• Structural Engineering and Mechanics
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• v.55 no.3
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• pp.495-510
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• 2015

Free vibration of non-uniform embedded nanoplates based on classical (Kirchhoff's) plate theory in conjunction with nonlocal elasticity theory has been studied. The nanoplate is assumed to be rested on two-parameter Winkler-Pasternak elastic foundation. Non-uniform material properties of nanoplates have been considered by taking linear as well as quadratic variations of Young's modulus and density along the space coordinates. Detailed analysis has been reported for all possible casesof such variations. Trial functions denoting transverse deflection of the plate are expressed in simple algebraic polynomial forms. Application of the present method converts the problem into generalised eigen value problem. The study aims to investigate the effects of non-uniform parameter, elastic foundation, nonlocal parameter, boundary condition, aspect ratio and length of nanoplates on the frequency parameters. Three-dimensional mode shapes for some of the boundary conditions have also been illustrated. One may note that present method is easier to handle any sets of boundary conditions at the edges.

• Smart Structures and Systems
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• v.17 no.2
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• pp.257-274
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• 2016

The nonlocal static bending, buckling, free and forced vibrations of graphene nanosheets are examined based on the Kirchhoff plate theory and Taylor expansion approach. The nonlocal nanoplate model incorporates the length scale parameter which can capture the small scale effect. The governing equations are derived using Hamilton's principle and the Navier-type solution is developed for simply-supported graphene nanosheets. The analytical results are proposed for deflection, natural frequency, amplitude of forced vibration and buckling load. Moreover, the effects of nonlocal parameter, half wave number and three-dimensional sizes on the static, dynamic and stability responses of the graphene nanosheets are discussed. Some illustrative examples are also addressed to verify the present model, methodology and solution. The results show that the new nanoplate model produces larger deflection, smaller circular frequencies, amplitude and buckling load compared with the classical model.

• Advances in nano research
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• v.7 no.3
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• pp.157-167
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• 2019

In this paper, a classical plate model is utilized to formulate the wave propagation problem of magnetostrictive sandwich nanoplates (MSNPs) while subjected to hygrothermal loading with respect to the scale effects. Herein, magnetostriction effect is considered and controlled on the basis of a feedback control system. The nanoplate is supposed to be embedded on a visco-Pasternak substrate. The kinematic relations are derived based on the Kirchhoff plate theory; also, combining these obtained equations with Hamilton's principle, the local equations of motion are achieved. According to a nonlocal strain gradient theory (NSGT), the small scale influences are covered precisely by introducing two scale coefficients. Afterwards, the nonlocal governing equations can be derived coupling the local equations with those of the NSGT. Applying an analytical solution, the wave frequency and phase velocity of propagated waves can be gathered solving an eigenvalue problem. On the other hand, accuracy and efficiency of presented model is verified by setting a comparison between the obtained results with those of previous published researches. Effects of different variants are plotted in some figures and the highlights are discussed in detail.

• Structural Engineering and Mechanics
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• v.68 no.2
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• pp.171-182
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• 2018

For the first time, nonlocal damped vibration and buckling analyses of arbitrary tapered bidirectional functionally graded solid circular nano-plate (BDFGSCNP) are presented by employing modified spectral Ritz method. The energy method based on Love-Kirchhoff plate theory assumptions is applied to derive neutral equilibrium equation. The Eringen's nonlocal continuum theory is taken into account to capture small-scale effects. The characteristic equations and corresponding first mode shapes are calculated by using a novel modified basis in spectral Ritz method. The modified basis is in terms of orthogonal shifted Chebyshev polynomials of the first kind to avoid employing adhesive functions in the spectral Ritz method. The fast convergence and compatibility with various conditions are advantages of the modified spectral Ritz method. A more accurate multivariable function is used to model two-directional variations of elasticity modulus and mass density. The effects of nanoscale, in-plane pre-load, distributed dashpot, arbitrary tapering, pinned and clamped boundary conditions on natural frequencies and buckling loads are investigated. Observing an excellent agreement between results of current work and outcomes of previously published works in literature, indicates the results' accuracy in current work.

• Advances in nano research
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• v.10 no.1
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• pp.25-42
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• 2021

In this article, the vibration behavior of embedded Functionally Graded Nanoplate (FGNP) employing nonlocal Kirchhoff's plate theory has been investigated under hygrothermal environment. The FGNP is considered to be supported by Winkler-Pasternak foundation. The Eringen's differential theory is used for size effect on the vibration of the FGNP. Rayleigh-Ritz method with orthogonal polynomials are employed for the governing equations and edge constraints. The advantage of this method is that it overcomes all the drawbacks of edge constraints and can easily handle any combinations of mixed edge constraints. The coefficients viz. moisture expansion, thermal expansion and elastic coefficients are considered to be transversely graded across the FGNP. The similarity of the calculated natural frequencies is examined with the previous research, and a good concurrency is seen. The objective of this article is to analyze the parameters' effect on the nondimensionalized frequency of embedded FGNP under hygrothermal environment subjected to all possible edge constraints. For this, uniform and linear rise of temperature and moisture concentration are considered. The study highlights that the nonlocal effect is pronounced for higher modes. Moreover, the effect of the Pasternak modulus is seen to be prominent compared to the Winkler modulus on non dimensionalized frequencies of FGNP.

• Structural Engineering and Mechanics
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• v.61 no.1
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• pp.65-76
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• 2017

Prediction of the characteristics of both in-plane and out-of-plane elastic waves within conducting nanoplates in the presence of bidirectionally in-plane magnetic fields is of interest. Using Lorentz's formulas and nonlocal continuum theory of Eringen, the nonlocal elastic version of the equations of motion is obtained. The frequencies as well as the corresponding phase and group velocities pertinent to the in-plane and out-of-plane waves are analytically evaluated. The roles of the strength of in-plane magnetic field, wavenumber, wave direction, nanoplate's thickness, and small-scale parameter on characteristics of waves are discussed. The obtained results show that the in-plane frequencies commonly grow with the in-plane magnetic field. However, the transmissibility of the out-of-plane waves rigorously depends on the magnetic field strength, direction of the propagated transverse waves, small-scale parameter, and thickness of the nanoplate. The criterion for safe transferring of the out-of-plane waves through the conducting nanoplate immersed in a bidirectional magnetic field is also explained and discussed.

• Steel and Composite Structures
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• v.33 no.5
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• pp.717-727
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• 2019

This paper is motivated by the lack of studies in the technical literature concerning to vibration analysis of a single-layered graphene sheet (SLGS) with corner cutout based on the nonlocal elasticity model framework of classical Kirchhoff thin plate. An isogeometric analysis (IGA) based upon non-uniform rational B-spline (NURBS) is employed for approximation of the L-shape SLGS deflection field. Trimming technique is employed to create the cutout in geometry of L-shape plate. The L-shape plate is assumed to be Free (F) in the straight edges of cutout while any arbitrary boundary conditions are applied to the other four straight edges including Simply supported (S), Clamped (C) and Free (F). The Numerical studies are carried out to express the influences of the nonlocal parameter, cutout dimensions, boundary conditions and mode numbers on the variations of the natural frequencies of SLGS. It is precisely shown that these parameters have considerable effects on the free vibration behavior of the system. In addition, numerical results are validated and compared with those achieved using other analysis, where an excellent agreement is found. The effectiveness and the accuracy of the present IGA approach have been demonstrated and it is shown that the IGA is efficient, robust and accurate in terms of nanoplate problems. This study serves as a benchmark for assessing the validity of numerical methods used to analyze the single-layered graphene sheet with corner cutout.