Velocity gradient elasticity for nonlinear vibration of carbon nanotube resonators

Abstract

In this study, for the first time, we investigate the nonlocality superimposed to the size effects on the nonlinear dynamics of an electrically actuated single-walled carbon-nanotube-based resonator. We undertake two models to capture the nanostructure nonlocal size effects: the strain and the velocity gradient theories. We use a reduced-order model based on the differential quadrature method (DQM) to discretize the governing nonlinear equation of motion and acquire a discretized-parameter nonlinear model of the system. The structural nonlinear behavior of the system assuming both strain and velocity gradient theories is investigated using the discretized model. The results suggest that nonlocal and size effects should not be neglected because they improve the prediction of corresponding dynamic amplitudes and, most importantly, the critical resonant frequencies of such nanoresonators. Neglecting these effects may impose a considerable source of error, which can be amended using more accurate modeling techniques.

Keywords

velocity gradient elasticity theory / nanotube resonators / differential-quadrature method / nonlinear vibration

Introduction

Owing to their amazing electro-mechanical features, carbon nanotubes (CNTs) and their respective compounds have shown potential applications in numerous electronic devices, energy storage, smart materials and composites, as well as sensors and actuators [¹–⁵]. In addition, they can be employed in probe tips and low-power electronic devices as miniature power plants owing to the sharpness of their tips. Consequently, many researchers and scientists worldwide have conducted investigations of their interesting physio-mechanical properties by introducing nonclassical models; such investigations are expected to disclose their outstanding behavior.

Recently, it was shown that classical continuum mechanics cannot predict the micro- and nanostructure-dependent behavior of sub-micron devices and materials. Size effects should be considered when comparing the internal structure to the structural dimension [⁶–¹¹]. Thus, several nonclassical continuum theories have been presented and developed to lay the ground for precise extraction of the size-dependent behavior of CNTs for future applications. For example, nonlocal theories [¹²] and strain gradient elasticity theory [¹³] are useful in explaining the nanoscale characteristics of nanostructures. In nonlocal theories, it is assumed that the stress at a specific point is a function of the strain at that point and its neighborhood [¹⁴]. It is demonstrated that nonlocal theories only consider the inter-atomic long-range force [¹⁵], and the gradient elasticity theory exclusively considers the higher-order microstructure deformation mechanism [¹⁶]. Apuzzo et al. [¹⁷] investigated the size-dependent behavior of nanobeams using the modified nonlocal strain gradient theory. They formulated and subsequently solved the elastostatic problem of an Euler-Bernoulli FG nanobeam for different boundary conditions. Barretta et al. [¹⁸] extended the nonlocal strain gradient theory and investigated the extensional behavior of nanorods by introducing the integral elasticity model. They obtained the closed-form solutions of elastic nanorods under different loadings and kinematic boundary conditions. Finally, they assessed Young’s moduli of single-walled carbon nanotubes (SWCNTs) and compared them with the predictions of molecular dynamics (MD) [¹⁹,²⁰]. In another research, Romano et al. [²¹] addressed the nonlocal elasticity in terms of integral convolutions for different structural models and investigated the boundary effects of convolutions on bounded domains. They eventually calculated the extreme values of the nonlocal parameters for the considered structures. Ebrahimi et al. [²²] analyzed the wave propagation of an inhomogeneous FG nanoplate under the influence of nonlinear thermal loading by employing the nonlocal strain gradient theory. They compared their results to the published works in the literature and discussed the impacts of a nonlocal parameter, length scale parameter, and temperature distribution on the wave dispersion characteristics of the mentioned nanoplate.

From another point of view, the gradient elasticity theory captures the hardening behavior of a nanoscale structure; on the contrary, it can only model the softening behavior of nanostructures such as CNTs [¹⁵]. Therefore, to predict two different behaviors simultaneously and combine both possible features in nanoscale structures, it is convenient to merge two theories to consider two distinct properties simultaneously. In addition, to capture the possible realistic characteristics of nanomaterials when studying the vibrational response of such structures, the elastodynamic formulation is generalized; therefore, the strain and kinetic energies should be reformulated. Yaghoubi et al. [²³], for the first time, formulated the strain and velocity gradient theory for the third-order shear deformable beam theory. They generalized the strain energy to account for the strain and its gradient. In addition, they generalized the kinetic energy to incorporate velocity and its gradient. Finally, Euler-Bernoulli and Timoshenko beam models were also introduced by simplifying the third-order theory.

This research aims to conduct a comprehensive study to account for the nonlocality of the stress in nanostructures incorporating the complete gradient elastic analysis of structures; the strain and velocity gradients are included in the generalized governing equations. Therefore, in this research, in the framework of nonlocal gradient elasticity theory by incorporating the strain and velocity gradients, the nonlinear vibrational governing equation and the associated boundary conditions of CNT nanobeams are derived using Hamilton’s principle. The differential quadrature method (DQM) is used to discretize and subsequently solve the equations of motion.

Problem formulation

Here, we derive the equations governing the vibrational behavior of an electrostatically actuated SWCNT by incorporating the strain/velocity gradient size effects jointly with the nonlocal elasticity. The boundary conditions of the CNT-based resonator are considered as fixed-fixed (see Fig. 1). The CNT is triggered by its lower substrate with an assumed initial gap width d. The CNT will be assumed as a cylindrical beam shape of radius \tilde{R}, and length L; thus, the areal moment of inertia can be expressed as I=\mathrm{\pi }{\tilde{R}}^{\mathrm{4}}/\mathrm{4}, whereas the resultant cross-sectional area is expressed as A=\mathrm{\pi }{\tilde{R}}^{\mathrm{2}}. It also has a Young’s modulus E = 1 TPa and a density \rho = 1.35 g/cm³ [²⁴].

According to the nonlocal elasticity theory, the nonclassical internal potential energy U₀ is expressed as [²⁵,²⁶]:

(1)U_{\mathrm{0}}\left({ϵ}_{ij},{{\displaystyle {ϵ}^{\prime }}}_{ij},{\alpha }_{\mathrm{0}}\right)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{ϵ}_{ij}{C}_{ijkl}{\displaystyle {\int }_V{\alpha }_{\mathrm{0}}\left(\left|x-x^{\prime }\right|,e_{\mathrm{0}}a\right){{\displaystyle {ϵ}^{\prime }}}_{kl}\mathrm{d}{V}^{\text{'}}}.

where {ϵ}_{ij},{{ϵ}^{\prime }}_{ij} denote the strain at a reference point x and at all other points x’ within the domain V€’, respectively, {\alpha }_0 represents the Kernel operator,

C_{ijkl}

is the elastic modulus tensor. Considering the nonlocal effects of higher-order strain gradients {ϵ}_{ij,k}, in which the index k after the comma denotes the differentiation with respect to x_k (The expressions {\sigma }_{ij}=C_{ijkl}{\displaystyle {\int }_V{\alpha }_{\mathrm{0}}\left(\left|x-x^{\prime }\right|,e_{\mathrm{0}}a\right){{\displaystyle {ϵ}^{\prime }}}_{kl}\mathrm{d}V} and {\sigma }_{ijm}^{\left(\mathrm{1}\right)}=l_s^{\mathrm{2}}{C}_{ijkl}{\displaystyle {\int }_V{\alpha }_{\mathrm{1}}\left(\left|x-x^{\prime }\right|,e_{\mathrm{1}}a\right){{\displaystyle {ϵ}^{\prime }}}_{kl,m}\mathrm{d}V} represent the nonlocal stress tensor and the higher-order nonlocal stress tensor, respectively). The extended Eringen’s model obtains the following expression for the internal potential energy over the domain V:

