Nonlinear vibration analysis of an electrostatic functionally graded nano-resonator with surface effects based on nonlocal strain gradient theory

https://doi.org/10.1016/j.ijmecsci.2018.11.030Get rights and content

Highlights

  • The nonlinear behavior of a nonlocal strain gradient FG nano resonator is studied.

  • The influence of the size-dependent parameters and FG fraction index is investigated.

  • Size-dependent parameters exert either stiffness softening or hardening behavior.

  • The rise in FG volume fraction index enhances the stiffness softening behavior.

  • Saddle-node bifurcation points displace by considering the surface effects.

Abstract

In this paper, a comprehensive analysis of the nonlinear vibration of an electrostatic nanobeam resonator is presented based on the nonlocal strain gradient theory (NSGT) and by incorporating the Gurtin-Murdoch surface elasticity theory. The Von-Kármán geometrical nonlinearity and inter-molecular dispersion forces, i.e., van der Waals and Casimir forces are included in the equation of motion. Both DC and AC components of the electrostatic actuation are regarded as the excitation terms. The nanobeam is considered to be composed of a power-law functionally graded (FG) material. Utilizing Hamilton’s principle, the size-dependent nonlinear equation of motion of the system is derived. Multiple scales technique in conjunction with the differential quadrature method (DQM) is adopted to analytically obtain the solution. Results obtained are shown to be in good agreement against available literature. Static deflection and fundamental natural frequency are obtained for different size-dependent and volume fraction index parameters. Meanwhile, the variation of the oscillation amplitude by the quality factor, excitation magnitude, and frequency is determined near the primary resonance. The acquired results revealed that the nonlocal and strain gradient parameters can significantly displace the multi-valued portions and instability thresholds of the dynamical response diagrams. It is shown that the increment of the volume fraction index reduces the pull-in voltage while increasing any of the size-dependent parameters enlarge the instability voltage. Moreover, the surface effects induce the stiffness hardening behavior, whereas the inter-molecular forces impose the stiffness softening effect.

Introduction

Electrostatic nano-resonators (or nanoelectromechanical resonators) are electrically driven oscillatory systems which are rapidly developing for high precision applications. Duo to their ultra-tiny dimensions, nano-resonators are capable of operating at high frequencies and hence perfectly suited for the state-of-the-art technologies such as sub-micrometer actuators, switches, sensors and transducers [1], [2], [3]. The structure of the electrostatic nano resonators typically consists of a flexible nanobeam actuated by the electrostatic and fringing field [4] via a stationary electrode with a particular gap. The nanobeam also undergoes inter-molecular interactions, namely van der Waals and Casimir forces. Besides, the structural uncertainties [5], [6], [7] such as rippling effect [8] and variable gap size [9] are indicated to be effective on the dynamical behavior of these systems. Attributed to their extremely small sizes being comparable to their structural inter-atomic distances, the nanobeams deflection encounters size-dependent effects. It is known that the classical continuum theory cannot shed light on these phenomena [10]. Hence, non-classical continuum methods have been widely employed to capture size effects of the nanostructures.

Initiated by Eringen [11], the nonlocal theory is one of the applicable non-classical theories which states that the stress of a point depends not only on the strain of that point but on the strain of the all points in the continuum. By applying nonlocal theory, inter-atomic long-range forces are considered in the formation of the constitutive equations. Numerous papers studied the static and dynamic behavior of the nanobeams exploiting nonlocal elasticity theory [12], [13], [14]. For instance, Fakhrabadi et al. in [15] demonstrated that the contribution of the nonlocal effect leads to higher values of the instability voltage for an electrostatic carbon nanotube. However, it is known that the nonlocal theory is incapable of identifying stiffness enhancement of the micro and nanostructures which have been observed through experiments [16], [17]. In other words, nonlocal elastic models can only account for the softening behavior with increasing of the size-dependent parameter. The stiffness enhancement of the micro and nanostructures could be justified by applying the strain gradient theory [18]. The strain gradient theory (SGT) extended by Mindlin [19] and modified by Aifantis [20] represents the small-scaled structures as atoms with higher-order deformation mechanism instead of collections of points and hence takes into account the stress with additional strain gradient terms. Relying on the strain gradient theory, investigating dynamical behavior of micro and nanostructures have been carried out in several studies [21], [22], [23]. As an example, Arani et al.[24] studied the nonlinear vibration of a nanobeam coupled with a piezoelectric nanobeam within an elastic medium, via strain gradient theory. They concluded that for lower small scale parameters, the effect of external electric voltage becomes more considerable. Nevertheless, In contrast to the nonlocal model, the strain gradient elasticity can only anticipate stiffness hardening by increasing the material characteristic length parameter.

