Penalty hexahedral element formulation for flexoelectric solids based on consistent couple stress theory

This work proposes a penalty 20-node hexahedral element for size-dependent electromechanical analysis based on the consistent couple stress theory. The nodal rotation degrees of freedom (DOFs) are used to approximate the mechanical rotation, meeting the C-1 requirement in a weak sense by using the penalty function method, and to effectively enhance the standard isoparametric interpolation for determining the displacement test function. The normalized stress functions that satisfy the relevant equilibrium equation and strain compatibility equation a priori are employed to formulate the stress trial function. Since the element has only three displacement, three rotation and the electric potential DOFs per node, it has relatively simple formulation and can be readily incorporated into the existing finite element program. Several benchmarks are examined and the results demonstrate that the new element has good numerical accuracy and captures the size dependence effectively. In addition, the influence of the micro-inertia on electromechanical dynamic responses of flexoelectric solids at small scale are analyzed using the proposed element. It is shown that the nature frequency decreases with the increase of micro-inertia and in general, the influences are more obvious on higher order modes.