# Viscoelastic Finite Line Contact of Thin Bonded Layered Solids of Low Elastic Modulus
## 1 Introduction  
In recent years, increasing use has been made of load bearing polymeric or elastomeric surfaces as compliant members, for example in journal bearings [1; 2] or as lip seals or O-rings [3; 4; 5]. In these applications, localised deformation of the polymeric soft layer increases the gap between the contacting surfaces, which can enhance the lubricant film separating them, thus reducing friction. In other applications, soft layers are intended to act as protective layers. Polymeric and elastomeric surfaces act in a viscoelastic manner with stress relaxation [6; 7]. In bearings, the compliant members are usually in conforming contacts. In other cases, the viscoelastic layers suffer higher contact pressures when subjected to non-conforming contacts. In these cases, the contact characteristics depend on the layer thickness. In general, the contact mechanics of thin bonded layers deviate from the classical Hertzian contact mechanics, which admits only small strain localised deformation for assumed semi-infinite solids [8]. For viscoelastic semi-infinite solids of polymeric and elastomeric surfaces, where the contact footprint dimensions are far smaller than the layer thickness, [PERSON] [6] provided an analytical solution for an assumed infinite line contact, based on the Maxwell model. [PERSON] et al. [7; 9] provided solutions for non-conforming point contacts of thin bonded viscoelastic solids, showing deviation from semi-infinite solutions. They showed that Hertzian theory underestimates the generated contact pressures for thin compliant layers whose contact dimensions exceed the layer thickness by a ratio greater than two. In many cases an analytical solution cannot be assumed and a numerical analysis technique, such as finite difference or finite element, must be developed. This is the main motivation of this paper. Furthermore, [PERSON] et al. [7; 9] solutions for point contact and those of [PERSON] [6] for infinite line contact do not apply to the finite line contact geometry of many non-conforming contacts, for example in rolling element bearingswith raceways [10; 11] or cam contact with a flat tappet [12]. These contacts with finite line footprint geometry have edge stress concentrations at the contact extremities, inducing pressure spikes [10; 11; 12; 13; 14]. Therefore, another motivation of the current work is the development of a generic numerical analysis technique able to predict the edge stress discontinuities, such as pressure spikes, which often determine the limit of structural integrity of contacting solids. The same is true when a flat indenter contacts a thin polymeric viscoelastic layer, as in the case studied here. Therefore, such an experiment is used to validate finite element analysis of the finite line contact of the indenter against the viscoelastic layer of low elastic modulus represented by the Highly Filled Carbon Polymer (HFCP). This approach enables the development of elements with the correct choice of constitutive (stress-strain) relationship. The current highlighted predictive finite element-based viscoelastic layered contact mechanics approach is original, not hitherto reported in literature. As noted above, the results of the predictive analysis are compared with experimental contact mechanics using a pressure sensitive sandwich sensor made of small pieces of a Highly Filled Carbon Polymer (HFCP), bonded to a thin, flexible printed circuit board tape. Such skin sensors have been developed for biomedical applications and for tactile devices, mostly for robotics, some with contact imaging, based on finite element analysis techniques [15; 16; 17; 18; 19; 20]. HFCP layers contain a certain high percentage of carbon whose conductance alters fairly linearly with applied pressure [16]. However, the polymer layer is usually quite thin and susceptible to tear with repetitive use. Thus, a sandwich layered structure is preferred. Such a layered solid can emulate the histological structure of the human skin, at least in parts [18].