Derivation of plate and rod equations for a piezoelectric body from a mixed three-dimensional variational principle

We use a mixed 3-dimensional variational principle to derive 2-dimensional equations for an anisotropic plate-like piezoelectric body and one-dimensio nal equations for an anisotropic beam-like piezoelectric body. The formulat ion accounts for double forces without moments which may change the thickne ss of the plate and deform the cross-section of the rod. The dependence of the bending rigidities of a transversely isotropic plate upon the angle bet ween the normal to the midsurface and the direction of transverse isotropy is exhibited. The plate equations are used to study the cylindrical deforma tions of a transversely isotropic plate due to equal and opposite charges a pplied to its top and bottom surfaces. It is also found that a piezoelectri c circular rod with axis of transverse isotropy not coincident with its cen troidal axis and subjected to electric charges at the end faces is deformed into a non-prismatic body.