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Poly silicon nanobeam model based on strain gradient theory
Accepted Manuscript Title: Poly Silicon Nanobeam Model Based on Strain Gradient Theory Author: Ehsan Maani Miandoab Aghil Yousefi-Koma Hossein Nejat Pishkenari PII: DOI: Reference:
S0093-6413(14)00122-0 http://dx.doi.org/doi:10.1016/j.mechrescom.2014.09.001 MRC 2896
To appear in: Received date: Revised date: Accepted date:
28-7-2013 21-8-2014 4-9-2014
Please cite this article as: Miandoab, E.M., Yousefi-Koma, A., Pishkenari, H.N.,Poly Silicon Nanobeam Model Based on Strain Gradient Theory, Mechanics Research Communications (2014), http://dx.doi.org/10.1016/j.mechrescom.2014.09.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights: An efficient method to find the static response of nanobeams
Poly silicon Young’s modulus and length scale parameters estimation
Effect of axial stress and mid-plane stretching on strain gradient microbeam
Quick method for natural frequency estimation based on SG theory
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Poly Silicon Nanobeam Model Based on Strain Gradient Theory a
Ehsan Maani Miandoaba, Aghil Yousefi-Komaa,1 and Hossein Nejat Pishkenarib
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Center of Advanced Systems and Technologies (CAST), School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran b Center of Excellence in Design, Robotics and Automation (CEDRA), Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran
Abstract
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Conventional continuum theory does not account for contributions from length scale effects. Failure to include size-dependent contributions can lead to underestimates of deflection and stresses of micro and nanobeams. This research aims to use strain gradient elasticity theory to model size-dependent behavior of small beams. In this regard, Young’s modulus and length scale parameters of poly silicon are estimated by fitting the predicted static pull-in voltages to the reported experimental results in the literature. The results demonstrate that decreasing the beam thickness results in higher pull-in voltage, lower deflection and lower sensitivity to axial stress and mid-plane stretching in comparison with classical model results. Keywords: Micro and Nanobeams, Strain gradient theory, Pull-in voltage.
1. Introduction
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Applications of micro- and nano-electro-mechanical systems (MEMS\NEMS) with electrical actuation have recently increased due to the many advantages such as fast response, low energy consumption and low cost. A typical electrostatically actuated MEMS\NEMS is comprised of two parallel conducting electrodes, one is fixed and the other is movable due to applied electrostatic force. Increasing the input voltage beyond a critical value leads to instability of the movable electrode and pull-in phenomena. In design and test of MEMS devices, pull-in phenomena must be considered due to its importance in MEMS\NEMS structural safety (Tilmans and Legtenberg, 1994). Extensive research has been carried out on static and dynamic pull-in analysis of MEMS\NEMS under electrostatic actuation (Abdel-Rahman et al., 2002; Hu, 2006; Lin and Zhao, 2005; Nayfeh et al., 2007; Younis et al., 2003; Zhang and Zhao, 2006). In all of these studies, the movable electrode was modeled as an elastic beam based on classical continuum theory. The thickness of the beams used in MEMS and NEMS are in the order of microns and submicrons. Numerous experiments showed when decreasing the size of the device, size-dependent behavior turns out to be important. This cannot be explained using conventional mechanics theories due to lack of any material length scale parameter (Aifantis, 2009; Chasiotis and Knauss, 2003; Fleck et al., 1994; Lam et al., 2003; Lam and Chong, 1999; McFarland and Colton, 2005; Nix, 1989; Poole et al., 1996). More recently, non-classical continuum theories have been developed to predict the size-dependent behavior of materials in small size, such as modified couple stress theory, nonlocal and strain gradient elasticity theories and modified continuum model incorporating surface elasticity. Based on nonlocal continuum theory, the stress state at a point is a function of the strain states of all points in the body, while classical continuum mechanics assumes the stress state at a given point is dependent only on the strain state at that same point. Peddieson et al. used nonlocal theory in nanotechnology for the first time and developed the static response of micro and nanobeams based on this non-classical theory (Peddieson et al., 2003). Lim developed a new model to capture the static and dynamic response of micro and nanobeams based on nonlocal theory and it was shown that in contrast with the previous model, his model results in higher stiffness for microstructures as was observed in empirical tests (Lim, 2010). Lubineau et al. used morphing strategy to couple non-local to local
1
Corresponding author, Email: [email protected] Tel: +98 2161114228Fax:+9821 61114228
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continuum mechanics which can be used in modeling of large scale materials with non-local effects (Lubineau et al., 2012). Strain gradient theory is another non-classical approach that can describe the size effect resulting from underlying microstructures (Fannjiang et al., 2002; Fleck et al., 1994; Lam et al., 2003). In this theory, new additional equilibrium equations are utilized to model the behavior of higher-order stresses with three independent higher-order materials length scale parameters for isotropic linear elastic materials. Kong et al. (Kong et al., 2009) studied the static and dynamic behavior of EulerBernoulli beams using strain gradient theory. They studied the size effects on the beam bending response and its natural frequencies for two kinds of boundary conditions and it was found that strain gradient theory results in higher stiffness for a beam in comparison with classical continuum theory. Furthermore, many studies have been performed on the analysis of static and dynamic behavior of beams considering size effect. Comparison of results obtained from non-classical techniques with classical continuum theory demonstrates the significance of the size effect; (Akgöz and Civalek, 2011, 2012a; Asghari et al., 2011; Kahrobaiyan et al., 