Stochastic Properties of Static Friction

The onset of frictional motion is mediated by rupture-like slip fronts, which nucleate locally and propagate eventually along the entire interface causing global sliding. The static friction coefficient is a macroscopic measure of the applied force at this particular instant when the frictional interface loses stability. However, experimental studies are known to present important scatter in the measurement of static friction; the origin of which remains unexplained. Here, we study the nucleation of local slip at interfaces with slip-weakening friction of random strength and analyze the resulting variability in the measured global strength. Using numerical simulations that solve the elastodynamic equations, we observe that multiple slip patches nucleate simultaneously, many of which are stable and grow only slowly, but one reaches a critical length and starts propagating dynamically. We show that a theoretical criterion based on a static equilibrium solution predicts quantitatively well the onset of frictional sliding. We develop a Monte-Carlo model by adapting the theoretical criterion and pre-computing modal convolution terms, which enables us to run efficiently a large number of samples and to study variability in global strength distribution caused by the stochastic properties of local frictional strength. The results demonstrate that an increasing spatial correlation length on the interface, representing geometric imperfections and roughness, causes lower global static friction. Conversely, smaller correlation length increases the macroscopic strength while its variability decreases. We further show that randomness in local friction properties is insufficient for the existence of systematic precursory slip events. Random or systematic non-uniformity in the driving force, such as potential energy or stress drop, is required for arrested slip fronts. Our model and observations..

Linear elastic fracture mechanics predicts the propagation distance of frictional slip

When a frictional interface is subject to a localized shear load, it is often (experimentally) observed that local slip events initiate at the stress concentration and propagate over parts of the interface by arresting naturally before reaching the edge. We develop a theoretical model based on linear elastic fracture mechanics to describe the propagation of such precursory slip. The model's prediction of precursor lengths as a function of external load is in good quantitative agreement with laboratory experiments as well as with dynamic simulations, and provides thereby evidence to recognize frictional slip as a fracture phenomenon. We show that predicted precursor lengths depend, within given uncertainty ranges, mainly on the kinetic friction coefficient, and only weakly on other interface and material parameters. By simplifying the fracture mechanics model we also reveal sources for the observed non-linearity in the growth of precursor lengths as a function of the applied force. The discrete nature of precursors as well as the shear tractions caused by frustrated Poisson's expansion are found to be the dominant factors. Finally, we apply our model to a different, symmetric set-up and provide a prediction of the propagation distance of frictional slip for future experiments

The existence of a critical length scale in regularised friction

We study a regularisation of Coulomb's friction law on the propagation of local slip at an interface between a deformable and a rigid solid. This regularisation, which was proposed based on experimental observations, smooths the effect of a sudden jump in the contact pressure over a characteristic length scale. We apply it in numerical simulations in order to analyse its influence on the behaviour of local slip. We first show that mesh convergence in dynamic simulations is achieved without any numerical damping in the bulk and draw a convergence map with respect to the characteristic length of the friction regularisation. By varying this length scale on the example of a given slip event, we observe that there is a critical length below which the friction regularisation does not affect anymore the propagation of the interface rupture. A spectral analysis of the regularisation on a periodic variation of Coulomb's friction is conducted to confirm the existence of this critical length. The results indicate that if the characteristic length of the friction regularisation is smaller than the critical length, a slip event behaves as if it was governed by Coulomb's law. We therefore propose that there is a domain of influence of the friction regularisation depending on its characteristic length and on the frequency content of the local slip event. A byproduct of the analysis is related to the existence of a physical length scale characterising a given frictional interface. We establish that the experimental determination of this interface property may be achieved by experimentally monitoring slip pulses whose frequency content is rich enough.Comment: 21 pages, 7 figure

An FFT-based framework for predicting corrosion-driven damage in fractal porous media

