Surface Roughness and Contact Connectivity. Ilya I. Kudish. Donald K. Cohen. Brenda Vyletel

Transcription

1 Surface Roughness and Contact Connectivity Ilya I. Kudish Professor of Mathematics, ASME Fellow, Department of Mathematics, Kettering University, Flint, MI, 48439, USA Donald K. Cohen Michigan Metrology, LLC, N. Laurel Park Dr., Suite 51, Livonia, MI 48152, USA Brenda Vyletel Electron Microbeam Analysis Laboratory, University of Michigan, Ann Arbor, Michigan 48109, USA Abstract Solution of a plane frictionless contact problem for two rough elastic solids is considered. An exact solution of the problem resulting in a singly connected contact region is considered and it is conveniently expressed in the form of a series in Chebyshev polynomials. A sufficient (not necessary) condition for a contact of the solids to be singly connected is derived. The singly connected contact condition depends on the surface micro-topography, material effective elastic modulus, solid shapes, and applied load. It is determined that under certain conditions a normal contact of three times differentiable rough surfaces with sufficiently small asperity amplitude and/or sufficiently large applied load is singly connected. KEY WORDS: rough elastic contacts, contact connectivity 1

2 Introduction Analysis of elastic dry rough contacts has a long history. Namely, some theoretical models of contact interaction of dry rough surfaces which are based on specific roughness topography as well as taking into account surface adhesion were proposed in [1]-[9] while in [10] a numerical study was performed. In [11], the authors presented an elegant and clear analytical way to show that if the surface roughness of elastic surfaces is a continuous but nowhere differentiable Weierstrass function then a contact of such surfaces must be multiply connected. The proof is based on the assumed specific fractal properties of the Weierstrass function. As will be discussed, the Weierstrass roughness distribution may not be a good approximation for a real surface topography. This paper as well as [11] consider conceptually the same contact problem of elasticity for rough surfaces based on continuum mechanics, i.e. the theory of elasticity but due to different assumption concerning the distribution of the roughness come to opposite conclusions. In [11] the conclusion is that a contact of elastic rough surfaces is always multiply connected, i.e. for any load there are discrete contacts of asperities while this paper comes to the conclusion that for sufficiently smooth roughness distributions and sufficiently high load a contact is singly connected. The reasonable question is, what makes the conclusions of these two papers so different? The difference between these two papers and their opposite conclusions is due to the assumption concerning the degree of smoothness/differentiability of the surface roughness distributions. To demonstrate the assumption of the present paper that the surface roughness can be described by a three times differentiable function the authors conducted a series of roughness measurements using an Optical Profilometer and Atomic Force Microscope on a real ground steel surface (for example, used for gears) with progressively increasing lateral resolution. The obtained images presented in the paper demonstrate that higher lateral resolution measurements did not reveal any significant increase in the irregularity of the surface texture as would be suggested by a Weierstrass fractal type (i.e. non-differentiable) texture distribution. Moreover, these images revealed that texture of industrial metal (for example, steel) surfaces can be described by a three times differentiable function. Therefore, the assumption concerning surface roughness in the current paper is substantiated by real measurements of a real surface. At the same 2

