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University of Cambridge, Department of Engineering
RESEARCH REPORTS
University of Cambridge, Department of Engineering K. L. Johnson* Research in tribology in the Engineering Department at Cambridge is mainly concerned with the mechanics of surfaces in contact including elastohydrodynamic lubrication. The main projects will now be outlined. Elastic contact stresses
Non-conforming surfaces of brittle solids pressed into contact usually fail by the formation of a crack round the periphery of the contact area, under the action of the radial tensile stress predicted by the Hertz theory. A theoretical study in progress shows that the magnitude of this stress is appreciably modified from the classical value when the two solids have different elastic moduli, for example, when a steel ball is pressed against a glass surface. The stress is reduced in the more elastic surface (glass) and increased in the more rigid surface (steel). In addition the point of maximum stress is located at a small distance outside the contact area. When two very smooth surfaces, one of which is highly elastic, are pressed into contact they are observed to adhere by the action of surface forces. These forces have the effect of pulling the surfaces into contact over an area which is greater than would be expected from the compressive load alone, and also of demanding a tensile force to be exerted to separate the surfaces subsequently. Using the concept of surface energy, the Hertz theory of elastic contact has been modified to take the action of surface forces into account. Theoretical predictions of the increased contact area under compression and of the force to pull the surfaces apart has been well supported by experiments with smooth soft spheres of rubber and gelatine 1. There has been a long standing interest in the contact stresses between elastic bodies which combine rolling motion with 'spin' (relative rotation about an axis perpendicular to their contact surface) 2. Recent experiments have been carried out with a pneumatically inflated ball. Although the behaviour is qualitatively similar to that of a solid ball, there is a significant quantitative difference arising from the difference in resistance to compression (which depends largely on the inflation pressure) and the resistance to shear of the skin. These experiments suggested that a hollow ball or a wheel with a pneumatic tyre would not roll in a stable manner in the trough of a grooved track. Instead it would creep up one side or other of the track until the angle of limiting friction was reached. An experiment with small diameter bicycle wheels at low-to-medium inflation pressures confirmed this behaviour. * University Engineering Dept, Trumpington Street, Cambridge CB2 1PZ, England
Plastic contact stresses
The permanent indentations produced in the surface of an elastic-plastic solid by compressive loads which exceed the elastic limit have been reassessed. It has been shown that the behaviour with different materials and different indenter shapes (e.g. ball, Vickers pyramid, etc) can be correlated in a plot of Pm/Y against Etan/3/Y, where Pra is the mean contact pressure, E is Young's modulus, Y is a representative yield stress and/3 is the slope of the indenter at the edge of the indentation 3 . The plastic deformation produced by the combined action of normal and tangential forces on a wedge-shaped indenter are being studied experimentally and theoretically using slip-line theory. The growth in area of contact when a plastic wedge is crushed by a rigid flat has been analysed 4 and shown to follow Tabor's measurements of 'junction growth'. Currently the ploughing action of a blunt rigid wedge on a soft metal surface is being analysed. With a ductile metal under lubricated conditions a plastic wave propagates along the surface in front of the wedge. High friction at the wedge surface, on the other hand, leads to a 'built-up nose' adhering to the wedge in the manner observed by Cocks and others. Vibration in rolling contact
A study is being made of the vibration of bodies in rolling contact arising from the inevitable roughness of their surfaces and, in particular, the circumstances in which vibration can lead to permanent surface corrugations developing s (see Fig 1). The significant mode of vibration is shown to arise from the elasticity of the bodies in the vicinity of the region of contact, which constitutes a nonlinear 'spring' coupling the masses of the two bodies. The frequency in this mode of vibration ('contact resonance') determines the wavelength of surface corrugations. It has been found possible to apply the methods of random vibration analysis to predict satisfactorily the dynamic response of the rolling bodies to measured surface roughness 6 . The amount of damping present is found to be very important in the dynamic response and in determining whether corrugation will develop. Elastohydrodynamic lubrication theory
The Grubin theory of the elastohydrodynamic lubrication of rolling cylinders is valuable in that it provides direct analysis of the flow and pressure distribution in the entry region without recourse to elaborate computations. The Grubin approach has been extended to the exit region, including the pressure spike (theoretically infinite) and the constriction which are features of full computer solutions.
TRIBOLOGY April 1972
83
the centre of the region is largely f'dled with a nondeforming plug of grease. The film thickness calculated on this basis is shown to be little different from that with the base oil. It was concluded that the reduction in thickness with time was due to degradation of the grease with shearing, probably aided by thermal effects.
