Analysis of the axisymmetric indentation of a semi-infinite piezoelectric material: The evaluation of the contact stiffness and the effective piezoelectric constant

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Abstract
A general solution of the axisymmetric indentation is obtained in the closed form for a semi-infinite, transverse isotropic piezoelectric material by a rigid-conducting indenter of arbitrary-axisymmetric profile. Explicit relationships are derived for dependences of the indentation depth and the indentation-induced charge on indentation force and applied electrical potential. Simple formulas are obtained for contact stiffness and effective piezoelectric constant, which can be used in indentation test and piezoresponse force microscopy to analyze the elastic and piezoelectric responses of piezoelectric materials. Depending on the direction of electric field (the potential difference), the electric field can either increase or suppress indentation deformation. The corresponding results are given for cylindrical, conical, and paraboloidal indenters.