U_{\mathrm{0}}\left({{\displaystyle ϵ}}_{ij},{{\displaystyle {ϵ}^{\prime }}}_{ij},{\alpha }_{\mathrm{0}}\right)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{ϵ}_{ij}{C}_{ijkl}{\displaystyle {\int }_V{\alpha }_{\mathrm{0}}\left(\left|x-x^{\prime }\right|,e_{\mathrm{0}}a\right){{\displaystyle {ϵ}^{\prime }}}_{kl}\mathrm{d}{V}^{\prime }},

U_{\mathrm{0}}\left({ϵ}_{ij},{{\displaystyle {ϵ}^{\prime }}}_{ij},{\alpha }_{\mathrm{0}}\mathrm{;}{ϵ}_{ij,m},{{\displaystyle {ϵ}^{\prime }}}_{ij,m},{\alpha }_{\mathrm{1}}\right)

={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{ϵ}_{ij}{C}_{ijkl}{\displaystyle {\int }_V{\alpha }_{\mathrm{0}}\left(\left|x-x^{\prime }\right|,e_{\mathrm{0}}a\right){{\displaystyle {ϵ}^{\prime }}}_{ij}\mathrm{d}V}

+{\displaystyle \frac{l_{\mathrm{s}}^{\mathrm{2}}}{\mathrm{2}}}{ϵ}_{ij,m}{C}_{ijkl}{\displaystyle {\int }_V{\alpha }_{\mathrm{1}}\left(\left|x-x^{\prime }\right|,e_{\mathrm{1}}a\right){{\displaystyle {ϵ}^{\prime }}}_{kl,m}\mathrm{d}V}

(2)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle {\int }_V\left({\sigma }_{ij}{ϵ}_{ij}+{\sigma }_{ijm}^{\left(\mathrm{1}\right)}{{\displaystyle ϵ}}_{ij,m}\right)\mathrm{d}V},

where e_{\mathrm{0}}a and e_{\mathrm{1}}a represent the influence of the inter atomic long range force, l_s represents the strain gradient length parameter, {\alpha }_1represents the Kernel function and {{ϵ}^{\prime }}_{kl,m} is the higher-order strain gradients. In the nonlocal strain gradient elasticity model [²⁷], the total stress tensor t_{ij} can be expressed in terms of the nonlocal stress tensor {\sigma }_{ij}and the higher-order stress tensor \nabla {\sigma }_{ijm}^{\left(\mathrm{1}\right)}; it can be defined as follows:

(3)t_{ij}={\sigma }_{ij}-\nabla {\sigma }_{ijm}^{\left(\mathrm{1}\right)}.

It is assumed that the nonlocal parameters are equal: e_{\mathrm{0}}=e_{\mathrm{1}}=ea [²⁸]. Accordingly, the general constitutive relation is written as:

\left[\mathrm{1}-{\left(ea\right)}^{\mathrm{2}}{\nabla }^{\mathrm{2}}\right]{{\displaystyle t}}_{xx}=\left(\mathrm{1}-{{\displaystyle l}}_{\mathrm{s}}^{\mathrm{2}}{\nabla }^{\mathrm{2}}\right)E\left(z\right){{\displaystyle ϵ}}_{xx},

(4)\left[\mathrm{1}-{\left(ea\right)}^{\mathrm{2}}{\nabla }^{\mathrm{2}}\right]{{\displaystyle t}}_{xz}=\left(\mathrm{1}-{{\displaystyle l}}_{\mathrm{s}}^{\mathrm{2}}{\nabla }^{\mathrm{2}}\right)G\left(z\right){{\displaystyle \gamma }}_{xz},

where {ϵ}_{xx}is the axial strain, E\left(z\right) stands for the modulus of elasticity, G\left(z\right) is shear modulus, {\gamma }_{xz}is shear strain, t_{xx}is the axial normal stress, and t_{xz}is shear stress. From the above relation, we can see the combination of two well-known theories. Setting l_s = 0, we obtain

(5)\left[\mathrm{1}-{\left(ea\right)}^{\mathrm{2}}{\nabla }^{\mathrm{2}}\right]t_{xx}=E\left(z\right){ϵ}_{xx},\left[\mathrm{1}-{\left(ea\right)}^{\mathrm{2}}{\nabla }^{\mathrm{2}}\right]t_{xz}=G\left(z\right){\gamma }_{xz},

which is the Eringen’s theory. Setting ea=\mathrm{0},

(6)t_{xx}=\left(\mathrm{1}-l_{\mathrm{s}}^{\mathrm{2}}{\nabla }^{\mathrm{2}}\right)E\left(z\right){ϵ}_{xx},t_{xz}=\left(\mathrm{1}-l_{\mathrm{s}}^{\mathrm{2}}{\nabla }^{\mathrm{2}}\right)G\left(z\right){\gamma }_{xz},

which defines the strain gradient theory by setting l_{\mathrm{2}}=l_{\mathrm{s}}^{},l_{\mathrm{0}}=l_{\mathrm{1}}=\mathrm{0}\mathrm{;} therefore, the above combination of the two statements is called nonlocal strain gradient theory.

Euler-Bernoulli size-dependent beam model

In the framework of an Euler-Bernoulli beam theory, the CNT displacements u₁, u₂, u₃ are described as:

(7)u_{\mathrm{1}}=u\left(x,t\right)-zw{\left(x,t\right)}_{,x},u_{\mathrm{2}}=\mathrm{0},u_{\mathrm{3}}=w\left(x,t\right).

where u\left(x,t\right),\hspace{0.17em}\hspace{0.17em}w\left(x,t\right)are components of displacement vector in x and z directions, respectively, and w{\left(x,t\right)}_{,x} represents its derivative wrt x. Next, considering the so-called von-Karman nonlinearity, which stands for the mid-plane stretching effects [²⁹] for the fixed-fixed boundary conditions, the nonlinear strain-displacement relation of the beam (to the first order) can be expressed as:

(8){{\displaystyle ϵ}}_{xx}={{\displaystyle u}}_{,x}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}w,x^{\mathrm{2}}-{{\displaystyle zw}}_{,xx}{{\displaystyle ϵ}}_{xy}={{\displaystyle ϵ}}_{xz}={{\displaystyle ϵ}}_{yz}={{\displaystyle ϵ}}_{yy}={{\displaystyle ϵ}}_{zz}=\mathrm{0.}

Hence, the variation of the total potential energy U_t is then extracted as:

\delta U_{\mathrm{t}}={\displaystyle \underset{V}{\int }\left({\sigma }_{xx}\delta {ϵ}_{xx}+{\sigma }_{xxx}^{\left(\mathrm{1}\right)}\nabla \delta {ϵ}_{xx}\right)\mathrm{d}V}

={\displaystyle \underset{V}{\int }\left({\sigma }_{xx}-\nabla {\sigma }_{xxx}^{\left(\mathrm{1}\right)}\right)\delta {ϵ}_{xx}\mathrm{d}V}+{\left[{\displaystyle \underset{A_{\mathrm{b}}}{\int }{\sigma }_{xxx}^{\left(\mathrm{1}\right)}\delta {ϵ}_{xx}\mathrm{d}{A}_{\mathrm{b}}}\right]}_{\mathrm{0}}^L

