Assessment of continuum mechanics models in predicting buckling strains of single-walled carbon nanotubes

Zhang, Yingyan; Wang, Chen; Duan, Wenhui; Xiang, Yang; Zong, Zhi

doi:10.1088/0957-4484/20/39/395707

Cited by 118 publications

(60 citation statements)

References 44 publications

(90 reference statements)

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“…This is a fact anticipated from the intuition of physics that nonlocal influence at the molecular level imposes additional constraints on molecular displacement and thus the stiffness is higher. It is also a fact which is congruent with many other molecular dynamic simulations on stiffness [46][47][48], experiments on stiffness [23,25,26] and other continuum nanomechanical models such as strain gradients on hardness [24] as well as stiffness [25] and couple stress on stiffness [30] analyses. It is noticed that the rate of change of extension, i.e.…”

Section: A Hard-fixed Soft-free Nanorod With An End Loadsupporting

confidence: 83%

Is a nanorod (or nanotube) with a lower Young’s modulus stiffer? Is not Young’s modulus a stiffness indicator?

Lim

2010

Sci. China Phys. Mech. Astron.

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It has been a known fact in classical mechanics of materials that Young's modulus is an indicator of material stiffness and materials with a higher Young's modulus are stiffer. At the nanoscale, within the scope and under specific circumstances described in this paper, however, a nanorod (or a nanotube) with a smaller Young's modulus (smaller stress-strain rate) is stiffer. In such a scenario, Young's modulus is not a stiffness indicator for nanostructures. Furthermore, the nonlocal stress-strain rate is dependent on types of load, boundary conditions and location. This is likely to be one of the many possible reasons why numerous experiments in the past obtained significantly varying values of Young's modulus for a seemingly identical nanotube, i.e. because the types of loading and/or boundary conditions in the experiments were different, as well as at which point the property was measured. Based on the nonlocal elasticity theory and within the scope of material and geometric linearity, this paper reports the strange and hitherto unrealized effect that a nanorod (or a nanotube) with a lower Young's modulus (smaller stress-strain rate) indicates smaller extension in tensile analysis. Similarly, it is also predicted that a nanorod (or a nanotube) with a lower Young's modulus results in smaller bending deflection, higher critical buckling load, higher free vibration frequency and higher wave propagation velocity, which are at all consequences of a stiffer nanostructure. nanorod, nanotube, nonlocal stress, nonlocal elasticity, stiffness, tensile, Young's modulus PACS: 62.25.-g, 62.20.D-, 62.23.-c

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Section: A Hard-fixed Soft-free Nanorod With An End Loadsupporting

confidence: 83%

Is a nanorod (or nanotube) with a lower Young’s modulus stiffer? Is not Young’s modulus a stiffness indicator?

Lim

2010

Sci. China Phys. Mech. Astron.

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“…The conclusion was further substantiated by experiments that showed (i) much higher tensile strengths of finely structured microlaminate films than the strengths of monolithic films [44] ; (ii) significant increased hardness of nanoindentation of crystalline materials [38] ; and (iii) significantly increased bending stiffness of a nano-cantilever with decreasing thickness [39,45] . Comparison with molecular dynamic simulations on nanotubes for wave propagation [46] and buckling [47] also indicates stiffness strengthening behavior for nanotubes with respect to classical solutions.…”

Section: Applicability Of An Approximate Nonlocal Strain Gradient Modelmentioning

confidence: 87%

On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection

Lim

2010

Appl. Math. Mech.-Engl. Ed.

158

101

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This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams: (i) why does the presence of increasing 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 intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam 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 nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, domain governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary conditions and the true basic static responses for bending of nanobeams. It is also concluded 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 nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.

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“…For an one dimensional Euler-Bernoulli beam, Eq. (6) is now written as follows ( Arash and Wang, 2012;Xu, 2006;Zhang et al, 2009;2010 ) …”

Section: Model Developmentmentioning

confidence: 99%

Measuring the nonlocal effects of a micro/nanobeam by the shifts of resonant frequencies

Zhang

Zhao

2016

International Journal of Solids and Structures

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a b s t r a c tA method of using the shifts of the beam resonant frequencies to determine the unknown parameters of nonlocal effects together with other effects such as surface elasticity, surface stress and residual stress is presented. The nonlocal effects are size-dependent, which only stand out when a specimen size diminishes. However, when the size of a specimen is small, other effects may also impact its mechanical properties and it is difficult to tell them apart. Unlike the static tests, the dynamic method presented in this study can differentiate various effects and thus determine the parameters. The parameters of different effects are solved as an inverse problem and the accuracy of the method is also demonstrated. So far, there are still no clear physical mechanisms to determine the parameters of nonlocal effects and different experiments yield different results. Because of the sensitivity to pre-existing defects, the mechanical properties of a small specimen may vary from one to another and the nonlocal effects can be disguised by other effects, which leads to different or even wrong interpretations on experimental data. With the capability of differentiating various effects, this study provides a more accurate and reliable method of determining the parameters of nonlocal effects, which should be of some help to the nonlocal theories.

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Assessment of continuum mechanics models in predicting buckling strains of single-walled carbon nanotubes

Cited by 118 publications

References 44 publications

Is a nanorod (or nanotube) with a lower Young’s modulus stiffer? Is not Young’s modulus a stiffness indicator?

Is a nanorod (or nanotube) with a lower Young’s modulus stiffer? Is not Young’s modulus a stiffness indicator?

On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection

Measuring the nonlocal effects of a micro/nanobeam by the shifts of resonant frequencies

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