• Advances in nano research
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• v.3 no.4
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• pp.193-206
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• 2015

The present paper investigate the elastic buckling of chiral double-walled carbon nanotubes (DWCNTs) under axial compression. Using the non-local elasticity theory, Timoshenko beam model has been implemented. According to the governing equations of non-local theory, the analytical solution is derived and the solution for non-local critical buckling loads is obtained. The numerical results show the influence of non-local small-scale coefficient, the vibrational mode number, the chirality of carbon nanotube and aspect ratio of the (DWCNTs) on non-local critical buckling loads of the (DWCNTs). The results indicate the dependence of non-local critical buckling loads on the chirality of single-walled carbon nanotube with increase the non-local small-scale coefficient, the vibrational mode number and aspect ratio of length to diameter.

• Structural Engineering and Mechanics
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• v.67 no.4
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• pp.417-425
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• 2018

The main aim of this paper is to investigate the bending of Euler-Bernouilli nano-beams made of bi-directional functionally graded materials (BDFGMs) using Eringen's non-local elasticity theory in the integral form with compare the differential form. To the best of the researchers' knowledge, in the literature, there is no study carried out into integral form of Eringen's non-local elasticity theory for bending analysis of BDFGM Euler-Bernoulli nano-beams with arbitrary functions. Material properties of nano-beam are assumed to change along the thickness and length directions according to arbitrary function. The approximate analytical solutions to the bending analysis of the BDFG nano-beam are derived by using the Rayleigh-Ritz method. The differential form of Eringen's non-local elasticity theory reveals with increasing size effect parameter, the flexibility of the nano-beam decreases, that this is unreasonable. This problem has been resolved in the integral form of the Eringen's model. For all boundary conditions, it is clearly seen that the integral form of Eringen's model predicts the softening effect of the non-local parameter as expected. Finally, the effects of changes of some important parameters such as material length scale, BDFG index on the values of deflection of nano-beam are studied.

• Advances in aircraft and spacecraft science
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• v.6 no.4
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• pp.273-282
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• 2019

This article is concerned with the investigation of geometrically non-linear vibration response of refined thick porous nanobeams. To this end, non-local theory of elasticity has been adopted to provide the nanobeam formulation. Voids or pores can affect the material characteristics of the nanobeam. So, their effects have been considered in this research and also there are various void distributions. The closed form solution of the non-linear problem has been used that is adopted from previous articles. Then, it is focused on the impacts of non-local field, void distribution, void amount and geometrical properties on non-linear vibrational characteristic of a nano-size beam.

• Structural Engineering and Mechanics
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• v.65 no.3
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• pp.335-342
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• 2018

The transverse free vibration of chiral double-walled carbon nanotube (DWCNTs) embedded in elastic medium is modeled by the non-local elasticity theory and Euler Bernoulli beam model. The governing equations are derived and the solutions of frequency are obtained. According to this study, the vibrational mode number, the small-scale coefficient, the Winkler parameter and chirality of double-walled carbon nanotube on the frequency ratio (xN) of the (DWCNTs) are studied and discussed. The new features of the vibration behavior of (DWCNTs) embedded in an elastic medium and the present solutions can be used for the static and dynamic analyses of double-walled carbon nanotubes.

• Advances in nano research
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• v.4 no.1
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• pp.31-44
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• 2016

This article is concerned with the free vibration problem for chiral double-walled carbon nanotube (DWCNTs) modelled using the non-local elasticity theory and Euler Bernoulli beam model. According to the governing equations of non-local Euler Bernoulli beam theory and the boundary conditions, the analytical solution is derived and two branches of transverse wave propagating are obtained. The numerical results obtained provide better representations of the vibration behaviour of double-walled carbon nanotube, where the aspect ratio of the (DWCNTs), the vibrational mode number, the small-scale coefficient and chirality of double-walled carbon nanotube on the frequency ratio (${\chi}^N$) of the (DWCNTs) are significant. In this work, the numerical results obtained can be used to predict and prevent the phenomenon of resonance for the forced vibration analyses of double -walled carbon nanotubes.