(9)={\displaystyle \underset{V}{\int }{t}_{xx}\delta {ϵ}_{xx}\mathrm{d}V}+{\left[{\displaystyle \underset{A_{\mathrm{b}}}{\int }{\sigma }_{xxx}^{\left(\mathrm{1}\right)}\delta {ϵ}_{xx}\mathrm{d}{A}_{\mathrm{b}}}\right]}_{\mathrm{0}}^L.

where {\sigma }_{xx},\hspace{0.17em}\hspace{0.17em}\nabla {\sigma }_{xxx}^{\left(1\right)} are nonlocal and higher-order stress tensors, respectively and A_b denotes the cross section area of the nanobeam. Substituting Eq. (8) into Eq. (9), we obtain:

\delta {{\displaystyle U}}_t={\displaystyle \underset{V}{\int }{{\displaystyle t}}_{xx}}\delta \left({{\displaystyle u}}_{,x}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{w_{,x}}^{\mathrm{2}}-{{\displaystyle zw}}_{,xx}\right)\mathrm{d}V

+{{\displaystyle \left[{\displaystyle \underset{A_{\mathrm{b}}}{\int }{{\displaystyle \sigma }}_{xxx}^{\left(\mathrm{1}\right)}}\delta \left({{\displaystyle u}}_{,x}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{w_{,x}}^{\mathrm{2}}-{{\displaystyle zw}}_{,xx}\right)\mathrm{d}{{\displaystyle A}}_{\mathrm{b}}\right]}}_{\mathrm{0}}^L

={\displaystyle \underset{V}{\int }\left({{\displaystyle t}}_{xx}\delta {{\displaystyle u}}_{,x}+{{\displaystyle t}}_{xx}{{\displaystyle w}}_{,x}\delta {{\displaystyle w}}_{,x}-{{\displaystyle zt}}_{xx}\delta {{\displaystyle w}}_{,xx}\right)}\mathrm{d}V

+{{\displaystyle \left[{\displaystyle \underset{A_{\mathrm{b}}}{\int }\left({{\displaystyle {{\displaystyle \sigma }}_{xxx}^{\left(\mathrm{1}\right)}\delta u}}_{,x}+{{\displaystyle \sigma }}_{xxx}^{\left(\mathrm{1}\right)}{{\displaystyle w}}_{,x}\delta {{\displaystyle w}}_{,x}-z{{\displaystyle {{\displaystyle \sigma }}_{xxx}^{\left(\mathrm{1}\right)}\delta w}}_{,xx}\right)}\mathrm{d}{{\displaystyle A}}_{\mathrm{b}}\right]}}_{\mathrm{0}}^L

={\displaystyle \underset{V}{\int }\left({{\displaystyle t}}_{xx}\mathrm{d}{{\displaystyle A}}_{\mathrm{b}}\delta {{\displaystyle u}}_{,x}+{{\displaystyle t}}_{xx}{{\displaystyle w}}_{,x}\mathrm{d}{{\displaystyle A}}_{\mathrm{b}}\delta {{\displaystyle w}}_{,x}-{{\displaystyle zt}}_{xx}\mathrm{d}{{\displaystyle A}}_{\mathrm{b}}\delta {{\displaystyle w}}_{,xx}\right)}\mathrm{d}x

+{{\displaystyle \left[N^{\left(\mathrm{1}\right)}\delta {{\displaystyle u}}_{,x}+N^{\left(\mathrm{1}\right)}{{\displaystyle w}}_{,x}\delta {{\displaystyle w}}_{,x}-M^{\left(\mathrm{1}\right)}\delta {{\displaystyle w}}_{,xx}\right]}}_{\mathrm{0}}^L

={\displaystyle {\int }_{\mathrm{0}}^L\left(N\delta {{\displaystyle u}}_{,x}+N{{\displaystyle w}}_{,x}\delta {{\displaystyle w}}_{,x}-M\delta {{\displaystyle w}}_{,xx}\right)}\mathrm{d}x

(10)+{{\displaystyle \left[N^{\left(\mathrm{1}\right)}\delta {{\displaystyle u}}_{,x}+N^{\left(\mathrm{1}\right)}{{\displaystyle w}}_{,x}\delta {{\displaystyle w}}_{,x}-M^{\left(\mathrm{1}\right)}\delta {{\displaystyle w}}_{,xx}\right]}}_{\mathrm{0}}^L,

where the total and higher-order axial forces N,\hspace{0.17em}\hspace{0.17em}{N}^{\left(1\right)}and bending moments M,\hspace{0.17em}\hspace{0.17em}{M}^{\left(1\right)}can be expressed as follows,

N={\displaystyle \underset{A_{\mathrm{b}}}{\int }{t}_{xx}\mathrm{d}{A}_{\mathrm{b}}},M={\displaystyle \underset{A_{\mathrm{b}}}{\int }z{t}_{xx}\mathrm{d}{A}_{\mathrm{b}}},

(11)N^{\left(\mathrm{1}\right)}={\displaystyle \underset{A_{\mathrm{b}}}{\int }{\sigma }_{xxx}^{\left(\mathrm{1}\right)}\mathrm{d}{A}_{\mathrm{b}}},M^{\left(\mathrm{1}\right)}={\displaystyle \underset{A_{\mathrm{b}}}{\int }z{\sigma }_{xxx}^{\left(\mathrm{1}\right)}\mathrm{d}{A}_{\mathrm{b}}}.

Now, substituting the constitutive relation, Eq. (3), into Eq. (11), the axial force N\left(x,t\right) and the moment M\left(x,t\right) can be obtained respectively as:

(12)N\left(x,t\right)={\displaystyle \underset{A_{\mathrm{b}}}{\int }{t}_{xx}\mathrm{d}{A}_{\mathrm{b}}}={\displaystyle \underset{A_{\mathrm{b}}}{\int }\left({\sigma }_{xx}-{\sigma }_{xxx,x}^{\left(\mathrm{1}\right)}\right)\mathrm{d}{A}_{\mathrm{b}}}=N^{\left(\mathrm{0}\right)}-N_{,x}^{\left(\mathrm{1}\right)},

(13)M\left(x,t\right)={\displaystyle \underset{A_{\mathrm{b}}}{\int }z{t}_{xx}\mathrm{d}{A}_{\mathrm{b}}}={\displaystyle \underset{A_{\mathrm{b}}}{\int }z\left({\sigma }_{xx}-{\sigma }_{xxx,x}^{\left(\mathrm{1}\right)}\right)\mathrm{d}{A}_{\mathrm{b}}}=M^{\left(\mathrm{0}\right)}-M_{,x}^{\left(\mathrm{1}\right)},

where, the superscripts ‘⁽⁰⁾’ and ‘⁽¹⁾’ stand for the nonlocal and higher-order terms and subscript ‘_,x’ represents the derivative with respect to x, we have

(14)M^{\left(\mathrm{0}\right)}={\displaystyle \underset{A_{\mathrm{b}}}{\int }z{\sigma }_{xx}\mathrm{d}{A}_{\mathrm{b}}},N^{\left(\mathrm{0}\right)}={\displaystyle \underset{A_{\mathrm{b}}}{\int }{\sigma }_{xx}\mathrm{d}{A}_{\mathrm{b}}}.

