Variational principles for polar piezoelectric media in elastic range

The fundamental equations of polar piezoelectric media in differential form are alternatively established in variational forms with their well-known features. First, a 3-field variational principle with some constraint conditions is deduced for a regular region of media from a general principle of physics. The principle is modified by using an involutory transformation and a 9-field variational principle operating on all the field variables is derived. Next, this principle is extended and a unified variational principle is obtained for the region with a fixed internal surface of discontinuity. The unified variational principle is further generalized for the equations of a laminated polar region. The generalized variational principle with the only constraint of initial conditions yields all the equations of the laminae region, including the interface conditions, as its Euler-Lagrange equations. The variational principles are shown to recover some of earlier variational principles, as special cases.