A straightforward approach to Eringen’s nonlocal elasticity stress model and applications for nanobeams

Koutsoumaris, C. Chr.; Eptaimeros, K.G.; Zisis, Th.; Tsamasphyros, G.

doi:10.1063/1.4968757

Cited by 5 publications

(3 citation statements)

References 14 publications

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“…Romano et al [43] demonstrated that an ill-posed structural-mechanics problem (bounded domain) arises with adoption of the Eringen nonlocal differential model. Furthermore, Koutsoumaris et al [44] demonstrated that the Eringen nonlocal differential model does not conform to the requirement of the quadratic energy functional form of elasticity. To obtain a more rational small-scale beam model, the strain-gradient elasticity theory has been employed as an alternative [38,[45][46][47][48][49].…”

Section: Introductionmentioning

confidence: 99%

Bending, Buckling and Free Vibration Analyses of Nanobeam-Substrate Medium Systems

et al. 2022

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This study presents a newly developed size-dependent beam-substrate medium model for bending, buckling, and free-vibration analyses of nanobeams resting on elastic substrate media. The Euler-Bernoulli beam theory describes the beam-section kinematics and the Winkler-foundation model represents interaction between the beam and its underlying substrate medium. The reformulated strain-gradient elasticity theory possessing three non-classical material constants is employed to address the beam-bulk material small-scale effect. The first and second constants is associated with the strain-gradient and couple-stress effects, respectively while the third constant is related to the velocity-gradient effect. The Gurtin-Murdoch surface elasticity theory is adopted to account for the surface-free energy. To obtain the system governing equation as well as corresponding boundary conditions, Hamilton’s principle is called for. Three numerical simulations are presented to characterize the influences of the material small-scale effect, the surface-energy effect, and the surrounding substrate medium on bending, buckling, and free vibration responses of nanobeam-substrate medium systems. The first simulation focuses on the bending response and shows the ability of the proposed model to eliminate the paradoxical characteristic inherent to nanobeam models proposed in the literature. The second and third simulations perform the sensitivity investigation of the system parameters on the buckling load and the natural frequency, respectively. All analytical results reveal that both material small-scale and surface-energy effects consistently stiffen the system response while the velocity-gradient effect weakens the system response. Furthermore, these sized-scale effects are more pronounced when the underlying substrate medium becomes softer.

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Section: Introductionmentioning

confidence: 99%

Bending, Buckling and Free Vibration Analyses of Nanobeam-Substrate Medium Systems

et al. 2022

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“…Romano et al [ 35 ] have thoroughly diagnosed the cause of these problematic responses and concluded that an ill-posed mathematical problem is encountered with the adoption of Eringen nonlocal differential model. In addition, the Eringen nonlocal differential model would not accept quadratic energy functional form of elasticity [ 36 ] and the work conjugate nature of stress and strain in this nonlocal constitutive model is ambiguous [ 37 ].…”

Section: Introductionmentioning

confidence: 99%

Strain-Gradient Bar-Elastic Substrate Model with Surface-Energy Effect: Virtual-Force Approach

et al. 2022

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This paper presents an alternative approach to formulating a rational bar-elastic substrate model with inclusion of small-scale and surface-energy effects. The thermodynamics-based strain gradient model is utilized to account for the small-scale effect (nonlocality) of the bar-bulk material while the Gurtin–Murdoch surface theory is adopted to capture the surface-energy effect. To consider the bar-surrounding substrate interactive mechanism, the Winkler foundation model is called for. The governing differential compatibility equation as well as the consistent end-boundary compatibility conditions are revealed using the virtual force principle and form the core of the model formulation. Within the framework of the virtual force principle, the axial force field serves as the fundamental solution to the governing differential compatibility equation. The problem of a nanowire embedded in an elastic substrate medium is employed as a numerical example to show the accuracy of the proposed bar-elastic substrate model and advantage over its counterpart displacement model. The influences of material nonlocality on both global and local responses are thoroughly discussed in this example.

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“…It was diagnosed by Romano et al [ 41 ] that adoption of the Eringen nonlocal differential model can result in an ill-posed structural mechanics problem (bounded domain). Furthermore, Koutsoumaris et al [ 42 ] showed that the quadratic energy functional form of elasticity does not exist for the Eringen nonlocal differential model, and Ma et al [ 43 ] pointed out that there is ambiguity in the conjugate stress–strain pairs in this nonlocal constitutive model. To deviate from skeptical and inconsistent features inherent to the Eringen nonlocal differential model, several researchers have paid attention to strain-gradient-type constitutive models.…”

Section: Introductionmentioning

confidence: 99%

Static and Free Vibration Analyses of Single-Walled Carbon Nanotube (SWCNT)–Substrate Medium Systems

et al. 2022

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This paper proposes a novel nanobar–substrate medium model for static and free vibration analyses of single-walled carbon nanotube (SWCNT) systems embedded in the elastic substrate medium. The modified strain-gradient elasticity theory is utilized to account for the material small-scale effect, while the Gurtin–Murdoch surface theory is employed to represent the surface energy effect. The Winkler foundation model is assigned to consider the interactive mechanism between the nanobar and its surrounding substrate medium. Hamilton’s principle is used to consistently derive the system governing equation, initial conditions, and classical as well as non-classical boundary conditions. Two numerical simulations are employed to demonstrate the essence of the material small-scale effect, the surface energy effect, and the surrounding substrate medium on static and free vibration responses of single-walled carbon nanotube (SWCNT)–substrate medium systems. The simulation results show that the material small-scale effect, the surface energy effect, and the interaction between the substrate and the structure led to a system-stiffness enhancement both in static and free vibration analyses.

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A straightforward approach to Eringen’s nonlocal elasticity stress model and applications for nanobeams

Cited by 5 publications

References 14 publications

Bending, Buckling and Free Vibration Analyses of Nanobeam-Substrate Medium Systems

Bending, Buckling and Free Vibration Analyses of Nanobeam-Substrate Medium Systems

Strain-Gradient Bar-Elastic Substrate Model with Surface-Energy Effect: Virtual-Force Approach

Static and Free Vibration Analyses of Single-Walled Carbon Nanotube (SWCNT)–Substrate Medium Systems

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