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We present a fully implicit formulation of coupled fluid flow and geomechanics for fluid injection/withdrawal in fractured reservoirs in the context of CO2storage. Utilizing a Galerkin finite-element approach, both flow and poroelasticity equations are discretized on a shared three-dimensional mesh. The fluid flow is assumed to be single-phase. The hydraulic behaviour of fractures is represented through a double-nodes flow element, which allows to efficiently model longitudinal and transversal fracture permeabilities. In addressing the mechanical subproblem, fractures are explicitly modelled using cohesive elements to account for contact, friction and opening phenomena. The nonlinear set of equations is solved implicitly through an iterative partitioned conjugate gradient procedure, extending its traditional application to continuous problems to those involving explicit discontinuities such as faults and fractures. The model's accuracy is verified against analytical solutions for different geomechanical problems, notably for the growth of a frictional slip rupture along a fault due to fluid injection. Such a particularly challenging benchmark for a critically stressed fault is here reproduced for the first time by a finite-element based scheme. The capabilities of the developed parallel solver are then illustrated through a scenario involving CO2 injection into a faulted aquifer. The original solver code, tutorials, and data visualization routines are publicly accessible.

To model rupture dynamics, a friction law must be assumed. Commonly used constitutive laws for modeling friction include slip-weakening laws which are characterized by a drop from static to dynamic frictional stress. Within this framework, the prevailing understanding asserts that the frictional behavior is solely controlled by the fracture energy -- the area beneath the frictional stress versus the cumulative slip curve. In particular, it is claimed that the curve's shape itself has no influence on the system's response. Here we perform fully dynamic rupture simulations to challenge prevailing beliefs by demonstrating that the constitutive law shape exerts an intimate control over rupture profiles. For a consistent fracture energy but varying constitutive law shapes, the velocity profile is different: each abrupt slope transition leads to the localization of a distinct velocity peak. For example, in the case of a bilinear slip-weakening law featuring two different slopes, the rupture exhibits two distinct velocity peaks. This phenomenon arises from the transition between a constant weakening rate to another. However, this distinction does not seem to influence how a rupture responds to a stress barrier or the cumulative radiated energy emitted. These results are derived through two separate numerical schemes (spectral boundary integral and finite element methods) ensuring their independence from the computational approach employed., Comment: 12 pages, 7 figures

The emergence of statistical complexity in frictional systems, manifested in broad distributions of various observables, is not yet understood. We study this problem in velocity-driven, homogeneous (no quenched disorder) unstable frictional systems of height $H$. The latter are described at the continuum scale within a realistic rate-and-state friction interfacial constitutive framework, where elasto-frictional instabilities emerge from rate-weakening friction. For large $H$, such frictional systems were recently shown to undergo continuous coarsening until settling into a spatially periodic traveling solution. We show that when the system's height-to-length ratio becomes small, coarsening is less effective and the periodic solution is dynamically avoided. Instead, and consistently with previous reports, the system settles into a stochastic, statistically stationary state. The latter features slip bursts, classified into predominantly non-propagating small bursts and propagating large bursts, which are non-trivially distributed. The statistical distributions emerge from dynamically self-generated heterogeneity, where both the non-equilibrium history of the interface and wave reflections from finite boundaries, mediated by material inertia, play central roles. Specifically, the dynamics and statistics of large bursts reveal a timescale $\sim\!H/c_{\rm s}$, where $c_{\rm s}$ is the shear wave-speed. We discuss the robustness of our findings against variations of the frictional parameters, most notably affecting the magnitude of frictional rate-weakening, as well as against different interfacial state evolution laws. Finally, we demonstrate a reverse transition in which statistical complexity disappears in favor of the spatially periodic traveling solution. Overall, our results elucidate how relatively simple physical ingredients can give rise to the emergence of slip complexity., Comment: v2: new results on the robustness of complexity and a reverse complexity-to-coarsening transition, including new section, figures (Figs. 4 & 5) and appendix

A viscous, lubrication-like response can be triggered in a thin film of fluid squeezed between a rigid and flat surface and the tip of an incoming projectile. We develop a comprehensive theory for this viscous approach stage of fluid-mediated normal impact, applicable to soft impactors. Under the assumption of mediating fluid being incompressible, the impacting solid displays two limit regimes: one dominated by elasticity and the other by inertia. The transition between the two is predicted by a dimensionless parameter, which can be interpreted as the ratio between two time scales that are the time that it takes for the surface waves to warn the leading edge of the impactor of the forthcoming impact, and the characteristic duration of the final viscous phase of the approach. Additionally, we assess the role of solid compressibility and elucidate why nearly-incompressible solids feature (a) substancial "gliding" prior to contact at the transition between regimes, (b) the largest size of entrapped bubble between the deformed tip of the impactor and the flat surface, and (c) a sudden drop in entrapped bubble radius past the transition between regimes. Finally, we argue that the above time scale ratio (a dimensionless number) can govern the different dynamics reported experimentally for a fluid droplet as a function of its viscosity and surface tension., Comment: 14 pages, 9 figures, 3 tables