To overcome the contradiction of the nonlocal and strain gradient theories, recently Lim et al. [25] elaborated on the nonlocal strain gradient theory (NSGT) in the framework of thermodynamics and reformed the classical continuum equations to a novel form featuring both the Eringen’s nonlocal and Aifantis’s strain gradient parameters. The NSGT formulation considers two entirely different effects, i.e. inter-atomic long-range forces and deformation gradient in accordance with the nonlocal and strain gradient theories respectively [26]. Nonetheless, A limited number of analyses have been performed to investigate the dynamical behavior of the nanostructures, based on the NSGT. For instance, Li et al. [27] addressed the axial vibration of the small-scaled rods relying on the NSGT. They figured out that the system can exert either stiffness hardening or softening if the SGT parameter is larger or smaller than the nonlocal parameter respectively. Xu et al.[28] studied the buckling and bending of the elastic NSGT beams and demonstrated that the bending, in contrast to the buckling, is independent of the higher-order boundary conditions. Ebrahimi et al.[29] studied the variation of the natural frequency of a nanobeam resting on a viscoelastic foundation based on the nonlocal strain gradient theory including thermal effects. Li et al.[30] analyzed the free vibration of a nonlocal strain gradient Timoshenko beam and compared the results by those of a Euler-Bernoulli beam. Guo et al. [31] studied the vibro-buckling and natural frequencies of an axially moving and rotating nanobeam on the basis of NSGT and found that the critical rotating velocity of the forward and backward waves decrease/increase when the nonlocal/strain gradient parameter increases. Lu et al. [32] demonstrated that the natural frequency of an NSGT beam is smaller than the corresponding value of the SGT beam and larger than the nonlocal beam. Li et al. in [33] authenticate that the nonlocal strain gradient theory can overcome the paradox which the nonlocal and classical beam formulation yields the same bending deformation when the beam is subjected to a concentrated force. Ouakad et al. [34] investigated the static deflection and natural frequencies of a carbon nanotube NEMS actuated by a dc voltage based on the NSGT and strain velocity theory. The frequency response of an NSGT nanotube near the fundamental natural frequency is explored in [35] by Ghayesh et al. considering a linear harmonic excitation. It is demonstrated that by increasing the SGT or nonlocal parameter, the resonance frequency increases or decreases respectively. It is also found that the SGT parameter, in contrast to the nonlocal parameter, does not affect the modal interaction.

On the other hand, by miniaturizing structures to the nanoscale, the elastic modeling should be altered to embrace the surface effects, namely the surface elasticity and surface stresses which is first proposed by Gurtin and Murdoch [36], [37]. Considered as a size-dependent effect, surface effects emerge due to the enlargement of the surface layer atoms with respect to all atoms present in micro/nanostructures. In light of the Gurtin-Murdoch surface elasticity theory, numerous articles have been published. However, the influence of the surface elasticity and surface residual stress on the nonlinear dynamical behavior of the NEMSs has not been widely reflected in the previous researches. For instance, Gheshlaghi et al. [38] studied the free vibration of the nanobeams incorporating surface effects. Miandoab et al. [39] demonstrated that the surface energy may lead to an inverse effect on nano-resonators frequency response by changing surface parameters. Wang et al. [40] studied the dynamical response of an electrically actuated nanobeam including surface energy. They demonstrated that the neglecting surface energy results in lower natural frequencies of the system. Zhao et al. [41] assessed the nonlinear dynamics of nanobeam incorporating surface effects and showed that for negative values of the surface residual stress, the system is unstable. In Ref[42]. Ma et al. investigated the effect of the surface elasticity and surface residual stress on the instability of NEMS. In Ref[43]. by Pourkiaee et al., it is shown that whether or not taking the surface effects into account could result in two opposing behavior in frequency response, namely stiffness hardening and softening phenomenon. Eltaher et al. [44] demonstrated that the surface effects may increase or decrease the fundamental natural frequency of a simply supported nanobeam according to the sign of the residual surface stress.