2010; Rajabi and Ramezani, 2012; Reddy, 2010; Xu and Jia, 2007). Some studies have been done on analysis of functionally graded materials (FGM) based on strain gradient theory (Chan et al., 2008; Fannjiang et al., 2002). For instance, Paulino et al. presented a theoretical framework to model antiplane shear cracks in FGM using strain gradient elasticity (Paulino et al., 2003). Despite considerable progress in the field of beam modeling, only a few works have been devoted to the investigation of size-dependent behavior for electrostatically actuated devices. Here, some of these works are reviewed. Fu and Zhang investigated the dynamic response and pull-in of electrically actuated nanobeam based on surface energy theory using the analogue equation method (Fu et al., 2010). To determine the deflection and static pull-in voltage of silicon micro-cantilevers, Rahaeifard et al. used the modified couple stress theory. Based on their results, the length scale parameter of silicon was determined as l 592nm (Rahaeifard et al., 2011). Baghani presented an analytical solution for size-dependent response of cantilever microbeams using the modified couple stress theory (Baghani, 2012). Wang et al. investigated the dynamic response of poly silicon microbeams under electrostatic actuation using strain gradient theory and determined the poly silicon length scale parameter as l 110nm by fitting the predicted normalized resonant frequency to experimental data (Wang et al., 2011). They used the generalized differential quadrature (GDQ) method to solve the sixth-order partial differential equation. Rokni et al. proposed an analytical closed-form solution for static pull-in voltage of microbeams based on modified couple stress theory and the value of length scale parameter of poly silicon was estimated to be in the order of 100 nm (Rokni et al., 2012). In this paper, an efficient method, the analogue equation method, is proposed to find the static response of clamped-clamped micro and nanobeams based on strain gradient theory under electrostatic actuation. In mathematical modeling of the system, fringing field effect, residual axial stress and mid-plane stretching are considered. The validation of the proposed method has been checked by comparison of the static pull-in voltages with the results obtained using the Differential Quadrature Method (DQM) (Kuang and Chen, 2004), Homotopy Analysis Method (HAM) (Moghimi Zand and Ahmadian, 2009) and the Fredholm Integral Equation method (FIE) (Rokni et al., 2012). An important drawback of the previous theoretical research using non-classical continuum theories is that they don’t account for effects of the size on the Young modulus of the beam and merely adjust the length scale parameters for small sizes to fit data with experimental results; while in the present study, to model poly silicon nanobeams based on strain gradient theory, the Young’s modulus and length scale parameters of poly silicon have been determined by fitting the static pull-in voltages of different MEMS to experimental results and it has been demonstrated that the proposed model based on strain gradient theory leads to an appropriate agreement with experimental observation. Moreover, to investigate the size effect on the axial stress and mid-plane stretching, we have varied the beam thickness and compared our results with classical continuum theory. 2. Mathematical Modeling Figure (1) shows a schematic of a traditional MEMS system under electrostatic actuation.
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L
x
Movable Beam h
g
z Fig.1. Schematic of an electrostatically actuated microbeam
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Fixed Beam
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As seen in this figure, this system is composed of one movable and one fixed conductive electrode. The dynamic behavior of microbeam under electrostatic force can be derived as (Akgöz and Civalek, 2012b; Kong et al., 2009):
6w 4w 2w EA w 2 2w bV 2 ( g w ) S A ( N ( ) dx ) (1 ) 6 4 2 2 2 b x x t x 2l 0 x 2( g w )
an
l
K where
S EI A (2l 02
M
4 K I (2l 02 l12 ) 5
(1)
(2)
8 2 2 l1 l 2 ) 15
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In derivation of this relation, the Euler-Bernoulli beam assumption based on strain gradient theory was used. The first term on the right hand of Eq. (1) represents the axial force and mid-plane stretching effects which vanish for a micro-cantilever. The parameters E , A , , I ,V , , N , g , and w are respectively Young’s modulus, beam cross section area, material density, cross section moment of inertia, applied voltage, dielectric constant, axial force, gap distance between two electrodes, shear modulus and beam deflection, which is a function of the beam axial position x and time t , meaning w( x, t ) . is the parameter that presents the fringing field effect and the independent
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length scale parameters l0 , l1 and l2 are respectively dilatation gradients, deviatoric stretch gradients and rotation gradients. In order to facilitate the study of the qualitative behavior of the system, the following relations are considered:
xˆ
(3)
x w t ; wˆ ; tˆ l g T
By substituting Eq. (3) in Eq. (1), Eq. (1) can be rewritten as
0
1 2 2 6wˆ 4wˆ 2wˆ ˆ ( wˆ )2 dxˆ ) wˆ 2V (1 g 1 wˆ ) ( N 1 xˆ 6 xˆ 4 tˆ 2 xˆ xˆ 2 (1 wˆ )2 b 0
(4)
Parameters appearing in Eqs. (3) and (4) are
0
2 K EAg 2 bl 4 ˆ l N, T , , , N 1 2 Sl 2 S 2S 2Sg 3
4
bhl 4 S
(5)
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It is worthy to note that the static response of the microbeam can be found by neglecting the time dependent term in Eq. (4). Boundary conditions corresponding to a clamped-clamped beam based on strain gradient theory are:
wˆ (0) wˆ (1) 0 dwˆ (0) dwˆ (1) 0 dxˆ dxˆ
(6a)
ip t
(6b)
d 3wˆ (0) d 3wˆ (1) 0 dxˆ 3 dxˆ 3
(6c)
(7a) (7b) (7c)
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an
d 4wˆ (1) d 2wˆ (1) 0 4 2 0 d xˆ d xˆ d 5wˆ (1) d 3wˆ (1) 0 5 3 0 d xˆ d xˆ d 3wˆ (1) 0 dxˆ 3
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The boundary condition for a micro-cantilever beam with a free end at x l is as follows.
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In order to analyze the static behavior of the microbeam, it is enough to solve the nonlinear boundary value problem presented here. In this regard, in the next section, the analogue equation method will be used to solve this problem.
