosities, we find that the effect of varying the speed, depth, and viscosity can be described by the empirical function μ_{e}=μ_{o}+kJ^{n}, where μ_{o} is the effective friction measured in the quasistatic limit, and k and n are material constants. The drag is then analyzed in terms of the effective viscosity η_{e} and found to decrease systematically as a function of J. We further show that η_{e} as a function of J is directly proportional to the fluid viscosity and the μ_{e} encountered by the rod.

Memory Distance for Interfacial Chemical Bond-Induced Friction at the Nanoscale

Macroscale rate and state friction (RSF) laws include a memory distance, D_c, which is considered to be the distance required for a population of frictional contacts to renew itself via slip, counteracting the effects of aging in slow or static contact. This concept connects static friction and kinetic friction. Here, we use atomic force microscopy to study interfacial chemical bond-induced kinetic friction and the memory distance at the nanoscale for single silica-silica nanocontacts. We observe a logarithmic trend of decreasing friction with sliding velocity (i.e., velocity-weakening) at low velocities and a transition to increasing friction with velocity at higher velocities (i.e., velocity-strengthening). We propose a physically based kinetic model for the nanoscale memory effect, the "activation-passivation loop" model, which accounts for the activation and passivation of chemical reaction sites and the formation of new chemical bonds from dangling bonds during sliding. In the model, we define the memory distance to be the average sliding distance that accrues before an activated reaction site becomes passivated. Results from numerical simulations based on this model match experimental friction data well in the velocity-weakening regime and show that D_c is sensitive to the surface chemistry, and nearly independent of sliding velocity. The simulations also show values of D_c that are consistent with those obtained from the experiments. We propose a semiquantitative physical explanation of the observed logarithmic velocity-weakening behavior based on the conservation of the number of interfacial bonds during sliding. We also extract from the experimental data physically reasonable values of the energy barriers to the activation of reaction sites. Our results provide one possible physical mechanism for the nanoscale memory distance.

Frictional weakening of slip interfaces

When two objects are in contact, the force necessary to overcome friction is larger than the force necessary to keep sliding motion going. This difference between static and dynamic friction is usually attributed to the growth of the area of real contact between rough surfaces in time when the system is at rest. We directly measure the area of real contact and show that it actually increases during macroscopic slip, despite the fact that dynamic friction is smaller than static friction. This signals a decrease in the interfacial shear strength, the friction per unit contact area, which is due to a mechanical weakening of the asperities. This provides a novel explanation for stick-slip phenomena in, e.g., earthquakes.

Simultaneous improvements of strength and toughness in topologically interlocked ceramics

Topologically interlocked materials (TIMs) are an emerging class of architectured materials based on stiff building blocks of well-controlled geometries which can slide, rotate, or interlock collectively providing a wealth of tunable mechanisms, precise structural properties, and functionalities. TIMs are typically 10 times more impact resistant than their monolithic form, but this improvement usually comes at the expense of strength. Here we used 3D printing and replica casting to explore 15 designs of architectured ceramic panels based on platonic shapes and their truncated versions. We tested the panels in quasi-static and impact conditions with stereoimaging, image correlation, and 3D reconstruction to monitor the displacements and rotations of individual blocks. We report a design based on octahedral blocks which is not only tougher (50×) but also stronger (1.2×) than monolithic plates of the same material. This result suggests that there is no upper bound for strength and toughness in TIMs, unveiling their tremendous potential as structural and multifunctional materials. Based on our experiments, we propose a nondimensional "interlocking parameter" which could guide the exploration of future architectured systems.

Nonmonotonic Aging and Memory in a Frictional Interface

We measure the static frictional resistance and the real area of contact between two solid blocks subjected to a normal load. We show that following a two-step change in the normal load the system exhibits nonmonotonic aging and memory effects, two hallmarks of glassy dynamics. These dynamics are strongly influenced by the discrete geometry of the frictional interface, characterized by the attachment and detachment of unique microcontacts. The results are in good agreement with a theoretical model we propose that incorporates this geometry into the framework recently used to describe Kovacs-like relaxation in glasses as well as thermal disordered systems. These results indicate that a frictional interface is a glassy system and strengthen the notion that nonmonotonic relaxation behavior is generic in such systems.

