Hindawi Advances in Tribology Volume 2018, Article ID 8431467, 8 pages https://doi.org/10.1155/2018/8431467 Research Article Partial Contact of a Rigid Multisinusoidal Wavy Surface with an Elastic Half-Plane Ivan Y. Tsukanov Ishlinsky Institute for Problems in Mechanics RAS, Prospekt Vernadskogo, 101-1, Moscow, 119526, Russia Correspondence should be addressed to Ivan Y. Tsukanov; ivan.yu.tsukanov@gmail.com Received 7 May 2018; Revised 24 July 2018; Accepted 4 October 2018; Published 18 October 2018 Academic Editor: Patrick De Baets Copyright © 2018 Ivan Y. Tsukanov. is Th is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The interaction eeff cts, arising at partial contact of rigid multisinusoidal wavy surface with an elastic half-plane, are considered in the assumption of continuous contact configuration. eTh analytical exact and asymptotic solutions for periodic and nonperiodic contact problems for wavy indenters are derived. Continuous contact configuration, appearing at small ratios of amplitude to wavelength for cosine harmonics, leads to continuous oscillatory contact pressure distribution and oscillatory relations between mean pressure and a contact length. Comparison of periodic and nonperiodic solutions shows that long-range elastic interaction between asperities does not depend on a number of cosine wavelengths. 1. Introduction stamps [10, 11] were generally used. The analytical solution of the pointed problem for those surface geometries was Real rough surfaces are three-dimensional and multiscale. obtained by different methods. For the cosine prole fi they Besides fully random rough surfaces [1] there are natural and are complex stress function [7], dual series equation [12], technical surfaces, having quasiregular character of asperities intercontact gaps method [13], variable transform method on several scales (e.g., periodic anisotropic waviness) [2]. For [11], and fracture mechanics approach [14]. these surfaces the geometric model of two-dimensional wavy Taking into account simple wavy geometry (cosine or 2D profile can be applied as a first approximation. Also, in squared cosine) the contact problems with more complicated some fields of engineering, the wavy textures of different boundary conditions were studied: sliding problem with shapes are used (e.g., in optical devices and MEMS) [3]. friction [15, 16], with a u fl id lubricant [17], with a partial slip Considering the elastic contact processes, occurring for soft [11, 18], with adhesion and sliding friction [19], for viscoelastic materials (polymers and biological materials), the various material [20], for Winkler model of viscoelastic material and analytical methods of plane elasticity can be applied. In adhesion [21], for elastic layer with presence of friction and the case of full contact, when the gap between surfaces is wear [22], and dynamic problem for anisotropic half-plane lfi led, the problem can be easily solved by Fourier transform [23]. method [4]. However, the very high applied pressure is The normal elastic problem for a two-dimensional non- required to reach the full contact condition even for soft sinusoidal wavy prole, fi where a shape of a waveform is materials, so partial contact is the more often case. Partial controlled by a parameter, was solved analytically [24]. contact between wavy surfaces is a problem with mixed It was established that the pressure distribution is highly boundary conditions, which was solved by different mathe- sensitive to the shape of a wavy surface, especially at large matical techniques. loads. The classic periodic contact problem in plane elasticity The presence of several scales of a wavy surface leads in