功能梯度悬臂梁弯曲问题的解析解

Analytical Bending Solution of Functionally Graded Cantilever-Beam

将功能梯度悬臂梁作为平面应力问题处理.根据正交各向异性弹性体的基本方程,引入应力函数,假设所有材料常数沿厚度方向按同一函数规律变化,采用弹性力学半逆解法,求得功能梯度悬臂梁在端部集中力和力矩作用下的解析解.所得到的解,对任意梯度函数均成立,且退化到各向同性均匀弹性情况下的结果,与已有的理论解相一致.对弹性模量分别按指数函数和幂函数梯度变化的算例进行了分析,结果显示功能梯度梁的轴向位移仍近似直线变化.

Based on the semi-inverse method, an analytical solution is obtained for a functionally graded cantilever-beam that is clamped at one end and subjected to a concentrated force and a couple at another end. The problem is treated as a plane stress case of an orthotropic elastic body. The mechanical properties of the material have been assumed to have the same dependence on the height-coordinate. This solution is valid for arbitrary gradient functions and it can play as a benchmark result for assessing oth- er approximate methodologies or as a basis for establishing simplified functionally graded beam theories. Degenerate results for isotropic homogeneous elastic case are coincided well with existing analytical solutions. Some numerical examples are also given by assuming an exponential gradient function.