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3. Analogue Equation method (AEM) AEM is a powerful method for solving nonlinear partial differential equations and has been previously utilized for solving fourth-order nonlinear partial differential equations of microbeam dynamic response under electrostatic actuation (Fu et al., 2010; Maani Miandoab et al., 2014). This paper aims to use this method to find the solution of the introduced nonlinear boundary value problem. Consider the following equation: d 6w ( x ) a (x ) dx 6
(8)
where a ( x ) is the nonlinear friction load. By dividing the beam into N equal elements with N nodal points, the following solution can be presented for Eq. (8).
w ( x ) H( x )b G( x )a
(9)
where
H(x ) [x 5 ....., x ,1] b [b1 , b 2 ...........b 6 ]T l 3l (2N 1)l T a [a ( ), a ( ),........a ( )] 2N 2N 2N
5
(10)
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1l N
G( x ) [ G (x )d ,.............
l
G (x )d ]
( N 1) l N
0
In each element, the friction load a ( x ) is substituted with its value at the corresponding nodal point in the middle of the element and b i ( i 1,2...,6 ) are constants. G (x ) is the Green’s function which is
ip t
d 6G (x , ) (x , ) . where ( x ) is the Dirac delta function. Integrating both sides dx 6 of this equation with respect to x yields 1 x (x )4 240
us
G (x )
cr
the solution of
(11)
By inserting Eq. (11) into the corresponding boundary conditions, Eq. (6) or Eq. (7), b can be found as a function of a as b K aa , where K a is 6 N matrix. Thus Eq. (9) can be rewritten as
an
w (x ) H(x )b G(x )a (H(x )K a G(x ))a M(x )a
(12)
M
M ( x ) is a 1 N vector. Substituting Eq. (12) into Eq. (4) leads to the following equation at the nodal point k. 2 2 d 4M k g ˆ ( dM k a)2 dxˆ ) d M k a 2V a ( (1 (1 M k a)) 0 a( k ) N 1 4 2 2 b dxˆ dxˆ dxˆ (1 M k a) 0
te
where Mk M(x k ) and
(13)
d
1
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dn Mk d n H(xˆ k ) d n G(xˆ k ) ( Ka )a dxˆ n dxˆ n dxˆ n
(14)
To find the solution for a , the Gauss-Seidel method is utilized. On the basis of initial guess, the following equation can be used to determine the new vector, a new .
( 0I
d4R )a new F(aold ) dxˆ 4
(15)
where F(aold ) is the N 1 vector obtained by calculating of the right-hand of Eq. (13) for all N nodal points, I is the N N unit matrix and the matrix R is defined as follows.
M1 M 2 R . . Mn
6
(16)
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is N 0.0009 Newton .
Length
scale
parameters
are
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4. Numerical Simulation 4-1Solution validation In order to validate the proposed numerical method, pull-in voltages based on strain gradient theory will be compared with the results of the classical model for a range of parameters in which the size effects are negligible. The classical model solutions existing in the literature are obtained by the Differential Quadrature Method (DQM) (Kuang and Chen, 2004), Homotopy Analysis Method (HAM) (Moghimi Zand and Ahmadian, 2009) and the Fredholm Integral Equation (FIE) (Rokni et al., 2012). Clamped-clamped microbeam properties used for validation are given as follows: Young’s modulus and Poisson’s ratio are E 166GPa and 0.3 , beam length, width, height and initial gap are L 210μm , b 100μm , h 1.5μm and g 1.18μm respectively. Residual axial load as l 0 l1 l 2 Cl 10nm .
set
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Because h / Cl 20 , the size effect can be ignored and predicted results from non-classical continuum theories converge to classical ones. This makes it possible to compare the proposed method with classical results without loss of generality. Maximum displacement of a microbeam is plotted versus the applied voltage in Fig.2. As seen in this figure, by increasing the applied voltage, the slope of the curve increases and approaches infinity nearV 28.12V indicating the pull-in phenomenon. Stability limit is w max .41which is in good agreement with the results reported in the literature (Abdel-Rahman et al., 2002). 0.45
M
0.4 0.35 0.3
0.2
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w
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0.25 x a m n
0.15
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0.1
0.05 0
0
5
10
15 V(volt)
20
25
30
Fig.2. Beam midpoint normalized deflection for 40 iterations
Table 1 compares the pull-in voltages obtained using AEM (this study) with the results conducted in previous research. There is good agreement between the pull-in voltage obtained by the proposed method and DQM, FI and HAM. Table 1.Comparison of pull in voltage based on different methods Method
DQM(Kuang and Chen, 2004)
FI(Rokni et al., 2012)
V PI (V)
28.1
28.24
HAM(Moghimi Zand and Ahmadian, 2009)
28.86
7
Present study
28.12
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4-2Young’s modulus and length scale parameter estimation of poly silicon In this paper, experimental data reported in the literature are used to identify the Young’s modulus and length scale parameters of poly silicon based on strain gradient theory. The geometric parameters of 15 cases used for identification are given in Table 2. Table 2. Geometric parameters of used microbeams Boundary Condition
L (μm)
b (μm)
h (μm)
g (μm)
Ref.
1-4
Clamped-Clamped
210-510
100
1.5
1.18
(Tilmans and Legtenberg, 1994)
5-14
Clamped-Free
100-500
40
2.1
2.4
15
Clamped-Free
100
15
1.8
5
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Case
(Gupta, 1998)
cr
(Ballestra et al., 2008)
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Cases 5-14 have a constant radius of curvature Rc 40mm (Gupta, 1998). Using the finite-difference script in (Gupta, 1998), it has been shown that the pull-in voltage of a curved cantilever can be calculated as a function of pull-in voltage of the ideal uniformly flat case using the following equation:
l2 l2 2 ) 0 0006347( ) ) gR c gR c
an
V Curved V Flat (1 0.5096(
(17)
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where V Curved and V Flat are the pull-in voltages of a curved and flat beam, respectively. By fitting the empirical and simulated results based on the classical Euler-Bernoulli beam model, effective Young’s modulus of poly silicon was estimated as E 166GPa in (Tilmans and Legtenberg, 1994)and E 153GPa in (Gupta, 1998) based on the empirical results of 1.5μm (cases 1-4) and 2.1μm (cases 5-14) microbeams respectively. Because of size effect, reducing the beam thickness leads to higher stiffness which cannot be modeled using classical continuum theory. Static pull-in voltages of cases 514 based on classical continuum theory ( E 166GPa and E 153GPa ) are compared with experimental ones (Gupta, 1998) in Fig. 3. As can been seen in Fig. 3, estimated value of Young’s modulus for a 1.5μm height microbeam ( E 166GPa ) leads to higher pull-in voltage for 2.1μm height microbeam because of size effect. For modeling the beam based on the strain gradient theory, Young’s modulus and length scale parameters of poly silicon should be calibrated using experimental results. In the present study, by varying the Young’s modulus and length scale parameter simultaneously and comparing the obtained pull-in voltages with the reported experimental data, it was found that the following parameters lead to the best agreement with the experimental results: E 156GPa and l 0 l1 l 2 Cl 90nm .
