A new model for stability analysis of electromechanical nanoactuator based on GurtinMurdoch and consistent couplestress theories
Maryam Keivani^{1} , Ali Koochi^{2} , Mohamadreza Abadyan^{3}
^{1}Shahrekord University of Medical Sciences, Shahrekord, Iran
^{2, 3}Mechanical Engineering Group, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran
^{3}Corresponding author
Journal of Vibroengineering, Vol. 18, Issue 3, 2016, p. 14061416.
https://doi.org/10.21595/jve.2015.16452
Received 17 September 2015; received in revised form 26 October 2015; accepted 3 November 2015; published 15 May 2016
JVE Conferences
Herein, the pullin instability of cantilever NEMS is studied considering the presence of dispersion forces and squeezed film damping. Recently developed consistent couple stress theory in combination with the GurtinMurdoch elasticity is employed to incorporate the coupled effects of size phenomena and surface energy. The governing equation was solved using RayleighRitz method. Effects of various parameters including surface layer, size dependency, dispersion forces and damping on the pullin characteristics of the nanoactuator are discussed.
Keywords: consistent couple stress theory (CCST), pullin, surface energy, damping.
1. Introduction
Beamtype nanoactuators have become one of the common components in developing nanoelectromechanical systems (NEMS). A typical NEMS actuator consists of a conductive beam suspended above a rigid conductive plate, by a dielectric spacer between the two components. By exceeding voltage beyond upper limit, the beam spontaneously collapses towards the plate and the pullin instability occurs. In recent decade, comprehensive studies have been conducted on modeling the pullin instability of microactuators [13]. With the decrease in device dimensions to the nanoscale the dispersion forces, i.e. Casimir and van der Waals (vdW) attractions, appear [4, 5]. At separations typically less than several tens of micrometers, the attraction between two surfaces could be described by the Casimir interaction [6]. Previous researchers studied the effect of the Casimir force on the instability of electromechanical systems [710]. However, when separation is less than several tens of nanometers, the Casimir force should be replaced by the vdW force. Some investigators have studied the effect of vdW attraction on the instability of electromechanical systems [1113].
In addition, with dispersion forces, the surface layer characteristics might highly affect the behavior of nanobeams. Gurtin and Murdoch [14, 15] developed a continuum theory for modeling both residual surface stress and surface elasticity. This theory has been previously applied to investigate the effect of surface energy on the buckling [16], bending [17] and vibration [18] of nanostructures. In recent years, some researchers have investigated the influence of surface energy on the pullin characteristics of electromechanical nanobridge [19], nanoswitches [20, 21], graphite NEMS [22] and microplates [23, 24].
Besides the surface energy, the effect of size i.e. microstructuredependency of material characteristics at small scale might be necessary to be considered in modeling the nanoactuators. Experimental works [2527] demonstrate that the size dependency is an inherent property of conductive metals. The classical continuum theory is not able to model the effect of microstructure. To overcome this shortcoming, the nonclassical theories such as couplestress theory [28], modified couplestress theory [29], consistent couple stress theory (CCST) [30] etc. have been developed to consider the size effect. According to the modified couple stress theory the materials are in equilibrium if the applied forces, classical couples and moments of couples equal zero. The last premise is just an assumption and was not considered in the classical elasticity theories. Although the theory predicts stiffer models of micro/nanostructures, its correctness is still under question. In addition, this assumption results in a symmetric stress and couplestress tensors that are not so reasonable due to the nature of the original theory especially for the couplestress tensor. Since this additional consideration might not be acceptable, some researchers tried to find another reasonable solution. This could be achieved by using CCST [30, 31].
In this work, a modified beam model is developed for investigating the pullin behavior of nanoactuators. The coupled effects of surface energy and microstructure are incorporated Analytical RayleighRitz method is employed to solve the nonlinear governing equation.
2. Theory
Fig. 1 shows the schematic representation of a nanoactuator. The NEMS actuator is modeled by a nanobeam of length $L$, and a uniform crosssection of thickness $h$ and width $b$.
