Pure bending behaviors of transversely isotropic piezoelectric beam with flexoelectric effect

Numerous studies have explored beam bending involving piezoelectric effect and flexoelectric effect. However, a higher-order bending theory for transversely isotropic piezoelectric beam has not yet been established, and the associated independent material parameters remain unclear. In this paper, the higher-order bending theory of transversely isotropic beams is presented based on the general dielectric theory including strain gradient and polarization gradient. The general constitutive equations of transversely isotropic dielectrics are detailed for the first time. A semi-inverse solution for a transversely isotropic beam under plane-strain conditions is developed, and meanwhile, the Bernoulli–Euler bending solution is also obtained. The plane-strain solution for a purely bending beam considering strain gradient and polarization gradient can reduce to that of the Bernoulli–Euler beam when the strain along thickness direction is neglected. The electric potential induced by piezoelectric and flexoelectric effects is examined. We hope that the results of this paper will serve as a reference for verifying the reliability of numerical calculation methods and contribute to a deeper understanding of electromechanical coupling effects.