8. Contact Mechanics DE2-EA 2.1: M4DE. Dr Connor Myant 2017/2018

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1 DE2-EA 2.1: M4DE Dr Connor Myant 2017/ Contact Mechanics Comments and corrections to Lecture resources may be found on Blackboard and at

2 Contents Introduction... 3 Contact Mechanics... 3 Types of contacts... 5 Bearing contacts... 6 Case of a cylinder-cylinder contact Negligible clearance and rigid bodies Negligible clearance and elastic bodies Clearance and elastic bodies... 8 Case of a conformal sphere-sphere contact... 9 Hertz theory of elastic contact Simplifying Assumptions to Hertz's Theory The tangent plane Contact Radii and Elastic Modulus Equations for various contacts The effect of surface roughness Adhesive contacts References Dr Connor Myant DE2-EA2.1 M4DE 2

3 Introduction In this part of the module we are going to look at the contact between solid bodies, such as bearings, gears, wheels on the road or human joints. For now we are going to limit ourselves to contacts where no lubricant is present or when the conditions do not give rise to a lubricating film. This may be typically either because there is no lubricant present ( dry contacts) or, because the surface speeds are too low ( static contacts). In particular, we shall examine the motion and deformation near the area of contact between solids ( contact mechanics ). In practice, many of the results can also be used for situations in which a thin lubricant film is present. Design Engineering Example: Some examples of everyday engineering or biological contacts where no lubricant is present: Road, rail, wheel contacts (no effective lubricant) Wood working Gripping a tennis racket Slow gears in a wrist watch Contact Mechanics Contact mechanics is the study of the deformation of solids that touch each other at one or more points [1, 2]. The physical and mathematical formulation of the subject is built upon the mechanics of materials and continuum mechanics and focuses on computations involving elastic, viscoelastic, and plastic bodies in static or dynamic contact. Central aspects in contact mechanics are the pressures and adhesion acting perpendicular to the contacting bodies' surfaces (known as the normal direction) and the frictional stresses acting tangentially between the surfaces. The original work in contact mechanics dates back to 1882 with the publication of the paper "On the contact of elastic solids"[3] by Heinrich Hertz. Hertz was attempting to understand how the optical properties of multiple, stacked lenses might change with the force holding them together. Hertzian contact stress refers to the localized stresses that develop as two curved surfaces come in contact and deform slightly under the imposed loads. This amount of deformation is dependent on the modulus of elasticity of the material in contact. It gives the contact stress as a function of the normal contact force, the radii of curvature of both bodies and the modulus of elasticity of both bodies. Hertzian contact stress forms the foundation for the equations for load bearing capabilities and fatigue life in bearings, gears, and any other bodies where two surfaces are in contact. It is recorded that Hertz completed his analysis whilst still a student, during the Christmas vacation of 1881! Dr Connor Myant DE2-EA2.1 M4DE 3

4 Design Engineering Example: Contact mechanics provides necessary information for the safe and energy efficient design of technical systems and is a key component in the study of Tribology. Principles of contacts mechanics can be applied in areas such as train wheel-rail contact, braking systems, road tires, bearings, gears, combustion engines, mechanical linkages, human joints, contact lenses, seals, metalworking, electrical contacts, sports ball and racket interactions and many others. Current challenges faced in the field may include stress analysis of contact and coupling members and the influence of lubrication and material design on friction and wear. Applications of contact mechanics further extend into the micro- and nano-technological realm. Design Engineering Example: Tribology is defined by the IMechE as; "science and technology of interacting surfaces in relative motion", Originating from the Greek, "tribos" which means rubbing or attrition. First coined in the 19060s and has since emerged as a multidisciplinary subject with important applications in Engineering and Design. In brief Tribology is the study of friction, wear and lubrication. You can break down nearly any assembly, part, or mechanical system and identify elements where friction, wear and lubrication are key factors in it s performance. Dr Connor Myant DE2-EA2.1 M4DE 4