general to a multizone periodic contact problem [9, 25]. The is an old problem [5, 6]. Concerning geometry of a wavy surface, considered in the previous studies, the cosine [7, 8], asymptotic approximate solution for initial contact for a two- the squared cosine [9], and evenly spaced parabolic or wedge scale wavy surface was obtained [24]. It was shown that even 2 Advances in Tribology p(x) 2Δ E, ] (a) (b) (c) Figure 1: Contact of a cosine wavy profile, having one (a), two (b), and three (c) harmonics with an elastic half-plane. for initial contact the interplay between harmonics exists. small amplitudes of subsequent cosine harmonics, is analyzed Considering the problem in a wide range of applied loads analytically for periodic and nonperiodic multisinusoidal it is necessary to use a numerical procedure. Such studies rigid indenters in contact with an elastic half-plane. were performed by different techniques: Fourier series and cotangent transform [26], full contact solution and iteration 2. Problem Formulation and Assumptions procedure [27], FFT and variational principle [28], nonlinear The general scheme of the problem on the single period 𝜆 boundary integral equation [29], boundary element method 1 for one, two, and three cosine harmonics prole fi is presented [30], and finite element method [31]. The equations for inter- in Figure 1. nal stresses for sinusoidal pressure distributions in 2D and The wavy surface is assumed to be rigid, and the elastic 3D cases were also derived [32]. The results of these studies half-plane is an isotropic semi-infinite body with two elastic show that multiscale character of a wavy surface at partial constants: Young’s modulus 𝐸 and Poisson’s coecffi ient ]. contact with an elastic half-plane leads to multiple peaks of Also the plain strain condition is applied. The amplitudes of high pressure. The pressure distribution is jagged in this case, cosine harmonics are much smaller than their periods (Δ ≪ and a load-area dependence tends to proportionality at large 𝑖 𝜆 ,where i =1, 2 ... N is a harmonic sequence number). number of harmonics [31]. This condition makes it possible to apply the linear elasticity In cases, considered in the previous studies, the contact theory. The Johnson parameter 𝜒 = 𝜋 EΔ /2p 𝜆 (where 𝑝 = was partial at all scales, because the amplitudes of differ- 𝑖 ℎ 𝑖 ℎ 2p 𝜆 /𝜋 a, p is an applied mean pressure, and a is a contact ent harmonics were comparable. This situation leads to a ∞ 𝑖 ∞ half-width) should be𝜒< 1 [33] for preserving the continuous discontinuous (discrete) contact congfi uration [33]. Besides contact configuration. numerical methods, the other way to solve these problems The two different problems with similar geometry of a is usage of multiasperity contact models. Based on the rigid surface are considered. For the problem with periodic nature of the surfaces models can be deterministic and boundary conditions the integral equation with Hilbert statistical. Review of statistical models, based on individual kernel is used [11]: asperity contact, in comparison with the Persson’s model and numerical simulations, is performed in [34]. For the nearly 𝑎 𝐸 𝜕ℎ (𝑥 ) 1 𝑥−𝜉 = ∫ 𝑝 (𝜉 )cot 𝑑𝜉, (1) complete contact case, when the ratio of the real area of 2(1 −] ) 2𝜋 2 −𝑎 contact to the nominal contact area approaches unity, the statistical model, based on fracture mechanics approach, was where h(x) is an initial gap between surfaces, and p(x)is developed [35]. For deterministic multiscale surfaces, (e.g., a contact pressure distribution. multisinusoidal self-affine surfaces), the Archard’s approach For a nonperiodic indenter the integral equation with was successfully implemented [36, 37]. Cauchy kernel is used [10]: However, if at a certain scale amplitude of the cosine 𝐸 𝜕ℎ 𝑥 1 𝑝 (𝜉 ) ( ) harmonic is much smaller than its period, full contact = ∫ 𝑑𝜉. (2) 2(1 −] ) 2𝜋 𝑥− 𝜉 −𝑎 on this scale occurs, and a continuous oscillating pressure distribution on a larger scale will be observed [33, 38]. For Choosing for simplicity the largest wavelength 𝜆 =2𝜋 , distinguishing these cases the Johnson parameter, coupling one can write the expression for the gap function derivative an amplitude, a period of cosine harmonic, and a reduced for the 𝑖 th cosine harmonic: modulus of elasticity with Hertzian pressure at the point, 𝜕ℎ (𝑥 ) 𝜕 where maximum pressure occurs, is used [33]. In the present 𝑖 (3) = (𝛿−Δ (1− cos 𝑛 𝑥 ))=−Δ 𝑛 sin 𝑛 𝑥, 𝑖 𝑖 𝑖 𝑖 𝑖 study the continuous contact configuration, observed at 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝜕𝑥 Advances in Tribology 3 where 𝛿 is a contact approach and 𝑛 = 𝜆 /𝜆 .In the given where 𝑈 – is a Chebyshev polynomial of the second kind 𝑖 1 𝑖 𝑛 formulation of the problem 𝑛 ∈ N. with a degree 𝑛 [39]. 𝑖 𝑖 So, on the basis of the superposition principle and taking Taking into account the considered assumptions the into account the assumed continuous contact congfi uration solution of equation (9) can be obtained by means of the the total contact pressure distribution can be obtained as a Chebyshev expansion of the left side and the known spectral sum of distributions of separate cosine harmonics: relations for the Chebyshev polynomials (Appendix). In initial variables the contact pressure distribution for the 𝑖 th harmonic is determined by 𝑝 (𝑥 )= ∑ 𝑝 (𝑥 ), (4) 2(1 −] ) 𝑖=1 𝑝 (𝑥 ) where 𝑁 is a number of wavelengths and 𝑝 (x)isa component of pressure distribution for the 𝑖 th wavelength. 2∞ (10) tan (𝑥/2 ) tan (𝑥/2 ) The vertical elastic displacements can be obtained via the =Δ 𝑛 1−( ) ∑ 𝐴 𝑈 ( ), 𝑖 𝑖 𝑗−1 following expression [9, 10]: tan (𝑎/2 ) tan (𝑎/2 ) 𝑗=1 2(1 −] ) 𝑢 (𝑥 )= ∫ 𝑝 (𝜉 )ln 𝑥−𝜉 d𝜉+𝐶 where −∞ (5) 𝑁 1 𝜑 (𝑠 )𝑇 (𝑠 ) 2(1 −] ) 2 𝑖 𝑗 ∫ (11) = ∑ 𝑢 (𝑥 )+𝐶, 𝐴 = ,𝑠𝑑 𝑗 = 1,2,...; 𝑧𝑖 −1 1− 𝑠 𝑖=1 where 𝐶 is a constant depending on the selected datum tan (𝑎/2 )𝑠 1− (tan (𝑎/2 )𝑠 ) 𝜑 (𝑠 )= 𝑈 ( ), (12) 𝑖 𝑛 −1 2 𝑖 2 point and 𝑢 (𝑥) are the surface vertical displacements for the 𝑧𝑖 1+ (tan (𝑎/2 )𝑠 ) 1+ (tan (𝑎/2 )𝑠 ) 𝑖 th wavelength. The mean (nominal) pressure p is determined by where 𝑇 – is a Chebyshev polynomial of the first kind invoking the equilibrium equation: with a degree j [39]. 𝑁 𝑁 𝑎 The total pressure distribution is obtained by using 𝐸 1 𝐸 𝑝 = ∑ 𝑝 = ∑ ∫ 𝑝 (𝑥 )𝑑𝑥, (6) equation (4). For numerical calculations it is necessary to ∞ ∞𝑖 𝑖 2 2 (1 −] ) 4𝜋 (1 −] ) −𝑎 𝑖=1 𝑖=1 hold finite terms of the infinite series in equation (10). For the arbitrary period, the variables 𝑥 and 𝑎 in equations (10) and where p is a component of mean pressure corresponding to ∞𝑖 (12) should be multiplied on 2𝜋 /𝜆 . The mean pressure p 1 ∞𝑖 the 𝑖 th wavelength. and the vertical displacements 𝑢 (𝑥) can be obtained using 𝑧𝑖 numerical integration in equations (5) and (6). The maximum 3. Solutions of the Problem for pressure is determined as the pressure at the point x =0. the 𝑖 th Harmonic 3.2. Solution for a Wavy Rigid Nonperiodic Indenter. Follow- 3.1. Solution for a Periodic Wavy Surface. Following equations ing equations (2) and (3) the integral equation for the 𝑖 th (1) and (3) the integral equation for the 𝑖 th harmonic is as harmonic is follows: 1 𝑥− 𝜉 −Δ 𝑛 sin 𝑛 𝑥= ∫ 𝑝 𝜉 cot 𝑑𝜉. (7) ( ) 1 𝑝 (𝜉 ) 𝑖 𝑖 𝑖 𝑖 𝜋 2 −𝑎 −Δ 𝑛 sin 𝑛 𝑥= ∫ 𝑑𝜉. (13) 𝑖 𝑖 𝑖 𝜋 𝑥−𝜉 −𝑎 The analytical solution for the contact pressure distribu- tion for the 𝑖 th harmonic can be obtained via the reduction The solution of the equation (13) can be obtained, using of equation (7) to the integral equation with Cauchy kernel an inversion without singularities on both endpoints [8, 11] using the following variable transform [11]: and the Chebyshev expansion of the left side [39], which can be written explicitly: 𝑢= tan ; 𝑥 𝑝 (𝑥 ) V = tan ; (8) ∞ (14) 𝑥 𝑥 𝑎 −𝑗 =Δ 𝑛 1−( ) ∑ (−1) J (𝑎𝑛 )𝑈 ( ), 𝛼= tan . 