Finite Coefficient Of Friction Research Articles

Friction between a plane strain circular indenter and a thick poroelastic substrate

This paper presents a computational study on the role of poroelasticity in gel friction. Motivated by recent experimental studies in the literature, we develop a plane strain finite element model to elucidate the contact mechanics between a circular indenter and a thick poroelastic substrate under both normal and shear loadings. Two cases are considered: i) steady state sliding under fixed normal displacements, and ii) relaxation under fixed normal and shear displacements. In steady state sliding, we find that a net friction force can arise even if no intrinsic adhesive or frictional interaction is implemented at the indenter/substrate interface. Such friction force exhibits a non-monotonic dependence on the sliding velocity and peaks at an intermediate velocity. Our model reveals that this friction force is induced by poroelastic diffusion in the gel substrate which can lead to considerable asymmetry in both the contact profile and contact pressure. In terms of relaxation, if the indenter/substrate interface is set to be frictionless, we find that the friction force induced by poroelasticity relaxes to zero with a characteristic time much faster than that of the normal force. When a finite friction coefficient is introduced at the interface, the normalized relaxation curve for the friction force approaches that for the normal force as the friction coefficient increases. These modeling results suggest that poroelasticity can be an important contributing mechanism for gel friction.

Calculation of toothed gear mechanisms in machines and assemblies considering the effect of lubricants

The basic criteria of operational efficiency of most toothed gear mechanisms is the contact endurance of the conjugated surfaces of the teeth and the flexural endurance of teeth legs. In this case, the calculation based on contact stresses is the principal one in terms of determining the geometrical dimensions of toothed gear mechanisms, while the calculation of tooth bending is performed for validation.It is known that in order to prolong longevity and improve operational efficiency and load capacity of toothed mechanisms, different lubricants are used. However, given the insufficient body of research into the influence of a lubricant on contact endurance of the active surfaces of teeth, traditional methods for the calculation of toothed gears (for example, GOST 21354-87) equate a lubricant influence coefficient to unity, that is, a perfect case is considered when friction is absent. Such an approach leads to the inaccurate evaluation of load capacity of toothed gears that can be a reason for both their premature failure and the overestimation of their geometrical dimensions.In this work, we have solved the contact problem on a contact between two bodies of arbitrary shape, close to half-planes, at the finite friction coefficient; it was found that the value of the resulting contact stress exceeds the stress, calculated according to the known Hertz solution, by 6 %.The proposed procedure for the calculation of toothed gears in terms of contact strength at the finite friction coefficient, without assumptions about the smallness of the contact area and the shape of borders, makes it possible to estimate load capacity of the toothed gears considering the influence of lubrication and the existence of friction between the conjugated surfaces of the teeth.We have derived the analytical expression for a lubricant influence coefficient based on the solution to the contact problem of pressure from a rigid stamp on the elastic half-plane in terms of the coefficient of friction between the conjugated surfaces of toothed gears. That allows the estimation of the true load capacity of toothed gears under the influence of various lubricants, which is of great theoretical and practical importance when designing machines and assemblies

МЕХАНИКА СТОЛКНОВЕНИЙ ТВЕРДЫХ ТЕЛ: ВЛИЯНИЕ ТРЕНИЯ И АДГЕЗИИ. II. ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ

This paper is the second part of the review on the physics of two-particle collisions of solids. The first one describes theoretical and experimental works on collisions of elastic and elastic-plastic solid bodies in the case when the energy dissipation is caused by the inner or interface friction, plasticity, adhesion, or some other damping mechanisms. In this second part, we concentrate on the case of collisions of elastic particles. Results of the analytical and numerical modelling within the dimensionality reduction (MDR) method are presented. MDR allows describing a three-dimensional contact by a reduction to an equivalent problem in a one-dimensional space. We consider three cases: a collision between bodies without slipping (equivalent to an infinite coefficient of friction), with a finite coefficient of friction in the interface, and with the presence of adhesion, which is described in the JKR limit. In all the considered cases, proper dimensionless variables have been identified which fully characterize the collision process, and the functions determining the transformation of the state before the impact into the state after the impact has been determined numerically. By using these dimensionless variables, it is possible to calculate the velocity components and the angular velocity, and hence the trajectory of the ball after the collision, in case if analogous parameters are known before the collision. Theoretical results are compared with the available results, the experimental data show agree well with the experiments.

