The Symplectic Method For Plane Problems Of Functionally Graded Materials

As a special kind of inhomogeneous multiphase composites, functionally graded materials (FGMs) with material constants varying continuously with spatial positions exhibit excellent physical and mechanical properties, and hence they have been widely used in aeronautics and astronautics, electronics, machine, optics and so on. Functionally graded piezoelectric materials (FGPMs) and functionally graded magneto-electric-elastic materials (FGMEEMs) have also been developed by integrating the concept of FGM with piezoelectric and magneto-electro-elastic materials. These new types of functional materials promote a prospective development and application in material science and smart structures.This paper extends the symplectic approach to investigate the plane problems of functionally graded elastic, piezoelectric and magneto-electric-elastic materials. The material properties are assumed to vary along the length direction in an identical exponential form. By introducing the new components of stress (electric displacement and magnetic induction), the modified state vector is constructed and the state equation is derived. Within the symplectic framework, the origin problem can be converted into analyzing the eigenvalues and eigensolutions through the method of separation of variables along with the eigenfunction expansion technique.In contrast to the homogenous materials, the operator matrices for FGM, FGPM and FGMEEM are not in a conventional Hamiltonian form. In this paper, a new concept, i.e. the Shift-Hamiltonian operator, is coined since its eigenvalues are symmetric with respect to-α/2, instead of zero as for the conventional Hamiltonian matrix. The zero eigensolutions bear definite physical interpretations; while the origin solutions corresponding to the particular eigenvalue -a exhibit some unique characteristics, which can not describe the Saint-Venant solutions directly because of the influence of material inhomogeneity. The origin solutions corresponding to other general eigenvalues (i.e. those excluding the particular eigenvalues zero and-α) represent the local effect and the decay exponentially with the distance.The symplectic method is also applied to analyze plane problems of two-dimensional functionally graded materials (2D-FGMs), whose elastic modulus varies exponentially both along the axial and transverse coordinates while the Poisson's ratio remains constant. The operator matrix exhibits somehow different characteristics when compared with the conventional Hamiltonian matrix as well as the Shift-Hamiltonian matrix. From the physical point of view, it is known that the Saint-Venant solutions still correspond to the particular eigenvalues zero and-α. The first-order eigenvector of Jordan normal form for the special eigenvalue-αis solved in a different way than that for the homogenous materials and the 1D axial FGM. The material inhomogeneity has an important effect on the general eigenvalues.According to the traditional formulation of the symplectic expansion series, enormous difference in magnitudes exists between two sets of general eigensolutions when a large number of eigensolutions are adopted to obtain accurate results. The large numbers can be avoided by rewriting the expansion formula so as to achieve the stability of numerical calculation. The numerical examples show that the treatment is simple and effective for both homogenous and functionally graded materials.Through the present study, it is seen that the symplectic method can be an alternative to solve problems of functionally graded materials. Its further development will greatly enrich the current analysis methodology for heterogeneous materials and structures.