基于Cosserat理论的四边简支自由振动微平板尺度效应研究

微观下材料内部结构将极大地影响材料的力学性能,对微纳米器件中典型的微平板结构尺度效应进行研究具有十分重要的意义.论文基于Cosserat理论推导出了微平板自由振动的微分方程,并根据四边简支边界条件假设振型函数,给出了固有频率的计算公式,对不同尺寸微平板固有频率的尺度效应进行了仿真分析.结果表明,考虑了尺度效应的微平板自由振动固有频率要高于经典理论中的固有频率.当特征长度与微平板厚度大小相当时,微平板固有频率表现出明显的尺度效应,并随着特征长度的增加而增大.同时,自由振动的尺度效应将随着微平板厚度的减小而逐渐增强,振动模态及长宽比不影响尺度效应.论文的研究将为微结构与系统的应用提供一定的理论基础.

ON THE SIZE EFFECTS IN A FREELY-VIBRATING MICRO-PLATE WITH THE FOUR EDGES SIMPLY-SUPPORTED BASED ON THE COSSERAT THEORY

The mechanical properties of a certain material are always dependent on its internal structures,and thus,the investigation on the size effects of microstructure dynamics is greatly significant.The differential equation of a micro-plate under free vibration is presented using the modified Cosserat theory in this paper,and the formula of natural frequencies is obtained that meet SSSS boundary conditions,then the size effects of natural frequencies of micro-plates with different dimensions are analyzed.The results show that the natural frequency taking the size effects into account is higher than that based on the classic elastic theory.The size effects becomes more significant on the free vibrating natural frequency when the characteristic length and the micro-plates thickness come closer,which results in the increase of the natural frequencies along with the increase of characteristic length.Additionally,the size effect of micro-plate gradually strengthens with decrease of thickness,and the aspect ratio and vibration modes don’t influence the size effects.The study presented is of fundament importance for the application of microstructure system.