• Advances in nano research
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• v.9 no.4
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• pp.221-235
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• 2020

The following composition establishes a nonlocal strain gradient plate model that is essentially related to mass sensors laying on Winkler-Pasternak medium for the vibrational analysis from graphene sheets. To achieve a seemingly accurate study of graphene sheets, the posited theorem actually accommodates two parameters of scale in relation to the gradient of the strain as well as non-local results. Model graphene sheets are known to have double variant shear deformation plate theory without factors from shear correction. By using the principle of Hamilton, to acquire the governing equations of a non-local strain gradient graphene layer on an elastic substrate, Galerkin's method is therefore used to explicate the equations that govern various partition conditions. The influence of diverse factors like the magnetic field as well as the elastic foundation on graphene sheet's vibration characteristics, the number of nanoparticles, nonlocal parameter, nanoparticle mass as well as the length scale parameter had been evaluated.

• Structural Engineering and Mechanics
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• v.75 no.1
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• pp.109-122
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• 2020

An asymptotic local plane strain elasticity theory is reformulated for the static analysis of a simply-supported, multiple graphene sheet system (MGSS) in cylindrical bending and resting on an elastic medium. The dimension of the MGSS in the y direction is considered to be much greater than those in the x and z directions, such that all the field variables are considered to be independent of the y coordinate. Eringen's nonlocal constitutive relations are used to account for the small length scale effects in the formulation examining the static behavior of the MGSS. The interaction between the MGSS and its surrounding foundation is modelled as a Winkler foundation with the parameter k_w, and the interaction between adjacent graphene sheets (GSs) is considered using another Winkler model with the parameter c_w. A parametric study with regard to some effects on the static behavior of the MGSS resting on an elastic medium is undertaken, such as the aspect ratio, the number of the GSs, the stiffness of the medium between the adjacent layers and that of the surrounding medium of the MGSS, and the nonlocal parameter.

• Structural Engineering and Mechanics
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• v.48 no.3
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• pp.351-365
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• 2013

In this paper, unified nonlocal shear deformation theory is proposed to study bending, buckling and free vibration of nanobeams. This theory is based on the assumption that the in-plane and transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. In addition, this present model is capable of capturing both small scale effect and transverse shear deformation effects of nanobeams, and does not require shear correction factors. The equations of motion are derived from Hamilton's principle. Analytical solutions for the deflection, buckling load, and natural frequency are presented for a simply supported nanobeam, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory and Reddy beam theories.

• Advances in nano research
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• v.8 no.1
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• pp.13-24
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• 2020

In this paper, for the first time based on the nonlocal strain gradient theory the effect of size dependency in torsional vibration of bi-direction functionally graded (FG) nonlinear nano-cone is study. The material properties were assumed to vary according to the arbitrary function in radial and axial directions. The Navier equation and boundary conditions of the size-dependent bidirectional FG nonlinear nano-cone were derived by Hamilton's principle. These equations were solved by employing the generalized differential quadrature method (GDQM). The presented model can turn into the classical model if the material length scale parameters are taken to be zero. The effects of some parameters, such as inhomogeneity constant, cross-sectional area parameter and small-scale parameters, were studied. As an essential result of this study can be stated that an FG nano-cone model based on the nonlocal elasticity theory behaves softer and based on the strain gradient theory behaves harder.

• Advances in nano research
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• v.4 no.2
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• pp.65-84
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• 2016

This paper investigates the buckling behavior of shear deformable piezoelectric (FGP) nanoscale beams made of functionally graded (FG) materials embedded in Winkler-Pasternak elastic medium and subjected to an electro-magnetic field. Magneto-electro-elastic (MEE) properties of piezoelectric nanobeam are supposed to be graded continuously in the thickness direction based on power-law model. To consider the small size effects, Eringen's nonlocal elasticity theory is adopted. Employing Hamilton's principle, the nonlocal governing equations of the embedded piezoelectric nanobeams are obtained. A Navier-type analytical solution is applied to anticipate the accurate buckling response of the FGP nanobeams subjected to electro-magnetic fields. To demonstrate the influences of various parameters such as, magnetic potential, external electric voltage, power-law index, nonlocal parameter, elastic foundation and slenderness ratio on the critical buckling loads of the size-dependent MEE-FG nanobeams, several numerical results are provided. Due to the shortage of same results in the literature, it is expected that the results of the present study will be instrumental for design of size-dependent MEE-FG nanobeams.

• Advances in nano research
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• v.12 no.1
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• pp.101-115
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• 2022

Some researchers pointed out that the nonlocal cantilever models do not predict the dynamic softening behavior for nanostructures (including nanobeams) with clamped-free (CF) ends. In contrast, some indicate that the nonlocal cantilever models can capture the stiffness softening characteristics. There are substantial differences on this issue between them. The vibration analysis of porosity-dependent functionally graded nanoscale tubes with variable boundary conditions is investigated in this study. Using a modified power-law model, the tube's porosity-dependent material coefficients are graded in the radial direction. The theory of nonlocal strain gradients is used. Hamilton's principle is used to derive the size-dependent governing equations for simply-supported (S), clamped (C) and clamped-simply supported (CS). Following the solution of these equations by the extended differential quadrature technique, the effect of various factors on vibration issues was investigated further. It can be shown that these factors have a considerable effect on the vibration characteristics. It also can be found that our numerical results can capture the unexpected softening phenomena for cantilever tubes.

