The Variational Principle for Multi-Layer Timoshenko Beam Systems Based on the Simplified
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摘要: 材料特征尺寸与其内禀尺寸相当时,材料表现出明显的尺寸效应.基于简化的应变梯度理论,通过半逆法,本文给出多层简化应变梯度Timoshenko梁的变分原理,通过最小总势能原理导出系统的边界条件并对其低阶和高阶边界条件进行讨论,随后给出简支梁系统屈曲载荷和振动频率的Rayleigh(瑞利)解.通过双层梁系统的振动分析算例得到内禀尺寸、长径比等因素对梁系统振动频率的影响.该文构造的Rayleigh解有望对其他数值方法,如有限元法、传递矩阵法等,提供一定的参考和对比.Abstract: The mechanical properties of member materials exhibit notable size effects when the characteristic sizes of the members are comparable to their instinct length parameters. A variational formulation of the nanosize multi-layer Timoshenko beam problem was developed via the semi-inverse method within the context of the simplified strain gradient theory. This method was fit for determining all the possible low-order and high-order boundary condtions directly from the governing equations of the system, according to the minimum total potential energy principle. In turn, the Rayleigh solutions of buckling load and free vibration frequencies of the simply supported beam system were given. The numerical simulations indicate the prominent effects of the instinct length parameters and aspect ratios on the free vibration frequencies of the double-layer beam systems. As a possible benchmark for the later numerical studies with the transfer matrix method or the finite element method, the present Rayleigh solutions of buckling load and free vibration frequencies of the multi-layer beam systems will make good sense.
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Key words:
- strain gradient theory /
- buckling /
- vibration /
- variational principle
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[1] Peddieson J, Buchanan G R, McNitt R P. Application of nonlocal continuum models to nanotechnology[J]. International Journal of Engineering Science,2003,41(3/5): 305-312. [2] Wang K F, Wang B L, Kitamura T. A review on the application of modified continuum models in modeling and simulation of nanostructures[J]. Acta Mechanica Sinica,2015: 1-18. doi: 10.1007/s10409-015-0508-4. [3] Adali S. Variational principles for transversely vibrating multiwalled carbon nanotubes based on nonlocal Euler-Bernoulli beam model[J]. Nano Letters,2009,9(5): 1737-1741. [4] Adali S. Variational formulation for buckling of multi-walled carbon nanotubes modelled as nonlocal Timoshenko beams[J]. Journal of Theoretical and Applied Mechanics,2012,50(1): 321-333. [5] Kucuk I, Sadek I S, Adali S. Variational principles for multiwalled carbon nanotubes undergoing vibrations based on nonlocal Timoshenko beam theory[J]. Journal of Nanomaterials,2010,2010(3): 461252. doi: 10.1155/2010/461252. [6] 李明, 郑慧明. 小尺度对振动简支单层碳纳米管边界条件的影响[J]. 固体力学学报, 2014,35(S): 6-8.(LI Ming, ZHENG Hui-ming. Small scale effect on boundary conditions of vibrating simply supported single-walled carbon nanotubes[J]. Chinese Journal of Solid Mechanics,2014,35(S): 6-8.(in Chinese)) [7] 姚征, 郑长良. 积分形式非局部本构关系的界带分析方法[J]. 应用数学和力学, 2015,36(4): 362-370.(YAO Zheng, ZHENG Chang-liang. Inter-belt analysis of the integral-form nonlocal constitutive relation[J]. Applied Mathematics and Mechanics, 2015,36(4): 362-370.(in Chinese)) [8] HE Ji-huan. Variational approach to (2+1)-dimensional dispersive long water equations[J]. Physics Letters A,2005,335(2/3): 182-184. [9] Kumar D, Heinrich C, Waas A M. Buckling analysis of carbon nanotubes modeled using nonlocal continuum theories[J]. Journal of Applied Physics,2008,103(7): 073521. [10] LI Xian-fang, WANG Bao-lin, LEE Kang-yong. Size effects of the bending stiffness of nanowires[J]. Journal of Applied Physics,2009,105(7): 074306. [11] WANG Bing-lei, ZHAO Jun-feng, ZHOU Shen-jie. A micro scale Timoshenko beam model based on strain gradient elasticity theory[J]. European Journal of Mechanics—A/Solids,2010,29(4): 591-599. [12] Nojoumian M A, Salarieh H. Comment on “A micro scale Timoshenko beam model based on strain gradient elasticity theory”[J]. European Journal of Mechanics—A/Solids,2013. doi: 10.1016/j.euromechsol.2013.12.003. [13] Challamel N. Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams[J]. Composite Structures,2013,105: 351-368. [14] XU Xiao-jian, DENG Zi-chen. Variational principles for the buckling and vibration of MWCNTs modeled by strain gradient theory[J]. Applied Mathematics and Mechanics(English Edition),2014,35(9): 1115-1128. [15] KONG Sheng-li, ZHOU Shen-jie, NIE Zhi-feng, WANG Kai. Static and dynamic analysis of micro beams based on strain gradient elasticity theory[J]. International Journal of Engineering Science,2009,47(4): 487-498. [16] Akgz B, Civalek . Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory[J]. Archive of Applied Mechanics,2012,82(3): 423-443. [17] WANG Li-feng, GUO Wan-lin, HU Hai-yan. Group velocity of wave propagation in carbon nanotubes[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science,2008,464(2094): 1423-1438.
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