Surface roughness is a key factor when it comes to friction and wear, as well as to other physical properties. These phenomena are controlled by mechanisms acting at small scales, in which the topography of apparently-flat surfaces is revealed. Roughness in natural surfaces has been reported to conform to self-affine statistics in a wide variety of settings (ranging from earthquake physics to micro-electro-mechanical devices), meaning that the height profile can be described using a spectrum where the amplitude is proportional to its wavelength raised to a constant power, which is related to a statistical parameter named Hurst exponent. We analyze the roughness evolution in atomistic surfaces during molecular dynamics simulations of wear. Both pairs of initially-flat and initially-rough surfaces in contact are worn by a third body formed by particles trapped between them during relative sliding. During the first sliding stages, the particles trapped between the first bodies scratch the surfaces. Once the former become coated with atoms from the latter, the wear process slows down and becomes "adhesive-like". The initial particle sizes are consistent with the minimum size to be expected for the debris, but tend to grow by material removal from the surfaces and to agglomerate. We show that, for the particular configurations under consideration, the surface roughness seems to converge to a steady state characterized by Hurst exponent close to 0.8, independently of the initial conditions.

We employ a novel discrete element method (DEM) force formulation to simulate adhesive wear and assess the effects of material and loading parameters on the properties of the third-body layer (TBL) formed during sliding motion. The study emphasizes the role of a material's critical length scale d* in the rheology of the TBL. This critical length scale is already known for controlling the size of smallest wear particles. We observe the emergence of a several wear regimes involving wear particle creation and aggregation, with limited effect from d* on TBL properties. Instead, material strength and surface energy have a profound influence. This study opens up new avenues for exploration of larger systems, three-dimensional setups, and other loading conditions., Comment: This version of the article has been accepted for publication, after peer review, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s11249-023-01772-x

Model-free data-driven computational mechanics (DDCM) is a new paradigm for simulations in solid mechanics. The modeling step associated to the definition of a material constitutive law is circumvented through the introduction of an abstract phase space in which, following a pre-defined rule, physically-admissible states are matched to observed material response data (coming from either experiments or lower-scale simulations). In terms of computational resources, the search procedure that performs these matches is the most onerous step in the algorithm. One of the main advantages of DDCM is the fact that it avoids regression-based, bias-prone constitutive modeling. However, many materials do display a simple linear response in the small-strain regime while also presenting complex behavior after a certain deformation threshold. Motivated by this fact, we present a novel refinement technique that turns regular elements (equipped with a linear-elastic constitutive law) into data-driven ones if they are expected to surpass the threshold known to trigger material non-linear response. We term this technique ``data refinement'', ``d-refinement'' for short. Starting from an initially regular FEM mesh, the proposed algorithm detects where the refinement is needed and iterates until all elements presumed to display non-linearity become data-driven ones. Insertion criteria are discussed. The scheme is well-suited for simulations that feature non-linear response in relatively small portions of the domain while the rest remains linear-elastic. The method is validated against a traditional incremental solver (i.e., Newton-Raphson method) and we show that the d-refinement framework can outperform it in terms of speed at no loss of accuracy. We provide an application that showcases the advantage of the new method: bridging scales in architected metamaterials., Comment: 24 pages, 12 figures, 4 tables

Prediction of ground motion triggered by earthquakes is a prime concern for both the seismology community and geotechnical earthquake engineering one. The subfield occupied with such a problem is termed site response analysis (SRA), its one-dimensional flavor (1D-SRA) being particularly popular given its simplicity. Despite the simple geometrical setting, a paramount challenge remains when it comes to numerically consider intense shaking in the soft upper soil strata of the crust: how to mathematically model the high-strain, dissipative, potentially rate-dependent soil behavior. Both heuristics models and phenomenological constitutive laws have been developed to meet the challenge, neither of them being exempt of either numerical limitations or physical inadequacies or both. We propose herein to bring the novel data-driven paradigm to bear, thus giving away with the need to construct constitutive behavior models altogether. Data-driven computational mechanics (DDCM) is a novel paradigm in solid mechanics that is gaining popularity; in particular, the multiscale version of it relies on studying the response of the microstructure (in the case of soil, representative volumes containing grains) to populate a dataset that is later used to inform the response at the macroscale. This manuscript presents the first application of multiscale DDCM to 1D-SRA: first, we demonstrate its capacity to handle wave propagation problems using discrete datasets, obtained via sampling grain ensembles using the discrete element method (DEM), in lieu of a constitutive law and then we apply it specifically to analyze the propagation of harmonic waves in a soft soil deposit that overlies rigid bedrock. We validate the implementation via comparison to regular finite elements analyses (FEA) and demonstrate that traditional amplification functions are recovered when using the DDCM., Comment: 14 pages, 7 figures

Building on an analogy to ductile fracture mechanics, we quantify the size of debris particles created during adhesive wear. Earlier work suggested a linear relation between tangential work and wear debris volume, assuming that the debris size is proportional to the micro contact size multiplied by the junction shear strength. However, the present study reveals deviations from linearity. These deviations can be rationalized with fracture mechanics and imply that less work is necessary to generate debris than what was assumed. Here, we postulate that the work needed to detach a wear particle is made of the surface energy expended to create new fracture surfaces, and also of plastic work within a fracture process zone of a given width around the cracks. Our theoretical model, validated by molecular dynamics simulations, reveals a super-linear scaling relation between debris volume ($V_d$) and tangential work ($W_t$): $V_d \sim W_t^{3/2}$ in 3D and $V_d \sim W_t^{2}$ in 2D. This study provides a theoretical foundation to estimate the statistical distribution of sizes of fine particles emitted due to adhesive wear processes.

