Dynamical one-dimensional models of passive piezoelectric sensors (Q2875347)

Starting with the three-dimensional equations of piezoelectricity in the quasi-electrostatic approximation together with proper boundary conditions, the authors deduce one-dimensional models that describe the behavior of slender rods used as passive sensors to measure the displacement in an underlying elastic half-space, excited by external sources. Asymptotic expansions and limiting processes are used to get the final forms of one-dimensional models. These models differ in the applied boundary conditions and aim at providing flexibility for the identification of the boundary displacement imposed at one end of the rod. The differential operators are first decomposed into partial derivatives along the transverse direction, scaled to a small parameter that will ultimately be taken to tend to zero, and along the longitudinal direction of the slender bar. Equations are then derived for the electric potential and for the mechanical displacement fields. The structure of the first-order electric and displacement fields as well as the associated coupled limit equations are determined. An analysis is provided for the effect of chosen boundary conditions on the homogenized material parameters. The obtained one-dimensional models of piezoelectric sensors are analyzed, and it is finally shown how they enable the identification of the boundary displacement associated with the probed elastic medium.