Numerous applications of micro- and nano-electromechanical systems (MEMS/NEMS) have motivated researchers to study their performance in various situations. Because the classical continuum theories cannot consider the size effects in the mechanical analysis of nanostructures,^[1–4] some size-dependent continuum theories like Eringenʼs nonlocal elasticity theory,^[5] the couple stress elasticity theory,^[6] the Gurtin-Murdoch continuum elasticity theory,^[7–11] the strain gradient elasticity theory,^[12] and the stress-driven nonlocal model^[13] were proposed to consider the size effects. In the classical theories, the stress state at a given point is determined only by the strain state at that point, but in Eringenʼs nonlocal elasticity theory, the stress state at a given point is determined by the strain states of all points in the body. The first use of Eringenʼs nonlocal elasticity theory to nanotechnology was proposed by Peddieson et al.,^[14] followed by many other researchers.^[15–19] One of the important designing considerations in MEMS/NEMS is the pull-in instability.^[20] The pull-in instability happens when the internal and applied external forces surpass the elastic restoring force of the nanostructures, leading to contact between the movable and substrate electrodes. When the rate of applied voltage variation is significant, the effect of inertia is considered. In this case, the pull-in instability is referred to as the dynamic pull-in instability.

Once the space of movable and bottom electrodes is less than the plasma wavelength or the absorption wavelength of the ingredient material of surfaces, the intermolecular force is considered as the van der Waals (vdW) attraction. In this situation, there is a small separation regime such that the vdW force is the dominant attraction and it is proportional to the inverse cube of the separation. Nevertheless, when the separation is adequately large (typically above 20 nm) the intermolecular interaction is referred to as the Casimir force.^[21] In this case, there is a large separation regime in which the Casimir force is dominant (typically above several tens of nanometers) and it is proportional to the inverse fourth power of the separation , in which h=1.055×10⁻³⁴ is Planckʼs constant divided by 2π and c=2.998×10⁸ m/s is the speed of light.^[21] The reader is referred to Refs.[22–30] as some important papers and books about the Casimir effect. One of the most remarkable predictions of quantum electrodynamics (QED), obtained by Casimir in 1948, is that two parallel, closely spaced, conducting plates will be mutually attracted.^[31] This measurement, as reported by Sparnaay in 1958, confirmed the formula.^[32] A closely related effect, the attraction of a neutral atom to a conducting plate, has been also measured.^[33]

In the past few years, many researchers have focused on the pull-in instability of nanoplates. For instance, based on a modified continuum model, Ansari et al.^[34] studied the size-dependent pull-in behavior of electrostatically and hydrostatically actuated rectangular nanoplates considering the surface stress effects. Ebrahimi and Hosseini^[35] investigated the effect of temperature on pull-in voltage and nonlinear vibration of nanoplate-based NEMS under hydrostatic and electrostatic actuations. Mirkalantari et al.^[36] studied the pull-in instability of rectangular nanoplates based on the strain gradient theory taking the surface stress effects into account. Shokravi^[37] analyzed the dynamic pull-in of viscoelastic nanoplates under the electrostatic and Casimir forces. The interested reader is referred to Refs.[38–44].

Moreover, different methods have been used for the vibration analysis of rectangular nanoplates. For example, Aghababaei and Reddy^[45] presented the Navier solutions for the vibrations of rectangular plates based on the nonlocal third-order shear deformation plate theory. Also, Pradhan and Phadikar^[46] used the same solution technique for addressing the vibration problem of rectangular plates with simply-supported boundary conditions in the context of Eringenʼs nonlocal model, the classical and first-order shear deformation plate theories. Another application of the Navier-type method to the vibration problem of nonlocal plates can be found in Ref. [47]. Aksencer and Aydogdu^[48] employed the Levy-type solution method for the vibration analysis of nanoplates based on the nonlocal elasticity theory. Ansari et al.^[49,50] used the generalized differential quadrature method to numerically solve the free vibration problem of rectangular Mindlin-type plates with various boundary conditions. The Galerkin method was applied by Shakouri et al.^[51] for the vibrational analysis of nonlocal Kirchhoff plates with different edge supports.