5 Types of contacts We can divide contacts into two distinct types; Conforming (or conformal) contacts: between a convex surface (male cylinder or sphere) and a concave surface (female cylinder or sphere: bore or hemispherical cup). Examples Shoe and floor Pin-in-hole Journal bearing Ball-in-socket Eye and eye lid Characteristics Low pressure between the surfaces (~ Pa) Large area of contact Small elastic deformations Figure 8.1 Classic example of a highly conformal contact from a universal ball joint. Non-conforming (or counterformal) contacts: between two convex surfaces. Examples Ball or roller bearings Gear teeth Cam and tappet Baseball bat and ball Characteristics High local pressures (~GPa) Area of contact small compared with other dimensions Significant elastic (or plastic) deformations Dr Connor Myant DE2-EA2.1 M4DE 5

6 Figure 8.2. The contact point between two touching snooker balls is a perfect example of highly nonconformal contact. Bearing contacts Bearing contacts are a particular case of contact mechanics often occurring in conformal contacts. A contact between a male part (convex) and a female part (concave) is considered when the radii of curvature are close to one another. There is no tightening and the joint slides with no friction therefore, the contact forces are normal to the tangent of the contact surface. Moreover, bearing pressure is restricted to the case where the charge can be described by a radial force pointing towards the centre of the joint. Case of a cylinder-cylinder contact In the case of a revolute joint or of a hinge joint, there is a contact between a male cylinder and a female cylinder. The complexity depends on the situation, Figure 8.3 shows the three cases which can be distinguished by: 1. the parts have negligible clearance and are rigid bodies, 2. the parts have negligible clearance and are elastic bodies; 3. the clearance cannot be ignored and the parts are elastic bodies. In all three cases the axes of the cylinders are along the z-axis, and two external forces apply to the male cylinder: - a force F along the y-axis, the load; - the action of the bore (contact pressure). The main concern is the contact pressure with the bore, which is uniformly distributed along the zaxis. Dr Connor Myant DE2-EA2.1 M4DE 6

7 Figure 8.3. Bearing pressure for a cylinder-cylinder contact 1. Negligible clearance and rigid bodies In this first modelling, the pressure is uniform. It is equal to; P = radial load projected area = F D L Where D is the nominal diameter of both male and female cylinders and L the guiding length. Figure 8.4. Uniform bearing pressure: case of rigid bodies when the clearing can be neglected. 2. Negligible clearance and elastic bodies If it is considered that the parts deform elastically, then the contact pressure is no longer uniform and transforms to a sinusoidal repartition; Dr Connor Myant DE2-EA2.1 M4DE 7

8 P(θ) = P max cosθ Where P max = 4 π F LD Figure 8.5. Bearing pressure with a sinusoid repartition: case of elastic bodies when the clearing can be neglected. 3. Clearance and elastic bodies In cases where the clearance cannot be neglected, the contact between the male part is no longer the whole half-cylinder surface but is limited to a 2θ 0 angle. The pressure follows Hooke's law: P(θ) = KΦ α (θ) Where; - K is the rigidity coefficient; a positive real number that represents the rigidity of the materials - Φ(θ) is the radial displacement of the contact point at the angle θ - α is a coefficient that represents the behaviour of the material: α = 1 for metals (purely elastic behaviour), α > 1 for polymers (viscoelastic or viscoplastic behaviour). The maximum pressure is defined as; P max = 4F LD 1 cosθ 0 2θ 0 sin2θ 0 Dr Connor Myant DE2-EA2.1 M4DE 8

9 the angle θ 0 is in radians. Figure 8.6. Bearing pressure in case of elastic bodies when the clearance must be taken into account. Case of a conformal sphere-sphere contact A sphere-sphere contact corresponds to a spherical joint (socket/ball), such as the one shown in Figure 8.1. It can also describe the situation of bearing balls. For cases of uniform pressure we can model the contact like the cases above: i.e. when the parts are considered as rigid bodies and the clearance can be neglected (Figure 8.7.a), then the mean pressure is supposed to be uniform. It can also be calculated considering the projected area; P = radial load projected area = F πr 2 As in the case of cylinder-cylinder contact, when the parts are modelled as elastic bodies with a negligible clearance (Figure 8.7.b), then the mean pressure can be modelled with a sinusoidal repartition, and simplified to; P = 3F 2πR 2 Dr Connor Myant DE2-EA2.1 M4DE 9

