On the Truth of Nanoscale for Nanobeams Based on Nonlocal Elastic Stress Field Theory: Equilibrium,Governing Equation and Static Deflection
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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.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.
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Key words:
- bending /
- effective nonlocal bending moment /
- nanobeam /
- nanomechanics /
- nanoscale /
- nonlocal elastic stress /
- strain gradient
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[1] Iijima S.Helical microtubules of graphitic carbon[J].Nature,1991,354: 56-58. doi: 10.1038/354056a0 [2] Treacy M M J,Ebbesen T W,Gibson T M.Exceptionally high Young’s modulus observed for individual carbon nanotubes[J].Nature,1996,381: 680-687. [3] Ball P.Roll up for the revolution[J].Nature,2001,414: 142-144. doi: 10.1038/35102721 [4] Iijima S,Brabec C,Maiti A,et al.Structural flexibility of carbon nanotubes[J].J Chem Phys,1996,104: 2089-2092. doi: 10.1063/1.470966 [5] Yakobson B I,Campbell M P,Brabec C J,et al.High strain rate fracture and C-chain unraveling in carbon nanotubes[J].Comput Mater Sci,1997,8: 341-348. [6] He X Q,Kitipornchai S,Liew K M.Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for Van der Waals interaction[J].J Mech Phys Solids,2005,53: 303-326. doi: 10.1016/j.jmps.2004.08.003 [7] Yakobson B I,Brabec C J,Bernholc J.Nanomechanics of carbon tubes: instabilities beyond linear range[J].Phys Rev Lett,1996,76: 2511-2514. doi: 10.1103/PhysRevLett.76.2511 [8] Ru C Q.Effective bending stiffness of carbon nanotubes[J].Phys Rev B,2000,62: 9973-9976. doi: 10.1103/PhysRevB.62.9973 [9] Ru C Q.Elastic buckling of single-walled carbon nanotubes ropes under high pressure[J].Phys Rev B,2000,62: 10405-10408. doi: 10.1103/PhysRevB.62.10405 [10] Zhang P,Huang Y,Geubelle P H,et al.The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials[J].I J Solids Struct,2002,39: 3893-3906. [11] Gurtin M E,Murdoch A.A continuum theory of elastic material surfaces[J].Archives of Rational Mechanics and Analysis,1975,57: 291-323. [12] Gurtin M E,Murdoch A I.Effect of surface stress on wave propagation in solids[J].J Appl Phys,1976,47: 4414-4421. doi: 10.1063/1.322403 [13] He L H,Lim C W.On the bending of unconstrained thin crystalline plates caused by change in surface stress[J].Surface Sci,2001,478(3): 203-210. doi: 10.1016/S0039-6028(01)00953-0 [14] He L H,Lim C W,Wu B S.A continuum model for size-dependent deformation of elastic films of nano-scale thickness[J].I J Solids Struct,2004,41: 847-857. [15] Lim C W,He L H.Size-dependent nonlinear response of thin elastic films with nano-scale thickness[J].I J Mech Sci,2004,46(11): 1715-1726. [16] Lim C W,Li Z R,He L H.Size dependent,nonuniform elastic field inside a nano-scale spherical inclusion due to interface stress[J].I J Solids Struct,2006,43: 5055-5065. [17] Wang Z Q,Zhao Y P,Huang Z P.The effects of surface tension on the elastic properties of nano structures[J].I J Engineering Science,2009.doi: 10.1016/j.ijengsci.2009.07.007 [18] Eringen A C.Linear theory of nonlocal elasticity and dispersion of plane waves[J].I J Engineering Science,1972,10(5): 425-435. [19] Eringen A C.Nonlocal polar elastic continua[J].I J Engineering Science,1972,10(1): 1-16. [20] Eringen A C.On nonlocal fluid mechanics[J].I J Engineering Science, 1972,10(6): 561-575. [21] Eringen A C,Edelen D G B.On nonlocal elasticity[J].I J Engineering Science,1972,10(3): 233-248. [22] Eringen A C.Linear theory of nonlocal microelasticity and dispersion of plane waves[J].Lett Appl Engng Sci,1973,1(2): 129-146. [23] Eringen A C.On nonlocal microfluid mechanics[J].I J Engineering Science,1973,11(2): 291-306. [24] Eringen A C.Theory of nonlocal