Self-Excited Oscillations of a Finite-Thickness Elastic Layer Sliding Against a Rigid Surface With a Constant Coefficient of Friction

This investigation considers the dynamic stability of the steady-state frictional sliding of a finite-thickness elastic layer pressed against a moving rigid and flat surface of infinite extent. The elastic layer is fixed on its bottom surface; on its entire top surface, the rigid surface slides with constant speed and with a constant friction coefficient. The plane-strain equations of motion for a linear isotropic elastic solid are solved analytically for small dynamic disturbances. The analysis shows that even with a constant (speed-independent) friction coefficient, the steady solution is dynamically unstable for any finite friction coefficient. Eigenvalues with positive real parts lead to self-excited vibrations which occur for any sliding speed and which increase with increasing coefficient of friction. This is in contrast to the behavior of an elastic half-space sliding against a rigid surface in which the instability only occurs if the coefficient of friction is greater than unity. This work and its extensions are expected to be relevant in the theoretical aspects of sliding friction as well as in a variety of areas such as earthquake motion and brake dynamics.

Relaxation damping and friction

Recent research has shown that energy losses named as ‘relaxation damping’ can occur even with an infinite coefficient of friction using the method of dimensional reduction (MDR) which is strictly applicable to axis-symmetric elastic contact problems. If the solution with infinite coefficient of friction is regarded as a limiting case of ‘very large’ friction, this limiting solution is often useful because it will typically be much simpler than that obtained assuming a finite coefficient of friction, and it is likely to give reasonable estimates for the actual dissipation in cases that are far from the gross slip (sliding) limit. Using the advantage of relaxation damping, we shall give a generalized proof that is independent of the MDR. We shall also extend the proof to problems in which the applied loads follow a general trajectory in three-dimensional vector space. Further, we shall demonstrate how the energy dissipation per cycle varies with an increasing normal force, showing that relaxation damping could be useful as a limit in cases where there is significant variation in a normal force.

An introduction to the mechanics of the lasso.

Trick roping evolved from humble origins as a cattle-catching tool into a sport that delights audiences all over the world with its complex patterns or 'tricks'. Its fundamental tool is the lasso, formed by passing one end of a rope through a small loop (the honda) at the other end. Here, we study the mechanics of the simplest rope trick, the Flat Loop, in which the rope is driven by the steady circular motion of the roper's hand in a horizontal plane. We first consider the case of a fixed (non-sliding) honda. Noting that the rope's shape is steady in the reference frame rotating with the hand, we analyse a string model in which line tension is balanced by the centrifugal force and the rope's weight. We use numerical continuation to classify the steadily rotating solutions in a bifurcation diagram and analyse their stability. In addition to Flat Loops, we find planar 'coat-hanger' solutions, and whirling modes in which the loop collapses onto itself. Next, we treat the more general case of a honda that can slide due to a finite coefficient of friction of the rope on itself. Using matched asymptotic expansions, we resolve the shape of the rope in the boundary layer near the honda where the rope's bending stiffness cannot be neglected. We use this solution to derive a macroscopic criterion for the sliding of the honda in terms of the microscopic Coulomb static friction criterion. Our predictions agree well with rapid-camera observations of a professional trick roper and with laboratory experiments using a 'robo-cowboy'.

From force distribution to average coordination number in frictional granular matter

We study the joint probability distribution of normal and tangential frictional forces in jammed granular media, P μ ( f t , f n ) , for various values of the friction coefficient μ , especially when μ = ∞ . A universal scaling law is found to collapse the data for μ = 0 to ∞ , demonstrating a link between the force distribution P μ ( f t , f n ) and the average coordination number, z c μ . The results determine z c μ for a finite friction coefficient, extending the constraint-counting argument of isostatic granular packing to finite frictional packings.

