Saint-Venant decay analysis of FGPM laminates and dissimilar piezoelectric laminates

Using a similar procedure to the Hamiltonian system based model [Wang, J.S., Qin, Q.H., 2007. A symplectic model for piezoelectric wedges and its application to analysis of electroelastic singularities, Philosophical Magazine 87 (2), 225–251], the mixed-variable state space formulation is developed for functionally graded piezoelectric material (FGPM) strips and laminates, in which the material inhomogeneity is considered. For dissimilar homogeneous piezoelectric laminates, the state space formulation degenerated to a Hamiltonian system. Applying the developed model, we analyzed the decay of Saint-Venant end effects in a single FGPM strip and a FGPM laminate. The numerical results show that the decay rate depended strongly on the eigenvalue of the proposed operator matrix for the single FGPM strip. By using the coordinate transformation technique and the continuity conditions on the interface between two dissimilar materials (different piezoelectric properties or different material inhomogeneous parameters), the decay rates are also determined for multi-layered FGPM laminates, including dissimilar piezoelectric laminates (without material inhomogeneity) as a special case. Numerical results are presented to show the applicability of the proposed state space model to piezoelectric and FGPM laminates. In addition, the variation of the decay rate with the thickness has also been investigated for the dissimilar homogeneous piezoelectric laminates. This study indicates that material inhomogeneity plays an important role in Saint-Venant end effects for FGPM laminates, as in the case of a single FGPM strip [Borrelli, A., Horgan, C.O., Patria, M.C., 2004. Exponential decay of end effects in anti-plane shear for functionally graded piezoelectric materials. Proceedings of the Royal Society of London Series A 460, 1193–1212].