Substituting the general constitutive relation, Eq. (4), and the von-Karman strain expression Eq. (8), into Eqs. (12), (13), and (14), we obtain the following relations:

(15)M\left(x,t\right)={\left(ea\right)}^{\mathrm{2}}{M}_{,xx}+EI\left(\mathrm{1}-l_s^{\mathrm{2}}{\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial x^{\mathrm{2}}}}\right)w_{,xx},

(16)N\left(x,t\right)={\left(ea\right)}^{\mathrm{2}}{{\displaystyle N}}_{,xx}+EA\left(\mathrm{1}-{{\displaystyle l}}_s^{\mathrm{2}}{\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial x^{\mathrm{2}}}}\right)\left[{{\displaystyle u}}_{,x}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{w_{,x}}^{\mathrm{2}}\right].

Velocity gradient theory

The generalized kinetic energy K in the framework of the elasticity gradient theory can be written as [³⁰]:

(17)K={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho u_{i,t}{u}_{i,t}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho l_k^{\mathrm{2}}{u}_{i,jt}{u}_{i,jt},

where \rho is the mass density, l_kis the velocity gradient kinetic internal length scale, and index t stands for the derivative with respect to time. Consequently, the kinetic energy K_{\mathrm{t}}of the elastically deformed material (at time t) in the domain V can be defined as

(18)K_{\mathrm{t}}={\displaystyle \underset{V}{\int }K\mathrm{d}V}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \underset{V}{\int }\rho \left(u_{i,t}{u}_{i,t}+l_k^{\mathrm{2}}{u}_{i,jt}{u}_{i,jt}\right)\mathrm{d}V},

and its respective variational can be obtained as

(19)\delta K_t={\displaystyle \underset{V}{\int }\rho \left(u_{i,t}\delta u_{i,t}+l_k^{\mathrm{2}}{u}_{i,jt}\delta u_{i,jt}\right)\mathrm{d}V}.

Using the displacement field of Eq. (7), the total kinetic energy of the system is then described as

(20){{\displaystyle K}}_t={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\displaystyle \underset{x=\mathrm{0}}{\overset{x=L}{\int }}{\displaystyle {\displaystyle \underset{A}{\int }\left[{u_{\mathrm{1},t}}^{\mathrm{2}}+{u_{\mathrm{3},t}}^{\mathrm{2}}+{{\displaystyle l}}_k^{\mathrm{2}}\left({u_{\mathrm{1},xt}}^{\mathrm{2}}+{u_{\mathrm{1},zt}}^{\mathrm{2}}+{u_{\mathrm{3},xt}}^{\mathrm{2}}\right)\right]}}}\mathrm{d}A\mathrm{d}x.

For thin structures, the kinetic energy caused by rotation is negligible in comparison with the other terms, i.e., {u_{\mathrm{1},xt}}^{\mathrm{2}}\mathrm{0}, and {u_{\mathrm{3},zt}}^{\mathrm{2}}\mathrm{0}. Thereby, the first variation of the total kinetic energy is expressed as follows.

\delta K_t=\rho {\displaystyle \underset{x=\mathrm{0}}{\overset{x=L}{\int }}{\displaystyle \underset{A}{\int }[z^{\mathrm{2}}{w}_{,xt}\mathrm{\delta }{w}_{,xt}+w_{,t}\mathrm{\delta }{w}_{,t}+u_{,t}\mathrm{\delta }{u}_{,t}}}

{\displaystyle {\displaystyle -\mathrm{2}z(w_{,xt}\mathrm{\delta }{u}_{,t}+u_{,t}\mathrm{\delta }{w}_{,xt})]\mathrm{d}A}\mathrm{d}x}

+\rho l_k^{\mathrm{2}}{\displaystyle \underset{x=\mathrm{0}}{\overset{x=L}{\int }}{\displaystyle \underset{A}{\int }\left[\left(z^{\mathrm{2}}{w}_{,xxt}\mathrm{\delta }{w}_{,xxt}+\mathrm{2}{w}_{,xt}\mathrm{\delta }{w}_{,xt}\right)\right]\mathrm{d}A}\mathrm{d}x}

=\rho I{\displaystyle \underset{x=\mathrm{0}}{\overset{x=L}{\int }}{w}_{,xt}\mathrm{\delta }{w}_{,xt}\mathrm{d}x}+\rho A{\displaystyle \underset{x=\mathrm{0}}{\overset{x=L}{\int }}(w_{,t}\mathrm{\delta }{w}_{,t}+u_{,t}\mathrm{\delta }{u}_{,t})\mathrm{d}x}

(21)+\rho l_k^{\mathrm{2}}I{\displaystyle \underset{x=\mathrm{0}}{\overset{x=L}{\int }}{w}_{,xxt}\mathrm{\delta }{w}_{,xxt}\mathrm{d}x}+\mathrm{2}\rho l_k^{\mathrm{2}}A{\displaystyle \underset{x=\mathrm{0}}{\overset{x=L}{\int }}{w}_{,xt}\mathrm{\delta }{w}_{,xt}\mathrm{d}x}.

Finally, the first variation of the work done by the external forces W_t can be written as:

(22)\delta W_t={\displaystyle \underset{x=\mathrm{0}}{\overset{x=L}{\int }}(F_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(x,t\right)-c{w}_{,t})\delta w\mathrm{d}x}.

where F_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(x,t\right) is Electrostatic force.

Extended Hamilton principle

To derive the equation of motion and the corresponding boundary conditions, Hamilton’s principle, which is an important variational approach, is employed. In the continuous systems, the state of the system is defined using continuous functions for space and time domains. The extended Hamilton principle is accordingly described as:

(23){\displaystyle {\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\delta L_{\mathrm{t}}\mathrm{d}t}={\displaystyle {\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left(\delta K_{\mathrm{t}}-\delta U_{\mathrm{t}}+\delta W_t\right)\mathrm{d}t=}\mathrm{0.}

In Eq. (23), L_t denotes the Lagrangian of the system, K_t represents the total kinetic energy, U_t denotes the elastic potential energy, W_t is the nonconservative work of the external loads, and t₁, t₂ symbolize the initial and final times, respectively. We substitute the relations for \delta U_{\mathrm{t}}, \delta K_{\mathrm{t}}, and \delta W_{\mathrm{t}} into Eq. (23), perform some mathematical computations, and then adopt that the variations of the variables are equal to zero at time t₁ and t₂. We can then obtain the following nonlinear governing equations of a fixed-fixed CNT resonator based on the nonlocal strain and velocity gradient theories [¹⁵]:

\left(\mathrm{1}-{\left(ea\right)}^{\mathrm{2}}{\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial x^{\mathrm{2}}}}\right)[\rho A{w}_{,tt}-\rho I{w}_{,xxtt}+c{w}_{,t}

-\left({\displaystyle \frac{EA}{\mathrm{2}L}}{\displaystyle \underset{x=\mathrm{0}}{\overset{x=L}{\int }}{{\displaystyle w}}_{,x}{}^{\mathrm{2}}\mathrm{d}x}\right)w_{,xx}-F_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(x,t\right)

(24)=\rho l_k^{\mathrm{2}}\left(\mathrm{2}A{w}_{,xxtt}-I{w}_{,xxxxtt}\right)-\left(\mathrm{1}-l_s^{\mathrm{2}}{\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial x^{\mathrm{2}}}}\right)EI{w}_{,xxxx}.