𝑖 𝑖 2𝑗+1 𝑖 2𝑗 𝑎 𝑎 𝑗=0 Considering the symmetry of the prole fi the integral equation (7) is reduced to where J (t) is the Bessel function of the rfi st kind of the integer order 𝑗 and the argument t [39]. 2 𝛼 2V 1− V 1 𝑝 (𝑢 ) The displacements within the contact zone 𝑥 ∈ [−𝑎,𝑎] −Δ 𝑛 𝑈 ( )= ∫ 𝑑𝑢, (9) 𝑖 𝑖 𝑛 −1 2 𝑖 2 1+ V 1+ V 𝜋 V−𝑢 −𝛼 can be determined analytically using equation (5) and the 𝜋𝐸 𝑖𝑗 𝜋𝐸 𝑖𝑗 4 Advances in Tribology relations for Chebyshev polynomials [40]. For the 𝑖 th har- Then, applying the Jacobi-Anger expansion [42], the monic the final relation is close-form relation is 𝑝 (𝑥 )≈Δ 𝑛 1−( ) [ J (𝑎𝑛 ) 𝑥 1 2 𝑖 𝑖 𝑖 1 𝑖 𝑢 (𝑥 )=−Δ 𝑛 𝑎 J (𝑎𝑛 )( − − ln ) 𝑎 𝑧𝑖 𝑖 𝑖 1 𝑖 𝑎 2 𝑎 (20) + J (𝑎𝑛 +1)cos (2 ( )) 2 𝑖 / (1−𝑗)/2 + ∑ (−1) J (𝑎𝑛 ) (15) 𝑗 𝑖 𝑗=3 +0.5 cos((𝑛𝑎 +1)sin ( )) − J (𝑎𝑛 +1)]. 𝑖 0 𝑖 𝑇 (𝑥/𝑎 ) 𝑇 (𝑥/𝑎 ) 𝑗+1 𝑗−1 ] The close-form integral relation for a maximum pressure ⋅( − ) , 𝑗+1 𝑗−1 (x = 0) can be determined exactly from equation (18): 𝑎𝑛 𝑝 =Δ 𝑛 ∑ J (𝑎𝑛 ) = 0.5Δ 𝑛 ∫ J (𝑡 )𝑑𝑡. (21) 𝑖 max 𝑖 𝑖 2𝑗+1 𝑖 𝑖 𝑖 0 where sign ∑ identifies the sum of terms with odd 𝑗 only. 𝑗=0 According to equation (6) the mean pressure for the 𝑖 th harmonic p is calculated by integration of equation (14) and ∞𝑖 resulting in a simple expression: 4. Results and Discussion The evolution of the dimensionless contact pressure dis- 𝑝 = 0.25Δ J (𝑎𝑛 ). (16) tribution p(x)/p∗ (p∗ = 𝜋 EΔ /𝜆 ) for a periodic problem ∞𝑖 𝑖 𝑖 1 𝑖 1 1 (equations ((6) and (10)-(12)), 𝜆 =2𝜋 , Δ = 0.5) for various 1 1 contact lengths (2a) and two different profiles f (x)is shown The approximate close-form relation for the contact in Figure 2. pressure distribution can be obtained assuming that the The exact (solid lines, equation (14)) and the approximate largest values of pressure are concentrated near the point x (dotted lines, equation (20)) graphs of the dimensionless con- = 0. Then equation (14) can be represented as tact pressure p(x)/p∗ for different profiles of a nonperiodic wavy indenter are shown in Figure 3. Figures 2 and 3 illustrate that, with increasing the num- 𝑥 𝜕 𝑝 (𝑥 )=Δ 𝑛 1−( ) ber of harmonics, the pressure distribution becomes more 𝑖 𝑖 𝑖 complex and the maximum pressure grows significantly. For a (17) ∞ single-scale periodic cosine prole fi (Figure 2(a)) the Wester- 𝑇 (sin (𝑥/𝑎 )) / (1−𝑗)/2 ⋅ ∑ (−1) 𝑎J (𝑎𝑛 ) , gaard’s solution is recovered. For a single-scale nonperiodic 𝑗 𝑖 𝑗=0 indenter (Figure 3(a)) the Hertz solution is observed, as the cosine function is very close to the quadratic parabola. Thereby, the distributions, presented in Figures 3(b) and where sign identifies the sum of terms with odd 𝑗 only. 3(c), correspond to wavy cylinder problem at small waviness Using the known relations for the Chebyshev polynomials [38]. Comparison of the exact and the approximate values of [41] the following expression can be written: pressure for a single indenter (Figure 3) shows that equation (20) satisfactorily describes the behavior of the pressure distribution. 𝑝 (𝑥 ) Comparing the periodic and the nonperiodic solutions the elastic interaction effect is of interest. The mean pressure 2 (18) 𝑥 𝑥 – contact length curves for two profiles, calculated from =Δ 𝑛 1−( ) ∑ J (𝑎𝑛 )cos ((2𝑗 + 1)( )) . 𝑖 𝑖 2𝑗+1 𝑖 𝑎 𝑎 periodic and nonperiodic solutions, are presented in Figure 4. 