The tapping-mode atomic force microscopy response has been analyzed by the Krylov–Bogolubov–Mitropolsky asymptotic method. Due to the presence of repulsive force, attractive force and damping in the tip-sample interaction, the response exhibits complicate nonlinear phenomena. The numerical and experimental results shows that the phenomena can be well described by this approximation solution when the driving frequency is close to the free resonance frequency and the setpoint amplitude ratio is larger than 0.4.

This paper, based on extensive finite element simulations and scaling analysis, presents scaling functions for the inverse problem in nanoindentation with sharp indenters to determine material properties from nanoindentation response. All the inverse scaling functions were directly compared with results calculated using the large deformation finite element method and are valid from the elastic to the full plastic regimes. To relate the material properties to measurable indentation parameters a new nondimensional experimental parameter Λ=P/(DS) was introduced, where P is load, D is indentation depth, and S is contact stiffness. This parameter is monotonically related to the ratio of yield stress to modulus. The modulus, hardness and yield stress are presented as explicit functions of Λ and the strain hardening exponent. The error in the inverse modulus, hardness, and yield stress due to uncertainty of the strain hardening exponent was studied and is compared with that of the traditional Oliver–Pharr method. The method of determining the strain hardening exponent from measurement with an additional indenter with a different cone apex angle is described. For this, a scaling function with the strain hardening exponent as the only unknown was obtained. In this way, the modulus, hardness, yield stress and strain hardening exponent may be determined. Experimental results show the inversion method permits the modulus and hardness to be accurately determined irrespective of the effects of pileup or sink-in.

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