Two-dimensional exact analysis of piezoelectric curved beam within symplectic framework

The plane problem of piezoelectric curved beam which is polarized in the radial direction is considered within the frame of state space. The piezoelectric material is assumed to be transverse isotropic and all the elastic stiffness, piezoelectric and dielectric constants are constants. The symplectic method uses the displacements, the electrical potential function and their conjugate variables (stresses, electric displacement) as the state vector, so that the state equation is formulated directly from the constitutive equations and equilibrium equations. The exact solutions are obtained by using the method of separation of variables along with the eigenfunction expansion technique. The origin problem is reduced to solve the eigenvalues and eigensolutions. Instead of dividing all the eigenvalues into several groups as before, we analyze the eigenfunctions of the general eigenvalue to start with and try to find the particular eigenvalue which has explicit physical interpretations. Similar to the work in the rectangular coordinate system, it can be found that the particular solutions for the curved beam with uniformly distribution forces on the lateral boundary conditions can be solved by using Jordan form eigensolutions. The particular solutions for arbitrary inhomogeneous lateral boundary conditions are also considered in present work, so that the symplectic analysis approach can be applied to solve more general problems of the piezoelectric curve beam. The symplectic approach can also be applied to analyze plane problems of the functional graded piezoelectric curved beams based on the present work.