静水压荷载下功能梯度圆环的弹性屈曲

Elastic Buckling of Functionally Graded Rings Under Hydrostatic Pressure

管道屈曲问题中,结构通常被考虑为平面应力圆环,但若存在轴向位移约束,其状态更接近平面应变圆环的状态.文中从理论上研究静水压作用下平面应力和平面应变功能梯度材料圆环的弹性屈曲问题.假定材料沿厚度方向弹性物性按照幂律分布连续变化,结合一阶剪切变形理论和Kárman几何关系,应用虚位移原理,导出结构的基本平衡方程;采用本征值方法得到了屈曲控制方程和结构屈曲临界荷载的解析解.通过算例比较了平面应力和平面应变状态下结构理论屈曲性能,同时考察了结构径厚比和材料组分参数的影响.结果表明:平面应变功能梯度材料圆环屈曲临界荷载略高于平面应力的结果;当材料组分参数介于0.1至10区间内时,两种圆环的屈曲临界荷载均下降得较为明显.

In the buckling issue of pipelines, the structure is usually considered as a plane stress ring, but if there exists axial displacement constrain, the status may be closer to the status of a plane strain ring. In this paper, the elastic buckling problem of the plane-stress and plane-strain functionally graded material rings under hydrostatic pressure is theoretically investigated. It is assumed that the elastic material properties along the thickness direction continually changes according to the power law distribution. In this case, by combining the first shear deformation theory and the K6rman geometry relation and by employing the virtual displacement principle, a basic equilibrium equation of structures is deduced. Then, the buckling governing equation and the analytic solution to the structural buckling critical load are obtained by means of the eigenvalue method. Finally, the theoretical buckling perfor- mances of the structure in the plane-stress and plane-strain states are compared, and the influences of the structural radius-to-thickness ratio and the material constituent parameter on the performance are discussed. The results show that the buckling critical load of the plane-stain functionally graded material ring is slightly higher than that of the plane-stress case, and that as the inhomogeneous parameter of the material ranges from O. 1 to 10, the buckling criti- cal loads of the two kinds of rings decrease more obviously.