The Contact Between Rough Surfaces 105 The results as generated from the simulation for cutting conditions covering the practical range of applications are used in a regression analysis to obtain the best fit for the equation parameters C, k,, kZ, k3, and k4. The values of the system parameters of Eq. (4.3) for the case of a chuck mass (M,) of 34kg and machine structures with stiffness K, greater than 10' N/m were found to be independent of K, and are only dependent on the uncoupled tool natural frequency, (5. All the simulated results were curve-fitted to give the following equations: where C;, = natural frequency of tool assembly (Hz) These equations are applicable for the following conditions when K, is greater than 10' N/m: 6lm/min < V < 305 m/min 0.127 mm/rev < f < 0.88 mm/rev 0.31 mm -= d < 0.71 mm 0.79 mm < t < 2.38 mm 4.3 THE REAL AREA OF CONTACT BETWEEN ROUGH SURFACES Analytical studies and measurements show that the real area of contact between surfaces occurs at isolated points where the asperities came together. This constitutes a very small fraction of the apparent area for flat surfaces (Fig. 4.3a) or the contour area for curved surfaces (Fig. 4.3b). For the case of steel on steel flats, the real area of contact is in the order of 0.0001 cm2 per kilogram load. This indicates that pressure on the microcontacts for any combination of materials is constant and is inde- pendent of load. The interactions between the two bodies at the real area are what determines the frictional resistance and wear when they undergo relative I06 Chapter 4 Apparent Area f Real Area (4 Contour Area Real Area (b) Figure 4.3 (a) Contact of flat surfaces. (b) Contact of spherical surfaces. The Contact Between Rough Surfaces 107 sliding. Even in the case of a Hertzian contact, the pressure distribution is not continuous. Due to surface roughness it occurs at discrete points, and the force between the bodies is the sum of the individual forces on contacting asperities which constitute the pressure distribution. An interesting investi- gation of this problem was conducted by Greenwood and Tripp [30]. They analyzed the contact between rough spheres by a physical model of a smooth sphere with the equivalent radius of both spheres pressed against a rough flat surface where the asperity heights follow a normal Gaussian distribution about a mean surface, as illustrated in Fig. 4.4a. They further assumed that the tops of the asperities are spherical with the same radius and that they deform elastically according to Hertz theory. The forces on the aspe- rities constitute the loading on the nominal smooth bodies whose deforma- tion controls the extent of the asperity contacts. The problem is solved iteratively until convergence occurs. Assuming that the radius of the asperities is p, the radius of the contact on top of an asperity due to a penetration depth @can be calculated as (Fig. 4.4b): and the corresponding area of contact From the Hertz theory, the load on the asperity can be calculated as: where E,, E2 = elastic moduli for the two materials ul, u2 = Poisson's ratios for the materials If z is the height of the asperity and U is the distance between the nominal surfaces at that location, it can be seen from Fig. 4.2b that o = z- U. Assuming a height distribution probability function $(z), the probability that an asperity is in contact at any location with nominal separation U can be expressed as: 00 prob(z =- U) = $(z)dz 108 Chapter 4 Figure 4.4 (a) Nominal surfaces and superimposed asperities. (b) Contact between rough surfaces - asperity contact. The expected force is: If the asperity density is assumed to be q, the expected number of asperities to be in contact over an element of the surface (&), where the separation between the nominal surfaces is U can be expressed as: The Contact Between Rough Surfaces 109 The expected area of contact: 00 i3A = m,$(&) 1 (z - u)~/~~(z) dz U and the expected load