Fig. 1. Schematic representation of a cantilever nanoactuator
2.1. Fundamentals of consistent couple stress theory (CCST)
In the CCST, the equations of the isotropic materials are formulated as [30]:
where ${\sigma}_{ij}$ and ${\mu}_{ij}$ represent the forcestress tensor (classical) and couplestress tensors, respectively. In addition, ${F}_{i}$ and ${C}_{i}$ are the body force and the body couple per unit volume of the body, respectively. Here ${e}_{ijk}$ is the permutation tensor or LeviCivita symbol.
The stress tensor is generally nonsymmetric. Thus, it can be decomposed to the symmetric and skewsymmetric components as following:
where ${\sigma}_{\left(ij\right)}$ is the symmetric part and ${\sigma}_{\left[ij\right]}$ is the skewsymmetric part of the forcestress tensor.
In order to define the elements of Eqs. (1) and (2), the kinematic parameters should be utilized. The displacement gradient can be decomposed into two distinct parts:
where the strain tensor and the rotation tensor can explain as:
The rotation tensor is skewsymmetrical and a vector can be defined dual to it as:
The gradient of rotation tensor can be decomposed into two subtensors as:
where:
The corresponding dual vector of the skewsymmetric curvature tensor $\left({\kappa}_{ij}\right)$ can be formulated as:
Substituting Eq. (6) in (8) one obtain:
The symmetrical part of the forcestress tensor in Eq. (2) is same as the forcestress tensor in classical elasticity and can be obtained as:
where $\lambda $ and $\mu $ are the Lame’s constants. The couplestress tensor is skewsymmetrical and a vector ${m}_{i}$ can be introduced dual to the tensor:
For the isotropic linear materials, the couple stress can be computed as [30]:
The parameter, $l$, varies from one scale to another scale. Hadjesfandiari and Dargush [30] showed that the skewsymmetric component of the stress tensor can be obtained as:
Therefore, the strain energy density can be written as [31]:
2.2. Fundamentals of surface elasticity
According to the surface elasticity the strain energy in the surface layer (${U}_{S}$) is [32]:
The governing equations for the surface layer of zero thickness can be explained as [32]:
where ${t}_{\alpha}$ are the components of the traction vector on the surface, ${\kappa}_{\alpha \beta}$ are the components of the surface curvature tensor, ${n}_{i}$ are the components of the outward unit normal to the surface ($n={n}_{i}{e}_{i}$) (with ${n}_{\beta}$ being the inplane components of $n$), and ${\tau}_{\alpha \beta}$ are given by:
where ${\mu}_{0}$ and ${\lambda}_{0}$ are the surface elastic constants, and ${\tau}_{0}$ is the residual surface stress and the outofplane components of the surface stress tensor are given by [15]:
2.3. Nonlinear constitutive equation
For a EulerBernoulli beam, the displacement field can be expressed as [33]:
where $w$ is the centerline deflection of the beam in the $Z$ direction and ${u}_{1}$, ${u}_{2}$ and ${u}_{3}$ are the displacement components in the $X$, $Y$ and $Z$ directions, respectively.
2.3.1. Size dependent strain energy of the bulk
Substituting Eq. (20) in Eqs. (4)(6) and (13) the nonzero components are obtained as:
By substituting Eq. (21) in Eq. (15) after some elaboration and integrating over the beam volume, the bending strain energy is obtained as the following:
2.3.2. Strain energy in the surface layer
By substituting Eq. (20) in Eqs. (17)(19), one obtains:
where ${E}_{0}={\lambda}_{0}+2{\mu}_{0}$ is the surface elastic modulus. By substituting Eq. (23) in Eq. (16) the surface energy conclude as:
For rectangular cross section we have:
2.3.3. Work of external forces
The work by the external forces can be obtained as:
The external force (${F}_{ext}$) is the summation of electrical and dispersion forces.
The electrostatic force per unit length of the nanobeam can be written as^{}[21]:
where ${\epsilon}_{0}=$ 8.85×10^{12} c^{2} N^{1}m^{2 }is the vacuum permittivity and ${\epsilon}_{r}$ is the permittivity of dielectric.
