Determination of contact area in 'cushion form' bearings for artificial hip joints

Current forms of artificial hip joints produce wear debris, which contributes to loosening of the prostheses. These 'hard' joints articulate with boundary or mixed lubrication, whereas the natural joint articulates with full fluid film lubrication. An artificial joint that articulates with full fluid film lubrication could greatly reduce wear and frictional torque and hence reduce the incidence of loosening and inflammatory tissue reaction. The use of a thin lining of low elastic modulus in the acetabular cup is one possible way of promoting full fluid film lubrication. In the design of such cushion forms of bearings, it is important to be able to predict the contact area, stress distribution and film thickness. This paper presents experimental techniques to determine the contact area in low elastic modulus layers and compares these measured areas with theoretical predictions using linear elasticity theory. At low loads experimental results and theoretical predictions were close. However, at loads above 300 N, the theory overestimated the width of the contact area by up to 8.5 per cent. This difference is mainly attributed to the non-linear behaviour of the elastomer at the higher levels of stress.     

Stress analysis of cushion form bearings for total hip replacements.

Cushion form bearings comprise a thin layer of low elastic modulus material on the articulating surface of the bearing, which can deform to help preserve a film of lubricant between the bearing surfaces and therefore reduce friction and wear. The long-term function of this type of bearing is dependent on the strength and durability of this compliant layer. Finite difference and finite element methods have been used to analyse the stress distribution in the compliant layer of cushion form bearing for artificial hip joints under physiological loading conditions. A good agreement between finite difference and finite element methods was found. Under normal loading, the highest value of the maximum shear stress was found to be at the interface between the compliant layer and the more rigid substrate close to the edge of the contact. The values of maximum shear stress in the centre of the contact close to the articulating surface were lower than in the equivalent Hertzian contact. A friction force acting at the surface had little effect on the stress distribution for coefficients of friction less than 0.05. However, for higher values of friction coefficient (larger than 0.2), corresponding to inadequate lubrication, the maximum shear stress increased by a factor of four and was found to be located at the surface. The analysis predicts that the mode of failure will be at the interface with the substrate under fluid film or mixed lubrication conditions and at the articulating surface when the bearing runs dry with higher levels of friction. Both failure modes have been observed experimentally under the conditions specified.     

The effect of porosity of articular cartilage on the lubrication of a normal human hip joint.

The effect of porosity of articular cartilage on the lubrication of a normal human hip joint has been studied. The poroelasticity equation of articular cartilage and the modified Reynolds equation for the synovial fluid lubricant have been successfully solved under squeeze-film motion and for the conditions experienced in a normal human hip joint. It has been shown that porosity of the articular cartilage depletes the lubricant film thickness, rather than increasing it, particularly when the lubricant film thickness becomes small. Furthermore, it has been shown that articular cartilage can be treated as a single-phase incompressible elastic material in the lubrication modelling under physiological walking conditions.     

Elastic deformation in thin film hydrodynamic lubrication

An elastohydrodynamic numerical simulation is conducted for one-dimensional fixed slider plane bearings. The numerical model takes into account the piezoviscous effect of the lubricant and elastic deformation of the bounding surfaces to solve the one-dimensional Reynolds equation. It is found that a small elastic deformation of less than 100 nm plays an important role in load capacity in thin film hydrodynamic lubrication. As the film thickness decreases, a flat film shape appears from the leading side of the contact area. The expansion of the flat film thickness over the contact area leads to considerably lower load capacity.     

Numerical modeling an elastohydrodynamic contact of shaped rollers with allowance for misalignment of their axes

A method of numerically solving an elastohydrodynamic (EHD) contact of shaped rollers with allowance for misalignment of their axes in a plane perpendicular to the rolling direction is advanced. The mode of EHD lubrication is typical of such friction assemblies as roller bearings and gearings, in which the contacting elastic bodies are separated by a lubricant film and deformed under the action of an external load. Results of numerical modeling demonstrate the significant effect of the misalignment angle on the distribution of pressure and thickness of the lubricant film in the EHD contact and can be used further to analyze friction in a contact area and the stress tensor in a subsurface layer. The mathematical model of the EHD contact is described through nonlinear integro-differential equations and inequalities. The computational algorithm is based on Newton’s method.     

