For beam bending in transversely isotropic piezoelectric media, the reciprocal theorem and the general solution of piezoelasticity are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all orders for the beam of general edge geometry and loadings. By generalizing the method developed by Gregory and Wan, a set of necessary conditions on the edge-data for the existence of a rapidly decaying solution is established. The prescribed edge-data of the beam must satisfy these conditions in order that they could generate a decaying state within the beam. When stress and mixed conditions are imposed on the beam edge, these decaying state conditions for the case of bending deformation of piezoelectric beam are derived explicitly. They are then used for the correct formulation of boundary conditions for the beam theory solution (or the interior solution). Besides, an analytical solution of elastic beam is formulated to verify validity of our boundary conditions. For the stress data, our boundary conditions coincide with those obtained in conventional forms of beam theories. More importantly, the appropriate boundary conditions with two sets of mixed edge-data are obtained for the first time.
GAO Yang1, XU SiPeng2 & ZHAO BaoSheng3 1 College of Science, China Agricultural University, Beijing 100083, China
This paper presents the forms of the general solution for general anisotropic piezoelectric media starting from the basic equations of piezoelasticity by using the operator method introduced by Lur’e (1964), and gives the analytical form of the general solution for special orthotropic piezoelectric media. This paper uses the non-uniqueness of the general solution to obtain the generalized LHN solution and the generalized E-L solution for special orthotropic piezoelectric media. When the special orthotropic piezoelectric media degenerate to transversely piezoelectric media, the solution given by this paper degenerates to the solution for transversely isotropic piezoelectric media accordingly, so that this paper generalized the results in transversely isotropic piezoelectric media.
