Coulomb Friction Law Research Articles

Inter-Ply Slipping Behaviors and Kinetic Equation of Carbon Fiber-Reinforced Epoxy Prepregs for Hot Diaphragm Preforming.

Wrinkles are urgent problems to be solved in the process of hot diaphragm preforming. Inter-ply slipping resistance is one of the causes of wrinkles. In this paper, based on the vertical inter-ply slipping test system, the inter-ply slipping behaviors of carbon fiber-reinforced epoxy resin composite prepregs were characterized. The mechanism of wrinkles caused by inter-ply slipping resistance was analyzed. According to the different characteristics expressed by the fiber and resin during the slip process, the inter-ply slipping behaviors of the prepregs were divided into three stages. The effect of temperature on the inter-ply slipping stresses was shown. The temperature will affect the viscosity of the prepregs. When the viscosity of the prepregs is low, the inter-ply slipping resistance will decrease. Based on the Coulomb friction law and the hydrodynamic equation, the inter-ply slipping kinetic equation of the prepregs was established. The inter-ply slipping kinetic equation was introduced into the ABAQUS main program by the 'vfriction' subroutine. The introduction of inter-ply slipping dynamics improved the accuracy of predicting the shape and position of wrinkles.

Application of a regularised Coulomb sliding law to Jakobshavn Isbræ, western Greenland

Abstract. Reliable projections of future sea level rise from the polar ice sheets depend on the ability of ice sheet models to accurately reproduce flow dynamics in an evolving ice sheet system. Ice sheet models are sensitive to the choice of the basal sliding law, which remains a significant source of uncertainty. In this study we apply a range of sliding laws to a hindcast model of Jakobshavn Isbræ, western Greenland, from 2009 to 2018. We demonstrate that a linear viscous sliding law requires the assimilation of regular velocity observations into the model in order to reproduce the observed large seasonal and inter-annual variations in flow speed. This requirement introduces a major limitation for producing accurate future projections. A regularised Coulomb friction law, in which basal traction has an upper limit, is able to more accurately reproduce the range of speeds from 2012 to 2015, the period of peak flow and maximal retreat, without the requirement for assimilating regular observations. Additionally, we find evidence that the speed at which sliding transitions between power-law and Coulomb regimes may vary spatially and temporally. These results point towards the possible form of an ideal sliding parameterisation for accurately modelling fast-flowing glaciers and ice streams, although determining this is beyond the scope of this study.

On Lubrication Regime Changes during Forward Extrusion, Forging, and Drawing

The tribological phenomena concerning the lubrication regime change (LRC) during bulk metal forming are comprehensively studied. A multi-step cold forward extrusion process shows the evolution of LRC and reveals the shortcomings of the traditional Coulomb friction law. The previous works of the specific author’s research group on friction are reviewed, focusing on the LRC during bulk metal forming. Various LRC phenomena from various examples are revealed. It has been found that the drawing and forward extrusion processes are vulnerable to LRC because of significant sliding motion at the material–die interface, and that when the strain hardening of the material is slight, the influence of friction increases, and as a result, the influence of LRC increases excessively. The new findings also include the impact of LRC on the macroscopic phenomena of the process and the reason for the sharp increase in friction coefficient via LRC, which is validated by the work of Wilson. This paper aims to make engineers and researchers think much of the tribology with lubricant in bulk metal forming with a focus on the dependence of tribological phenomena on the state of the lubricants and the irrationality of traditional friction law, especially in the forging of materials with a low strain hardening capability.

Analysis of a dynamic contact problem with long memory

We examine a mathematical framework detailing a dynamic frictional contact involving thermo-viscoelastic materials characterized by long memory and damage. The contact is modeled by the normal compliance condition with Coulomb's friction law, while surface wear is also factored in. The model was in the form of a coupled system which includes an evolution equation written in inclusion form for the frictional contact, parabolic variational inequality for the damage field, nonlinear differential equation for the temperature field and an equation for the wear particles and an evolution. A variational formulation for the model was derived and the existence of the unique weak solution established, under smallness assumptions on a part of the problem data. The proof used various results from the theory of evolution inequalities and repeated fixed-point arguments.

Dissipative solutions to the model of a general compressible viscous fluid with the Coulomb friction law boundary condition

We study a model of a general compressible viscous fluid subject to the Coulomb friction law boundary condition. For this model, we introduce a dissipative formulation and prove the existence of dissipative solutions. The proof of this result consists of a three level approximation method: A Galerkin approximation, the classical parabolic regularization of the continuity equation as well as convex regularizations of the potential generating the viscouss stress and the boundary terms incorporating the Coulomb friction law into the dissipative formulation. This approach combines the techniques already known from the proof of the existence of dissipative solutions to a model of general compressible viscous fluids under inflow-outflow boundary conditions as well as the proof of the existence of weak solution to the incompressible Navier-Stokes equations under the Coulomb friction law boundary condition. It is the first time that this type of boundary condition is considered in the case of compressible flow.