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80 Exp. E=166GPa E=153GPa
70
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60
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50 40
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30
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20
0 100
150
200
250
M
10
300
350
400
450
500
d
Fig.3. Simulated pull-in voltage of cases 5-14 based on classical continuum theory in comparison with experimental results
te
Comparison of the average relative errors of pull-in voltages of different cases based on strain gradient (SG) theory with the classical (CL) ones is demonstrated in Table 3 .
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Table 3. Average relative errors of pull-in voltages of different cases based on classical and strain gradient continuum theories
h (μm)
1.5 (cases 1-4) 1.8 (cases 15) 2.1 (case 5-14)
CL( E 153GPa )
Error (%) CL( E 166GPa )
2.27 4.28 1.48
1.94 0.7 4.34
SG( E 156GPa, Cl =90nm ) 1.89 1.3 2.21
As can be seen in Table 3, predicted pull-in voltages based on strain gradient theory with calibrated parameters show the best agreement with the experimental ones. It should be noted that all of the cases investigated in this study are narrow beams ( b 5h ), thus estimated Young’s modulus for these cases is plate modulus ( E E / (1 ) ). By setting 0.23 (Tilmans and Legtenberg, *
2
1994), Young’s modulus of poly silicon can be found as E * 147.7GPa . 4-2-1 Natural frequencies of micro and nanobeams based on strain gradient theory In this section, the natural frequency of microbeams based on strain gradient theory using calibrated material parameters ( E * 147.7GPa and Cl 90nm ) is compared with experimental results. Using the least square fitting to experimental results, the following equation was presented for natural resonant frequency of cases 1-4 in (Tilmans and Legtenberg, 1994).
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(18)
( f 1l ) 2 7.5031 10 7 ( h l ) 2 814.94
where f 1 is the natural frequency of microbeams.
1
(xˆ )T (wˆ )dx 0
Mu K 1u K 3u 3 0
The parameters in Eq. (19) are
0
K 1 ( 0 ( xˆ ) 0
d (xˆ ) d 4 (xˆ ) d 2 (xˆ ) ˆ ˆ ( x ) N ( x ) )dxˆ dxˆ 6 dxˆ 4 dxˆ 2 6
1
K 3 1 ( xˆ )
d 2 ( xˆ ) d ( xˆ ) 2 ( ( ) dxˆ )dxˆ 2 dxˆ dxˆ 0 1
M
0
(20)
an
1
us
1
M ( xˆ ) 2 dxˆ ,
(19)
cr
0
ip t
To find the natural frequency of microbeam using strain gradient theory, we set V 0 in Eq. (4). By applying Galerkin’s projection method, the governing differential equation of beam midpoint, u (tˆ) , can be expressed as
te
d
where (xˆ ) is the first mode shape of the microbeam based on strain gradient theory. To find the resonant frequencies experimentally, a.c. actuation voltage was kept small so that the hard spring effect can be ignored (Tilmans and Legtenberg, 1994). Thus to compare the simulated results with experimental ones, the nonlinear term in Eq. (19) is neglected ( K 3 0 ) and the dimensional natural frequency of a microbeam is found as
1 2T
(21)
K1 M
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f sg
where T is defined in Eq. (5).
Table 4 compares the frequencies of microbeams using the proposed model (Eq. (19)) with experimental results (Tilmans and Legtenberg, 1994). As can be seen in this table, the predicted results using the proposed model for poly silicon microbeam show the excellent agreement with experimental results. Table 4. Comparison of microbeams frequencies (KHz) based on strain gradient theory with experimental ones Case 1 2 3 4
Exp.(Tilmans and Legtenberg, 1994) 324.48 163.58 104.03 75.02
SG 324.58 163.65 104.08 75.06
4-3Effect of size on the static response of micro- and nano-beams To show the effect of size on the static response of nanobeams, a clamped-clamped beam (case 1) is considered in this section and the effect of size is investigated on the deflection and pull-in voltage of the beam through the variation of the beam thickness by keeping the cross-sectional shape (b / h ) , aspect ratio (l / h ) and normalized voltage (V / h ) of the beam the same. The beam midpoint deflection is plotted versus the beam height in Fig.4 for three axial stresses based on strain gradient and classical theories which are presented in this figure as SG and CL.As can be
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seen, the effect of size increases when decreasing the beam height for all three axial stresses and diminishes when the beam height is far greater than the length scale parameter. This figure also indicates that applying axial tension stress leads to lower deflection in contrast with compressive stress. However, based on strain gradient theory, the axial stress has no effect when the thickness of the beam is approximately equal to the material length scale parameter.
ip t
0.25
cr
0.2
us
0.15 x a m n
CL =0
0.1
0
5
10
M
0.05
0
CL = 0
an
w
15 h/Cl
20
CL =- 0 SG =- 0 SG =0 SG = 0 25
30
te
d
Fig.4. Normalized Beam midpoint deflection as a function of beam height for different axial stresses
2.6
Non-linear CL Non-linear SG Linear CL Linear SG
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2.5 2.4 2.3 2.2
)l C / h( /I P V
2.1 2
1.9 1.8 1.7 1.6
5
10
15
20
25
30
h/Cl
Fig.5. Pull-in voltage as a function of beam height for different axial stress
Pull-in voltage of a beam based on strain gradient and classical theories is plotted in Fig.5 with and without mid-plane stretching. Fig.5 indicates that when the nanobeam thickness is in the order of
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length scale parameters, pull-in voltage of NEMS predicted by strain gradient is about 1.6 times that from classical theory and the two theories converge to each other when the height of the beam becomes far greater than the length scale parameter. The figure also reveals that considering midplane stretching leads to higher pull-in voltage for both theories; however, it has no effect for beam thicknesses in the order of the length scale parameter as for the axial stress depicted in Fig. 4.