The classical analytical methods cannot handle the strongly nonlinear differential equations. In this regard, Liao^[52] developed an efficient technique called the homotopy analysis method (HAM), which can be adopted for solving ordinary and partial differential equations with different nonlinearities. For example, Samadani et al.^[53] applied HAM for the pull-in and nonlinear vibration analysis of nanobeams using a nonlocal Euler-Bernoulli beam model. Moghimi Zand and Ahmadian^[54] used HAM in studying the dynamic pull-in instability of microsystems. Also, Miandoab et al.^[55] utilized this method for the forced vibration analysis of a nano-resonator with cubic nonlinearities.

In the present paper, HAM is used to study the static and dynamic pull-in instabilities of rectangular nanoplates using the nonlocal Kirchhoff plate theory. The rest of paper is organized as follows: In Sec. 2, using Eringenʼs nonlocal elasticity and the Kirchhoff plate theory, the nonlinear equation of motion subjected to fully clamped boundary condition (CCCC) is derived. In secs. 3 and 4, the governing equation of motion is reduced to an NLODE in the time domain by the Galerkin method. Then, HAM is adopted to solve the obtained nonlinear equation. The effects of intermolecular, hydrostatic, and thermal actuations as well as the nonlocal parameter on the deflection, nonlinear frequency, and the critical voltage of dynamic pull-in instability (V_pdyn) are investigated in Sec. 5. At the end, the main findings of the paper is given in Sec. 6.

A schematic of nanoelectromechanical rectangular plate with length l_a and width l_b, including a pair of parallel electrodes with the distance g is given in Fig. 1. The upper movable electrode is assumed to be under the influence of intermolecular, hydrostatic, and thermal actuations.

Fig. 1.

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	Fig. 1. Schematic of fully clamped nanoelectromechanical rectangular plates under intermolecular, hydrostatic and thermal actuations.

It is noted that the movable electrode pulls down the fixed electrode by applying the DC voltage between two electrodes. Once the applied voltage approaches the critical point (pull-in voltage), the structure becomes unstable.^[57]

The electrostatic force per unit area of nanoplate can be described as^[39]
where is the vacuum permittivity, g is the air initial gap of nanoplates, and V_dc is the direct current voltage as illistrated in Fig. 1. The van dar Waals effect per unit area of nanoplate can be written as^[39]
where A_h is the Hamaker constant in the range of .

In the following analyses, it is assumed that
where F_h stands for the hydrostatic actuation. By substituting Eqs. (15) and (16) into Eq. (17) and then inserting the resulting equation in Eq. (14), the following governing equation of motion is obtained
with the following fully clamped boundary conditions
By considering the following nondimensional variables
and using the Taylor expansion (see Appendix), the nondimensional form of governing equation can be derived as
with the following boundary conditions
Here, GM is utilized to reduce Eq. (21) to an NLODE. To this end, it is considered that
where is the first eigenmode of fully clamped nanoplate and .^[58]

By inserting in Eq. (21), multiplying it by and then integrating twice from zero to one, the following NLODE is obtained
where the parameters a₀, a₁, a₂, a₃, a₄, and M are given in Appendix.

Now, using the transformation , the existing reduced problem
is changed into
where the oscillation nonlinear frequency is expressed as
In a manner similar to that performed in Ref. [53], one can obtain Ω and u(T) for vibrating actuated fully clamped nanoplate as below

The current section provides numerical results to show the effects of intermolecular, hydrostatic, and thermal actuations as well as the nonlocal parameter μ on the deflection, nonlinear frequency, and V_pdyn. For producing the results, the following parameters are selected: h = 21 nm, , g=1.2h, ϑ=0.35, and E=68.5 GPa (Al alloy).

In Fig. 2, the nondimensional center point deflection of nanoplate obtained using HAM is compared to that calculated using the Runge-Kutta method. It is seen that there is an excellent agreement between the results of two methods.

Fig. 2.