Simulation of impacts between spherical rigid bodies with frictional effects

AbstractThis work studies the impact between spherical rigid bodies in the frame of nonsmooth contact dynamics considering friction effects. A new impact element formulation based on the classical instantaneous local Newton impact law is presented. The kinematics properties of the spheres are described by a rigid body formulation with translational and rotational degrees of freedom referred to an inertial frame. In addition, an extension of the nonsmooth generalized‐ time integration scheme applied to collisions with multiple impacts including Coulomb's friction law is given. Six numerical examples are presented to evaluate the robustness and the performance of the proposed methodology.

The dynamic behaviour of flexible oscillators rocking and sliding on concentrated springs

AbstractThis study presents the Flexible Rocking Model on Concentrated Springs (FRMCS), developed to investigate 2D laterally flexible oscillators rocking and sliding on deformable support media during ground excitations. In this model, concentrated vertical springs and viscous dampers simulate the contact forces from support medium at the corners of the body; the tensionless vertical contact element is linear in compression. Horizontal concentrated springs and linear viscous dampers simulate the frictional behaviour at the corners; the constitutive law for the springs models elastic deformations and sliding (according to Coulomb's friction law). With these elements, FRMCS can model the response of a rocking body which can experience sliding and free‐flight phases of motion. The consideration of the flexibility of the support medium enables the evaluation of the forces exerted by the support medium on the structure during an impact. In this study, the FRMCS response is first compared to a previous model where the support medium deformability and the effects of sliding and free‐flight are ignored. Then, the responses of four configurations, which feature either stiff or soft lateral springs and stiff or soft high‐grip support media, are examined under the influence of pulse excitations. Finally, to understand the potential influence of sliding, a configuration with a low‐grip support medium is explored. The comparative influence of lateral flexibility and support medium deformability and sliding is quantified with stability diagrams and various response spectra, describing structural force and moment demands.

A novel physics-based modeling approach for tangential fretting contact behavior of jointed interface considering multi-scale effects

A novel physics-based constitutive model has been developed to map the tangential fretting behavior of joint surfaces. The model integrates the fractal normal contact model, which considers multi-scale effects, and the Iwan model through the Coulomb friction law. In this model, a new distribution of yield force is proposed for Jenkins elements, which is determined by fractal topography and material parameters and related to the scale of asperity. The effects of fractal topography, material parameters, and the normal load applied to the joint surface on the tangential responses such as tangential force, tangential stiffness, energy dissipation, and the distribution of yield force have been discussed. It has been found that the fractal parameters D and G have opposite effects on the tangential responses and yield force distribution.

Responses of the Pine Island and Thwaites glaciers to melt and sliding parameterizations

Abstract. The Pine Island and Thwaites glaciers are the two largest contributors to sea level rise from Antarctica. Here we examine the influence of basal friction and ice shelf basal melt in determining projected losses. We examine both Weertman and Coulomb friction laws with explicit weakening as the ice thins to flotation, which many friction laws include implicitly via the effective pressure. We find relatively small differences with the choice of friction law (Weertman or Coulomb) but find losses to be highly sensitive to the rate at which the basal traction is reduced as the area upstream of the grounding line thins. Consistent with earlier work on Pine Island Glacier, we find sea level contributions from both glaciers to vary linearly with the melt volume averaged over time and space, with little influence from the spatial or temporal distribution of melt. Based on recent estimates of melt from other studies, our simulations suggest that the combined melt-driven and sea level rise contribution from both glaciers may not exceed 10 cm by 2200, although the uncertainty in model parameters allows for larger increases. We do not include other factors, such as ice shelf breakup, that might increase loss, or factors such as increased accumulation and isostatic uplift that may mitigate loss.

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.

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.

Moving contact problem of a functionally graded orthotropic coated half plane

This 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.

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.

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.

Method of solution composition in contact problems with friction of deformable bodies

A new iterative method for solving static contact problems of two deformable bodies is proposed. The method is based on alternately solving the unilateral contact problem for the first body and the linear elasticity problem with natural boundary conditions for the second body. Fulfillment of Coulomb's friction law involves correction of tangential nodal forces in the sliding area and setting kinematic boundary conditions in the sticking area for the contact boundary of the first body. The goal of solving the linear elasticity problem for the second body is to gradually equalize contact loads on the interacting surfaces. The advantages of the method are demonstrated by solving a number of model examples, including unilateral contact of a linear-elastic plate with a solid foundation, bilateral contact of pressing a deformable block into the foundation, the Hertz problem of contact of two deformable cylinders etc. The method can solve problems on flat and curvilinear contact boundaries.

