A Mixed Finite Element Formulation for Piezoelectric Materials

A geometrically nonlinear finite element formulation to analyze multi‐field problems as they arise e.g. in piezoelectric or magnetostrictive materials is presented. Here we focus on piezoelectric problems. The formulation is based on a Hu‐Washizu functional considering six independent fields. These are displacements u , electric potential ϕ , strains E , stresses S , electric field $\vec E$ , and the electric flux density $\vec D$ . The finite element approximation leads to an 8‐node hexahedral element with u and ϕ as nodal degrees of freedom. The fields E , S , $\vec E$ , and $\vec D$ are interpolated on element level by employing some internal degrees of freedom. These fields do not require continuity across the element boundaries, thus the internal degree of freedoms are eliminated on element level by a static condensation. The geometrically non‐linear theory allows large deformations and accounts for stability problems. To fulfill the charge conservation law in bending dominated situations exactly a quadratic approximation of the electric potential is necessary. This leads in general to additional nodal degrees of freedom, which is circumvented by the presented formulation by employing appropriate interpolations of $\vec E$ and $\vec D$ . Numerical examples show that the locking effect which arise in low order elements are significantly reduced and that the element provides good accuracy with respect to experimental data. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)