Determination of surface deformation of a bonded elastic layer indented by a rigid cylinder using the Chebyshev series method

The surface deformations due to the pressure applied by a frictionless rigid cylindrical punch upon an elastic layer, of finite thickness, attached to a rigid substrate (bonded layer) are calculated numerically. The numerical results show excellent agreement with the analytic solution when the layer becomes a half-plane. The influence of the layer thickness and Poisson ratio on the obtained results is examined.     

Asymptotic solutions for axisymmetric contact of a thin,transversely isotropic elastic layer

The problem of a rigid sphere in normal contact with a thin, transversely isotropic elastic layer is investigated. The thin elastic layer rests on a rigid foundation and both the limiting cases, wherein the interface is either ideally bonded or unbonded and frictionless, are considered. Approximate analytical solutions for the contact pressure beneath the spherical indenter and for the contact patch radius are presented. By comparing these with those predicted by the finite element method, the accuracy of the solution methods is determined.     

Frictionless contact between an elastic layer on a rigid base and a circular flat-ended punch with rounded edge or a conical punch with rounded tip

The axi-symmetric frictionless indentation problem involving an elastic layer on a rigid base and a rigid indenter (punch) having either a rounded edge or a rounded tip profile has been investigated. The layer is either firmly attached to a rigid base (bonded layer) or resting without friction on a rigid base (unbonded layer). A set of asymptotic solutions for the contact pressure is derived for thin layers. Numerical and theoretical results showed good agreement.     

On the microelastohydrodynamic lubrication of an elastomeric bonded layer

A numerical solution to the line EHL problem of an incompressible elastic layer with sinusoidal roughness and attached firmly (bonded) to a rigid substrate is presented. A modified version of the numerical method developed by the author for smooth layered contacts is used. Effects of the layer thickness, elasticity, roughness amplitude, wavelength and phase angle parameters on the obtained results are considered. When the layer becomes thin (contact width is significantly larger than the layer thickness) a formula for the predication of the minimum film thickness was derived using the first order approximation. The formula showed similar features to the numerical data.     

Pressure-viscosity effect on the solutions of a lubricated thick elastic bonded strip

Utilizing the numerical method developed by the author for isoviscous, fully flooded, elastohydrodynamic lubrication of a rigid cylinder rolling or sliding on an elastic strip (layer) which is attached to a rigid substrate (bonded strip), the influence on the pressure-viscosity coefficient, α, upon solutions is investigated. The present solutions are obtained for contacts operating in the transition region betweeen isoviscous-elastic and piezoviscous-elastic regimes where a pressure spike can be expected in some sense.New sets of results are presented for central and minimum film thicknesses in dimensionless form when the bonded strip is thick (0γ1, where γ = a/t is the ratio of the half contact width to strip thickness). It is shown that the film thickness depends not only on the values of α but also is influenced by Poisson's ratio, v.     

Formulae for the squeeze film between a rigid cylinder and a thin elastic layer with sinusoidal waviness

A set of formulae for the variation of the film thickness with time when a rigid cylinder approaching a thin elastic layer with sinusoidal waviness is derived. The shapes of the solids are assumed to be the same as the elastically deformed surfaces under dry conditions. Two geometrical configurations are investigated. First, the layer is resting without friction on a rigid substrate (unbonded layer). Second, the layer is attached firmly to a rigid substrate (bonded layer). The obtained results showed a good agreement with those reported for a smooth bonded layer.     

A finite element study of the indentation mechanics of an adhesively bonded layered solid

The objective of this work is to study the mechanics of indentation of an adhesively bonded layered solid. To this end, several (plane strain) finite element simulations of wedge indentation of a ductile strip which is adhesively bonded to a rigid substrate are conducted by varying the properties of the adhesive layer. The stress fields below the indenter tip and at the strip-adhesive interface are examined for various depths of indentation. The effects of the adhesive properties on the above features of the finite element solution, as well as on the hardness versus penetration characteristics, are investigated. The above results are also compared with those for an unbonded strip resting on a frictionless surface. It is found that once yielding commences in the adhesive layer, the state of stress in it is comprised of a shear stress and a superposed hydrostatic compression. Also, it is observed that increasing the yield strength of the adhesive layer significantly delays the onset of the decreasing phase of the hardness versus penetration curve, whereas, changing the elastic modulus of the adhesive has negligible effect on it.     

