Flexoelectric effect on bandgap properties of periodic bi-directional-graded curved nanoshells

Abstract

With the miniaturization of devices, the size effect of structures becomes increasingly apparent and it becomes crucial to consider flexoelectricity in MEMS/NEMS. Given that functionally graded materials (FGMs) can generate strain gradients that enhance flexoelectricity, it is both novel and crucial to investigate the synergistic effects of FGMs and flexoelectricity on bandgap characteristics. Understanding these interactions is essential for advancing materials science and could lead to significant innovations in nanotechnology. In this work, a theorectical model for the periodic flexoelectric doubly-curved nanoshells with bi-directional functionally graded (BDFG) and porosity considered are constructed. As the governing equations for longitudinal FG problems are more difficult to be solved analytically, their precise analysis is more challenging. Thus, an improved transfer matrix method based on state-space is proposed innovatively to obtain the complex band structures for longitudinal FG problems. Subsequently, the influences of the flexoelectric effect, strain gradient effect, BDFG indices, and porosity distributions on bandgaps are systematically discussed. The results indicate that the flexoelectric effect is non-negligible at small scales, which generally widens the bandgaps, with higher frequency ranges and larger attenuation capacity. Bandgaps are sensitive to the variation of functionally graded indices. Particularly, flexoelectricity brings about the appearance and disappearance of some bandgaps during the variation of FG index in the thickness direction $n_3$, and complicates the bandgaps variation. Besides, the porosity distribution patterns and porosity coefficients enrich the regulation of bandgaps. Our investigation offers valuable insights into the application of flexoelectric electrical components combined with BDFG in MEMS/NEMS to the wave propagation domain.