Flexoelectric effects in second-order extension of rods

We study flexoelectric couplings in the second-order extension of rods when a cross section does not remain planar during extension. This happens when the rod is not very thin or, in dynamic problems, when the extensional wave length is not much larger than the characteristic dimension of the cross section of the rod. To exhibit flexoelectric couplings in such deformations or motions, a system of approximate, one-dimensional equations is derived from the three-dimensional theory of flexoelectricity for a rod of a rectangular cross section. The equations take into account extensional, thickness-stretch and symmetric thickness-shear deformations. They are used to analyze a few static and dynamic problems, showing that flexoelectric coupling is accompanying second-order extension through symmetric thickness shear where the cross section is no longer planar.