Free Vibrations of Homogeneous and Layered Piezoelectric Hollow Spheres

A finite element formulation in spherical coordinates is presented for the study of the vibrations of piezoelectric homogeneous and layered hollow spheres. The finite element model is based on nine-node Lagrangian interpolation functions. Representative cases are considered including solid elastic spheres and hollow homogeneous and laminated piezoelectric spheres. The accuracy of the proposed formulation is verified by comparison with existing solutions showing excellent agreement. Several new results are presented for different piezoelectric materials in both tabular and graphical format.