𝑗=0 Figure 4 shows that, at small contact lengths (2a < 0.25𝜆 ),thesolutionsareclose.Thepieceofgraphsagreement With the use of an approximate relation between zeros of does not depend on profile geometry. With increase of load Bessel functions of integer order [42] the following expres- the periodic solution gives the smaller contact length due to sion can be written: elastic interaction on the largest scale. For the prole fi with two cosine harmonics and continuous contact configuration, 2 presented in this study, the oscillations of mean pressure – √ [ 𝑝 𝑥 ≈Δ 𝑛 1−( ) J (𝑎𝑛 ) ( ) contact length curves are observed (Figure 4(b)). Curves in 𝑖 𝑖 𝑖 1 𝑖 Figure 4(a) correspond to Westergaard’s (curve 1)and Hertz (19) (curve 2) solutions, recovered for profile with one wavelength. Graphs of the mean and the maximum pressures versus + ∑ J (𝑎𝑛 +1)cos (2𝑗( )) . 2𝑗 𝑖 contact length on the interval 2a< 0.25𝜆 for different profiles 𝑗=2 ] of a wavy nonperiodic indenter are shown in Figure 5. 𝜕𝑥 𝑎𝑛 Advances in Tribology 5 p(x)/p p(x)/p 1.65 1.65 1.10 1.10 0.55 0.55 1 1 −0.8 −0.5 −0.3 0 0.3 0.5 0.8 −0.8 −0.5 −0.3 0 0.3 0.5 0.8 x/a x/a (a) (b) Figure 2: Evolution of contact pressure distribution for wavy periodic profile ( 𝜆 =2𝜋 , Δ =0.5): f (x)= Δ cos(𝑥) (a); f (x)= 1 1 1 Δ cos(𝑥) +0.02Δ cos(11𝑥) (b); 2a =0.05𝜆 (1); 2a =0.2𝜆 (2); 2a =0.3𝜆 (3); 2a =0.5𝜆 (4); 2a =0.8𝜆 (5). 1 1 1 1 1 1 1 ∗ ∗ ∗ p(x)/p p(x)/p p(x)/p 1.0 1.0 1.0 0.5 0.5 0.5 0 0 −0.25 0 0.25 −0.25 0 0.25 −0.25 0 0.25 x/a x/a x/a (a) (b) (c) Figure 3: Evolution of contact pressure distribution for a nonperiodic wavy indenter ( 𝜆 =2𝜋 , Δ =0.5): f (x)= Δ cos(𝑥) (a); f (x)= 1 1 1 Δ cos(𝑥) +0.02Δ cos(11𝑥) (b); f (x)= Δ cos(𝑥) +0.02Δ cos(11𝑥)+0.0015Δ cos(40𝑥) (c); 2a =0.05𝜆 (1); 2a =0.2𝜆 (2); 2a =0.25𝜆 (3). 1 1 1 1 1 1 1 1 p /p ∞ p /p 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.2 0.4 0.6 2a/ 0 0.2 0.4 0.6 2a/ 1 1 (a) (b) Figure 4: Graphs of dimensionless mean pressure as a function of dimensionless contact length (𝜆 =2𝜋 , Δ =0.5): f (x)= Δ cos(𝑥) (a); 1 1 1 f (x)= Δ cos(𝑥) +0.02Δ cos (11x)(b); 1 – periodic solution, 2 – nonperiodic solution. 1 1 6 Advances in Tribology p /p max p /p 0.15 0.10 0.5 0.05 0 0.07 0.13 0.19 0 0.07 0.13 0.19 2a/ 2a/ 1 1 (a) (b) Figure 5: Dimensionless mean pressure (a) and maximum pressure (b) as a function of dimensionless contact length for profiles with different numbers of cosine harmonics ( 𝜆 =2𝜋 , Δ =0.5): f (x)= Δ cos(𝑥) (1); f (x)= Δ cos(𝑥) +0.02Δ cos(11𝑥) (2); f (x)= 1 1 1 1 1 Δ cos(𝑥) +0.02Δ cos(11𝑥)+0.0015Δ cos(40𝑥) (3). 1 1 1 p /p 5. Conclusions max The continuous contact configuration is one of the two possible configurations, arising at indentation of a multi- sinusoidal 2D wavy surface into an elastic half-plane. This 1.0 configuration leads to continuous oscillatory contact pressure distribution. Comparison of the derived periodic and nonpe- riodic solutions shows that the long-range elastic interaction between asperities does not depend on a number of cosine 0.5 wavelengths and can be neglected at small loads (contact lengths) for arbitrary wavy profile geometry. The assumption of neglecting the long-range periodicity leads to exact equa- tions for determining the remote and the maximum pressures from the contact length, described by oscillatory functions. 