Friction Law Research Articles

Constraining Parameter Uncertainties in the Coulomb Rate–State Approach: A Case Study of Seismicity in the Longmenshan Region after the 2008 Mw 7.9 Wenchuan Earthquake, China

Abstract The rate–state frictional law, coupled with the Coulomb failure stress changes (ΔCFS), is one of the most popular physics-based models to forecast seismicity rate changes following a major earthquake. However, its effectiveness is hampered by parameter uncertainties. To seek possible solutions for such uncertainties, this article carried out retrospective forecasts of the decade-long seismicity in the Longmenshan region, China, after the 2008 Mw 7.9 Wenchuan earthquake, and proposed methods to constrain parameter uncertainties. First, we derived spatially variable ta and Aσ from fault-slip rates. This method not only provides observational constraints for these two parameters but also reflects spatial variations of fault rate–state properties. Second, although both complete and declustered background catalogs are common in Coulomb rate–state forecasts, this study demonstrated that declustering avoids false alerts of seismicity rate increase that resulted from temporary seismicity fluctuations in the background catalog. Finally, we extended the model from its typical application based on a stress step (the coseismic stress change) to a calculation that allows a more complex stress evolution (the postseismic viscoelastic stress change). With these methods to constrain parameter uncertainties, we are able to obtain more reliable forecasts.

A generalization of B-differentiable equations method for elastoplastic contact problems with dual mortar discretization

In this study, a novel method based on the B-differentiable equations (BDEs) is proposed to solve the three-dimensional (3D) static and dynamic elastoplastic contact problems with friction. The contact conditions are formulated as the BDEs system, which can precisely satisfy Coulomb’s law of friction. Based on the mortar segment-to-segment contact model, the contact constraints are enforced by dual Lagrange multipliers for nonconforming meshes of the contact boundary. The non-smooth and nonlinear system equations composed of equilibrium and contact equations are simultaneously solved by the B-differentiable damping Newton method (BDNM), which has global convergence property. In each iteration, firstly, some contact displacements and contact forces, called substituted contact variables, are expressed by others according to the different contact states and contact conditions. The equilibrium equation is then solved, where the stiffness matrix is updated by substituting the contact variables. At last, the substituted contact variables will be restored. In the proposed method, the contact flexibility matrix is no longer needed, which saves the time for calculating the contact flexibility matrix due to changes in the stiffness matrix resulting from the update of stresses in the iteration step. By comparing with the results obtained by finite element commercial software ANSYS, the accuracy of the proposed algorithm is verified, and the property of convergence and efficiency are demonstrated for the elastoplastic contact problems by several numerical examples.

On the dynamic Coulomb's frictional contact problem for thermo‐piezoelectric materials

AbstractIn this work, we study, from a variational point of view, a dynamic contact problem between a thermo‐piezoelectric body and a thermally conductive foundation. The normal compliance contact condition and Coulomb's friction law are employed to model the contact. We provide existence and uniqueness results of a weak solution to the model using adequate auxiliary problems, an abstract result on nonlinear first‐order evolution inequalities and Banach fixed point argument. Finally, the continuous dependence of the solution on the surface traction force and the surface electrical charge is studied.

The earthquake probabilities of the Liupanshan fault zone based on the Coulomb stress and friction constitutive law

As the forefront of the northeastern margin of the Tibetan Plateau, the Liupanshan area is subjected to significant Cenozoic deformation. Considering the background of historical earthquakes and the current seismic activity, the Liupanshan fault poses a risk of strong earthquakes. This paper analyzes the evolution of Coulomb stress induced by the coseismic and postseismic viscoelastic deformation of six historical strong earthquakes that occurred around the Liupanshan area since the twentieth century. Additionally, based on the disturbance in Coulomb stress and the background seismicity rate, the probability of earthquake occurrence is quantitatively evaluated using the frictional constitutive law. The results indicate that northern section of the Liupanshan fault has higher earthquake potential than the southern section, warranting close attention. These findings serve as a reference for medium- and long-term earthquake forecast and risk assessment in the vicinity of the Liupanshan fault.

