## Abstract

In this paper, the contact of a rigid sinusoid sliding on a viscoelastic half-space is studied. The solution of the problem is obtained by following the path drawn by Hunter for cylindrical contacts. Results show that depending on the remote applied load, a transition from full contact conditions to partial contact may occur depending on the sliding velocity. This effect, which is not observed in smooth single asperity contacts, is related to the viscoelastic stiffening of the material and to the periodicity of the contacts. Frictional properties as well as contact area, displacement and pressure distributions are discussed in detail.

## 1. Introduction

Current advances in polymer technology make available a large class of new rubber and rubber-made materials for many engineering components, including either traditional products, such as tyres, belts and seals, and innovative applications, such as microelectro-mechanical systems and artificial scaffolds for biological applications. However, the smart design of such elements requires an accurate analysis of the viscoelastic material response and, in particular, of the viscoelastic dissipation, which is always entailed when using these materials. This is a crucial point in current applied mechanics research: as examples, a more efficient design of tyres or an improved sealing action of mechanical seals may have a prominent impact in terms of significant energy savings and enhanced wear resistance. This optimization research effort requires the accurate understanding of the viscoelastic properties of contact problems. The intrinsic difficulty of the viscoelastic contact mechanics, whose solution is characterized by strongly dissipative phenomena [1], is marked by the huge number of different approaches, including theories [2–8] and numerical methodologies [1,9–19] developed to investigate rolling, sliding and lubricated contacts of viscoelastic materials.

Furthermore, the complexity is boosted by the roughness between the contacting surfaces. Indeed, the sliding or rolling contact between rough surfaces involves a very large number of spatial and time scales (covering more than six orders of magnitude), provoking a huge increase of computational cost and making conventional numerical techniques, including finite-element (FE) solvers, unfeasible for this type of investigations. For these reasons, specially designed numerical boundary element methodologies have been developed to address the rough contact problem [1,17,20–25]. On the other hand, different approximate analytic approaches have been proposed to deal with this type of problems, which belong mainly to two categories: (i) mean-field theories [3,5,6,26] and (ii) multi-asperity models [27–32].

In this paper, we deal with a simpler problem consisting of a viscoelastic half-space sliding in steady-state adhesionless contact with a rigid sinusoidal indenter. While the problem is interesting in itself, because it generalizes, to the case of viscoelastic contact, the Westergaard's solution [33], it also put the basis for the development of a more complicated theory of viscoelastic contact of rough profiles, because randomly rough profiles may be regarded as a sum of a multitude of sinusoid.

By following a path similar to the one proposed by Hunter [2], for the case of a rigid cylinder rolling on a viscoelastic half-space, we reformulate the problem in terms of a Fredholm integral equation of the first kind, where the logarithmic Kernel is exactly the same found for two-dimensional elastic problems [34–37], plus a couple of first-order differential equations and the corresponding boundary conditions. This formulation allows to exploit some of the solutions already known for elastic materials [34], thus leading to a strong simplification of the viscoelastic contact problem at hand.

We analyse the effect of sliding velocity and remote applied load on contact area size and shape of the deformed material and pressure distribution. Moreover, the viscoelastic frictional properties of the system are investigated. A detailed analysis is devoted to the transition from full contact to partial contact, which occurs, for load in between two limiting values, at a certain speed.

## 2. Formulation

We consider the problem of a sinusoidal rigid indenter in sliding contact, at constant velocity *V* , with a linear viscoelastic half-space. We assume no tangential stresses at the interface between the rigid indenter and solid, thus only normal stresses are present. Figure 1 shows the geometrical quantities involved in the problem. Given the wavelength λ and the amplitude *h* of the moving rigid sinusoidal indenter, of shape *s* as the distance between the mean lines of the rigid indenter and of the deformed viscoelastic body. The penetration of the sinusoidal indenter into the viscoelastic half-space is then defined as Δ=*h*−*s*. The quantity *v*(*x*) is the profile measured from the mean line of the deformed viscoelastic body.