Fig 1 The profile of a 100 mm disc brass disc after rolling contact in the disc machine. Corrugations are well developed over about ¼ periphery. Incipient corrugations can be observed in other places adjacent to surface irregularities s
Measurements of film thickness are observed to fall below the predictions of isothermal e.h.1, theory with increasing rolling speed. A Grubin-type of analysis of the inlet region has been used to estimate the temperature rise in pure rolling 7. The results agree with Crook's estimate of I°C for the case he considered, but give much larger values (20°C) in other cases. A modified Reynolds equation which includes thermal effects is solved approximately to find the effect of shear-heating on film thickness. Assuming that viscosity ~ depends upon temperature 0 according to Slotte's equation: = A(Oo + O)-n
the film thickness is reduced by the factor 1 + 0-2n rT0 U2/K(Oo + O) where U is the rolling speed. This gives appreciable reductions at 3-5 m/s and goes a long way towards explaining the experimental results.
Grease in e.h.I. Measurements, using a magnetic reluctance technique ~, of the thickness of grease films in a disc machine showed a time dependent behaviour also observed by Dyson. With a fresh sample of grease the film was appreciably thicker than that obtained with the base oil, but it decreased over a period of several minutes to a steady value which was thinner than that with the base oil. Previous working of grease in a Klein mill reduced the initial film thickness obtained in the disc machine but did not change the ultimate thickness 9 . An analysis has been made of the flow in the inlet region m, assuming that the grease follows a relation of the fornl r
/clu~n = ro + ~ / 1 / \dy]
where ro and r/ are constants and n -~ 1. This implies that
84
T R I B O L O G Y April 1972
Traction in e.h.I. The traction developed in an e.h.1, contact when sliding accompanies rolling depends upon the rheological properties of the lubricant at very high pressure. It is fairly well established, for example Ref 11, that the behaviour of the fluid under these conditions is non-Newtonian. Experiments are in progress to elucidate this behaviour. In most applications of this work, such as toothed gear lubrication, low friction (i.e. traction) is desirable, but variable speed drives such as the Perbury gear required high traction for satisfactory operation. This aspect of lubrication has been examined ~2. Rough surfaces Practical surfaces operating under conditions of e.h.1, have surface irregularities whose height is commonly of the same order of magnitude as the thickness of the oil film. In these circumstances the load is shared between hydrodynamic pressure in the film and deformation pressure of the asperities. Evidence is growing that the fraction of the load carried by the asperities is a significant parameter in surface breakdown. A simple theory has recently been developed to enable the division of load to be estimated from the measured surface roughness and the specified conditions of lubrication 13. The theory should assist in the interpretation of both laboratory experiements and field experience of surface failures under e.h.l, conditions. References I Johnson, K. L., Kendal, K. and Roberts, A. D. 'Surface energy and the contact of elastic solids', Proc R Soc, Ser A, Vol 324 (1971)pp 301-313 2 Johnson, K. k. 'Influence on elastic deformation upon the rolling motion of a ball between two surfaces', Proc Instn mech Engrs, Vol 173 (1959) pp 795-806 3 Johnson, K. L. 'The correlation of indention experiments', JMech PhysSolids, Vol 18 (1970) pp 115-126 4 Johnson, K. L. 'Deformation of a plastic wedge by a rigid flat under the action of a tangential force', JMech Phys Solids, Vol 16 (1968) pp 395-402 5 (?arson,R. M. and Johnson, K. L. "Surface corrugations spontaneously generated in a rolling disc machine', Wear, Vol 17, No 1 (1971) pp 59-62 6 Gray, C. G. and Johnson, K. L. 'The dynamic response of elastic bodies in rolling contact to random roughness of their surfaces', Submitted for publication 7 Greenwood, J. A. and Kauzlarich, J. J. 'Inlet shear heating in elastohydrodynamic lubrication' Submittcd for publication 8 Cameron, R. and Geogroy, R. W. 'Measurement of oil-fihn thickness between rolling discs using a variablc reluctance technique', Proc Instn mech Engrs. Vol 182, pt 3N (1967-68) p 24 9 Poon, S.Y. "An experimental study of grcasc in e.h.l.', Submitted for publication It) Kauzlarich, J. J. and Greenwood, J. A. 'Elastohydrodynanaic lubrication with Herschel-Bulkeymodel greases', Accepted fo publication by ASLE 1I Johnson, K. L. and Cameron, R. 'Shear behaviour of e.h.l, oil films at high rolling contact pressures', Proc Insln mech Engrs, Vol 182, Pt 1 (1967-8) pp 307-319 12 Poon, S. Y. "Some calculations to assess the effect of spin of the tractive capacity of rolling contact drives', Proc lnst mech Engrs, Vol 185, Pt 1 (1970-71) pp 1015-1022 13 Johnson, K. L., Greenwood, J. A. and Poon, S. Y. "A simple theory of asperity contact in e.h.l.', Wear, Vol 19, No 1 (1971) pp91-108
University of Cambridge, Department of Engineering K. L. Johnson* Research in tribology in the Engineering Department at Cambridge is mainly concerned with the mechanics of surfaces in contact including elastohydrodynamic lubrication. The main projects will now be outlined. Elastic contact stresses
