微细观尺度下欧拉梁的力学模型及有限元分析

Mechanical model and finite element analysis of Euler beam in a micro scale

基于修正偶应力理论建立了仅含一个尺度参数且适用于各种边界条件的微尺度欧拉梁模型。基于哈密顿原理推导了微尺度欧拉梁的平衡微分方程,该方程与经典梁的平衡微分方程具有相似的形式,只是在弯曲刚度中多了一项与尺度效应有关的项,可直接用于分析和解释多尺度问题。提出了一种模量折算策略,从而利用经典梁单元即可完成对微细观尺度下的梁的弯曲、动力和稳定问题的求解。算例结果表明,在微细观尺度下梁结构将表现出比宏观状态下更强的抗弯刚度,即本文模型能捕捉到尺度效应。进一步的研究则指出,几何尺寸的大小是尺度效应的决定性因素。

A model of micro Euler beam containing only one internal material length scale parameter and applieing to arbitrary boundary conditions was proposed based on the modified couple stress theory. Equilibrium differential equation of micro Euler beam was deduced using the Hamilton＇s principle,of which the form was similar to the classical model. The only difference was an additional term in bending rigidity associated with material length scale parameter. The equation could be used directly to analyze and explain multi-scale problems. Therefore,a strategy reducing modulus was proposed to solve the problems such as bending,vibration and buckling of micro beams by classical beam elements. The numerical results showthat the bending rigidity of the Euler beam in a micro scale is higher than that in a macro scale,indicating that model presented in this paper can capture the scale effects. It is found that the geometric size is a significant factor of the scale effects.