3 time, these measurements do not support the assumption that a real ground steel surface roughness can be described by a Weierstrass type function made in [11]. It means that the conclusions of the current paper are correct and correspond to the practice while the conclusions of [11] are based on the assumption concerning roughness distribution which is not supported by experimental data and, therefore, the conclusions of [11] for high applied loads may not correspond to practice. In [12], the exact solution of a problem for a rigid indenter with a flat bottom contacting an elastic surface with a triangular shaped concavity/bump with a sharp (non-differential) vertex shows that no matter how high the applied load is the contact will always be multiply connected. The classic models [2, 3] and similar to them assume that asperities are represented by semi-spheres or semi-ellipsoids creating concavities at their bases with an elastic half-space making the rough surfaces not differentiable across these border curves. Therefore, in the light of the results from [12] it is not surprising that in the cases described by these models the contact area always (for any load) consists of discrete contacts of asperities. However, the measurements presented in [12] and in this paper show that the roughness distribution is everywhere a differential function, i.e. it does not possess points/curves at/across which this roughness distribution is not differentiable. Therefore, it would be improper to refer to these classic papers [2, 3] and compare their multiply connected contacts with singly connected contacts of this paper for sufficiently smooth rough surfaces and high applied load as the surface roughness distributions are very different. There is a dichotomy in the understanding of the contacts of rough elastic solids. On the one hand, there is a seminal model of Greenwood and Williamson [3] leading to the conclusion that for any applied load a contact of two rough surfaces is always multiply connected (i.e. there is no perfect seal). On the other hand, among researches and practitioners there is still a common widely shared and deeply rooted misconception that it is obvious that if two solids are pressed together hard enough the contact would always be singly connected. To clearly see that it is not obvious it is sufficient to take a look at the two examples presented in [12] where based on the exact contact problem solution it is shown that when there is a dimple with a non-differentiable bottom the contact will be always (i.e. for any applied load) multiply connected. In some cases researchers accept the assumptions of Greenwood and Williamson and, at the same time, are convinced that for sufficiently high load any two rough surfaces would make a perfect contact. 3

4 It is a contradiction. Significant strides in understanding of rough contacts were made over more than 50 years of research but still there are some fundamental questions to be answered such as: Is it possible to have a perfect contact of two rough elastic surfaces? and if it then What are the conditions for this to occur? These questions were addressed in [1, 13] for a specific roughness topography represented by a sinusoidal wave. In practice, depending on a number of factors such as the loading, initial surface roughness topography, material stress yield, and residual stress distribution the running-in stage results in a reduction in asperity curvature and overall roughness amplitude. Therefore, even if the original surface topography is very irregular (non-differentiable), due to material plasticity and wear the asperity distribution quickly (after few loading cycles) becomes more regular, i.e. mathematically speaking several times differentiable. Of course, the whole notion of differentiability can be applied only on the scale much larger then the atomic one which, obviously, coincides with the range of continuum mechanics applicability. In addition to that, after running-in stage all material deformations occur in the elastic range of parameters, i.e. plastic deformations no longer take place. Therefore, the paper considers connectivity of a contact of rough elastic surfaces with relatively smooth texture described by a differentiable distribution. The goal of this paper is to answer these questions, i.e. to rigorously consider the conditions under which a contact of elastic surfaces is always singly connected if the solid surface roughness is described by a general sufficiently smooth (not necessarily sinusoidal) function, i.e. sufficient number of times differential function. To do that a simple analytical analysis was conducted and an estimate of the minimal force applied to the solids producing a singly connected contact is presented. The whole analysis is based on application of continuum mechanics equations which are supported by the presented measurements where the height resolution is approximately 6 nm and 0.1 nm for the Optical Profiler and Atomic Force Microscope, respectively. Note that the lateral resolution for the Optical Profiler is approximately 1 µm while the lateral resolution for the Atomic Force Microscope is approximately 1 nm. 1 Contact of Two Rough Elastic Solids Let us consider a contact of two rough elastic solids infinitely extended in the z direction. Suppose the elastic moduli and Poisson s ratios of the solid 4

5 P y y(x)=f 2 (x)+φ 2 (x) E 2, ν 2 a b x E 1, ν 1 y(x)=-f 1 (x)-φ 1 (x) Figure 1: Basic geometry for a contact problem for two elastic solids with surface with textures. materials are E 1, E 2 and ν 1, ν 2, respectively. In a coordinate system related to the contact the shapes of the lower and upper surfaces are described by functions y = f 1 (x) φ 1 (x) and y = f 2 (x) + φ 2 (x), where functions f 1 (x) and f 2 (x) describe the smooth shapes of the elastic solids while functions φ 1 (x) and φ 2 (x) reflect the roughness topography of the two surfaces. The solids are loaded with a normal load P. Let us assume that the contact of these surfaces is frictionless and singly connected, i.e. it occupies the interval [a, b] on which the contact pressure q(x) > 0 for all a < x < b (see Fig. 1). Assuming that the elastic displacements of our solid surfaces can be approximated by the corresponding displacements of elastic half-planes with the same elastic parameters the contact problem can be reduced to the 5

