The boundary layer phenomena in two‐dimensional transversely isotropic piezoelectric media by exact symplectic expansion

Abstract

SUMMARYA separable variable method is introduced to find the exact homogeneous solutions of a two-dimensional transversely isotropic piezoelectric media to handle general boundary conditions. The usual method of separable variables for partial differentiation equations cannot be readily applicable due to the tangling of the unknowns and their derivatives. Introducing dual variables of stresses, we obtain a set of first-order Hamiltonian equations whose eigensolutions are symplectic spanning over the solution space to cover all possible boundary conditions. The solutions consist of two parts. The first part is the derogative zero-eigenvalue solutions of the Saint Venant type together with all their Jordan chains. The second part is the decaying non-zero-eigenvalue solutions describing the boundary layer effects. The classical solutions are actually the zero-eigenvalue solutions representing the simple extension, bending, equipotential field, and the uniform electric displacement. On the other hand, the non-zero-eigenvalue solutions represent the localized solutions, which are sensitive to the boundary conditions and are decaying rapidly with respect to the distance from the boundaries. Some rate-of-decay curves of the newly found non-zero-eigenvalue solutions are shown by numerical examples. Finally, the complete boundary layer effects are quantified for the first time.

The mode III stress/electric intensity factors and singularities analysis for edge-cracked circular piezoelectric shafts

“…In addition, Zhang and Zhong (2003) obtained analytical formula for an unbounded domain containing a plane crack. Gu et al (2005) and Leung et al (2007) obtained 2D solutions for transversely isotropic media with the symplectic method. Moreover, Xu et al (2008) applied the symplectic expansion method to solve three-dimensional problem for transversely isotropic piezoelectric media and introduced a 3D sub-symplectic structure for transversely isotropic piezoelectric media further in Xu et al (2005).…”

Symplectic Method for the Thin Piezoelectric Plates

“…Therefore, many researchers use the symplectic method to study problems in mechanics and engineering science. The 2D and 3D symplectic structures of transversely isotropic piezoelectric media have been derived by Gu [11][12][13][14]. Xu et al [15] studied the problem of interfacial cracks in non-ideal interfacial magnetoelectroelastic bimaterials.…”

Analytical stress intensity factors for edge-cracked cylinder

“…In addition, Zhang and Zhong [21] obtained analytical formula near crack corners of multi-material junctions. Xu et al [22][23][24] applied the method to solve the stress intensity factors for finite elastic disk and generalized the symplectic expansion method to solve the stress distribution for piezoelectric medium.…”