0 0.1 0.2 0.3 p /p However, the dependences of the maximum pressure from the mean pressure are not oscillatory for the prolfi es with two Figure 6: Dimensionless maximum pressure as a function of and three wavelengths and resemble those for a simple cosine dimensionless mean pressure for different profiles ( 𝜆 =2𝜋 , Δ = 1 1 profile of indenter. The inu fl ence of the additional cosine 0.5): f (x)= Δ cos(𝑥) (1); f (x)= Δ cos(𝑥) +0.02Δ cos(11𝑥) (2); f (x) 1 1 1 = Δ cos(𝑥) +0.02Δ cos(11𝑥)+0.0015Δ cos(40𝑥) (3). harmonics on the maximum pressure is significantly larger 1 1 1 than on the mean pressure for the same contact zone length. The derived equations can be used at the analysis of contact characteristics of deterministic prole fi s of arbitrary geometry and also at the validation of more complex numerical models Figure 5 shows that the maximum pressure depends on of rough surfaces contact. prole fi geometry stronger than the mean pressure. However, adding the third harmonic leads to insignificant change of the graphs character. Continuous contact congfi uration at the Appendix presence of several cosine wavelengths leads to oscillatory character of the mean and the maximum pressure graphs. Derivation of Contact Pressure Distribution Combining these two graphs numerically one can obtain the for the Periodic Problem dependence of peak pressure from mean pressure (Figure 6). Figure 6 shows that dependences of the maximum pres- The main integral equation of the considered contact problem sure from the mean pressure are not oscillatory for the for the 𝑖 th harmonic in transformed variables (8) is proles fi with two and three wavelengths, and additional cosine harmonics change the graph considerably in value but 2 𝛼 2V 1− V 1 𝑝 (𝑢 ) not in character. This statement can be useful in the analysis −Δ 𝑛 𝑈 ( )= ∫ 𝑑𝑢, (A.1) 𝑖 𝑖 𝑛 −1 2 𝑖 2 1+ V 1+ V 𝜋 V −𝑢 −𝛼 of contact surfaces fracture processes [33]. Advances in Tribology 7 where 𝑈 is aChebyshev polynomial of asecond kind Data Availability with a degree 𝑛 . No data were used to support this study. The appropriate inversion of this integral equation has to be nonsingular on both endpoints [11]: Conflicts of Interest 𝛼 2 Δ 𝑛 V 1− V 𝑖 𝑖 2 2 𝑝 (V)=− 𝛼 − V ∫ 𝑈 ( ) 𝑖 𝑛 −1 2 𝑖 2 𝜋 1+ V 1+ V The author declares that there are no conflicts of interest −𝛼 (A.2) regarding the publication of this paper. 1 1 ⋅ 𝑑𝑢. 2 2 √ 𝑢− V 𝛼 −𝑢 Acknowledgments By introducing the new variables, The research was supported by RSF (project no. 14-29-00198). 𝑟= , (A.3) References 𝑠= , [1] B.N. J.Persson,“eTh oryof rubber frictionand contact mechanics,” eTh Journalof ChemicalPhysics , vol. 115, no. 8, pp. the expression (A.2) can be written in the following form: 3840–3861, 2001. [2] A.C.Rodr´ıguez Urribarr´ı, E. van der Heide, X. Zeng, and M. Δ 𝑛 1 1 𝑖 𝑖 𝑝 (𝑟 )=− 1− 𝑟 ∫ 𝜑 (𝑠 ) 𝑑𝑠, (A.4) B. de Rooij, “Modelling the static contact between a fingertip 𝑖 𝑖 𝜋 √ 𝑠− 𝑟 −1 1−𝑠 and a rigid wavy surface,” Tribology International, vol.102,pp. 114–124, 2016. where the function 𝜑 (s)is [3] A. S.Adnan,V. Ramalingam, J. H.Ko,and S.Subbiah,“Nano 2 2 1−𝛼 𝑠 texture generation in single point diamond turning using 𝜑 (𝑠 )= 𝑈 ( ). (A.5) 𝑖 𝑛 −1 2 2 𝑖 2 2 backside patterned workpiece,” Manufacturing Letters,vol. 2, 1+𝛼 𝑠 1+𝛼 𝑠 no. 1, pp. 44–48, 2013. Since the integrand function is defined on the interval [-1; [4] Y. Ju and T. N. Farris, “Spectral Analysis of Two-Dimensional 1] and satisefi s the H older condition, it can be represented as Contact Problems,” Journal of Tribology,vol.118,no. 2,p. 320, an expansion in Chebyshev polynomials of the first kind [41]. 1996. [5] M. A. Sadowski, “Zwiedimensionale probleme der elastizitat- 𝑖0 shtheorie,” ZAMM—Zeitschrift f u¨r Angewandte Mathematik (A.6) 𝜑 (𝑠 )= +𝐴 𝑇 (𝑠 )+𝐴 𝑇 (𝑠 )+ ..., 𝑖 𝑖1 1 𝑖2 2 und Mechanik, vol.8,no.2,pp.107–121,1928. [6] N. I. Muskhelishvili, Some Basic Problems of the Mathematical where 𝑇 is a Chebyshev polynomial of the rfi st kind with Theory of Elasticity , Springer, Dordrecht, Netherlands, 1977. adegree j [39]. [7] H. M. Westergaard, “Bearing pressures and cracks,” Journal of The coefficients 𝐴 in equation (A.6) are defined by the Applied Mechanics, vol.6,pp.49–53, 1939. following expression [43]: [8] K.L.Johnson, Contact Mechanics, Cambridge University Press, 𝐴 =0, Cambridge, UK, 1987. 