electromagnetic elastic solids[J].J Math Phys,1973,14(6): 733-740. doi: 10.1063/1.1666387 [25] Eringen A C.Theory of nonlocal thermoelasticity[J].I J Engineering Science,1974,12: 1063-1077. [26] Eringen A C.Memory-dependent nonlocal thermoelastic solids[J].Lett Appl Engng Sci,1974,2(3): 145-149. [27] Eringen A C.Memory dependent nonlocal elastic solids[J].Lett Appl Engng Sci,2(3): 145-159. [28] Eringen A C.Nonlocal elasticity and waves[C]Thoft-Christensen P.Continuum Mechanics Aspect of Geodynamics and Rock Fracture Mechanics.Netherlands: Kluwer Academic Publishers Group,1974:81-105. [29] Eringen A C.Continuum Physics[M].Vol Ⅱ,Sect 1.3.New York: Academic Press,1975. [30] Eringen A C.Nonlocal Polar Field Theories[M].New York: Academic,1976. [31] Eringen A C.Mechanics of Continua[M].2nd ed.Melbourne,FL: Krieger,1980. [32] Eringen A C.On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves[J].J Appl Phys,1983,54(9): 4703-4710. doi: 10.1063/1.332803 [33] Eringen A C.Theory of nonlocal piezoelectricity[J].J Math Phys,1984,25(3): 717-727. doi: 10.1063/1.526180 [34] Eringen A C.Point charge,infrared dispersion and conduction in nonlocal piezoelectricity[C]Maugin G A.The Mechanical Behavior of Electromagnetic Solid Continua.North-Holland: Elsevier Science,1984:187-196. [35] Eringen A C.Nonlocal Continuum Field Theories[M].New York: Springer,2002. [36] Peddieson J,Buchanan G R,McNitt R P.Application of nonlocal continuum models to nanotechnology[J].I J Engineering Science,2003,41(3/5): 305-312. [37] Sudak L J.Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics[J].J Appl Phys,2003,94(11): 7281-7287. doi: 10.1063/1.1625437 [38] Nix W,Gao H.Indentation size effects in crystalline materials: a law for strain gradient plasticity[J].Journal of the Mechanics and Physics of Solids,1998,46(3): 411-425. doi: 10.1016/S0022-5096(97)00086-0 [39] Lam D C C,Yang F,Chong A C M,et al.Experiments and theory in strain gradient elasticity[J].Journal of the Mechanics and Physics of Solids,2003,51: 1477-1508. doi: 10.1016/S0022-5096(03)00053-X [40] Li C Y,Chou T W.Vibrational behaviors of multi-walled carbon nanotube-based nanomechanical resonators[J].Appl Phys Lett,2004,84: 121-123. doi: 10.1063/1.1638623 [41] Park S K,Gao X-L.Bernoulli-Euler beam model based on a modified couple stress theory[J].Journal of Micromechanics and Microengineering,2006,16: 2355-2359. doi: 10.1088/0960-1317/16/11/015 [42] Park S K,Gao X-L.Variational formulation of a modified couple stress theory and its application to a simple shear problem[J].Z Angew Math Phys,2008,59: 904-917. doi: 10.1007/s00033-006-6073-8 [43] Ma H M,Gao X-L,Reddy J N.A microstructure-dependent Timoshenko beam model based on a modified couple stress theory[J].Journal of the Mechanics and Physics of Solids,2008,56(12): 3379-3391. doi: 10.1016/j.jmps.2008.09.007 [44] Was G S,Foecke T.Deformation and fracture in microlaminates[J].Thin Solid Films,1996,286: 1-31. doi: 10.1016/S0040-6090(96)08905-5 [45] McFarland A W,Colton J S.Role of material microstructure in plate stiffness with relevance to microcantilever sensors[J].Journal of Micromechanics and Microengineering,2005,15: 1060-1067. doi: 10.1088/0960-1317/15/5/024 [46] Liew K M,Hu Y G,He X Q.Flexural wave propagation in single-walled carbon nanotubes[J].J Computational and Theoretical Nanoscience,2008,5: 581-586. [47] Zhang Y Y,Wang C M,Duan W H,et al.Assessment of continuum mechanics models in predicting buckling strains of single-walled carbon nanotubes[J].Nanotechnology,2009,20, 395707. doi: 10.1088/0957-4484/20/39/395707 [48] Lim C W,Wang C M.Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams[J].Journal of Applied Physics,2007,101, 054312. doi: 10.1063/1.2435878
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