Compliant mechanism design of combined aircraft wing for stable separation

Abstract Stable separation is a crucial condition that must be met in order for combined aircraft to successfully engage in cooperative flight. In order to achieve the desired fast and controlled separation, this paper proposes a novel design for a torque-driven compliant separation mechanism. By taking into account the compliance characteristics of a sinusoidal acceleration function curve, a mechanical model for the separation mechanism is developed. By utilising the Coulomb friction law, an accurate determination of the aerodynamic load distribution under various conditions is achieved. Subsequently, the relationship between the unlocking moment and the aerodynamic load is derived based on these findings. Through the utilisation of the finite element method, a model of the separation mechanism is generated. To ensure the safety and reliability of the compliant separation mechanism, the mechanical properties of the structural materials are thoroughly analysed under the maximum aerodynamic load. Subsequently, the separation mechanism structure is constructed and subjected to testing in order to showcase the compliance characteristics. In addition, this paper conducts a simulation to analyse the impact of flight speed and angle-of-attack on the separation process. By doing so, the optimal conditions for separation are determined. The methods and findings presented in this study have the potential to contribute valuable insights to the design of combined aircraft.

Tribological properties and evolutionary mechanisms of a nanocrystalline Al-Mg-Si alloy at room and cryogenic temperatures

A nanocrystalline Al-Mg-Si alloy model was established to reveal the temperature effect in nanotribology using atomistic simulations. Compared to room temperature, the samples at cryogenic temperature (CT) exhibited an increase in both the coefficient of friction (COF) and wear resistance. At CT, the thermal activation of interface atoms reduced, thereby inhibiting surface grain rotation and sliding within the atomistic model. There was grain coarsening (suppressed at CT) instead of grain refinement typically observed for surface grains. The results also indicated that changes in load influenced the COF response, with higher loads resulting in higher COF values. The friction laws in simulation and experiment are similar.

Moving contact problem of a functionally graded orthotropic coated half plane

AbstractThis paper develops a frictional moving contact model for a functionally graded (FG) orthotropic layer pressed by a rigid cylindrical punch. The FG orthotropic layer is fully bonded to the isotropic half-plane. The punch moves to the left on the layer at a constant subsonic velocity and a shear stress arises in the contact zone according to the Coulomb friction law. General expressions of displacements and stresses are derived with the help of the Fourier transform and Galilean transformation. Using boundary conditions, the moving contact problem is reduced to a Cauchy-type singular integral equation, the unknowns of which are contact stress and contact width. Gauss–Jacobi integration formula is used to solve the obtained singular integral equation. The effect of some parameters and material properties on the contact width, contact stress and in-plane stress are given in graphical forms and detailed numerical interpretations are presented.

Vibration characteristics of a bolted joint rotor-bearing system with rubbing and pedestal looseness coupled fault: Simulation and experiment

Rotor-stator rub and pedestal looseness are common faults in rotating machinery, such as large gas turbines that typically use bolted joint structures to fasten adjacent disks together. It is of great significance to investigate the vibration behaviors of the bolted joint rotor-bearing system with pedestal looseness and rubbing coupled fault, which can provide proofs for the fault diagnosis. In the present work, a dynamic model of the bolted joint rotor-bearing system with coupled fault is developed by using the finite element (FE) method, and then the vibration characteristic analysis is carried out. The rotor shafts that connected by the bolted joint element (BJE) are discretized as two sets of Timoshenko beam elements. The BJE is composed with four springs (two linear translational springs and two rotational springs with piecewise-linear stiffness). The interactions of rotor-stator contact are then modeled based on the Hertz contact theory and Coulomb's friction law, and the bearing forces are also calculated based on the Hertz contact theory. Moreover, the pedestal looseness fault is modeled through the piecewise-linear stiffness spring. The equations of motions of the bolted joint rotor-bearing system with coupled fault are then developed through FE theory and solved by the Newmark method. The dynamic responses are then calculated, and the frequency spectrums, bending stiffness of bolted joint, and vibration responses under different rotation speeds under the coupled fault are analyzed. The experimental study is finally conducted based on a bolted joint rotor test rig. Simulation results are in agreement with experiment results. The research results can provide a basis for understanding the rotor dynamic features of a bolted joint rotor system with coupled faults of rubbing and pedestal looseness.

