准三维功能梯度微梁的尺度效应模型及微分求积有限元

A Size⁃Dependent Quasi⁃3D Functionally Graded Microbeam Model and Related Differential Quadrature Finite Elements

基于修正的偶应力理论与四参数高阶剪切⁃法向伸缩变形理论,提出了一种具有尺度依赖性的准三维功能梯度微梁模型,并应用于小尺度功能梯度梁的静力弯曲和自由振动分析中.采用第二类Lagrange方程,推导了微梁的运动微分方程及边界条件.针对一般边值问题,构造了一种融合Gauss⁃Lobatto求积准则与微分求积准则的2节点16自由度微分求积有限元.通过对比性研究,验证了理论模型以及求解方法的有效性.最后,探究了梯度指数、内禀特征长度、几何参数及边界条件对微梁静态响应与振动特性的影响.结果表明,该文所发展的梁模型及微分求积有限元适用于研究各种长细比的功能梯度微梁的静/动力学问题,引入尺度效应会显著地改变微梁的力学特性.

A size⁃dependent quasi⁃3D functionally graded(FG)microbeam model was presented within the combined framework of the modified couple stress theory and a 4⁃unknown higher⁃order shear and normal de⁃formation theory.Then the model was applied to analyze the static bending and free vibration of FG mi⁃crobeams.With the 2nd Lagrange equation,the corresponding motion equations and the appropriate boundary conditions were obtained.A 2⁃node 16DOF differential quadrature finite element combining the Gauss⁃Lobatto quadrature rule with the differential quadrature rule was constructed to handle the general static/dynamic boundary value problems of FG microbeams.A comparison study was performed to show the efficacy of the proposed theoretical model and solution method.Finally,the effects of the gradient index,the intrinsic length scale parameter,the geometrical parameters and the boundary conditions on the static and dynamic characteris⁃tics of FG microbeams were examined.Numerical results reveal that the developed beam model and element are applicable to the analysis of mechanical behaviors of FG microbeams with various slenderness ratios.Besides,introduction of the couple stress effect can significantly change the static and dynamic characteristics of FG mi⁃crobeams.