Crack fronts deform due to heterogeneities, and inspecting these deformations can reveal local variations of material properties, and help predict out of plane damage. Current models neglect the influence of a finite dissipation length-scale behind the crack tip, called the process zone size. The latter introduces scale effects in the deformation of the crack front, that are mitigated by the dynamics of the crack. We provide a theoretical framework for dynamic crack front deformations in heterogeneous cohesive materials and validate its predictions using numerical simulations.

When a crack interacts with material heterogeneities, its front distorts and adopts complex tortuous configurations that are reminiscent of the energy barriers encountered during crack propagation. As such, the study of crack front deformations is key to rationalize the effective failure properties of micro-structured solids and interfaces. Yet, the impact of a localized dissipation in a finite region behind the crack front, called the process zone, has often been overlooked. In this work, we derive the equation ruling 3D coplanar crack propagation in heterogeneous cohesive materials where the opening of the crack is resisted by some traction in its wake. We show that the presence of a process zone results in two competing effects on the deformation of crack fronts: (i) it makes the front more compliant to small-wavelength perturbations, and (ii) it smooths out local fluctuations of strength and process zone size, from which emerge heterogeneities of fracture energy. Their respective influence on front deformations is shown to strongly impact the stability of perturbed crack fronts, as well as their stationary shapes when interacting with arrays of tough obstacles. Overall, our theory provides a unified framework to predict the variety of front profiles observed in experiments, even when the small-scale yielding hypothesis of linear elastic fracture mechanics breaks down.

The use of molecular dynamics (MD) simulations has led to promising results to unravel the atomistic origins of adhesive wear, and in particular for the onset of wear at nanoscale surface asperities. However, MD simulations come with a high computational cost and offer access to only a narrow window of time and length scales. We propose here to resort to the discrete element method (DEM) to mitigate the computational cost. Using DEM particles with contact and cohesive forces, we reproduce the key mechanisms observed with MD, while having particle diameters and system sizes an order of magnitude higher than with MD. The pairwise forces are tuned to obtain a solid with reasonably approximated elastic and fracture properties. The simulations of single asperity wear performed with MD are successfully reproduced with DEM using a range particle sizes, validating the coarse-graining procedure. More complex simulations should allow the study of wear particles and the evolution of worn surfaces in an adhesive wear context, while reaching scales inaccessible to MD., Comment: 17 pages, 20 figures

We describe benchmark experiments to evaluate the frictional properties of laser patterned low-density polyethylene as a function of sliding velocity, normal force and humidity. The pattern is a square lattice of square cavities with sub-mm spacing. We find that dynamic friction decreases compared to non-patterned surfaces, since stress concentrations lead to anticipated detachment, and that stick-slip behavior is also affected. Friction increases with humidity, and the onset of stick-slip events occurs in the high humidity regime. Experimental results are compared with numerical simulations of a simplified 2-D spring-block model. A good qualitative agreement can be obtained by introducing a deviation from the linear behavior of the Amontons-Coulomb law with the load, due to a saturation in the effective contact area with pressure. This also leads also to the improvement of the quantitative results of the spring-block model by reducing the discrepancy with the experimental results, indicating the robustness of the adopted simplified approach, which could be adopted to design patterned surfaces with controlled friction properties.

Volume increase between the reactants and the products of alkali silica reaction could reach up to 100%. Taking place inside the aggregates, ASR imposes internal pressure on the surrounding material. In the current paper, we study the possibility of crack growth due to such internal loading. This study is done by employing a semi-analytical mechanical model comprising an elastic solution to a well-known Eshelby problem and a linear elastic fracture mechanics solution to a ring-shaped crack encircling a spheroidal inclusion. The proposed method implies the availability of pre-existing micro-fissures within the aggregate. The study reveals dependence of the crack growing potential on the spheroid's shape: the larger is the ASR pocket - the longer crack it can open. Two most critical shapes, causing a highest stress intensity factor and developing the longest crack, are a sphere and a spheroid with 1/4 aspect ratio respectively. The size analysis of the problem suggests a critical spheroid's radius below which no crack growth is expected. For a chosen material properties and expansion value, such radius lays in the range between 0.1 and 1 micrometer. Independently of the expansion value and the shape of the pocket of ASR product, the maximum crack length has a power-law dependence on the size of a spheroid. All the theoretical predictions are confirmed by a numerical model based on the combination of the finite element method and cohesive element approach.

