Distributed Modal Signals of Conical Shells Based on Flexoelectric Effect

Flexoelectricity is known as the electromechanical coupling effect between the strain gradient and the polarization. It is the only contribution of polarization from inhomogeneous mechanical deformation in nonpiezoelectric materials. Conical shells are commonly used as injectors, sprays and rocket nozzles, etc, which are generally clamped at the minor end and free at the major end when mounted. In this study, a flexoelectric layer is laminated on conical shells with clamped-free boundary conditions (BCs) to monitor the natural modal signal distributions. The direct flexoelectric effect defined in a tri-orthogonal coordinate system is presented first, followed by the sensing mechanism of a generic flexoelectric sensor patch. The mode shape functions of conical shells obtained by using the Rayleigh-Ritz method are briefly reviewed. The spatially distributed microscopic sensing signal with respect to position coordinates is evaluated in detail to reveal the modal signal distributions. Due to the gradient effect, the bending strain component is the only contribution to the total sensing signal. The total signal consists of two components resulting from the two bending strain components: circumferential bending strains and longitudinal bending strains. Analytical results show that, the flexoelectric sensing signal induced by the circumferential bending strain is the dominant contribution to the total signal for lower order modes. The optimal location of flexoelectric sensors is discussed for selected vibration modes.