Rate and State Friction Relation for Nanoscale Contacts: Thermally Activated Prandtl-Tomlinson Model with Chemical Aging

Rate and state friction (RSF) laws are widely used empirical relationships that describe macroscale to microscale frictional behavior. They entail a linear combination of the direct effect (the increase of friction with sliding velocity due to the reduced influence of thermal excitations) and the evolution effect (the change in friction with changes in contact "state," such as the real contact area or the degree of interfacial chemical bonds). Recent atomic force microscope (AFM) experiments and simulations found that nanoscale single-asperity amorphous silica-silica contacts exhibit logarithmic aging (increasing friction with time) over several decades of contact time, due to the formation of interfacial chemical bonds. Here we establish a physically based RSF relation for such contacts by combining the thermally activated Prandtl-Tomlinson (PTT) model with an evolution effect based on the physics of chemical aging. This thermally activated Prandtl-Tomlinson model with chemical aging (PTTCA), like the PTT model, uses the loading point velocity for describing the direct effect, not the tip velocity (as in conventional RSF laws). Also, in the PTTCA model, the combination of the evolution and direct effects may be nonlinear. We present AFM data consistent with the PTTCA model whereby in aging tests, for a given hold time, static friction increases with the logarithm of the loading point velocity. Kinetic friction also increases with the logarithm of the loading point velocity at sufficiently high velocities, but at a different increasing rate. The discrepancy between the rates of increase of static and kinetic friction with velocity arises from the fact that appreciable aging during static contact changes the energy landscape. Our approach extends the PTT model, originally used for crystalline substrates, to amorphous materials. It also establishes how conventional RSF laws can be modified for nanoscale single-asperity contacts to provide a physically based friction relation for nanoscale contacts that exhibit chemical bond-induced aging, as well as other aging mechanisms with similar physical characteristics.

Stick-Slip Instabilities for Interfacial Chemical Bond-Induced Friction at the Nanoscale

Earthquakes are generally caused by unstable stick-slip motion of faults. This stick-slip phenomenon, along with other frictional properties of materials at the macroscale, is well-described by empirical rate and state friction (RSF) laws. Here we study stick-slip behavior for nanoscale single-asperity silica-silica contacts in atomic force microscopy experiments. The stick-slip is quasiperiodic, and both the amplitude and spatial period of stick-slip increase with normal load and decrease with the loading point (i.e., scanning) velocity. The peak force prior to each slip increases with the temporal period logarithmically, and decreases with velocity logarithmically, consistent with stick-slip behavior at the macroscale. However, unlike macroscale behavior, the minimum force after each slip is independent of velocity. The temporal period scales with velocity in a nearly power law fashion with an exponent between -1 and -2, similar to macroscale behavior. With increasing velocity, stick-slip behavior transitions into steady sliding. In the transition regime between stick-slip and smooth sliding, some slip events exhibit only partial force drops. The results are interpreted in the context of interfacial chemical bond formation and rate effects previously identified for nanoscale contacts. These results contribute to a physical picture of interfacial chemical bond-induced stick-slip, and further establish RSF laws at the nanoscale.

A phase-plane analysis of localized frictional waves

Sliding frictional interfaces at a range of length scales are observed to generate travelling waves; these are considered relevant, for example, to both earthquake ground surface movements and the performance of mechanical brakes and dampers. We propose an explanation of the origins of these waves through the study of an idealized mechanical model: a thin elastic plate subject to uniform shear stress held in frictional contact with a rigid flat surface. We construct a nonlinear wave equation for the deformation of the plate, and couple it to a spinodal rate-and-state friction law which leads to a mathematically well-posed problem that is capable of capturing many effects not accessible in a Coulomb friction model. Our model sustains a rich variety of solutions, including periodic stick–slip wave trains, isolated slip and stick pulses, and detachment and attachment fronts. Analytical and numerical bifurcation analysis is used to show how these states are organized in a two-parameter state diagram. We discuss briefly the possible physical interpretation of each of these states, and remark also that our spinodal friction law, though more complicated than other classical rate-and-state laws, is required in order to capture the full richness of wave types.

Load and Time Dependence of Interfacial Chemical Bond-Induced Friction at the Nanoscale

Rate and state friction (RSF) laws are widely used empirical relationships that describe the macroscale frictional behavior of a broad range of materials, including rocks found in the seismogenic zone of Earth's crust. A fundamental aspect of the RSF laws is frictional "aging," where friction increases with the time of stationary contact due to asperity creep and/or interfacial strengthening. Recent atomic force microscope (AFM) experiments and simulations found that nanoscale silica contacts exhibit aging due to the progressive formation of interfacial chemical bonds. The role of normal load (and, thus, normal stress) on this interfacial chemical bond-induced (ICBI) friction is predicted to be significant but has not been examined experimentally. Here, we show using AFM that, for nanoscale ICBI friction of silica-silica interfaces, aging (the difference between the maximum static friction and the kinetic friction) increases approximately linearly with the product of the normal load and the log of the hold time. This behavior is attributed to the approximately linear dependence of the contact area on the load in the positive load regime before significant wear occurs, as inferred from sliding friction measurements. This implies that the average pressure, and thus the average bond formation rate, is load independent within the accessible load range. We also consider a more accurate nonlinear model for the contact area, from which we extract the activation volume and the average stress-free energy barrier to the aging process. Our work provides an approach for studying the load and time dependence of contact aging at the nanoscale and further establishes RSF laws for nanoscale asperity contacts.