Frictional sliding is an intrinsically complex phenomenon, emerging from the interplay between driving forces, elasto-frictional instabilities, interfacial nonlinearity and dissipation, material inertia and bulk geometry. We show that homogeneous rate-and-state dependent frictional systems, driven at a prescribed boundary velocity -- as opposed to a prescribed stress -- in a range where the frictional interface is rate-weakening, generically host self-healing slip pulses, a sliding mode not yet fully understood. Such velocity-driven frictional systems are then shown to exhibit coarsening dynamics saturated at the system length in the sliding direction, independently of the system's height, leading to steadily propagating pulse trains. The latter may be viewed as a propagating phase-separated state, where slip and stick characterize the two phases. While pulse trains' periodicity is coarsening-limited by the system's length, the single pulse width, characteristic slip velocity and propagation speed exhibit rich properties, which are comprehensively understood using theory and extensive numerics. Finally, we show that for sufficiently small system heights, pulse trains are accompanied by periodic elasto-frictional instabilities., Comment: 18 page, 9 figures (including appendices)

Frictional contacts lead to the formation of a surface layer called the third body, consisting of wear particles and structures resulting from their agglomerates. Its behavior and properties at the nanoscale control the macroscopic tribological performance. It is known that wear particles and surface topography evolve with time and mutually influence one another. However, the formation of the mature third body is largely uncharted territory and the properties of its early stages are unknown. Here we show how a third body initially consisting of particles acting as roller bearings transitions into a shear-band-like state by forming adhesive bridges between the particles. Using large-scale atomistic simulations on a brittle model material, we find that this transition is controlled by the growth and increasing disorganization of the particles with increasing sliding distance. Sliding resistance and wear rate are at first controlled by the surface roughness, but upon agglomeration wear stagnates and friction becomes solely dependent on the real contact area in accordance with the plasticity theory of contact by Bowden and Tabor.

The transition from quasi-static slip growth to dynamic rupture propagation constitutes one possible scenario to describe earthquake nucleation. If this transition is rather well understood for homogeneous faults, how the friction properties of multiscale asperities may influence the overall stability of seismogenic faults remains largely unclear. Combining classical nucleation theory and concepts borrowed from condensed matter physics, we propose a comprehensive analytical framework that predicts the influence of heterogeneities of weakening rate on the nucleation length $L_c$ for linearly slip-dependent friction laws. Model predictions are compared to nucleation lengths measured from 2D dynamic simulations of earthquake nucleation along heterogeneous faults. Our results show that the interplay between frictional properties and the asperity size gives birth to three instability regimes (local, extremal, and homogenized), each related to different nucleation scenarios, and that the influence of heterogeneities at a scale far lower than the nucleation length can be averaged.

We develop an analytical model of adhesive wear between two unlubricated rough surfaces, forming micro-contacts under normal load. The model is based on an energy balance and a crack initiation criteria. We apply the model to the problem of self-affine rough surfaces under normal load, which we solve using the boundary element method. We discuss how self-affinity of the surface roughness, and the complex morphology of the micro-contacts that emerge for a given contact pressure, challenge the definition of contact junctions. Indeed, in the context of adhesive wear, we show that elastic interactions between nearby micro-contacts can lead to wear particles whose volumes enclose the convex hull of these micro-contacts. We thereby obtain a wear map describing the instantaneous produced wear volume as a function of material properties, roughness parameters and loading conditions. Three distinct wear regimes can be identified in the wear map. In particular, the model predicts the emergence of a severe wear regime above a critical contact pressure, when interactions between micro-contacts are favored., Comment: 16 pages, 14 figures, published in Tribology Letters

Reversibility is of paramount importance in the correct representation of surface peeling in various physical settings, ranging from motility in nature, to gripping devices in robotic applications, and even to sliding of tectonic plates. Modeling the detachment-reattachment sequence, known as stick-slip, imposes several challenges in a continuum framework. Here we exploit customized reversible cohesive elements in a hybrid finite element model that can handle occurrence of snap-through instabilities. The simulations capture various peeling phenomena that emerge in experimental observations, where layers are pulled from a flat, rigid substrate in the direction parallel to the surface. For long layers, periodicity in reattachment is shown to develop and is linked to the concept of Schallamach waves. Further, the connection between surface properties and stick-slip behavior is investigated: we find that stick-slip is linked to the propensity of the interface to localize deformation and damage. Beyond elucidating the various peeling behaviors and the detachment modes, the computational framework developed here provides a straightforward approach for investigation of complex delamination processes, which can guide development of future applications across different scales and in various settings.

We conduct boundary element simulations of a contact problem consisting of an elastic medium subject to tangential load. Using a particle swarm optimization algorithm, we find the optimal shape and location of the micro-contacts to maximize for a given load the stored elastic energy contributing to the removal of a spherical particle contained in between the micro-contacts. We propose an ice scream scoop as an application of this optimization process., Comment: 7 pages

Friction and wear are important phenomena occurring in all devices with moving parts. While their origin and the way they evolve over time are not fully understood, they are both intimately linked to surface roughness. Guided by pin-on-disc experiments, we present the steps giving rise to the formation of surface roughness on silica, first by the creation of roughly spherical wear particles whose size is related to a critical length scale governing the transition between ductile and brittle behavior, then by the accumulation of these small particles into a larger third body layer, or gouge. We show that, for the explored range of loading conditions, the surface roughness evolves toward a common steady state under unlubricated wear regardless of the initial surface roughness, hinting toward the possible predictability of roughness evolution in wearing components. The friction coefficient is shown to be related to the surface roughness and its time evolution is discussed as well., Comment: 12 pages, 10 figures, published in Wear

