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Elasticity solution of the bending of beams with the flexoelectric and piezoelectric effects

This paper studies the bending behavior of a nanosclae elastic beam with flexoelectric and piezoelectric effects. Based on a two-dimensional theory of piezoelectricity and flexoelectricity, an exact elasticity solution is derived when applied transverse mechanical loading and/or electric potential difference between the top and bottom surfaces are prescribed. Obtained results may apply to not only pure piezoelectric beams without flexoelectricity only if setting all flexoelectric coefficients to zero but also beams of isotropic material with flexoelectricity only if setting all piezoelectric coefficients to zero. The solution can be taken as a benchmark solution of related models of beams as actuators, generators, etc. The influences of piezoelectricity and flexoelectricity on the transverse deflection, stress distribution and electric polarization are analyzed. Flexoelectricity plays a crucial role in altering bending deflection, electric polarization for a single homogeneous simply supported beam. The deflection and electric polarization of a beam with both piezoelectric and flexoelectric effects are not the sum of the corresponding ones of a purely piezoelectric beam and a purely flexoelectric beam.