Elasticity solution of functionally graded beams with consideration of the flexoelectric effect

This article studies the electroelastic response in bending behavior of a functionally graded elastic beam with consideration of the flexoelectric effect. The variational principle is employed to deduce the governing differential equations along with appropriate boundary conditions including nonclassic conditions based on a two-dimensional theory of elasticity and flexoelectricity. An exact elasticity solution is derived for a simply supported functionally graded beam subjected to mechanical loading and/or electric voltage between the top and bottom surfaces. Numerical computations are carried out for a SrTiO₃ functionally graded beam of cubic symmetry subjected to typical sinusoidally mechanical and electric loading, respectively. Obtained results show the influence of the gradient parameter on the deflection, stress distribution, and electric polarization of the functionally graded beams. The suitable choice of the gradient parameter may decrease the stress or enhance the polarization, and improve the performance of the flexoelectricity of the beam. (paper)