Wear is well known for causing material loss in a sliding interface. Available macroscopic approaches are bound to empirical fitting parameters, which range several orders of magnitude. Major advances in tribology have recently been achieved via Molecular Dynamics, although its use is strongly limited by computational cost. Here, we propose a study of the physical processes that lead to wear at the scale of the surface roughness, where adhesive junctions are formed between the asperities on the surface of the materials. Using a brittle formulation of the variational phase-field approach to fracture, we demonstrate that the failure mechanisms of an adhesive junction can be linked to its geometry. By imposing specific couplings between the damage and the elastic energy, we further investigate the triggering processes underlying each failure mechanism. We show that a large debris formation is mostly triggered by tensile stresses while shear stresses lead to small or no particle formation. We also study groups of junctions and discuss how microcontact interactions can be favored in some geometries to form macro-particles. This leads us to propose a classification in terms of macroscopic wear rate. Although based on a continuum approach, our phase-field calculations are able to effectively capture the failure of adhesive junctions, as observed through discrete Molecular Dynamics simulations., Comment: 20 pages, 13 figures

The alkali-silica reaction (ASR) in concrete is a chemical reaction, which produces an expansive product, generally called ''ASR gel'', and causes cracking and damage in concrete over time. Affecting numerous infrastructures all around the world, ASR has been the topic of much research over the past decades. In spite of that, many aspects of this reaction are still unknown. In this numerical-investigation paper, a three-dimensional concrete meso-structure model is simulated using the finite-element method. Coarse aggregates, cement paste, and ASR gel are explicitly represented. A temperature dependent eigen-strain is applied on the simulated ASR gel pockets to capture their expansive behavior. This applies pressure on the surrounding aggregates and the cement paste, leading to cracks initiation and propagation. Free expansion of concrete specimens due to ASR is modeled and validated using experimental data. Influence of different key factors on damage generation in aggregates and paste and macroscopic expansion are discussed.

A finite-element approach based on the first-order FE 2 homogenisation technique is formulated to analyse the alkali-silica reaction-induced damage in concrete structures, by linking the concrete degradation at the macro-scale to the reaction extent at the meso-scale. At the meso-scale level, concrete is considered as a heterogeneous material consisting of aggregates embedded in a mortar matrix. The mechanical effects of the Alkali-Silica Reaction (ASR) are modelled through the application of temperature-dependent eigenstrains in several localised spots inside the aggregates and the mechanical degradation of concrete is modelled using continuous damage model, which is capable of reproducing the complex ASR crack networks. Then, the effective stiffness tensor and the effective stress tensor for each macroscopic finite element are computed by homogenising the mechanical response of the corresponding representative volume element (RVE), thus avoiding the use of phenomenological constitutive laws at the macro-scale. Convergence between macro- and meso-scales is achieved via an iterative procedure. A 2D model of an ASR laboratory specimen is analysed as a proof of concept. The model is able to account for the loading applied at the macro-scale and the ASR-product expansion at the meso-scale. The results demonstrate that the macroscopic stress state influences the orientation of damage inside the underlying RVEs. The effective stiffness becomes anisotropic in cases where damage is aligned inside the RVE.

In order to develop predictive wear laws, relevant material parameters and their influence on the wear rate need to be identified. Despite decades of research, there is no agreement on the mathematical form of wear equations and even the simplest models, such as Archard's, contain unpredictable fit parameters. Here, we propose a simple model for adhesive wear in dry sliding conditions that contains no fit parameters and is only based on material properties and surface parameters. The model connects elastoplastic contact solutions with the insight that volume detachment from sliding surfaces occurs in the form of wear particles, the minimum size of which can be estimated. A novelty of the model is the explicit tracking of the sliding process, which allows us to meaningfully connect particle emission rates and sizes to the macroscopic wear rate. The results are qualitatively promising, but we identify the necessity for more controlled wear experiments and the parameters needed from such work in order to fully verify and improve our model.

Adhesion between two bodies is a key parameter in wear processes. At the macroscale, strong adhesive bonds are known to lead to high wear rates, as observed in clean metal-on-metal contact. Reducing the strength of the interfacial adhesion is then desirable, and techniques such as lubrication and surface passivation are employed to this end. Still, little is known about the influence of adhesion on the microscopic processes of wear. In particular, the effects of interfacial adhesion on the wear particle size and on the surface roughness evolution are not clear, and are therefore addressed here by means of molecular dynamics simulations. We show that, at short timescales, the surface morphology and not the interfacial adhesion strength dictates the minimum size of wear particles. However, at longer timescales, adhesion alters the particle motion and thus the wear rate and the surface morphology.

