基于非局部弹性应力场理论的纳米尺度效应研究:纳米梁的平衡条件、控制方程以及静态挠度

C·W·林

doi:10.3879/j.issn.1000-0887.2010.01.005

基于非局部弹性应力场理论的纳米尺度效应研究:纳米梁的平衡条件、控制方程以及静态挠度

doi: 10.3879/j.issn.1000-0887.2010.01.005

C·W·林

香港城市大学建筑系,香港

基金项目: 香港特别行政区研究资助委员会资助项目(RGCHKSAR,CityU117406)

详细信息

中图分类号: O343
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出版历程
- 收稿日期: 2009-09-29
- 修回日期: 2009-11-19
- 刊出日期: 2010-01-15

On the Truth of Nanoscale for Nanobeams Based on Nonlocal Elastic Stress Field Theory: Equilibrium,Governing Equation and Static Deflection

C. W. Lim

Department of Building and Construction, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, P. R. China

摘要

摘要: 该文成功地解答了3个关于非局部应力理论用于纳米梁的问题：(ⅰ) 在绝大多数研究中，非局部效应增加导致纳米结构体刚度下降，其现象表现为弯曲挠度增加，固有频率减少，屈曲载荷下降，但为什么Eringen 的非局部弹性理论给出了完全相反的结论；(ⅱ) 为什么在某些研究结果中，非局部效应消失或是对研究结果无影响，比如纳米悬臂梁在集中载荷作用下的弯曲挠度； (ⅲ) 在高阶控制方程中，为什么高阶边界条件不存在．通过应用非局部弹性理论和精确变分原理分析纳米梁的弯曲问题，推导出全新的平衡条件、控制方程、边界条件和静态响应．这些方程和条件包含了与之前的相关研究结果符号相反的高阶微分项，这一差别导致了纳米效应对结构体的影响结果完全相反．还证明之前为大家所公认的纳米梁静态或动态平衡条件实际上没有达到平衡，只有用等效弯矩代替非局部弯矩时，才可达到平衡．这些结论通常是可以被其它方法，比如应变梯度理论、耦合应力模型以及相关实验所证明．
- 弯曲 /
- 等效非局部弯矩 /
- 纳米梁 /
- 纳米力学 /
- 纳米尺度 /
- 非局部应力 /
- 应变梯度
Abstract: This paper has success fully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams: (ⅰ) why does the presence of in creasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i. e. increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise? (ⅱ) the in triguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, eg. bending deflection of a cantilevernanobeam with a point load at its tip; and (ⅲ) the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocale lasticity field theory in nanomechanics and an exact variational principal approach, the new equilibrium conditions, domain governing differential equation and boundary conditions for bending of nanobeams were derived. These equations and conditions involved essential higher-orderd ifferential terms which were oppositein sign with respect to the previous studies in statics and dynamics of nonlocal nano-structures. The difference in higher-order terms resulted in reverse trends of nanoscale effects with respect to the conclusion. Effectively, this paper reported new equilibrium conditions, governing differential equation and boundary conditions and the true basic static responses for bending of nanobeams. It also concludes that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective non local bending moment. The conclusions above were substantiated, in a general sense, by other approaches in nanostructuralmodels such as strain gradient theory, modified couple stress models and experiments.
- bending /
- effective nonlocal bending moment /
- nanobeam /
- nanomechanics /
- nanoscale /
- nonlocal elastic stress /
- strain gradient

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