Coulomb Friction Law Research Articles

Vibro-Impact Problems with Dry Friction---Part I: Existence Result

We consider a discrete mechanical system with a nontrivial inertia operator subjected to a unilateral constraint with soft and frictional contact. Its dynamics is described in generalized coordinates by a measure-differential inclusion generalizing the usual formulation of Coulomb's friction law. An existence result for this problem is established. The key tool is the study of a time-stepping algorithm which allows one to construct a sequence of approximate trajectories, reproducing at the discrete level the main features of Coulomb's law.

Earthquake response of adjacent structures with viscoelastic and friction dampers

We study the seismic response of two adjacent structures connected with a dry friction damper. Each of them consists of a viscoelastic rod and a rigid block, which can slide without friction along the moving base. A simplified earthquake model is used for modeling the horizontal ground motion. Energy dissipation is taken by the presence of the friction damper, which is modeled by the set-valued Coulomb friction law. Deformation of viscoelastic rods during the relative motion of the blocks represents another way of energy dissipation. The constitutive equation of a viscoelastic body is described by the fractional Zener model, which includes fractional derivatives of stress and strain. The problem merges fractional derivatives as non-local operators and theory of set-valued functions as the non-smooth ones. Dynamical behaviour of the problem is governed by a pair of coupled multi-valued differential equations. The posed Cauchy problem is solved by use of the Gr?nwald-Letnikov numerical scheme. The behaviour of the system is analyzed for different values of system parameters.

On the limits of quasi-static analysis for a simple Coulomb frictional oscillator in response to harmonic loads

Due to the nonlinearity of the Coulomb friction law, even the simplest models of interfaces in contact show a very rich dynamic solution. It is often desirable, especially if the frequency of loading is only a fraction of the first natural frequency of the system, to replace a full dynamic analysis with a quasi-static one, which obviously is much simpler to obtain. In this work, we study a simple Coulomb frictional oscillator with harmonic tangential load, but with constant normal load. It is found that the quasi-static solution (which has only 2 stops) captures approximately the displacement peak as long as the forcing frequency is low enough for the dynamic solution to have 2 or, even better, more than 2 stops. Instead, the velocity peak is not correctly estimated, since the velocity becomes highly irregular due to the stick–slip stops, whose number increases without limit for zero frequency. In this sense, the classical quasi-static solution, obtaining by cancelling inertia terms in the equilibrium equations, does not coincide with the limit of the full dynamic solution at low frequencies. The difference is not eliminated by adding a small amount of viscous damping, as only with critical damping, the dynamic solution is very close to the quasi-static one. Additional discrepancies arise above a limit frequency whose value depends on the ratio of the tangential load to the limit one for sliding, and correspond to when the dynamic solution turns from 2 to 0 stop per cycle.

Fatigue hysteresis behavior of unidirectional C/SiC ceramic–matrix composite at room and elevated temperatures

Abstract In this paper, the tensile fatigue hysteresis behavior of unidirectional C/SiC composite at room and elevated temperatures in air atmosphere has been investigated. The fatigue hysteresis modulus and fatigue hysteresis loss energy corresponding to different number of applied cycles have been analyzed. Based on the damage mechanism of fiber slipping relative to matrix in the interface debonded region, the fatigue hysteresis loops models based upon the Coulomb friction law instead of a constant fiber/matrix interface shear stress usually assumed in the hysteresis analysis, have been developed. The relationships between the fatigue hysteresis loss energy, fatigue hysteresis loops, interface frictional slip and interface frictional coefficient have been established. When the fiber/matrix interface frictional coefficient degrades, the fatigue hysteresis loss energy first increases to the maximum value, and then decreases to zero; the fatigue hysteresis loops correspond to different interface frictional slip cases. By comparing the experimental fatigue hysteresis loss energy with theoretical computational values, the fiber/matrix interface frictional coefficient corresponding to different number of applied cycles, has been obtained. The variations of fatigue hysteresis modulus, fatigue hysteresis loss energy and interface frictional coefficient as a function of cycle number, have been analyzed for different fatigue peak stresses and test conditions. The fatigue hysteresis loops predicted using the hysteresis loops models and estimated fiber/matrix interface frictional coefficient agreed with experimental results.

The R-linear convergence rate of an algorithm arising from the semi-smooth Newton method applied to 2D contact problems with friction

The goal is to analyze the semi-smooth Newton method applied to the solution of contact problems with friction in two space dimensions. The primal-dual algorithm for problems with the Tresca friction law is reformulated by eliminating primal variables. The resulting dual algorithm uses the conjugate gradient method for inexact solving of inner linear systems. The globally convergent algorithm based on computing a monotonously decreasing sequence is proposed and its R-linear convergence rate is proved. Numerical experiments illustrate the performance of different implementations including the Coulomb friction law.