where c is the damping coefficient. Its respective boundary conditions are

\begin{array}{l}w\left(x=\mathrm{0},t\right)=w\left(x=L,t\right)=\mathrm{0},\hfill \\ w_{,x}\left(x=\mathrm{0},t\right)=w_{,x}\left(x=L,t\right)=\mathrm{0},\hfill \end{array}

(25)w_{,xx}\left(x=\mathrm{0},t\right)=w_{,xx}\left(x=L,t\right)=\mathrm{0.}

The nonlinear actuation force applied on the CNT-based resonator is expressed as [³¹]:

(26)F_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}}\left(x,t\right)={\displaystyle \frac{\mathrm{\pi }{ϵ}_{\mathrm{0}}{V}^{\mathrm{2}}}{\sqrt{\left(d-w\right)\left(d-w+\mathrm{2}\tilde{R}\right)}{(\cos {\mathrm{h}}^{-\mathrm{1}}\left(\mathrm{1}+{\displaystyle \frac{d-w}{\tilde{R}}}\right))}^{\mathrm{2}}}}.

ƒTo generalize the nonlinear equation of motion described by Eq. (24), the following nondimensional parameters are introduced:

(27)\widehat{w}={\displaystyle \frac{w}{d}},\widehat{x}={\displaystyle \frac{x}{L}},\widehat{t}={\displaystyle \frac{t}{T}},

in which T is a constant defined by T=\sqrt{\rho A{L}^{\mathrm{4}}/EI}. Subsequently, by omitting the symbol “^{€^€}”, the normalized governing equation and corresponding boundary conditions of a doubly-clamped CNT can be obtained as:

\left(\mathrm{1}-{\mu }_{\mathrm{s}}{\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial x^{\mathrm{2}}}}\right){\displaystyle \frac{{\partial }^{\mathrm{4}}w}{\partial x^{\mathrm{4}}}}+\left(\mathrm{1}-{\mu }_{\mathrm{0}}{\displaystyle \frac{{\partial }^{\mathrm{2}}}{\partial x^{\mathrm{2}}}}\right)

({\displaystyle \frac{{\partial }^{\mathrm{2}}w}{\partial t^{\mathrm{2}}}}+{\alpha }_{\mathrm{d}}{\displaystyle \frac{\partial w}{\partial t}}-{\alpha }_{\mathrm{r}}{\displaystyle \frac{{\partial }^{\mathrm{4}}w}{\partial x^{\mathrm{2}}\partial t^{\mathrm{2}}}}-{\alpha }_{\mathrm{s}}\left({\displaystyle \underset{\mathrm{0}}{\overset{\mathrm{1}}{\int }}{\left({\displaystyle \frac{\partial w}{\partial x}}\right)}^{\mathrm{2}}}\mathrm{d}x\right){\displaystyle \frac{{\partial }^{\mathrm{2}}w}{\partial x^{\mathrm{2}}}}

-{\alpha }_{\mathrm{e}}\Gamma \left(w\right)={\mu }_{\mathrm{k}}\left(\mathrm{2}{\displaystyle \frac{{\partial }^{\mathrm{4}}w}{\partial x^{\mathrm{2}}\partial t^{\mathrm{2}}}}-{\alpha }_{\mathrm{r}}{\displaystyle \frac{{\partial }^{\mathrm{6}}w}{\partial x^{\mathrm{4}}\partial t^{\mathrm{2}}}}\right),

(28)\begin{array}{l}w\left(x=\mathrm{0},t\right)=w\left(x=L,t\right)=\mathrm{0},\hfill \\ w_{,x}\left(x=\mathrm{0},t\right)=w_{,x}\left(x=L,t\right)=\mathrm{0},\hfill \\ w_{,xx}\left(x=\mathrm{0},t\right)=w_{,xx}\left(x=L,t\right)=\mathrm{0},\hfill \end{array}

where

{\alpha }_{\mathrm{s}}={\displaystyle \frac{A{d}^{\mathrm{2}}}{\mathrm{2}I}},{\alpha }_{\mathrm{e}}={\displaystyle \frac{\mathrm{\pi }{ϵ}_{\mathrm{0}}{L}^{\mathrm{4}}}{EI{d}^{\mathrm{2}}}},{\alpha }_{\mathrm{r}}={\displaystyle \frac{I}{A{L}^{\mathrm{2}}}},{\alpha }_{\mathrm{d}}=\tilde{c}{\displaystyle \frac{L^{\mathrm{4}}}{EI}},{\mu }_{\mathrm{0}}={\left({\displaystyle \frac{ea}{L}}\right)}^{\mathrm{2}},

(29){\mu }_{\mathrm{s}}={\left({\displaystyle \frac{l_s^{}}{L}}\right)}^{\mathrm{2}},{\mu }_{\mathrm{k}}={\left({\displaystyle \frac{l_k^{}}{L}}\right)}^{\mathrm{2}},R={\displaystyle \frac{\tilde{R}}{d}},

where \tilde{R} is the CNT cross sectional radius, and

(30)\Gamma \left(w\right)={\displaystyle \frac{V^{\mathrm{2}}}{\sqrt{\left(\mathrm{1}-w\right)\left(\mathrm{1}-w+\mathrm{2}R\right)}{\left({\cosh }^{-\mathrm{1}}\left(\mathrm{1}+{\displaystyle \frac{\mathrm{1}-w}{R}}\right)\right)}^{\mathrm{2}}}}.

Discretization of the problem

Spatial discretization using the differential quadrature method

Owing to the complication in the nonlinear terms of the normalized equation of motion, Eq. (28), it is essential to implement a mathematical technique to simulate the nanoresonator dynamical response. Several numerical/computational approaches, such as an artificial neural network [32], collocation method [33–35], nonlocal operator [36], DQM [37], finite element method [38–42], finite difference method, and Isogeometric analysis [43,44], have been employed by many researchers to solve the boundary/initial value problems. Below, we suggest how to implement the DQM superimposed to the finite-difference method (FDM). The basic idea of DQM is to approximate the derivative of a function w\left(x\right) with respect to a space variable x at a given sampling point as a weighted linear combination of the function values at all sampling points in the domain of x. Differential equations will then be transformed into a set of algebraic equations for time-independent problems, and a set of ordinary differential equations in time for initial-value problems. For the accuracy of the numerical results, the subsequent lattice distribution is assumed [17]:

(31)x_i={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left[\mathrm{1}-\cos \left({\displaystyle \frac{i-\mathrm{1}}{n-\mathrm{1}}}\mathrm{\pi }\right)\right],i=\mathrm{1},\mathrm{2},\mathrm{...},n\mathrm{.}

ƒTherefore, for a normalized space variable x in the interval (0,1) and defining n discretization points in the space domain, the pth-sequence derivative of w\left(x\right) at point x=x_iis expressed as [³⁷]:

(32){{\displaystyle \frac{{\partial }^{p}w}{\partial x^p}}}_{x=x_i}={\displaystyle \sum _{j=\mathrm{1}}^n{D}_{ij}^{\left(p\right)}{w}_j},

where the off-diagonal expression for the weighting coefficient matrix of the first sequence derivative D_{ij}^{p=\mathrm{1}} is defined as:

(33)D_{ij}^{(p=\mathrm{1})}={\displaystyle \frac{{\displaystyle \prod _{k=\mathrm{1},k\ne i}^n(x_i-x_k)}}{(x_i-x_j){\displaystyle \prod _{k=\mathrm{1},k\ne j}^n(x_j-x_k)}}}\begin{array}{l}i,j=\mathrm{1},\mathrm{2},\mathrm{...},n,i\ne j\hfill \end{array}.