• Advances in aircraft and spacecraft science
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• v.6 no.1
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• pp.1-18
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• 2019

This present article represents the study of the forced vibration of nanobeam of a single-walled carbon nanotube (SWCNTs) surrounded by a polymer matrix. The modeling was done according to the Euler-Bernoulli beam model and with the application of the non-local continuum or elasticity theory. Particulars cases of the local elasticity theory have also been studied for comparison. This model takes into account the different effects of the interaction of the Winkler's type elastic medium with the nanobeam of carbon nanotubes. Then, a study of the influence of the amplitude distribution and the frequency was made by variation of some parameters such as (scale effect ($e_0{^a}$), the dimensional ratio or aspect ratio (L/d), also, bound to the mode number (N) and the effect of the stiffness of elastic medium ($K_w$). The results obtained indicate the dependence of the variation of the amplitude and the frequency with the different parameters of the model, besides they prove the local effect of the stresses.

• Structural Engineering and Mechanics
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• v.70 no.3
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• pp.269-277
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• 2019

Using the non-local elasticity theory, Timoshenko beam model is developed to study the non- local buckling of Triple-walled carbon nanotubes (TWCNTs) embedded in an elastic medium under axial compression. The chirality and small scale effects are considered. The effects of the surrounding elastic medium based on a Winkler model and van der Waals' (vdW) forces between the inner and middle, also between the middle and outer nanotubes are taken into account. Considering the small-scale effects, the governing equilibrium equations are derived and the critical buckling loads under axial compression are obtained. The results show that the critical buckling load can be overestimated by the local beam model if the small-scale effect is overlooked for long nanotubes. In addition, significant dependence of the critical buckling loads on the chirality of zigzag carbon nanotube is confirmed. Furthermore, in order to estimate the impact of elastic medium on the non-local critical buckling load of TWCNTs under axial compression, the use of these findings are important in mechanical design considerations, improve and reinforcement of devices that use carbon nanotubes.

• Structural Engineering and Mechanics
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• v.72 no.2
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• pp.229-244
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• 2019

Weak form integral equations are developed to investigate the flapwise bending vibration of the rotating beams. Rayleigh and Eringen nonlocal elasticity theories are used to investigate the rotatory inertia and Size-dependency effects on the flapwise bending vibration of the rotating cantilever beams, respectively. Through repetitive integrations, the governing partial differential equations are converted into weak form integral equations. The novelty of the presented approach is the approximation of the mode shape function by a power series which converts the equations into solvable one. Substitution of the power series into weak form integral equations results in a system of linear algebraic equations. The natural frequencies are determined by calculation of the non-trivial solution for resulting system of equations. Accuracy of the proposed method is verified through several numerical examples, in which the influence of the geometry properties, rotatory inertia, rotational speed, taper ratio and size-dependency are investigated on the natural frequencies of the rotating beam. Application of the weak form integral equations has made the solution simpler and shorter in the mathematical process. Presented relations can be used to obtain a close-form solution for quick calculation of the first five natural frequencies of the beams with flapwise vibration and non-local effects. The analysis results are compared with those obtained from other available published references.

• Structural Engineering and Mechanics
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• v.32 no.6
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• pp.811-833
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• 2009

In the present article bending, buckling and vibration analyses of tapered beams using Eringen non-local elasticity theory are being carried out. The associated governing differential equations are solved employing Rayleigh-Ritz method. Both Euler-Bernoulli and Timoshenko beam theories are considered in the analyses. Present results are in good agreement with those reported in literature. Beam material is considered to be made up of functionally graded materials (fgms). Non-local analyses for tapered beam with simply supported - simply supported, clamped - simply supported and clamped - free boundary conditions are carried out and discussed. Further, effect of length to height ratio on maximum deflections, vibration frequencies and critical buckling loads are studied.

• Steel and Composite Structures
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• v.35 no.1
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• pp.147-157
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• 2020

In the present research, thermo-elastic buckling of small scale functionally graded material (FGM) nano-size plates with clamped edge conditions rested on an elastic substrate exposed to uniformly, linearly and non-linearly temperature distributions has been investigated employing a secant function based refined theory. Material properties of the FGM nano-size plate have exponential gradation across the plate thickness. Using Hamilton's rule and non-local elasticity of Eringen, the non-local governing equations have been stablished in the context of refined four-unknown plate theory and then solved via an analytical method which captures clamped boundary conditions. Buckling results are provided to show the effects of different thermal loadings, non-locality, gradient index, shear deformation, aspect and length-to-thickness ratios on critical buckling temperature of clamped exponential graded nano-size plates.