A class of dynamic contact problems with Coulomb friction in viscoelasticity

The aim of this work is to study a class of nonsmooth dynamic contact problems which model several surface interactions, including relaxed unilateral contact conditions, adhesion and Coulomb friction laws, between two viscoelastic bodies of Kelvin–Voigt type. An abstract formulation which generalizes these problems is considered and the existence of a solution is proved by using Ky Fan’s fixed point theorem, suitable approximation properties, several estimates and compactness arguments.

DEM Simulation of the Impact of Ultrafine Glass Particles on the Partition Wall of the Multichannel Cyclone

This article focuses on the simulation of the impact of ultrafine particles on the rigid partition wall of the multichannel cyclone by applying the discrete element method (DEM). Ultrafine particles with diameters ranging from 0.1 to 10 µm are assumed to be most dangerous pollutants because of their property to easily penetrate the human circulatory system through the respiratory tract. An emphasis is placed on demonstrating the stick-rebound behavior of glass pollutant particles. The numerical experiment simulating the oblique impact load induced by mechanically generated inlet air flow has been numerically conducted. Hertz contact and short-range van der Waals attraction are the main forces describing normal contact, while their tangential counterpart is defined according to Coulomb friction law and by considering normal compression and attraction. Both normal and tangential models are modified to comprise the effect of energy dissipation due to adhesion hysteresis. Simulation results provide a theoretical framework for designing of the cyclone when a device is applied to capture ultrafine pollutant particles.

Effect of Fiber Poisson Contraction on Matrix Multicracking Evolution of Fiber-Reinforced Ceramic-Matrix Composites

An analytical methodology has been developed to investigate the effect of fiber Poisson contraction on matrix multicracking evolution of fiber-reinforced ceramic-matrix composites (CMCs). The modified shear-lag model incorporated with the Coulomb friction law is adopted to solve the stress distribution in the interface slip region and intact region of the damaged composite. The critical matrix strain energy criterion which presupposes the existence of an ultimate or critical strain energy limit beyond which the matrix fails has been adopted to describe matrix multicracking of CMCs. As more energy is placed into the composite, matrix fractures and the interface debonding occurs to dissipate the extra energy. The interface debonded length under the process of matrix multicracking is obtained by treating the interface debonding as a particular crack propagation problem along the fiber/matrix interface. The effects of the interfacial frictional coefficient, fiber Poisson ratio, fiber volume fraction, interface debonded energy and cycle number on the interface debonding and matrix multicracking evolution have been analyzed. The theoretical results are compared with experimental data of unidirectional SiC/CAS, SiC/CAS-II and SiC/Borosilicate composites.

Interface-resolved direct numerical simulation of the erosion of a sediment bed sheared by laminar channel flow

A numerical method based upon the immersed boundary technique for the fluid–solid coupling and on a soft-sphere approach for solid–solid contact is used to perform direct numerical simulation of the flow-induced motion of a thick bed of spherical particles in a horizontal plane channel. The collision model features a normal force component with a spring and a damper, as well as a damping tangential component, limited by a Coulomb friction law. The standard test case of a single particle colliding perpendicularly with a horizontal wall in a viscous fluid is simulated over a broad range of Stokes numbers, yielding values of the effective restitution coefficient in close agreement with experimental data. The case of bedload particle transport by laminar channel flow is simulated for 24 different parameter values covering a broad range of the Shields number. Comparison of the present results with reference data from the experiment of Aussillous et al. (2013) yields excellent agreement. It is confirmed that the particle flow rate varies with the third power of the Shields number once the known threshold value is exceeded. The present data suggests that the thickness of the mobile particle layer (normalized with the height of the clear fluid region) increases with the square of the normalized fluid flow rate.

Positioning Error Analysis Modeling of the Aircraft Skin

The low-stiffness aircraft skin is easy to deform during the positioning process. This paper presents a method of positioning error analysis for the aircraft skin. The positioning error analysis model is divided into two parts: rigid error analysis and flexible error analysis. For the rigid part, the deterministic locating method is applied to analysis the position error of the key characteristic point according to the production errors. For the flexible part, a static equilibrium equation of the skin is used to describe the relationship of the contact forces. And based on the principle of minimum complementary energy and Coulomb friction law, a nonlinear programming function is built to solve the passive contact forces and the skin deformation in each step. This method is more effective to describe the positioning process and obtain the positioning error.