Creep to inertia dominated stick-slip behavior in sliding friction modulated by tilted non-uniform loading

Comprehension of stick-slip motion is very important for understanding tribological principles. The transition from creep-dominated to inertia-dominated stick-slip as the increase of sliding velocity has been described by researchers. However, the associated micro-contact behavior during this transition has not been fully disclosed yet. In this study, we investigated the stick-slip behaviors of two polymethyl methacrylate blocks actively modulated from the creep-dominated to inertia-dominated dynamics through a non-uniform loading along the interface by slightly tilting the angle of the two blocks. Increasing the tilt angle increases the critical transition velocity from creep-dominated to inertia-dominated stick-slip behaviors. Results from finite element simulation disclosed that a positive tilt angle led to a higher normal stress and a higher temperature on blocks at the opposite side of the crack initiating edge, which enhanced the creep of asperities during sliding friction. Acoustic emission (AE) during the stick-slip has also been measured, which is closely related to the different rupture modes regulated by the distribution of the ratio of shear to normal stress along the sliding interface. This study provided a more comprehensive understanding of the effect of tilted non-uniform loading on the local stress ratio, the local temperature, and the stick-slip behaviors.

Friction and nonlinear dynamics

The nonlinear dynamics associated with sliding friction forms a broad interdisciplinary research field that involves complex dynamical processes and patterns covering a broad range of time and length scales. Progress in experimental techniques and computational resources has stimulated the development of more refined and accurate mathematical and numerical models, capable of capturing many of the essentially nonlinear phenomena involved in friction.

Universal Aging Mechanism for Static and Sliding Friction of Metallic Nanoparticles

The term "contact aging" refers to the temporal evolution of the interface between a slider and a substrate usually resulting in increasing friction with time. Current phenomenological models for multiasperity contacts anticipate that such aging is not only the driving force behind the transition from static to sliding friction, but at the same time influences the general dynamics of the sliding friction process. To correlate static and sliding friction on the nanoscale, we show experimental evidence of stick-slip friction for nanoparticles sliding on graphite over a wide dynamic range. We can assign defined periods of aging to the stick phases of the particles, which agree with simulations explicitly including contact aging. Additional slide-hold-slide experiments for the same system allow linking the sliding friction results to static friction measurements, where both friction mechanisms can be universally described by a common aging formalism.

Colloidal particles driven across periodic optical-potential-energy landscapes

We study the motion of colloidal particles driven by a constant force over a periodic optical potential energy landscape. First, the average particle velocity is found as a function of the driving velocity and the wavelength of the optical potential energy landscape. The relationship between average particle velocity and driving velocity is found to be well described by a theoretical model treating the landscape as sinusoidal, but only at small trap spacings. At larger trap spacings, a nonsinusoidal model for the landscape must be used. Subsequently, the critical velocity required for a particle to move across the landscape is determined as a function of the wavelength of the landscape. Finally, the velocity of a particle driven at a velocity far exceeding the critical driving velocity is examined. Both of these results are again well described by the two theoretical routes for small and large trap spacings, respectively. Brownian motion is found to have a significant effect on the critical driving velocity but a negligible effect when the driving velocity is high.

Self-Amplification of Solid Friction in Interleaved Assemblies

It is nearly impossible to separate two interleaved phone books when held by their spines. A full understanding of this astonishing demonstration of solid friction in complex assemblies remains elusive. In this Letter, we report on experiments with controlled booklets and show that the force required increases sharply with the number of sheets. A model captures the effect of the number of sheets, their thickness, and the overlapping distance. Furthermore, the data collapse onto a self-similar master curve with one dimensionless amplification parameter. In addition to solving a long-standing familiar enigma, this model system provides a framework with which one can accurately measure friction forces and coefficients at low loads, and that has relevance to complex assemblies from the macro- to the nanoscale.

Are there reliable constitutive laws for dynamic friction?

