• Computers and Concrete
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• v.26 no.1
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• pp.75-94
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• 2020

The aim of this research is to investigate free vibration of a novel five layer Timoshenko microbeam which consists of a transversely flexible porous core made of Al-foam, two graphen platelets (GPL) nanocomposite reinforced layers to enhance the mechanical behavior of the structure as well as two piezo-magneto-electric face sheets layers. This microbeam is subjected to a thermal load and resting on Pasternak's foundation. To accomplish the analysis, constitutive equations of each layer are derived by means of nonlocal strain gradient theory (NSGT) to capture size dependent effects. Then, the Hamilton's principle is employed to obtain the equations of motion for five layer Timoshenko microbeam. They are subsequently solved analytically by applying Navier's method so that discretized governing equations are determined in form of dynamic matrix giving the possibility to gain the natural frequencies of the Timoshenko microbeam. Eventually, after a validation study, the numerical results are presented to study and discuss the influences of various parameters such as nonlocal parameter, strain gradient parameter, aspect ratio, porosity, various volume fraction and distributions of graphene platelets, temperature change and elastic foundation coefficients on natural frequencies of the sandwich microbeam.

• 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.

• Advances in nano research
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• v.15 no.3
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• pp.215-224
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• 2023

Nonlinearity plays an important role in control systems and the application of design. For this reason, in addition to linear vibrations, nonlinear vibrations of the stepped nanobeam are also discussed in this manuscript. This study investigated the vibrations of stepped nanobeams according to Eringen's nonlocal elasticity theory. Eringen's nonlocal elasticity theory was used to capture the nanoscale effect. The nanoscale stepped Euler Bernoulli beam is considered. The equations of motion representing the motion of the beam are found by Hamilton's principle. The equations were subjected to nondimensionalization to make them independent of the dimensions and physical structure of the material. The equations of motion were found using the multi-time scale method, which is one of the approximate solution methods, perturbation methods. The first section of the series obtained from the perturbation solution represents a linear problem. The linear problem's natural frequencies are found for the simple-simple boundary condition. The second-order part of the perturbation solution is the nonlinear terms and is used as corrections to the linear problem. The system's amplitude and phase modulation equations are found in the results part of the problem. Nonlinear frequency-amplitude, and external frequency-amplitude relationships are discussed. The location of the step, the radius ratios of the steps, and the changes of the small-scale parameter of the theory were investigated and their effects on nonlinear vibrations under simple-simple boundary conditions were observed by making comparisons. The results are presented via tables and graphs. The current beam model can assist in designing and fabricating integrated such as nano-sensors and nano-actuators.

• Advances in nano research
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• v.16 no.5
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• pp.473-487
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• 2024

The current study employs the nonlocal Timoshenko beam (NTB) theory and von-Kármán's geometric nonlinearity to develop a non-classic beam model for evaluating the nonlinear free vibration of bi-directional functionally-graded (BFG) nanobeams. In order to avoid the stretching-bending coupling in the equations of motion, the problem is formulated based on the physical middle surface. The governing equations of motion and the relevant boundary conditions have been determined using Hamilton's principle, followed by discretization using the differential quadrature method (DQM). To determine the frequencies of nonlinear vibrations in the BFG nanobeams, a direct iterative algorithm is used for solving the discretized underlying equations. The model verification is conducted by making a comparison between the obtained results and benchmark results reported in prior studies. In the present work, the effects of amplitude ratio, nanobeam length, material distribution, nonlocality, and boundary conditions are examined on the nonlinear frequency of BFG nanobeams through a parametric study. As a main result, it is observed that the nonlinear vibration frequencies are greater than the linear vibration frequencies for the same amplitude of the nonlinear oscillator. The study finds that the difference between the dimensionless linear frequency and the nonlinear frequency is smaller for CC nanobeams compared to SS nanobeams, particularly within the α range of 0 to 1.5, where the impact of geometric nonlinearity on CC nanobeams can be disregarded. Furthermore, the nonlinear frequency ratio exhibits an increasing trend as the parameter µ is incremented, with a diminishing dependency on nanobeam length (L). Additionally, it is established that as the nanobeam length increases, a critical point is reached at which a sharp rise in the nonlinear frequency ratio occurs, particularly within the nanobeam length range of 10 nm to 30 nm. These findings collectively contribute to a comprehensive understanding of the nonlinear vibration behavior of BFG nanobeams in relation to various parameters.

