• Structural Engineering and Mechanics
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• v.66 no.2
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• pp.237-248
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• 2018

This study presents the investigation of wave dispersion characteristics of a magneto-electro-elastic functionally graded (MEE-FG) nanosize beam utilizing nonlocal strain gradient theory (NSGT). In this theory, a material length scale parameter is propounded to show the influence of strain gradient stress field, and likewise, a nonlocal parameter is nominated to emphasize on the importance of elastic stress field effects. The material properties of heterogeneous nanobeam are supposed to vary smoothly through the thickness direction based on power-law form. Applying Hamilton's principle, the nonlocal governing equations of MEE-FG nanobeam are derived. Furthermore, to derive the wave frequency, phase velocity and escape frequency of MEE-FG nanobeam, an analytical solution is employed. The validation procedure is performed by comparing the results of present model with results exhibited by previous papers. Results are rendered in the framework of an exact parametric study by changing various parameters such as wave number, nonlocal parameter, length scale parameter, gradient index, magnetic potential and electric voltage to show their influence on the wave frequency, phase velocity and escape frequency of MEE-FG nanobeams.

• Steel and Composite Structures
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• v.28 no.1
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• pp.13-24
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• 2018

A size-dependent novel hyperbolic shear deformation theory of simply supported functionally graded beams is presented in the frame work of the non-local strain gradient theory, in which the stress accounts for only the nonlocal strain gradients stress field. The thickness stretching effect (${\varepsilon}_z{\neq}0$) is also considered here. Elastic coefficients and length scale parameter are assumed to vary in the thickness direction of functionally graded beams according to power-law form. The governing equations are derived using the Hamilton principle. The closed-form solutions for exact critical buckling loads of nonlocal strain gradient functionally graded beams are obtained using Navier's method. The derived results are compared with those of strain gradient theory.

• Structural Engineering and Mechanics
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• v.54 no.4
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• pp.693-710
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• 2015

This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler-Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton's principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

• Advances in materials Research
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• v.6 no.3
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• pp.279-301
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• 2017

Thermo-mechanical vibration characteristics of in homogeneousporous functionally graded (FG) micro/nanobeam subjected to various types of thermal loadings are investigated in the present paper based on modified couple stress theory with consideration of the exact position of neutral axis. The FG micro/nanobeam is modeled via a refined hyperbolic beam theory in which shear deformation effect is verified needless of shear correction factor. A modified power-law distribution which contains porosity volume fraction is used to describe the graded material properties of FG micro/nanobeam. Temperature field has uniform, linear and nonlinear distributions across the thickness. The governing equations and the related boundary conditions are derived by Extended Hamilton's principle and they are solved applying an analytical solution which satisfies various boundary conditions. A comparison study is performed to verify the present formulation with the known data in the literature and a good agreement is observed. The parametric study covered in this paper includes several parameters such as thermal loadings, porosity volume fraction, power-law exponents, slenderness ratio, scale parameter and various boundary conditions on natural frequencies of porous FG micro/nanobeams in detail.

• Advances in nano research
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• v.8 no.1
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• pp.37-47
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• 2020

This work focuses on the behavior of non-local shear deformation beam theory for the vibration of functionally graded (FG) nanobeams with porosities that may occur inside the functionally graded materials (FG) during their fabrication, using the non-local differential constitutive relations of Eringen. For this purpose, the developed theory accounts for the higher-order variation of transverse shear strain through the depth of the nanobeam. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton's principle. Analytical solutions are presented for a simply supported FG nanobeam with porosities. The validity of this theory is verified by comparing some of the present results with other higher-order theories reported in the literature, the influence of material parameters, the volume fraction of porosity and the thickness ratio on the behavior mechanical P-FGM beam are represented by numerical examples.

• Structural Engineering and Mechanics
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• v.71 no.6
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• pp.669-682
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• 2019

We in this article study nonlinear thermal buckling of bi-directional functionally graded beams in the theoretical frameworks of nonlocal strain graded theory. To begin with, it is assumed that the effective material properties of beams vary continuously in both the thickness and width directions. Then, we utilize a higher-order shear deformation theory that includes a physical neutral surface to derive the size-dependent governing equations combining with the Hamilton's principle and the von $K{\acute{a}}rm{\acute{a}}n$ geometric nonlinearity. It should be pointed out that the established model, containing a nonlocal parameter and a strain gradient length scale parameter, can availably account for both the influence of nonlocal elastic stress field and the influence of strain gradient stress field. Subsequently, via using a easier group of initial asymptotic solutions, the corresponding analytical solution of thermal buckling of beams is obtained with the help of perturbation method. Finally, a parametric study is carried out in detail after validating the present analysis, especially for the effects of a nonlocal parameter, a strain gradient length scale parameter and the ratio of the two on the critical thermal buckling temperature of beams.

