Contact mechanics for coated spheres that includes the transition from weak to strong adhesion
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Recently published results for a rigid spherical indenter contacting a thin, linear elastic coating on a rigid planar substrate have been extended to include the case of two contacting spheres, where each sphere is rigid and coated with a thin, linear elastic material. This is done by using an appropriately chosen effective radius and coating modulus. The earlier work has also been extended to provide analytical results that span the transition between the previously derived Derjaguin–Müller–Toporov (DMT)-like (work of adhesion/coating-modulus ratio is small) and Johnson–Kendall– Roberts (JKR)-like (work of adhesion/coating-modulus ratio is large) limits.
I. INTRODUCTION
Thin coatings are often applied to contacting materials. For example, the polycrystalline silicon surfaces in microelectromechanical systems (MEMS) are sometimes coated with a molecular monolayer to reduce adhesion, friction, and wear.1 The presence of such coatings can have a significant effect on contact stress and contact area.2–4 In the case of nanoscale contacts, the effect of adhesion forces must also be included when analyzing bodies with a thin, compliant coating.5–8 A relatively simple-to-use contact mechanics theory that explicitly accounts for the presence of the coating and includes adhesion forces would be useful to those studying thin coatings. For example, such a theory could be of use when interpreting scanning probe microscope results, particularly when attempting to isolate coating properties from composite data that includes the constraint imposed by the substrate.9,10 A thin-coating contact mechanics (TCCM) theory would also be useful in multiasperity analysis of adhesion and friction.11,12 The range of applicability of a continuum analysis to molecular coatings has yet to be established, but would seem most appropriate for coatings that are at least many atoms thick (>1 nm thick). The need for an elementary TCCM theory was the motivation for one recent study.13 That study presented analytic results for a rigid spherical indenter contacting a thin, linear elastic coating on a rigid substrate. As shown in Fig. 1, the coating has a thickness h and linear elastic properties defined by Young’s modulus E and Poisson’s ratio . The coating is assumed to be compliant (i.e.,
rubbery, incompressible coatings are not considered) and subjected to a maximum nominal coating strain of less than 20% (likely an upper bound for small strain, linear elastic response). The rigid spherical indenter has a radius R and is pushed into the elastic coating with frictionless contact. The compressive force P and normal indenter displacement U are defined as positive. The contact radius is denoted by a. Both h/R and a/R are assumed to be much less than one, while a/h is assumed to be much greater than one. Under such conditions, a simple approximate analysis, analogous to that described by Johnson for cylindrical contact with an elastic layer,14 is applicable. The fundamental assumption in this TCCM theory is that both the contact radius and the compres
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