• Structural Engineering and Mechanics
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• v.90 no.1
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• pp.1-18
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• 2024

The present study conducts a thorough analysis of thermal vibrations in functionally graded porous nanocomposite beams within a thermal setting. Investigating the temperature-dependent material properties of these beams, which continuously vary across their thickness in accordance with a power-law function, a finite element approach is developed. This approach utilizes a nonlocal strain gradient theory and accounts for a linear temperature rise. The analysis employs four different patterns of porosity distribution to characterize the functionally graded porous materials. A novel two-variable shear deformation beam nonlocal strain gradient theory, based on trigonometric functions, is introduced to examine the combined effects of nonlocal stress and strain gradient on these beams. The derived governing equations are solved through a 3-nodes beam element. A comprehensive parametric study delves into the influence of structural parameters, such as thicknessratio, beam length, nonlocal scale parameter, and strain gradient parameter. Furthermore, the study explores the impact of thermal effects, porosity distribution forms, and material distribution profiles on the free vibration of temperature-dependent FG nanobeams. The results reveal the substantial influence of these effects on the vibration behavior of functionally graded nanobeams under thermal conditions. This research presents a finite element approach to examine the thermo-mechanical behavior of nonlocal temperature-dependent FG nanobeams, filling the gap where analytical results are unavailable.

• Advances in nano research
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• v.7 no.3
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• pp.145-155
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• 2019

In this article, the influence of small scale effects on the free vibration response of curved magneto-electro-elastic functionally graded (MEE-FG) nanobeams has been investigated considering nonlocal elasticity theory. Power-law is used to judge the through thickness material property distribution of MEE nanobeams. The Euler-Bernoulli beam model has been adopted and through Hamilton's principle the Nonlocal governing equations of curved MEE-FG nanobeam are obtained. The analytical solutions are obtained and validated with the results reported in the literature. Several parametric studies are performed to assess the influence of nonlocal parameter, magnetic potential, electric voltage, opening angle, material composition and slenderness ratio on the dynamic behaviour of MEE curved nanobeams. It is believed that the results presented in this article may serve as benchmark results in accurate analysis and design of smart nanostructures.

• Smart Structures and Systems
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• v.19 no.2
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• pp.115-126
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• 2017

In this paper, size dependent bending and free flexural vibration behaviors of functionally graded (FG) nanobeams are investigated using a nonlocal quasi-3D theory in which both shear deformation and thickness stretching effects are introduced. The nonlocal elastic behavior is described by the differential constitutive model of Eringen, which enables the present model to become effective in the analysis and design of nanostructures. The present theory incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect, and furthermore accounts for both shear deformation and thickness stretching effects by virtue of a hyperbolic variation of all displacements through the thickness without using shear correction factor. The material properties of FG nanobeams are assumed to vary through the thickness according to a power law. The neutral surface position for such FG nanobeams is determined and the present theory based on exact neutral surface position is employed here. The governing equations are derived using the principal of minimum total potential energy. The effects of nonlocal parameter, aspect ratio and various material compositions on the static and dynamic responses of the FG nanobeam are discussed in detail. A detailed numerical study is carried out to examine the effect of material gradient index, the nonlocal parameter, the beam aspect ratio on the global response of the FG nanobeam. These findings are important in mechanical design considerations of devices that use carbon nanotubes.

• Steel and Composite Structures
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• v.42 no.6
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• pp.805-826
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• 2022

The free and live load-forced vibration behaviour of porous functionally graded (PFG) higher order nanobeams in the thermal and magnetic fields is investigated comprehensively through this work in the framework of nonlocal strain gradient theory (NLSGT). The porosity effects on the dynamic behaviour of FG nanobeams is investigated using four different porosity distribution models. These models are exploited; uniform, symmetrical, condensed upward, and condensed downward distributions. The material characteristics gradation in the thickness direction is estimated using the power-law. The magnetic field effect is incorporated using Maxwell's equations. The third order shear deformation beam theory is adopted to incorporate the shear deformation effect. The Hamilton principle is adopted to derive the coupled thermomagnetic dynamic equations of motion of the whole system and the associated boundary conditions. Navier method is used to derive the analytical solution of the governing equations. The developed methodology is verified and compared with the available results in the literature and good agreement is observed. Parametric studies are conducted to show effects of porosity parameter; porosity distribution, temperature rise, magnetic field intensity, material gradation index, non-classical parameters, and the applied moving load velocity on the vibration behavior of nanobeams. It has been showed that all the analyzed conditions have significant effects on the dynamic behavior of the nanobeams. Additionally, it has been observed that the negative effects of moving load, porosity and thermal load on the nanobeam dynamics can be reduced by the effect of the force induced from the directed magnetic field or can be kept within certain desired design limits by controlling the intensity of the magnetic field.

