The stiffness of micron-sized sphere–plate contacts was studied by employing high frequency, tangential excitation of variable amplitude (0–20 nm). The contacts were established between glass spheres and the surface of a quartz crystal microbalance (QCM), where the resonator surface had been coated with either sputtered SiO2 or a spin-cast layer of poly(methyl methacrylate) (PMMA). The results from experiments undertaken in the dry state and in water are compared. Building on the shifts in the resonance frequency and resonance bandwidth, the instrument determines the real and the imaginary part of the contact stiffness, where the imaginary part quantifies dissipative processes. The method is closely analogous to related procedures in AFM-based metrology. The real part of the contact stiffness as a function of normal load can be fitted with the Johnson–Kendall–Roberts (JKR) model. The contact stiffness was found to increase in the presence of liquid water. This finding is tentatively explained by the rocking motion of the spheres, which couples to a squeeze flow of the water close to the contact. The loss tangent of the contact stiffness is on the order of 0.1, where the energy losses are associated with interfacial processes. At high amplitudes partial slip was found to occur. The apparent contact stiffness at large amplitude depends linearly on the amplitude, as predicted by the Cattaneo–Mindlin model. This finding is remarkable insofar, as the Cattaneo–Mindlin model assumes Coulomb friction inside the sliding region. Coulomb friction is typically viewed as a macroscopic concept, related to surface roughness. An alternative model (formulated by Savkoor), which assumes a constant frictional stress in the sliding zone independent of the normal pressure, is inconsistent with the experimental data. The apparent friction coefficients slightly increase with normal force, which can be explained by nanoroughness. In other words, contact splitting (i.e., a transport of shear stress across many small contacts, rather than a few large ones) can be exploited to reduce partial slip.
Complex Shepard Operators and Their Summability