For the rest of the higher order derivatives, they are all obtained through the following recursive expressions:

(34)\{\begin{array}{l}{A}_{ij}^{\left(p\right)}=p\left[A_{ii}^{p-\mathrm{1}}{A}_{ij}^{\mathrm{1}}-{\displaystyle \frac{A_{ij}^{p-\mathrm{1}}}{x_i-x_j}}\right],\hfill \\ \hfill \\ i,j=\mathrm{1},\mathrm{2},\mathrm{...},n,\begin{array}{l}i\ne j\hfill \end{array},\mathrm{2}\le p\le n-\mathrm{1},\hfill \\ \hfill \\ A_{ii}^{\left(p\right)}=\begin{array}{ll}-{\displaystyle A_{ik}^p},\hfill & i=\mathrm{1},\mathrm{2},\mathrm{...},n,\hfill \end{array}\mathrm{1}\le p\le n-\mathrm{1}.\hfill \end{array}

Note that before using the DQM, it is necessary to perform some mathematical computations, such as integration by parts, and to subsequently utilize the associated boundary conditions to refine the integral terms. In particular, the von-Karman nonlinearity in Eq. (28), is re-written as:

{\displaystyle \underset{x=\mathrm{0}}{\overset{x=\mathrm{1}}{\int }}{\left({\displaystyle \frac{\partial w}{\partial x}}\right)}^{\mathrm{2}}}\mathrm{d}x={\displaystyle \underset{\mathrm{0}}{\overset{\mathrm{1}}{\int }}\left({\displaystyle \frac{\partial w}{\partial x}}\right)\left({\displaystyle \frac{\partial w}{\partial x}}\right)}\mathrm{d}x

(35)={\left[w{\displaystyle \frac{\partial w}{\partial x}}\right]}_{x=\mathrm{0}}^{x=\mathrm{1}}-{\displaystyle \underset{\mathrm{0}}{\overset{\mathrm{1}}{\int }}{\displaystyle \frac{{\partial }^{\mathrm{2}}w}{\partial x^{\mathrm{2}}}}}w\mathrm{d}x,

where {\left[w{\displaystyle \frac{\partial w}{\partial x}}\right]}_{x=\mathrm{0}}^{x=\mathrm{1}}=\mathrm{0}.

Next, the above-mentioned term is approximated by the Newton-Cotes formula [³⁷] using the following relation:

{\displaystyle \underset{\mathrm{0}}{\overset{\mathrm{1}}{\int }}{w}^{\prime }w}\mathrm{d}x={\displaystyle \sum _{i=\mathrm{1}}^n{C}_i{w^{\prime }}_i{w}_i},

(36)C_i=\left({\displaystyle \underset{\mathrm{0}}{\overset{\mathrm{1}}{\int }}{\displaystyle \prod _{k=\mathrm{1},k\ne i}^n{\displaystyle \frac{x-x_k}{x_i-x_k}}\mathrm{d}x}}\right).

Note that Eq. (36) uses the same grid points described in Eq. (31). Using the integral approximation and the differential quadrature discretization method, Eq. (28) can be re-written as follows:

\left(w_i-{\mu }_{\mathrm{0}}{\displaystyle \sum _{j=\mathrm{1}}^n{D}_{ij}^{\left(\mathrm{2}\right)}{w}_j}\right)+\left({\displaystyle \sum _{j=\mathrm{1}}^n{D}_{ij}^{\left(\mathrm{4}\right)}{w}_j}-{\mu }_s{\displaystyle \sum _{j=\mathrm{1}}^n{D}_{ij}^{\left(\mathrm{6}\right)}{w}_j}\right)

+{\alpha }_d\left(w_i-{\mu }_{\mathrm{0}}{\displaystyle \sum _{j=\mathrm{1}}^n{D}_{ij}^{\left(\mathrm{2}\right)}{w}_j}\right)-{\alpha }_r\left({\displaystyle \sum _{j=\mathrm{1}}^n{D}_{ij}^{\left(\mathrm{2}\right)}{w}_j}-{\displaystyle \sum _{j=\mathrm{1}}^n{D}_{ij}^{\left(\mathrm{4}\right)}{w}_j}\right)

+{\alpha }_s\left({\displaystyle \sum _{j=\mathrm{1}}^n{\displaystyle \sum _{k=\mathrm{1}}^n{C}_j{D}_{ij}^{\left(\mathrm{2}\right)}{w}_j{w}_k}}\right)\left({\displaystyle \sum _{j=\mathrm{1}}^n{D}_{ij}^{\left(\mathrm{2}\right)}{w}_j}-{\mu }_{\mathrm{0}}{\displaystyle \sum _{j=\mathrm{1}}^n{D}_{ij}^{\left(\mathrm{4}\right)}{w}_j}\right)=

+{\mu }_k\left(\mathrm{2}{\displaystyle \sum _{j=\mathrm{1}}^n{D}_{ij}^{\left(\mathrm{2}\right)}{w}_j}-{\alpha }_r{\displaystyle \sum _{j=\mathrm{1}}^n{D}_{ij}^{\left(\mathrm{4}\right)}{w}_j}\right)

(37)+{\alpha }_e\left(\Gamma \left(w_i\right)-{\mu }_{\mathrm{0}}{\Gamma }_{,xx}\left(w_i\right)\right),i=\mathrm{1},\mathrm{2},\mathrm{...},n-\mathrm{3.}

ƒNext, to extract all equations of motion, the following boundary conditions are utilized:

(38)\{\begin{array}{l}{w}_{\mathrm{1}}=w_n=\mathrm{0},\hfill \\ \hfill \\ {\displaystyle D_{\mathrm{1}j}^{\left(\mathrm{1}\right)}{w}_j}={\displaystyle D_{nj}^{\left(\mathrm{1}\right)}{w}_j}=\mathrm{0},\hfill \\ \hfill \\ {\displaystyle D_{\mathrm{1}j}^{\left(\mathrm{2}\right)}{w}_j}={\displaystyle D_{nj}^{\left(\mathrm{2}\right)}{w}_j}=\mathrm{0.}\hfill \end{array}

Time discretization using finite difference method

To discretize the time domain and formulate the governing Eq. (37) using the FDM, the following steps are implemented.

Step 1. Discretizing the domain.

A uniformly partitioned meshing scheme is employed to discretize the time domain. This implies that we used the constant time step \mathrm{\Delta }t=T/N_t and then defined the nth discretized point as t_n=n\mathrm{\Delta }t,\hspace{0.17em}\hspace{0.17em}n=\mathrm{0},\mathrm{1},\mathrm{...},N_t. The mesh function w^n is then introduced for the approximated solution at the nth discretized point, which can be computed from the algebraic equations extracted from the governing equation.

Step 2. Satisfying the equation of motion at discretized time points.

The governing equation described in Eq. (37) has to be fulfilled at each discretized point in which the approximate solution should be found.

Step 3. Replacing the derivatives by finite difference approximation.