within (da) is: 00 dP = $ qE,B'/2(da) jU (z - u)~/~~(z) dz Assuming the standard deviation of the asperity heights to be equal to cr, the following dimensionless relationships were used in developing a general for- mulation for the problem: 0 0 Asperity penetration U* = - Separation of nominal surface at any location h = fi d Minimum separation between the nominal surface d* = - 0 0 Radial distance in the contact region p = Pressure p* = rn E', Jrn P Height of asperity S = 0 Accordingly, the equations for the contact conditions within an elementary area (da) can be written as: dP - = <! q/?1/2a3/2)F3/2(h) = average pressure (d4 where F,(h) = (S - h)"@*(s) ds r The two main independent variables are the total load and the surface roughness. These were used in a dimensionless form as follows to evaluate the contact conditions: 110 Chapter 4 T = dimensionless total load = 2 P/o E,m p = dimensionless roughness parameter = qa,/ZX$ Accordingly: T = JDm 2nPP*(P)dP U* = dimensionless parameter representing the spread of pressure over the contact region (Le., affective radius), of the contact area = q; = dimensionless maximum value of the average contact pressure 8a* I /3 Tc qf, = dimensionless average contact pressure = Numerical results are presented in Greenwood and Tripp [30] based on the previous analysis from which the following conclusions can be stated: 1. Load has remarkably small effect on the mean real pressure on top of the asperities. This is illustrated by the numerical results given in Fig. 4.5. Consequently the mean real area of contact is approximately linearly dependent on the applied load. The proportionality constant between the real area and load increases with increased root mean square (r.m.s.) roughness (a) decreased asperity density and decreased raidus of the aspe- rities. The effective radius of the area over which the pressure is spread is considerably larger than the Hertzian contact radius for low loads and approaches the Hertzian contact condition for high loads. Consequently, the average mean pressure is considerably lower than the Hertzian pressure for low loads and approaches it for high loads. This is illustrated in Fig. 4.6. 2. 3. 4. It is interesting to note that the first two conclusions are the same as those noted by Bowden and Tabor [31] and the electric contact resistance measurements reported by Holm [32]. The constant value of the average pressure on the real area of asperity contact was assumed to be the yield stress at the asperity contacts. However, the analysis presented by Greenwood and Tripp discussed in this chapter provides a rational proce- The Contact Between Rough Surfaces 0.2 - 0.1 111 - I I I I 0.8 0.7 0.8 1s 0.8 - w Figure 4.5 Effect of load on mean real pressure. A: q = 500/mm2, 0 = 5 x 10-~m~, fi = 0.2mm; B: q = 940/mm2, 0 = 5 x 10-~mm, B = 0.2mm; C: q = 500/mm2, B = 9.4 x 10-4 mm, /I = 0.2mm. (From Ref. 30.) dure based on elastic deformation of the asperities for calculating this con- stant stress value from the surface roughness data and the elastic constants of the surface layer. Later investigations showed that a combination of elastic and plastic asperity contacts can occur for typical surface finishing processes depending on the load and the thickness of the lubricating film. This will be discussed later in the book. 4.4 THE INTERACTION BETWEEN ROUGH SURFACES DURING RELATIVE MOTION It has been shown in the last section that the contact between elastic bodies with rough surfaces occurs at discrete points on the top of the asperities. The interaction takes place at surfaces covered with thin layers of materials, which have different chemical, physical, and thermal characteristics from the bulk material. These surface layers which unite under pressure due to the influence of molecular forces, are damaged when the contact is broken by relative movement. During the making and breaking of the contacts, the Chapter 4 \ \ \ I \ I Rough (Greenwood) \/- I Dimensionless Radius Rough (Greenwood) Dimensionless Radius Figure 4.6 Comparison of pressure distribution for rough and smooth surfaces: (a) low load; (b) high load. The Contact Between Rough Surfaces I13 underlying material deforms. The forces necessary to the making and break- ing of the contacts, in deforming the underlying material constitute the frictional resistance to relative motion. It can therefore be concluded that friction has a dual molecular-mechanical nature. The relative contribution of these two components to the resistance to movement depends on the types of materials, surface geometry, roughness, physical and chemical properties of the surface layer, and the environmental conditions in which the frictional pair operates. 4.5 A MODEL FOR THE MOLECULAR RESISTANCE Molecular resistance or adhesion between surfaces is a function of the real area of contact and molecular forces which take place there. A theoretical relationship describing the effect of the molecular forces can be given as: where h = Planck’s constant = 6.625 x 10-27 erg-sec c = speed of light I = distance between the contacting surfaces m, n, e = mass, charge, and volume density of electrons in the solid Adhesive forces are generally not significant in metal-to-metal contacts where the surfaces generally have thin chemical or oxide layers. It can be significant, however, in contacts between nonmetals or metals with thin wetted layers on the surface as well as in the contacts between microma- chined surfaces. 4.6 A MODEL FOR THE MECHANICAL RESISTANCE The role of roughness in the frictional phenomena has been a central issue since Leonardo da Vinci’s first attempt to rationalize the frictional resis- tance. His postulation that frictional forces are the result of dragging one body up the surface roughness of another was later articulated by Coulomb. This rationale is based on the assumption that both bodies are rigid and that no deformation takes place in the process. I I4 Cii up I c r 4 Figure 4.7 Surface waves generated by asperity penetration. A modern interpretation of the mechanical role of roughness is based on the elastic deformation of the contacting surfaces due to asperity penetra- tion. The penetrating asperity moving in a tangential direction deforms the underlying material and gives rise to a semi-cylindrical bulge in front of the identor which is lifted up and also spreads sideways as elastic waves. This is diagrammatically illustrated in Fig. 4.7. The size of the bulge depends on the relative depth of penetration w/p. where w is the penetration depth and B is the radius of the asperity. The process is analogous to that of the movement of a boat creating waves on the water surface. According to this theory. the energy dissipated in the process of deforming the surface is the source of the mechanical frictional resistance and the surface waves generated are the source of frictional noise. 4.7 FRICTION AND SHEAR Both friction and shear represent resistance to tangential displacement. In the first case, the traction resistance is on the surface or "external" to the [...]... Output-Multi Independent Variable Turning Research by Response Surface Methodology,” Int J Prod Res., 19 74, Vol 12 (2), pp 233-245 Wu, S M., “Tool Life Testing by Response Surface Methodology Parts I and 11 ,” Trans ASME, 19 64 , Vol 86, pp 10 5 The Contact Between Rough Surfaces 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 11 9 Hasegawa, M., Seireg, A., and Lindberg, R A., “Surface Roughness Model for Turning,”... a solid lubricant at the interface 1 2 3 4 5 6 7 Albrecht, A B., “How to Secure Surface Finish in Turning Operations,” Am Machin., 19 56, Vol 10 0, pp 13 3 -13 6 Chandiramani, K L., and Cook, N H., “Investigations on the Nature of Surface Finish and Its Variations with Cutting Speed,” Trans ASME, 19 70, Vol 86, pp 13 4 -14 0 Olsen, K V., “Surface Roughness in Turned Steel Components