Conclusion
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References
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In this paper, based on the analogue equation method and Gauss-Seidel iteration technique, an efficient method is proposed to find the static response of micro- and nano-beams under electrical actuation. The Euler-Bernoulli beam assumption based on strain gradient theory has been used for micro and nano-beam modeling. The accuracy of the proposed method is verified by comparing the pull-in voltages with those available in the literature. Poly silicon Young’s modulus and length scale parameters are determined as E * 147.7GPa and l 0 l1 l 2 Cl 90nm by fitting the predicted pull-in voltages of different sets of MEMS reported experimental data. The Galerkin decomposition method is used to convert the partial differential equation to an ordinary one and a closed form expression is presented to find the natural frequency of micro and nano-beams based on strain gradient theory. Analytically obtained natural frequencies based on the proposed model for poly silicon showed the best agreement with the reported empirical data in the literature The effect of the size, beam thickness, axial stress and mid-plane stretching have been investigated and the following results have been deduced: Reducing the beam thickness decreases the normalized deflection of nano- and micro-beams resulting in higher pull-in voltages in comparison to classical continuum theory results. For both classical and non-classical theories, tension axial load leads to lower deflection and higher pull-in voltage in contrast with compression force, however, non-classical theory indicates that nano-beam sensitivity to different axial loads reduces when decreasing the beam thickness. Based on classical theory, mid-plane stretching has the same effect for different beam thicknesses; however, based on non-classical theory, its effect diminishes when the beam thickness decreases.
Abdel-Rahman, E.M., Younis, M.I., Nayfeh, A.H., 2002. Characterization of the mechanical behavior of an electrically actuated microbeam. Journal of Micromechanics and Microengineering 12, 759. Aifantis, E.C., 2009. Exploring the applicability of gradient elasticity to certain micro/nano reliability problems. Microsystem Technologies 15, 109-115. Akgöz, B., Civalek, Ö., 2011. Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. International Journal of Engineering Science 49, 1268-1280. Akgöz, B., Civalek, Ö., 2012a. Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory. Archive of Applied Mechanics 82, 423-443. Akgöz, B., Civalek, Ö., 2012b. Comment on “Static and dynamic analysis of micro beams based on strain gradient elasticity theory” by S. Kong, S. Zhou, Z. Nie, and K. Wang,(International Journal of Engineering Science, 47, 487–498, 2009). International Journal of Engineering Science 50, 279-281. Asghari, M., Kahrobaiyan, M., Rahaeifard, M., Ahmadian, M., 2011. Investigation of the size effects in Timoshenko beams based on the couple stress theory. Archive of Applied Mechanics 81, 863-874.
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Baghani, M., 2012. Analytical study on size-dependent static pull-in voltage of microcantilevers using the modified couple stress theory. International Journal of Engineering Science 54, 99-105. Ballestra, A., Brusa, E., Munteanu, M.G., Somà, A., 2008. Experimental characterization of electrostatically actuated in-plane bending of microcantilevers. Microsystem Technologies 14, 909-918. Chan, Y.-S., Paulino, G.H., Fannjiang, A.C., 2008. Gradient elasticity theory for mode III fracture in functionally graded materials—part II: crack parallel to the material gradation. Journal of Applied Mechanics 75, 061015. Chasiotis, I., Knauss, W.G., 2003. The mechanical strength of polysilicon films: Part 2. Size effects associated with elliptical and circular perforations. Journal of the Mechanics and Physics of Solids 51, 1551-1572. Fannjiang, A.C., Paulino, G.H., Chan, Y.-s., 2002. Strain gradient elasticity for antiplane shear cracks: a hypersingular integrodifferential equation approach. SIAM Journal on Applied Mathematics 62, 1066-1091. Fleck, N., Muller, G., Ashby, M., Hutchinson, J., 1994. Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia 42, 475-487. Fu, Y., Zhang, J., Jiang, Y., 2010. Influences of the surface energies on the nonlinear static and dynamic behaviors of nanobeams. Physica E: Low-dimensional Systems and Nanostructures 42, 2268-2273. Gupta, R.K., 1998. Electrostatic pull-in test structure design for in-situ mechanical property measurements of microelectromechanical systems (MEMS). Citeseer. Hu, Y.-C., 2006. Closed form solutions for the pull-in voltage of micro curled beams subjected to electrostatic loads. Journal of Micromechanics and Microengineering 16, 648. Kahrobaiyan, M., Asghari, M., Rahaeifard, M., Ahmadian, M., 2010. Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory. International Journal of Engineering Science 48, 19851994. Kong, S., Zhou, S., Nie, Z., Wang, K., 2009. Static and dynamic analysis of micro beams based on strain gradient elasticity theory. International Journal of Engineering Science 47, 487-498. Kuang, J.-H., Chen, C.-J., 2004. Dynamic characteristics of shaped micro-actuators solved using the differential quadrature method. Journal of Micromechanics and Microengineering 14, 647. Lam, D., Yang, F., Chong, A., Wang, J., Tong, P., 2003. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids 51, 1477-1508. Lam, D.C., Chong, A.C., 1999. Indentation model and strain gradient plasticity law for glassy polymers. JOURNAL OF MATERIALS RESEARCH-PITTSBURGH- 14, 3784-3788. Lim, C., 2010. Is a nanorod (or nanotube) with a lower Young’s modulus stiffer? Is not Young’s modulus a stiffness indicator? SCIENCE CHINA Physics, Mechanics & Astronomy 53, 712-724. Lin, W.-H., Zhao, Y.-P., 2005. Casimir effect on the pull-in parameters of nanometer switches. Microsystem Technologies 11, 80-85. Lubineau, G., Azdoud, Y., Han, F., Rey, C., Askari, A., 2012. A morphing strategy to couple non-local to local continuum mechanics. Journal of the Mechanics and Physics of Solids 60, 1088-1102. Maani Miandoab, E., Nejat Pishkenari, H., Yousefi-Koma, A., 2014. Dynamic Analysis of Electostatically Actuated Nano-Beam Based on Strain Gradient Theory. International Journal of Structural Stability and Dynamics, In Press.
