功能梯度多层石墨烯片增强复合材料梁的非线性自由振动  

Nonlinear free vibration of functionally graded multilayer graphene platelet-reinforced composite beams

作　　者：吕颖 张静 李联和[1,2] LU Ying;ZHANG Jing;LI Lianhe(College of Mathematics Science,Inner Mongolia Normal University,010022 Hohhot,China;Inner Mongolia Center of Applied Mathematics,010022 Hohhot,China)

机构地区：[1]内蒙古师范大学数学科学学院,呼和浩特010022 [2]内蒙古应用数学中心,呼和浩特010022

出　　处：《应用力学学报》2025年第2期415-424,共10页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金资助项目(No.11962026);内蒙古自然科学基金资助项目(No.2021MS01013);内蒙古师范大学基本科研业务费专项资金资助项目(No.2022JBZD010,2022JBXC013);内蒙古师范大学研究生科研创新基金资助项目(No.CXJJS22101)。

摘　　要：为了研究非线性弹性基础上功能梯度多层石墨烯片增强复合材料(graphene platelets reinforced composite,GPLRC)梁在轴向力和温度场共同作用下的非线性自由振动问题。基于虚功原理,采用一阶剪切变形梁理论和冯卡门非线性应变位移关系,建立了梁的非线性控制方程,并将3个方程简化为2个关于挠度和转角的非线性方程,利用二次摄动法进行离散,得到各阶摄动方程,逐阶求出渐进解,获得两端简支边界条件下频率的表达式。通过参数分析,探究了石墨烯片(graphene platelets,GPLs)的分布模式、几何形状、重量分数,以及弹性基础刚度参数、温度和轴向力对非线性自由振动的影响。计算结果表明:GPL质量分数越高增强效果越明显,采用GPL表面积更大、单层GPL较少、靠近顶部和底部的位置添加更多的GPL、增大线性弹性基础刚度都可以使固有频率增大,从而使梁的非线性与线性频率之比减小。非线性弹性基础刚度增大、温度升高和轴向压力增大会使非线性与线性频率之比增大。We aimed to study the nonlinear free vibration of a functionally graded multilayer graphene platelet-reinforced composite(GPLRC)beam on a nonlinear elastic foundation under the combined effect of axial forces and temperature field.Based on the principle of virtual displacements,the nonlinear governing equations of beams are established by using the first-order shear deformation beam theory and von Kármán type nonlinear strain-displacement relationship.The three equations are simplified into two nonlinear equations about deflection and rotation,which are discretized by using the two-step perturbation method to obtain the perturbation equations of each order,and the asymptotic solutions are derived step by step to obtain the expressions of the frequencies under the simply supported boundary conditions at both ends.Through the parameter analysis,the effects of the graphene platelet(GPLs)distribution pattern,geometry and size,weight fraction,elastic foundation stiffness parameter,temperature and axial force on the nonlinear free vibration are explored.The calculation results show that the higher GPL weight fraction,then the more obvious the enhancement effect.Using larger surface area of GPL and less single layer GPL,adding more GPL near at the top and the bottom,and increasing the linear elastic foundation stiffness parameter can improve the natural frequency better,thus reducing nonlinear frequency-to-linear frequency ratio.The ratio of nonlinear frequency to linear frequency increases with the increase of nonlinear elastic foundation stiffness,temperature,and axial compression force.

关 键 词：非线性自由振动 二次摄动法 石墨烯片 功能梯度材料 

分 类 号：O343[理学—固体力学]