• Structural Engineering and Mechanics
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• v.63 no.3
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• pp.401-415
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• 2017

In this article, hygro-thermo-mechanical vibration and buckling of exponentially graded (EG) nanoplates resting on two-parameter Pasternak foundations are studied using the four-unknown shear deformation plate theory. The material properties are presumed to change only in the thickness direction of the EG nanoplate according to two exponential laws distribution. The boundary conditions of the nanoplate may be simply supported, clamped, free or combination of them. To consider the small scale effect on forced frequencies and buckling, Eringen's differential form of nonlocal elasticity theory is employed. The accuracy of the present study is investigated considering the available solutions in literature. A detailed analysis is executed to study the influences of the plate aspect ratio, side-to-thickness ratio, temperature rise, moisture concentration and volume fraction distributions on the vibration and buckling of the nanoplates.

• Steel and Composite Structures
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• v.22 no.2
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• pp.277-299
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• 2016

In this paper, free vibration, forced vibration, resonance and stress wave propagation behavior in nanocomposite plates reinforced by wavy carbon nanotube (CNT) are studied by a mesh-free method based on first order shear deformation theory (FSDT). The plates are resting on Winkler-Pasternak elastic foundation and subjected to periodic or impact loading. The distributions of CNTs are considered functionally graded (FG) or uniform along the thickness and their mechanical properties are estimated by an extended rule of mixture. In the mesh-free analysis, moving least squares (MLS) shape functions are used for approximation of displacement field in the weak form of motion equation and the transformation method is used for imposition of essential boundary conditions. Effects of CNT distribution, volume fraction, aspect ratio and waviness, and also effects of elastic foundation coefficients, plate thickness and time depended loading are examined on the vibrational and stresses wave propagation responses of the nanocomposite plates reinforced by wavy CNT.

• Steel and Composite Structures
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• v.11 no.6
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• pp.489-504
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• 2011

This paper presents a theoretical investigation in free vibration of sigmoid functionally graded beams with variable cross-section by using Bernoulli-Euler beam theory. The mechanical properties are assumed to vary continuously through the thickness of the beam, and obey a two power law of the volume fraction of the constituents. Governing equation is reduced to an ordinary differential equation in spatial coordinate for a family of cross-section geometries with exponentially varying width. Analytical solutions of the vibration of the S-FGM beam are obtained for three different types of boundary conditions associated with simply supported, clamped and free ends. Results show that, all other parameters remaining the same, the natural frequencies of S-FGM beams are always proportional to those of homogeneous isotropic beams. Therefore, one can predict the behaviour of S-FGM beams knowing that of similar homogeneous beams.

• Steel and Composite Structures
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• v.32 no.5
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• pp.687-699
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• 2019

In this paper, free and force vibration behaviors of graphene-reinforced composite functionally graded (GRC-FG) cylindrical shells in thermal environments are investigated based on Reddy's third-order shear deformation theory (HSDT). The GRC-FG cylindrical shells are composed of piece-wise pattern graphene-reinforced layers which have different volume fraction. Based on the extended Halpin-Tsai micromechanical model, the effective material properties of the resulting nanocomposites are evaluated. Using the Hamilton's principle and the assumed mode method, the motion equation of the GRC-FG cylindrical shells is formulated. Using the time- and frequency-domain methods, free and force vibration properties of the GRC-FG cylindrical shell are analyzed. Numerical cases are provided to study the effects of distribution of graphene, shell radius-to-thickness ratio and temperature changes on the free and force vibration responses of GRC-FG cylindrical shells.

• Structural Engineering and Mechanics
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• v.52 no.4
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• pp.663-686
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• 2014

This paper deals with free vibration analysis of bidirectional functionally graded annular plates resting on a two-parameter elastic foundation. The formulations are based on the three-dimensional elasticity theory. This study presents a novel 2-D six-parameter power-law distribution for ceramic volume fraction of 2-D functionally graded materials that gives designers a powerful tool for flexible designing of structures under multi-functional requirements. Various material profiles along the thickness and in the in-plane directions are illustrated by using the 2-D power-law distribution. The effective material properties at a point are determined in terms of the local volume fractions and the material properties by the Mori-Tanaka scheme. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The fast rate of convergence of the method is shown and the results are compared against existing results in literature. Some new results for natural frequencies of the plates are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. The interesting results indicate that a graded ceramic volume fraction in two directions has a higher capability to reduce the natural frequency than conventional 1-D functionally graded materials.

• Computers and Concrete
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• v.26 no.1
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• pp.31-52
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• 2020

This paper investigates the size dependent effect on the vibration analysis of a porous nanocomposite viscoelastic plate reinforced by functionally graded-single walled carbon nanotubes (FG-SWCNTs) by considering nonlocal strain gradient theory. Therefore, using energy method and Hamilton's principle, the equations of motion are derived. In this article, the effects of nonlocal parameter, aspect ratio, strain gradient parameter, volume fraction of carbon nanotubes (CNTs), damping coefficient, porosity coefficient, and temperature change on the natural frequency are perused. The innovation of this paper is to compare the effectiveness of each mentioned parameters individually on the free vibrations of this plate and to represent the appropriate value for each parameter to achieve an ideal nanocomposite plate that minimizes vibration. The results are verified with those referenced in the paper. The results illustrate that the effect of damping coefficient on the increase of natural frequency is significantly higher than the other parameters effect, and the effects of the strain gradient parameter and nonlocal parameter on the natural frequency increase are less than damping coefficient effect, respectively. Furthermore, the results indicate that the natural frequency decreases with a rise in the nonlocal parameter, aspect ratio and temperature change. Also, the natural frequency increases with a rise in the strain gradient parameter and CNTs volume fraction. This study can be used for optimizing the industrial and medical designs, such as automotive industry, aerospace engineering and water purification system, by considering ideal properties for the nanocomposite plate.