The Casimir force per unit length of beam can be obtained as [32]:
where $\mathrm{\hslash}=\mathrm{}$1.055×10^{34} Js is the reduced Planck’s constant and $c=$2.998×10^{8} m/s is the light speed.
The vdW force per unit length of beam can be explained as [4]:
where $\stackrel{}{A}$ is the Hamaker constant.
2.3.4. Kinetic energy and squeezed film damping
The kinetic energy of the beam can be expressed as:
The squeezed film damping of parallel plates per unit length can be defined as [34]:
where ${\kappa}_{0}$ is viscosity coefficient. Therefore, the virtual work ${W}_{sd}$ performed by damping effects can be expressed as:
2.4. Dimensionless energy of system
The total energy of system can be summarized as:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\frac{1}{2}{\int}_{0}^{L}\left\{{E}_{0}{I}_{0}{\left(\frac{{\partial}^{2}W}{\partial {X}^{2}}\right)}^{2}+{\tau}_{0}{S}_{0}{\left(\frac{\partial W}{\partial X}\right)}^{2}\right\}dX$
$\frac{1}{2}\underset{0}{\overset{L}{\int}}\left[\left(EI+4\mu A{l}^{2}\right){\left(\frac{{\partial}^{2}W}{\partial {X}^{2}}\right)}^{2}\right]dX+\underset{0}{\overset{L}{\int}}\underset{0}{\overset{W}{\int}}{F}_{ext}\left(X\right)dWdX.$
Now, by substituting Eqs. (27)(29) in Eq. (33), considering $x=X/L$ and $w=W/g$ and some mathematical elaboration the dimensionless total energy can be explained as:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\frac{1}{2}\underset{0}{\overset{1}{\int}}\left[\left(1+\frac{24}{1+\upsilon}{\left(\frac{l}{h}\right)}^{2}+{e}_{0}\right){\left(\frac{{\partial}^{2}w}{\partial {x}^{2}}\right)}^{2}\right]dx$
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+\underset{0}{\overset{1}{\int}}\underset{0}{\overset{w}{\int}}\left(\frac{{\alpha}^{2}\left[1+0.65\gamma \left(1w\right)\right]}{(1w{)}^{2}}+\frac{{\beta}_{n}}{(1w{)}^{n}}\right)dwdx,$
where the dimensionless parameters are identified as:
$\widehat{c}={c}_{d}{L}^{2}\sqrt{\frac{1}{\rho bhEI}\mathrm{}},\gamma =\frac{g}{b},\mathrm{}\mathrm{}\mathrm{}{\beta}_{3}=\frac{\stackrel{}{A}b{L}^{4}}{6\pi {g}^{4}EI},{\beta}_{4}=\frac{{\pi}^{2}\mathrm{\hslash}cb{L}^{4}}{240{g}^{5}EI}.$
3. Solution methods
To solve the governing equation using RayleighRitz method the displacement is expressed as a combination of independent basis functions ${\phi}_{i}\left(x\right)$ in the form of:
where the index $i$ refers to the number of modes included in the simulation.
We use the linear mode shapes (based on the classic theory) as basic functions:
where ${\omega}_{i}$ is the $i$th root of characteristic equation of the cantilever beams in the classical theory. For minimize the total energy of the system we must have:
This leads to a system of timedependent equations which can be solved numerically. Substituting Eqs. (34) and (36) into Eq. (38), assuming the orthogonality of ${\phi}_{i}\left(x\right)$ and following some mathematical operations, a system of governing equations is found as:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\int}_{0}^{1}{\varphi}_{i}\sum _{k=0}^{\mathrm{\infty}}{A}_{k}{\left(\sum _{j=1}^{N}{q}_{j}{\varphi}_{j}\right)}^{k}dx+{t}_{0}{\left.\frac{\partial}{\partial x}\left(\sum _{j=1}^{N}{q}_{j}{\varphi}_{j}\right)\frac{d{\varphi}_{i}}{dx}\right}_{x=1}=0,\mathrm{}\mathrm{}\mathrm{}\mathrm{}i=1,\mathrm{}2,\dots ,N,$
where:
The Maple software is employed to numerically solve the system of equations.
