Perfect Mechanical Sealing in Rough Elastic Contacts: Is It Possible?

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1 Perfect Mechanical Sealing in Rough Elastic Contacts: Is It Possible? Ilya I. Kudish Professor, Fellow of ASME Department of Mathematics Kettering University Flint, MI, Donald K. Cohen Owner of Michigan Metrology, LLC N. Laurel Park Dr., Suite 51 Livonia, MI Brenda Vyletel Electron Microbeam Analysis Laboratory University of Michigan Ann Arbor, MI ABSTRACT Generally, it is assumed that under any applied force there will always be some gap between the surfaces in a contact of rough elastic surfaces resulting in a discontinuous (i.e. multiply connected) contact. The presence of gaps along the line contact relates to the ability to form an adequate mechanical seal across an interface. This paper will demonstrate that for a twice continuously differentiable rough surface with sufficiently small asperity amplitude and/or sufficiently large applied load and/or sufficiently low material elastic modulus singly connected contacts exist. Solution of a contact problem for a rough elastic half-plane and a perfectly smooth rigid indenter with sharp edges is considered. First, a problem with artificially created surface irregularity is considered and it is shown that for such a surface the contact region is always multiply connected. An exact solution of the problem for an indenter with sharp edges 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 an indenter with sharp edges and a rough elastic surface to be singly connected is derived. The singly connected contact condition depends on the surface micro-topography, material effective elastic modulus, and applied load. It is determined that in most cases a normal contact of a twice continuously differentiable rough surface with sufficiently small asperity amplitude, sufficiently low material elastic modulus, and/or sufficiently large applied load is singly connected. Nomenclature x point coordinate, [m], q(x) contact pressure, [N/m ], a contact half-width, [m], φ (x) geometry of surface roughness, [m], φ(x) dimensionless geometry of surface roughness, E effective elastic modulus of material, [N/m ], P applied normal load, [N/m], ω dimensionless parameter characterizing the overall asperity height amplitude, dimensionless coefficients in the series in Chebyshev polynomials, ξ k Address all correspondence related to ASME style format and figures to this author.

2 y -a a x δ Fig. 1. Basic geometry for the problem of an indenter with sharp edges making contact with a surface with texture. T k (x) Chebyshev polynomial of the first kind of degree k, U k (x) Chebyshev polynomial of the second kind of degree k 1 Introduction Analysis of elastic dry rough contacts has a long history. For dry elastic rough contacts there are some analytical models which are based on specific assumptions related to roughness topography and localized interactions of asperities [1, ] and adhesion effects [3,4] as well as some numerical studies, for example, [5]. In [6,7] contact fatigue of rough elastic surfaces is modeled based on analytical asymptotic solutions obtained for stress intensity factors at tips of subsurface fatigue cracks [8]. In [9], the authors presented an elegant and clear analytical way to show that if the surface roughness of elastic surfaces is a Weierstrass function (i.e. everywhere continuous but nowhere differentiable function) then a contact of such surfaces must be multiply connected. 1 The proof is based on the assumed specific fractal properties of the Weierstrass function. The Weierstrass roughness distribution is a pretty demanding assumption which is hard to validate by any particular measurements or other considerations. In practice, depending on a number of factors such as loading, initial surface roughness topography, material stress yield, and residual stress distribution the running-in stage reduces roughness and make sharp asperities smoother due to material plasticity and wear. Therefore, even if the original surface roughness is as irregular as the Weierstrass type distribution analyzed in [9] due to material plasticity and wear the asperity distribution