A Piezoelectric Mechanical-Like Beam Element with Superconvergence Property

Abstract

A finite element is developed for a piezoelectric beam theory whose displacement varies according to the Timoshenko assumption, and the electric potential varies quadratically through each piezoelectric patch thickness. The application of Gauss’s law to patches in the sensing mode is treated as a constraint in order to eliminate electric degrees of freedom from the weak form of the problem and, thus, formulate a two-node element with the same degrees of freedom as its purely mechanical counterpart (i.e., two displacements and one rotation per node). As a result, the element can be easily included in any computer program by simply replacing the existing purely mechanical element with this piezoelectric one. Since the approximation space is the same as that of the general homogeneous solution of the piezoelectric beam theory, the element enjoys the superconvergence property and is completely free of locking. Simple numerical tests on linear static problems are presented to demonstrate the ability of the element to give exact nodal values for the displacements, rotations, and reactions, irrespective of the mechanical and electrical loadings applied and mesh used. Also, the ability of a postprocessing scheme, which is set up to recover the exact sensing voltages at the nodes, is demonstrated. The element is further employed in shape-control problems.