• Advances in nano research
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• v.10 no.2
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• pp.151-163
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• 2021

The main target of this study is to investigate nonlinear transient responses of moving polymer nano-size plates fortified by means of Graphene Platelets (GPLs) and resting on a Winkler-Pasternak foundation under a transverse pressure force and a temperature variation. Two graphene spreading forms dispersed through the plate thickness are studied, and the Halpin-Tsai micro-mechanics model is used to obtain the effective Young's modulus. Furthermore, the rule of mixture is employed to calculate the effective mass density and Poisson's ratio. In accordance with the first order shear deformation and von Karman theory for nonlinear systems, the kinematic equations are derived, and then nonlocal strain gradient scheme is used to reflect the effects of nonlocal and strain gradient parameters on small-size objects. Afterwards, a combined approach, kinetic dynamic relaxation method accompanied by Newmark technique, is hired for solving the time-varying equation sets, and Fortran program is developed to generate the numerical results. The accuracy of the current model is verified by comparative studies with available results in the literature. Finally, a parametric study is carried out to explore the effects of GPL's weight fractions and dispersion patterns, edge conditions, softening and hardening factors, the temperature change, the velocity of moving nanoplate and elastic foundation stiffness on the dynamic response of the structure. The result illustrates that the effects of nonlocality and strain gradient parameters are more remarkable in the higher magnitudes of the nanoplate speed.

• Steel and Composite Structures
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• v.49 no.4
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• pp.361-380
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• 2023

A size dependent bending behavior of piezoelectrical flexoelectric layered perforated functionally graded (FG) composite nanobeam rested on an elastic foundation is investigated analytically. The composite beam is composed of regularly cutout FG core and two piezoelectric face sheets. The material characteristics is graded through the core thickness by power law function. Regular squared cutout perforation pattern is considered and closed forms of the equivalent stiffness parameters are derived. The modified nonlocal strain gradient elasticity theory is employed to incorporate the microstructure as well as nonlocality effects into governing equations. The Winkler as well as the Pasternak elastic foundation models are employed to simulate the substrate medium. The Hamiltonian approach is adopted to derive the governing equilibrium equation including piezoelectric and flexoelectric effects. Analytical solution methodology is developed to derive closed forms for the size dependent electromechanical as well as mechanical bending profiles. The model is verified by comparing the obtained results with the available corresponding results in the literature. To demonstrate the applicability of the developed procedure, parametric studies are performed to explore influences of gradation index, elastic medium parameters, flexoelectric and piezoelectric parameters, geometrical and peroration parameters, and material parameters on the size dependent bending behavior of piezoelectrically layered PFG nanobeams. Results obtained revealed the significant effects both the flexoelectric and piezoelectric parameters on the bending behavior of the piezoelectric composite nanobeams. These parameters could be controlled to improve the size dependent electromechanical as well as mechanical behaviors. The obtained results and the developed procedure are helpful for design and manufacturing of MEMS and NEMS.

• Advances in nano research
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• v.9 no.3
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• pp.133-145
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• 2020

In this study, nonlinear vibrations and dynamic instabilities of a smart embedded micro shell conveying varied fluid flow and subjected to the combined electro-thermo-mechanical loadings are investigated. With the aim of designing new hydraulic sensors and actuators, the piezoelectric materials are employed for the body and the effects of applying electric field on the stability of the system as well as the induced voltage due to the dynamic behavior of the system are studied. The nonlocal piezoelasticity theory and the nonlinear cylindrical shell model in conjunction with the energy approach are utilized to mathematically modeling of the structure. The fluid flow is assumed to be isentropic, incompressible and fully develop, and for more generality of the problem both steady and time dependent flow regimes are considered. The mathematical modeling of fluid flow is also carried out based on a scalar potential function, time mean Navier-Stokes equations and the theory of slip boundary condition. Employing the modified Lagrange equations for open systems, the nonlinear coupled governing equations of motion are achieved and solved via the state space problem; forth order numerical integration and Bolotin's method. In the numerical results, a comprehensive discussion is made on the dynamical instabilities of the system (such as divergence, flutter and parametric resonance). We found that applying positive electric potential field will improve the stability of the system as an actuator or vibration amplitude controller in the micro electro mechanical systems.

• Advances in materials Research
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• v.3 no.2
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• pp.77-89
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• 2014

In the present article, the thermal buckling of zigzag single-walled carbon nanotubes (SWCNTs) is studied using a nonlocal refined shear deformation beam theory and Von-Karman geometric nonlinearity. The model developed simulates both small scale effects and higher-order variation of transverse shear strain through the depth of the nanobeam. Furthermore the present formulation also accommodates stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. The equivalent Young's modulus and shear modulus for zigzag SWCNTs are derived using an energy-equivalent model. The present study illustrates that the thermal buckling properties of SWCNTs are strongly dependent on the scale effect and additionally on the chirality of zigzag carbon nanotube. Some illustrative examples are also presented to verify the present formulation and solutions. Good agreement is observed.

• Structural Engineering and Mechanics
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• v.66 no.6
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• pp.737-748
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• 2018

The current study presents a new technique in the framework of the nonlocal elasticity theory for a comprehensive buckling analysis of Euler-Bernoulli nano-beams made up of bidirectional functionally graded material (BDFGM). The mechanical properties are considered by exponential and arbitrary variations for axial and transverse directions, respectively. The various circumstances including tapering, resting on two-parameter elastic foundation, step-wise or continuous variations of axial loading, various shapes of sections with various distribution laws of mechanical properties and various boundary conditions like the multi-span beams are taken into account. As far as we know, for the first time in the current work, the buckling analyses of BDFGM nano-beams are carried out under mentioned circumstances. The critical buckling loads and mode shapes are calculated by using energy method and a new technique based on calculus of variations and collocation method. Fast convergence and excellent agreement with the known data in literature, wherever possible, presents the efficiency of proposed technique. The effects of boundary conditions, material and taper constants, foundation moduli, variable axial compression and small-scale of nano-beam on the buckling loads and mode shapes are investigated. Moreover the analytical solutions, for the simpler cases are provided in appendices.

• Structural Engineering and Mechanics
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• v.66 no.2
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• pp.249-262
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• 2018

In this paper, reaction of functionally graded (FG) thick nanoplates resting on a viscoelastic foundation to a moving nanoparticle/load is investigated. Nanoplate is assumed to be thick by using second order shear deformation theory and small-scale effects are taken into account in the framework of Eringen's nonlocal theory. Material properties are varied through the thickness using FG models by having power-law, sigmoid and exponential functions for material changes. FG nanoplate is assumed to be on a viscoelastic medium which is modeled using Kelvin-Voight viscoelastic model. Galerkin, state space and fourth-order Runge-Kutta methods are employed to solve the governing equations. A comprehensive parametric study is presetned to show the influence of different parameters on mechanical behavior of the system. It is shown that material variation in conjunction with nonlocal term have a significant effect on the dynamic deformation of nanoplate which could be used in comprehending and designing more efficient nanostructures. Moreover, it is shown that having a viscoelastic medium could play an important role in decreasing these dynamic deformations. With respect to the fresh studies on moving atoms, molecules, cells, nanocars, nanotrims and point loads on different nanosctructures using scanning tunneling microscopes (STM) and atomic force microscopes (AFM), this study could be a step forward in understanding, predicting and controlling such kind of behaviors by showing the influence of the moving path, velocity etc. on dynamic reaction of the plate.