Fundamental size dependent natural frequencies of non-uniform orthotropic nano scaled plates using nonlocal variational principle and finite element method

Nonlocal effect is one of the critical reasons which cause an extraordinary vibration response in small-scale structures. In the present work, vibration characteristics of functionally graded nonlocal plate are studied using Eringen’s nonlocal classical elasticity theory. A computationally efficient numerical method has been proposed in this study by reformulating the classical plate theory and Rayleigh–Ritz method using nonlocal differential relationship of Eringen’s theory in conjugation with algebraic polynomial displacement functions. The reformulated method helps to evaluate the natural frequencies of functionally graded nonlocal plates subjected to all possible combinations of edge conditions. The material properties are assumed to vary through the thickness of the plate following the power law. The computed results of natural frequencies are first tested for convergence and then validated with the published one. A parametric study has been thoroughly conducted focusing on the effects of aspect ratio, nonlocal parameter, material property index and Young’s modulus ratio on the natural frequency parameters of the functionally graded nonlocal plate. It has been observed that the material property index and aspect ratio affect the vibration behaviour of the functionally graded plate. The study also establishes that nonlocal effect has a pronounced influence on the higher modes of vibration of functionally graded plate. 3D mode shapes of functionally graded material nonlocal plate have also been reported.

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“…The expression of kinetic energy of the orthotropic nanoplate using nonlocal theory is given by 27,29 …”

Section: Mathematical Modellingmentioning

confidence: 99%

Size-dependent natural frequencies of functionally graded plate with out of plane material inhomogeneity using Eringen’s theory of nonlocal elasticity

Singh

Azam

Ranjan

2019

Proceedings of the Institution of Mechanical Engineers, Part L:

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“…It was observed that the frequencies and the buckling loads depend on the structural damping, loading ratio, Pasternak foundation, and the small-scale effect. Shahidi et al [6] used Eringen's theory to study the vibrations of orthotropic Kirchhoff nano-plates. The effects of several factors including the scale parameter on the frequencies were conducted.…”

Section: Introductionmentioning

confidence: 99%

Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

2020

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The natural vibration behavior of axially functionally graded (AFG) double nanobeams is studied based on the Euler–Bernoulli beam and Eringen’s non-local elasticity theory. The double nanobeams are continuously connected by a layer of linear springs. The oscillatory differential equation of motion is established using the Hamilton’s principle and the constitutive relations. The Chebyshev spectral collocation method (CSCM) is used to transform the coupled governing differential equations of motion into algebraic equations. The discretized boundary conditions are used to modify the Chebyshev differentiation matrices, and the system of equations is put in the matrix-vector form. Then, the dimensionless transverse frequencies and the mode shapes are obtained by solving the standard eigenvalue problem. The effects of the coupling springs, Winkler stiffness, the shear stiffness parameter, the breadth and taper ratios, the small-scale parameter, and the boundary conditions on the natural transverse frequencies are carried out. Several numerical examples were conducted, and the authors believe that the results may be interesting in designing and analyzing double and multiple one-dimensional nano structures.

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“…It was shown that the buckling loads are strongly dependent on the small-scale coefficients and the stiffness of the surrounding elastic medium. Shahidi et al [15] developed a nonlocal continuum model based on Eringen's theory for vibration analysis of orthotropic nanoplates (modelled as Kirchhoff plates) with arbitrary variation in thickness. The variational principle and Ritz functions were employed, and the effect of thickness variation on natural frequencies was investigated for different nonlocal parameters, mode numbers, geometries and boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Buckling Analysis of Axially Functionally Graded Tapered Nano-Beams Resting on Elastic Foundation Based on Nonlocal Elasticity Theory

Sari¹,

Al‐Kouz²,

Atieh³

2018

SV-JME

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The stability analysis of nonlocal axially functionally graded tapered beams has been investigated. Euler-Bernoulli beams at the micro-or nanoscale are modeled using Eringen's nonlocal elasticity theory. The governing equations are derived using the differential constitutive relations, and the Chebyshev collocation method is utilized to convert the differential equation of motion into a set of algebraic equations. Next, the boundary conditions are applied, and the resulting eigenvalue problem is solved to obtain the critical buckling loads. The effects of the Winkler modulus parameter, the shear modulus parameter, the breadth taper ratio, the height taper ratio, the nonlocal scale coefficient, and the boundary conditions on the critical buckling loads have been studied.

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Fundamental size dependent natural frequencies of non-uniform orthotropic nano scaled plates using nonlocal variational principle and finite element method

Cited by 31 publications

References 43 publications

Size-dependent natural frequencies of functionally graded plate with out of plane material inhomogeneity using Eringen’s theory of nonlocal elasticity

Size-dependent natural frequencies of functionally graded plate with out of plane material inhomogeneity using Eringen’s theory of nonlocal elasticity

Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

Buckling Analysis of Axially Functionally Graded Tapered Nano-Beams Resting on Elastic Foundation Based on Nonlocal Elasticity Theory

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