quickly becomes more regular (i.e. mathematically speaking at least twice differentiable). Therefore, it makes sense to consider connectivity of a contact of rough elastic surface with relatively smooth topography described by a differentiable distribution. Generally, it is assumed that under any applied force there will always be some gap between the surfaces in a contact of rough elastic surfaces resulting in a discontinuous (i.e. multiply connected) contact. This perception is strongly supported by lots of numerical studies and studies based on various models of rough contacts (see, for example, [1, 5, 9, 10]). In [11] (see p. 118) for face seals it is directly stated that due to seal roughness Most of the face... may not be contacting at all. At the same time, any theoretical substantiation of the impossibility of the existence of singly connected contacts is absent. The presence of gaps along the line contact relates to the ability to form an adequate mechanical seal across an interface. The goal of this paper is to shed light on the question as to whether a contact of elastic surfaces is always multiply connected or it can be singly connected in the case of relatively smooth rough surfaces (i.e. a differentiable texture profile) with relatively low elastic moduli subjected to relatively high load. A specific simple example of a surface contact with which results in a multiply connected region is given. A simple analytical analysis was conducted to determine the possibility of a contact of a rigid indenter with sharp edges with an elastic half-plane to be singly connected. Contact of a Rigid Indenter with Sharp Edges with a Rough Elastic Half-Plane Let us consider a contact problem for rough surfaces (see Figure 1). More specifically, let us assume that a rigid indenter with a flat bottom y = 0 for a < x < a and sharp edges at x = ±a is normally indented with load P in a dry elastic halfplane (with modulus E and Poisson s ratio ν) bounded by a rough horizontal line. Adhesion of asperities is neglected. The roughness profile is described by the function φ (x). In most cases frictional stress in a contact causes very small changes in contact pressure. Therefore, the contact is considered to be frictionless. Let us analyze the case of a singly connected contact region. Therefore, we will assume that the applied load P is sufficiently large while the surface roughness profile amplitude and spatial properties are small enough to provide a continuous 1 According to the accepted mathematical definition a singly connected region is such a region that any two points of it can be connected by a continuous curve all points of which belong to the region. Otherwise, the region is multiply connected.

3 contact, i.e. the contact pressure q(x) > 0 for all x < a. This leads to the following formulation of the contact problem [1] φ (x) + a πe a q(t)ln a x t dt = δ, a q(t)dt = P, (1) where q(x) is the contact pressure, a and a are contact boundaries, E = E is the effective elastic modulus, δ is a constant 1 ν related to the rigid normal displacement of the indenter and δ differs from it by just a constant. In dimensionless variables a x = a x, t = a t, q (x ) = q(x) q 0, ωφ (x ) = πe P φ (x), δ = πe P δ, () problem (1) can be rewritten in the form (for simplicity primes are omitted) ωφ(x) + π q(t)ln 1 x t dt = δ, q(t)dt = π, (3) where ω is a dimensionless constant characterizing the amplitude of roughness profile which is directly proportional to the characteristic asperity height and inverse proportional to the applied characteristic pressure q 0 = P πa. The exact solution of problem (3) has the classic form [1] 1 q(x) = {1 + ω 1 x π dφ(t) 1 t dt dt t x }. (4) 3 Contact of a Rigid Indenter with Sharp Edges with a Surface with an Artificial Irregularity Let consider the case of an elastic half-plane with the boundary which is described as follows φ(x) = 0, x > c; φ(x) = α(1 x c ), 0 x c; φ(x) = α(1 + x c ), c x 0, (5) where α is a positive or negative constant and c is a constant, 0 < c 1. Obviously, if α = 0 then we have a classic singly connected contact region (, 1). Therefore, we will assume that α 0. If α > 0 we have a dimple while if α < 0 we have a bump. For simplicity we will consider α = ±1. That does not diminish the generality of the case because the value of the parameter ω > 0 controls the actual shape of this surface irregularity. The center of the bump/dimple is at x = 0. If we have a singly connected contact then pressure is positive at every point of the contact and the expression for pressure is determined by formula (4). Let us assume that our contact is singly connected. Then substituting (5) into (4) and using the substitution t = τ to calculate the integral in (4) we obtain 1+τ 1 α q(x) = [1 ω 1 x πc (1 1 c )] + ω πc α Q(x), Q(x) = ln ( 1 1 x 1+ 1 x ) c (1+ c (1 1 x ) x (1 1 x ) x (1 1 c ) 1 c. ) (6) It is important to keep in mind that Q(x) = ln c (1 1 c ) x +... as x 0, (7) Q(x) = ln c 1 c c c x as x ±c, 0 < c < 1, (8) Q(x) = 1 1 c 1 c 1 x as x ±1. (9)

4 Based on (4) and (9) we can conclude that at the end points of the contact (i.e., as x ±1) the pressure q(x) behaves similar to a pressure in a classic contact of a flat rigid indenter with a smooth elastic half-plane. From (4), (7), and (8) we can see that at x = ±c and x = 0 pressure q(x) has logarithmic singularities. Let us consider these singularities in more detail. In case of a bump (α = ) at its top (at x = 0) we have positive infinite pressure while at x = ±c the pressure reaches a negative infinite value. Therefore, for any value of ω > 0 in some vicinity of x = ±c the pressure remains negative which makes the contact region multiply connected. In a sense, the situation with a dimple (α = 1) is similar. At the bottom of the dimple (at x = 0) the pressure attains a negative infinite value while at x = ±c it reaches a positive infinite value. It means that for any value of ω > 0 in some vicinity of x = 0 the pressure remains negative. That makes the contact region multiply connected. This example shows that if the elastic surface has a concavity which is described by a function which is differentiable but its derivative is not continuous then we can expect to have negative pressure in some vicinity of such points of discontinuity of the surface shape derivatives. Also, in order to get a multiply connected contact region it is not necessary to use such a special roughness distribution as the Weierstrass distribution [9]. In particular, for an artificial surface irregularity described by φ(x) = 0, x > c; φ(x) = α(1 x c ), x c; α = ±1, 0 < c < 1 (10) the pressure q(x) also has a singularity at the points x = ±c similar to the described above and the contact is multiply connected if the irregularity is a bump (α = ). In this case points x = ±c belong to the surface concavity. This pressure behavior at the points x = ±c explains why modeling of asperities by spheres or ellipsoids (see, for example, [1]) cannot produce a singly connected contact. 4 Contact of a Rigid Indenter with Sharp Edges with an Elastic Half-plane with Smooth Roughness Let us assume that function φ(x) describing surface roughness profile can be represented by a series in Chebyshev orthogonal polynomials of the first kind [13] as follows φ(x) = ξ k T k (x), x 1, ξ k = π φ(t)t k (t)dt 1 t, k = 1,,... (11) Assuming that the coefficients ξ k vanish sufficiently fast as k (the details of this requirement will be considered later) the series (11) can be differentiated as follows [13] dφ(x) dx = T ξ k (x) k dx = kξ k U k (x), x 1. (1) Substituting (1) in the first equation of (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 1 q(x) = 1 x {1 ω kξ k T k (x)}, x 1. (13) 4.1 Analysis of Contact Connectivity for the Case of an Indenter with Sharp Edges 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 (13) this requirement translates into the inequality 1 ω kξ k T k (x) > 0, < x < 1. (14) Chebyshev orthogonal polynomials of the first kind T k (x) and of the second kind U k (x) [13] are defined as follows T k (cosθ) = coskθ and U k (cosθ) = sin(k+1)θ sinθ, respectively. These polynomials satisfy the following properties 1 T k (t)t m(t)dt Tk = 0 if k m, 1 t (t)dt 1 t = π if k 0 and the integral is equal to π if k = 0; 1 t U k (t)u m (t)dt = 0 if k m, 1 t Uk (t)dt = π if k 0, and 1 U k (t) 1 t dt t x = πt k (x), k = 1,,...