6 following equations φ (x) + 2 πe b a q(t) ln a 0 dt = δ f(x), q(a) = q(b) = 0, x t b a q(t)dt = P, (1.1) where E 1 is the effective elastic modulus, = 1 ν2 E 1 E ν2 2 E 2, δ is a constant related to the rigid normal displacement of the solid centers and it differs only by a constant, a 0 is a characteristic semi-width of the contact, φ (x) is the function describing the distribution of the combined roughness of the two surfaces, φ (x) = φ 1 (x) + φ 2 (x), f(x) is the clearance between the two smooth surfaces, f(x) = f 1 (x) + f 2 (x). Obviously, the same way we can formulate and, therefore, analyze a plane contact problem for a rigid indenter with perfectly smooth surface and smooth edges and an elastic half-plane with rough surface. In dimensionless variables (x, t, a, b ) = 1 a 0 (x, t, a, b), q (x ) = q(x) q 0, (ωφ (x ), f (x )) = πe 2P (φ (x), f(x)), δ = πe 2P δ, (1.2) problem (1.1) can be rewritten in the form (for simplicity primes are omitted) ωφ(x) + 2 π b a q(t) ln 1 dt = δ f(x), q(a) = q(b) = 0, x t b a q(t)dt = π 2, (1.3) where ω is a dimensionless constant characterizing the amplitude of the combined surface roughness profiles which is directly proportional to the characteristic combined asperity height and inversely proportional to the applied characteristic pressure q 0 = 2P πa 0. The analysis of this problem is more convenient to conduct using a different independent variable u = 2 a+b [x ]. b a 2 In this variable (i.e. in u) the exact solution of problem (1.3) for a singly 6

7 connected contact has the classic form [14] q(u) = 1 u 2 π(b a) { 1 f ( a+b 2 t)dt (t u) + ω 1 φ ( a+b 2 t)dt (t u) }, 1 1 π f ( a+b 2 t)tdt = ω π φ ( a+b 2 t)tdt, (1.4) 1 π f ( a+b 2 t)dt = ω π φ ( a+b 2 t)dt, where differentiation in (1.4) is done with respect to the variable x. The value of the constant δ can be easily determined from the first equation in (1.3) after the values of q(u), a, and b are calculated from equations (1.4). Let us assume that the function φ ( a+b + b a u) can be represented by a 2 2 series in Chebyshev orthogonal polynomials of the first kind [15] 1 as follows ξ 0 = 1 π φ ( a+b 2 + b a 2 u) = φ ( a+b 2 t)dt, ξ k = 2 π k=0 ξ k T k (x), x 1, φ ( a+b 2 t)t k(t)dt, k = 1, 2,... (1.5) The representation (1.5) of the surface roughness is very closely related (can be reduced) to a Fourier representation which can be used successfully, for example, for any function with a piece-wise continuous first derivative. As it can be seen from presented below pictures of a real rough surface its roughness distribution is certainly differentiable which justifies the choice for the distribution of (1.5). 1 Chebyshev orthogonal polynomials of the first kind T k (x) and of the second kind U k (x) [15] are defined as follows T k (cos θ) = cos kθ and U k (cos θ) = sin(k+1)θ sin θ, respectively. These polynomials satisfy the following properties k (t)t m (t)dt T 1 t = 0 if k m, 2 T 2 k (t)dt = π if k = 0, k = 0; T 2 k (t)dt = π 2 1 t2 U k (t)u m (t)dt = 0 if k m, T k (t)dt 1 t2 (t x) = πu k(x), k = 1, 2,... if k 0 and the integral is equal to π if 1 t2 U 2 k (t)dt = π 2 if k 0, and 7