Force network ensemble for the triangular lattice: A tale of tiles

In granular materials, contact forces between neighboring grains are organized in disordered force networks. At present, it is unclear whether or not theories based on statistical mechanics can correctly predict the statistics of these force networks. The force network ensemble is a convenient model system for studying force networks. This ensemble comprises all sets of noncohesive contact forces on a fixed underlying contact network and a fixed load, i.e., global stress tensor, for which all grains are in static force and torque balance. Each force network is assigned an equal a priori probability. The corresponding phase space can be sampled using the Monte Carlo method. Figure 1 a shows a typical force network for the frictionless triangular lattice. Each force network has a complementary representation known as a reciprocal tiling Fig. 1 b in which each grain maps to a particular tile and each contact force to a tile face. The face is oriented at a /2 rotation to the contact force f with a length proportional to f . Due to local force balance, the tile faces form closed loops and, because forces come in action-reaction pairs, all tiles fit together without any gaps. The local pressure of a single grain is equal to its tile perimeter. One can show that fixing the global stress tensor  leads to a conservation of the total tile area A. In Ref. 2, we derived an analytical expression for the distribution of the local pressure on individual grains, by maximizing the entropy while conserving  and A. Figure 1 c shows that this theoretical prediction is in excellent agreement with the numerical result from the force network ensemble. Due to these constraints, large local stresses obey Gaussian statistics, in sharp contrast to the common belief that exponential force statistics are characteristic for granular materials. This observation is robust to changes in contact network including disordered networks and finite friction coefficient.

Micromechanics of a Compressed Fiber Mass

A theory is presented for the rate modeling of flexible granular solids based on affine average motion of interparticle contacts. We allow contacts to form and break continually but assume the existence of a finite friction coefficient rendering contacts force free as they form or break. The resulting constitutive equations are of the hypoelastic type. A specific model for the deformation of a fiber mass is then developed. The model improves on previous theories for fiber masses in at least two respects: First, it is more general in that it is not restricted to uniaxial compression, although it is restricted to predominantly compressive deformations histories, due to neglect of frictional dissipation. Second, by allowing torsion as well as bending of fibers, this theory covers a larger deformation range. Compression experiments are performed on carded slivers of PA6 fibers under various conditions. The measured response is found to be in close agreement with that predicted by the model.

High speed indentation of an elastic half-space by conical or wedge-shaped indentors

Self-similar problems of indentation of an elastic half-space by rigid cones or wedges are solved, assuming perfect adhesion, when the velocity of indentation is large enough for the area of contact to spread faster than the speed of P-waves. In contrast to the earlier study of the wholly subsonic case [2], the present problems can be solved in closed form without approximation. It emerges, too, that the no slip condition would be satisfied for a range of values of a finite coefficient of friction, in contrast to the situation in [2], where any finite friction is bound to allow some slip. A variety of wave fronts exist in the present problems and all of their amplitudes are found explicitly and discussed.

Effect of Sliding Friction on Contact Stresses in a Rectangle Compressed by Rigid Planes

The plane-strain compression of an elastic rectangle by rigid, rough planes with finite coefficient of friction between the surfaces is considered. The contact area is divided into an inner adhesive region in which the surface displacements are known, surrounded by regions in which the friction is limiting and the displacement parallel to the interface is not known. The remaining set of parallel edges of the rectangle is free from tractions. The problem is formulated in terms of Papkovich-Fadle eigenfunctions which lead to the solution of a set of two integral equations of the second kind. Solution of the integral equations which satisfies the finiteness of stresses at the point which separates the adhesive from the slip zone, determines the extent of adhesion. This is found to be independent of the magnitude of load, but depends on the values of frictional coefficients, Poisson’s ratio and the aspect ratio. Numerical results of the quantities of practical interest are reported.

An eigenvalue problem for elastic contact with finite friction

AbstractThe two-dimensional indentation of an elastic half space by a rigid punch under a slowly applied normal load is considered, for the case in which there is a finite coefficient of friction μ between the surfaces. The contact area is then divided into an inner adhesive region −c<x<cin which the surface displacements are known, surrounded by regionsc< |x| < 1 in which the friction is limiting and the lateral displacement (which must increase in proportion to the overall load) is not known in advance. The problem is formulated in terms of a coupled pair of singular integral equations for the normal and shear stresses σ and τ at the surface; these are combined to give a single homogeneous Fredholm equation with positive kernel for a quantity π proportional to the difference τ – μσ in the adhesive region. The largest eigenvalue of this equation, for which π > 0, gives the adhesive boundarycin terms of μ and Poisson's ratio ν. A similarity transformation shows thatchas the same value for both flat-faced and power law punches.

Finite-element analysis of alternating axial loading of an elastic plate pressed between two elastic rectangular blocks with finite friction

A method has been developed, making use of finite-element techniques, for calculating stress-distributions, slip amplitudes, and damping capacities in a number of simple clamped joints under alternating loading conditions with a finite coefficient of friction on the interface. Typical results are given for one geometry.