Steady-state elastohydrodynamic lubrication analysis of a metal-on-metal hip implant employing a metallic cup with an ultra-high molecular weight polyethylene backing

The elastohydrodynamic lubrication (EHL) analysis was carried out in this study for a 28 mm diameter metal-on-metal hip prosthesis employing a metallic cup with an ultra-high molecular weight polyethylene (UHMWPE) backing under a simple steady state rotation representing the flexion/extension during walking. Both Reynolds and elasticity equations were coupled and solved numerically by the finite difference method. The elastic deformation was determined by means of the fast Fourier transform (FFT) technique using the displacement coefficients obtained from the finite element method. Excellent agreement of the predicted elastic deformation was obtained between the FFT technique and the conventional direct summation method. The number of grid points used in the lubrication analysis was found to be important in predicting accurate film thicknesses, particularly at low viscosities representative of physiological lubricants. The effect of the clearance between the femoral head and the acetabular cup on the predicted lubricant film thickness was shown to be significant, while the effect of load was found to be negligible. Overall, the UHMWPE backing was found not only to reduce the contact pressure as identified in a previous study by the authors (Liu et al., 2003) but also significantly to increase the lubricant film thickness for the 28 mm diameter metal-on-metal hip implant, as compared with a metallic mono-block cup.     

Investigation into the Normal Contact Stiffness of Rough Surface in Line Contact Mixed Elastohydrodynamic Lubrication

In this work, the statistical asperity microcontact models in combination with the acoustic spring model and the load sharing concept are utilized to study the interfacial normal contact stiffness for a rough surface in line contact elastohydrodynamic lubrication (EHL). Two different statistical microcontact models of Greenwood and Williamson (GW) and Kogut and Etsion (KE) are employed to derive the normal contact stiffness expressions for a dry rough line contact considering the purely elastic contact and the multiple regimes elastic–elastoplastic–fully plastic contact, respectively. The liquid film stiffness is calculated based on the relationship between film thickness and bulk modulus of the lubricant. The lubricant film thickness equations are employed in conjunction with the load sharing concept and the empirical formulas for the maximum contact pressure in a dry rough contact are fitted for the GW model and the KE model, to evaluate the relationship between film thickness and motion velocity for the purely elastic GW microcontact model and the multiregime KE microcontact model, respectively. The comparison with experimental results shows that the KE model predicts closer total contact stiffness results than the GW model. The stiffness contributions from the solid asperity contact and lubricant film are obtained and effects of surface roughness, applied load, motion velocity, and type of lubricant on the normal contact stiffness are analyzed.     

涂层材料特性对滚子轴承润滑性能的影响*

涂层材料被广泛应用于滚子轴承中以改善界面性能和提高疲劳寿命,为了探讨涂层材料性能对滚子轴承润滑性能的影响,基于流体力学与接触力学理论,建立带涂层的有限长线接触弹流润滑模型,探讨不同载荷、速度以及涂层材料特性对油膜压力、油膜厚度的影响。研究表明：随着涂层厚度的增加,硬涂层使得最小油膜厚度先增加后减小,而软涂层轻载时使得最小油膜厚度先减小后增加,重载时最小油膜厚度一直减小;随着速度的增加,出口区二次压力峰值增加,硬涂层尤为明显,并且油膜厚度也增加,油膜平坦区域减小,出口区油膜紧缩值增加。为提高润滑性能,当使用较厚软涂层时应考虑增加滚子凸度量,而使用较厚硬涂层时应考虑减小滚子凸度量。     

Elastomer-lined bearings: An analysis of a heavily loaded cylinder sliding on a compliant layer

An analytical solution to the elastohydrodynamic lubrication of a rigid cylinder sliding on an elastic layer which is bonded to a rigid substrate is presented. The central film thickness is predicted by assuming that the lubricant film may be divided into a curved inlet region and a central tilted pad region. Expressions for pressure and deformation derived from the dry contact formulae are used to enable compatibility of pressure between the two regions to be established. The computed lubricant pressure distribution is compared with that for indentation in dry contact under the same load. The film thickness results under heavy loads indicate similar trends to those obtained by alternative solutions.     