Analysis of Fluid-Injection-Induced Seismicity Using a Dynamic Sliding Model Incorporating the Rate- and State-Dependent Friction Law

Summary Earthquakes can be triggered by fluid injection into underground formations. Fluid injection can cause large changes in the underground volume that exert stresses on nearby preexisting faults, leading to seismic activity. Assuming an increase in underground development activities in the future, our understanding of the mechanism underlying induced seismicity must be improved, and methods must be developed to properly assess the risk of seismic events. The objective of this study is to develop a seismicity prediction model that calculates the magnitude and timing of triggered earthquakes or seismic events occurring during various subsurface fluid injection activities. We developed an injection-induced seismicity analysis model that predicts the dynamic earthquake nucleation caused by changes in stress and pore pressure that occur during various subsurface activities. The governing equations consisting of the dynamic motion of the poroelastic spring-slider system, rate and state friction laws and pore pressure diffusion equation were solved using the embedded semi-implicit Runge-Kutta (SIRK) method. The dynamic sliding model was also incorporated into the finite element method (FEM) model, considering the variations in the stresses and pore pressures in the formation. A field case study was also conducted to compare the model results with typical microseismicity responses observed from hydraulic fracturing treatments in shale fields. Contrary to the popular understanding derived from Amonton’s law, the dynamic friction model revealed that a large normal stress on the fault leads to rapid sliding. A larger normal stress accumulates a large amount of elastic energy until it slips owing to fluid injection, nucleating large seismic waves. The poroelastic spring-slider model estimated reasonable microseismic magnitudes for hydraulic fracturing treatment but overestimated the time required to trigger a microseismic event under field conditions. To improve the analysis results, the poroelastic spring-slider model was coupled with a linear elastic FEM that considered the complex interplay of stress changes from hydraulic fracturing and the associated pore pressure variation in the formation. Compared with the field data, the coupled simulation model estimated a reasonable timing for the induced microseismic events when the increasing pore pressure during hydraulic fracturing penetrated deep into the formation. These findings suggest the existence of permeable natural fractures in the formation, which intensify early frictional sliding during treatment. The seismicity prediction model presented in this study simulates the magnitude and timing of seismic nucleation, helping to manage and mitigate the environmental impacts of induced seismicity during various subsurface development activities, such as oil and gas extraction, hydraulic fracturing, geothermal, and carbon dioxide sequestration. Moreover, the case study results imply that the time series of seismic events predicted by the model can be used to understand the possible fracture geometry and extent of fluid invasion for field applications.

Measurement and comparison of transient thermal boundary layer of a Local-Heated Mini-Channel under sub- and super-critical conditions

This study is focused on experimentally measuring the transient thermal boundary layer following a sudden heat addition to turbulent flow boundary in supercritical mini-channels. Interferometric diagnostics were performed using a visualization channel to capture the time-evolution boundary layer. The measured pressure drop shows good agreement with numerical iterations according to the friction laws. Transient scenarios have been extracted from the raw interferograms to facilitate quantitative comparison of boundary conditions, such as bulk temperature and heat flux: (1) Localized heat accumulation and a slowly thermal response have been observed in subcritical state; (2) Relatively high density of 1.2 kg/m3 and low temperature changes of 0.008 K are observed in cases with relatively shorter critical distances to the critical point, attributed to the sharp increase in heat capacity and thermal expansion coefficients. (3) A relatively fast thermal convective field is established at larger heat fluxes due to the rapid density shifts of 1.2 kg/m3 and associated flow acceleration in the boundary layer.

