Figure 43. Some common mechanical systems involving contact.

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1 33 Demonstration: experimental surface measurement ADE PhaseShift Whitelight Interferometer Surface measurement Surface characterization - Probability density function - Statistical analyses - Autocorrelation - Power spectrum - Fractal dimension Surface contact Contact of engineering surfaces - Power and motion are typically transmitted through an interface between mechanical components (power is lost and failure originates) - Some common examples: Gear teeth Cam and follower Rolling element bearing Figure 43. Some common mechanical systems involving contact. - Two major contact categories: 1. Conformal (highly conformal contacts have complicated mechanics usually solved through numerical computations by means of the finite element method) 2. Non-conformal, or counter-formal (solved analytically) Conformal contact Non-conformal contact Figure 44. Illustrations of conformal and non-conformal contact.

2 34 Contact of elastic bodies Based on Hertzian theories (1882) 1. Contact between two elastic cylinders (line contact) For no load, the contact footprint is a line (becomes a rectangle whose width is 2a as the load increases) Load P per unit length Figure 45. Schematic depiction of a line contact (Wang, 2002). - Contact half width: a = 4PR E * - Effective modulus: E* = E 1 - Effective radius: R = R 1 R 2 - Surface normal stress (pressure): (P /a) 1/2 1 E Pressure distribution: p(x) = 2P a 1 x 2 a 2 1/2 = PE * R 1 x 2 a 2 1/2 p(x) = 0 at x = a - Maximum pressure (at x = 0): p 0 = 2P a = PE * R 1/2 - a p(x) a x

3 35 - Stress distribution (3D within the body): Figure 46. Von Mises stress distribution due to the combined action of a contact pressure and frictional shear (Liu and Wang, 1999). - Maximum shear stress: 1 = 0.30p 0 at x = 0, z = 0.78a

4 36 2. Contact between two elastic spheres (point contact) For no load, the contact footprint is a point (becomes a circular area whose radius is a as the load increases) Figure 47. Schematic depiction of a point contact (Wang, 2002). - Contact radius: a = 3WR 4E * - Effective modulus: E* = E 1 1/3 E Effective radius: R = R 1 R 2 1 p(x,y) - Surface normal stress (pressure): (W /a 2 ) x - Pressure distribution: p(x, y) = 3W 2a 1 x 2 2 a y 2 2 a 2 1/2 p(x,y) = 0 at x = y = a y z W - Maximum pressure (at x = y = 0): p 0 = 3W 6WE *2 = 2 2a 3 R 2 - Normal approach (deformation): = a2 R = 9 W 2 16 RE * 2 1/3 1/3 2a

5 37 - Maximum shear stress: 1 = 0.31p 0 at r = 0, z = 0.48a - Maximum tensile stress: r = 1 3 (1 2) p 0 at r = a, z = 0 3. Extension of Hertzian theories - The above theories can also be used for the contact of a cylinder or sphere against a flat surface (R 2 = ) - The theories can also be used for conformal contact (R 2 is negative) Real area of contact Surface roughness significantly affects the contact between two bodies The true area of contact << the apparent area of contact The contact pressure (stresses) >> than the nominal contact pressure Figure 48. Contact stresses between asperities (Stachowiak and Batchelor, 2001). Contact of rough surfaces Greenwood and Williamson statistical model of multiple asperity contact (1966) z d Figure 49. Model for contact between a rough surface and a smooth rigid plane (Hutchings, 1992).

6 38 - All asperities are assumed to have spherical surfaces of the same radius r - Height of an individual asperity above the reference plane: z - Separation between the reference plane and the flat surface: d - If d < z, the asperity will be elastically compressed and will support a load w which can be predicted from Hertz s theory: w = 4 3 Er1/2 (z d) 3/2 - Asperity heights are statistically distributed The probability that a particular asperity has a height between z and z + dz will be (z)dz, where (z) is the probability density function describing the distribution of asperity heights (asperities are identified by neighboring height values) *Note that (z) is different from p(z), where p(z) is the probability density function of all surface heights - Probability that an asperity makes contact with the opposing plane surface = probability that its height z > plane of separation d: prob (z > d) = d (z)dz - For a total of N asperities on the surface, the expected number of contacts n is: n = N d (z)dz - The total load carried by all the asperities is W = 4 3 NEr1/2 (z d) 3/2 (z)dz d - The load W is linearly proportional to the total real area of contact Figure 50. Theoretical curve of true area of contact versus load for steel flats of 10 cm 2 nominal area (Hutchings, 1992).