Effect of wear of bearing surfaces on elastohydrodynamic lubrication of metal-on-metal hip implants

The effect of geometry change of the bearing surfaces owing to wear on the elastohydrodynamic lubrication (EHL) of metal-on-metal (MOM) hip bearings has been investigated theoretically in the present study. A particular MOM Metasul bearing (Zimmer GmbH) was considered, and was tested in a hip simulator using diluted bovine serum. The geometry of the worn bearing surface was measured using a coordinate measuring machine (CMM) and was modelled theoretically on the assumption of spherical geometries determined from the maximum linear wear depth and the angle of the worn region. Both the CMM measurement and the theoretical calculation were directly incorporated into the elastohydrodynamic lubrication analysis. It was found that the geometry of the original machined bearing surfaces, particularly of the femoral head with its out-of-roundness, could lead to a large reduction in the predicted lubricant film thickness and an increase in pressure. However, these non-spherical deviations can be expected to be smoothed out quickly during the initial running-in period. For a given worn bearing surface, the predicted lubricant film thickness and pressure distribution, based on CMM measurement, were found to be in good overall agreement with those obtained with the theoretical model based on the maximum linear wear depth and the angle of the worn region. The gradual increase in linear wear during the running-in period resulted in an improvement in the conformity and consequently an increase in the predicted lubricant film thickness and a decrease in the pressure. For the Metasul bearing tested in an AMTI hip simulator, a maximum total linear wear depth of approximately 13 microm was measured after 1 million cycles and remained unchanged up to 5 million cycles. This resulted in a threefold increase in the predicted average lubricant film thickness. Consequently, it was possible for the Metasul bearing to achieve a fluid film lubrication regime during this period, and this was consistent with the minimal wear observed between 1 and 5 million cycles. However, under adverse in vivo conditions associated with start-up and stopping and depleted lubrication, wear of the bearing surfaces can still occur. An increase in the wear depth beyond a certain limit was shown to lead to the constriction of the lubricant film around the edge of the contact conjunction and consequently to a decrease in the lubricant film thickness. Continuous cycles of a running-in wear period followed by a steady state wear period may be inevitable in MOM hip implants. This highlights the importance of minimizing the wear in these devices during the initial running-in period, particularly from design and manufacturing points of view.     

Effects of consistency variation of power law lubricants in squeeze films

The characteristics of squeeze film bearings with power law lubricants have been investigated by considering the effects of consistency variation. Various bearing geometries have been considered with rigid surfaces as well as with compliant layers. With stiff solids, a high consistency layer adjacent to the bearing surface increases the load capacity and time of squeezing and this increase is enhanced by the pseudoplastic behaviour of the lubricant. For squeeze film bearings with compliant layers, the film thickness increases with load, compliancy and conformity of the surfaces even with power law lubricants. It also increases with the consistency of the layer adjacent to the bearing surface.     

Numerical Analysis of the Oil-Supply Condition in Isothermal Elastohydrodynamic Lubrication of Finite Line Contacts

In order to investigate the effect of oil-supply condition on the lubrication performance of machine components, such as gears and roll bearings, a full numerical solution of the isothermal finite line contact elastohydrodynamic lubrication (EHL) problem under different oil-supply conditions was obtained. The supplied oil quantity was given with the thicknesses of layers of oil films on both solid surfaces, and an equivalent thickness of such supplied oil films was introduced. An algorithm similar to that proposed by Elrod in 1981 was developed to determine the pressure starting position automatically. The pressure field was solved with a multi-grid solver which enables the difficulty of the huge mesh differences in two directions be overcome easily. The surface deformation produced by pressure was calculated with a multilevel multi-integration method. Based on the Newtonian lubricant assumption, comparisons of solutions between the starved and fully flooded contacts have been made. Results show that the pressure starting position and the central and minimum film thicknesses vary with different oil-supply thicknesses. In addition, the influence of the thickness of the oil-supply layer, the end profile radius, the entrainment velocity, and the maximum Hertzian pressure on the starved fluid film thickness has been investigated. In conclusion, the optimum quantity of the supplied oil is very important for the discussed problem.     

