功能梯度纯弯曲梁弹塑性问题的解析解

Exact elastoplastic analysis of a functionally graded beam subjected to pure bending

在小变形前提下研究了功能梯度材料纯弯曲梁的弹塑性受力变形特征.假定功能梯度材料为理想弹塑性材料,且其弹性模量与屈服强度沿梁高度方向按照指数函数变化,根据Mises屈服条件导出了纯弯曲梁的弹性极限弯矩、截面弹塑性应力以及塑性极限弯矩的解析表达式.算例分析表明功能梯度材料梁的弹塑性性能与均匀材料梁不同,材料屈服不一定首先产生于截面最大应力点,塑性变形的产生、扩展具有多种不同的模式,材料弹性模量与屈服强度沿截面高度的梯度变化对纯弯曲梁的应力分布规律及极限承载能力均有较大影响.研究结果可为功能梯度材料梁的弹塑性分析提供验证的考题,也可为简化理论的建立提供一定的依据.

The elastoplastic behavior of a functionally graded beam subjected to pure bending is studied under the framework of small deformation theory. The modulus of elasticity and the yield limit of the material are assumed to vary exponentially along the thickness of the beam. And the material is a desirable elastoplastic model. The elastic limit bending moment, the elastoplastic stress and plastic limit bending moment are obtained by Mises yield criterion. The numerical examples show that the elastoplastic performance of a functionally graded beam subjected to pure bending is different from the homogeneous material beam. The yield of the material do not definitely derive from the maximum stress. The plastic deformation and the development of plastic zones are related to the variation of material properties along the thickness and the graded variations of the elastic modulus and the yield limit of the material have great effect on the stress distribution and limit carrying capacity. The analytical results presented here can serve as benchmarks for verifying numerical solutions of problems and will be helpful in establishing the simplified beam theories.