A semi-continuum-based bending analysis for extreme-thin micro/nano-beams and new proposal for nonlocal differential constitution

Shen, Jianqiang; Li, C.

doi:10.1016/j.compstruct.2017.03.070

Cited by 47 publications

(21 citation statements)

References 42 publications

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“…It is also demonstrated that the bending deflection may increase or decrease with increasing nonlocal effect. Therefore, the numerical result validates both the nonlocal stiffness-weakening and stiffness-strengthening models, which is consistent with the previous work [30]. This also verifies the correctness of the proposed model in this paper qualitatively.…”

Section: Numerical Examplessupporting

confidence: 91%

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Analytical Solutions for Bending of Nanoscaled Bars Based on Eringen’s Nonlocal Differential Law

Zhang

et al. 2019

Journal of Nanomaterials

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The one-dimensional nanoscaled bar is commonly seen in current nanodevices, and it plays a significant role in nanoelectromechanical system (NEMS) and related nanotechnology. Motivated by this, the paper is concerned with the bending and stability of a nanoscaled bar especially a flexible bar subjected to vertical concentrated, vertical linearly distributed, and horizontal concentrated forces simultaneously. The theoretical model is developed by employing Eringen’s nonlocal differential law. Hence, the nonlocal differential constitutions in terms of stress and bending moment at a nanoscale are applied to the classical equilibrium formulation in order to construct the nonlocal constitutive model and then determine the analytical solutions for bending of such a nanoscale bar. Subsequently, the effect of a nonlocal scale on the midpoint deflections and internal forces is shown numerically, and the internal and external characteristic length scales are also examined. It is revealed that the capacity of resisting compression decreases with increasing the nonlocal effect since the internal characteristic scale cannot be neglected by comparing with the external characteristic scale. There exists a threshold value for bending stiffness. Both the midpoint deflection and bending moment vary monotonously when the material bending stiffness exceeds the threshold value, while the mechanical quantities fluctuate when the bending stiffness is less than its threshold value, namely, for a flexible nanoscaled bar. Accordingly, two different kinds of nonlocal predictions and the related disputes are resolved in the context of Eringen’s nonlocal differential law. The work is expected to be useful for the design, application, and optimization of nanoscaled bars sustaining bending with linear vertical and horizontal forces.

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Section: Numerical Examplessupporting

confidence: 91%

“…People were puzzled with such different nonlocal results. In recent years, Shen and Li [30] revealed the physical meanings of both nonlocal differential models and proved 2…”

Section: Introductionmentioning

confidence: 99%

Analytical Solutions for Bending of Nanoscaled Bars Based on Eringen’s Nonlocal Differential Law

Zhang

et al. 2019

Journal of Nanomaterials

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“…In the literature, nonlocal elasticity proposed by Eringen [23], Modified Couple stress Theory (MCT) of Yang et al [24] and Modified Strain gradient Theory (MST) of Lam et al [25] are widely used to investigate the behaviour of micro/nano structures. Recent studies on the developments and applications of nonlocal theory to nanobeams and nanopaltes could be found in the works of Li et al [26][27][28][29], Nguyen et al [30], and Phung-Van et al [31]. Based on the MCT, numerous studies have been also carried out to investigate the behaviour of micro beams and plates [32][33][34][35][35][36][37][38][39][40][41][42][43][44][45].…”

Section: Introductionmentioning

confidence: 99%

Post-buckling of functionally graded microplates under mechanical and thermal loads using isogeomertic analysis

Thai

et al. 2017

Engineering Structures

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The present study uses the isogeometric analysis (IGA) to investigate the post-buckling behavior of functionally graded (FG) microplates subjected to mechanical and thermal loads. The modified a strain gradient theory with three length scale parameters is used to capture the size effect. The Reddy third-order shear deformation plate theory with the von Kámán nonlinearity (i.e., small strains and moderate rotations) is employed to describe the kinematics of the microplates. Material variations in the thickness direction of the plate are described using a rule of mixtures. In addition, material properties are assumed to be either temperature-dependent or temperature-independent. The governing equations are derived using the principle of virtual work, which are then discretized using the IGA approach, whereby a C 2-continuity requirement is fulfilled naturally and efficiently. To trace the post-buckling paths, Newton's iterative technique is utilized. Various parametric studies are conducted to examine the influences of material variations, size effects, thickness ratios, and boundary conditions on the post-buckling behavior of microplates.

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“…As for the behavior of the same micro-beam when electrically loaded by (V DC = 50 V and V AC = 2 V), Figure 9c and d show that the increase of the length scale parameter decreases the softening type of nonlinearity since the bending of the curves to the left is lessened, with a noticeable increase of the saddle-node bifurcation points (SN), as it is clear in the second column of Table 7. The reader is referred to similar trends in [41] and [42]. Figure 10a and b show the effect of the axial load on the frequency response curves for a micro-beam with an axial velocity v = 0.4 and a length scale parameter l = 0.5 h. It is observed that as the axial load increases, the frequency response curves tend to shift to the right, i.e., when the micro-beam is electrically loaded by (V DC = 20 V and V AC = 2 V) gaining a hardening behavior, or when it is electrically loaded by (V DC = 50 V and V AC = 2 V) gaining a softening behavior.…”

Section: Effect Of the Axial Velocitymentioning

confidence: 84%

Nonlinear behavior of a vibrating axially moving small‐size beam under an electrostatic force

Alshaqaq

Hawwa

2020

Z Angew Math Mech

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Small size (micro/nano)‐scale beams constitute important building blocks of microelectromechanical systems (MEMS)/nanoelectromechanical systems (NEMS). Emerging roll‐based, high rate, manufacturing processes can make these small size‐beams vibrate, while they are axially moving. In this paper, an analytical‐numerical study on the nonlinear transverse vibration of the representative case of axially moving micro‐beam under an electrostatic force is conducted. The analytical model is realized by employing Hamilton's principle together with Galerkin discretization. The method of multiple time‐scales and Runge‐Kutta based numerical scheme are utilized to investigate the nonlinear dynamic behavior of the micro‐beam. Results are obtained for the influence of axial beam velocity and modified couple stress theory length scale parameter (i) on the values of pull‐in instability voltage of the small‐size beam, and (ii) on the small‐size beam nonlinear softening/hardening characteristics. The effect of axial load on the frequency response is investigated.

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A semi-continuum-based bending analysis for extreme-thin micro/nano-beams and new proposal for nonlocal differential constitution

Cited by 47 publications

References 42 publications

Analytical Solutions for Bending of Nanoscaled Bars Based on Eringen’s Nonlocal Differential Law

Analytical Solutions for Bending of Nanoscaled Bars Based on Eringen’s Nonlocal Differential Law

Post-buckling of functionally graded microplates under mechanical and thermal loads using isogeomertic analysis

Nonlinear behavior of a vibrating axially moving small‐size beam under an electrostatic force

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