𝑖0 [9] I. Y. Schtaierman, Contact Problem of Theory of Elasticity , 1 (A.7) 𝜑 (𝑠 )𝑇 (𝑠 ) 2 𝑖 𝑗 Gostekhizdat, Moscow, Russia, 1949. 𝐴 = ∫ ,𝑠𝑑 𝑗 = 1,2,...; 2 [10] L. A. Galin, Contact problems,Springer, Netherlands, Dor- 𝜋 √ −1 1−𝑠 drecht, 2008. With the use of integral relation between the Chebyshev [11] J. M. Block and L. M. Keer, “Periodic contact problems in plane polynomials of the first and the second kind [41] elasticity,” Journal of Mechanics of Materials and Structures,vol. 3, no.7,pp.1207–1237,2008. 𝑇 (𝑠 ) 1 1 (A.8) [12] J. Dundurs, K. C. Tsai, and L. M. Keer, “Contact between elastic ∫ = 𝑈 (𝑟 ), 𝑗 = 0,1,2,... 𝑗−1 𝜋 √ 𝑠−𝑟 −1 1−𝑠 bodies with wavy surfaces,” Journal of Elasticity, vol.3,no. 2,pp. 109–115, 1973. and equation (A.4) the expression for the contact pressure [13] A. A. Krishtafovich, R. M. Martynyak, and R. N. Shvets, “Con- distribution for the 𝑖 th harmonic is tact between anisotropic half-plane and rigid body with regular microrelief,” Journal of Friction and Wear,vol.15, pp.15–21, 𝑝 (𝑟 )=−Δ 𝑛 1−𝑟 ∑ 𝐴 𝑈 (𝑟 ). (A.9) 𝑖 𝑖 𝑖 𝑗−1 𝑗=1 [14] Y. Xu and R. L. Jackson, “Periodic Contact Problems in Plane Elasticity: eTh Fracture Mechanics Approach,” Journal of Returning to the original variables, and bearing in mind Tribology,vol.140,no.1,p.011404, 2018. positive pressures notation, one can obtain [15] E. A. Kuznetsov, “Periodic contact problem for half-plane 𝑝 (𝑥 ) allowing for forces of friction,” Soviet Applied Mechanics,vol. 12, no. 10, pp. 1014–1019, 1976. 2∞ (A.10) [16] M. Nosonovsky and G. G. Adams, “Steady-state frictional tan (𝑥/2 ) tan (𝑥/2 ) =Δ 𝑛 1−( ) ∑ 𝐴 𝑈 ( ). 𝑖 𝑖 𝑗−1 sliding of two elastic bodies with a wavy contact interface,” tan (𝑎/2 ) tan (𝑎/2 ) 𝑗=1 Journal of Tribology, vol.122,no.3, pp. 490–495,2000. 𝑖𝑗 𝑖𝑗 𝑑𝑠 𝑖𝑗 𝑖𝑗 𝛼𝑠 8 Advances in Tribology [17] Y. A. Kuznetsov, “Eeff ct of u fl id lubricant on the contact [35] Y.Xu,R.L.Jackson, and D.B.Marghitu, “Statistical model characteristics of rough elastic bodies in compression,” Wear, of nearly complete elastic rough surface contact,” International vol. 102, no. 3, pp. 177–194, 1985. Journal of Solids and Structures,vol. 51,no. 5,pp. 1075–1088, [18] M. Ciavarella, “eTh generalized Cattaneo partial slip plane con- tact problem. II. Examples,” International Journal of Solids and [36] M. Ciavarella, G. Murolo, G. Demelio, and J. R. Barber, “Elastic Structures, vol.35, no.18,pp.2363–2378, 1998. contact stiffness and contact resistance for the Weierstrass profile,” Journal of the Mechanics and Physics of Solids,vol. 52, [19] G. Carbone and L. Mangialardi, “Adhesion and friction of an no. 6, pp. 1247–1265, 2004. elastic half-space in contact with a slightly wavy rigid surface,” Journal of the Mechanics and Physics of Solids, vol.52, no.6,pp. [37] R. L. Jackson, “An Analytical solution to an archard-type fractal 1267–1287, 2004. rough surface contact model,” T ribology Transactions,vol. 53, no. 4, pp. 543–553, 2010. [20] N. Menga, C. Putignano, G. Carbone, and G. P. Demelio, “The sliding contact of a rigid wavy surface with a viscoelastic half- [38] Hills D. A., Nowell D., and Sackfield A., Mechanics of elastic space,” Proceedings of the Royal Society A Mathematical, Physical contacts, Butterworth-Heinemann, Oxford, UK, 1993. and Engineering Sciences,vol.470, no. 2169,pp. 20140392- [39] I.S.Gradshteyn and I.M.Ryzhik, Table of Integrals, Series, and 20140392, 2014. Products,Elsevier, 8th edition, 2015. [21] I. G. Goryacheva and Y. Y. Makhovskaya, “Modeling of friction [40] A. Arzhang, H. Derili, and