5 Fig.. 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. If the series in (14) 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 (14) is a continuous function on a closed interval [, 1] where it reaches its maximum. Therefore, in this case by choosing sufficiently small ω it is always possible to satisfy the inequality (14) on the entire interval (,1) and make pressure q(x) from (13) nonnegative in the entire contact (,1). That would make the contact singly connected. The value of the parameter ω can be made smaller by using surfaces with smaller asperity heights, smaller effective elasticity modulus E, and/or by applying higher load P (see ()). Let us consider now the most interesting case of an arbitrary rough surface represented by an infinite series in (13) and (14). Due to the fact that T k (x) 1 for all x, for the inequality (14) to be valid it is sufficient to require that It is important to realize that it is a sufficient but not necessary condition. 1 ω k ξ k > 0. (15) The condition (15) takes place if and only if the series k ξ k converges to a finite number and it is possible to find such a positive number ω that 0 ω < ω, ω = [ k ξ k ] > 0. (16) Therefore, in this case according to (16) for sufficiently small roughness profile amplitude, low material elastic modulus, and sufficiently high applied load a contact of a rigid perfectly smooth indenter with sharp edges with a rough elastic half-plane is singly connected. Now, let us analyze the conditions which should be imposed on the coefficients ξ k in order for all assumptions made to be valid. In particular, we have to specify the particular conditions under which the series in (11)-(13) and (16) are convergent and are continuous functions of x. To do that it is sufficient to consider the behavior of the coefficients ξ k from (11) as k. Making the substitution t = cosθ in the second equation in (11) we obtain the integral ξ k = π π φ(cosθ)coskθdθ, k = 1,,..., (17) 0

6 Fig. 3. Measurement of a Ground surface using an Atomic Force Microscope over a field of view of µm µm, 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. the asymptotic of which for k can be determined taking the real part of the Fourier integral π π f (θ)e ikθ dθ, f (θ) = φ(cosθ). (18) 0 Assuming that φ(x) is N times differentiable function and integrating integral in (18) by parts one gets [15] b a f (θ)e ikθ dθ = N ( k i )n+1 [e ika f (n) (a) e ikb f (n) (b)] + O(k N ), k 1. (19) n=0 It can be shown that if f (θ) = φ(cosθ) then any derivative of f (θ) of odd order has the form of f (k+1) (θ) = h(cosθ)sinθ, 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 a = 0, b = π we get that any derivative of f (θ) = φ(cosθ) of odd order at θ = 0 and θ = π is equal to zero. In addition to that, for a = 0, b = π we have e ika = 1 and e ikb = () k. Therefore, every term of the sum in (19) with even value of index n is purely imaginary while the terms with odd values of index n are real. It means that if function f (θ) = φ(cosθ) is N times continuously differentiable then ξ k = O( 1 k N+1 ), k 1. (0) Therefore, the assumption that function f (θ) = φ(cosθ) is continuously differentiable just once (i.e. N = 1) leads to the estimate ξ k = O( 1 k ), k 1. (1) Estimate (1) means that the series in (11) converges uniformly to the continuous function φ(x) while series in (1) and (13) may converge or diverge depending on the behavior of Chebyshev polynomials T k (x) and function dφ(x) dx. On the other hand, if we assume that φ(x) is a twice continuously differentiable function (i.e. N = ) then from (19) follows that ξ k = O( 1 k 3 ), k 1, ()

7 Fig. 4. Measurement of a Ground surface using an Atomic Force Microscope over a field of view of µm µm, 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. because f (0) = f (π) = 0. Estimate () means that the series in (11)-(13) converge uniformly to continuous functions as well as the interchange of integration and summation made to obtain (13) is legitimate. In particular, it means that the nonnegative pressure q(x) is a continuous function for < x < 1. Also, it means that for ω < ω the contact is singly connected (see (16)). Therefore, a contact of a rigid flat indenter with sharp edges with an elastic rough half-plane is singly connected as long as the roughness profile function φ(x) is at least twice continuously differentiable and 0 ω < ω (see (16)). 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 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 [9] 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. The case of a rigid indenter of an arbitrary convex 3 shape with sharp edges can be considered in a similar fashion. In particular, if the shape of the indenter is described by a twice continuously differentiable function f (x) and the contact occupies the interval [a,b] then in dimensionless variables () and (x,a,b ) = a 1 0 (x,a,b), f (x ) = πe P f (x) (as before further primes at dimensionless variables are omitted) and a new independent variable u = b a b+a [x ] the problem is formulated as follows ωφ( b+a + b a u) + b a π q(t)ln 1 b+a u t dt = δ f ( + b a u), q(t)dt = b a π, (3) 3 A region is convex when any of its two points can be connected by a segment of a straight line all points of which belong to the region.

