具有尺度效应的微梁静态弯曲分析  

Micro-beam Static Flex with Scale Effect

作　　者：黄文君 卿海 Huang Wenjun;Qing Hai(State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China)

机构地区：[1]南京航空航天大学机械结构力学及控制国家重点实验室,江苏南京210016

出　　处：《甘肃科学学报》2019年第1期5-11,共7页Journal of Gansu Sciences

基　　金：机械结构力学及控制国家重点实验室开放基金(MCMS-0217G02)

摘　　要：基于修正偶应力理论和表面弹性理论,提出了一种微尺度下的均匀梁模型,通过表面弹性理论和广义Young-Laplace方程引入剪切变形。微梁的总应变能除了基于经典弹性理论的应变能外,还考虑了由旋转梯度和表面效应引起的应变能。利用Hamilton原理,推导得到了微梁的平衡方程和边界条件。使用微分求积单元法研究了微梁在不同边界条件下的静态弯曲问题,并把简支条件下微梁弯曲挠度的解析解与数值解进行对比。结果表明,由微分求积单元法得到的数值解与解析解得到的结果基本一致,验证了数值解的正确性。分析了偶应力、表面效应和微梁的厚度对微梁弯曲挠度的影响。该模型得到的微梁的弯曲挠度与经典弹性理论得到的结果相比具有显著的不同,证明了微梁尺度效应的存在。A uniform beam mode under micro-scale is put forward based on modified couple stress theory and surface elasticity theory,and shear deformation is introduced based on surface elasticity theory and generalized Young-Laplace formula.Except for the strain energy based on classical elastic theory,the strain energy caused by rotation gradient and surface effect is also considered in the total strain energy of the micro-beam.The equilibrium equation and boundary condition of the micro-beam is deduced by making use of Hamilton principle.The static flex problem of the micro-beam under different boundary conditions is researched by making use of differential quadrature element method(DQEM),and the analytical solution of the micro-beam bending deflection under simply support condition is compared with arithmetic solution.The result shows that,the results gained by arithmetic solution gotten by differential quadrature element method and analytical solution are basically the same,and the correctness of the arithmetic solution is verified.The effects of couple stress,surface effect and micro-beam to micro-beam bending deflection is analyzed.Compare the bending deflection of the micro-beam gained by this model and the result gained by classical elasticity theory,there's a significant difference,which verify the existence of micro-beam scale effect.

关 键 词：尺度效应 抛物线剪切变形梁理论 修正偶应力理论 表面弹性理论 微分求积单元法 

分 类 号：O342[理学—固体力学]