Non-conforming surfaces of brittle solids pressed into contact usually fail by the formation of a crack round the periphery of the contact area, under the action of the radial tensile stress predicted by the Hertz theory. A theoretical study in progress shows that the magnitude of this stress is appreciably modified from the classical value when the two solids have different elastic moduli, for example, when a steel ball is pressed against a glass surface. The stress is reduced in the more elastic surface (glass) and increased in the more rigid surface (steel). In addition the point of maximum stress is located at a small distance outside the contact area. When two very smooth surfaces, one of which is highly elastic, are pressed into contact they are observed to adhere by the action of surface forces. These forces have the effect of pulling the surfaces into contact over an area which is greater than would be expected from the compressive load alone, and also of demanding a tensile force to be exerted to separate the surfaces subsequently. Using the concept of surface energy, the Hertz theory of elastic contact has been modified to take the action of surface forces into account. Theoretical predictions of the increased contact area under compression and of the force to pull the surfaces apart has been well supported by experiments with smooth soft spheres of rubber and gelatine 1. There has been a long standing interest in the contact stresses between elastic bodies which combine rolling motion with 'spin' (relative rotation about an axis perpendicular to their contact surface) 2. Recent experiments have been carried out with a pneumatically inflated ball. Although the behaviour is qualitatively similar to that of a solid ball, there is a significant quantitative difference arising from the difference in resistance to compression (which depends largely on the inflation pressure) and the resistance to shear of the skin. These experiments suggested that a hollow ball or a wheel with a pneumatic tyre would not roll in a stable manner in the trough of a grooved track. Instead it would creep up one side or other of the track until the angle of limiting friction was reached. An experiment with small diameter bicycle wheels at low-to-medium inflation pressures confirmed this behaviour. * University Engineering Dept, Trumpington Street, Cambridge CB2 1PZ, England
Plastic contact stresses
The permanent indentations produced in the surface of an elastic-plastic solid by compressive loads which exceed the elastic limit have been reassessed. It has been shown that the behaviour with different materials and different indenter shapes (e.g. ball, Vickers pyramid, etc) can be correlated in a plot of Pm/Y against Etan/3/Y, where Pra is the mean contact pressure, E is Young's modulus, Y is a representative yield stress and/3 is the slope of the indenter at the edge of the indentation 3 . The plastic deformation produced by the combined action of normal and tangential forces on a wedge-shaped indenter are being studied experimentally and theoretically using slip-line theory. The growth in area of contact when a plastic wedge is crushed by a rigid flat has been analysed 4 and shown to follow Tabor's measurements of 'junction growth'. Currently the ploughing action of a blunt rigid wedge on a soft metal surface is being analysed. With a ductile metal under lubricated conditions a plastic wave propagates along the surface in front of the wedge. High friction at the wedge surface, on the other hand, leads to a 'built-up nose' adhering to the wedge in the manner observed by Cocks and others. Vibration in rolling contact
A study is being made of the vibration of bodies in rolling contact arising from the inevitable roughness of their surfaces and, in particular, the circumstances in which vibration can lead to permanent surface corrugations developing s (see Fig 1). The significant mode of vibration is shown to arise from the elasticity of the bodies in the vicinity of the region of contact, which constitutes a nonlinear 'spring' coupling the masses of the two bodies. The frequency in this mode of vibration ('contact resonance') determines the wavelength of surface corrugations. It has been found possible to apply the methods of random vibration analysis to predict satisfactorily the dynamic response of the rolling bodies to measured surface roughness 6 . The amount of damping present is found to be very important in the dynamic response and in determining whether corrugation will develop. Elastohydrodynamic lubrication theory
The Grubin theory of the elastohydrodynamic lubrication of rolling cylinders is valuable in that it provides direct analysis of the flow and pressure distribution in the entry region without recourse to elaborate computations. The Grubin approach has been extended to the exit region, including the pressure spike (theoretically infinite) and the constriction which are features of full computer solutions.
TRIBOLOGY April 1972
83
the centre of the region is largely f'dled with a nondeforming plug of grease. The film thickness calculated on this basis is shown to be little different from that with the base oil. It was concluded that the reduction in thickness with time was due to degradation of the grease with shearing, probably aided by thermal effects.
