梯形载荷作用下功能梯度简支梁弯曲的解析解

摘要

基于应力函数法，对梯形分布载荷作用下、材料属性在厚度上任意变化的功能梯度简支梁弯曲问题的解析解进行了研究。首先引入了一个应力函数，根据平面应力问题的基本方程，得出了功能梯度梁的应力函数应满足的偏微分方程，并根据应力边界条件得出了应力函数及各向应力的表达式；进而根据功能梯度材料的本构方程和位移边界条件，得出了各向应变与位移的显式解析表达式。在算例中，分别采用文中方法和经典理论对均质各向同性梁进行求解，验证了文中方法的正确性；并求解了材料组分呈幂律分布的功能梯度梁的应力和位移分布，分析了上下表层材料的弹性模量比λ与组分材料体积分数指数 n 对应力和位移分布的影响。%Based on the stress function method, an analytical solution of bending is presented for simply sup-ported functionally graded beam subjected to trapeziform pressure with arbitrary property distribution across the thickness. A stress function Ф is introduced, and the system of partial differential equations for stress function is established based on the fundamental equations for plane stress states. The expressions of stress function and stresses are given according to the boundary conditions for stresses for the simply sup-ported functionally graded beam subjected to hydrostatic pressure. Then, the explicit analytical expressions of strains and displacements are presented according to the constitutive relations of functionally graded materials and displacement boundary conditions, In the example, the proposed solution is validated by com-paring the results with the classical theory on the homogeneous isotropic beam. This paper also studies the distributions of the stresses and displacements of the functionally graded beam whose material properties obey a power law of distribution of the volume fraction of the constituents. The effects of top-bottom sur-faces’ Young’s modulus ratio λ and volume fraction exponent n on the variation of the stresses and dis-placements of the functionally graded beam are also examined.