Structural vibration controlled by interfacial friction is widespread, ranging from friction dampers in gas turbines to the motion of violin strings. To predict, control or prevent such vibration, a constitutive description of frictional interactions is inevitably required. A variety of friction models are discussed to assess their scope and validity, in the light of constraints provided by different experimental observations. Three contrasting case studies are used to illustrate how predicted behaviour can be extremely sensitive to the choice of frictional constitutive model, and to explore possible experimental paths to discriminate between and calibrate dynamic friction models over the full parameter range needed for real applications.

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.

Dynamics of intermittent force fluctuations in vesicular nanotubulation

Irregular force fluctuations are seen in most nanotubulation experiments. The dynamics behind their presence has, however, been neither commented upon nor modeled. A simple estimate of the mean energy dissipated in force drops turns out to be several times the thermal energy. This coupled with the rate dependent nature of the deformation reported in several experiments point to a dynamical origin of the serrations. We simplify the whole process of tether formation through a three-stage model of successive deformations of sphere to ellipsoid, neck-formation, and tubule birth and extension. Based on this, we envisage a rate-softening frictional force at the neck that must be overcome before a nanotube can be pulled out. Our minimal model includes elastic and visco-elastic deformation of the vesicle, and has built-in dependence on pull velocity, vesicle radius, and other material parameters, enabling us to capture various kinds of serrated force-extension curves for different parameter choices. Serrations are predicted in the nanotubulation region. Other features of force-extension plots reported in the literature such as a plateauing serrated region beyond a force drop, serrated flow region with a small positive slope, an increase in the elastic threshold with pull velocity, force-extension curves for vesicles with larger radius lying lower than those for smaller radius, are all also predicted by the model. A toy model is introduced to demonstrate that the role of the friction law is limited to inducing stick-slip oscillations in the force, and all other qualitative and quantitative features emerging from the model can only be attributed to other physical mechanisms included in the deformation dynamics of the vesicle.

The acoustics of the violin: a review

To understand the design and function of the violin requires investigation of a range of scientific questions. This paper presents a review: the relevant physics covers the nonlinear vibration of a bowed string, the vibration of the instrument body, and the consequent sound radiation. Questions of discrimination and preference by listeners and players require additional studies using the techniques of experimental psychology, and these are also touched on in the paper. To address the concerns of players and makers of instruments requires study of the interaction of all these factors, coming together in the concept of 'playability' of an instrument.

Novel friction law for the static friction force based on local precursor slipping

The sliding of a solid object on a solid substrate requires a shear force that is larger than the maximum static friction force. It is commonly believed that the maximum static friction force is proportional to the loading force and does not depend on the apparent contact area. The ratio of the maximum static friction force to the loading force is called the static friction coefficient µ_M, which is considered to be a constant. Here, we conduct experiments demonstrating that the static friction force of a slider on a substrate follows a novel friction law under certain conditions. The magnitude of µ_M decreases as the loading force increases or as the apparent contact area decreases. This behavior is caused by the slip of local precursors before the onset of bulk sliding and is consistent with recent theory. The results of this study will develop novel methods for static friction control.

Kinetics of the coefficient of friction of elastomers

We study theoretically and numerically the kinetics of the coefficient of friction of an elastomer due to abrupt changes of sliding velocity. Numerical simulations reveal the same qualitative behavior which has been observed experimentally on different classes of materials: the coefficient of friction first jumps and then relaxes to a new stationary value. The elastomer is modeled as a simple Kelvin body and the surface as a self-affine fractal with a Hurst exponent in the range from 0 to 1. Parameters of the jump of the coefficient of friction and the relaxation time are determined as functions of material and loading parameters. Depending on velocity and the Hurst exponent, relaxation of friction with characteristic length or characteristic time is observed.

Slow slip and the transition from fast to slow fronts in the rupture of frictional interfaces

The failure of the population of microjunctions forming the frictional interface between two solids is central to fields ranging from biomechanics to seismology. This failure is mediated by the propagation along the interface of various types of rupture fronts, covering a wide range of velocities. Among them are the so-called slow fronts, which are recently discovered fronts much slower than the materials' sound speeds. Despite intense modeling activity, the mechanisms underlying slow fronts remain elusive. Here, we introduce a multiscale model capable of reproducing both the transition from fast to slow fronts in a single rupture event and the short-time slip dynamics observed in recent experiments. We identify slow slip immediately following the arrest of a fast front as a phenomenon sufficient for the front to propagate further at a much slower pace. Whether slow fronts are actually observed is controlled both by the interfacial stresses and by the width of the local distribution of forces among microjunctions. Our results show that slow fronts are qualitatively different from faster fronts. Because the transition from fast to slow fronts is potentially as generic as slow slip, we anticipate that it might occur in the wide range of systems in which slow slip has been reported, including seismic faults.