8 Substituting (1.5) in the first equation of (1.4) and assuming that it is legitimate to interchange the order of summation and integration in an infinite series (the details of that are presented below) we obtain q(u) = 1 u 2 π(b a) { 1 f ( a+b 2 t)dt (t u) + πω ξ k U k (u)}, u 1. (1.6) k=1 2 Analysis of Contact Connectivity In order for the rough contact to be singly connected it is necessary that at all points of the contact segment (, 1) pressure would be positive, i.e. q(x) > 0 for all < x < 1. Based on formula (1.6) this requirement translates into the inequality f ( a+b 2 t)dt (t u) + πω ξ k U k (u) > 0, < u < 1. (2.1) k=1 The further analysis is done for the case when the original elastic surfaces are perfectly smooth (i.e. φ ( a+b + b a u) = 0) and create a singly connected 2 2 contact. The latter is represented by the inequality f ( a+b 2 t)dt (t u) > 0, < u < 1. (2.2) In most practical cases the inequality (2.2) is satisfied. The rare exceptions to that are the cases when the contact of the associated smooth surfaces with the gap f(x) is multiply connected. If the series in (2.1) is finite (i.e. there exists such a number K > 0 that ξ k = 0 for all k > K) then the sum of the series in (2.1) is a continuous function on a closed interval [, 1]. Taking into account the fact that for < u < 1 the integral in (1.6) and (2.1) is a continuous function as long as the function g(u) = f ( a+b + b au) satisfies the inequality g(u 2 2 2) g(u 1 ) u 2 u 1 µ for all u 1, u 2 [, 1] and some constant 0 < µ < 1, the whole expression in the left-hand side of the inequality (2.1) is a continuous function on the interval [, 1] where it reaches its minimum. Therefore, assuming that (2.2) is true it is sufficient to choose a small enough positive ω for the inequality (2.1) to be valid on the entire interval (, 1) and make pressure q(x) from (1.6) positive in the entire contact (, 1), making the contact singly connected. The value of the parameter ω can be made smaller 8

9 by using surfaces with smaller asperity heights, smaller effective elasticity modulus E, and/or by applying higher load P (see (1.2)). Now, let us consider the most interesting case of an arbitrary rough surface represented by an infinite series in (1.6) and (2.1). Due to the fact that U k (x) k for all x 1, for the inequality (2.1) to be valid it is sufficient to require that f ( a+b 2 t)dt (t u) πω k ξ k > 0, < u < 1. (2.3) k=1 It is important to realize that (2.3) is a sufficient but not necessary condition on the minimal load, i.e. on the minimal load sufficient to make the contact singly connected.. Now, let us analyze the conditions which should be imposed on the coefficients ξ k in order for all assumptions made above to be valid. In particular, we have to specify the particular conditions under which the series in (1.5)- (2.1) and (2.3) are convergent and are continuous functions of u. To do that it is sufficient to consider the behavior of the coefficients ξ k from (1.5) as k. Making the substitution t = cosθ in the second equation in (1.5) we obtain the integral ξ k = 2 π π 0 φ ( a+b + b a cos θ) cos kθdθ, k = 1, 2, (2.4) the asymptotic of which for k can be determined taking the real part of the Fourier integral 2 π π f(θ)e ikθ dθ, f(θ) = φ ( a+b + b a cosθ). 2 2 (2.5) 0 Assuming that φ(x) is N times differentiable function and integrating integral in (2.5) by parts one gets [16] = N n=0 β f(θ)e ikθ dθ α ( i k )n+1 [e ikα f (n) (α) e ikβ f (n) (β)] + O(k N ), k 1, (2.6) It can be shown that if f(θ) = φ ( a+b + b a cosθ) (where differentiation is 2 2 done with respect to x) then any derivative of f(θ) of odd order is a derivative 9