The effect of sinusoidal roughness profile upon rolling plane layered contact

A numerical solution to the free rolling layered contact problem of a regular wavy surface is presented. Both bonded and unbonded layers are investigated. The governing dual elasticity equations are reduced to a system of linear equations where the solution provides a set of results for the unknown parameters. Sliding and complete adhesion cases are treated. The influence of roughness, coefficient of friction, layer thickness and layer compressibility on the results are examined. It is found that the stability of the present method is strongly dependent on the ratio of the amplitude of the wave to its wavelength. In the case of adhesive rolling, the shear traction becomes smooth for an incompressible layer. An asymptotic solution for the surface tractions is obtained when an unbonded layer in full sliding becomes thin. The numerical data showed good agreement with the existing analytical solutions for smooth surfaces and with the derived formula.     

Minimum and central film thickness formulae for an elastic layer firmly bonded to a rigid cylindrical substrate under entraining motion

Analytical minimum and central film thickness formulae have been derived for elastic layers firmly bonded to rigid cylindrical substrates under entraining motion when the contact width is significantly larger than the layer thickness. The elastic layer has been assumed to be either a compressible or an incompressible material; the corresponding Poisson's ratios are assumed to be 0.4 and 0.5 respectively. Good agreement has been found between the film thicknesses predicted by these film thickness formulae and some numerical solutions for similar conditions. Good agreement has also been found with the general full numerical solution for incompressible elastic layered surfaces.     

Effects of surface coating on elastohydrodynamic lubrication of line contacts

A numerical solution to the elastohydrodynamic lubrication (EHL) problem of two coated elastic bodies in line contact is introduced. The non‐Newtonian behavior of the lubricant is incorporated into EHL analysis using Eyring's nonlinear viscous model. The surface elastic deformations are computed from full elasticity analysis of layered elastic half‐space. The iterative Newton–Raphson technique is used in the numerical solution. Subsurface stresses are calculated using two different techniques, the numerical integral scheme based on Fourier transformation and the finite element method. The effects of surface coating (material and thickness) on the pressure profile and subsurface stresses are presented and discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

Stresses in hard coating due to a rigid spherical indenter on a layered elastic half-space

The axisymmetrical contact problem of elasticity connected with an indentation of a rigid spherical indenter in an elastic semi-space covered by an elastic layer is considered. Stress tensor components in interior points of the non-homogeneous half-space by numerical calculation of some integrals was obtained. Detailed analysis of the maximal tensile stress distributions and Huber-von Mises reduced stress distributions produced by contact pressure is presented. The dependence between these stresses and the ratio between the layer thickness and contact area width is explored. The obtained results for stresses are compared with results obtained for half-space loaded by the Hertz pressure.     

Asymptotic behaviour of thin elastic layers bonded and unbonded to a rigid foundation

Formulae for pressure distributions are derived for axisymmetric contacts involving thin layers bonded and unbonded to a rigid foundation and indented by a frictionless rigid sphere. Accuracy and validity of these formulae are compared with results reported in the literature.     

Plastic deformation in rough surface line contacts—a finite element study

The paper describes an elastic-plastic finite element (EPFE) analysis of line contact between a cylinder and rigid plane using commercial software. The range of loading demonstrates the transition from purely elastic to fully plastic contact behaviour, revealing the residual deformations and stress fields upon unloading. A multiple contact configuration was analysed in the form of sinusoidal roughness. Results obtained under elastic conditions were validated by comparison with theoretical solutions. This model was extended by replacing the sinusoidal surface with a real roughness profile. Modelling multiple contacts indicates the influence of adjacent surface “asperities” on contact pressure and residual stress distributions.     

Normal contact between rough surfaces by the Discrete Element Method

This paper presents the study of nonadhesive, frictionless contact between elastic solids realized by the Discrete Element Method. This numerical method dedicated to multi-contact problems is applied to the field of tribology by studying the normal contact between a rigid rough surface and an elastic body modeled by spheres. A specific interparticle stiffnesses derived from homogenization techniques is implemented. From numerical tests carried out on spheres packings, we observe that the desired main macroscopic elastic constants are correctly modeled. Concerning the study of normal contact between rough surfaces, the obtained results are in accordance to existing theoretical models and numerical results from the literature, thereby demonstrating the potential of the Discrete Element Method to study the normal contact between contacting elastic bodies with rough surfaces. In particular, we recover the linear dependence of the real contact area with the normal load. In addition, we show that decreasing the surface roughness increases the average contact pressure.     