Fig 1 The profile of a 100 mm disc brass disc after rolling contact in the disc machine. Corrugations are well developed over about ¼ periphery. Incipient corrugations can be observed in other places adjacent to surface irregularities s
Measurements of film thickness are observed to fall below the predictions of isothermal e.h.1, theory with increasing rolling speed. A Grubin-type of analysis of the inlet region has been used to estimate the temperature rise in pure rolling 7. The results agree with Crook's estimate of I°C for the case he considered, but give much larger values (20°C) in other cases. A modified Reynolds equation which includes thermal effects is solved approximately to find the effect of shear-heating on film thickness. Assuming that viscosity ~ depends upon temperature 0 according to Slotte's equation: = A(Oo + O)-n
the film thickness is reduced by the factor 1 + 0-2n rT0 U2/K(Oo + O) where U is the rolling speed. This gives appreciable reductions at 3-5 m/s and goes a long way towards explaining the experimental results.
Grease in e.h.I. Measurements, using a magnetic reluctance technique ~, of the thickness of grease films in a disc machine showed a time dependent behaviour also observed by Dyson. With a fresh sample of grease the film was appreciably thicker than that obtained with the base oil, but it decreased over a period of several minutes to a steady value which was thinner than that with the base oil. Previous working of grease in a Klein mill reduced the initial film thickness obtained in the disc machine but did not change the ultimate thickness 9 . An analysis has been made of the flow in the inlet region m, assuming that the grease follows a relation of the fornl r
/clu~n = ro + ~ / 1 / \dy]
where ro and r/ are constants and n -~ 1. This implies that
84
T R I B O L O G Y April 1972
Traction in e.h.I. The traction developed in an e.h.1, contact when sliding accompanies rolling depends upon the rheological properties of the lubricant at very high pressure. It is fairly well established, for example Ref 11, that the behaviour of the fluid under these conditions is non-Newtonian. Experiments are in progress to elucidate this behaviour. In most applications of this work, such as toothed gear lubrication, low friction (i.e. traction) is desirable, but variable speed drives such as the Perbury gear required high traction for satisfactory operation. This aspect of lubrication has been examined ~2. Rough surfaces Practical surfaces operating under conditions of e.h.1, have surface irregularities whose height is commonly of the same order of magnitude as the thickness of the oil film. In these circumstances the load is shared between hydrodynamic pressure in the film and deformation pressure of the asperities. Evidence is growing that the fraction of the load carried by the asperities is a significant parameter in surface breakdown. A simple theory has recently been developed to enable the division of load to be estimated from the measured surface roughness and the specified conditions of lubrication 13. The theory should assist in the interpretation of both laboratory experiements and field experience of surface failures under e.h.l, conditions. References I Johnson, K. L., Kendal, K. and Roberts, A. D. 'Surface energy and the contact of elastic solids', Proc R Soc, Ser A, Vol 324 (1971)pp 301-313 2 Johnson, K. k. 'Influence on elastic deformation upon the rolling motion of a ball between two surfaces', Proc Instn mech Engrs, Vol 173 (1959) pp 795-806 3 Johnson, K. L. 'The correlation of indention experiments', JMech PhysSolids, Vol 18 (1970) pp 115-126 4 Johnson, K. L. 'Deformation of a plastic wedge by a rigid flat under the action of a tangential force', JMech Phys Solids, Vol 16 (1968) pp 395-402 5 (?arson,R. M. and Johnson, K. L. "Surface corrugations spontaneously generated in a rolling disc machine', Wear, Vol 17, No 1 (1971) pp 59-62 6 Gray, C. G. and Johnson, K. L. 'The dynamic response of elastic bodies in rolling contact to random roughness of their surfaces', Submitted for publication 7 Greenwood, J. A. and Kauzlarich, J. J. 'Inlet shear heating in elastohydrodynamic lubrication' Submittcd for publication 8 Cameron, R. and Geogroy, R. W. 'Measurement of oil-fihn thickness between rolling discs using a variablc reluctance technique', Proc Instn mech Engrs. Vol 182, pt 3N (1967-68) p 24 9 Poon, S.Y. "An experimental study of grcasc in e.h.l.', Submitted for publication It) Kauzlarich, J. J. and Greenwood, J. A. 'Elastohydrodynanaic lubrication with Herschel-Bulkeymodel greases', Accepted fo publication by ASLE 1I Johnson, K. L. and Cameron, R. 'Shear behaviour of e.h.l, oil films at high rolling contact pressures', Proc Insln mech Engrs, Vol 182, Pt 1 (1967-8) pp 307-319 12 Poon, S. Y. "Some calculations to assess the effect of spin of the tractive capacity of rolling contact drives', Proc lnst mech Engrs, Vol 185, Pt 1 (1970-71) pp 1015-1022 13 Johnson, K. L., Greenwood, J. A. and Poon, S. Y. "A simple theory of asperity contact in e.h.l.', Wear, Vol 19, No 1 (1971) pp91-108