10 Figure 8.7. Bearing pressure in the case of a sphere-sphere contact. When the clearance cannot be neglected (Figure 8.7.c), it is then necessary to know the value of the half contact angle θ 0, which cannot be determined in a simple way and must be measured. When this value is not available, the Hertz contact theory can be used. Hertz theory of elastic contact Hertz was concerned with the contact between optical lenses, but his analysis, the results of which are summarised here, is widely used for most non-conformal contacts and is especially important for those in rolling bearings, gears, railways (rail-and-wheel) and engine valve-gear (cam-and-tappet). The Hertz theory is normally only valid when the surfaces are non-conformal; one surface must be convex, the other one must be also convex or plane. However, we can, and do, apply Hertz theory for conformal contacts but the results must be considered with great care. It is an approximation only, and only valid when the inner radius of the container R 1 is far greater than the outer radius of the content R 2, in which case the surface container is then seen as flat by the content. However, in all cases, the pressure that is calculated with the Hertz theory is greater than the actual pressure (because the contact surface of the model is smaller than the real contact surface), which affords designers with a safety margin for their design. Dr Connor Myant DE2-EA2.1 M4DE 10

11 Figure 8.8. Hertz contact stress in the case of a male cylinder-female cylinder contact. Figure 8.9. Hertz contact stress in the case of a male sphere-female sphere contact. Simplifying Assumptions to Hertz's Theory Hertz's model of contact stress is based on the following simplifying assumptions: The materials in contact are homogeneous and the yield stress is not exceeded, Contact stress is caused by the load which is normal to the contact tangent plane, The contact area is very small compared with the dimensions of the contacting solids, The contacting solids are at rest and in equilibrium, The effect of surface roughness is negligible, There is now adhesive force acting between the two contacting surfaces. Dr Connor Myant DE2-EA2.1 M4DE 11

12 The tangent plane At the point of contact the surfaces are tangent to a plane and have a common normal. This means the stress experienced by the contacting bodies is caused by the load which is normal to the contact tangent plane which effectively means that there are no tangential forces acting between the solids. Figure Contact tangent plane for a non-conformal contact. Contact Radii and Elastic Modulus In general, if the elements in contact have three-dimensional shapes, such as balls in a rolling-element bearing, the contact can be simplified into that of an equivalent sphere and a rigid half space (flat plane). The contact in this idealised model is said to have a radius or relative curvature, R, and a contact elastic modulus of E. Suffices 1, 2 refer to the two bodies. x and y, refer to the directions of principal (i.e. maximum or minimum) curvature, which are assumed to coincide for the two bodies. Thus R x1 is the radius of body 1 in the x direction etc. We define the following: Radii of relative curvature (convex positive, concave negative): 1 = R x R x1 R x2 And, Dr Connor Myant DE2-EA2.1 M4DE 12

13 1 = R y R y1 R y2 1 R = 1 R x + 1 R y If either R x or R y =, then the contact patch is rectangular (a line contact). If R x = R y, then the contact patch is circular. We will not be considering elliptical contacts here. Contact modulus: E = ( 1 v v 2 2 ) E 1 E 2 1 where v is the Poisson's ratio of the material and E its Young's modulus. Equations for various contacts Semi contact width or radius, a Contact between a ball (sphere) and flat surface (a half-space) a = ( 3FR 4E ) 1 3 Maximum contact pressure, P max P max = 3F 2πa 2 = ( 1 π ) (6FE 2 R 2 ) Mean pressure, p 1 p = F 3 πa 2 Depth of indentation, δ δ = a2 R = ( 9F2 16E 2 R ) 1 3 Contact between two spheres For contact between two spheres of radii R 1 and R 2, the area of contact is a circle of radius a. The equations are the same as for a ball on flat (above) except that the effective radius R has changed. Dr Connor Myant DE2-EA2.1 M4DE 13

