On the free vibration characteristics of postbuckled third-order shear deformable FGM nanobeams including surface effects
On the free vibration characteristics of postbuckled third-order shear deformable FGM nanobeams including surface effects
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DOI：
10.1016/j.compstruct.2014.11.033
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发表时间：
2015-03
影响因子：
6.3
通讯作者：
S. Sahmani;M. M. Aghdam-M.;M. Bahrami
中科院分区：
文献类型：
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作者：
S. Sahmani;M. M. Aghdam-M.;M. Bahrami
基于一种有效的数值求解方法，考虑表面自由能的影响，研究了功能梯度材料(FGM)三阶剪切变形纳米棒在屈曲后区域的自由振动响应。将Gurtin-Murdoch弹性理论与von Karman几何非线性相结合应用于经典的三阶剪切变形梁理论。为了考虑纳米梁表面的平衡条件，假定体法向应力沿厚度呈三次分布。假定材料性能在厚度方向上按幂分布进行分级。利用哈密顿原理，导出了非经典的非线性运动控制微分方程和相应的边界条件。然后根据Chebyshev-Gauss-Lobatto网格点，用广义微分求积法离散控制方程和求解区域。随后，利用伪弧长延拓技术对非线性问题进行求解。结果表明，与前屈曲区域相比，在后屈曲区域，随着材料性质梯度指数的增大，功能梯度梁的固有频率降低。此外，表面效应对厚度较小的屈曲功能梯度材料的振动特性有较大的影响。
On the basis of an efficient numerical solution methodology, the free vibration response of third-order shear deformable nanobeams made of functionally graded materials (FGMs) around the postbuckling domain is investigated incorporating the effects of surface free energy. Gurtin–Murdoch elasticity theory in conjunction with von Karman geometric nonlinearity is implemented into the classical third-order shear deformation beam theory. In order to consider balance conditions on the surfaces of nanobeam, it is assumed that the bulk normal stress is distributed cubically through the thickness. The material properties are assumed to be graded in the thickness direction based on power-law distribution. Using Hamilton’s principle, the non-classical nonlinear governing differential equations of motion and associated boundary conditions are derived. Then generalized differential quadrature method is employed to discretize the governing equations and solution domain according to the Chebyshev–Gauss–Lobatto grid points. Subsequently, the pseudo-arc length continuation technique is utilized to solve the nonlinear problem. It is demonstrated that in contrast to the prebuckling domain, in the postbuckling domain, the natural frequency of FGM nanobeam decreases by increasing the value of material property gradient index. Also, it is revealed that the surface effect plays more important role on the vibration characteristics of the buckled FGM nanobeams with lower thicknesses.