A systematic finite element procedure for flexoelectric piezoelectric nanobeams using the asymptotic numerical method

Analysis of small-scale beam structures under the combined effects of flexoelectric and piezoelectric responses is of interest in this work. The problem is tackled by the derivation of a mixed finite element model accounting for geometric nonlinearity and large strains through a total Lagrangian representation, so that the von Karman strain can be considered. In addition, a Penalty method is introduced to account for higher order stress tensors. The asymptotic numerical method (ANM) is implemented for the first time in this context to deal with nonlinearities. To avoid tedious manual calculation of the ANM power series terms, a systematic procedure is proposed in this work with an application to a clamped–clamped nanobeam. The obtained model is applied for different flexoelectric-piezoelectric materials, and the results indicate that the applied electric voltage can lead to buckling of the beam. However, this instability is observed for negative voltages for some materials, and positive voltages for others, depending on a combination of flexoelectric and piezoelectric coefficients of the material.