The first and second derivatives of the mesh function, i.e., \dot{w}\left(t_n\right) and \ddot{w}\left(t_n\right), are replaced by finite difference approximations as follows:

\dot{w}\left(t_n\right){\displaystyle \frac{w^{n+\mathrm{1}}-w^{n-\mathrm{1}}}{\mathrm{2}\mathrm{\Delta }t}},

(39)\ddot{w}\left(t_n\right){\displaystyle \frac{w^{n+\mathrm{1}}-\mathrm{2}{w}^n+w^{n-\mathrm{1}}}{\mathrm{\Delta }{t}^{\mathrm{2}}}}\mathrm{.}

The above relations are then inserted into the equation of motion.

Step 4. Re-arranging the governing equation by a recursive formulation

To compute the approximate solution using a computational scheme while substituting the derivatives defined in Eq. (39) into the governing Eq. (37), it is necessary to re-arrange the formulation at each time step to calculate the unknown mesh function w^{n+\mathrm{1}} as we have already computed the mesh functions w^{n-\mathrm{1}} and w^n. The computational algorithm is then successively applied on Eq. (37) for n=\mathrm{1},\mathrm{2},\mathrm{...},N_t-\mathrm{1}.

Dynamic analysis

Validation of the present analysis

As a case study, different parameters of the considered SWCNT are summarized in Table 1. In our previous work, it was demonstrated that using 11 discretization points in the DQM is appropriate to obtain accurate results and ensure the desirable convergence [⁴⁵]. However, in this problem, one should deal with two additional nonclassical boundary conditions. Therefore, 19 discretization points are used in the DQM to obtain the converged results. To verify the convergence of the present analysis, the authors simulated the governing equation of motion in Eq. (28) by employing the long-time integration model. In the latter method, the contribution of 5 modes is included in the integration process. From Fig. 2, one can easily determine that the results of FDM-DQM analysis are in an ideal agreement with those obtained by the long-time integration model.

Effect of the nonlocal parameter

The variation of the steady-state maximum deflection W_max versus the forcing AC frequency Ω is outlined in Fig. 3 for different values of the nonlocal parameter µ₀. According to the illustrated results in Fig. 3(a), when DC electrostatic voltage V_DC = 1 V, an increase in the nonlocal parameter resulted in decreasing maximum dynamic amplitude of the CNT. Furthermore, when considering the nonlocality effect, a softening-like behavior of the CNT has been replaced by a hardening-type behavior. Note that the presence of a nonlocal parameter resulted into a drastic decrease in the nonlinear resonant frequency of the nanosystem. From Figs. 3(b) and 3(c), it is established that, by increasing the electrostatic voltage while keeping the dynamic voltage V_AC constant, the softening behavior vanishes and the resonator exhibits only hardening behavior. Finally, by comparing the results of Figs. 3(a), 3(b), and 3(c), one can comprehend that the maximum dynamic deflection of the CNT resonator is increased by increasing the electrostatic actuation voltage and decreased by increasing the nonlocal effect.

Effect of the strain gradient parameter

Figure 4 displays the frequency response of CNT at various DC voltages as a function of driving frequency and for different values of the strain gradient parameter μ_s. Following the obtained results, it should be noted that considering the strain gradient parameter will significantly decrease the hardening effects of the nanoresonator. It is concluded that, by setting the DC voltage equal to μ_s = 0.2², the nonlinear effects may vanish and the backbone curve becomes straight accordingly (see Fig. 4(c)). It is also evident that the maximum amplitude of the system is increased by increasing the value of DC voltage and the hardening behavior as the nonlinear frequency does not change in principle. Moreover, one can determine that the bandwidth expansion is increased as the DC excitation voltage increases.

Effect of the velocity gradient parameter

Here, we examine the impact of velocity gradient parameter μ_k on the dynamic behavior of CNT resonators. Note that, when studying the vibrational response of nanostructures, the elastodynamic formulation may be generalized; therefore, the strain and kinetic energies should be considered. Consequently, the effects of static and kinetic internal length scale parameters are both considered in modeling the dynamical and structural behavior of the nanostructure. Figure 5 shows three different frequency-response curves for three different values of the velocity gradient parameter μ_k. As seen from Fig. 5(a), the maximum dynamic deflection of the structure is considerably amplified by increasing the velocity gradient parameter; consequently, the bandwidth expansion increases as the velocity gradient parameter increases. Note that any increase in the bandwidth is a desirable phenomenon in nanoresonators as it causes a larger signal-to-noise ratio through a broader domain of driving frequency [⁴⁷]. By comparing the simulated results of Figs. 5(a), 5(b), and 5(c), it is established that the nonlinear resonant frequency of CNT nanobeam is reduced by increasing the parameter μ_k. In contrast, it should be noted that the velocity gradient parameter has no meaningful effect on the backbone curve of CNT and thereby does not change the hardening behavior of the considered structure.

Finally, it would be useful to investigate the influence of DC voltage in the presence of the velocity gradient parameter. As can be observed from the plotted results in Fig. 6, varying the electrostatic voltage does not change the hardening behavior of CNT, and no significant effect has been reported on the backbone curve.

Conclusions

In this study, a parametric investigation into the nonlinear dynamics of an electrically actuated doubly-clamped SWCNT resonator was performed. An Euler-Bernoulli beam model was adopted by incorporating the nonlocal and strain/velocity gradient theories. Two size-dependent theories, namely, the strain and the velocity gradient elasticity were assumed in this work. The derived nonlinear governing equation was discretized using a reduced-order technique through DQM and then solved numerically by assuming FDM. Moreover, the illustrated results in this work demonstrated that neglecting the nonlocal and the size effects imposed considerable errors in modeling the dynamic behavior of SWCNT, and, consequently, in accurately determining its dynamical parameters, such as its resonant deflection (10%–20% error) and fundamental resonant frequency (15%–20% error). Therefore, these effects should not be ignored in estimating the dynamical behavior of electrically actuated SWCNT if accurate estimations of all its structural parameters are required.

References

[1] Mohamed N, Mohamed S, Eltaher M A. Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model. Engineering with Computers, 2020 (in press)

[2] Eltaher M A, Almalki T A, Almitani K H, Ahmed K I E, Abdraboh A M. Modal participation of fixed-fixed single-walled carbon nanotube with vacancies. International Journal of Advanced Structural Engineering, 2019, 11: 151–163

[3] Eltaher M A, Mohamed N, Mohamed S, Seddek L F. Postbuckling of curved carbon nanotubes using energy equivalent model. Journal of Nano Research, 2019, 57: 136–157

[4] Mohamed N, Eltaher M A, Mohamed S A, Seddek L F. Energy equivalent model in analysis of postbuckling of imperfect carbon nanotubes resting on nonlinear elastic foundation. Structural Engineering and Mechanics, 2019, 70(6): 737–750

[5] Esmaeili M, Tadi Beni Y. Vibration and buckling analysis of functionally graded flexoelectric smart beam. Journal of Applied and Computational Mechanics, 2019, 5(5): 900–917

[6] Barretta R, Marotti de Sciarra F. A nonlocal model for carbon nanotubes under axial loads. Advances in Materials Science and Engineering, 2013, 2013: 360935

[7] Barretta R, Faghidian A A, Luciano R. Longitudinal vibrations of nano-rods by stress-driven integral elasticity. Mechanics of Advanced Materials and Structures, 2019, 26(15): 1307–1315