8 Fig. 5. Measurement of a Ground surface using an Atomic Force Microscope over a field of view of µm µm, with lateral resolution of 8nm and height resolution of 0.1nm. Note that the finer spaced parallel line features are scanning related artifacts. and its classic solution for pressure q(u) has the form [1] q(u) = 1 u 1 { 1 b a + π 1 d f ( b+a + b a t) dt 1 t dt t u + π ω dφ( b+a + b a t) dt 1 t dt t u }. (4) Here we have to assume that the indenter shape is such that its contact with a perfectly smooth and flat elastic half-plane is singly connected, i.e. we assume that 1 + b a π d f ( b+a + b a t) dt 1 t dt t u > 0, < u < 1. (5) Then assuming that φ( b+a + b a u) = ξ k T k (u), u 1, ξ k = π φ( b+a + b a k(t)dt, (6) 1 t (k = 1,,...) the sufficient condition for the contact of this indenter with a rough elastic half-plane to be singly connected is represented by an inequality (similar to (16)) 0 ω < ω, ω = {1 + b a π min u 1 d f ( b+a + b a t) dt 1 t dt t u }{ k ξ k } > 0. (7) It has to be noted that the above analysis does not cover the cases of indenters with smooth (contrary to the case of sharp) edges which warrant a separate consideration. An example of the above analysis is the case of a parabolic indenter with f (x) = x with sharp edges at x = ±b, i.e. with contact region ( b,b). The fact that 0 < b < 1 provides for a positive pressure q(u) in the entire contact < u < 1 with a

9 Fig. 6. Measurement of a Ground surface using an Atomic Force Microscope over a field of view of µm µm, with lateral resolution of 4nm and height resolution of 0.1nm. Note that the finer spaced parallel line features are scanning related artifacts. perfectly smooth (flat) elastic half-plane. Then the condition which follows from (7) has the form 0 ω < ω, ω = 1 b b k ξ k > 0. (8) It is important to remember that the established condition on the roughness profile to be twice continuously differentiable for a contact to be singly connected is sufficient but not necessary. It means that there may be some less smooth rough surfaces which nonetheless provide singly connected contacts. In practice, manufacturing of surfaces required for functions such as friction, wear, and sealing may involve expensive finishing operations. If singly connected contacts are desired (for example, in a sealing application) then based on the criteria (7) established in this paper, manufacturers have flexibility in producing the surface roughness as related to other characteristics of the mating interface such as elastic modulus and applied loads (see the definition of parameter ω in ()). Thus the need to provide the absolute lowest surface roughness amplitudes (typically requiring higher cost) may not be necessary. Some examples of the topography of a ground surface measured with an Optical Profiler (WYKO NT8000) and Atomic Force Microscope (Veeco Dimension Icon AFM) are presented in Figures -6. The WYKO measurements were performed with a field of view 100 µm 100 µm with lateral resolutions of the order of 1 µm. The AFM measurements were performed in the tapping mode providing lateral resolution of 4 nm 16 nm over a measurement area of µm µm. Both the WYKO and AFM offer height resolution of the order of 1 nm. Figures -6 demonstrate for a ground surface, that as the lateral resolution is improved the nature of the texture does not change appreciably with the introduction of any additional structures. Thus over a field of view of µm µm as illustrated with AFM, the surface appears to be twice differentiable. Note that if the surface were not twice differentiable, as the lateral resolution was improved, additional structures would emerge such as fractal structure such as for a surface described by a Weierstrass function. Furthermore, depending on the material properties, the surface texture features that may originally not be twice differentiable are likely to plastically deform under initial contact leaving a twice differentiable distribution of asperities. Note that the horizontal line features that appear on the AFM figures are artifacts from the scanning process. The AFM measurement consists of a series of high lateral/height resolution line traces that are juxtaposed to create a 3D image. Thus, slight misalignments of the parallel line traces result in slight registration errors as displayed in Figures 3-6. Moreover, with increased resolution the number of these traces as well as the artifacts in the pictures increases.