Recently, functionally graded materials have attracted considerable interest owing to their well-known advantages over the conventional isotropic materials and laminated composites. Exploiting FG materials, the required demands of the nanosystems performance could be accomplished by synthesizing the material distribution law[45]. In this regard, several studies can be found in the literature. For example, in Ref[46]. Yang et al. studied the instability of a NEMS made of an NSGT functionally graded nanotube considering the inter-molecular forces. the nonlinear bending and free vibrations of an FG nanobeam are studied in Ref[47]. by Li et al. based on the nonlocal strain gradient theory. In this paper, it is deduced that the bending deflections can be generally decreased by increasing/decreasing material length scale parameter/the nonlocal parameter. In Ref[48]. by Şimşek, the nonlocal strain gradient theory involving the mid-plane stretching effect and a novel Hamiltonian approach is utilized to estimate the natural frequencies of an FGM nanobeam. Shanab et al. [49] investigated the nonlinear bending response of functionally graded nanobeams including surface effects and demonstrated the high-dependency of the deflection to the values of the material distribution law and gradient index. Sharabiani et al. [50] studied the nonlinear free vibration of an FG nanobeam considering the surface elasticity and stresses. They reported that at higher mode numbers, in contrast with the homogeneous nanobeam, the surface effects are less/more dominant in small/large amplitude ratios. Sedighi et al. [51] showed that enlargement of the gradient power decreases the fundamental frequency of electrostatic functionally graded nano-actuators. Hosseini-Hashemi et al.[52]analytically studied the free vibration of FG nanobeams considering surface effects. They figured out that ascending the volume fraction index decreases influence of the surface energy on the natural frequency.

To the authors’ best knowledge, the vibrational analysis of the nonlocal strain gradient FG nano-resonators considering the surface effects has not been studied so far and the nonlinear dynamics of such systems in view of both static (dc) and harmonic (ac) components of electrostatic actuation has not yet been reported in the literature. Accordingly, in this paper, a comprehensive study on the free and forced vibration of an FG nanoelectromechanical system based on the nonlocal strain gradient theory and surface elasticity theory is conducted for the first time. The considered nono resonator is assumed to comprise a power-law functionally graded Euler-Bernoulli NSGT beam, which is clamped at both ends and vibrates laterally under the influence of the electrostatic field, fringing field, and inter-molecular interactions incorporating surface elasticity and tension. The equation of motion is derived on the basis of Hamilton’s principle. The geometric nonlinearity due to the von-Kármán’s strain relations is taken into consideration. The FG nanobeam elasticity and density are varied according to the power law through the thickness. The multiple scales method is applied to analytically determine a uniform approximation of the nonlinear partial differential equation of motion. Then, the differential quadrature method (DQM) is implemented to find the solution of the static and dynamic nonlinear boundary value and eigenvalue problems. Subsequently, the frequency and force response of the system near the primary resonance is obtained. The acquired results are validated by comparing the obtained variation of the natural frequencies with those of an available study in the literature. Eventually, the influence of the size-dependent parameters, gradient index, inter-molecular interactions, dissipation force, surface elasticity, and surface residual stress are surveyed.

