• Smart Structures and Systems
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• v.23 no.3
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• pp.215-225
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• 2019

In the present paper, the nonlocal strain gradient refined model is used to study the thermal stability of sandwich nanoplates integrated with piezoelectric layers for the first time. The influence of Kerr elastic foundation is also studied. The present model incorporates two small-scale coefficients to examine the size-dependent thermal stability response. Elastic properties of nanoplate made of functionally graded materials (FGMs) are supposed to vary through the thickness direction and are estimated employing a modified power-law rule in which the porosity with even type of distribution is approximated. The governing differential equations of embedded sandwich piezoelectric porous nanoplates under hygrothermal loading are derived through Hamilton's principle where the Galerkin method is applied to solve the stability problem of the nanoplates with simply-supported edges. It is indicated that the thermal stability characteristics of the porous nanoplates are obviously influenced by the porosity volume fraction and material variation, nonlocal parameter, strain gradient parameter, geometry of the nanoplate, external voltage, temperature and humidity variations, and elastic foundation parameters.

• Steel and Composite Structures
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• v.32 no.4
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• pp.509-519
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• 2019

In this paper, an analytical approach for the free vibration analysis of spiral stiffened functionally graded (SSFG) cylindrical shells is investigated. The SSFG shell is resting on linear and non-linear elastic foundation with damping force. The elastic foundation for the linear model is according to Winkler and Pasternak parameters and for the non-linear model, one cubic term is added. The material constitutive of the stiffeners is continuously changed through the thickness. Using the Galerkin method based on the von $K\acute{a}rm\acute{a}n$ equations and the smeared stiffeners technique, the non-linear vibration problem has been solved. The effects of different geometrical and material parameters on the free vibration response of SSFG cylindrical shells are adopted. The results show that the angles of stiffeners and elastic foundation parameters strongly effect on the natural frequencies of the SSFG cylindrical shell.

• Structural Engineering and Mechanics
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• v.71 no.3
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• pp.257-269
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• 2019

This paper develops a nonlocal strain gradient plate model for damping vibration analysis of smart magneto-electro-viscoelastic nanoplates resting on visco-Pasternak medium. For more accurate analysis of nanoplate, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. Viscoelastic effect which is neglected in all previous papers on magneto-electro-viscoelastic nanoplates is considered based on Kelvin-Voigt model. Governing equations of a nonlocal strain gradient smart nanoplate on viscoelastic substrate are derived via Hamilton's principle. Galerkin's method is implemented to solve the governing equations. Effects of different factors such as viscoelasticity, nonlocal parameter, length scale parameter, applied voltage and magnetic potential on damping vibration characteristics of a nanoplate are studied.

• Advances in aircraft and spacecraft science
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• v.6 no.3
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• pp.257-272
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• 2019

Numerical study of the flutter of a plate on a viscoelastic foundation is carried out in the paper. Critical velocity of the flutter of a plate on an elastic and viscoelastic foundation is determined. The mathematical model for the investigation of viscoelastic plates is based on the Marguerre's theory applied to the study of the problems of strength, rigidity and stability of thin-walled structures such as aircraft wings. Aerodynamic pressure is determined in accordance with the A.A. Ilyushin's piston theory. Using the Bubnov - Galerkin method, the basic resolving systems of nonlinear integro-differential equations (IDE) are obtained. At wide ranges of geometric and physical parameters of viscoelastic plates, their influence on the flutter velocity has been studied in detail.

• Coupled systems mechanics
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• v.8 no.3
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• pp.247-257
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• 2019

Fee vibrational characteristics of porous steel double-coupled nanoplate system in thermo-elastic medium is studied via a refined plate model. Different pore dispersions called uniform, symmetric and asymmetric have been defined. Nonlocal strain gradient theory (NSGT) containing two scale parameters has been adopted to stablish size-dependent modeling of the system. Hamilton's principle has been adopted to stablish the governing equations. Obtained results from Galerkin's method are verified with those provided in the literature. The effects of nonlocal parameter, strain gradient, foundation parameters, porosity distributions and porosity coefficient on vibration frequencies of metal foam nanoscale plates have been examined.

• Kyungpook Mathematical Journal
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• v.61 no.3
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• pp.591-612
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• 2021

In this paper, a uniformly convergent numerical scheme is designed for solving singularly perturbed reaction-diffusion problems. The problem is converted to an equivalent weak form and then a Galerkin finite element method is used on a piecewise uniform Shishkin mesh with linear basis functions. The convergence of the developed scheme is proved and it is shown to be almost fourth order uniformly convergent in the maximum norm. To exhibit the applicability of the scheme, model examples are considered and solved for different values of a singular perturbation parameter ε and mesh elements. The proposed scheme approximates the exact solution very well.

• Structural Engineering and Mechanics
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• v.78 no.1
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• pp.103-116
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• 2021

In this work, a model of a functionally graded (FG) nanotube conveying fluid embedded in an elastic medium is developed based on the nonlocal strain gradient theory (NSGT) in conjunction with Euler-Bernoulli beam theory (EBT). The main objective of this research is to investigate the nonlinear vibration and stability analysis of fluid-conveying nanotubes. The governing equations of motion are derived by means of Hamiltonian principle. The analytical expressions of nonlinear frequencies and critical flow velocities for two different types of boundary conditions including pinned-pinned (P-P) and clamped-clamped (C-C) conditions are obtained by employing Galerkin method as well as Hamiltonian Approach (HA). Comparison of the obtained results with the published works show the acceptable accuracy of the current solutions. The effects of the power-law index, the nonlocal and material length scale parameters and the elastic medium on the stability and nonlinear responses of FG nanotubes are thoroughly investigated and discussed.

• Structural Engineering and Mechanics
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• v.78 no.1
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• pp.15-22
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• 2021

An investigation of the nonlinear thermal buckling behavior of a nano-sized beam constructed from intelligent materials called piezo-magnetic materials has been presented in this article. The nano-sized beam geometry has been considered based on two assumptions: an ideal straight beam and an imperfect beam. For incorporating nano-size impacts, the nano-sized beam formulation has been presented according to nonlocal elasticity. After establishing the governing equations based on classic beam theory and nonlocal elasticity, the nonlinear buckling path has been obtained via Galerkin's method together with an analytical trend. The dependency of buckling path to piezo-magnetic material composition, electro-magnetic fields and geometry imperfectness has been studied in detail.

• Kyungpook Mathematical Journal
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• v.61 no.4
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• pp.859-888
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• 2021

This paper is devoted to the study of a nonlinear Kirchhoff-Carrier wave equation in an annulus associated with Robin-Dirichlet conditions. At first, by applying the Faedo-Galerkin method, we prove existence and uniqueness results. Then, by constructing a Lyapunov functional, we prove a blow up result for solutions with a negative initial energy and establish a sufficient condition to obtain the exponential decay of weak solutions.

• Advances in nano research
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• v.12 no.4
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• pp.353-363
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• 2022

This paper presents an investigation about superharmonic and subharmonic resonances of a carbon nanotube reinforced composite beam subjected to lateral harmonic load with damping effect based on the modified couple stress theory. As reinforcing phase, three different types of single walled carbon nanotubes (CNTs) distribution are considered through the thickness in polymeric matrix. The governing nonlinear dynamic equation is derived based on the von Kármán nonlinearity with using of Hamilton's principle. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. Effects of different patterns of reinforcement, volume fraction, excitation force and the length scale parameter on the frequency-response curves of the carbon nanotube reinforced composite beam are investigated. The results show that volume fraction and the distribution of CNTs play an important role on superharmonic and subharmonic resonances of the carbon nanotube reinforced composite beams.