Semi-analytical solutions for static and dynamic responses of bi-directional functionally graded nonuniform nanobeams with surface energy effect

Shanab, Rabab A.; Attia, Mohamed A.

doi:10.1007/s00366-020-01205-6

Cited by 20 publications

(11 citation statements)

References 132 publications

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“…Power law was used to express the gradation of the material properties of the bulk continuum in the thickness and length directions, whereas the surface elastic constants were assumed to vary in thickness direction only, and the nonlocal parameter was assumed to be spatially-independent. Shanab and Attia [68][69][70] studied the bending, buckling, and vibration characteristics of uniform and tapered BDFG Euler-Bernoulli and Timoshenko micro/nanobeams in the framework of the MCST and GM-SET. Unlike [67], all the material properties of the bulk and surface continuums were assumed to vary in both the thickness and length directions via power law.…”

Section: Introductionmentioning

confidence: 99%

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

Attia

Shanab

2022

Acta Mech

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This paper presents an investigation of the dynamic behavior of bi-directionally functionally graded (BDFG) micro/nanobeams excited by a moving harmonic load. The formulation is established in the context of the surface elasticity theory and the modified couple stress theory to incorporate the effects of surface energy and microstructure, respectively. Based on the generalized elasticity theory and the parabolic shear deformation beam theory, the nonclassical governing equations of the problem are obtained using Lagrange’s equation accounting for the physical neutral plane concept. The material properties of the beam smoothly change along both the axial and thickness directions according to power-law distribution, accounting for the gradation of the material length scale parameter and the surface parameters, i.e., residual surface stress, two surface elastic constants, and surface mass density. Using trigonometric Ritz method (TRM), the trial functions denoting transverse, axial deflections, and rotation of the cross sections of the beam are expressed in sinusoidal form. Then, with the aid of Lagrange’s equation, the system of equations of motion are derived. Finally, Newmark method is employed to find the dynamic responses of BDFG subjected to a moving harmonic load. To validate the present formulation and solution method, some comparisons of the obtained fundamental frequency and dynamic response with those available in the literature are performed. A parametric study is performed to extensively explore the impact of the key parameters such as the gradient indices in both directions, moving speed, and excitation frequency of the acting load on the dynamic response of BDFG nanobeams. The obtained results can serve as a guideline for assessing the multi-functional and optimal design of micro/nanobeams acted upon by a moving load.

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Section: Introductionmentioning

confidence: 99%

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

Attia

Shanab

2022

Acta Mech

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“…Hence, there is a necessity to consider the multi-directional FGMs whose properties are customized and varied in two or three directions to obtain materials with optimized properties. 7…”

Section: Introductionmentioning

confidence: 99%

“…It was shown that factors such as the initial amplitude, elastic and mass density ratios, and the FG indices have noticeable influence on the frequencies and the damping behavior of the beams. Shanab and Attia 7 carried out the bending, buckling, and vibration behaviors of 2D-FG tapered micro and nano Euler–Bernoulli beams with simply supported conditions. The modified couple stress and the Gurtin–Murdoch surface elasticity theories are both employed to take into account the small scale and surface energy effects, and the GDQM is used to calculate the static bending deflections, critical buckling loads, and the fundamental frequencies of 2D-FG nonuniform beams at the micro/nano scales.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, there is a necessity to consider the multi-directional FGMs whose properties are customized and varied in two or three directions to obtain materials with optimized properties. 7 Mirzabeigy el al. 8 investigated the natural vibrations of double Euler-Bernoulli beam system joined through variable Winkler elastic translational springs.…”

Section: Introductionmentioning

confidence: 99%

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Natural vibrations of double bi-directional functionally graded Euler–Bernoulli beams connected by a variable Winkler elastic layer

Sari

2022

Journal of Low Frequency Noise, Vibration and Active Control

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The free transverse vibrations of two-dimensional functionally graded double Euler–Bernoulli beam system connected through a variable Winkler elastic layer are presented. The Modulus of elasticity and the density of the material of the beams are assumed to vary along the thickness and the length of the beams according to power-law and exponential functions, respectively. The governing differential equations of motion are established by the means of Hamilton’s principle. The Chebyshev Spectral Collocation Method (CSCM) is used to calculate the natural frequencies of the two-dimensional functionally graded (2D-FG) double beams. The effects of several parameters such as the height ratio of the beams, the stiffness of the coupling layer, the axial and transverse functionally graded indexes, and the boundary conditions on the natural frequencies of the 2D-FG double beams have been investigated. The frequencies obtained from the CSCM have been validated by comparing them with those available in previous literature.

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“…Remarkable features of CNTs, however, cannot be precisely realized applying the classical approach of the elasticity theory. A great attention is, accordingly, devoted in the recent literature to the generalized elasticity theories, such as nonlocal elasticity approach [6][7][8][9][10][11][12][13][14][15][16], strain gradient theory [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], and unified elasticity frameworks [32][33][34][35][36][37][38][39][40][41][42][43][44], for nanoscopic study of the field quantities.…”

Section: Introductionmentioning

confidence: 99%

Modified couple stress flexure mechanics of nanobeams

Sedighi¹,

Abouelregal²,

Faghidian³

2021

Phys. Scr.

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In view of numerous contributions accessible in the literature associated with the modified couple stress theory, its deficiency in precise description of the material response of continua with nanostructural features is investigated. The flexure mechanics of nano-scale beams in the framework of the modified couple stress theory is reassessed applying a consistent variational scheme. The physically motivated constitutive laws of the stress resultant fields, flexural moment and shear force, associated with the modified couple stress beam are introduced. The well-established differential conditions of equilibrium equipped with the proper mathematical form of the higher-order boundary conditions are reinstated. The exact analytical solutions of the kinematic fields of the nano-sized beam are derived by direct solving of the governing coupled differential equations. The flexure response of the modified couple stress beam is numerically illustrated wherein the effects of the symmetric rotation gradients along with the transverse shear deformation are examined. Closed-form analytical formulae of the elastic modulus of Carbon Nanotubes are detected and thoroughly discussed. Serious suspicions on the applicability of the modified couple stress beam model in accurately capturing the size-effects in nano-materials are emerged as evinced by rigorous examination of the elastic modulus of CNTs.

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Semi-analytical solutions for static and dynamic responses of bi-directional functionally graded nonuniform nanobeams with surface energy effect

Cited by 20 publications

References 132 publications

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

Natural vibrations of double bi-directional functionally graded Euler–Bernoulli beams connected by a variable Winkler elastic layer

Modified couple stress flexure mechanics of nanobeams

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