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

• Published in: Applied mathematics and mechanics 2012-10, Vol.33 (10), p.1225-1238
• Main Author: 赵莉 陈伟球 吕朝锋
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Classical Mechanics; Fluid- and Aerodynamics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Partial Differential Equations
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-012-1617-8

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 the Poisson＇s ratio remains con- stant. The state equation with a shift-Hamiltonian operator matrix has been established in the previous work, which is 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 prin- ciple. 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 mate- rial/structure 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.