Partial Contact of a Rigid Multisinusoidal Wavy Surface with an Elastic Half-Plane

The interaction effects, arising at partial contact of rigid multisinusoidal wavy surface with an elastic half-plane, are considered in the assumption of continuous contact configuration. The 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.


Introduction
Real rough surfaces are three-dimensional and multiscale.Besides fully random rough surfaces [1] there are natural and technical surfaces, having quasiregular character of asperities on several scales (e.g., periodic anisotropic waviness) [2].For these surfaces the geometric model of two-dimensional wavy 2D profile can be applied as a first approximation.Also, in some fields of engineering, the wavy textures of different shapes are used (e.g., in optical devices and MEMS) [3].Considering the elastic contact processes, occurring for soft materials (polymers and biological materials), the various analytical methods of plane elasticity can be applied.In the case of full contact, when the gap between surfaces is filled, the problem can be easily solved by Fourier transform method [4].However, the very high applied pressure is required to reach the full contact condition even for soft materials, so partial contact is the more often case.Partial contact between wavy surfaces is a problem with mixed boundary conditions, which was solved by different mathematical techniques.
The classic periodic contact problem in plane elasticity is an old problem [5,6].Concerning geometry of a wavy surface, considered in the previous studies, the cosine [7,8], the squared cosine [9], and evenly spaced parabolic or wedge stamps [10,11] were generally used.The analytical solution of the pointed problem for those surface geometries was obtained by different methods.For the cosine profile they are complex stress function [7], dual series equation [12], intercontact gaps method [13], variable transform method [11], and fracture mechanics approach [14].
Taking into account simple wavy geometry (cosine or squared cosine) the contact problems with more complicated boundary conditions were studied: sliding problem with friction [15,16], with a fluid lubricant [17], with a partial slip [11,18], with adhesion and sliding friction [19], for viscoelastic material [20], for Winkler model of viscoelastic material and adhesion [21], for elastic layer with presence of friction and wear [22], and dynamic problem for anisotropic half-plane [23].
The normal elastic problem for a two-dimensional nonsinusoidal wavy profile, where a shape of a waveform is controlled by a parameter, was solved analytically [24].It was established that the pressure distribution is highly sensitive to the shape of a wavy surface, especially at large loads.
The presence of several scales of a wavy surface leads in general to a multizone periodic contact problem [9,25].The asymptotic approximate solution for initial contact for a twoscale wavy surface was obtained [24].It was shown that even for initial contact the interplay between harmonics exists.

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Considering the problem in a wide range of applied loads it is necessary to use a numerical procedure.Such studies were performed by different techniques: Fourier series and cotangent transform [26], full contact solution and iteration procedure [27], FFT and variational principle [28], nonlinear boundary integral equation [29], boundary element method [30], and finite element method [31].The equations for internal stresses for sinusoidal pressure distributions in 2D and 3D cases were also derived [32].The results of these studies show that multiscale character of a wavy surface at partial contact with an elastic half-plane leads to multiple peaks of high pressure.The pressure distribution is jagged in this case, and a load-area dependence tends to proportionality at large number of harmonics [31].
In cases, considered in the previous studies, the contact was partial at all scales, because the amplitudes of different harmonics were comparable.This situation leads to a discontinuous (discrete) contact configuration [33].Besides numerical methods, the other way to solve these problems is usage of multiasperity contact models.Based on the nature of the surfaces models can be deterministic and statistical.Review of statistical models, based on individual asperity contact, in comparison with the Persson's model and numerical simulations, is performed in [34].For the nearly complete contact case, when the ratio of the real area of contact to the nominal contact area approaches unity, the statistical model, based on fracture mechanics approach, was developed [35].For deterministic multiscale surfaces, (e.g., multisinusoidal self-affine surfaces), the Archard's approach was successfully implemented [36,37].
However, if at a certain scale amplitude of the cosine harmonic is much smaller than its period, full contact on this scale occurs, and a continuous oscillating pressure distribution on a larger scale will be observed [33,38].For distinguishing these cases the Johnson parameter, coupling an amplitude, a period of cosine harmonic, and a reduced modulus of elasticity with Hertzian pressure at the point, where maximum pressure occurs, is used [33].In the present study the continuous contact configuration, observed at small amplitudes of subsequent cosine harmonics, is analyzed analytically for periodic and nonperiodic multisinusoidal rigid indenters in contact with an elastic half-plane.

Problem Formulation and Assumptions
The general scheme of the problem on the single period  1 for one, two, and three cosine harmonics profile is presented in Figure 1.
The wavy surface is assumed to be rigid, and the elastic half-plane is an isotropic semi-infinite body with two elastic constants: Young's modulus  and Poisson's coefficient ].Also the plain strain condition is applied.The amplitudes of cosine harmonics are much smaller than their periods (Δ  ≪   , where i = 1, 2 . . .N is a harmonic sequence number).This condition makes it possible to apply the linear elasticity theory.The Johnson parameter  = EΔ  /2p ℎ   (where  ℎ = 2p ∞   /a, p ∞ is an applied mean pressure, and a is a contact half-width) should be  < 1 [33] for preserving the continuous contact configuration.
The two different problems with similar geometry of a rigid surface are considered.For the problem with periodic boundary conditions the integral equation with Hilbert kernel is used [11]: where h(x) is an initial gap between surfaces, and p(x) is a contact pressure distribution.
For a nonperiodic indenter the integral equation with Cauchy kernel is used [10]: Choosing for simplicity the largest wavelength  1 = 2, one can write the expression for the gap function derivative for the th cosine harmonic: where  is a contact approach and   =  1 /  .In the given formulation of the problem   ∈ N.
So, on the basis of the superposition principle and taking into account the assumed continuous contact configuration the total contact pressure distribution can be obtained as a sum of distributions of separate cosine harmonics: where  is a number of wavelengths and   (x) is a component of pressure distribution for the th wavelength.
The vertical elastic displacements can be obtained via the following expression [9,10]: where  is a constant depending on the selected datum point and   () are the surface vertical displacements for the th wavelength.
The mean (nominal) pressure p ∞ is determined by invoking the equilibrium equation: where p ∞ is a component of mean pressure corresponding to the th wavelength.