Analysis and Numerical Simulation of Time-Fractional Derivative Contact Problem with Friction in Thermo-Viscoelasticity

Abstract The objective of this study is to analyze a quasistatic frictional contact problem involving the interaction between a thermo-viscoelastic body and a thermally conductive foundation. The constitutive relation in our investigation is constructed using a fractional Kelvin–Voigt model to describe displacement behavior. Additionally, the heat conduction aspect is governed by a time-fractional derivative parameter that is associated with temperature. The contact is modeled using the Signorini condition, which is a version of Coulomb’s law for dry friction. We develop a variational formulation for the problem and establish the existence of its weak solution using a combination of techniques, including the theory of monotone operators, Caputo derivative, Galerkin method, and the Banach fixed point theorem. To demonstrate the effectiveness of our approach, we include several numerical simulations that showcase the performance of the method.

A proxy implementation of thermal pressurization for earthquake cycle modelling on rate-and-state faults

SUMMARY The reduction of effective normal stress during earthquake slip due to thermal pressurization of fault zone pore fluids is a significant fault weakening mechanism. Explicit incorporation of this process into frictional fault models involves solving the diffusion equations for fluid pressure and temperature outside the fault at each time step, which significantly increases the computational complexity. Here, we propose a proxy for thermal pressurization implemented through a modification of the rate-and-state friction law. This approach is designed to emulate the fault weakening and the relationship between breakdown energy and slip resulting from thermal pressurization and is appropriate for fully dynamic simulations of multiple earthquake cycles. It preserves the computational efficiency of conventional rate-and-state friction models, which in turn can enable systematic studies to advance our understanding of the effects of fault weakening on earthquake mechanics. In 2.5-D simulations of pulse-like ruptures on faults with finite seismogenic width, based on our thermal pressurization proxy, we find that the spatial distribution of slip velocity near the rupture front is consistent with the conventional square-root singularity, despite continued slip-weakening within the pulse, once the rupture has propagated a distance larger than the rupture width. An unconventional singularity appears only at shorter rupture distances. We further derive and verify numerically a theoretical estimate of the breakdown energy dissipated by our implementation of thermal pressurization. These results support the use of fracture mechanics theory to understand the propagation and arrest of very large earthquakes.

Simplified homogenization technique for nonlinear finite element analysis of in-plane loaded masonry walls

The present investigation focuses on the nonlinear modelling of masonry walls under in-plane loading. To conduct this macro-scale study, a homogenization technique is applied to derive the equivalent secant moduli according to damage and friction sliding at the micro scale level for both bed mortar joint and brick material constituents. In the homogenization process the representative unit cell of masonry panel with a regular arrangement is first defined. Then a simplified kinematical model is proposed to describe the behaviour of the homogenized unit-cell under combined axial and shear stresses. This is based on the cohesive model for pure mode I and II and by allowing for the Coulomb's friction law for bed joint and the scalar damage model is adopted for the brick material. In the first stage, the proposed homogenization technique is verified for various unit cell constituents at micro-scale level. In the second step, this homogenization procedure was incorporated into proposed in-house finite elements software. To achieve this end, a new beam finite element is formulated in the context of the fibre higher order shear deformation theory HSDT. Using the proposed finite elements model, a numerical investigation is conducted to check the validity of the proposed model against the available experimental and numerical investigations. This allowed for the study of the effect of the applied compressive load on the lateral ultimate load.

