Symplectic elasticity for bi-directional functionally graded materials

This paper suggests a symplectic framework for the analysis of plane problems of bi-directional functionally graded materials (FGMs), in which the elastic modulus varies exponentially both along the longitudinal and transverse coordinates while the Poisson’s ratio remains constant. First, by a procedure which has been used in symplectic elasticity for homogeneous materials, along with the introduction of some stress variables, the governing equations are rewritten in an appropriate state-space form. The method of separation of variables is then adopted to transform the original problem to an eigenproblem, in which the eigenvalues and eigensolutions are determined subsequently. From the physical essence, we can know that the Saint–Venant solutions correspond to two particular eigenvalues (e.g. zero and −α, where α is the gradient index along the longitudinal coordinate). The first-order eigenvector of Jordan normal form for the special eigenvalue −α must be solved in a different way than that for the homogenous materials or the uni-directional FGMs. The presented general eigensolutions, which are usually covered up by the Saint–Venant principle, play a significant role in the local behavior and may be crucial to the onset of failure of the materials/structures. Two numerical examples are considered to show the stress distributions in bi-directional FGM rectangular beams, and to indicate the importance of the developed solution in the local behavior.

► We obtain the exact solutions of 2D-FGM plane beams with the symplectic framework. ► We obtain the accurate local behavior covered up by the Saint–Venant principle. ► The operator matrix for 2D-FGM is somehow different from the Hamiltonian matrix. ► The eigenfunctions of the eigenproblem is solved in a quite different way.