• Advances in nano research
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• v.6 no.4
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• pp.399-422
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• 2018

Unwanted vibration is an issue in many industrial systems, especially in nano-devices. There are many ways to compensate these unwanted vibrations based on the results of the past researches. Elastic medium and smart material etc. are effective methods to restrain unnecessary vibration. In this manuscript, dynamic analysis of viscoelastic nanosensor which is made of functionally graded (FGM) nanobeams is investigated. It is assumed that, the shaft is flexible. The system is modeled based on Timoshenko beam theory and also environmental condition, external linear varying loads and thermal loading effect are considered. The equations of motion are extracted by using energy method and Hamilton principle to describe the translational and shear deformation's behavior of the system. Governing equations of motion are extracted by supplementing Eringen's nonlocal theory. Finally vibration behavior of system especially the frequency of system is developed by implementation Semi-analytical differential transformed method (DTM). The results are validated in the researches that have been done in the past and shows good agreement with them.

• Steel and Composite Structures
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• v.29 no.5
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• pp.579-590
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• 2018

Size-dependent free vibration responses and magneto-electro-elastic bending results of a three layers piezomagnetic curved beam rest on Pasternak's foundation are presented in this paper. The governing equations of motion are derived based on first-order shear deformation theory and nonlocal piezo-elasticity theory. The curved beam is containing a nanocore and two piezomagnetic face-sheets. The piezomagnetic layers are imposed to applied electric and magnetic potentials and transverse uniform loadings. The analytical results are presented for simply-supported curved beam to study influence of some parameters on vibration and bending results. The important parameters are spring and shear parameters of foundation, applied electric and magnetic potentials, nonlocal parameter and radius of curvature of curved beam. It is concluded that the increase in radius of curvature tends to an increase in the stiffness of curved beam and consequently natural frequencies increase and bending results decrease. In addition, it is concluded that with increase of nonlocal parameter of curved beam, the stiffness of structure is decreased that leads to decrease of natural frequency and increase of bending results.

• Steel and Composite Structures
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• v.38 no.5
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• pp.583-597
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• 2021

This article develops a nonclassical model to analyze bending response of squared perforated microbeams considering the coupled effect of microstructure and surface stress under different loading and boundary conditions, those are not be studied before. The corresponding material and geometrical characteristics of regularly squared perforated beams relative to fully filled beam are obtained analytically. The modified couple stress and the modified Gurtin-Murdoch surface elasticity models are adopted to incorporate the microstructure as well as the surface energy effects. The differential equations of equilibrium including the Poisson's effect are derived based on minimum potential energy. Exact closed form solution is obtained for bending behavior of the proposed model considering the classical and nonclassical boundary conditions for both uniformly distributed and concentrated loads. The proposed model is verified with results available in the literature. Influences of the microstructure length scale parameter, surface energy, beam thickness, boundary and loading conditions on the bending behavior of perforated microbeams are investigated. It is observed that microstructure and surface parameters are vital in investigation of the bending behavior of perforated microbeams. The obtained results are supportive for the design, analysis and manufacturing of perforated nanobeams that commonly used in nanoactuators, nanoswitches, MEMS and NEMS systems.

• Advances in nano research
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• v.12 no.2
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• pp.139-150
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• 2022

This paper presents sets of explicit analytical equations that compute the static displacements of nanobeams by adopting the nonlocal elasticity theory of Eringen within the framework of Euler Bernoulli and Timoshenko beam theories. Castigliano's theorem is applied to an equivalent Virtual Local Beam (VLB) made up of linear elastic material to compute the displacements. The first derivative of the complementary energy of the VLB with respect to a virtual point load provides displacements. The displacements of the VLB are assumed equal to those of the nonlocal beam if nonlocal effects are superposed as additional stress resultants on the VLB. The illustrative equations of displacements are relevant to a few types of loadings combined with a few common boundary conditions. Several equations of displacements, thus derived, matched precisely in similar cases with the equations obtained by other analytical methods found in the literature. Furthermore, magnitudes of maximum displacements are also in excellent agreement with those computed by other numerical methods. These validated the superposition of nonlocal effects on the VLB and the accuracy of the derived equations.