• Steel and Composite Structures
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• v.20 no.5
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• pp.1119-1131
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• 2016

As a first attempt, an inverse hybrid numerical method for small scale parameter estimation of functionally graded (FG) nanobeams using measured frequencies is presented. The governing equations are obtained with the Eringen's nonlocal elasticity assumptions and the first-order shear deformation theory (FSDT). The equations are discretized by using the differential quadrature method (DQM). The discretized equations are transferred from temporal domain to frequency domain and frequencies of the nanobeam are obtained. By applying random error to these frequencies, measured frequencies are generated. The measured frequencies are considered as input data and inversely, the small scale parameter of the beam is obtained by minimizing a defined functional. The functional is defined as root mean square error between the measured frequencies and calculated frequencies by the DQM. Then, the conjugate gradient (CG) optimization method is employed to minimize the functional and the small scale parameter is obtained. Efficiency, convergence and accuracy of the presented hybrid method for small scale parameter estimation of the beams for different applied random error, boundary conditions, length-to-thickness ratio and volume fraction coefficients are demonstrated.

• Advances in nano research
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• v.7 no.2
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• pp.135-143
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• 2019

The important effect of porosity on the mechanical behaviors of a continua makes it necessary to account for such an effect while analyzing a structure. motivated by this fact, a new two-step porosity dependent homogenization scheme is presented in this article to investigate the wave propagation responses of functionally graded (FG) porous nanobeams. In the introduced homogenization method, which is a modified form of the power-law model, the effects of porosity distributions are considered. Based on Hamilton's principle, the Navier equations are developed using the Euler-Bernoulli beam model. Thereafter, the constitutive equations are obtained employing the nonlocal elasticity theory of Eringen. Next, the governing equations are solved in order to reach the wave frequency. Once the validity of presented methodology is proved, a set of parametric studies are adapted to put emphasis on the role of each variant on the wave dispersion behaviors of porous FG nanobeams.

• Steel and Composite Structures
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• v.36 no.6
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• pp.643-656
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• 2020

This article presents a comprehensive static analysis of simply supported cross-ply carbon nanotubes reinforced composite (CNTRC) laminated nanobeams under various loading profiles. The nonlocal strain gradient constitutive relation is exploited to present the size-dependence of nano-scale. New higher shear deformation beam theory with hyperbolic function is proposed to satisfy the zero-shear effect at boundaries and parabolic variation through the thickness. Carbon nanotubes (CNTs), as the reinforced elements, are distributed through the beam thickness with different distribution functions, which are, uniform distribution (UD-CNTRC), V- distribution (FG-V CNTRC), O- distribution (FG-O CNTRC) and X- distribution (FG-X CNTRC). The equilibrium equations are derived, and Fourier series function are used to solve the obtained differential equation and get the response of nanobeam under uniform, linear or sinusoidal mechanical loadings. Numerical results are obtained to present influences of CNTs reinforcement patterns, composite laminate structure, nonlocal parameter, length scale parameter, geometric parameters on center deflection ad stresses of CNTRC laminated nanobeams. The proposed model is effective in analysis and design of composite structure ranging from macro-scale to nano-scale.

• Structural Engineering and Mechanics
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• v.86 no.6
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• pp.731-738
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• 2023

This article presents an analytical approach to explore the bending behaviour of of two-dimensional (2D) functionally graded (FG) nanobeams based on a two-variable higher-order shear deformation theory and nonlocal strain gradient theory. The kinematic relations are proposed according to novel trigonometric functions. The material gradation and material properties are varied along the longitudinal and the transversal directions. The equilibrium equations are obtained by using the virtual work principle and solved by applying Navier's technique. A comparative evaluation of results against predictions from literature demonstrates the accuracy of the proposed analytical model. Moreover, a detailed parametric analysis checks for the sensitivity of the bending and stresses response of (2D) FG nanobeams to nonlocal length scale, strain gradient microstructure scale, material distribution and geometry.

• Advances in nano research
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• v.6 no.3
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• pp.201-217
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• 2018

In the present research, wave propagation characteristics of a rotating FG nanobeam undergoing rotation is studied based on nonlocal strain gradient theory. Material properties of nanobeam are assumed to change gradually across the thickness of nanobeam according to Mori-Tanaka distribution model. The governing partial differential equations are derived for the rotating FG nanobeam by applying the Hamilton's principle in the framework of Euler-Bernoulli beam model. An analytical solution is applied to obtain wave frequencies, phase velocities and escape frequencies. It is observed that wave dispersion characteristics of rotating FG nanobeams are extremely influenced by angular velocity, wave number, nonlocal parameter, length scale parameter, temperature change and material graduation.

• Advances in materials Research
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• v.6 no.2
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• pp.93-128
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• 2017

In this paper, thermal vibration behavior of nanoscale beams made of functionally graded (FG) materials subjected to various types of thermal loading are investigated. A Reddy shear deformation beam theory which captures both the microstructural and shear deformation effects without the need for any shear correction factors is employed. Material properties of FG nanobeam are assumed to be temperature-dependent and vary gradually along the thickness according to the power-law form. The influence of small scale is captured based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying analytical solution. The comparison of the obtained results is conducted with those of nonlocal Euler-Bernoulli beam theory and it is demonstrated that the proposed modeling predict correctly the vibration responses of FG nanobeams. The effects of nonlocal parameter, material graduation, mode number, slenderness ratio and thermal loading on vibration behavior of the nanobeams are studied in detail.