The symplectic method for plane elasticity problem of functionally graded materials

There are several typical methods that have been widelyemployed to analyze static and dynamic behavior of functionally gradedmaterial (FGM) structures, such as the simplified analytical methods (basedon one or two-dimensional structural theories), three-dimensional exactelasticity methods, approximate elasticity methods based on laminatedmodels, semi-analytical methods, and the numerical methods. In this paper,the symplectic approach, originally developed for homogeneous orpiece-wisely homogeneous materials, is extended to consider the planeelasticity problem with a rectangular domain of FGM. In the present FGM,Young's modulus varies exponentially with the axial coordinate, whilePoisson ratio remains unaltered. After introducing new stress components,the problem is formulated within the frame of state space, and solved usingthe method of separation of variables along with the eigenfunction expansiontechnique. The operator matrix, called shift-Hamiltonian matrix, is not inan exact Hamiltonian form, since the eigenvalues are symmetric with respectto - \alpha / 2, rather than zero in the standard Hamilton matrix. Inthis case, the symplectic adjoint eigenvalue of zero is induced as - \alpha.The Saint-Venant solutions derived in the paper exhibit some uniquecharacteristics, but they can be degenerated to the ones for homogeneousmaterials after imposing certain rigid motions.The symplectic method enriches the analysis methodology for heterogeneousmaterial. Furthermore, it can indicate the certain physical essence of theproblem that can not be revealed by other methods.