梯度型非局部高阶梁理论与非局部弯曲新解法

陈玲; 沈纪苹; 李成; 刘鑫培

doi:10.6052/0459-1879-15-170

梯度型非局部高阶梁理论与非局部弯曲新解法

doi: 10.6052/0459-1879-15-170

苏州大学城市轨道交通学院, 江苏苏州 215131

基金项目: 国家自然科学基金(11202145),苏州大学“车吴学者计划”项目和江苏省高校自然科学基金(14KJB460026)资助项目.

详细信息

通讯作者:
李成,副教授,主要研究方向:微纳米力学、复合材料力学.E-mail:licheng@suda.edu.cn

中图分类号: O331
计量
- 文章访问数: 983
- HTML全文浏览量: 70
- PDF下载量: 914
- 被引次数: 0
出版历程
- 收稿日期: 2015-05-12
- 修回日期: 2015-08-17
- 刊出日期: 2016-01-18

GRADIENT TYPE OF NONLOCAL HIGHER-ORDER BEAM THEORY AND NEW SOLUTION METHODOLOGY OF NONLOCAL BENDING DEFLECTION

School of Urban Rail Transportation, Soochow University, Suzhou 215131, Jiangsu, China

摘要

摘要: 针对文献中关于纳米结构刚度受非局部效应影响趋势的不一致预测,基于梯度型的非局部微分本构模型,利用迭代法及泰勒展开法求得了非局部高阶应力的无穷级数表达,非局部应力相当于经典弯曲应力与非局部挠度的逐阶梯度之和. 然后推导了梯度型非局部高阶梁弯曲的挠曲轴微分方程,并结合正则摄动思想,求解了非局部挠度的表达式. 最后给出数值算例,具体量化挠度受非局部尺度因子的影响大小. 研究表明:相比于其经典值,纳米结构的非局部弯曲挠度可呈现出或增大或减小或不变的趋势,考虑梯度型非局部高阶应力降低或提高或不影响纳米结构的刚度,具体结果依赖于外载和边界约束的类型. 算例显示外载形式和边界约束条件均各自独立地影响着纳米结构的非局部弯曲挠度,同时首次观察到非局部最大弯曲挠度的位置可能受非局部尺度因子的影响. 研究结论解决了非局部弹性力学应用于纳米结构的若干疑点,并为理论的发展和优化提供支持.
- 纳米结构 /
- 高阶应力 /
- 梯度型非局部理论 /
- 正则摄动法
Abstract: Di erent predictions were found in di erent literatures about the trend of nonlocal e ects on nanostructural sti ness. By employing an iterative method and the Taylor expansion method, an infinite series for nonlocal high-order stress is achieved based on the gradient-type of nonlocal di erential constitutive model. The nonlocal stress consists of the classical bending stress and each order gradient of nonlocal deflection. Consequently, the di erential equation of the bending deflection curve for nonlocal high-order beam is derived. The nonlocal deflection is determined via the regular perturbation method. Some numerical examples are provided to reveal and quantize the e ects of nonlocal scale factor on bending deflection. It is shown that compared with the corresponding classical results, the nonlocal bending deflections of nanostructure may increase, decrease or remain unchanged. The sti ness of nanostructures can be reduced or enhanced or the same as classical structures by considering the gradient-type of nonlocal high-order stress e ect, and the trend depends on external loads and boundary constraints which are found to play significant roles independently in nonlocal bending of nanostructures. Moreover, it is observed for the first time that the position of maximum nonlocal deflection may be influenced by nonlocal scale factor. The present studies are expected to solve the problems in the application of nonlocal elasticity theory to nanostructures, and further provide supports for the development and optimization of such theory.
- nanostructure /
- high-order stress /
- gradient-type of nonlocal theory /
- regular perturbation method

HTML全文

参考文献(21)