10 of the function φ ( a+b + b a cosθ) with respect to θ of even order and it has the 2 2 form of f (2k+1) (θ) = h(cos θ) sin θ, where h(cosθ) is some bounded function of θ while any derivative of f(θ) of even order is a certain function of just cosθ. Using this fact and taking α = 0, β = π we get that all derivatives of f(θ) of odd order at θ = 0 and θ = π are equal to zero. In addition to that, for α = 0 and β = π we have e ikα = 1 and e ikβ = () k. Also, it is important to remember that the values of the derivatives f (n) (α) and f (n) (β) are independent of k. Therefore, every term of the sum in (2.6) with even value of index n is purely imaginary while the terms with odd values of index n are real. It means that if the function f(θ) is N times continuously differentiable then 1 ξ k = O( ), k 1. (2.7) k N+1 Therefore, the assumption that the function f(θ) = φ ( a+b + b a cosθ) is 2 2 continuously differentiable just once (i.e. N = 1) leads to the estimate ξ k = O( 1 k 2 ), k 1. (2.8) Estimate (2.8) means that the series in (1.5) converges uniformly to the continuous function φ ( a+b+ b a u) while series in (1.6) and (2.1) may converge 2 2 or diverge depending on the behavior of Chebyshev polynomials T k (u) and function φ ( a+b + b au). On the other hand, if we assume that 2 2 φ ( a+b + b au) 2 2 is a twice continuously differentiable function (i.e. N = 2) then from (2.6) follows that ξ k = O( 1 ), k 1, (2.9) k 3 because f (0) = f (π) = 0. Estimate (2.9) means that the series in (1.5)- (2.1) converge uniformly to continuous functions as well as the interchange of integration and summation made to obtain (1.6) is legitimate. In particular, it means that the nonnegative pressure q(x) is a continuous function for < x < 1. Also, it means that for sufficiently small ω the contact is singly connected (see, for example, (2.18)). Thus, a contact of two elastic solids with rough surfaces or a rigid smooth indenter with smooth edges with an elastic rough half-plane are singly connected as long as the roughness profile function φ(x) is at least three times continuously differentiable and the parameter ω is sufficiently small (see, for example, (2.18)). In practice, the initial roughness profile can be a significantly irregular function, for example, a function similar to the Weierstrass function. However, depending on the number of factors such as loading, initial surface 10

11 Figure 2: Measurement of a Ground surface using an optical profiler over a field of view of 100µm 100µm, with diffraction limited lateral resolution of smaller than 1µm and height resolution of 6nm. Reprinted with permission from the ASME roughness topography, material yield, and residual stress distribution the running-in stage reduces the roughness profile amplitude and spacing characteristics and, in particular, makes sharp asperities smoother due to material plasticity and wear. Therefore, even if the original surface roughness profile is as irregular as the Weierstrass type distribution analyzed in [11] due to material plasticity and wear the asperity distribution quickly becomes more regular (i.e. mathematically speaking at least twice differentiable). Moreover, as soon as a contact becomes singly connected it tends to maintain this status because the pressure and frictional stresses become distributed more uniformly and material plastic deformations diminish as well as the rate of wear decreases. The latter is true until some other competing mechanisms (for example, such as pitting) take dominance. 11

12 Figure 3: Measurement of a Ground surface using an atomic force microscope over a field of view of 2µm 2µm, with lateral resolution of 16nm and height resolution of 0.1nm. Note the image is displayed on the same gray scale as for optical profiler measurements. Reprinted with permission from the ASME Let us consider an example of a rigid parabolic indenter with smooth surface (f(x) = x 2 ) in contact with an elastic half-plane with rough boundary described by the function φ(x). In this case the scaling parameter a 0 can be taken equal to the Hertzian contact half-length a H = 2 P R and the integral πe involved in (2.3) is equal to π(b a) 2 /2 and it is positive as long as b > a. Substituting the latter expression in (2.3) we get the sufficient condition for a singly connected contact in the form (b a) 2 2 ω k=1 k ξ k > 0. (2.10) Moreover, the last two equations in (1.4) for constants a and b are reduced 12

13 Figure 4: Measurement of a Ground surface using an atomic force microscope over a field of view of 2µm 2µm, with lateral resolution of 16nm and height resolution of 0.1nm. Note the image is displayed on a different gray scale as for the optical profiler measurements to demonstrate the surface features. Note that the finer spaced parallel line features are scanning related artifacts. Reprinted with permission from the ASME to the following system 1 ( b a 2 )2 = ω π φ ( a+b 2 t)tdt b, 2 a 2 = ω 2 π φ ( a+b 2 t)dt. (2.11) while the solution for pressure q(u) has the form (see (1.6)) q(u) = 1 u 2 { (b a)2 + ω ξ b a 2 k U k (u)}. (2.12) k=1 Assuming that φ ( b+a + b a u) can be represented by a series from (1.5) 2 2 and that the function φ(x) is three times differentiable we easily get the 13