Model of fractured medium and nondestructive control of composite materials

Non-destructive testing of composites is an important issue in the modern aircraft industry. Composites are susceptible to the barely visible impact damage which can affect the residual strength of the material and occurs both during production and operation. The continuum model for describing the damaged zone is presented. The slip theory relations used for a continuous distribution of slip planes are applied. At the initial stage, the isotropic background model is used. This model allows the material slippage along the fractures based on the Coulomb friction law with the small viscous addition. In this regime, the govern system of equations becomes rigid. To overcome this difficulty, the explicit–implicit grid-characteristic scheme is proposed. The standard ultrasound diagnostic procedure of damaged composite materials is successfully simulated. Compared with the trivial free-surface fracture model, different reactions on the compression and stretch waves are registered. This approach provided an effective way for the simulation of complex dynamic behavior of damage zones.

Grounding-line flux conditions for marine ice-sheet systems under effective-pressure- dependent and hybrid friction laws

Flux conditions are semi-analytical expressions that can be used to determine the flux at the grounding line of marine ice sheets. In the glaciology literature, such flux conditions have initially been established for the Weertman and Coulomb friction laws. However, the extension to more general and complex friction laws, such as the Budd friction law, for which basal friction depends on both the basal velocity and the effective pressure, is a topic of recent research. Several studies have also shown that hybrid friction laws, which consider a transition between a power-law friction far from the grounding line and a plastic behaviour close to it, were good candidates for improved modelling of marine ice sheets. In this article, we show that the flux conditions previously derived for the Weertman and Coulomb friction laws can be generalised to flux conditions for the Budd friction law with two different effective-pressure models. In doing so, we build a bridge between the results obtained for these two friction laws. We provide a justification for the existence and uniqueness of a solution to the boundary-layer problem based on asymptotic developments. We also generalise our results to hybrid friction laws, based on a parametrisation of the flux condition. Finally, we discuss the validity of the assumptions made during the derivation, and we provide additional explicit expressions for the flux that stay valid when the bedrock slopes are important or when the friction coefficients are relatively small.

Application of Collins’ upper bound theorem to upsetting processes

The standard formulation of the upper bound theorem is incompatible with the Coulomb friction law. However, this law is widely used in metal forming processes analysis. An alternative formulation based on non-kinematically admissible velocity fields can evaluate the load required to deform the material. The present paper adopts this formulation for evaluating the compression force in solid and hollow cylinder compression processes. The trial velocity fields are continuous. The field for the solid cylinder involves no parameter for minimization, and the field for the hollow cylinder involves one parameter. Parametric analyses of the solutions are provided. The compression forces found are compared to available results at small friction coefficients. The solution is also in agreement with physical expectations. In particular, the solution for the hollow cylinder compression predicts that the radial velocity is everywhere positive in the friction coefficient is small enough. For larger friction coefficients, the radius where the radial velocity vanishes propagates from the inner radius as the friction coefficient increases. The final result can be used in conjunction with standard procedures for evaluating the friction coefficient.

A semi-analytical matrix formalism for stress singularities in anisotropic multi-material corners with frictional boundary and interface conditions

A new general semi-analytic procedure for the characterisation of singular asymptotic elastic states in the vicinity of the apex of linearly elastic anisotropic multi-material corners, including frictional contact, is developed and tested. The corners can consist of any finite number of homogeneous wedges defined by polar sectors. The variability of configurations covered is enormous as frictional contact can be considered on one or more outer boundary surfaces or interfaces, in addition to a large variety of homogeneous boundary conditions and perfect bonding or frictionless sliding interface conditions between wedges in the corner. The Coulomb rate-independent and dry frictional contact law is assumed. One of the novelties is that, in addition to the singularity exponent λ, the angle ω of the friction tangential stress vector on each frictional contact surface is an a priory unknown to be determined by solving a nonlinear corner eigensystem. The procedure, which considers power-law stress singularities, is based on the Stroh formalism of anisotropic elasticity, assuming generalised plane strain (2.5D) conditions, and on the semi-analytic matrix formalism for wedge transfer-matrices and boundary and interface condition matrices. This makes it, firstly, very suitable for computational implementations, secondly, very efficient especially in cases with several perfectly bonded homogeneous wedges, and, thirdly, very accurate due to its fully semi-analytic nature. The code developed is tested by solving a large variety of examples, comparing the present results with those obtained by solving closed-form corner-eigenequations deduced by previous authors for specific useful practical configurations, confirming the extremely high accuracy of the present code in the computation of λ and ω. The differences observed in some cases with anisotropic materials are explained by the fact that some of the previous authors did not take the true 3D Coulomb friction law into account.