碳纳米管增强型复合材料功能梯度板的自由振动模型与尺度效应

摘要

A free vibration model of orthotropic functionally graded carbon nanotubes-reinforced com-posite plates is developed based on a new modified couple stress theory.The present model can describe the scale effects with only one material length parameter incorporated.The equations of motion are derived based on the Hamilton's principle in conjunction with the first-order shear deformation theory.A simply supported square plate is taken as the illustrative example and analytical solved.The effects of plate thickness,volume fraction and distribution of carbon nanotubes on the natural frequencies of the plate are investigated.Numerical results indicate that the natural frequencies predicted by the present model are always higher than those by the Mindlin plate model of classical elasticity theory.The differences between the natural frequencies predicted by the two models are significant when the plate thickness approaches the material length parameter,but are diminishing with the increase of the plate thickness.%基于一种新的各向异性修正偶应力理论,建立了碳纳米管增强复合材料功能梯度板的自由振动模型.该模型能够描述尺度效应,且仅包含一个尺度参数.基于一阶剪切变形理论和哈密顿原理推演了板的运动微分方程,并以四边简支板为例给出了自振频率的解析解.讨论了板的几何尺寸、碳纳米管体分比含量和分布方式等因素对板的自振频率的影响.结果表明,本文模型所预测的板的自振基频总是高于经典弹性理论的Mindlin板模型的预测结果,两者间的差异在板的几何尺寸接近尺度参数的值时非常明显,且会随着板的几何尺寸的增大而逐渐消失.