Effect of coating layer for spheres under Hertzian contact

A model is constructed to analyze the effects of layer hardness and thickness upon contact stresses for the coated elastic sphere under normal loading. It is assumed that the layer is perfectly bonded to the elastic substrate and the radius of contact is very small compared to the radius of indenter. By following a linear theory of elasticity, Fredholm integral equation is developed and it is solved numerically. The resulting contact stresses are calculated at the layer surface as well as the layer-substrate interface. Also, the second invariant of the deviatoric stress tensor, are calculated for various layer substrate combinations and for several layer thickness.     

Analysis of the dynamic stress of planar flexible-links parallel robots

This paper presents a method for the dynamic stress analysis of planar parallel robots with flexible links and a rigid moving platform. The finite element-based dynamic model of flexible parallel robots is proposed. The relation between elastic deformations and elastic displacements of the flexible links is investigated, considering the coupling effects of elastic motion and rigid motion. The elastic deformations of links are calculated. Considering the effects of bending-shearing strain and tensile-compression strain, the dynamic stress of the links and its position are derived by using the Kineto-Elastodynamics theory and the Timoshenko beam theory. Due to the flexibility of the links, the dynamic stresses are well illustrated through numerical simulation. Compared with the results of the finite element software SAMCEF, the numerical simulation results show the good coherence and advantages of the analysis method. The dynamic stress analysis is demonstrated to have a significant impact on the analysis, design and control of flexible parallel robots.     

Indentation of an elastic layer(s) bonded to a rigid cylinder—I. Quasistatic case without friction

A quasistatic, frictionless indentation problem for an incompressible elastic layer(s) bonded to a rigid cylinder is solved by use of a stress function in the form of a series. The method of Yau is then applied to obtain the solution of the dual series equations resulting from the mixed-type boundary condition which categorizes the problem. Although the analysis is given in detail only for a single-layered cylinder, comments on the many-layered case are made and some numerical results are given for a two-layered covering. In addition, some observations are made concerning a specific numerical example typical of rubber-covered steel rolls of the sort used in paper mills.     

Indentation of an elastic layer bonded to a rigid cylinder—II. Unidirectional slipping with Coulomb friction

A quasistatic, frictionless indentation problem for an incompressible elastic layer(s) bonded to a rigid cylinder is solved by use of a stress function in the form of a series. The method of Yau is then applied to obtain the solution of the dual series equations resulting from the mixed-type boundary condition which categorizes the problem. Although the analysis is given in detail only for a single-layered cylinder, comments on the many-layered case are made and some numerical results are given for a two-layered covering. In addition, some observations are made concerning a specific numerical example typical of rubber-covered steel rolls of the sort used in paper mills.     

Squeeze films between a rigid cylinder and an elastic layer bonded to a rigid foundation

A numerical solution to the elastohydrodynamic lubrication problem of a rigid cylinder and an elastic layer firmly bonded to a rigid substrate in normal approach and separated by a fluid of constant viscosity is presented. The rate of change of the deformation with time was neglected in the present investigation. The governing equations were solved via an iterative method in order to compute the pressure distribution and the corresponding film profile.Influences of the layer thickness, the layer compressibility and the central squeeze-film velocity on the results were investigated. In the case of the Hertzian contact, the present method was validated against the results reported in Herrebrugh [Elastohydrodynamic squeeze films between two cylinders in normal approach, Trans ASME, J Lubric Technol Ser F 1970; 92:292–302] where the results showed a very good agreement.     

The depth of propagation of elastic deformation in rough bodies under frictional contact

The problem of the depth of propagation of the perturbations of the stress-deformed state in an elastic half-space produced when a periodic sinusoidal indenter slides along the surface to model the roughness has been studied by using the solution of the periodic contact problem of elasticity theory. Equations are derived for the stresses and deformations on the axes of symmetry at the centre of the contacting area and between the projections of the rough surface. Calculations indicate that roughness and friction forces only affect the thin near-surface layer of the elastic half-space and the effects rapidly decrease with distance from the surface. The thickness of this perturbed layer is independent of the friction forces, the radius of curvature and heights of the roughnesses and the elasticity properties. It depends only on the distance between the roughnesses. At a depth equal to a factor of 1.5 of the distance between the projections the deflection of the stressed state does not exceed 0.4% and the deformation is 1%. This has been shown to be characteristic for roughnesses in general. The results of the calculations are presented as graphs.