History-dependent friction and slow slip from time-dependent microscopic junction laws studied in a statistical framework

To study how macroscopic friction phenomena originate from microscopic junction laws, we introduce a general statistical framework describing the collective behavior of a large number of individual microjunctions forming a macroscopic frictional interface. Each microjunction can switch in time between two states: a pinned state characterized by a displacement-dependent force and a slipping state characterized by a time-dependent force. Instead of tracking each microjunction individually, the state of the interface is described by two coupled distributions for (i) the stretching of pinned junctions and (ii) the time spent in the slipping state. This framework allows for a whole family of microjunction behavior laws, and we show how it represents an overarching structure for many existing models found in the friction literature. We then use this framework to pinpoint the effects of the time scale that controls the duration of the slipping state. First, we show that the model reproduces a series of friction phenomena already observed experimentally. The macroscopic steady-state friction force is velocity dependent, either monotonic (strengthening or weakening) or nonmonotonic (weakening-strengthening), depending on the microscopic behavior of individual junctions. In addition, slow slip, which has been reported in a wide variety of systems, spontaneously occurs in the model if the friction contribution from junctions in the slipping state is time weakening. Next, we show that the model predicts a nontrivial history dependence of the macroscopic static friction force. In particular, the static friction coefficient at the onset of sliding is shown to increase with increasing deceleration during the final phases of the preceding sliding event. We suggest that this form of history dependence of static friction should be investigated in experiments, and we provide the acceleration range in which this effect is expected to be experimentally observable.

Friction behavior of a microstructured polymer surface inspired by snake skin

The aim of this study was to understand the influence of microstructures found on ventral scales of the biological model, Lampropeltis getula californiae, the California King Snake, on the friction behavior. For this purpose, we compared snake-inspired anisotropic microstructured surfaces to other microstructured surfaces with isotropic and anisotropic geometry. To exclude that the friction measurements were influenced by physico-chemical variations, all friction measurements were performed on the same epoxy polymer. For frictional measurements a microtribometer was used. Original data were processed by fast Fourier transformation (FFT) with a zero frequency related to the average friction and other peaks resulting from periodic stick-slip behavior. The data showed that the specific ventral surface ornamentation of snakes does not only reduce the frictional coefficient and generate anisotropic frictional properties, but also reduces stick-slip vibrations during sliding, which might be an adaptation to reduce wear. Based on this extensive comparative study of different microstructured polymer samples, it was experimentally demonstrated that the friction-induced stick-slip behavior does not solely depend on the frictional coefficient of the contact pair.

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.

Plastic and viscous dissipations in foams: cross-over from low to high shear rates

Soft glassy materials made of deformable cells, such as liquid foams, simultaneously display elastic, plastic and viscous behaviours. Bubble deformation is elastic until the material plastically yields and bubbles swap neighbours, then bubbles relax dissipatively towards a new energy minimum. This relaxation occurs in a finite time, and shearing a foam at a fast strain rate compared to that time leads to a viscous flow. To describe such an elastic, plastic and viscous behaviour we introduce a simplified scalar model of foam deformation and flow with a periodic pinning potential. The continuum mechanics behaviour of the foam emerges as an ensemble average over disordered units without requiring that they are coupled. Our model captures surprisingly well various features of the viscous dissipation during plastic deformation. At low shear rates, the time averaged stress is smaller than the static yield stress. A critical shear rate exists: any flow at fixed stress has a shear rate above this critical value. Moreover, the model only involves measurable parameters, which enables us to compare it with existing experiments and simulations.

Fracture path in an anisotropic material in the light of a friction experiment

A slider is pulled by means of a flexible link on a flat solid surface which exhibits anisotropic frictional properties. The resulting trajectory of the slider is assessed experimentally. First, we check that the experimental results are in excellent agreement with a theoretical description of the problem based on an expression of the frictional forces. Second, we point out that the trajectory of the slider can be recovered by the use of a "maximum of energy release rate" criterion which is generally used to predict the path of a fracture even if the validity of the principle is difficult to verify in the latter complex systems.

Quantitative prediction of effective toughness at random heterogeneous interfaces

The propagation of an adhesive crack through an anisotropic heterogeneous interface is considered. Tuning the local toughness distribution function and spatial correlation is numerically shown to induce a transition between weak to strong pinning conditions. While the macroscopic effective toughness is given by the mean local toughness in the case of weak pinning, a systematic toughness enhancement is observed for strong pinning (the critical point of the depinning transition). A self-consistent approximation is shown to account very accurately for this evolution, without any free parameter.