Young’s modulus of nanocrystalline metal coatings is measured using the oscillating, that is, tapping, mode of a cantilever with a diamond tip. The resonant frequency of the cantilever changes when the diamond tip comes in contact with a sample surface. A Hertz-contact-based model is further developed using higher-order terms in a Taylor series expansion to determine a relationship between the reduced elastic modulus and the shift in the resonant frequency of the cantilever during elastic contact between the diamond tip and sample surface. The tapping mode technique can be used to accurately determine Young’s modulus that corresponds with the crystalline orientation of the sample surface as demonstrated for nanocrystalline nickel, vanadium, and tantalum coatings. 1. Introduction A variety of indentation-based test methods are used to evaluate strengthening effects [1–4] in materials at the nanoscale. Nanoindentation normal to the surface is routinely used to measure the hardness and Young’s modulus. Triboindentation tests are used [5–7] to measure both hardness and shear strength as well as quantify strain-rate sensitivity [8, 9] effects in the evaluation of deformation mechanisms in nanocrystalline alloys. The standard approach [10–12] to determine the elastic modulus during nanoindentation evaluates the load ( ) versus displacement ( ) curve during unloading after plastic deformation. It remains difficult to collect a sufficient quantity of versus data during elastic loading since elastic displacements ( ) are commonly limited to depths of only a few nanometers or less in many hard materials. The initial linear slope ( ) of the power-law-shaped unloading curve is used [11, 12] to determine the reduced elastic modulus ( ) as The contact area ( ) equals the square of the contact depth ( ) multiplied by the tip area ( ) coefficient, and the shape parameter equals 1.00 for flat punch, 1.034 for Berkovich, and 1.012 for Vickers indenter tips. The reduced elastic modulus ( ) of the indenter tip and sample surface system is related to the elastic moduli of the indenter tip and sample surface as Here, the subscripts and represent the probe tip and sample, respectively, for the Poisson ratio ( ) and Young’s modulus ( ) values. Indentation size effects (ISEs) are found with the directional loading of the indenter tip. For example, beyond an indentation depth, that is, 10% of the film thickness, the use of a Meyer plot indicates [13, 14] that the substrate material contributes to the elastic and plastic property measurements of the coating. Also, the sensitivity
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