A novel 3D mixed finite element for flexoelectricity in piezoelectric materials

Flexoelectricity is the intrinsic length-scale dependent higher-order electromechanical response of all centro- and non-centro-symmetric dielectrics, including piezoelectrics. Direct flexoelectricity is defined as the appearance of an electric field due to induced strain gradients. The numerical modeling of flexoelectricity is largely carried out using mixed FE, which has its historical foundations in strain gradient theories. However, existing finite elements are either limited to 2D or have inherited numerical instabilities due to the known saddle-point structuring. The current work presents a numerically robust three-dimensional mixed FE for higher-order electromechanical applications without the use of stabilization or penalty parameters. After its verification, the new finite element is applied to the new problem of truncated semicone torsion, taking into account flexoelectricity in piezoelectric solids, and the original findings are reported. Current research reveals the complex interaction between first-order (piezoelectricity) and higher-order (flexoelectricity) electromechanical coupling.