Understanding fracture in cementitious materials caused by the deposition and growth of corrosion products requires scale-bridging approaches due to the large length-scale difference between the micro-pores, where deposition occurs, and the structure, where deterioration manifests. Cementitious materials bear a highly heterogeneous micro-structure owing to the fractal nature of micro-pores. Simultaneously, a corrosion-driven fracture is a multi-physics problem involving ionic diffusion, chemical reactions, and stress development. This multi-scale and multi-physical character makes scale-bridging studies computationally costly, often leading to the use of simplified fractal porous media, which has important consequences for the quantitative interpretation of the results. Recent advances in homogenization approaches using Fast-Fourier-Transform (FFT) based methods have raised interest due to their ease of implementation and low computational cost. This paper presents an FFT-based framework for solving corrosion-driven fractures within fractal porous media. We demonstrate the effectiveness of the Fourier-based spectral method in resolving the multiple corrosion-driven mechanisms such as ionic diffusion, stress development, and damage within a fractal porous microstructure. Based on the presented methodology, we analyze the impact of simplifying fractal porous media with simple Euclidean geometry on corrosion-driven fracture. Our results demonstrate the importance of preserving both the porosity and fractal nature of pores for precise and reliable modeling of corrosion-driven failure mechanisms

Transonic and supershear crack propagation driven by geometric nonlinearities

Linear elastic fracture mechanics theory predicts that the speed of crack growth is limited by the Rayleigh wave speed. Although many experimental observations and numerical simulations have supported this prediction, some exceptions have raised questions about its validity. The underlying reasons for these discrepancies and the precise limiting speed of dynamic cracks remain unknown. Here, we demonstrate that tensile (mode I) cracks can exceed the Rayleigh wave speed and propagate at supershear speeds. We show that taking into account geometric non-linearities, inherent in most materials, is sufficient to enable such propagation modes. These geometric non-linearities modify the crack-tip singularity, resulting in different crack-tip opening displacements, cohesive zone behavior, and energy flows towards the crack tip.Comment: 8 pages, 4 figure

A discretization-convergent Level-Set-DEM

The recently developed level-set-DEM is able to seamlessly handle arbitrarily shaped grains and their contacts through a discrete level-set representation of grains' volume and a node-based discretization of their bounding surfaces. Heretofore, the convergence properties of LS-DEM with refinement of these discretizations have not been examined. Here, we examine these properties and show that the original LS-DEM diverges upon surface discretization refinement due to its force-based discrete contact formulation. Next, we fix this issue by adopting a continuum-based contact formulation wherein the contact interactions are traction-based, and show that the adapted LS-DEM is fully discretization convergent. Lastly, we discuss the significance of convergence in capturing the physical response, as well as a few other convergence-related topics of practical importance

Beam-like topologically interlocked structures with hierarchical interlocking

Topologically interlocked materials and structures, which are assemblies of unbonded interlocking building blocks, are a promising concept for versatile structural applications. They have been shown to display exceptional mechanical properties including outstanding combinations of stiffness, strength, and toughness, beyond those achievable with common engineering materials. Recent work established the theoretical upper limit for the strength and toughness of beam-like topologically interlocked structures. However, this theoretical limit is only achievable for structures with unrealistically high friction coefficients and, therefore, it remains unknown if it is achievable in actual structures. Here, we propose, inspired by biological systems, a hierarchical approach for topological interlocking which overcomes these limitations and provides a path toward optimized mechanical performance. We consider beam-like topologically interlocked structures with geometrically designed surface morphologies, which increases the effective frictional strength of the interfaces, and hence enables us to achieve the theoretical limit with realistic friction coefficients. Using numerical simulations, we examine the effect of sinusoidal surface morphology with controllable amplitude and wavelength on the maximum load-carrying capacity of the structure. Our study discusses various effects of architecturing the surface morphology of beam-like topological interlocked structures, and most notably, it demonstrates its ability to significantly enhance the structure's mechanical performance

The key to the enhanced performance of slab-like topologically interlocked structures with non-planar blocks