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	Fig. 2. HAM results versus those of the Runge-Kutta method.

Figure 3 shows the change in amplitude of vibration against the nondimensional time. In this case, the dynamic pull-in voltage is 22.540. The V_pdyn obtained in the absence of the intermolecular and thermal parameters agrees well with those reported in Refs.[34,59] (in Ref. [34] V_pdyn=22.5 and in Ref. [59] V_pdyn=22.38). The amplitude enhances with the increase of time and the nanoplate experiences a harmonic motion. Also, the nanoplate collapses onto the bottom, when the pull-in happens.

Fig. 3.

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	Fig. 3. Centerpoint deflection of a fully clamped nanoplate at .

Figure 4 presents the normalized fundamental frequency of nanoplate with respect to the electrostatic parameter β. It is observed that the normalized fundamental frequency becomes zero when the applied voltage reaches V_pdyn.

Fig. 4.

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	Fig. 4. Relation between the normalized fundamental frequency and the electrostatic force parameter.

Figure 5 indicates that the pull-in time decreases (11 percent) by increasing μ (0.01 per unit). By decreasing μ, the pull-in phenomenon occurs later in this model.

Fig. 5.

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	Fig. 5. Effect of the parameter μ on the pull-in and deflection time.

The variations of fundamental frequencies against V_pdyn are illustrated in Fig. 6 for different values of nonlocal parameter. It is observed that by increasing μ (0.02 per unit), the fundamental frequency decreases (2 percent). For example, when μ is considered to be 0.06, V_pdyn occurs at 21.

Fig. 6.

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	Fig. 6. Effect of the parameters μ and Vpdyn on the fundamental frequency.

Figure 7 demonstrates the variations of fundamental frequencies against the hydrostatic pressure parameter for different values of nonlocal parameter. The increase of N_hydro leads to the decrease of fundamental frequency.

Fig. 7.

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	Fig. 7. Effect of the nondimensional hydrostatic pressure on the fundamental frequency.

The variations of fundamental frequencies against the nondimensional thermal parameter are illustrated in Fig. 8 for various values of μ. One can find that via increasing N_thermal, the fundamental frequency gets larger.

Fig. 8.

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	Fig. 8. Effect of the nondimensional thermal actuation on the pull-in frequency.

Figure 9 shows the variations of fundamental natural frequency versus A for a number of vdW parameters. It is seen that by increasing A, the nonlinear frequency of vibration diminishes. For instance, when R₃ is equal to 5, the pull-in phenomenon happens at A = 1.

Fig. 9.

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	Fig. 9. Effect of the intermolecular actuation on the pull-in frequency.

The influence of on the static nondimensional deflection versus the nonlocal parameter is highlighted in Fig. 10. According to this figure, the pull-in instability is suspended as λ increases. The influences of distance of parallel electrodes, thermal and nonlocal parameters on the static and dynamic pull-in voltage parameter are compared in Figs. 11–13, respectively. Based on Fig. 11, the pull-in voltage is postponed as the distance increases. Also, Fig. 12 indicates that as the thermal parameter increases, the pull-in voltage is delayed. Finally, Fig. 13 shows that by increasing the nonlocal parameter, the pull-in voltage decreases.

Fig. 10.

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	Fig. 10. Variations of static pull-in deflection of the nanoplate versus the μ.

Fig. 11.

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	Fig. 11. Comparison of the static and dynamic pull-voltage parameter versus g in μ = 0.01.

Fig. 12.

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	Fig. 12. Evaluation of the static and dynamic pull-voltage against Nthermal in μ = 0.01.

Fig. 13.

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	Fig. 13. Comparision of dynamic and static pull-in voltage parameter of the nanoplate versus the μ.

In the present research, the dynamic pull-in of CCCC plate-type nanosensor, subjected to electrostatic, intermolecular, hydrostatic and thermal actuations was analyzed based on the nonlocal theory. GM was utilized for reducing the governing NLPDE to an NLODE in the time domain. HAM was also applied for solving the NLODE. The outcomes reveal that: