Shear behaviour of elastohydrodynamic oil films
24 Aug 1977-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (The Royal Society)-Vol. 356, Iss: 1685, pp 215-236
TL;DR: In this article, a simple constitutive equation is proposed for the isothermal shear of lubricant films in rolling/sliding contacts. But the model may be described as nonlinear Maxwell, since it comprises nonlinear viscous flow superimposed on linear elastic strain.
Abstract: A simple constitutive equation is proposed for the isothermal shear of lubricant films in rolling/sliding contacts. The model may be described as nonlinear Maxwell, since it comprises nonlinear viscous flow superimposed on linear elastic strain. The nonlinear viscous function can take any convenient form. It has been found that an Eyring 'sinh law' fits the measurements on five different fluids, although the higher viscosity fluids at high pressure are well described by the elastic/perfectly plastic equations of Prandtl-Reuss. The proposed equation covers the complete range of isothermal behaviour: linear and nonlinear viscous, linear viscoelastic, nonlinear viscoelastic and elastic/plastic under any strain history. Experiments in support of the equations are described. The nonlinear Maxwell constitutive equation is expressed in terms of three independent fluid parameters: the shear modulus $G$, the zero-rate viscosity $\eta $ and a reference stress $\tau _{0}$. The variations of these parameters with pressure and temperature, deduced from the experiments, are found to be in broad agreement with the Eyring theory of fluid flow.
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TL;DR: An alternative numerical method for solving contact problems for real rough surfaces using a single-loop iteration scheme based on the conjugate gradient method, which converges for arbitrary rough surfaces.
530 citations
TL;DR: The celebrated equations due to Fick and Darcy are approximations that can be obtained systematically on the basis of numerous assumptions within the context of mixture theory; the equations however not having been developed in such a manner by Fick or Darcy.
Abstract: The celebrated equations due to Fick and Darcy are approximations that can be obtained systematically on the basis of numerous assumptions within the context of mixture theory; the equations however not having been developed in such a manner by Fick or Darcy. Relaxing the assumptions made in deriving these equations via mixture theory selectively leads to a hierarchy of mathematical models and it can be shown that popular models due to Brinkman, Biot and many others can be obtained via various approximations. It is shown that a variety of other generalizations are possible in addition to those that are currently in favor, and these might be appropriate for describing interesting technological applications.
265 citations
TL;DR: It is shown that the superpositioning of relaxation times for various glass-forming materials when expressed as a function of TV(gamma), where the exponent gamma is a material constant, can be extended to the viscosity.
Abstract: Viscosities η and their temperature T and volume V dependences are reported for seven molecular liquids and polymers. In combination with literature viscosity data for five other liquids, we show that the superpositioning of relaxation times for various glass-forming materials when expressed as a function of TVγ, where the exponent γ is a material constant, can be extended to the viscosity. The latter is usually measured to higher temperatures than the corresponding relaxation times, demonstrating the validity of the thermodynamic scaling throughout the supercooled and higher T regimes. The value of γ for a given liquid principally reflects the magnitude of the intermolecular forces (e.g., steepness of the repulsive potential); thus, we find decreasing γ in going from van der Waals fluids to ionic liquids. For some strongly H-bonded materials, such as low molecular weight polypropylene glycol and water, the superpositioning fails, due to the nontrivial change of chemical structure (degree of H bonding) wi...
250 citations
01 Sep 1986
TL;DR: In this paper, a non-linear Maxwell rheological model based on the Eyring theory of fluid flow was proposed for disc machine traction tests of EHD lubricants at constant mean film temperature, and the model incorporates three fluid properties: shear modulus G, viscosity τ and Eyring stress τ 0.
Abstract: The methods of measuring the rheological properties of EHD lubricants are reviewed, but for pressures in excess of 1.0 GPa there is currently no simple alternative to the disc machine. A technique has been developed which enables disc machine traction tests to be carried out at constant mean film temperature. Isothermal tests provide further evidence for the existence of a limiting shear stress τc at which the fluid shears in the manner of a plastic solid. At stresses below this limit the experimental data are found to be in very good accord with the non-linear Maxwell rheological model based on the Eyring theory of fluid flow proposed by Johnson and Tevaarwerk and by Hirst and Moore. The model incorporates three fluid properties: shear modulus G, viscosity τ and Eyring stress τ0. Disc machine measurements of τ, τ0 and the limiting shear stress τc for three fluids—a mineral oil HVI 650, a synthetic polyphenyl ether 5P4E and a traction fluid Santotrac 50—are presented for a range of pressures (0.6–2.5 GPa)...
244 citations
TL;DR: Elastohydrodynamic lubrication is the dominant mode of lubrication in many critical, highly stressed machine elements such as gears, rolling bearings, cams and followers as discussed by the authors, and it also governs the effective operation of many highly deformable or "soft" bearing systems such as elastomeric bearings, seals and synovial joints.
191 citations
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TL;DR: In this paper, a review of the yield behavior of polymers is presented, focusing on the importance of the hydrostatic component of stress, the viscoelastic nature of yield process, and the behaviour of oriented polymers.
Abstract: Recent research on the yield behaviour of polymers is reviewed. Particular attention is given to the importance of the hydrostatic component of stress, the viscoelastic nature of the yield process, and the behaviour of oriented polymers.
256 citations
01 Jun 1967
TL;DR: In this article, the authors measured the apparent viscosity of E.H.L. oil films at very low sliding speeds, under isothermal conditions, and found that the variation with temperature at high pressure was exponential with an index similar to that at low pressure.
Abstract: The variation with sliding speed of the traction transmitted by E.H.L. oil films has been measured in a rolling contact disc machine, at contact pressures up to 260,000 lbf/in2 and at various rolling speeds and disc temperatures. From the traction measurements at very low sliding speeds, under isothermal conditions, the variation in apparent viscosity of the oil with pressure and temperature has been found. The variation with temperature at high pressure was exponential with an index similar to that at low pressure. The variation with pressure showed a striking reduction in the rate of increase with pressure in the high pressure region (> 105 lbf/in2).At high sliding speeds the traction coefficient was found to approach a common ‘ceiling’, which was largely independent of contact pressure, rolling speed and disc temperature. The ceiling appears to be a function of sliding speed only; it falls from 0.07 at 5 in/s to 0.04 at 60 in/s.Using the measurements of apparent viscosity obtained at low rolling speed,...
161 citations
TL;DR: In this article, it is shown that in the elasto-hydrodynamic regime T R is independent of load and simply proportional to the thickness of the hydrodynamic film.
Abstract: Measurements of friction with disk machines are described. The measurements have been made under conditions which fit them particularly for comparison with the theory of hydrodynamic lubrication of rollers and with the theory of frictional heating which has been developed in part III. The frictions due to rolling and sliding have been measured separately. It is shown that the frictional tractions T 1 and T 2 acting upon the surfaces of two disks can be expressed by — _ T _ where T R is the traction due to rolling and T s is that due to sliding. If the disk 1 runs freely in frictionless bearings T 1 must be zero and the disk runs in the condition T R = T s . A four-disk machine is described in which the central disk, since it has no bearings, runs in the above condition unless some external torque is imposed upon it. External torques were applied by a band brake and from the curves of traction against sliding thereby obtained both and the way in which T s increases with sliding speed were deduced. It is shown that in the elasto-hydrodynamic regime T R is independent of load and simply proportional to the thickness of the hydrodynamic film. It is also shown that this behaviour follows from the hydrodynamic theory and that the magnitudes of the rolling friction as predicted by theory and as deduced from experiment are substantially in accord. From experimental determinations of film thickness and from the linear increase of T s with sliding speed over the range of sliding speeds employed ( ~ 1 cm s- 1 ) values of the effective viscosity ( rjm) at the rolling point were deduced and are presented as a function of rolling speed {u as | ( mj +«2)). By effective viscosity is meant that constant viscosity throughout the conjunction of the disks which would give rise to the observed friction. It is shown that the values of so obtained when extrapolated to zero rolling speed are consistent with published results of experiments with dropping ball viscometers in high-pressure apparatus ( Pressure Viscosity Report 1953) both with respect to the effects of pressure and of temperature. But the more important feature of the results from the disk experiments is that they show the effects of pressure and temperature upon the apparent viscosity of the oil to diminish as the rolling speed is increased, i.e. as the time for which the oil is under stress diminishes. This behaviour is interpreted in terms of a Maxwellian fluid and the required values of the elastic modulus in shear are deduced. However, although the visco-elastic hypothesis accounts for the observations it is stressed that it cannot yet be taken as the definitive explanation. In a further series of experiments a two-disk machine adapted for the direct measurement of friction independently of bearing frictions was used to explore sliding speeds up to 400 cm s- 1 . In contrast with the previous measurements, at such speeds of sliding frictional heating has a major effect upon effective viscosity, for example, in a particular instance the introduction of 400 cm s- 1 sliding caused the effective viscosity to fall from ~ 3000 P at the rolling point to ~ 20 P. It is shown that the frictions and effective viscosities predicted by the theory of frictional heating (part III) and the measurements now reported are, in their larger aspects, substantially in accord. For example, in a particular instance theory predicted a coefficient of friction of 0-05 whereas experiment gave a coefficient of 0.03 and both experiment and theory show that as the sliding speed increases the friction rises to a maximum and then falls. But the theory of part III when applied to the experimental results leads to a value of the thermal conductivity of the oil of about half that to be expected from Bridgman’s work (1949). However, measurements in finer detail of friction up to a sliding speed of 30 cm s- 1 indicate that an intrinsic effect (i.e. an effect at constant temperature) of rate of strain upon viscosity exists. By taking this into account the anomaly with respect to thermal conductivity can be resolved. The experimental results show clearly that in a lubricating system of widespread type (e.g. ballraces, gears) a mineral oil exhibits distinctive dynamic characteristics which are of significance with respect both to friction and to the thickness of the hydrodynamic oil film. The comparison of experiment and theory also emphasizes the importance of the thermal conductivity of the oil in relation to friction and to the temperatures in the oil film.
143 citations
TL;DR: In this paper, a new theoretical model is proposed from which the viscoelastic relaxational behaviour of pure liquids in the supercooled region can be predicted within experimental error, which is regarded as a limiting case for liquids which follow the free volume description of steady flow viscosity.
Abstract: In addition to the six liquids for which the results of viscoelastic measurements are reported in part I, studies have also been made on a further eight liquids which differ widely in molecular composition, namely: squalane, tri($\beta$-chloroethyl) phosphate, tri(o-tolyl) phosphate, tri(m-tolyl) phosphate, tris(2-ethyl hexyl) phosphate, tetra(2-ethyl hexyl) silicate, m-bis(m-phenoxy phenoxy) benzene and bis(m-(m-phenoxy phenoxy) phenyl) ether. It has been confirmed for each liquid that the limiting shear modulus, G$\_\infty$, varies with temperature according to the relation previously found, $\frac{1}{G\_\infty}=\frac{1}{G\_0}+C(T-T\_0).$ Plotting the quantities $R\_L/(\rho G\_\infty)^\frac{1}{2}$ and $X\_L/(\rho G\_\infty)^\frac{1}{2}$ against log$\_{10}$($\omega\eta$/G$\_\infty$), where R$\_L$ and X$\_L$ are the resistive and reactive components of the shear mechanical impedance, shows that the results for six of these liquids are indistinguishable within experimental error. Moreover, the curves drawn through the experimental points represented in this manner are identical with the corresponding curves found for five of the liquids for which results are described in part I. The results for these eleven liquids show a striking agreement. A new theoretical model is put forward from which the viscoelastic properties of pure liquids in the supercooled region can be predicted within experimental error. This is regarded as a limiting case for liquids which follow the free volume description of steady flow viscosity. In the light of available evidence, it is envisaged that the viscoelastic relaxational behaviour approaches that represented by a single (Maxwell) relaxation process as the degree of co-operative motion of the molecules decreases. Only three of the fourteen liquids investigated show a behaviour which does not follow the theoretical prediction. The observed deviations are not great but are outside the limit of experimental error. Measurements on mixtures of liquids which individually exhibit behaviour in agreement with theory show that an impurity content of less than 2% can result in similar departures from theory. The lack of agreement for the three liquids in question is therefore attributed to impurities in the samples measured.
87 citations
TL;DR: In this article, a rolling contact experiment has been devised with the object of making direct observations of viscoelastic behaviour of elastohydrodynamic lubricant films, and a 'point contact' disk machine has been used which permits the axes of the disks to be either tilted or skewed through small angles relative to each other.
Abstract: A novel rolling-contact experiment has been devised with the object of making direct observations of viscoelastic behaviour of elastohydrodynamic lubricant films. A 'point contact' disk machine has been used which permits the axes of the disks to be either tilted or skewed through small angles relative to each other. Tilt subjects the film to cyclic shear while skew subjects it to unidirectional shear. By measuring and comparing the resultant shear forces developed by the film in each case, its viscoelastic properties can be deduced. Experiments at 23 degrees C with a high-viscosity mineral oil revealed viscoelastic behaviour above a contact pressure of about 0.5 GPa. In the pressure range 1.0 to 2.0 GPa the effective elastic shear modulus of the oil was found to vary from 0.1 to 1.0 GPa. At a pressure of 0.8 GPa the oil exhibited a viscoelastic transition in the temperature range 70 to 30 degrees C. The investigation was restricted to small shear strains for which the behaviour is linear.
82 citations