The wear volume is known to keep increasing during frictional processes, and Archard notably proposed a model to describe the probability of wear particle formation upon asperity collision in a two-body contact configuration. While this model is largely adopted in the investigations of wear, the presence of wear debris trapped between the surfaces changes the system into a three-body contact configuration already since the early stages of the process. In such a configuration, a significant amount of wear is produced at the interface between the trapped debris and the sliding bodies. Here, relying on analytical models, we develop a framework that describes crack growth in a three-body configuration at the particle-surface interface. We then show that crack growth is favoured within the sliding surfaces, instead of within the debris particle, and test such result by means of numerical simulations with a phase-field approach to fracture. This leads to an increase in the wear volume and to debris particle accretion, rather than its break down. The effects of adhesion, coefficient of friction, and ratio of the applied global tangential and normal forces are also investigated.

When two solids start rubbing together, frictional sliding initiates in the wake of slip fronts propagating along their surfaces in contact. This macroscopic rupture dynamics can be successfully mapped on the elastodynamics of a moving shear crack. However, this analogy breaks down during the nucleation process, which develops at the scale of surface asperities where microcontacts form. Recent atomistic simulations revealed how a characteristic junction size selects if the failure of microcontact junctions either arises by brittle fracture or by ductile yielding. This work aims at bridging these two complementary descriptions of the onset of frictional slip existing at different scales. We first present how the microcontacts failure observed in atomistic simulations can be conveniently "coarse-grained" using an equivalent cohesive law. Taking advantage of a scalable parallel implementation of the cohesive element method, we study how the different failure mechanisms of the microcontact asperities interplay with the nucleation and propagation of macroscopic slip fronts along the interface. Notably, large simulations reveal how the failure mechanism prevailing in the rupture of the microcontacts (brittle versus ductile) significantly impacts the nucleation of frictional sliding and, thereby, the interface frictional strength. This work paves the way for a unified description of frictional interfaces connecting the recent advances independently made at the micro- and macroscopic scales., Comment: Postprint (minor changes)

The detachment of material in an adhesive wear process is driven by a fracture mechanism which is controlled by a critical length-scale. Previous efforts in multi-asperity wear modeling have applied this microscopic process to rough elastic contact. However, experimental data shows that the assumption of purely elastic deformation at rough contact interfaces is unrealistic, and that asperities in contact must deform plastically to accommodate the large contact stresses. We therefore investigate the consequences of plastic deformation on the macro-scale wear response using novel elastoplastic contact simulations. The crack nucleation process at a rough contact interface is analyzed in a comparative study with a classical $J_2$ plasticity approach and a saturation plasticity model. We show that plastic residual deformations in the $J_2$ model heighten the surface tensile stresses, leading to a higher crack nucleation likelihood for contacts. This effect is shown to be stronger when the material is more ductile. We also show that elastic interactions between contacts can increase the likelihood of individual contacts nucleating cracks, irrespective of the contact constitutive model. This is supported by a statistical approach we develop based on a Greenwood--Williamson model modified to take into account the elastic interactions between contacts and the shear strength of the contact junction., Comment: 18 pages, 12 figures

Numerous laboratory experiments have demonstrated the dependence of the friction coefficient on the interfacial slip rate and the contact history, a behavior generically called rate and state friction. Although numerical models have been widely used for analyzing rate and state friction, in general they consider infinite elastic domains surrounding the sliding interface and rely on boundary integral formulations. Much less work has been dedicated to modeling finite size systems to account for interactions with boundaries. This paper investigates rate and state frictional interfaces in the context of finite size systems with the finite element method in explicit dynamics. We investigate the long term behavior of the sliding interface for two different friction laws: a velocity weakening law, for which the friction monotonously decreases with increasing sliding velocity, and a velocity weakening-strengthening law, for which the friction coefficient first decreases but then increases above a critical velocity. We show that for both friction laws at finite times, that is before wave reflections from the boundaries come back to the sliding interface, a temporary steady state sliding is reached, with a well-defined stress drop at the interface. This stress drop gives rise to a stress concentration and leads to an analogy between friction and fracture. However, at longer times, that is after multiple wave reflections, the stress drop is essentially zero, resulting in losing the analogy with fracture mechanics. Finally, the simulations reveal that velocity weakening is unstable at long time scales, as it results in an acceleration of the sliding blocks. On the other hand, velocity weakening-strengthening reaches a steady state sliding configuration.

Current engineering wear models are often based on empirical parameters rather than built upon physical considerations. Here, we look for a physical description of adhesive wear at the microscale, at which the interaction between two surfaces comes down to the contact of asperities. Recent theoretical work has shown that there is a critical micro-contact size above which it becomes energetically favorable to form a wear particle. We extend this model by taking into consideration the elastic interaction of multiple closely-spaced micro-contacts in 2D, with different sizes and separation distances. Fundamental contact mechanics solutions are used to evaluate the elastic energy stored by shearing the micro-contacts, and the stored energy is compared to the energy needed to detach a single joined debris particle or multiple debris particles under the micro-contacts. Molecular dynamics simulations are used to test the predictions of the outcome for various sets of parameters. Our model provides simple criteria to evaluate the energetic feasibility of the different wear formation scenarios. Those criteria can be used to rationalize the transition between mild and severe wear regimes and help define the notion of asperity., Comment: 9 pages, 11 figures

