Size effect on the electromechanical coupling factor of a thin piezoelectric film due to a nonlocal polarization law

We analyze thickness deformation of a thin piezoelectric film under surface tractions. The film has a nonlocal constitutive relation between polarization and electric field. An exact solution is obtained in closed form. In contrast to the corresponding solution from piezoelectricity, the nonlocal solution shows an electromechanical boundary layer near the film surface and that the electromechanical coupling factor is dependent on the film thickness.