Two-dimensional complete rational analysis of functionally graded beams within symplectic framework

doi:10.1007/s10483-012-1617-8

Abstract

Abstract: Exact solutions for generally supported functionally graded plane beams are given within the framework of symplectic elasticity. The Young’s modulus is assumed to exponentially vary along the longitudinal direction while Poisson’s ratio remains constant. The state equation with a shift-Hamiltonian operator matrix has been established in the previous work, which are limited to the Saint-Venant solution. Here, a complete rational analysis of the displacement and stress distributions in the beam is presented by exploring the eigensolutions that are usually covered up by the Saint-Venant principle. These solutions play a significant role in the local behavior of materials that is usually ignored in the conventional elasticity methods but possibly crucial to the materials/structures failures. The analysis makes full use of the symplectic orthogonality of the eigensolutions. Two illustrative examples are presented to compare the displacement and stress results with those for homogenous materials, demonstrating the effects of material inhomogeneity.

Key words: functionally graded material (FGM), exact solution, expansion of eigenfunction, symplectic elasticity, semi-infinite orthotropic strip, stress singularity at clamped end, generalized Cauchy type singular integral equation

2010 MSC Number:

Li ZHAO;Wei-qiu CHEN;Chao-feng L�. Two-dimensional complete rational analysis of functionally graded beams within symplectic framework. Applied Mathematics and Mechanics (English Edition), 2012, 33(10): 1225-1238.

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