Post-derailment dynamic behavior of railway vehicles travelling on a railway bridge during an earthquake

This study developed a post-derailment dynamic model of railway vehicles to investigate their post-derailment dynamic behavior when travelling on a railway bridge during an earthquake. The model is composed of three parts: multi-body dynamic model of the railway vehicle, a collision detection model, and a contact force model. The dynamic model considered all the various kinds of contacts that may occur during the derailment, such as wheel/track slab contacts, wheel/fastener contacts, and brake-disc/rail contacts. A collision detection model that was based on the tight-fitting oriented bounding box trees approach was adopted to detect the collision situations during the derailment. The contact force model was used to calculate the contact force based on a Hertz spring-damper and the Coulomb friction law. Using this model, the post-derailment dynamic behavior of railway vehicles travelling on a railway bridge experiencing earthquake excitations was investigated. First, a seismic response analysis of the railway bridge was conducted in order to obtain the seismic response of the railway bridge; it was subsequently used as the input to the post-derailment dynamic model. Then the derailment postures of the vehicle and contact situations were fully analyzed to understand the post-derailment dynamic behavior of railway vehicles during the earthquake. The obtained results show that the brake disc adopted in the model has the ability to limit the deviation of the wheelsets after the derailment.

Existence and uniqueness of attractors in frictional systems with uncoupled tangential displacements and normal tractions

We consider the class of two or three-dimensional discrete contact problems in which a set of contact nodes can make frictional contact with a corresponding set of rigid obstacles. Such a system might result from a finite element discretization of an elastic contact problem after the application of standard static reduction operations. The Coulomb friction law requires that the tractions at any point on the contact boundary must lie within or on the surface of a ‘friction cone’, but the exact position of any ‘stuck’ node (i.e., a node where the tractions are strictly within the cone) depends on the initial conditions and/or the previous history of loading. If the long-term loading is periodic in time, we anticipate that the system will eventually approach a steady periodic cycle.Here we prove that if the elastic system is ‘uncoupled’, meaning that changes in slip displacements alone have no effect on the instantaneous normal contact reactions, the time-varying terms in this steady cycle are independent of initial conditions. In particular, we establish the existence of a unique ‘permanent stick zone’ T comprising the set of all nodes that do not slip after some finite number of cycles. We also prove that the tractions and slip velocities at all nodes not contained in T approach unique periodic functions of time, whereas the (time-invariant) slip displacements in T may depend on initial conditions.Typical examples of uncoupled systems include those where the contact surface is a plane of symmetry, or where the contacting bodies can be approximated locally as half spaces and Dundurs’ mismatch parameter β=0. An important consequence of these results is that systems of this kind will exhibit damping characteristics that are independent of initial conditions. Also, the energy dissipated at each slipping node in the steady state is independent of initial conditions, so wear patterns and the incidence of fretting fatigue failure should also be so independent.

SPH procedures for modeling multiple intersecting discontinuities in geomaterial

SummaryA methodology is developed in SPH framework to analyze the behavior of preexisting multiple intersecting discontinuities or joints in rock material. The procedure does not require any additional unknowns to represent discontinuities and to capture velocity jump across them. Instead, a discontinuity is represented by a set of joint particles placed along the discontinuity plane, in which relative velocity and traction vector is evaluated, obeying the Mohr–Coulomb friction law with zero tension constrain. For failure of continuous rock material, the Drucker–Prager yield criterion with tensile cracking is employed in the elastic‐plastic constitutive model. Free‐sip, no‐slip, and symmetric boundary conditions are also implemented in SPH framework for proper representation of physical system.The paper analyzes behavior of a rock sample having a discontinuity plane under uniaxial loading and compares velocity and stress with a theoretical solution derived considering effective vertical stiffness of the joint planes. The efficacy of the proposed method is successfully demonstrated by solving another two problems of jointed rock mass under uniaxial and gravitational loading conditions.Copyright © 2014 John Wiley & Sons, Ltd.

Determination of Friction Coefficients of Interstice of a Shaft-Hub-Connection Manufactured by Lateral Extrusion

A combination of several manufacturing process steps in a simultaneous manner allows savings in energy costs, reduced investment and economizing logistical efforts. In particular, this investigation deals with manufacturing and joining of a frictionally engaged shaft-hub-connection by lateral extrusion. One key challenge of such new process management is the prior layout of both optimal friction conditions during manufacturing process on the one hand and highly expected static friction interaction of joined components in order to transmit tangential and axial forces on the other hand. Therefore, cylindrical, thin-walled hubs have been manufactured and joined to shafts. Several parameters describing the tribological system between shaft and hub such as contact surface topographies and lubricant, have been varied within this study. By measuring the radial deformation of the hub, the contact pressure is determined and a normal force applied to the contact surface of shaft and hub is calculated. When separating hub and shaft in a destructive manner, an extensive axial tensile force is applied and measured. According to Coulomb's friction law, specific friction coefficients are calculated depending on manufacturing process parameters as mentioned above.