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McFarland, A.W., Colton, J.S., 2005. Role of material microstructure in plate stiffness with relevance to microcantilever sensors. Journal of Micromechanics and Microengineering 15, 1060. Moghimi Zand, M., Ahmadian, M.T., 2009. Application of homotopy analysis method in studying dynamic pull-in instability of microsystems. Mechanics Research Communications 36, 851-858. Nayfeh, A.H., Younis, M.I., Abdel-Rahman, E.M., 2007. Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dynamics 48, 153-163. Nix, W.D., 1989. Mechanical properties of thin films. Metallurgical and Materials Transactions A 20, 2217-2245. Paulino, G., Fannjiang, A., Chan, Y.-S., 2003. Gradient elasticity theory for mode III fracture in functionally graded materials—part I: crack perpendicular to the material gradation. Journal of Applied Mechanics 70, 531-542. Peddieson, J., Buchanan, G.R., McNitt, R.P., 2003. Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science 41, 305-312. Poole, W., Ashby, M., Fleck, N., 1996. Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Materialia 34, 559-564. Rahaeifard, M., Kahrobaiyan, M., Ahmadian, M., Firoozbakhsh, K., 2011. Size-dependent pull-in phenomena in nonlinear microbridges. International Journal of Mechanical Sciences. Rajabi, F., Ramezani, S., 2012. A nonlinear microbeam model based on strain gradient elasticity theory with surface energy. Archive of Applied Mechanics 82, 363-376. Reddy, J., 2010. Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. International Journal of Engineering Science 48, 1507-1518. Rokni, H., Seethaler, R.J., Milani, A.S., Hosseini-Hashemi, S., Li, X.-F., 2012. Analytical closed-form solutions for size-dependent static pull-in behavior in electrostatic microactuators via Fredholm integral method. Sensors and Actuators A: Physical. Tilmans, H.A., Legtenberg, R., 1994. Electrostatically driven vacuum-encapsulated polysilicon resonators: Part II. Theory and performance. Sensors and Actuators A: Physical 45, 67-84. Wang, B., Zhou, S., Zhao, J., Chen, X., 2011. Size-dependent pull-in instability of electrostatically actuated microbeam-based MEMS. Journal of Micromechanics and Microengineering 21, 027001. Xu, L., Jia, X., 2007. Electromechanical coupled nonlinear dynamics for microbeams. Archive of Applied Mechanics 77, 485-502. Younis, M.I., Abdel-Rahman, E.M., Nayfeh, A., 2003. A reduced-order model for electrically actuated microbeam-based MEMS. Microelectromechanical Systems, Journal of 12, 672-680. Zhang, Y., Zhao, Y.-p., 2006. Numerical and analytical study on the pull-in instability of micro-structure under electrostatic loading. Sensors and Actuators A: Physical 127, 366-380.
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S0093-6413(14)00122-0 http://dx.doi.org/doi:10.1016/j.mechrescom.2014.09.001 MRC 2896
To appear in: Received date: Revised date: Accepted date:
28-7-2013 21-8-2014 4-9-2014
Please cite this article as: Miandoab, E.M., Yousefi-Koma, A., Pishkenari, H.N.,Poly Silicon Nanobeam Model Based on Strain Gradient Theory, Mechanics Research Communications (2014), http://dx.doi.org/10.1016/j.mechrescom.2014.09.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights: An efficient method to find the static response of nanobeams
Poly silicon Young’s modulus and length scale parameters estimation
Effect of axial stress and mid-plane stretching on strain gradient microbeam
Quick method for natural frequency estimation based on SG theory
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Poly Silicon Nanobeam Model Based on Strain Gradient Theory a
Ehsan Maani Miandoaba, Aghil Yousefi-Komaa,1 and Hossein Nejat Pishkenarib
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Center of Advanced Systems and Technologies (CAST), School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran b Center of Excellence in Design, Robotics and Automation (CEDRA), Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran
Abstract
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Conventional continuum theory does not account for contributions from length scale effects. Failure to include size-dependent contributions can lead to underestimates of deflection and stresses of micro and nanobeams. This research aims to use strain gradient elasticity theory to model size-dependent behavior of small beams. In this regard, Young’s modulus and length scale parameters of poly silicon are estimated by fitting the predicted static pull-in voltages to the reported experimental results in the literature. The results demonstrate that decreasing the beam thickness results in higher pull-in voltage, lower deflection and lower sensitivity to axial stress and mid-plane stretching in comparison with classical model results. Keywords: Micro and Nanobeams, Strain gradient theory, Pull-in voltage.
