Unified theory of beam bending within flexoelectricity with including piezoelectricity

Abstract

The behaviour of small size dielectric elastic beams is described within higher-grade theory with including electric polarization. The coupling between strain gradients and polarization is incorporated into the constitutive laws in the form of flexoelectricity, while piezoelectricity is involve in the classical form. Both the governing equations and boundary conditions are derived using variational formulation for electro-elastic continuous media and deformation assumptions employed in three various beam bending theories such as the classical theory (Euler-Bernoulli theory), the 1st order shear deformation theory (Timoshenko theory) and 3rd order shear deformation theory. The unified formulation allows switching between theories with various bending assumptions by a proper selection of two key factors.