10 5 Conclusions Solution of a contact problem for a rigid indenter with sharp edges and a rough elastic half-plane is considered. An exact solution of the problem for singly connected contact regions is considered and it is conveniently expressed in the form of a series in Chebyshev polynomials. The conditions for a rough contact to be singly connected are analyzed. It is determined that for not continuously differentiable concave roughness profiles near the points of discontinuity pressure is negative for any asperity height, elastic parameters of the material, and load, i.e. the contact regions are multiply connected. On the other hand, it is shown that for sufficiently small asperity amplitudes and sufficiently large applied load contacts of rough surfaces are singly connected for twice differentiable roughness profiles. In particular, for a given roughness profile and material elastic parameters a normal load can be found which guarantees a singly connected contact or for a given normal load and material elastic parameters the roughness profile amplitude can be chosen which would produce a singly connected contact, etc. Some examples of roughness measurements for real ground rough surfaces are given. These real measured surfaces seem sufficiently smooth, i.e. they seem at least twice differentiable. References [1] Greenwood, J.A. and Williamson, J.B.P., 1966, Contact of Nominally Flat Surface, Proc. R. Soc. Lond., Ser. A, 95, pp [] Ciavarellaa, M., Delfine, V., and Demelio, V.G., 006, A Re-Vitalized Greenwood and Williamson Model of Elastic Contact Between Fractal Surfaces J. of the Mech. and Phys. of Solids, 54(1), pp [3] Johnson, K.L., Kendall, K., and Roberts, A.D., 1971, Surface Energy and the Contact of Elastic Solids, Proc. R. Soc. Lond., Ser. A, 34, pp [4] Derjaguin, B.V., Muller, V.M., and Toporov, Y.P., 1975, J. Coll. Interf. Sci., 53, pp [5] Kalker, J.J., T hree Dimensional Elastic Bodies in Rolling Contact, Solid Mechanics and Its Applications,, Kluwer Academic Publisher. [6] Kudish, I.I. and Covitch, M.J., 010. Modeling and Analytical Methods in Tribology, Chapman & Hall/CRC, Taylor & Francis Group, Boca Raton, London, New York. [7] Kudish, I.I., 011. Contact Fatigue o f Elastic Sur f aces with Small Roughness, ASME Journal of Tribology, 133, July, pp [8] Kudish, I.I Contact Problem of the Theory of Elasticity for Pre-Stressed Bodies with Cracks. J. Appl. Mech. and Techn. Phys. 8, pp [9] Ciavarella, M., Demelio, G., Barber, J.R., and Jang, Y.H., 000, Linear Elastic Contact of the Weierstrass Profile, Proc. Roy. Soc. London, Ser. A, 456, pp [10] Snidle, R.W. and Evans, H.P., 1994, A Simple Method of Elastic Contact Simulation, Proc. Instn. Mech. Engrs., Part J, J. Engineering Tribology, 08, pp [11] Lebeck, A.O., Principles and Design o f Mechanical Face Seals, John Wiley & Sons, Inc., New York. [1] Galin, L.A., Contact Problems in Elasticity and Visco Elasticity, Moscow: Nauka. [13] Handbook o f Mathematical Functions with Formulas, Graphs and Mathematical Tables, Eds. M. Abramowitz and I.A. Stegun, National Bureau of Standards, 55, [14] Szegö, G Orthogonal Polynomials. American Mathematical Society, Colloquim Publications, Vol. XXIII, New York. [15] Wong, R., 001. Asymptotic Approximations o f Integrals. Academic Press, Inc., New York.