Stress Balance in Synthetic Serpentinized Peridotites Deformed at Subduction Zone Pressures

AbstractWeak serpentine minerals affect the mechanical behavior of serpentinized peridotites at depth, and may play a significant role in deformation localization within subduction zones, at local or regional scale. Mixtures of olivine with 5, 10, 20 and 50 vol. % fraction of antigorite, proxies for serpentinized peridotites, were deformed in axial shortening geometry under high pressures (ca. 2–5 GPa) and moderate temperatures (ca. 350°C), with in situ stress and strain measurements using synchrotron X‐rays. We evaluate the average partitioning of stresses at the grains scale within each phase (mineral) of the aggregate and compare with pure olivine aggregates in the same conditions. The in situ stress balance is different between low antigorite contents up to 10 vol. %, and higher contents above 20 vol. %. Microstructure and stress levels suggest the deformation mechanisms under these experimental conditions are akin to (semi)brittle and frictional processes. Unlike when close to dehydration temperatures, hardening of the aggregate is observed at low serpentine fractions, due to an increase in local stress concentrations. Below and above the 10–20 vol. % threshold, the stress state in the aggregate corresponds to friction laws already measured for pure olivine aggregates and pure antigorite aggregates respectively. As expected, the behavior of the two‐phase aggregate does not evolve as calculated from simple iso‐stress or iso‐strain bounds, and calls for more advanced physical models of two‐phase mixtures.

On an indentation of a circular cylindrical punch into an elastic half‐space under friction

AbstractThe paper discusses an axisymmetric contact problem of indentation of a circular cylindrical punch with a flat base into a homogeneous elastic half‐space with friction. It is assumed that the shear contact stresses are related to the normal contact pressure by the law of dry friction, in which the coefficient of friction depends on the radial coordinates of the points of the surfaces in contact and is directly proportional to them. With the help of rotation operators, the solution of the stated problem is reduced to the solution of a singular integral equation of the second kind with variable coefficients, and its closed solution in quadratures is constructed. The numerical calculations display the patterns of change in contact stresses and rigid displacement of the punch depending on the maximum value of the friction coefficient.

Rupture Jumping and Seismic Complexity in Models of Earthquake Cycles for Fault Stepovers with Off-Fault Plasticity

ABSTRACT Fault stepovers are prime examples of geometric complexity in natural fault zones that may affect seismic hazard by determining whether an earthquake rupture continues propagating or abruptly stops. However, the long-term pattern of seismicity near-fault stepovers and underlying mechanisms of rupture jumping in the context of earthquake cycles are rarely studied. Leveraging a hybrid numerical scheme combining the finite element and the spectral boundary integral methods, FEBE, we carry out fully dynamic simulations of sequences of earthquakes and aseismic slip for both compressive and tensile stepovers with off-fault plasticity. We consider a rate-and-state friction law for the fault friction and pressure-sensitive Drucker–Prager plasticity for the off-fault bulk response. We observe that the accumulation of plastic deformation, an indication of off-fault damage, is significantly different in the two cases, with more plastic deformation projected in the overlapping region for the tensile stepover. The seismic pattern for a tensile stepover is more complex than for a compressive stepover, and incorporating plasticity also increases complexity, relative to the elastic case. A tensile stepover with off-fault plasticity shows rupture segmentation, temporal clustering, and frequent rupture jumping from one fault to another. These results shed light on possible mechanisms of rupture jumping in fault stepovers as well as the long-term evolution of the fault zone.

Validating ensemble historical simulations of Upernavik Isstrøm (1985–2019) using observations of surface velocity and elevation

Abstract The future of tidewater glaciers in response to climate warming is one of the largest sources of uncertainty in the contribution of the Greenland ice sheet to global sea-level rise. In this study, we investigate the ability of an ice-sheet model to reproduce the past evolution of the velocity and surface elevation of a tidewater glacier, Upernavik Isstrøm, by prescribing front positions. To achieve this, we run two ensembles of simulations with a Weertman and a regularised-Coulomb friction law. We show that the ice-flow model has to include a reduction in friction in the first 15 km upstream of the ice front in fast-flowing regions to capture the trends observed during the 1985–2019 period. Without this process, the ensemble model overestimates the ice flow before the retreat of the front in 2005 and does not fully reproduce its acceleration during the retreat. This results in an overestimation of the total mass loss between 1985 and 2019 of 50% (300 vs 200 Gt). Using a variance-based sensitivity analysis, we show that uncertainties in the friction law and the ice-flow law have a greater impact on the model results than surface mass balance and initial surface elevation.

