考虑尺度依赖的平面正交各向异性功能梯度微梁的自由振动分析

基于一种新修正偶应力理论建立了微尺度平面正交各向异性功能梯度梁的自由振动模型。模型中包含两个材料尺度参数,能够分别描述两个正交方向上不同程度的尺度效应。当梁的几何尺寸远大于材料尺度参数时,本文模型亦可自动退化为相应的传统宏观模型。基于哈密顿原理推导了运动控制方程并以简支梁的自由振动为例分析了几何尺寸、功能梯度变化指数等对尺度效应产生的影响。算例结果表明:采用本文模型所预测的梁自振频率总是大于传统理论的结果,即捕捉到了尺度效应。尺度效应会随着梁几何尺寸的增大而逐渐减弱并在几何尺寸远大于尺度参数时消失;高阶自振频率所体现出的尺度效应较低阶自振频率更加明显。此外,功能梯度变化指数对尺度效应也有一定的影响。

Size-dependent free vibration analysis of plane orthotropic functionally graded micro-beams

A size-dependent model for the free vibration of plane orthotropic functionally graded micro-beams was developed on the basis of a new modified couple stress theory.The model contains two material length scale parame-ters,which enables it to separately represent the different scale effects in two orthogonal directions.The present model can be degenerated to classical macroscopic model when the geometrical size of the beam is much larger than the material length parameter.The governing equations were derived through Hamilton's principle.A simply sup-ported micro-beam was taken as the illustrative example and analytical solved.The influences of geometrical size and power law index on the scale effects were analyzed.Numerical results indicate that the natural frequencies of the mi-cro-beam predicted by the present model are always greater than those predicted by the classical FG beam model, i.e.the scale effects are captured.The scale effects will be gradually weaken with the increasing of the geometrical size of the beam,and diminish when the geometrical size is much larger than the material length parameter.The scale effects reflected by higher order natural frequencies are more apparent than that reflected by lower order natu-ral frequencies.In addition,the power law index also has a specific influence on the scale effects.