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

C. W. Lim

doi:10.3879/j.issn.1000-0887.2010.01.005

Volume 31 Issue 1

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C. W. Lim. On the Truth of Nanoscale for Nanobeams Based on Nonlocal Elastic Stress Field Theory: Equilibrium,Governing Equation and Static Deflection[J]. Applied Mathematics and Mechanics, 2010, 31(1): 35-50. doi: 10.3879/j.issn.1000-0887.2010.01.005

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

doi: 10.3879/j.issn.1000-0887.2010.01.005

C. W. Lim

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

Received Date: 2009-09-29
Rev Recd Date: 2009-11-19
Publish Date: 2010-01-15

Abstract

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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References(48)

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