Now, assuming *h*/λ≪1, the linearity of the viscoelastic material allows us to relate the time-dependent displacement field *v*(*x, t*) with the time-dependent normal stress distribution *σ*(*x, t*) through the constitutive equation
*k*=2*π*/λ, whereas *J*(*t*) is the viscoelastic creep function. By adopting the linear standard model formulation for a one relaxation time viscoelastic material, the creep function takes the following form
*H*(*t*) is the Heaviside step function, and *E*_{0} is the low frequency elastic modulus. Because we are interested in determining the long-time response of the system, when any information about the initial state has been completely lost, we can write that *v*(*x, t*)=*v*(*x*−*V* *t*) and *σ*(*x, t*)=*σ*(*x*−*V* *t*). Therefore, using the substitution *Φ*_{V}(*x*) is the new Green's function for steady-state sliding contacts, which parametrically depends on the constant sliding speed *V*

where
*p*(*x*) and the elastic displacement by *q*(*x*). Therefore, the analytical elastic solution obtained in [34] can be successfully exploited to solve the problem at hand. Note that, during the sliding at finite speed *V* , because of the viscoelastic response of the material, the contact area, of size 2*a* (figure 2), will show a certain degree of asymmetry, quantified by the eccentricity parameter *e* (figure 2). Thus, recalling that in the contact domain *Ω*=[−*a, a*] the elastic displacement is

As shown in reference [34], equations (2.9) and (2.10) can be analytically solved (see appendix A), and allow to calculate, for any given sliding velocity *V* , the quantities *a*, *e*, and the remote applied pressure *a* of the contact area and the eccentricity *e* are not independent quantities and must be determined as a part of the solution. To accomplish this purpose, once known

Equations (2.12) and (2.13) enable us to determine the two constants of integration of equations (2.8), and to write the following additional closure equations
*a* and the contact eccentricity *e* as a function of the remote applied pressure

## 3. Results

Here, we discuss the main peculiarities of the viscoelastic problem under investigation in terms of contact area, penetration and viscoelastic friction. We also show how these quantities depend on the sliding velocity *V* . In the calculations, we use *V*_{0}=(*τk*)^{−1}, and the dimensionless remote applied pressure

Figure 3*a* shows the shape of the deformed profile given the applied load *V* , the Westergaard's elastic solution is recovered [33,34,36]. This is because the viscoelastic material shows an elastic behaviour at very small and very high frequencies, with two different values of the elastic modulus: *E*_{0} and *a*) and extremely low sliding velocity, the system behaves as a soft elastic material, with elastic modulus *E*_{0}, and no viscoelastic dissipation takes places, making the interfacial stress and displacement distributions perfectly symmetric. As *V* is increased, the material stiffens and, given the same applied load, it penetrates less into the substrate. In such conditions, the hysteretic behaviour of the viscoelastic materials leads to a clear asymmetry of the contact region, characterized by a marked shrinkage at the trailing edge of the contact. At very high sliding velocity, the material behaves again elastically but it is much stiffer, with elastic modulus *b*), the displacement fields, obtained in these two limiting elastic cases, are exactly the same. This is not surprising, because dimensional arguments (as confirmed by the Westergaard's elastic solution) lead to the conclusion that the relation between the penetration Δ, the contact area 2*a*, the amplitude *h* of the sinusoidal rigid substrate and its wavelength λ, cannot involve the remote load *V*/*V*_{0}. Of course, Δ increases with the applied pressure *V* causes a significant reduction of the penetration Δ. Similarly, in figure 5, the dependence of the semi-contact area *a* on the dimensionless load *V*/*V*_{0}. As expected, because of velocity induced viscoelastic stiffening, at fixed applied load *a* decreases as the sliding velocity is increased. On the other hand, in figure 6, we plot the semi-contact area *a* as a function of the penetration Δ. We note that, at extremely slow and extremely high velocities the curves *a* versus Δ perfectly overlap to each other. Different is the case of intermediate sliding velocity, when viscoelasticity plays a significant role causing the *a* versus Δ curve to deviate from the perfectly elastic cases. This is even more clear in figure 7 where, given the value of penetration Δ, the contact area first decreases as the sliding velocity *V* is increased, then reaches a minimum value corresponding to a sliding velocity *V* ≈*V*_{0}, at which viscoelastic effects are maximized, and then increases again towards the high-speed elastic solution. This behaviour is expected because, at intermediate velocity, the effect of internal viscous losses is mainly to strongly reduce the contact pressure at the trailing edge, thus leading to a reduction of the contact area size.