Discussion of “Viscoelastic-Plastic Model of Asphalt-Roller Interaction” by Fares Beainy, Sesh Commuri, Musharaf Zaman, and Imran Syed

Discussion of “Viscoelastic-Plastic Model of Asphalt-Roller Interaction” by Fares Beainy, Sesh Commuri, Musharaf Zaman, and Imran Syed

The Influence of the Elastoplastic Behavior and the Load Pattern on the Tribological Properties of Two-Dimensional Frictional Contact Problems

The macromechanical tribological mechanism describes the friction phenomenon by considering the stress and the strain distributions, and the total elastic and plastic deformations. Based on the finite element method (FEM), the elastoplastic frictional contact problem is formulated as an incremental convex programming model (CPM). The Lagrange multiplier approach is adopted for imposing the inequality contact constraints. The Coulomb's friction law and the Prandtl–Reuss flow rule are used for the friction conditions and the elastoplastic behavior, respectively. The frictional contact examples are analyzed using the developed adaptive incremental procedure to elucidate the tribological behavior of the contact bodies and the model applicability.

ADAPTIVE INCREMENTAL FINITE ELEMENT PROCEDURE FOR SOLVING ELASTOPLASTIC FRICTIONAL CONTACT PROBLEMS SUBJECTED TO NORMAL AND TANGENTIAL LOADS

This paper presents a numerical model for analyzing the stresses and displacements of deformable bodies in contact with the presence of friction and material nonlinearity. Based on the finite element method (FEM), the elastoplastic frictional contact problem is formulated as an incremental convex programming model (ICPM) under inequality contact constraints and friction conditions. The classical Coulomb's friction law and the Prandtl–Reuss flow rule with the von Mises yield criterion are used to simulate the interface friction conditions and the elastoplastic behavior of the contacting bodies, respectively. The Lagrange multiplier approach is adopted for imposing the contact constraints. Furthermore, an effective adaptive incremental procedure is developed for solving the elastoplastic frictional contact problems. Examples for the frictional contact having advancing and receding nature are analyzed. The obtained results prove the ability of the developed procedure to investigate the sequence of different events during monotonic application of external loads. In addition, the results elucidate the effect of external side force on the friction behavior in the presence of plastic deformation. Good agreement has been found with published results.

Effect of surface roughness on sliding friction of micron-sized glass beads

Abstract In order to understand the contact phenomena of micron-sized particles, which have a tremendous impact on a variety of applications in industry and technology, direct access to the loads as well as the displacements accompanying such contacts are mandatory. Typical particle ensembles show a size variation ranging from the nanometer to the tenths of micron scale. Especially the contact behavior of particles featuring radii of several up to several tenths of microns is scarcely studied as these particles are typically too large for atomic force microscopy (AFM) based approaches and too small for conventional macroscopic testing setups. In this work a nanoindenter based approach is introduced to gain insight into the contact mechanics of micron-sized glass beads sliding on rough silicon surfaces at various constant low normal loads. The results are analyzed by a simple modified Coulomb friction law, as well as Hertz, JKR, and DMT contact theory.

Lagrangian Methods of Finite Element Simulations in Sheet Metal Forming Process

Simulation method has become an increasing important problem to establish the trustworthiness of models in the finite element simulations. Finite element models in sheet metal forming, which involves large deformations, large displacement and friction, are uncertainly with different methods. Finite element models based on different Lagrangian methods have been used to analyze the deep drawing process. The tools are modeled as rigid surfaces. The blank is modeled as a deformable body. The interaction of sliding surfaces is modeled with a modified Coulomb friction law. The Finite element results of strain and thickness distributions are validated by comparison with the experiment. It is shown that the main difference of Total Lagrangian formulation (T.L.) and Updated Lagrangian formulation (U.L.) lies in the large displacement and deformation, whilst agree well with small displacement and deformation.

Frictional Indentation of Anisotropic Magneto-Electro-Elastic Materials by a Rigid Indenter

An exact analysis on frictional contact between a rigid punch and anisotropic magneto-electro-elastic materials is performed, within the framework of the fully coupled theory. The indenter moves relative to magneto-electro-elastic materials, and Coulomb friction law is used. The mixed boundary value problem is reduced to singular integral equations of the second kind with analytical solution presented. For a triangular or semiparabolic indenter, explicit expression for surface physical in-plane stress, electrical displacement and magnetic induction are obtained. Influences of the friction coefficient and the volume fraction on contact behaviors are detailed under the prescribed contact loading conditions. Under either a triangular or semiparabolic indenter, the surface in-plane stress, electric displacement and magnetic induction are discontinuous and unbounded around the leading edge, and such a serious near-edge response can be relieved through adjusting the values of the friction coefficient or the volume fraction.