[8] Abazid M A. The nonlocal strain gradient theory for hygrothermo-electromagnetic effects on buckling, vibration and wave propagation in piezoelectromagnetic nanoplates. International Journal of Applied Mechanics, 2019, 11(7): 1950067

[9] Chen W, Wang L, Dai H. Stability and nonlinear vibration analysis of an axially loaded nanobeam based on nonlocal strain gradient theory. International Journal of Applied Mechanics, 2019, 11(7): 1950069

[10] Sedighi H M. The influence of small scale on the pull-in behavior of nonlocal nanobridges considering surface effect, Casimir and Van der Waals Attractions. International Journal of Applied Mechanics, 2014, 6(3): 1450030

[11] Sedighi H M. Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory. Acta Astronautica, 2014, 95: 111–123

[12] Eringen A C. Nonlocal Continuum Field Theories. 1st ed. New York: Springer-Verlag, 2002

[13] Lam D C C, Yang F, Chong A C M, Wang J, Tong P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 2003, 51(8): 1477–1508

[14] Picu C R. The Peierls stress in non-local elasticity. Journal of the Mechanics and Physics of Solids, 2002, 50(4): 717–735

[15] Ouakad H M, El-Borgi S, Mousavi S M, Friswell M I. Static and dynamic response of CNT nanobeam using nonlocal strain and velocity gradient theory. Applied Mathematical Modelling, 2018, 62: 207–222

[16] Zhu X, Li L. Closed form solution for a nonlocal strain gradient rod in tension. International Journal of Engineering Science, 2017, 119: 16–28

[17] Apuzzo A, Barretta R, Faghidian S A, Luciano R, Marotti de Sciarra F. Nonlocal strain gradient exact solutions for functionally graded inflected nano-beams. Composites. Part B, Engineering, 2019, 164: 667–674

[18] Barretta R, Čanadija M, Marotti de Sciarra F. Modified nonlocal strain gradient elasticity for nano-rods and application to carbon nanotubes. Applied Sciences (Basel, Switzerland), 2019, 9(3): 514

[19] Liu R, Zhao J, Wang L, Wei N. Nonlinear vibrations of helical graphene resonators in the dynamic nano-indentation testing. Nanotechnology, 2020, 31(2): 025709

[20] Liu R, Wang L, Zhao J. Nonlinear vibrations of circular single-layer black phosphorus resonators. Applied Physics Letters, 2018, 113(21): 211901

[21] Romano G, Luciano R, Barretta R, Diaco M. Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours. Continuum Mechanics and Thermodynamics, 2018, 30(3): 641–655

[22] Ebrahimi F, Barati M R, Dabbagh A. A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. International Journal of Engineering Science, 2016, 107: 169–182

[23] Yaghoubi S T, Mousavi S M, Paavola J. Strain and velocity gradient theory for higher-order shear deformable beams. Archive of Applied Mechanics, 2015, 85(7): 877–892

[24] Yu M F. Fundamental mechanical properties of carbon nanotubes: current understanding and the related experimental studies. Journal of Engineering Materials and Technology, 2004, 126(3): 271–278

[25] Eringen A C. Nonlocal polar elastic continua. International Journal of Engineering Science, 1972, 10(1): 1–16

[26] Eringen A C. Linear theory of nonlocal elasticity and dispersion of plane waves. International Journal of Engineering Science, 1972, 10(5): 425–435

[27] Lim C W, Wang C M. Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams. Journal of Applied Physics, 2007, 101(5): 054312

[28] Eringen A C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 1983, 54(9): 4703–4710

[29] Alkharabsheh S A, Younis M I. Dynamics of MEMS arches of flexible supports. Journal of Microelectromechanical Systems, 2013, 22(1): 216–224

[30] Mindlin R D. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 1964, 16(1): 51–78

[31] Pradiptya I, Ouakad H M. Size-dependent behavior of slacked carbon nanotube actuator based on the higher-order strain gradient theory. International Journal of Mechanics and Materials in Design, 2018, 14(3): 393–415

[32] Anitescu C, Atroshchenko C, Alajlan N, Rabczuk T. Artificial neural network methods for the solution of second order boundary value problems. Computers, Materials & Continua, 2019, 59(1): 345–359

[33] Guo H, Zhuang X, Rabczuk T A. Deep collocation method for the bending analysis of Kirchhoff plate. Computers, Materials & Continua, 2019, 59(2): 433–456

[34] Almoaeet M K, Shamsi M, Khosravian-Arab HTorres D F M. A collocation method of lines for two-sided space-fractional advection-diffusion equations with variable coefficients. Mathematical Methods in the Applied Sciences, 2019, 42(10): 3465–3480

[35] Fang J, Wu B, Liu W. An explicit spectral collocation method using nonpolynomial basis functions for the time-dependent Schrödinger equation. Mathematical Methods in the Applied Sciences, 2019, 42(1): 186–203

[36] Rabczuk T, Ren H, Zhuang X. A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Computers, Materials & Continua, 2019, 59(1): 31–55

[37] Tomasiello S. Differential quadrature method: Application to initial-boundary-value problems. Journal of Sound and Vibration, 1998, 218(4): 573–585

[38] Wacker B. Two variants of magnetic diffusivity stabilized finite element methods for the magnetic induction equation. Mathematical Methods in the Applied Sciences, 2019, 42(13): 4554–4569

[39] Trochimczuk R, Łukaszewicz A, Mikołajczyk T, Aggogeri F, Borboni A. Finite element method stiffness analysis of a novel telemanipulator for minimally invasive surgery. Simulation, 2019, 95(11): 1015–1025

[40] Alizadeh V. Finite element analysis of controlled low strength materials. Frontiers of Structural and Civil Engineering, 2019, 13(5): 1243–1250

[41] Yu Y, Chen Z, Yan R. Finite element modeling of cable sliding and its effect on dynamic response of cable-supported truss. Frontiers of Structural and Civil Engineering, 2019, 13(5): 1227–1242

[42] Abdul Nariman N, Ramadan A M, Mohammad I I. Application of coupled XFEM-BCQO in the structural optimization of a circular tunnel lining subjected to a ground motion. Frontiers of Structural and Civil Engineering, 2019, 13(6): 1–15

[43] Vu-Bac N, Duong T X, Lahmer T, Zhuang X, Sauer R A, Park H S, Rabczuk T. A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Computer Methods in Applied Mechanics and Engineering, 2018, 331: 427–455

[44] Vu-Bac N, Duong T X, Lahmer T, Areias P, Sauer R A, Park H S, Rabczuk T. A NURBS-based inverse analysis of thermal expansion induced morphing of thin shells. Computer Methods in Applied Mechanics and Engineering, 2019, 350: 480–510

[45] Ouakad H M, Najar F, Hattab O. Nonlinear analysis of electrically actuated carbon nanotube resonator using a novel discretization technique. Mathematical Problems in Engineering, 2013, 2013: 517695

[46] Pugno N, Ke C H, Espinosa H D. Analysis of doubly clamped nanotube devices in the finite deformation regime. Journal of Applied Mechanics, 2005, 72(3): 445–449

[47] Pallay M, Daeichin M, Towfighian S. Dynamic behavior of an electrostatic MEMS resonator with repulsive actuation. Nonlinear Dynamics, 2017, 89(2): 1525–1538

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