1 Eringen AC, Edelen DGB. On nonlocal elasticity. International Journal of Engineering Science, 1972, 10: 233-248

2 Eringen AC. On di erential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 1983, 54: 4703-4710

3 郑长良. 非局部弹性直杆振动特性及Eringen 常数的一个上限. 力学学报, 2005, 37(6): 796-798 (Zheng Changliang. The free vibration characteristics of nonlocal continuum bar and an upper bound of material constant in Eringen's nonlocal model. Chinese Journal of Theoretical and Applied Mechanics, 2005, 37(6): 796-798 (in Chinese))

4 Liang J,Wu SP, Du SY. The nonlocal solution of two parallel cracks in functionally graded materials subjected to harmonic anti-plane shear waves. Acta Mechanica Sinica, 2007, 23(4): 427-435

5 易大可, 王自强. 能量非局部模型和新的应变梯度理论. 力学学报, 2009, 41(1): 60-66 (Yi Dake, Wang Tzuchiang. Energy nonlocal model and new strain gradient theory. Chinese Journal of Theoretical and Applied Mechanics, 2009, 41(1): 60-66 (in Chinese))

6 Peddieson J, Buchanan GG, McNitt RP. Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 2003, 41: 305-312

7 Wang Q, Varadan VK. Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Materials & Structures,2006, 15: 659-666

8 Lu P, Lee HP, Lu C, et al. Application of nonlocal beam models for carbon nanotubes. International Journal of Solids and Structures,2007, 44: 5289-5300

9 Hu YG, Liew KM, Wang Q, et al. Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes. Journal of the Mechanics and Physics of Solids, 2008, 56: 3475-3485

10 Kiani K. Vibration analysis of elastically restrained double-walled carbon nanotubes on elastic foundation subjected to axial load using nonlocal shear deformable beam theories. International Journal of Mechanical Sciences, 2013, 68: 16-34

11 Li YS, Cai ZY, Shi SY. Buckling and free vibration of magnetoelectroelastic nanoplate based on nonlocal theory. Composite Structures,2014, 111: 522-529

12 Ke LL, Wang YS, Yang J, et al. Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory. Acta Mechanica Sinica, 2014, 30(4): 516-525

13 Hosseini-Hashemi S, Kermajani M, Nazemnezhad R. An analytical study on the buckling and free vibration of rectangular nanoplates using nonlocal third-order shear deformation plate theory. European Journal of Mechanics—A/Solids, 2015, 51: 29-34

14 Lim CW. On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection. Applied Mathematics and Mechanics, 2010, 31:37-54

15 Lim CW, Yang Y. Wave propagation in carbon nanotubes: Nonlocal elasticity-induced sti ness and velocity enhancement e ects. Journal of Mechanics of Materials and Structures, 2010, 5: 459-476

16 Lim CW, Yang Q. Nonlocal thermal-elasticity for nanobeam deformation: exact solutions with sti ness enhancement e ects. Journal of Applied Physics, 2011, 110: 013514

17 Wang L. A modified nonlocal beam model for vibration and stability of nanotubes conveying fluid. Physica E, 2011, 44: 25-28

18 Yang Q, Lim CW. Thermal e ects on buckling of shear deformable nanocolumns with von Kármán nonlinearity based on nonlocal stress theory. Nonlinear Analysis: Real World Applications, 2012,13: 905-922

19 Lim CW, Xu R. Analytical solutions for coupled tension-bending of nanobeam-columns considering nonlocal size e ects. Acta Mechanica,2012, 223: 789-809

20 Yu YM, Lim CW. Nonlinear constitutive model for axisymmetric bending of annular graphene-like nanoplate with gradient elasticity enhancement e ects. Journal of Engineering Mechanics-ASCE,2013, 139: 1025-1035

21 Islam ZM, Jia P, Lim CW. Torsional wave propagation and vibration of circular nanostructures based on nonlocal elasticity theory. International Journal of Applied Mechanics, 2014, 6: 1450011

施引文献

资源附件(0)

访问统计