Planar four node piezoelectric elements with drilling degrees of freedom

Several new planar four node piezoelectric elements with drilling degrees of freedom are presented. We begin by deriving two families of variational formulations accounting for piezoelectricity and in-plane rotations. The first family retains the skew-symmetric part of the stress tensor, while in the second, the skew part of stress is eliminated from the functional. The finite elements derived from two of the variational formulations derived in this paper are investigated. The first element is based on an ‘irreducible’ form, while the other is based on a fully mixed functional, with both stress and electric flux density assumed. Our new elements are shown to be accurate and robust in comparison with a number of existing elements, for several benchmark test problems