辛求解体系下功能梯度材料平面梁的完整解析解

Two-Dimensional Complete Rational Analysis of Functionally Graded Beams Within the Symplectic Framework

采用辛弹性力学解法,求取弹性模量沿轴向指数变化,而Poisson比保持不变的功能梯度材料平面梁的完整解析解.通过求解被Saint-Venant原理覆盖的一般本征解,建立起完整的解析分析过程,进而给出平面梁位移和应力的精确分布规律.传统的弹性力学分析方法常常忽略被Saint-Venant原理覆盖的解,但这些衰减的本征解对材料的局部效应起着较大的影响作用,可能导致材料或结构的突然失效.采用辛求解方法,充分利用本征向量之间的辛共轭正交关系,得到了功能梯度材料梁的完整解析解.两个数值算例分别将功能梯度材料平面梁的位移和应力分布与相应均匀材料情形的结果进行比较,研究了材料非均匀性对位移和应力解的影响.

Exact solutions for generally supported functionally graded plane beams were given within the framework of symplectic elasticity.The Young’s modulus was assumed to vary exponentially along the longitudinal direction while Poisson’s ratio remained constant.The state equation with a shift-Hamiltonian operator matrix had been established in our previous work,but limited to the Saint-Venant solution.Here it was presented that a complete rational analysis of the displacement and stress distributions in the beam by exploring the eigensolutions which were usually covered up by the Saint-Venant principle.These solutions played a significant role on local behavior of materials that was usually ignored by the conventional elasticity methods but may be crucial to the failure of the materials/structures.The analysis made full use of symplectic orthogonality of the eigensolutions.Two illustrative examples were presented to compare the displacement and stress results with those for homogenous materials,and to demonstrate the effect of material inhomogeneity.