7 39 - Plasticity index (proportion of asperity contacts at which plastic flow occurs) = E * H * r 1/2 where: E* = effective modulus of the two surfaces H = indentation hardness of the rough surface (a measure of the plastic flow stress of the asperities) * = standard deviation of the distribution of asperity heights Figure 51. Dependence of asperity deformation mode on plasticity index for aluminum surfaces with different roughness values (Hutchings, 1992). Thermoelastic deformation - Sliding between two interfaces creates frictional heating - The temperature increase on the surfaces can be calculated through thermodynamic equations - This frictional heating can cause thermoelastic surface deformations (when combined with the contact stresses, possibly resulting in thermally induced surface fatigue) - Frictional heating is usually locally concentrated at asperity contacts

8 40 Figure 52. Concentration of frictional energy at asperity contacts (Stachowiak and Batchelor, 2001). Example: Evolution of surface contact From: M.T. Siniawski, S.J. Harris, Q. Wang and S. Liu, Wear initiation of steel sliding against a thin boron carbide coating, Tribology Letters 15 n1, 2003, Background: A three-dimensional thermo-mechanical asperity contact model developed by Liu and Wang (2001) was used to model the contact evolution of a steel ball sliding against a B 4 C coated disc. The model takes into account steady-state heat transfer and asperity distortion due to thermoelastic deformations. Discrete convolution and FFT (DC-FFT) and a conjugate gradient method (CGM) were employed as the solution methods. By neglecting the thermal conductivity of the B 4 C coating (the thermal conductivity of the B 4 C coating is 92 W/m-K, whereas the thermal conductivity of steel is 46.6 W/m-K), the model treats the ball-disk contact as a steel surface contacting a rigid adiabatic plane, which gives an upper bound on the frictional heating.

9 41 Steel ball surface evolution Figure 53. Steel ball wear scar evolution for the first six data files (sliding distances).

10 42 Contact pressure evolution Figure 54. Evolution of the contact pressure contours for the first six data files (sliding distances), where the black dots represent regions of high contact pressure.

11 43 Evolution of the plastically deformed area Figure 55. Plastic deformation area as a function of sliding distance. Observations: Prior to wear initiation, the contact pressure is a highly concentrated circular patch. As the wear begins and sliding increases, the contact pressure becomes distributed over a larger surface area, corresponding to the increasing size of the wear scar. Although the overall size of the contact area increases with sliding distance, the size of the contact area experiencing plastic deformation remains nearly constant once wear begins. This result, that the nature of the contact is independent of the nominal contact area except for a spreading out of the contact points, is consistent with the Greenwood Williamson theory.

12 44 References Books K.L. Johnson, Contact Mechanics, Cambridge University Press, G.W. Stachowiak and A. W. Batchelor, Engineering Tribology, 2nd ed., Butterworth-Heinemann, Boston, I.M. Hutchings, Tribology: Friction and Wear of Engineering Materials, Edward Arnold, London, Articles G. Liu and Q. Wang, Thermoelastic Asperity Contacts, Frictional Shear and Parameter Correlation, ASME Journal of Tribology v122 n1, 1999, J.A. Greenwood and J.B.P. Williamson, Contact of nominally flat rough surface, Proceedings of the Royal Society of London A295, 1966, M.T. Siniawski, S.J. Harris, Q. Wang and S. Liu, Wear initiation of steel sliding against a thin boron carbide coating, Tribology Letters 15 n1, 2003, S. Liu and Q. Wang, A Three-Dimensional Thermomechanical Model of Contact Between Non-Conforming Rough Surfaces, ASME Journal of Tribology 123, 2001, Other Q. Wang, Introduction to Tribology, Lecture Notes, Northwestern University, 2002.

13 45 Homework 3 (Due 09/21/2006) Please complete the two problems listed below. Turn in a brief, typed report along with any created MATLAB programs, if appropriate. Problem 1 The contact stresses for gears in contact are typically determined through the AGMA formulas. However, the contact stress can also be directly determined through the methods discussed above. The diagram below shows the basic geometry of a pair of spur gears, where the base-circle radii are r b1 and r b2, the pitch-circle radii are r 1 and r 2. AP and BP are the radii of curvature for two gear teeth meshing at the pitch point. Figure 56. Basic geometry for a pair of spur gears in contact (Wang, 2002). If the pressure angle is = 20, the gear module is m = 5 mm, the numbers of teeth for the gears are N 1 = 20 and N 2 = 40, the materials are gear steel grade 1 (E = 210 GPa and hardness of 300 BHN), please determine the following: 1. The radii of curvature of the teeth at the pitch point. 2. The equivalent elastic modulus E* and the effective radius of curvature R, if we treat the tooth contact as a cylinder-plane contact. 3. The contact half width a, the maximum contact pressure p 0 and the normal approach, if the normal load is 40 kn/m. 4. The average pressure in the contact region. 5. How serious is the plastic deformation under this load? Does the material yield at all, or yield in the substrate only? Has surface flow been initiated?

14 46 Problem 2 Use the ASCII test surface data file for the worn surface, which is available from the course website. Find the asperity with the maximum height value and create a 2-D profile that includes this maximum asperity from the 3-D data set. The asperities can be approximated by spheres as shown below, where the radius of curvature is determined by means of the second-order differentiation of the data with the finite difference method: 1 R = 2z i z i1 z i+1 x 2 Figure 57. Finite difference approximation of the asperity tip radius (Wang, 2002). Please determine the following: 1. Identify the highest asperity and simplify it as a sphere whose height is that of the original asperity h and radius is that of the radius of curvature at the tip of the asperity R. 2. If this rough 2-D surface profile is for a typical steel (E = 210 GPa) that is now in contact with an ideally smooth surface of the same material, calculate the maximum contact pressure on this asperity when the normal approach is 1/5 of its height.