Free transverse vibrations of nano-to-micron scale beams

Xu, Mingtian

doi:10.1098/rspa.2006.1712

Cited by 101 publications

(60 citation statements)

References 26 publications

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“…Since the first publication of nonlocal elasticity theory by Eringen and his associate, 1-3 many articles have been published on the application of this model in nanomechanics, particularly in the early 21st century, such as Peddieson et al, 4 Sudak, 5 Zhang et al, 6 Wang, 7,8 Lu et al, 9 Xu, 10 Wang and Hu, 11 etc. Almost all articles presented a simplified nonlocal beam model by assuming that the beam midplane is governed by a second-order ordinary differential equation, while in the transverse direction the classical Euler-Bernoulli beam model or the Timoshenko beam model applies.…”

Section: Introductionmentioning

confidence: 99%

Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams

Lim

Wang

2007

Journal of Applied Physics

126

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This article presents a complete and asymptotic representation of the one-dimensional nanobeam model with nonlocal stress via an exact variational principle approach. An asymptotic governing differential equation of infinite-order strain gradient model and the corresponding infinite number of boundary conditions are derived and discussed. For practical applications, it explores and presents a reduced higher-order solution to the asymptotic nonlocal model. It is also identified here and explained at length that most publications on this subject have inaccurately employed an excessively simplified lower-order model which furnishes intriguing solutions under certain loading and boundary conditions where the results become identical to the classical solution, i.e., without the small-scale effect at all. Various nanobeam examples are solved to demonstrate the difference between using the simplified lower-order nonlocal model and the asymptotic higher-order strain gradient nonlocal stress model. An important conclusion is the discovery of significant over-or underestimation of stress levels using the lower-order model, particularly at the vicinity of the clamped end of a cantilevered nanobeam under a tip point load. The consequence is that the design of a nanobeam based on the lower-order strain gradient model could be flawed in predicting the nonlocal stress at the clamped end where it could, depending on the magnitude of the small-scale parameter, significantly over-or underestimate the failure criteria of a nanobeam which are governed by the level of stress.

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Section: Introductionmentioning

confidence: 99%

Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams

Lim

Wang

2007

Journal of Applied Physics

126

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“…Challenging applications of MB theory may be framed in the context of bottom-up multi-scale methods that, starting from experimental measures at nanoscales and prescribed geometrical hierarchy, yield the macroscopic MB theory with proper scaling of material mechanical parameters. Such an approach may be used in several nano-/micro-applications in materials science [48,49] to provide an analytical framework to the so-called bio-inspired materials as well as in medical sciences to predict the efficiency of modern nano-based therapies for solid tumours as in the very rent oncophysics [50].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

The mechanically based non-local elasticity: an overview of main results and future challenges

Paola¹,

Failla²,

Pirrotta³

et al. 2013

Phil. Trans. R. Soc. A.

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The mechanically based non-local elasticity has been used, recently, in wider and wider engineering applications involving small-size devices and/or materials with marked microstructures. The key feature of the model involves the presence of non-local effects as additional body forces acting on material masses and depending on their relative displacements. An overview of the main results of the theory is reported in this pape

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“…Similar analysis of Euler-Bernoulli [10][11][12] and Timoshenko beams [13,14] have also been performed using the nonlocal elasticity theory. Analytical solutions for bending, buckling and vibration of Euler-Bernoulli, Timoshenko, Reddy, and Levinson beams were derived by using Hamilton's principle and the nonlocal elasticity theory [15].…”

Section: Introductionmentioning

confidence: 99%

Vibration of Timoshenko Beams Using Non-classical Elasticity Theories

Santos

Reddy

2012

Shock and Vibration

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Abstract. This paper presents a comparison among classical elasticity, nonlocal elasticity, and modified couple stress theories for free vibration analysis of Timoshenko beams. A study of the influence of rotary inertia and nonlocal parameters on fundamental and higher natural frequencies is carried out. The nonlocal natural frequencies are found to be lower than the classical ones, while the natural frequencies estimated by the modified couple stress theory are higher. The modified couple stress theory results depend on the beam cross-sectional size while those of the nonlocal theory do not. Convergence of both non-classical theories to the classical theory is observed as the beam global dimension increases.

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Free transverse vibrations of nano-to-micron scale beams

Cited by 101 publications

References 26 publications

Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams

Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams

The mechanically based non-local elasticity: an overview of main results and future challenges

Vibration of Timoshenko Beams Using Non-classical Elasticity Theories

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