• Advances in nano research
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• v.14 no.2
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• pp.141-154
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• 2023

Effects of viscoelastic foundation on vibration of curved-beam structure with clamped and simply-supported boundary conditions is investigated in this study. In doing so, a micro-scale laminate composite beam with two piezoelectric face layer with a carbon nanotube reinforces composite core is considered. The whole beam structure is laid on a viscoelastic substrate which normally occurred in actual conditions. Due to small scale of the structure non-classical elasticity theory provided more accurate results. Therefore, nonlocal strain gradient theory is employed here to capture both nano-scale effects on carbon nanotubes and microscale effects because of overall scale of the structure. Equivalent homogenous properties of the composite core is obtained using Halpin-Tsai equation. The equations of motion is derived considering energy terms of the beam and variational principle in minimizing total energy. The boundary condition is assumed to be clamped at one end and simply supported at the other end. Due to nonlinear terms in the equations of motion, semi-analytical method of general differential quadrature method is engaged to solve the equations. In addition, due to complexity in developing and solving equations of motion of arches, an artificial neural network is design and implemented to capture effects of different parameters on the inplane vibration of sandwich arches. At the end, effects of several parameters including nonlocal and gradient parameters, geometrical aspect ratios and substrate constants of the structure on the natural frequency and amplitude is derived. It is observed that increasing nonlocal and gradient parameters have contradictory effects of the amplitude and frequency of vibration of the laminate beam.

• Advances in nano research
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• v.14 no.5
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• pp.475-493
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• 2023

In the context of nonclassical nonlocal strain gradient elasticity, this article studies the free and forced responses of functionally graded material (FGM) porous nanoplates exposed to thermal and magnetic fields under a moving load. The developed mathematical model includes shear deformation, size-scale, miscorstructure influences in the framework of higher order shear deformation theory (HSDT) and nonlocal strain gradient theory (NSGT), respectively. To explore the porosity effect, the study considers four different porosity models across the thickness: uniform, symmetrical, asymmetric bottom, and asymmetric top distributions. The system of quations of motion of the FGM porous nanoplate, including the effects of thermal load, Lorentz force, due to the magnetic field and moving load, are derived using the Hamilton's principle, and then solved analytically by employing the Navier method. For the free and forced responses of the nanoplate, the effects of nonlocal elasticity, strain gradient elasticity, temperature rise, magnetic field intensity, porosity volume fraction, and porosity distribution are analyzed. It is found that the forced vibrations of FGM porous nanoplates under thermal and live loads can be damped by applying a directed magnetic field.

• Advances in nano research
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• v.16 no.1
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• pp.27-40
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• 2024

This work is the first of its kind to integrate Mindlin's theory with analytical methods in order to produce an exact solution to a specific vibration issue as well as a bending problem involving a nanoplate that is supported by a viscoelastic foundation. The plate is exposed to the simultaneous effects of a compressive load in the plate plane and a force operating perpendicular to the plane of the nanoplate. In addition, the flexoelecity effect is included into the plate. The strain gradient component is taken into consideration while calculating the plate equilibrium equation using the nonlocal theory and Hamilton's principle. The free vibration and static responses of the nanoplate seem to be both real and imaginary components because of the appearance of the viscoelastic drag coefficient of the viscoelastic foundation. This study also shows that when analyzing the mechanical response for nanostructure, taking into account the flexoelectricity effect and the influence of the nonlocal parameter, the results will be completely different from the case in which this parameter is ignored. This indicates that it is vital to take into consideration the effects of nonlocal parameters on the nanosheet structure while also taking into consideration the effect of flexoelectricity.

• Smart Structures and Systems
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• v.20 no.6
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• pp.709-728
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• 2017

This disquisition proposes a nonlocal strain gradient beam theory for thermo-mechanical dynamic characteristics of embedded smart shear deformable curved piezoelectric nanobeams made of porous electro-elastic functionally graded materials by using an analytical method. Electro-elastic properties of embedded curved porous FG nanobeam are assumed to be temperature-dependent and vary through the thickness direction of beam according to the power-law which is modified to approximate material properties for even distributions of porosities. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Since variation of pores along the thickness direction influences the mechanical and physical properties, so in this study thermo-mechanical vibration analysis of curve FG piezoelectric nanobeam by considering the effect of these imperfections is performed. Nonlocal strain gradient elasticity theory is utilized to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field. The governing equations and related boundary condition of embedded smart curved porous FG nanobeam subjected to thermal and electric field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is utilized to achieve the natural frequencies of porous FG curved piezoelectric nanobeam resting on Winkler and Pasternak foundation. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, electric voltage, coefficient of porosity, elastic foundation parameters, thermal effect, gradient index, strain gradient, elastic opening angle and slenderness ratio on the natural frequency of embedded curved FG porous piezoelectric nanobeam are successfully discussed. It is concluded that these parameters play important roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.