M. Yousefi, “eTh approximate at different scale levels,” Mechanics of Solids,vol.45,no. 3, pp. solution of a class of Fredholm integral equations with a 390–398, 2010. logarithmic kernel by using Chebyshev polynomials,” Global [22] I. A. Soldatenkov, “eTh contact problem for an elastic strip Journal of Computer Sciences,vol.3,no.2, pp. 37–48,2013. and a wavy punch under friction and wear,” Journal of Applied [41] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Mathematics and Mechanics,vol. 75, no.1, pp. 85–92, 2011. Chapman and Hall, London, UK, 2003. [23] Y.-T. Zhou and T.-W. Kim, “Analytical solution of the dynamic [42] K. Oldham, J. Myland, and J. Spanier, An Atlas of Functions, contact problem of anisotropic materials indented with a rigid Springer, New York, NY, USA, 2nd edition, 2008. wavy surface,” Meccanica, vol.52, no. 1-2,pp. 7–19,2017. [43] D. Elliott, “eTh evaluation and estimation of the coefficients in [24] I. Y. Tsukanov, “Eeff cts of shape and scale in mechanics of the Chebyshev series expansion of a function,” Mathematics of elastic interaction of regular wavy surfaces,” Proceedings of the Computation, vol.18, pp.274–284,1964. Institution of Mechanical Engineers, Part J: Journal of Engineer- ing Tribology, vol. 231, no. 3, pp. 332–340, 2017. [25] I. G. Goryacheva, Contact mechanics in tribology,vol.61,Kluwer Academic Publishers, Dordrecht, 1998. [26] W. Manners, “Partial contact between elastic surfaces with periodic profiles,” Proceedings of the Royal Society,vol.454,no. 1980, pp. 3203–3221, 1998. [27] O. G. Chekina and L. M. Keer, “A new approach to calculation of contact characteristics,” Journal of Tribology, vol.121,no. 1,pp. 20–27, 1999. [28] H. M. Stanley and T. Kato, “An fft-based method for rough surface contact,” Journal of Tribology,vol. 119, no.3, pp.481–485, [29] F. M. Borodich and B. A. Galanov, “Self-similar problems of elastic contact for non-convex punches,” Journal of the Mechanics and Physics of Solids, vol.50, no.11,pp.2441–2461, [30] M. Ciavarella, G. Demelio, and C. Murolo, “A numerical algorithm for the solution of two-dimensional rough contact problems,” Journal of Strain Analysis for Engineering Design,vol. 40,no.5,pp.463–476, 2005. [31] M. Paggi and J. Reinoso, “A variational approach with embed- ded roughness for adhesive contact problems,” 2018, https:// arxiv.org/abs/1805.07207. [32] J. H. Tripp, J. Van Kuilenburg, G. E. Morales-Espejel, and P. M. Lugt, “Frequency response functions and rough surface stress analysis,” Tribology Transactions, vol.46,no.3,pp.376–382, [33] C.Paulin,F. Ville,P.Sainsot, S. Coulon,and T.Lubrecht, “Eec ff t of rough surfaces on rolling contact fatigue theoretical and experimental analysis,” Tribology and Interface Engineering Series, vol.43, pp.611–617, 2004. [34] R. L. Jackson and I. Green, “On the modeling of elastic contact between rough surfaces,” Tribology Transactions, vol.54, no.2, pp. 300–314, 2011. International Journal of Advances in Rotating Machinery Multimedia Journal of The Scientific Journal of Engineering World Journal Sensors Hindawi Hindawi Publishing Corporation Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 http://www www.hindawi.com .hindawi.com V Volume 2018 olume 2013 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 Submit your manuscripts at www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Hindawi Hindawi Hindawi Volume 2018 Volume 2018 Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com www.hindawi.com www.hindawi.com Volume 2018 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018
Advances in Tribology – Hindawi Publishing Corporation
Published: Oct 18, 2018
You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.