14 Figure 5: Measurement of a Ground surface using an atomic force microscope over a field of view of 2µm 2µm, with lateral resolution of 8nm and height resolution of 0.1nm. Note that the finer spaced parallel line features are scanning related artifacts. Reprinted with permission from the ASME expressions for the integrals involved in (2.11) and equations (2.11) can be rewritten in the form 1 ( b a 2 )2 = ω 2 ξ 1, b 2 a 2 2 = ωξ 0. (2.13) The latter equations can be easily solved for a and b as follows a = 2+ω(ξ 0 ξ 1 ) 2 ωξ 1 1 ω ξ 2 1, b = 2 ω(ξ 0+ξ 1 ) 2 ωξ 1 1 ω ξ 2 1. (2.14) However, to determine the coefficients ξ 0 and ξ 1 the expressions from (2.14) need to be substituted in the formulas (1.5) for ξ 0 and ξ 1 which lead to a system of nonlinear equations ξ 0 = 1 π φ ((t ωξ 0 ) 1 ω 2 ωξ 1 2 ξ 1)dt, ξ 1 = 2 π 14 φ ((t ωξ 0 2 ωξ 1 ) 1 ω 2 ξ 1)tdt. (2.15)

15 Figure 6: Measurement of a Ground surface using an atomic force microscope over a field of view of 2µm 2µm, with lateral resolution of 4nm and height resolution of 0.1nm. Note that the finer spaced parallel line features are scanning related artifacts. Reprinted with permission from the ASME The system (2.15) can be solved iteratively using the fixed-point iteration method or using a perturbation method for ω 1. The latter method for ω 1 gives the following approximate solutions for coefficients ξ 0 and ξ 1 : ξ 0 = 1 π φ (t)dt + O(ω), ξ k = 2 π k = 1, 2,..., ω 1. φ (t)t k (t)dt + O(ω), (2.16) Using the first equation in (2.13) the sufficient condition (2.10) can be rewritten in the form 2 ωξ 1 ω k ξ k > 0. (2.17) k=1 15

16 Therefore, for a parabolic indenter the sufficient condition for a singly connected contact can be expressed in the form 0 ω < ω, ω = 2{ξ 1 + k ξ k } > 0. (2.18) In this case according to (2.18) for sufficiently small roughness profile amplitude, sufficiently small effective elastic modulus, and sufficiently high applied load a contact of two rough elastic solids is singly connected. For practical applications the infinite series in (2.3) and (2.18) can be truncated and replaced by sufficiently precise and convenient finite series. A similar problem for a rigid indenter with sharp edges was considered in [12]. Let us consider a simple and transparent example for a parabolic rigid indenter and a symmetric about the origin x = 0 roughness profile φ (x), i.e. when φ (x) is an even function of x. Then b = a and in the dimensional variables we have f(x) = x2, where R is the effective radius. At the 2R same time, we assume that the dimensional roughness profile is described by just one harmonic. Taking into account the accepted assumption that φ (x) is an even function of x we assume that φ (x) is described by just one Chebyshev polynomial (see (1.5)) k=1 φ (x) = A k T k ( x ), x b, (2.19) b where A k is a dimensionless constant characterizing the roughness amplitude and k is an odd positive integer representing the order of the harmonic, k > 1. Then in the dimensionless variables (1.2) we have f(x) = x 2, φ (x) = sign(a k )T k (x), ξ k = sign(a k ), ω = π A k E b 2P. (2.20) Therefore, the sufficient requirement (2.18) for a singly connected contact is reduced to the following inequality ω < 2 k, (2.21) which in dimensional variables is equivalent to the inequality on the force P applied to the contact P > k A k πe b. (2.22) 4 The latter inequality is similar to the one obtained by K.L. Johnson [13]. Based on formulas (2.14) we can conclude that b = a H [1+O(ω)] = 2 P R [1+ πe 16