1. Introduction
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Applications of micro- and nano-electro-mechanical systems (MEMS\NEMS) with electrical actuation have recently increased due to the many advantages such as fast response, low energy consumption and low cost. A typical electrostatically actuated MEMS\NEMS is comprised of two parallel conducting electrodes, one is fixed and the other is movable due to applied electrostatic force. Increasing the input voltage beyond a critical value leads to instability of the movable electrode and pull-in phenomena. In design and test of MEMS devices, pull-in phenomena must be considered due to its importance in MEMS\NEMS structural safety (Tilmans and Legtenberg, 1994). Extensive research has been carried out on static and dynamic pull-in analysis of MEMS\NEMS under electrostatic actuation (Abdel-Rahman et al., 2002; Hu, 2006; Lin and Zhao, 2005; Nayfeh et al., 2007; Younis et al., 2003; Zhang and Zhao, 2006). In all of these studies, the movable electrode was modeled as an elastic beam based on classical continuum theory. The thickness of the beams used in MEMS and NEMS are in the order of microns and submicrons. Numerous experiments showed when decreasing the size of the device, size-dependent behavior turns out to be important. This cannot be explained using conventional mechanics theories due to lack of any material length scale parameter (Aifantis, 2009; Chasiotis and Knauss, 2003; Fleck et al., 1994; Lam et al., 2003; Lam and Chong, 1999; McFarland and Colton, 2005; Nix, 1989; Poole et al., 1996). More recently, non-classical continuum theories have been developed to predict the size-dependent behavior of materials in small size, such as modified couple stress theory, nonlocal and strain gradient elasticity theories and modified continuum model incorporating surface elasticity. Based on nonlocal continuum theory, the stress state at a point is a function of the strain states of all points in the body, while classical continuum mechanics assumes the stress state at a given point is dependent only on the strain state at that same point. Peddieson et al. used nonlocal theory in nanotechnology for the first time and developed the static response of micro and nanobeams based on this non-classical theory (Peddieson et al., 2003). Lim developed a new model to capture the static and dynamic response of micro and nanobeams based on nonlocal theory and it was shown that in contrast with the previous model, his model results in higher stiffness for microstructures as was observed in empirical tests (Lim, 2010). Lubineau et al. used morphing strategy to couple non-local to local
1
Corresponding author, Email: [email protected] Tel: +98 2161114228Fax:+9821 61114228
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continuum mechanics which can be used in modeling of large scale materials with non-local effects (Lubineau et al., 2012). Strain gradient theory is another non-classical approach that can describe the size effect resulting from underlying microstructures (Fannjiang et al., 2002; Fleck et al., 1994; Lam et al., 2003). In this theory, new additional equilibrium equations are utilized to model the behavior of higher-order stresses with three independent higher-order materials length scale parameters for isotropic linear elastic materials. Kong et al. (Kong et al., 2009) studied the static and dynamic behavior of EulerBernoulli beams using strain gradient theory. They studied the size effects on the beam bending response and its natural frequencies for two kinds of boundary conditions and it was found that strain gradient theory results in higher stiffness for a beam in comparison with classical continuum theory. Furthermore, many studies have been performed on the analysis of static and dynamic behavior of beams considering size effect. Comparison of results obtained from non-classical techniques with classical continuum theory demonstrates the significance of the size effect; (Akgöz and Civalek, 2011, 2012a; Asghari et al., 2011; Kahrobaiyan et al., 2010; Rajabi and Ramezani, 2012; Reddy, 2010; Xu and Jia, 2007). Some studies have been done on analysis of functionally graded materials (FGM) based on strain gradient theory (Chan et al., 2008; Fannjiang et al., 2002). For instance, Paulino et al. presented a theoretical framework to model antiplane shear cracks in FGM using strain gradient elasticity (Paulino et al., 2003). Despite considerable progress in the field of beam modeling, only a few works have been devoted to the investigation of size-dependent behavior for electrostatically actuated devices. Here, some of these works are reviewed. Fu and Zhang investigated the dynamic response and pull-in of electrically actuated nanobeam based on surface energy theory using the analogue equation method (Fu et al., 2010). To determine the deflection and static pull-in voltage of silicon micro-cantilevers, Rahaeifard et al. used the modified couple stress theory. Based on their results, the length scale parameter of silicon was determined as l 592nm (Rahaeifard et al., 2011). Baghani presented an analytical solution for size-dependent response of cantilever microbeams using the modified couple stress theory (Baghani, 2012). Wang et al. investigated the dynamic response of poly silicon microbeams under electrostatic actuation using strain gradient theory and determined the poly silicon length scale parameter as l 110nm by fitting the predicted normalized resonant frequency to experimental data (Wang et al., 2011). They used the generalized differential quadrature (GDQ) method to solve the sixth-order partial differential equation. Rokni et al. proposed an analytical closed-form solution for static pull-in voltage of microbeams based on modified couple stress theory and the value of length scale parameter of poly silicon was estimated to be in the order of 100 nm (Rokni et al., 2012). In this paper, an efficient method, the analogue equation method, is proposed to find the static response of clamped-clamped micro and nanobeams based on strain gradient theory under electrostatic actuation. In mathematical modeling of the system, fringing field effect, residual axial stress and mid-plane stretching are considered. The validation of the proposed method has been checked by comparison of the static pull-in voltages with the results obtained using the Differential Quadrature Method (DQM) (Kuang and Chen, 2004), Homotopy Analysis Method (HAM) (Moghimi Zand and Ahmadian, 2009) and the Fredholm Integral Equation method (FIE) (Rokni et al., 2012). An important drawback of the previous theoretical research using non-classical continuum theories is that they don’t account for effects of the size on the Young modulus of the beam and merely adjust the length scale parameters for small sizes to fit data with experimental results; while in the present study, to model poly silicon nanobeams based on strain gradient theory, the Young’s modulus and length scale parameters of poly silicon have been determined by fitting the static pull-in voltages of different MEMS to experimental results and it has been demonstrated that the proposed model based on strain gradient theory leads to an appropriate agreement with experimental observation. Moreover, to investigate the size effect on the axial stress and mid-plane stretching, we have varied the beam thickness and compared our results with classical continuum theory. 2. Mathematical Modeling Figure (1) shows a schematic of a traditional MEMS system under electrostatic actuation.