Topologically interlocked structures are assemblies of interlocking blocks that hold together solely through contact. Such structures have been shown to exhibit high strength, energy dissipation, and crack arrest properties. Recent studies on beam-like topologically interlocked structures have shown that, with non-planar blocks, it is possible to reach levels of strength and work-to-failure which are otherwise possible only with unrealistically high friction coefficients. While non-planar blocks have been extensively used for slab-like assemblies, many questions in that context are still not fully understood. Specifically, it is unclear what are the exact characteristics of non-planar surface morphologies which can potentially improve the enhanced mechanical response of slab-like assemblies. In addition, it is unclear if slab-like structures with non-planar surface blocks can reach a saturated response with realistic friction coefficient values, as is the case with beam-like ones. Here, we investigate such fundamental questions using numerical simulations. We show that, by using non-planar blocks, it is possible to reach saturation to the response capacity of the structure with a realistic friction coefficient. Furthermore, we show that the key morphology parameter responsible for the enhanced performance is the local angle of inclination at the top of the loaded block. Lastly, we show that non-planar morphologies lead to improved work-to-failure and ultimate deflection, which cannot be attained with planar-faced blocks. These findings shed new light on topologically interlocked structures with non-planar blocks, allowing for a better understanding of their strengths and potential applications

Dynamic fields at the tip of sub-Rayleigh and supershear frictional rupture fronts

The onset of frictional motion at the interface between two distinct bodies in contact is characterized by the propagation of dynamic rupture fronts. We combine friction experiments and numerical simulations to study the properties of these frictional rupture fronts. We extend previous analysis of slow and sub-Rayleigh rupture fronts and show that strain fields and the evolution of real contact area in the tip vicinity of supershear ruptures are well described by analytical fracture-mechanics solutions. Fracture-mechanics theory further allows us to determine long sought-after interface properties, such as local fracture energy and frictional peak strength. Both properties are observed to be roughly independent of rupture speed and mode of propagation. However, our study also reveals discrepancies between measurements and analytical solutions that appear as the rupture speed approaches the longitudinal wave speed. Further comparison with dynamic simulations illustrates that, in the supershear propagation regime, transient and geometrical (finite sample thickness) effects cause smaller near-tip strain amplitudes than expected from the fracture-mechanics theory. By showing good quantitative agreement between experiments, simulations and theory over the entire range of possible rupture speeds, we demonstrate that frictional rupture fronts are classic dynamic cracks despite residual friction.Comment: 20 pages including 11 figure

Nucleation of frictional sliding by coalescence of microslip

The onset of frictional motion is mediated by the dynamic propagation of a rupture front, analogous to a shear crack. The rupture front nucleates quasi-statically in a localized region of the frictional interface and slowly increases in size. When it reaches a critical nucleation length it becomes unstable, propagates dynamically and eventually breaks the entire interface, leading to macroscopic sliding. The nucleation process is particularly important because it determines the stress level at which the frictional interface fails, and therefore, the macroscopic friction strength. However, the mechanisms governing nucleation of frictional rupture fronts are still not well understood. Specifically, our knowledge of the nucleation process along a heterogeneous interface remains incomplete. Here, we study the nucleation of localized slip patches on linear slip-weakening interfaces with deterministic and stochastic heterogeneous friction properties. Using numerical simulations, we analyze the process leading to a slip patch of critical size for systems with varying correlation lengths of the local friction strength. Our deterministic interface model reveals that the growth of the critical nucleation patch at interfaces with small correlation lengths is non smooth due to the coalescence of neighboring slip patches. Existing analytical solutions do not account for this effect, which leads to an overestimation of global interface strength. Conversely, when the correlation length is large, the growth of the slip patch is continuous and our simulations match the analytical solution. Furthermore, nucleation by coalescence is also observed on stochastic interfaces with small correlation length. In this case, the applied load for a given slip patch size is a random variable. We show that its expectation follows a logistic function, which allows us to predict the strength of the interface well before failure occurs. Our model and observations provide new understanding of the nucleation process and its effect on the static frictional strength