17 O(ω)] and, therefore, from formula (2.22) we obtain P > P min, P min = k 2 A 2 k πe R 4 [1 + O(ω)], ω 1. (2.23) Obviously, the higher the order of the roughness harmonic k, the higher is the force P which is sufficient to flatten the rough surface. Moreover, the minimum force P min increases quadratically with the order of the harmonic k. Let us consider a specific example of an EPDM seal (with elastic modulus E = 9 MP a and Poisson s ratio ν = 0.4) between a piston made of 1215 steel (with elastic modulus E = 200 GP a and Poisson s ratio ν = 0.29) and a bore made of aluminum Al 6061-T5 (with elastic modulus E = 70 GP a and Poisson s ratio ν = 0.33). Simple calculations show that the effective elastic modulus E in the contacts of the seal with the piston and bore is practically equal to the one calculated for the relatively soft seal. The above formulas show that for a specific sealing macro geometry and the same roughness height parameter Rt = 0.4 µm (peak to valley) for the roughness harmonic k = 5 the minimum load providing a perfect seal is P min = N/m while for the roughness harmonic k = 297 the minimum load providing a perfect seal P min = 166, 804 N/m is much higher. Obviously, the ability to take into account the actual texture of a surface roughness (not only its amplitude) provides the opportunity for a manufacturer to control the operating sealing conditions. In case of a rigid indenter with a flat bottom considered in [12] if we take φ (x) = A k T k ( x a ) (A k is a dimensional constant characterizing the roughness amplitude) then we get the force P sufficient to flatten the rough half-plane surface P > P min, P min = ka k πe 2. (2.24) The difference between the values of the minimum load P min from (2.23) and (2.24) sufficient to flatten the surface of the rough half-plane in terms of their dependence on the harmonic order k and roughness amplitude A k is due to a relatively low pressure q(x) near the end points x = ±b of the contact with the parabolic indenter without sharp edges. Generally, the force P min for the case of a parabolic indenter is higher than for the case of an indenter with a flat bottom. When manufacturing critical components, various potentially expensive finishing operations are used to minimize the surface roughness. For bearing surfaces the roughness amplitude might be minimized to reduce pressure 17

18 spikes. For seal related applications, the drive towards perfectly smooth surfaces may not be necessary. As discussed above, a criteria similar to (2.18) may reduce manufacturing cost by indicating the roughness level sufficient for creating singly connected contacts in intended applications, i.e. for given loads or vice versa for the given load a level of surface roughness amplitude can be determined below which contacts of such surfaces would be singly connected. Similar predictions can be made concerning the influence of the effective elastic modulus on contact connectivity (see the definition of parameter ω in (1.2)). Some examples of the topography of a ground surface measured on a WYKO NT8000 and Veeco Dimension Icon AFM are presented in Fig The WYKO NT8000 measurements were performed with a field of view of 100 µm 100 µm with a lateral resolution of 1 µm and height resolution of 6 nm. The AFM measurements were performed in the tapping mode providing lateral resolution of 4 nm 16 nm over a measurement area of 2 µm 2 µm. The height resolution of the AFM measurements was 0.1 nm. Figures 4-6 demonstrate that over the same field of view and sequentially increasing lateral resolutions the texture appears to be described by essentially the same at least three times differentiable functions. Would the surface be of fractal nature for each higher lateral resolutions the pictures would show new surface features which is not the case here. Furthermore, depending on the material properties, the surface texture features that may not be three times differentiable are likely to plastically deform under initial contact leaving at least a three times differentiable distribution of asperities. These measurements of a real surface demonstrate the assumption that the roughness profile can be represented by a sufficiently smooth (i.e. sufficient number of times differentiable) function and also demonstrate the fact that the continuum mechanics equations describing the phenomenon of rough elastic surface interaction are still valid on the considered scale. Conclusion Solution of a contact problem for two elastic solids with rough surfaces and a rigid smooth indenter with smooth edges and a rough elastic half-plane are considered. An exact solution of the problem for singly connected contact regions is obtained and it is conveniently expressed in the form of a series in Chebyshev polynomials. The conditions for a rough contact to be 18

