Free vibration analysis of nonlocal nanobeams: a comparison of the one-dimensional nonlocal integral Timoshenko beam theory with the two-dimensional nonlocal integral elasticity theory
Abstract:Beam theories such as the Timoshenko beam theory are in agreement with the elasticity theory. However, due to the different nonlocal averaging processes, they are expected to yield different results in their nonlocal forms. In the present work, the free vibration behavior of nonlocal nanobeams is studied using the nonlocal integral Timoshenko beam theory (NITBT) and two-dimensional nonlocal integral elasticity theory (2D-NIET) with different kernels and their results are compared. A new kernel, termed the comp… Show more
“…Danesh and Javanbakht investigated the free vibration behavior of nonlocal nanobeams using the nonlocal Timoshenko beam theory and two-dimensional nonlocal integral elasticity theory. They found that the natural frequencies obtained from the two-dimensional nonlocal integral elasticity theory were comparable with those obtained from the nonlocal Timoshenko beam theory for any boundary condition [43].…”
Graphene sheets are the basis of nano-electromechanical switches, which offer a unique insight into the world of quantum mechanics. In this paper, we proposed a new size-dependent multi-beam shear model for investigating the pull-in instability of multilayer graphene/substrate nano-switches within the context of the Timoshenko beam theory. As the graphene/substrate bemas bent toward the graphene layer due to the thermomechanical mismatch, the impact of curvature is considered in the proposed model. Also, the impact of the Casimir attraction is incorporated in the developed model by taking into account the limited conductivity of interacting surfaces. The scale dependency of the materials is considered in the framework of the nonlocal elasticity. To simulate the nano-switch and explore the pull-in instability, a finite element procedure is developed. The proposed approach is verified by comparing the pull-in voltage to published data. Finally, the role of influential parameters, including size dependency, length, initial gap, curvature, and the number of graphene layers on instability voltage of nano-switch, are investigated.
“…Danesh and Javanbakht investigated the free vibration behavior of nonlocal nanobeams using the nonlocal Timoshenko beam theory and two-dimensional nonlocal integral elasticity theory. They found that the natural frequencies obtained from the two-dimensional nonlocal integral elasticity theory were comparable with those obtained from the nonlocal Timoshenko beam theory for any boundary condition [43].…”
Graphene sheets are the basis of nano-electromechanical switches, which offer a unique insight into the world of quantum mechanics. In this paper, we proposed a new size-dependent multi-beam shear model for investigating the pull-in instability of multilayer graphene/substrate nano-switches within the context of the Timoshenko beam theory. As the graphene/substrate bemas bent toward the graphene layer due to the thermomechanical mismatch, the impact of curvature is considered in the proposed model. Also, the impact of the Casimir attraction is incorporated in the developed model by taking into account the limited conductivity of interacting surfaces. The scale dependency of the materials is considered in the framework of the nonlocal elasticity. To simulate the nano-switch and explore the pull-in instability, a finite element procedure is developed. The proposed approach is verified by comparing the pull-in voltage to published data. Finally, the role of influential parameters, including size dependency, length, initial gap, curvature, and the number of graphene layers on instability voltage of nano-switch, are investigated.
“…where p i , τ 1 ijk and m s ij are the work conjugate of γ i , η 1 ijk and χ s ij , respectively. We note that there are several possible formulations available in literature [22,27,[46][47][48][49][50].…”
Section: Representation Of Gradient Effects In Terms Of Length Scale ...mentioning
This work presents the development of a unified gradient electromechanical theory for thin flexoelectric beams considering both direct and converse flexoelectric effects. The two-way coupled electromechanical theory is developed starting from 3D variational formulation by considering an electric field-strain based free energy function. The formulation incorporates mechanical as well as electrical size effects. The coupled 3D theory is specialized to isotropic materials and a 1D beam theory for composite flexoelectric curved beams is derived using the classical Kirchhoff assumptions. The beam theory is solved using a novel C 2 continuous finite element framework for different loading and boundary conditions. Our finite element results are verified with analytical solutions for a simply-supported flexoelectric beam operating in both actuator and sensor modes. The results are also compared with existing literature for the special case of a passive micro-beam. Our computational framework is subsequently used to perform various parametric studies to analyze the effect of electrical and mechanical length scale parameters, geometric parameters like the radius of curvature, flexoelectric layer thickness etc., on the response of the beam. Also, contribution of converse flexoelectricity in the overall response of the flexoelectric beam is compared with that of the direct effect. Our simulation results predict that the converse effect is significant (≈ 10-25% of the direct effect) for a wide range of thickness and length scale parameter values. It is also observed that the effective electromechanical coupling coefficient, calculated in terms of the voltage developed across the flexoelectric layer thickness, is higher in flexoelectric materials compared to piezoelectric materials at smaller length scales (thickness of the order of a few microns). Our simulation results also agree well with the trends observed in recent experimental work [1].
“…These include nonlocal elasticity theory [16][17][18][19][20], micropolar [21][22][23], strain gradient theory [24][25][26], surface elasticity theory [27,28]and modified couple stress theory [29][30][31][32][33]. Based on the aforementioned theories, significant attention has been dedicated to studying the mechanical behavior of Micro/Nano Structures, particularly in the context of bending [34][35][36], buckling [13,37,38], vibration [39][40][41][42], and wave propagation [43][44][45][46]. Among these studies, particular attention has been given to examining the free vibration behavior of nanoscale structures.…”
Magneto-Electro-Elastic (MEE) Composites, as an innovative functional material blend, are composed of multiple materials, boasting exceptional strength, rigidity, and an extraordinary magneto-electric interaction effect. This paper establishes a nonlocal modified couple stress (NL-MCS) magneto-electro-elastic nanobeam dynamic model. To accurately capture the intricate influences of scale effects on nanostructures, This model meticulously examines scale effects from two distinct perspectives: leveraging nonlocal elasticity theory to elucidate the softening phenomena in nanostructures stemming from long-range particle interactions, and employing modified couple stress theory to reveal the hardening effects attributed to the rotational behavior of particles within the structure. By incorporating Von Karman geometric nonlinearity, Reddy's third-order shear deformation theory and Maxwell's equations, the governing equations for the nonlinear free vibration of MEE nanobeams are derived using Hamilton's principle. Finally, a two-step perturbation method is employed to solve these equations. Two-step perturbation method disintegrates the solution process into two stages, iteratively approximating and refining the solution, thereby progressively unraveling the intricate details and enhancing the precision of the solution in a systematic manner. Finally, the nonlinear free vibration behavior of MEE nanobeams is explored under the coupled magnetic-electric-elastic fields, with a focus on the effects of various factors that including length scale parameters, nonlocal parameters, Winkler-Pasternak coefficients, span-to-thickness ratios, applied voltages, and magnetic potentials.
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