Exploring the effect of geometric coupling on friction and energy dissipation in rough contacts of elastic and viscoelastic coatings
Introduction
Nowadays, a large number of systems in several application fields involve thin solid films. Soft coatings are one of the most frequent examples of such cases: a thin layer of compliant material with specific characteristics is deposited onto a significantly stiffer substrate (thus considered as rigid) in order to tailor the overall system behavior (e.g. chemical resistance to corrosion, enhanced or reduced stiffness, damping, frictional behavior). Possible applications range from engines, where specific low friction coatings are adopted to reduce energy dissipation in key contacting pairs (e.g. valve train systems and crankshafts, Dahotre and Nayak, 2005, Kano, 2006), to robotic clamps for objects manipulation (Voigt et al., 2012) or anti-skid tapes for ramps and stairs, where high frictional coatings are instead required to increase the grip. Coatings are also present in the case of biological systems such as human hands and feet, where the covering skin (which may locally be constituted by very thin layers) concurs in developing the high interfacial friction sustaining, for instance, the firm hand grip on the tennis racket handle, or the barefoot walking on different grounds.
For these reasons, a constantly rising interest on the tribological behavior of solid thin films, often studied as compliant layers of materials bonded to rigid bodies and indented by other rigid or deformable rough countersurfaces, has been reported in the last decades. Indeed, besides the theoretical (Greenwood and Williamson, 1966, Bush et al., 1975, Persson, 2001, Persson et al., 2004, Yang and Persson, 2008, Menga et al., 2014, Menga et al., 2018c, Menga et al., 2019, Menga and Carbone, 2019), numerical (Hyun et al., 2004, Campana et al., 2008, Pastewka and Robbins, 2016, Medina and Dini, 2014, Müser et al., 2017, Menga et al., 2018b) and experimental (Homola et al., 1990, Chateauminois and Fretigny, 2008, Krick et al., 2012, Ben-David et al., 2010) studies focusing on contact problems of semi-infinite bodies, detailed investigations have been also devoted to the case of contacts involving thin bodies (Carbone and Mangialardi, 2008, Putignano et al., 2015, Menga et al., 2016a, Menga et al., 2016b, Menga et al., 2018a, Menga et al., 2017, Menga et al., 2020).
To this regard, it is well known that dealing with half-space contacts, a certain degree of coupling between the normal and tangential displacement fields occurs in the case of material dissimilarity (Sackfield et al., 2013, Barber, 2018). Such a material coupling is governed by the Dundurs’ second constant, often referred to as , which if one of the bodies is rigid takes the value , with being the Poisson’s ratio. This effect has been explored in several studies, mostly focusing on stick–slip fretting problems associated to homogeneous (Nowell et al., 1988, Chen and Wang, 2008, Chen and Wang, 2009) and graded (Wang et al., 2010, Elloumi et al., 2010) elastic materials. Interestingly, in Nowell et al. (1988) it was reported that, in the case of dissimilar cylinders contacts, a non-negligible influence of the material coupling occurs on both the normal stiffness of the contact and the contact pressure distribution. However, a few pioneeristic studies (Bentall and Johnson, 1968, Nowell and Hills, 1988a, Nowell and Hills, 1988b), dealing with thin deformable layers, have shown that such a simple coupling representation is no longer valid as, since the normal deformation cannot be accommodated remotely as in the case of half-space, two possible independent sources of normal–tangential interactions exist: (i) the material coupling, due to material dissimilarity, governed by the Dundurs’ second constant ; and (ii) an additional geometric (or domain shape) coupling, which depends on the layer thickness (i.e. it vanishes for thick layers) and still occurs even in the case of similar contact pairs (i.e. ). Significantly less effort has been made to investigate the effect of the latter on the contact behavior of thin films. Indeed, moving from the pioneeristic study of Bentall and Johnson (1968), only a few authors have approached the problem (Nowell and Hills, 1988a, Nowell and Hills, 1988b, Jaffar, 1993, Jaffar, 1997) focusing on smooth single asperity contacts and showing a significant contact pressure asymmetry arising from the coupling. Furthermore, in a recent study (Menga, 2019), the rough contact behavior of elastic thin layers in the presence of interfacial friction has been investigated, showing that, even in the case of , the geometric coupling between the normal and tangential elastic fields may lead to a significant increase of the effective contact area, with non-negligible implication on contact-related phenomena such as interfacial hydraulic impedance, electrical conductivity (Kogut and Komvopoulos, 2003), and wear process evolution (Menga and Ciavarella, 2015). Interestingly, the geometric coupling may play an even more dramatic role in determining the frictional performance of interfaces in relative motion due to the asymmetry of the contact pressure distribution observed in Nowell and Hills, 1988a, Nowell and Hills, 1988b and Menga (2019). Focusing, for instance, on viscoelastic contact of thin layers, one can reasonably expect different energy dissipation due to bulk viscoelasticity and, in turn different frictional behavior of the interface, depending on the specific geometric coupling effect on the contact pressure and contact spots distribution which alter the effective excitation spectra during sliding. To the best of the authors knowledge, an investigation on this effect is currently missing in the specialized literature, and this work aims at filling this gap.
In this study we focus on the case of a thin coating, sufficiently softer than the underlying substrate so that the latter can be assumed as rigid, in frictional sliding contact with a rigid profile with self-affine roughness. We consider both elastic and viscoelastic coating materials. We investigate in details the effect of normal–tangential coupling in thin films on both the overall contact behavior and frictional response of the system, with further focus on the energy dissipation. As already mentioned, the system configuration studied here covers several technological applications related to the grip performance of bio-inspired or natural system for handling of objects as well as many other interesting problems, including protein-coated interfaces, paints and soft coatings for industrial use, finger tip contact with touch screens.
Section snippets
The contact problem formulation
The system under investigation is shown in Fig. 1, where a thin soft coating bonded to a rigid substrate is sketched. The free surface of the coating layer is indented by a rigid profile with roughness . According to Fig. 1, we define the coating thickness, the roughness fundamental wavelength, and the profile sliding speed. In our formulation, we assume with being the sound speed into the coating material; furthermore, we focus on long time observations so that steady state
Results and discussion
The presence of a deformable layer of finite thickness gives rise to coupling between the normal and tangential displacement fields (Sackfield et al., 2013, Barber, 2018). Indeed, in agreement with Bentall and Johnson (1968) and Menga (2019), focusing on the cross-coupled Green’s function given in Eq. (7), we observe two coupling terms: the first right-hand side term represents the material coupling, taking into account for the normal–tangential interactions in contact pairs of dissimilar
Conclusions
In this work we have investigated the frictional behavior of thin coatings bonded to rigid substrates in sliding rough contact. The analysis aims at exploring the effect of the peculiar coupling between the normal and tangential displacement fields arising in the case of thin bodies, even for similar contacting materials. The presence of Coulomb friction interactions, through non-null interfacial tangential stresses, activate the coupling effects, which instead vanishes in frictionless contacts.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska- Curie grant agreement no. 845756 (N.M. Individual Fellowship). D.D. acknowledges the support received from the Engineering and Physical Science Research Council, United Kingdom (EPSRC) through his Established Career Fellowship EP/N025954/1. This work was partly supported by the Italian Ministry of Education, University and Research under the Programme “Progetti
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All authors have contributed equally to this work.