Elastohydrodynamic Lubrication of Rough Surfaces under Oscillatory Line Contact with Non-Newtonian Lubricant

This paper presents the results of a transient analysis of elastohydrodynamic lubrication (EHL) of two parallel cylinders in line contact with a non-Newtonian lubricant under oscillatory motion. Effects of the transverse harmonic surface roughness are also investigated in the numerical simulation. The time-dependent Reynolds equation uses a power law model for viscosity. The simultaneous system of modified Reynolds equation and elasticity equation with initial conditions was solved using the multigrid, multilevel method with full approximation technique. The film thickness and the pressure profiles were determined for smooth and rough surfaces in the oscillatory EHL conjunctions, and the film thickness predictions were verified experimentally.

For an increase in the applied load on the cylinders or a decrease in the lubricant viscosity, there is a reduction in the minimum film thickness, as expected. The predicted film thickness for smooth surfaces is slightly higher than the film thickness obtained experimentally, owing primarily to cavitation that occurred in the experiments. The lubricant film under oscillatory motion becomes very thin near the ends of the contact when the velocity goes to zero as the motion direction changes, but a squeeze film effect keeps the fluid film thickness from decreasing to zero. This is especially true for surfaces of low elastic modulus. Harmonic surface roughness and the viscosity and power law index of the non-Newtonian lubricant all have significant effects on the film thickness and pressure profile between the cylinders under oscillatory motion.     

点接触弹流润滑供油条件退化的乏油分析

在点接触弹流润滑中,如果不能及时补充新油,则接触区的供油条件会随着润滑次数而退化。分析了供油油膜厚度、中心膜厚、最小膜厚和润滑油膜压力区形成位置与润滑次数的关系。结果表明:润滑开始时,由于供油油膜厚度较大,系统处于充分供油状态;随着润滑次数的增加,有一部分油从两侧泄漏,系统逐渐转到乏油状态,供油油膜厚度、中心膜厚和最小膜厚均逐渐变小,压力区形成位置则逐渐向Hertz接触区靠近;最终供油油膜厚度、中心膜厚和最小膜厚趋于定值,压力区趋于Hertz接触区,从而达到一种稳定乏油状态。     

Elastohydrodynamic lubrication analysis of ultra-high molecular weight polyethylene hip joint replacements under squeeze-film motion

Elastohydrodynamic lubrication was analysed under squeeze-film or normal approach motion for artificial hip joint replacements consisting of an ultra-high molecular weight polyethylene (UHMWPE) acetabular cup and a metallic or ceramic femoral head. A simple ball-in-socket configuration was adopted to represent the hip prosthesis for the lubrication analysis. Both the Reynolds equation and the elasticity equations were solved simultaneously for the lubricant film thickness and hydrodynamic pressure distribution as a function of the squeeze-film time was solved using the Newton-Raphson method. The elastic deformation of the UHMWPE cup was calculated by both the finite element method and a simple equation based upon the constrained column model. Good agreement of the predicted film thickness and pressure distribution was found between these two methods. A simple analytical method based upon the Grubin-Ertel-type approximation developed by Higginson in 1978 [1] was also applied to the present squeeze-film lubrication problem. The predicted squeeze-film thickness from this simple method was found to be remarkably close to that from the full numerical solution. The main design parameters were the femoral head radius, the radial clearance between the femoral head and the acetabular cup, and the thickness and elastic modulus for the UHMWPE cup; the effects of these parameters on the squeeze-film thickness generated in current hip prostheses were investigated.     

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.     