19 singly connected are analyzed. It is shown that for sufficiently small asperity amplitudes and sufficiently large applied load contacts of rough surfaces are singly connected for three times differentiable roughness profiles. A few simple examples of rough contacts are considered. Some examples of Atomic Force Microscope roughness measurements for the same surface piece and increasing lateral resolution are presented. These measurements support the assumption that real ground rough surfaces can be represented by sufficiently smooth (i.e. differentiable) functions. Nomenclature a, b - coordinates of the end-points of the contact, [m], a 0 - characteristic half-width of the contact, [m], a H - half-width of the Hertzian contact, [m], f(x) - combined clearance between the smooth surfaces corresponding to the rough surfaces in contact, [m], q(x) - contact pressure, [N/m 2 ], q 0 - characteristic pressure in the contact, [N/m 2 ], x - point coordinate, [m], A k - dimensionless or dimensional constant characterizing surface roughness amplitude, [1] or [m], respectively, E - effective elastic modulus of material, [N/m 2 ], P - applied normal load, [N/m], P min - minimum load sufficient to flatten the surface of a rough half-plane, [N/m], R - effective indenter radius, [m], T k (x) and U k (x) - Chebyshev polynomials, ξ k - dimensionless coefficients in the series in Chebyshev polynomials, φ (x) - surface roughness amplitude, [m], φ(x) - dimensionless surface roughness amplitude, ω - dimensionless parameter characterizing the overall asperity height amplitude 19

20 References [1] Westergaard, H.M. Bearing Pressure and Cracks. Thans. ASME J. Appl. Mech., 1939; Vol. 49, No. 6, A49-A53. [2] Bowden F.P. and Tabor D. F riction. An Introduction to T ribology, 2 nd ed. Malabar, Florida: Robert E. Krieger Publishing Co., [3] Greenwood J.A. and Williamson J.B.P. Contact of Nominally Flat Surface, Proc. R. Soc. Lond., 1966; Ser. A, Vol. 295, [4] Johnson, K.L., Greenwood, J.A., and Higginson, J.G. The Contact of Elastic Regular Wavy Surfaces, Intern. Journal of Mechanical Sciences, 1985; Vol. 27, Issue 6, [5] Ciavarella M., Delfine V., and Demelio V.G. A Re-Vitalized Greenwood and Williamson Model of Elastic Contact Between Fractal Surfaces J. of the Mech. and Phys. of Solids, 2006; Vol. 54, No. 12, [6] Johnson K.L., Kendall K., and Roberts A.D. Surface Energy and the Contact of Elastic Solids, Proc. R. Soc. Lond., 1971; Ser. A, Vol. 324, [7] Derjaguin B.V., Muller V.M., and Toporov Y.P. J. Effect of Contact Deformations on the Adhesion of Particles. Coll. Interf. Sci., 1975; Vol. 53, [8] Maugis D. On the Contact and Adhesion of Rough Surfaces, J. of Adhesion Sci. and Techn., 1996; Vol. 10, [9] Persson B.N.J. and Tosatti E. The Effect of Surface Roughness on the Adhesion of Elastic Solids, J. of Chem. Phys., 2001; V. 115, No. 12, [10] Kalker J.J. T hree Dimensional Elastic Bodies in Rolling Contact. 1 st ed. Netherlands: Kluwer Academic Publishers, 1990; Vol. 2. [11] Ciavarella M., Demelio G., Barber J.R., and Jang Y.H. Linear Elastic Contact of the Weierstrass Profile, Proc. Roy. Soc. London, 2000; A456,

21 [12] Kudish I.I., Cohen D.K., and Vyletel B. Perfect Mechanical Sealing in Rough Elastic Contacts: Is It Possible?, ASME J. of Applied Mechanics, 2013; V. 80, [13] Johnson K.L. Contact Mechanics. 1 st ed. Cambridge: Cambridge University Press, [14] Galin L.A. Contact P roblems in Elasticity and V isco Elasticity, Moscow: Nauka Publishing, [15] Abramowitz, M. and Stegun, I.A., Eds. Handbook of M athematical F unctions with F ormulas, Graphs and Mathematical T ables, 1 st ed. Mineola, New York: Dover Publications, Inc., [16] Wong R. Asymptotic Approximations of Integrals. 1 st ed. New York: Academic Press, Inc.,