Determination of Penetration Depth and Excited Volume of Rubber in Klüppel Friction Theory from Friction Law

ABSTRACT This article aims to determine the excited volume and penetration depth in the theoretical friction model of rubber sliding on a corundum surface. The theoretical procedure of the Klüppel friction theory was implemented using the power spectral density of the corundum surface and viscoelastic model for rubber. The power spectral density was obtained with a power-law mode using the height difference correlation function parameters calculated from a surface measurement taken with a profilometer. Viscoelastic model parameters for the rubber were derived from a dynamic mechanical analyzer. Empirical law for friction coefficient obtained from the side force experiments and simulations of a laboratory abrasion tester (LAT 100) were used in this work. The friction coefficient from the theoretical procedure was matched with the empirical friction coefficient to estimate the penetration depth and excited volume of rubber. The correlation between the theoretical and empirical model was satisfactory. Estimating the penetration depth and excited volume of sliding rubber provides an insight into the contact conditions near surface asperities and the volume of rubber contributing to energy dissipation during the frictional process.

Initiation of motion for an isolated finite size particle in a rotating tank flow

The initiation of motion of an isolated finite size particle within a rotating fluid in a tank is investigated experimentally. The study leverages on the absence of slip velocity to explore the impact rotation has on the initiation of the motion of a particle. Results show that the motion, when it occurs, is purely radial. It is initiated when the centrifugal force acting on the particle overcomes the friction force exerted on it by the tank bottom. A model of friction force that reproduces the experimental data quite well is proposed. It is based on the classical Coulomb’s friction law and pressure considerations. Denoting rc as the radial position of a particle when its motion starts, the critical radius r∗c=rc/d essentially depends on the density ratio γ, the Froude number Frω=ω2d/g but also the ratio of the tank radius to the particle size: α=R/d . Frω is based on the particle diameter d, the shear velocity ΔU=ωd with ω the rotation rate and g the acceleration due to gravity. The agreement of the model with the experimental data is obtained for the same friction coefficient λ = 0.019 whatever the particles. The value is not far from the value 0.05 generally expected for lubricated smooth plastic/plastic friction coefficients. The model also shows that for a given tank radius a critical rotation speed exists below which, no initiation of motion can be expected.

Transient and Steady‐State Friction in Non‐Isobaric Conditions

AbstractThe frictional properties of faults control the initiation and propagation of earthquakes and the associated hazards. Although the ambient temperature and instantaneous slip velocity controls on friction in isobaric conditions are increasingly well understood, the role of normal stress on steady‐state and transient frictional behaviors remains elusive. The friction coefficient of rocks exhibits a strong dependence on normal stress at typical crustal depths. Furthermore, rapid changes in normal stress cause a direct effect on friction followed by an evolutionary response. Here, we derive a constitutive friction law that consistently explains the yield strength of rocks from atmospheric pressure to gigapascals while capturing the transient behavior following perturbations in normal stress. The model explains the frictional strength of a variety of sedimentary, metamorphic, and igneous rocks and the slip‐dependent response upon normal stress steps of Westerly granite bare contact and synthetic gouges made of quartz and a mixture of quartz and smectite. The nonlinear normal stress dependence of the frictional resistance may originate from the distribution of asperities that control the real area of contact. The direct and transient effects may be important for induced seismicity by hydraulic fracturing or for naturally occurring normal stress perturbations within fault zones in the brittle crust.