In appendix B, the minimum value *p*_{FC} of the remote applied load which causes the system to snap in full contact is calculated (see also figure 8). Of course, *p*_{FC} is minimal in the limit of extremely small velocities [*p*_{FC}=(*p*_{FC})_{0}], and it monotonously increases until its maximum value [*p*_{FC}=(*p*_{FC})_{1}] is reached at very large velocities. In our case, we calculated *V*_{th}, and then undergoes a transition to partial contact conditions because of the material stiffening. This particular behaviour cannot be observed in smooth single asperity contacts. It is peculiar of viscoelastic periodic contacts, where viscoelastic interaction between asperities in contact plays a fundamental role. For each given value of the remote pressure *V*_{th} can be calculated by solving the equation
*E*(*ω*) is the complex viscoelastic modulus defined as *E*(*ω*)^{−1}=*iωJ*(*ω*), with

Figure 9 shows, for *V* <*V*_{th}, the system in full contact (i.e. *ka*=*π*). However, as soon as the sliding velocity exceeds the value, *V*_{th} partial contact conditions are established, and the contact area continuously shrinks as the sliding velocity *V* *a* is increased.

The transition from full contact to partial contact affects the eccentricity *e* of the contact area and the eccentricity

We observe that in full contact, because of linearity, both displacement and pressure distributions are sinusoidal. In this case, the quantity *e* can no longer be defined, whereas

For remote pressure values *e*, which can be defined only in partial contact conditions, i.e. for *V* >*V*_{th}, is always smaller than *V* =*V*_{th} when the two quantities take the same value.

We also calculated the mean frictional stress
*V* , for different values of the dimensionless remote load *μ* is described by bell-shaped curves, with almost no friction in the limiting cases of extremely small and extremely high sliding velocities, when the material behaves elastically. It is noteworthy to observe that, when the system is in full contact conditions, a further increase of the load leads to a decrease of the friction coefficient *μ*. This is expected, because, once full contact is reached, *τ*_{f} no longer changes with the normal load *μ*, defined above, must decrease as

## 4. Conclusion

We have studied the sliding contact of a rigid sinusoid over a viscoelastic half-space. We reformulate the problem into an equivalent elastic contact by means of ad hoc transformations of displacement and pressure distributions. This allows us to exploit existing elastic solutions of sinusoidal contact [34]. Results show that, depending on the remote applied load, particular conditions may occur which cause the contact to undergo a transition from full contact to partial contact as the sliding speed is increased. This effect cannot be observed in smooth single asperity contacts, as it is strongly related to the viscoelastic interaction between asperities in contact which necessarily occurs in periodic contacts. We also investigate how the shape of the contact, the contact area and the penetration, as well as the coefficient of friction are affected by the sliding speed and the remote load.

## Funding statement

The authors thank the Italian Ministry of Education, University and Research for supporting the research activity within the projects PON01-02238 and PON02 00576-3333604. C.P. also gratefully acknowledges the support of Marie Curie IEF project SOFT-MECH (grant no. 622632).

## Appendix A. The elastic contact solution

The viscoelastic solution, presented in the paper, was obtained by using the elastic-viscoelastic correspondence principle [38] which allowed us to exploit the results found, for the elastic case, by one of the authors in [34], where the adhesive non-symmetric contact between an elastic half-space and a rigid sinusoidal substrate has been studied. Here, we briefly summarize the main results obtained in [34]. The main problem is to solve the integral equation (2.9) and calculate the integral equation (2.10), given the contact domain *Ω*= ]−*a, a*[. The aforementioned problem can be solved by invoking the superposition principle and dual series approach [39]. The solution is, then, the sum of an asymmetric solution, denoted with the subscript 1 and a symmetric one, denoted with the subscript 2.
*χ*(*x*) is the characteristic function of the contact domain: *χ*(*x*)=1 for *x*∈*Ω* and *χ*(*x*)=0 otherwise.

The symmetric solution is found by superposing two simpler problems. The first one deals with the non-adhesive contact of an elastic half-space indented by a sinusoidal surface of amplitude *a*, under an apparent asymptotic tensile load

## Appendix B. Friction in full contact conditions

In full contact, one may easily determine the solution of the viscoelastic contact problem taking into consideration that, because of linearity, the pressure distribution will be sinusoidal as well, but, because of internal dissipation, pressure and displacement distributions will have different phases. In Fourier space, the relation between the stress *σ* and the displacement *u* can be written as [5,35,41]
*V* on a rigid surface, we can write *σ*(*x, t*)=*σ*(*x*−*V* *t*) and *u*(*x, t*)=*u*(*x*−*V* *t*). In Fourier space, we obtain

- Received May 15, 2014.
- Accepted June 19, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.