Section snippets

Theory and formulation

Consider a nanoelectromechanical system as illustrated in Fig. 1. This system consists of a flexible FG nanobeam of length L, width b and thickness h located in the vicinity of a fixed electrode with a gap denoted by g. For the functionally graded material, the material property variation through the thickness is governed by a power-law as:P(z)=(P2P1)(zh+12)k+P1in which P2 and P1 are material properties on the top and bottom of the beam, respectively, and k denotes the volume fraction index

Solve methodology

It is assumed that the total beam deflection consists of two parts, namely static (ws(x)) and dynamic (wd(x, t)), the former owing to stationary and the latter due to the harmonic electrostatic excitation terms:w(x,t)=ws(x)+wd(x,t).Since ws is only dependent on dc voltage and does not alternate by time, all the time dependent terms of Eq. (24) are disregarded so that the equilibria is obtained as the following boundary value problem (BVP):β2ws(6)=ws(4)[Γ1(ws,ws)+Γ2(ws,ws)+ζ2+η(ws)](ws(2)α2ws(4

Results and discussion

In order to validate the acquired results based on the proposed method, the variation of the natural frequency by the electrostatic actuation voltage is compared with two available results in literature[51], [58]. The considered systems are comprised of an FG nanobeam resonator without surface effect. While both systems are subjected to Casimir interaction, the former has no size-dependent effect (α=β=0) and the latter is subjected to the nonlocal effect (α0,β=0). in Fig. 3, the variation of

Summary and conclusion

In this work, the static and dynamic behavior of a functionally graded nano-resonator including the surface effects and inter-molecular forces has been studied via the nonlocal strain gradient theory. The considered system comprises a power-law FG Euler-Bernoulli nanobeam which is subjected to the electrostatic field including both static and harmonic components. The nonlinearity of the proposed model stems from the mid-plane stretching and also presence of the nonlinear forces namely the

References (58)

  • N. Fleck et al.

    Strain gradient plasticity: theory and experiment

    Acta Metall Mater

    (1994)
  • R.D. Mindlin

    Second gradient of strain and surface-tension in linear elasticity

    Int J Solids Struct

    (1965)
  • E.C. Aifantis

    On the role of gradients in the localization of deformation and fracture

    Int J Eng Sci

    (1992)
  • A. Nikpourian et al.

    On the nonlinear dynamics of a piezoelectrically tuned micro-resonator based on non-classical elasticity theories

    Int J Mech Mater Des

    (2016)
  • R. Vatankhah et al.

    Nonlinear forced vibration of strain gradient microbeams

    Appl Math Model

    (2013)
  • A.G. Arani et al.

    Nonlinear vibration of a nanobeam elastically bonded with a piezoelectric nanobeam via strain gradient theory

    Int J Mech Sci

    (2015)
  • L. Li et al.

    Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects

    Int J Mech Sci

    (2017)
  • L. Li et al.

    Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

    Int J Mech Sci

    (2016)
  • X.-J. Xu et al.

    Bending and buckling of nonlocal strain gradient elastic beams

    Compos Struct

    (2017)
  • F. Ebrahimi et al.

    Vibration analysis of viscoelastic inhomogeneous nanobeams resting on a viscoelastic foundation based on nonlocal strain gradient theory incorporating surface and thermal effects

    Acta Mech

    (2017)
  • S. Guo et al.

    Dynamic transverse vibration characteristics and vibro-buckling analyses of axially moving and rotating nanobeams based on nonlocal strain gradient theory

    Microsyst Technol

    (2018)
  • X. Li et al.

    Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory

    Compos Struct

    (2017)
  • H.M. Ouakad et al.

    Static and dynamic response of CNT nanobeam using nonlocal strain and velocity gradient theory

    Appl Math Model

    (2018)
  • M.E. Gurtin et al.

    A continuum theory of elastic material surfaces

    Arch Ration Mech Anal

    (1975)
  • B. Gheshlaghi et al.

    Surface effects on nonlinear free vibration of nanobeams

    Compos Part B: Eng

    (2011)
  • E.M. Miandoab et al.

    Effect of surface energy on nano-resonator dynamic behavior

    Int J Mech Sci

    (2016)
  • D. Zhao et al.

    Nonlinear free vibration of a cantilever nanobeam with surface effects: semi-analytical solutions

    Int J Mech Sci

    (2016)
  • J.B. Ma et al.

    Influence of surface effects on the pull-in instability of NEMS electrostatic switches

    Nanotechnology

    (2010)
  • O. Rahmani et al.

    Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory

    Int J Eng Sci

    (2014)
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