The failure of frictional interfaces -- the process of frictional rupture -- is widely assumed to feature crack-like properties, with far-reaching implications for various disciplines, ranging from engineering tribology to earthquake physics. An important condition for the emergence of a crack-like behavior is the existence of stress drops in frictional rupture, whose basic physical origin has been recently elucidated. Here we show that for generic and realistic frictional constitutive relations, and once the necessary conditions for the emergence of an effective crack-like behavior are met, frictional rupture dynamics are approximately described by a crack-like, fracture mechanics energy balance equation. This is achieved by independently calculating the intensity of the crack-like singularity along with its associated elastic energy flux into the rupture edge region, and the frictional dissipation in the edge region. We further show that while the fracture mechanics energy balance equation provides an approximate, yet quantitative, description of frictional rupture dynamics, interesting deviations from the ordinary crack-like framework -- associated with non-edge-localized dissipation -- exist. Together with the recent results about the emergence of stress drops in frictional rupture, this work offers a comprehensive and basic understanding of why, how and to what extent frictional rupture might be viewed as an ordinary fracture process. Various implications are discussed., Comment: v2: slightly revised single-edge singularity analysis (see revised Figs. 3 & S2 and Sect. S-1), new Fig. S4, new discussion of an analytic expression for the effective fracture energy (see Eq. S10). See also arXiv:1906.11533

The process of frictional rupture, i.e. the failure of frictional systems, abounds in the technological and natural world around us, ranging from squealing car brake pads to earthquakes along geological faults. A general framework for understanding and interpreting frictional rupture commonly involves an analogy to ordinary crack propagation, with far-reaching implications for various disciplines, from engineering tribology to geophysics. An important feature of the analogy to cracks is the existence of a reduction in the stress-bearing capacity of the ruptured interface, i.e. of a drop from the applied stress, realized far ahead of a propagating rupture, to the residual stress left behind it. Yet, how and under what conditions such finite and well-defined stress drops emerge from basic physics are not well understood. Here we show that for rapid rupture a stress drop is directly related to wave radiation from the frictional interface to the bodies surrounding it and to long-range bulk elastodynamics, and not exclusively to the physics of the contact interface. Furthermore, we show that the emergence of a stress drop is a transient effect, affected by the wave travel time in finite systems and by the decay of long-range elastic interactions. Finally, we supplement our results for rapid rupture with predictions for slow rupture. All of the theoretical predictions are supported by available experimental data and by extensive computations. Our findings elucidate the origin of stress drops in frictional rupture, i.e. they offer a comprehensive and fundamental understanding of why, how and to what extent frictional rupture might be viewed as an ordinary fracture process., Comment: v2: a new analysis of the spatiotemporal integral contribution to the stress drop (Sect. IIB and Appendix A7). See also: arXiv:1907.04376

Engineering wear models are generally empirical and lack connections to the physical processes of debris generation at the nanoscale to microscale. Here, we thus analyze wear particle formation for sliding interfaces in dry contact with full and reduced adhesion. Depending on the material and interface properties and the local slopes of the surfaces, we find that colliding surface asperities can either deform plastically, form wear particles, or slip along the contact junction surface without significant damage. We propose a mechanism map as a function of material properties and local geometry, and confirm it using quasi-two-dimensional and three-dimensional molecular dynamics and finite-element simulations on an amorphous, siliconlike model material. The framework developed in the present paper conceptually ties the regimes of weak and strong interfacial adhesion together and can explain that even unlubricated sliding contacts do not necessarily lead to catastrophic wear rates. A salient result of the present paper is an analytical expression of a critical length scale, which incorporates interface properties and roughness parameters. Therefore, our findings provide a theoretical framework and a quantitative map to predict deformation mechanisms at individual contacts. In particular, contact junctions of sizes above the critical length scale contribute to the debris formation., Comment: 17 pages, 19 figures

The contact of solids with rough surfaces plays a fundamental role in physical phenomena such as friction, wear, sealing, and thermal transfer. However, its simulation is a challenging problem due to surface asperities covering a wide range of length-scales. In addition, non-linear local processes, such as plasticity, are expected to occur even at the lightest loads. In this context, robust and efficient computational approaches are required. We therefore present a novel numerical method, based on integral equations, capable of handling the large discretization requirements of real rough surfaces as well as the non-linear plastic flow occurring below and at the contacting asperities. This method is based on a new derivation of the Mindlin fundamental solution in Fourier space, which leverages the computational efficiency of the fast Fourier transform. The use of this Mindlin solution allows a dramatic reduction of the memory imprint (as the Fourier coefficients are computed on-the-fly), a reduction of the discretization error, and the exploitation of the structure of the functions to speed up computation of the integral operators. We validate our method against an elastic-plastic FEM Hertz normal contact simulation and showcase its ability to simulate contact of rough surfaces with plastic flow.

