Coupling effect of surface energy and dispersion forces on nonlinear size-dependent pull-in instability of functionally graded micro-/nanoswitches

Attia, Mohamed A.; Mohamed, S.A.

doi:10.1007/s00707-018-2345-6

Cited by 21 publications

(6 citation statements)

References 96 publications

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“…Gurtin and Murdoch ([23,24]) proposed a surface elasticity theory to approximate the contribution of surface energy by assuming an elastic body with elastic surface layers of zero thickness that is perfectly bonded to the surface of the bulk continuum. The Gurtin and Murdoch surface elasticity theory (GM-SET) has been successfully employed in conjunction with the other nonclassical continuum theories in the analysis of homogeneous and one-dimensional FG structures, i.e., GM-SET combined with the DNET ( [47][48][49]), GM-SET combined with the SGT ( [50]), GM-SET combined with the DNSGT ( [51]), GM-SET combined with the MCST ( [52][53][54][55][56][57][58][59][60][61][62][63][64]), and GM-SET combined with the MCST in the framework of DNET ( [65,66]). Based on these studies, it has been shown that incorporating the influence of surface energy may show a stiffness-hardening or stiffness-softening of the studied nanostructures depending on whether the signs of the surface elastic constants are positive or negative.…”

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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“…So, it transforms the differential equations into a set of analogous algebraic equations in terms of the unknown function values at the discrete set of points in the problem domain. DQM has been widely applied in many areas of engineering and science 50–55 . In fact, DQM belongs to the spectral and pseudospectral methods; the DQM is identical to the Chebyshev collocation method whenever the grid points are chosen to be the same for both methods 49 .…”

Section: Introductionmentioning

confidence: 99%

“…DQM has been widely applied in many areas of engineering and science. [50][51][52][53][54][55] In fact, DQM belongs to the spectral and pseudospectral methods; the DQM is identical to the Chebyshev collocation method whenever the grid points are chosen to be the same for both methods. 49 The main advantage of DQM is its capability of producing highly accurate solutions with minimal computational effort especially when the method is applied to problems with globally smooth solutions.…”

Section: Introductionmentioning

confidence: 99%

A novel differential‐integral quadrature method for the solution of nonlinear integro‐differential equations

Mohamed

Abo-Hashem

2021

Math Methods in App Sciences

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In this work, we introduce an integration matrix operator that is fully consistent with the differentiation matrix operator defined by the differential quadrature method (DQM). Using these operators, a generic differential-integral quadrature method "DIQM" is proposed to directly discretize an integro-differential equation as a system of algebraic equations. To extend the applicability of the proposed DIQM to solve nonlinear and/or variable coefficients integrodifferential equation, some matrix manipulations are introduced. A stability analysis for Volterra integro-partial-differential equations is presented and the exponential convergence of the proposed method is examined. Various types of integro-differential equations are solved including ordinary/partial, linear/ nonlinear, Volterra parabolic/hyperbolic integro-differential equations with different boundary and initial conditions. Numerical results demonstrate the exponential convergence and the applicability of DIQM.

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“…The advantage of the differential quadrature method appears when it is used to solve boundaryvalue, initial-value, linear or nonlinear differential equations that DQM requires fewer grid points to obtain acceptable accuracy unlike finite difference method (FDM), finite element method (FEM) and finite volume method (FVM) which may need more number of grid points to obtain the solution. The authors in [41][42][43] succeeded to develop the DQM to solve problems that have an integral term in its differential equation by merging a new integrating matrix operator in the method and obtained differential integral quadrature method (DIQM). Also, by applying DIQM to other applications like nano-beams [44][45][46][47], it succeeds to obtain accurate solutions.…”

Section: Introductionmentioning

confidence: 99%

Computational DIQM Scheme for Solving Nonlinear Volterra Integro-Differential Equations

Mohamed

hashem

Mohamed

2020

The Egyptian International Journal of Engineering Sciences and

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The authors in this work develop the differential quadrature method (DQM) to solve the nonlinear Volterra integro-differential equation by introducing an integration matrix operator that is fully combined with the differentiation matrix in (DQM) method. The obtained method (DIQM) transforms the discretized nonlinear Volterra integro-differential equation into a nonlinear algebraic system of equations which is solved iteratively by Newton's method. The efficiency of the (DIQM) method is examined by solving three examples where the error norms and convergence rates achieve the expected exponential behaviour.

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Coupling effect of surface energy and dispersion forces on nonlinear size-dependent pull-in instability of functionally graded micro-/nanoswitches

Cited by 21 publications

References 96 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

A novel differential‐integral quadrature method for the solution of nonlinear integro‐differential equations

Computational DIQM Scheme for Solving Nonlinear Volterra Integro-Differential Equations

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