The State Vector Method For Layered Space Piezoelectric Media

Applications of the state vector method into the analysis of lamina and the current research state of piezoelectric materials are briefly reviewed based upon literature search. Solutions for axisymmetric and non-axisymmetric layered space piezoelectric media, and simply supported and multilayered magneto-electro-elastic plates via the state vector method is investigated. We also present the application of Mathematica in the study.Axisymmetric problem: The state vector equation of transversely isotropic axisymmetric space piezoelectric media is established in a cylindrical coordinate system. Using the Hankel transform, the state vector equations are reduced to a set of ordinary differential equations. According to the theory of ordinary differential equations and the Cayley-Hamilton theorem, an analytical solution of the problem is presented in the form of the product of initial state vector and transfer matrix, which is given for the three distinct eigenvalues each. Applications of the solutions are discussed. By the use of the continuity conditions at the layer interfaces, the general solution formulation of multilayered axisymmetric space piezoelectric media is given. A three layered piezoelectric half-space subjected to vertical mechanical loads or charge is analyzed as the numerical example, and the calculative results of two stacking sequences are compared.Non-axisymmetric problem: Two independent sets of the state vector equations are derived from its governing equations in a cylindrical coordinate system by introducing a set of auxiliary variables. Using the Fourier transform and following the similar reasoning steps of the axisymmetric problem, we obtain the general solution formulation of multilayered non-axisymmetric space piezoelectric media. The solutions are degenerated to the axisymmetric case as a checkout. The behavior of three layered piezoelectric half-space subjected to horizontal mechanical loads is emphatically analyzed.Simply supported and multilayered magneto-electro-elastic plates: The state vector equation of transversely isotropic magneto-electro-elastic solid is established from its governing equations in a Cartesian coordinate system.Using the Fourier transform due to boundary conditions of simply support, solutions of a single magneto-electro-elastic layer in the form of initial state vector and transfer matrix are obtained in Fourier transform space from the state vector equations. Using the perfectly bonded contact conditions between the layers, the general solution formulation of multilayered magneto-electro -elastic plates is given. As a numerical illustration, sandwich plates with two stacking sequences under a mechanical load or charge are analyzed.In addition, experiences in Mathematica used in both aided design and programming are introduced, in order to offer a reference mode for theoretical deduction and its programming in related fields.