The onset of rapid slip along initially quiescent frictional interfaces, the process of `earthquake nucleation', and dissipative spatiotemporal slippage dynamics play important roles in a broad range of physical systems. Here we first show that interfaces described by generic friction laws feature stress-dependent steady-state slip pulse solutions, which are unstable in the quasi-1D approximation of thin elastic bodies. We propose that such unstable slip pulses of linear size $L^*$ and characteristic amplitude are `critical nuclei' for rapid slip in a non-equilibrium analogy to equilibrium first-order phase transitions, and quantitatively support this idea by dynamical calculations. We then perform 2D numerical calculations that indicate that the nucleation length $L^*$ exists also in 2D, and that the existence of a fracture mechanics Griffith-like length $L_G\!<\!L^*$ gives rise to a richer phase-diagram that features also sustained slip pulses., Comment: Updated Figs. 3, 5 and S3, added scaling theory for $L^*$, added references

We study frictionless and non-adhesive contact between elastic half-spaces with self-affine surfaces. Using a recently suggested corrective technique, we ensure an unprecedented accuracy in computation of the true contact area evolution under increasing pressure. This accuracy enables us to draw conclusions on the role of the surface's spectrum breadth (Nayak parameter) in the contact area evolution. We show that for a given normalized pressure, the contact area decreases logarithmically with the Nayak parameter. By linking the Nayak parameter with the Hurst exponent (or fractal dimension), we show the effect of the latter on the true contact area. This effect, undetectable for surfaces with poor spectral content, is quite strong for surfaces with rich spectra. Numerical results are compared with analytical models and other available numerical results. A phenomenological equation for the contact area growth is suggested with coefficients depending on the Nayak parameter. Using this equation, the pressure-dependent friction coefficient is deduced based on the adhesive theory of friction. Some observations on Persson's model of rough contact, whose prediction does not depend on Nayak parameter, are reported. Overall, the paper provides a unifying picture of rough elastic contact and clarifies discrepancies between preceding results.

We introduce a corrective function to compensate errors in contact area computations coming from mesh discretization. The correction is based on geometrical arguments and requires only one additional quantity to be computed: the length of contact/non-contact interfaces. The new technique enables us to evaluate accurately the true contact area using a coarse mesh for which the shortest wavelength in the surface spectrum reaches the grid size. The validity of the approach is demonstrated for surfaces with different fractal dimensions and different spectral content using a properly designed mesh convergence test. In addition, we use a topology preserving smoothing technique to adjust the morphology of contact clusters obtained with a coarse grid., Comment: 17 pages

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.

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 figures

We carry out a statistically meaningful study on self-affine rough surfaces in elastic frictionless non-adhesive contact. We study the evolution of the true contact area under increasing squeezing pressure. Rough surfaces are squeezed from zero up to full contact, which enables us to compare the numerical results both with asperity based models at light pressures and with Persson's contact model for the entire range of pressures. Through the contact perimeter we estimate the error bounds inherent to contact area calculation in discrete problems. A large number of roughness realizations enables us to compute reliably the derivative of the contact area with respect to the pressure. In contrast to Persson's model and in agreement with asperity based models, we demonstrate that at light pressures it is a decreasing convex function. The nonlinearity of the contact area evolution, preserved for the entire range of pressures, is especially strong close to infinitesimal contact. This fact makes difficult an accurate investigation of contact at light pressure and prevents the evaluation of the proportionality coefficient, which is predicted by analytical models. A good agreement of numerical results with Persson's model is obtained for the shape of the area-pressure curve especially near full contact. We investigate the effects of the cutoff wavenumbers in surface spectrum onto the contact area evolution. Nayak's parameter is one of the central characteristics of roughness, it also plays an important role in rough contact mechanics, but its effect is significantly weaker than predicted by asperity based models. We give a detailed derivation of a new phenomenological contact evolution law; also we derive formulae that link Nayak's parameter and density of asperities with Hurst exponent and cutoffs wavenumbers., Comment: 47 pages, 17 figures, 72 references

A numerical analysis of mechanical frictionless contact between rough self-affine elastic manifolds was carried out. It is shown that the lower cutoff wavenumber in surface spectra is a key parameter controlling the representativity of the numerical model. Using this notion we demonstrate that for representative surfaces the evolution of the real contact area with load is universal and independent of the Hurst roughness exponent. By introducing a universal law containing three constants, we extend the study of this evolution beyond the limit of infinitesimal area fractions., Comment: 18 pages, 29 references, 4 figures

The dynamic initiation of sliding at planar interfaces between deformable and rigid solids is studied with particular focus on the speed of the slip front. Recent experimental results showed a close relation between this speed and the local ratio of shear to normal stress measured before slip occurs (static stress ratio). Using a two-dimensional finite element model, we demonstrate, however, that fronts propagating in different directions do not have the same dynamics under similar stress conditions. A lack of correlation is also observed between accelerating and decelerating slip fronts. These effects cannot be entirely associated with static local stresses but call for a dynamic description. Considering a dynamic stress ratio (measured in front of the slip tip) instead of a static one reduces the above-mentioned inconsistencies. However, the effects of the direction and acceleration are still present. To overcome this we propose an energetic criterion that uniquely associates, independently on the direction of propagation and its acceleration, the slip front velocity with the relative rise of the energy density at the slip tip., Comment: 15 pages, 6 figures