A Multiscale Framework for EHL and Micro-EHL

In this article, a heterogeneous multiscale method is introduced to analyze the microelastohydrodynamic lubrication (micro-EHL) of bearings with topological features. Two scales are adopted in the analysis: the large-scale simulations describe the entire bearing domain, and the small-scale simulations describe the fluid–structure interaction (FSI) at the small-scale features. Conservation of mass and momentum of the lubricant and the bearing's elastic deformation are solved for. The relationship between the pressure gradient and mass flow is obtained from homogenized small-scale FSI simulations and applied on a global scale via a scattered data interpolation method. When the micro structure is periodic the exact model at micro scale is replaced by an effective derived equation, i.e., homogenized model. The elastic deformation of the textured bearing surface is addressed at both the large and small scales, by decomposing the displacement influence matrix into the diagonal terms and nondiagonal terms (sorted at the small scale and large scale, respectively).

The multiscale method was demonstrated as being capable of modeling the global pressure and film thickness for a bearing with surface texture while maintaining the accuracy of the small-scale modeling features. The illustrative geometry was that of a linear converging pad bearing in two dimensions. The solutions were compared with those obtained using lubrication theory for the smooth surface case, and good agreement was obtained. The method was then demonstrated for geometries incorporating topographical features.     

Rebounding lubricant layers

Most research in elastohydrodynamic lubrication has concentrated on the immediate contact area between the two elements. In the case of ball or cylindrical roller bearings, the finitely separated rolling elements may have, between them, a free-surface lubricant film on each of their races. This film, formed by the wake emanating from one pair of elements, becomes the inlet boundary condition for the following pair. Its possible variation in shape with time is thus important when estimating the lubricant film thickness. With this practical situation in mind, the variation with time of the wake formed behind a ball sliding in a thin viscous lubricant layer on a plate has been studied. Using thin-film theory and an initial wake cross-section similar to those observed experimentally, the Navier Stokes equations are solved to yield solutions which describe the subsequent alteration in the wake shape with time. It was found that the track centre of the wake remains at, or slightly below, its chosen initial height for some time before commencing rebound towards the undisturbed layer condition. This behaviour is confirmed by some simple experiments carried out on a ball-and-plate machine which also indicate that the track centre height, measured just downstream of the ball, is initially less than the minimum film thickness under the ball and remains at or near this condition for some time before rebound commences. The experiments have also shown that time taken for the track centre height to regain the minimum height under the ball is found to depend on the lubricant properties, the undisturbed layer thickness and the depth of immersion of the ball.     

Inlet protein aggregation: a new mechanism for lubricating film formation with model synovial fluids

This paper reports a fundamental study of lubricant film formation with model synovial fluid components (proteins) and bovine serum (BS). The objective was to investigate the role of proteins in the lubrication process. Film thickness was measured by optical interferometry in a ball-on-disc device (mean speed range of 2-60 mm/s). A commercial cobalt-chromium (CoCrMo) metal femoral head was used as the stationary component. The results for BS showed complex time-dependent behaviour, which was not representative of a simple fluid. After a few minutes sliding BS formed a thin adherent film of 10-20 nm, which was attributed to protein absorbance at the surface. This layer was augmented by a hydrodynamic film, which often increased at slow speeds. At the end of the test deposited surface layers of 20-50 nm were measured. Imaging of the contact showed that at slow speeds an apparent 'phase boundary' formed in the inlet just in front of the Hertzian zone. This was associated with the formation of a reservoir of high-viscosity material that periodically moved through the contact forming a much thicker film. The study shows that proteins play an important role in the film-forming process and current lubrication models do not capture these mechanisms.     

Contact mechanics and lubrication analyses of ceramic-on-metal total hip replacements

Ceramic-on-metal (CoM) total hip replacements have shown reduced wear and friction. Lubrication and contact mechanics analyses play an important role in providing an overall understanding for the tribological performance of CoM bearings. In the present study, the steady-state contact mechanics and elastohydrodynamic lubrication (EHL) and transient EHL of CoM bearings were analyzed. The dry and lubricated contact pressures of CoM bearings showed typical characteristics of hard-on-hard hip bearings. The effects of head radius and radial clearance on the lubrication performance were predicted. CoC and CoM bearings are more likely to benefit full fluid film lubrication than MoM bearings.