4. Results and discussion
Fig. 2a shows the variation of static deflection of the actuator. The deflection of the actuator increases by increasing the applied voltage from zero to the pullin value, ${\alpha}_{PI}$. The time history and phase plane is shown in Fig. 2b and 2c. By increasing the voltage, the maximum amplitude of the tip deflection increases. If the applied voltage exceeds its critical value, ${\alpha}_{PI}$, the pullin occurs. The phase plane has two fixed points; the stable center point and the unstable saddle node.
The impact of surface energy on the pullin voltage is shown in Fig. 3a. As seen, by increasing the surface residual stress (${t}_{0}$), the pullin voltage enhances. Surface stresses induce hardening or softening effect depends on its sign; when it is positive, surface effect increase the pullin voltage and if it is negative, the surface effect reduces the pullin voltage. The dynamic pullin voltage is smaller than the static pullin voltage due to inertia forces.
The influences of microstructure and dispersion forces on the pullin voltage is shown in Fig. 3b. This figure reveals that the size parameter ($l/h$) results in increasing the pullin voltage. While for lower values of the size parameter the presence of dispersion forces reduces the pull in voltage, for larger size parameter the effects of dispersion force are negligible.
Fig. 2. Behavior of nanoactuator by neglecting size, surface and damping effect for different values of $\alpha $ from zero to pullin voltage, a) Static deflection, b) Dynamic behavior, c) Phase plane
a)
b)
c)
Fig. 3. Influence of small scale effect on the pullin voltage
a) Surface effect
b) Size effect
Fig. 4 shows the influence of size parameter on the pullin voltage when the positive and negative surface stresses, ${t}_{0}=1$ and ${t}_{0}=1$, are considered as well as dispersion forces. This figure reveals that consideration of the size effect increases the pullin voltage and has hardening effect. Fig. 4 reveals that the size effect always has hardening effect i.e. increases the pullin voltage. For positive surface stress the surface effect increases the pullin voltage (hardening effect) but for negative surface stress this effect decreases the pullin voltage (softening effect).
Fig. 4. Influence of size effect on pullin voltage
a) vdW regime
b) Casimir regime
Fig. 5. Influence of damping on the dynamic behavior
a) Time history
b) Phase plane
To investigate the effect of damping, the time history and phase plane are plotted for ${t}_{0}=$ 0.1, ${e}_{0}=$0.05 ${\beta}_{n}=$ 0.5, $\gamma =$ 1, $l/h=$ 0.2 and $\widehat{c}=$ 0.5. As seen in Fig. 5, the system returns to the stable focus point. By considering the damping effect, the actuator will oscillate with decreasing amplitude and converges to equilibrium. For the pullin voltage, the trajectories which are attracted to the stable focus point, diverge and the actuator becomes unstable.
4.1. Validation
Consider an actuator with $L$ = 1 μm, $b$ = 5 h, $g$ = 50 nm and $h$ = 50–100 nm made of silver where the $\upsilon $, $l$, $E$, ${\tau}_{0}$, ${E}_{0}$ and $A$ values are 0.37, 25 nm 76 GPa, 0.89 N/m, 1.22 N/m, 3.5×10^{19} J respectively. Fig. 6 represents the comparison between available results in the literature and those of our model. According to the present model, surface energy increases the pullin voltage. However, the model by Ma et al. [20] shows reversed trends due to ignoring the surface stress in the boundary conditions. The results of the present model for classic state are in good agreement with Ma et al. [20]. By considering the surface effect the results of present model are close to those of Rokni and Lu [22] and Shaat and Mohamed [35].
Fig. 6. Comparison of results from the present model with those from literatures
5. Conclusions
The coupled effects of surface energy and microstructure on pullin behavior of the nanoactuators were studied. The obtained results revealed that the surface layer might significantly affect the pullin of the actuators and cannot be ignored in theoretical model. For positive surface stress, the surface layer increases the pullin voltage. However, if the surface stress be negative, the surface effect reduces the pullin voltage. The size effect always increases the pullin voltage, while the dispersion forces decreases the pullin voltage.
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