• Steel and Composite Structures
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• v.36 no.6
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• pp.711-727
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• 2020

In the present research, the free vibration analysis of functionally graded (FG) nanocomposite deep spherical shells reinforced by graphene platelets (GPLs) on elastic foundation is performed. The elastic foundation is assumed to be Winkler-Past ernak-type. It is also assumed that graphaene platelets are randomly oriented and uniformly dispersed in each layer of the nanocomposite shell. Volume fraction of the graphene platelets as nanofillers may be different in the layers. The modified HalpinTsai model is used to approximate the effective mechanical properties of the multilayer nanocomposite. With the aid of the first order shear deformation shell theory and implementing Hamilton's principle, motion equations are derived. Afterwards, the generalized differential quadrature method (GDQM) is utilized to study the free vibration characteristics of FG-GPLRC spherical shell. To assess the validity and accuracy of the presented method, the results are compared with the available researches. Finally, the natural frequencies and corresponding mode shapes are provided for different boundary conditions, GPLs volume fraction, types of functionally graded, elastic foundation coefficients, opening angles of shell, and thickness-to-radius ratio.

• Structural Engineering and Mechanics
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• v.50 no.6
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• pp.773-796
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• 2014

This paper deals with free vibration analysis of continuous grading fiber reinforced (CGFR) and bi-directional FG annular sector plates on two-parameter elastic foundations under various boundary conditions, based on the three-dimensional theory of elasticity. The plates with simply supported radial edges and arbitrary boundary conditions on their circular edges are considered. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. Some new results for the natural frequencies of the plate are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. Results indicate that the non-dimensional natural frequency parameter of a functionally graded fiber volume fraction is larger than that of a discrete laminated and close to that of a 2-layer. It results that the CGFR plate attains natural frequency higher than those of traditional discretely laminated composite ones and this can be a benefit when higher stiffness of the plate is the goal and that is due to the reduction in spatial mismatch of material properties. Moreover, it is shown that a graded ceramic volume fraction in two directions has a higher capability to reduce the natural frequency than conventional one-dimensional functionally graded material. The multidirectional graded material can likely be designed according to the actual requirement and it is a potential alternative to the unidirectional functionally graded material. The new results can be used as benchmark solutions for future researches.

• Advances in nano research
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• v.16 no.4
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• pp.413-426
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• 2024

This paper investigates the dynamic behavior of a simply supported viscoelastic plate made of functionally graded carbon nanotube reinforced composite under dynamic loading. Carbon nanotubes are distributed in 5 different shapes: U, V, A, O and X, depending on the shape they form through the thickness of the plate. The displacement fields are derived in the Laplace domain using a higher-order shear deformation theory. Equations of motion are obtained through the application of the energy method and Hamilton's principle. The resulting equations of motion are solved using Navier's method. Transforming the Laplace domain displacements into the time domain involves Durbin's modified inverse Laplace transform. To validate the accuracy of the developed algorithm, a free vibration analysis is conducted for simply supported plate made of functionally graded carbon nanotube reinforced composite and compared against existing literature. Subsequently, a parametric forced vibration analysis considers the influence of various parameters: volume fractions of carbon nanotubes, their distributions, and ratios of instantaneous value to retardation time in the relaxation function, using a linear standard viscoelastic model. In the forced vibration analysis, the dynamic distributed load applied to functionally graded carbon nanotube reinforced composite viscoelastic plate is obtained in terms of double trigonometric series. The study culminates in an examination of maximum displacement, exploring the effects of different carbon nanotube distributions, volume fractions, and ratios of instantaneous value to retardation times in the relaxation function on the amplitudes of maximum displacements.

• Smart Structures and Systems
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• v.22 no.1
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• pp.105-120
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• 2018

In this paper, the nonlinear free and forced vibration responses of sandwich nano-beams with three various functionally graded (FG) patterns of reinforced carbon nanotubes (CNTs) face-sheets are investigated. The sandwich nano-beam is resting on nonlinear Visco-elastic foundation and is subjected to thermal and electrical loads. The nonlinear governing equations of motion are derived for an Euler-Bernoulli beam based on Hamilton principle and von Karman nonlinear relation. To analyze nonlinear vibration, Galerkin's decomposition technique is employed to convert the governing partial differential equation (PDE) to a nonlinear ordinary differential equation (ODE). Furthermore, the Multiple Times Scale (MTS) method is employed to find approximate solution for the nonlinear time, frequency and forced responses of the sandwich nano-beam. Comparison between results of this paper and previous published paper shows that our numerical results are in good agreement with literature. In addition, the nonlinear frequency, force response and nonlinear damping time response is carefully studied. The influences of important parameters such as nonlocal parameter, volume fraction of the CNTs, different patterns of CNTs, length scale parameter, Visco-Pasternak foundation parameter, applied voltage, longitudinal magnetic field and temperature change are investigated on the various responses. One can conclude that frequency of FG-AV pattern is greater than other used patterns.