Solutions of the Problem for the 𝑖th Harmonic
3.1.Solution for a Periodic Wavy Surface.Following equations ( 1) and (3) the integral equation for the th harmonic is as follows: The analytical solution for the contact pressure distribution for the th harmonic can be obtained via the reduction of equation (7) to the integral equation with Cauchy kernel using the following variable transform [11]: Considering the symmetry of the profile the integral equation ( 7) is reduced to where    -is a Chebyshev polynomial of the second kind with a degree   [39].
Taking into account the considered assumptions the solution of equation ( 9) can be obtained by means of the Chebyshev expansion of the left side and the known spectral relations for the Chebyshev polynomials (Appendix).In initial variables the contact pressure distribution for the th harmonic is determined by where (12) where   -is a Chebyshev polynomial of the first kind with a degree j [39].
The total pressure distribution is obtained by using equation (4).For numerical calculations it is necessary to hold finite terms of the infinite series in equation (10).For the arbitrary period, the variables  and  in equations ( 10) and ( 12) should be multiplied on 2/ 1 .The mean pressure p ∞ and the vertical displacements   () can be obtained using numerical integration in equations ( 5) and (6).The maximum pressure is determined as the pressure at the point x = 0.

Solution for a Wavy
Rigid Nonperiodic Indenter.Following equations ( 2) and (3) the integral equation for the th harmonic is The solution of the equation ( 13) can be obtained, using an inversion without singularities on both endpoints [8,11] and the Chebyshev expansion of the left side [39], which can be written explicitly: where J  (t) is the Bessel function of the first kind of the integer order  and the argument t [39].
The displacements within the contact zone  ∈ [−, ] can be determined analytically using equation ( 5) and the relations for Chebyshev polynomials [40].For the th harmonic the final relation is where sign ∑ / identifies the sum of terms with odd  only.
According to equation ( 6) the mean pressure for the th harmonic p ∞ is calculated by integration of equation ( 14) and resulting in a simple expression: The approximate close-form relation for the contact pressure distribution can be obtained assuming that the largest values of pressure are concentrated near the point x = 0. Then equation ( 14) can be represented as where sign ∑ / identifies the sum of terms with odd  only.
Using the known relations for the Chebyshev polynomials [41] the following expression can be written: With the use of an approximate relation between zeros of Bessel functions of integer order [42] the following expression can be written: Then, applying the Jacobi-Anger expansion [42], the close-form relation is The close-form integral relation for a maximum pressure (x = 0) can be determined exactly from equation (18):
The exact (solid lines, equation ( 14)) and the approximate (dotted lines, equation ( 20)) graphs of the dimensionless contact 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 number of harmonics, the pressure distribution becomes more complex and the maximum pressure grows significantly.For a single-scale periodic cosine profile (Figure 2(a)) the Westergaard's solution is recovered.For a single-scale nonperiodic 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 3(c), correspond to wavy cylinder problem at small waviness [38].Comparison of the exact and the approximate values of pressure for a single indenter (Figure 3) shows that equation (20) satisfactorily describes the behavior of the pressure distribution.
Comparing the periodic and the nonperiodic solutions the elastic interaction effect is of interest.The mean pressure -contact length curves for two profiles, calculated from periodic and nonperiodic solutions, are presented in Figure 4.
Figure 4 shows that, at small contact lengths (2a < 0.25 1 ), the solutions are close.The piece of graphs agreement does not depend on profile geometry.With increase of load the periodic solution gives the smaller contact length due to elastic interaction on the largest scale.For the profile with two cosine harmonics and continuous contact configuration, presented in this study, the oscillations of mean pressurecontact length curves are observed (Figure 4(b)).Curves in Figure 4(a) correspond to Westergaard's (curve 1) and Hertz (curve 2) solutions, recovered for profile with one wavelength.
Graphs of the mean and the maximum pressures versus contact length on the interval 2a < 0.25 1 for different profiles of a wavy nonperiodic indenter are shown in Figure 5.  Figure 5 shows that the maximum pressure depends on profile geometry stronger than the mean pressure.However, adding the third harmonic leads to insignificant change of the graphs character.Continuous contact configuration at the presence of several cosine wavelengths leads to oscillatory character of the mean and the maximum pressure graphs.Combining these two graphs numerically one can obtain the dependence of peak pressure from mean pressure (Figure 6).
Figure 6 shows that dependences of the maximum pressure from the mean pressure are not oscillatory for the profiles with two and three wavelengths, and additional cosine harmonics change the graph considerably in value but not in character.This statement can be useful in the analysis of contact surfaces fracture processes [33].

Figure 1 :
Figure 1: Contact of a cosine wavy profile, having one (a), two (b), and three (c) harmonics with an elastic half-plane.