and the Relevant mathematical... and Static Characteristics of a Wide Speed Range Machine Tool Spindle,” Precis Eng., 19 83, Vol 83, pp 17 5 -18 3 Nassipour, F., and Wu, S M., “Statistical Evaluation of Surface Finish and Its Relationship to Cutting Parameters in Turning,” Int J Mach Tool Des Res., 19 77, Vol 17 , pp 19 7-208 Zhang, G M., and Kapoor, S G., “Dynamic Generation of Machined Surfaces,” J Indust Eng., ASME, Parts I and 11 , 19 91, ... Tribol Int., 19 76, pp 285-289 Sata, T., “Surface Roughness in Metal Cutting,” CIRP, Ann Alen Band, 19 64 , Vol 4, pp 19 0 -19 7 Kronenberg, M., Machining Science and Application, Pergamon Press, London, England, 19 67 Tobias, S A., Machine Tool Vibration, Blackie and Son, London, England, 19 65 Kondo, Y., Kawano, O., and Soto, H., “Behavior of Self Excited Chatter due to Multiple Regenerative Effect,” J Eng Indust.,... Engr, 19 68 , pp 59 360 6 Solaja, V., “Wear of Carbide Tools and Surface Finish Generated in Finish Turning of Steel,” Wear, 19 58, Vol 2, pp 40-58 Ansell, C T., and Taylor, J., “The Surface Finish Properties of a Carbide and Ceramic tool,” Advances in Machine Tool Design and Research, Proceedings of 3rd International MTDR Conference, Pergamon Press, New York, NY, 19 62 , pp 235-243 Taraman, K., “Multi-Machining... Kim, K J., and Ha, J Y., “Suppression of Machine Tool Chatter Using a Viscoelastic Dynamic Damper,” J Indust Eng., Trans ASME, 19 87, Vol 10 9, pp 58 -65 Nakayama, K., and Ari, M., On the Storage of Data on Metal Cutting Forces,” Ann CIRP, 19 76, Vol 25, pp 13 -18 I20 Chapter 4 26 Rao, P N., Rao, U R K., and Rao, J S., “Towards Improved Design of Boring Bars Part I and 11 ,” Int J Mech Tools Manuf., 19 88, Vol... March, 19 76, pp 15 3 -15 9 Bowden, F P., and Tabor, D., Friction and Lubrication of Solids, Oxford University Press, 19 54 Kragelski, I V., Friction and Wear, Butterworths, Washington, 19 65 Holm, R., Ekctric Contacts Handbook, Springer, Berlin, 19 58 58 28 29 30 31 32 33 5 Thermal Considerations in Tribology 5 .1 INTRODUCTION This chapter gives a brief review of the fundamentals of transient heat transfer and. .. ASME, 19 65 , Vol 10 3, pp 447454 Sisson, T r., and Kegg, R L., “An Explanation of Low Speed Chatter Effects,” J Eng Indust., Trans ASME, 19 69 , Vol 91, pp 9 51- 955 f Armarego, E J A., and Brown, R H., The Machining o Metals, Prentice Hall, Englewood Cliffs, NJ, 19 69 Rakhit, A K., Sankar, T S., and Osman, M 0 M., “The Influence of Metal Cutting Forces on the Formation of Surface Texture in Turning,” MTDR, 19 70,... 11 , 19 91, Vol 11 3, pp 13 7 -15 9 Olgac, N., and Zhao, G., “A Relative Stability Study on the Dynamics of the Turning Mechanism,” J Dyn Meas Cont., Trans ASME, 19 87, Vol 10 9, pp 16 4 -17 0 Jemlielniak, K., and Widota, A., “Numerical Simulation of Non-Linear Chatter Vibration in Turning,” Int J Mach Tool Manuf., 19 89, Vol 29, pp 239-247 Tlusty, J., Machine Tool Structures, Pergamon Press, New York, NY, 19 70... Vol 28, pp 34- 27 Tlusty, J., and Ismail, F., “Special Aspects of Chatter in Milling,” ASME Paper No 18 -Det -18 , 19 81 Skelton, R C., “Surface Produced by a Vibrating Tool,” Int J Mech tools Manuf., 19 69 , Vol 9, pp 375-389 Jang, D Y., and Seireg, A., “Tool Natural Frequency as the Control Parameter for Surface Roughness,” Mach Vibr., 19 92, Vol 1, pp 14 7 -15 4 Greenwood, J A., and Tripp, J., “The Elastic . Testing by Response Surface Methodology Parts I and 11 ,” Trans. ASME, 19 64 , Vol. 86, pp. 10 5. 12 (2), pp. 233-245. The Contact Between Rough Surfaces 11 9 8. 9. 10 . 11 . 12 . 13 . 14 the interface. 1. 2. 3. 4. 5. 6. 7. Albrecht, A. B., “How to Secure Surface Finish in Turning Operations,” Am. Machin., 19 56, Vol. 10 0, pp. 13 3 -13 6. Chandiramani, K. L., and Cook,. Roughness in Metal Cutting,” CIRP, Ann. Alen Band, 19 64 , Vol. 4, pp. 19 0 -19 7. Kronenberg, M., Machining Science and Application, Pergamon Press, London, England, 19 67 . Tobias, S. A., Machine