简支功能梯度变厚度梁的弹性力学解

Elasticity Solution of Simply-supported Functionally Graded Beams with Variable Thickness

该文假设弹性模量为沿厚度变化的指数函数,泊松比为常数,利用平面应力问题的基本方程,导出满足控制微分方程和两端简支边界条件的位移函数的一般解,对上下表面的边界方程作傅里叶级数展开确定待定系数,结果具有很好的收敛性,精度可达三位有效数字。考察了弹性模量变化对功能梯度梁位移和应力的影响,为检验其它功能梯度梁近似理论和数值结果的有效性提供了依据。该文的方法可应用于对应力分析有较高精度要求的航空工程以及微型机械仪器设计等工程。

In this paper, the Young＇s modulus is graded through the thickness following the exponential-law and the Poisson＇s ratio keeps constant. According to the governing equations of plane stress problems, the general expressions of displacements, which exactly satisfy the governing differ- ential equations and the simply-supported boundary conditions at two ends of the beam, are deduced. The unknown coefficients in the solution are determined by using the Fourier sinusoidal series expansion to the boundary equations on the upper and lower surfaces of the beams. The excellent convergence of the solution is demonstrated. The results are accurately up to the third significant digit. The effect of the Young＇s modulus varying rules on the displacements and stresses of functionally graded beams is investigated. The proposed elasticity solution can be used to assess the validity of various approximate solutions and numerical methods for functionally graded beams. It is applicable in aerospace engineering and other projects with highly accurate demand on stress analysis such as the design of micro-mechanical apparatus.