Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

Li, Li; Li, Xiaobai

doi:10.1016/j.ijmecsci.2016.06.011

Cited by 220 publications

(61 citation statements)

References 55 publications

(111 reference statements)

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“…The majority of size-dependent continuum models of nanorods have been developed via the nonlocal theory of elasticity. However, more recently, the NSGT has been employed for describing the longitudinal vibration [103] and tension [104] of nanorods; the modified rod model was successfully calibrated employing MD results. 11…”

Section: 1b Size-dependent Mechanics Of Nanorodsmentioning

confidence: 99%

A review on the mechanics of nanostructures

Farajpour

Ghayesh

Farokhi

2018

International Journal of Engineering Science

201

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Understanding the mechanical behaviour of nanostructures is of great importance due to their applications in nanodevices such as in nanomechanical resonators, nanoscale mass sensors, electromechanical nanoactuators and nanogenerators. Due to the difficulties of performing accurate experimental measurements at nanoscales and the high computational costs associated with the molecular dynamics simulations, the continuum modelling of nanostructures has attracted a considerable amount of attention. Since size influences have a crucial role in the mechanics of structures at nanoscale levels, classical continuum-based theories have been modified to incorporate these effects. Among various modified continuum-based theories, the nonlocal elasticity and the nonlocal strain gradient elasticity have been employed to estimate the mechanical behaviour of nanostructures. In this review paper, first these two modified elasticity theories are briefly explained. Then, the nonlocal motion equations for different nanostructures including nanorods, nanorings, nanobeams, nanoplates and nanoshells are derived. Several papers which reported on the size-dependent mechanical behaviour of nanostructures using modified continuum models are reviewed. Furthermore, important results reported on the vibration, bending and buckling of nanostructures as well as the results of size-dependent wave propagation analyses are discussed.

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Section: 1b Size-dependent Mechanics Of Nanorodsmentioning

confidence: 99%

A review on the mechanics of nanostructures

Farajpour

Ghayesh

Farokhi

2018

International Journal of Engineering Science

201

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“…Hence, the solution of the differential equation (Equation 29) with the boundary conditions in Equations (30) and (31) provides the axial displacement…”

Section: Case I: Cf Fg Nano-rod With a Concentrated Load At The Free Endmentioning

confidence: 99%

“…Recently, the Eringen's integral law [1] has been combined with the strain gradient elasticity in [27] to formulate a higher-order nonlocal theory, thus collecting nonlocal theory and strain gradient theory into a single model. Using such a model, many contributions have been provided to model the size-dependent behavior of nano-rods and beams (see, e.g., [28][29][30][31][32][33][34][35]) and plates [36,37].…”

Section: Introductionmentioning

confidence: 99%

Modified Nonlocal Strain Gradient Elasticity for Nano-Rods and Application to Carbon Nanotubes

2019

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Nowadays, the modified nonlocal strain gradient theory provides a mathematically well-posed and technically reliable methodology to assess scale effects in inflected nano-structures. Such an approach is extended in this paper to investigate the extensional behavior of nano-rods. The considered integral elasticity model, involving axial force and strain fields, is conveniently shown to be equivalent to a nonlocal differential problem equipped with constitutive boundary conditions. Unlike treatments in the literature, no higher-order boundary conditions are required to close the nonlocal problem. Closed-form solutions of elastic nano-rods under selected loadings and kinematic boundary conditions are provided. As an innovative implication, Young’s moduli of Single-Walled Carbon Nanotubes (SWCNT) weare assessed and compared with predictions of Molecular Dynamics (MD). New benchmarks for numerical analyses were also detected.

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“…Some theoretical approaches were employed to predict and analyze the mechanical properties of CNT structures. These analytical theories and methods included with the nonlocal continuum mechanics, in which stress field at a reference position depends not only on the strain at that position but also on strains at all other points in the domain [33][34][35][36]. Barretta et al presented a stress-driven two-phases constitutive mixture by a convex combination of local and nonlocal phases [37].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Vibration Frequency of Carbon Nanotubes used as Nano-Force Sensors Considering Clamped Boundary Condition

Natsuki

Urakami

2019

Electronics

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Carbon nanotubes (CNTs) can be used as atomic force microscope (AFM) probes since they are ideal tip materials with a small diameter, high aspect ratio, and stiffness. In this study, a model of CNTs clamped in an elastic medium is proposed as nanoscale force sensing AFM probes. The relationship between vibration frequency and axial force of the CNT probe clamped in an elastic medium is analyzed based on the Euler-Bernoulli beam model and the Whitney-Riley model. The clamped length of CNTs, and the elastic modulus of elastic medium affect largely on the vibration and the buckling stability of a CNT AFM probe. The result showed that the sensitivity to vibration increases as the applied loads increase. The critical load in which the vibration frequency decreases rapidly, moving to large ones with decreasing ratio of length to diameter of CNTs. The theoretical investigation on the vibration frequency of CNT loaded in the axial direction would give a useful reference for designing a CNT used as a nano-force sensor.

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Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

Cited by 220 publications

References 55 publications

A review on the mechanics of nanostructures

A review on the mechanics of nanostructures

Modified Nonlocal Strain Gradient Elasticity for Nano-Rods and Application to Carbon Nanotubes

Analysis of Vibration Frequency of Carbon Nanotubes used as Nano-Force Sensors Considering Clamped Boundary Condition

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