14 Contact between two crossed cylinders This is equivalent to contact between a ball on flat. Consider rotating the image until you are looking straight down one cylinder! Contact between a rigid cylinder with flatended and an elastic half-space a is the radius of the cylinder P max = 1 π E δ a p = F F δ = πa2 2aE Contact between a rigid conical indenter and an elastic half-space a = ztanθ Where z is the depth of the contact region The stress has a logarithmic singularity at the tip of the cone p = 4F πa 2 δ = π 2 z Dr Connor Myant DE2-EA2.1 M4DE 14

15 Contact between two parallel cylinders a = ( 4FR πle ) 1 2 P max = 2F πal = ( FE 1 p = F πlr ) 2 2aL δ = 2P π [1 v 1 2 ln ( 4R 1 E 1 a 0.5) + 1 v 2 2 ln ( 4R 2 E 2 a 0.5)] The effect of surface roughness The above solutions and methods for calculating contact size/pressure/stress is only applicable to smooth surfaces. In practice all surfaces are rough, to some degree. This causes the actual area of contact to be less than that predicted by Hertz and the local pressures and subsurface stresses to be higher - often much higher. Normally surface roughness is ignored and for many bearing contacts (which have very smooth surface) this is fine. Adhesive contacts There are several engineering systems that involve the rubbing contact of highly elastic bodies, including windscreen wipers, tyres and elastomeric seals. In biological systems this situation is almost universal since one or both of the contacting bodies is generally formed of human, animal or plant tissue. With a highly elastic body in contact, the contact area becomes large, mainly because the two surfaces are able to fully conform despite their roughness. When this occurs, adhesive forces become significant compared to the applied load and so must be included in the contact model. Dr Connor Myant DE2-EA2.1 M4DE 15

16 The Hertzian model of contact does not consider adhesion possible. However, in the late 1960s, several contradictions were observed when the Hertz theory was compared with experiments involving contact between rubber and glass spheres. It was observed [5] that, though Hertz theory applied at large loads, at low loads; the area of contact was larger than that predicted by Hertz theory, the area of contact had a non-zero value even when the load was removed, and there was strong adhesion if the contacting surfaces were clean and dry. Design Engineering Example: There are numerous biological contacts, or bio-tribological systems, that can be studied using the contact mechanics discussed in this chapter. Many of these are of great interest to design engineers. Human-related: Natural and replacement human joints Tooth wear Hair conditioners Skin creams Contact lens lubrication Friction and feel of fabrics Oral processing (eating) Catheters and surgical instruments Non human-related: Mobility of simple organisms Gecko adhesion and motion Snail lubrication Leaf cleansing processes References 1. Johnson, K. L, 1985, Contact mechanics, Cambridge University Press. 2. Popov, Valentin L., 2010, Contact Mechanics and Friction. Physical Principles and Applications, Springer-Verlag, 362 p., ISBN H. Hertz, Über die berührung fester elastischer Körper (On the contact of rigid elastic solids). In: Miscellaneous Papers. Jones and Schott, Editors, J. reine und angewandte Mathematik 92, Macmillan, London (1896), p. 156 English translation: Hertz, H 4. Bradley, RS., 1932, The cohesive force between solid surfaces and the surface energy of solids, Philosophical Magazine Series 7, 13(86), pp K. L. Johnson and K. Kendall and A. D. Roberts, Surface energy and the contact of elastic solids, Proc. R. Soc. London A 324 (1971) Derjaguin, BV and Muller, VM and Toporov, Y.P., 1975, Effect of contact deformations on the adhesion of particles, Journal of Colloid and Interface Science, 53(2), pp Dr Connor Myant DE2-EA2.1 M4DE 16

17 7. Muller, VM and Derjaguin, BV and Toporov, Y.P., 1983, On two methods of calculation of the force of sticking of an elastic sphere to a rigid plane, Colloids and Surfaces, 7(3), pp Tabor, D., 1977, Surface forces and surface interactions, Journal of Colloid and Interface Science, 58(1), pp Bearing pressure images attributed to: Cdang (Own work) [CC BY-SA 4.0 ( via Wikimedia Commons Dr Connor Myant DE2-EA2.1 M4DE 17