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L
x
Movable Beam h
g
z Fig.1. Schematic of an electrostatically actuated microbeam
ip t
Fixed Beam
us
cr
As seen in this figure, this system is composed of one movable and one fixed conductive electrode. The dynamic behavior of microbeam under electrostatic force can be derived as (Akgöz and Civalek, 2012b; Kong et al., 2009):
6w 4w 2w EA w 2 2w bV 2 ( g w ) S A ( N ( ) dx ) (1 ) 6 4 2 2 2 b x x t x 2l 0 x 2( g w )
an
l
K where
S EI A (2l 02
M
4 K I (2l 02 l12 ) 5
(1)
(2)
8 2 2 l1 l 2 ) 15
te
d
In derivation of this relation, the Euler-Bernoulli beam assumption based on strain gradient theory was used. The first term on the right hand of Eq. (1) represents the axial force and mid-plane stretching effects which vanish for a micro-cantilever. The parameters E , A , , I ,V , , N , g , and w are respectively Young’s modulus, beam cross section area, material density, cross section moment of inertia, applied voltage, dielectric constant, axial force, gap distance between two electrodes, shear modulus and beam deflection, which is a function of the beam axial position x and time t , meaning w( x, t ) . is the parameter that presents the fringing field effect and the independent
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length scale parameters l0 , l1 and l2 are respectively dilatation gradients, deviatoric stretch gradients and rotation gradients. In order to facilitate the study of the qualitative behavior of the system, the following relations are considered:
xˆ
(3)
x w t ; wˆ ; tˆ l g T
By substituting Eq. (3) in Eq. (1), Eq. (1) can be rewritten as
0
1 2 2 6wˆ 4wˆ 2wˆ ˆ ( wˆ )2 dxˆ ) wˆ 2V (1 g 1 wˆ ) ( N 1 xˆ 6 xˆ 4 tˆ 2 xˆ xˆ 2 (1 wˆ )2 b 0
(4)
Parameters appearing in Eqs. (3) and (4) are
0
2 K EAg 2 bl 4 ˆ l N, T , , , N 1 2 Sl 2 S 2S 2Sg 3
4
bhl 4 S
(5)
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It is worthy to note that the static response of the microbeam can be found by neglecting the time dependent term in Eq. (4). Boundary conditions corresponding to a clamped-clamped beam based on strain gradient theory are:
wˆ (0) wˆ (1) 0 dwˆ (0) dwˆ (1) 0 dxˆ dxˆ
(6a)
ip t
(6b)
d 3wˆ (0) d 3wˆ (1) 0 dxˆ 3 dxˆ 3
(6c)
(7a) (7b) (7c)
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an
d 4wˆ (1) d 2wˆ (1) 0 4 2 0 d xˆ d xˆ d 5wˆ (1) d 3wˆ (1) 0 5 3 0 d xˆ d xˆ d 3wˆ (1) 0 dxˆ 3
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The boundary condition for a micro-cantilever beam with a free end at x l is as follows.
d
In order to analyze the static behavior of the microbeam, it is enough to solve the nonlinear boundary value problem presented here. In this regard, in the next section, the analogue equation method will be used to solve this problem.
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3. Analogue Equation method (AEM) AEM is a powerful method for solving nonlinear partial differential equations and has been previously utilized for solving fourth-order nonlinear partial differential equations of microbeam dynamic response under electrostatic actuation (Fu et al., 2010; Maani Miandoab et al., 2014). This paper aims to use this method to find the solution of the introduced nonlinear boundary value problem. Consider the following equation: d 6w ( x ) a (x ) dx 6
(8)
where a ( x ) is the nonlinear friction load. By dividing the beam into N equal elements with N nodal points, the following solution can be presented for Eq. (8).
w ( x ) H( x )b G( x )a
(9)
where
H(x ) [x 5 ....., x ,1] b [b1 , b 2 ...........b 6 ]T l 3l (2N 1)l T a [a ( ), a ( ),........a ( )] 2N 2N 2N
5
(10)
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1l N
G( x ) [ G (x )d ,.............
l
G (x )d ]
( N 1) l N
0
In each element, the friction load a ( x ) is substituted with its value at the corresponding nodal point in the middle of the element and b i ( i 1,2...,6 ) are constants. G (x ) is the Green’s function which is
ip t
d 6G (x , ) (x , ) . where ( x ) is the Dirac delta function. Integrating both sides dx 6 of this equation with respect to x yields 1 x (x )4 240
us
G (x )
cr
the solution of
(11)
By inserting Eq. (11) into the corresponding boundary conditions, Eq. (6) or Eq. (7), b can be found as a function of a as b K aa , where K a is 6 N matrix. Thus Eq. (9) can be rewritten as
an
w (x ) H(x )b G(x )a (H(x )K a G(x ))a M(x )a
(12)
M
M ( x ) is a 1 N vector. Substituting Eq. (12) into Eq. (4) leads to the following equation at the nodal point k. 2 2 d 4M k g ˆ ( dM k a)2 dxˆ ) d M k a 2V a ( (1 (1 M k a)) 0 a( k ) N 1 4 2 2 b dxˆ dxˆ dxˆ (1 M k a) 0
te
where Mk M(x k ) and
(13)
d
1
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dn Mk d n H(xˆ k ) d n G(xˆ k ) ( Ka )a dxˆ n dxˆ n dxˆ n
(14)
To find the solution for a , the Gauss-Seidel method is utilized. On the basis of initial guess, the following equation can be used to determine the new vector, a new .
( 0I
d4R )a new F(aold ) dxˆ 4
(15)
where F(aold ) is the N 1 vector obtained by calculating of the right-hand of Eq. (13) for all N nodal points, I is the N N unit matrix and the matrix R is defined as follows.
M1 M 2 R . . Mn
6
(16)
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is N 0.0009 Newton .
Length
scale
parameters
are
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4. Numerical Simulation 4-1Solution validation In order to validate the proposed numerical method, pull-in voltages based on strain gradient theory will be compared with the results of the classical model for a range of parameters in which the size effects are negligible. The classical model solutions existing in the literature are obtained by the Differential Quadrature Method (DQM) (Kuang and Chen, 2004), Homotopy Analysis Method (HAM) (Moghimi Zand and Ahmadian, 2009) and the Fredholm Integral Equation (FIE) (Rokni et al., 2012). Clamped-clamped microbeam properties used for validation are given as follows: Young’s modulus and Poisson’s ratio are E 166GPa and 0.3 , beam length, width, height and initial gap are L 210μm , b 100μm , h 1.5μm and g 1.18μm respectively. Residual axial load as l 0 l1 l 2 Cl 10nm .
set
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Because h / Cl 20 , the size effect can be ignored and predicted results from non-classical continuum theories converge to classical ones. This makes it possible to compare the proposed method with classical results without loss of generality. Maximum displacement of a microbeam is plotted versus the applied voltage in Fig.2. As seen in this figure, by increasing the applied voltage, the slope of the curve increases and approaches infinity nearV 28.12V indicating the pull-in phenomenon. Stability limit is w max .41which is in good agreement with the results reported in the literature (Abdel-Rahman et al., 2002). 0.45
M
0.4 0.35 0.3
0.2
te
w
d
0.25 x a m n
0.15
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0.1
0.05 0
0
5
10
15 V(volt)
20
25
30
Fig.2. Beam midpoint normalized deflection for 40 iterations
Table 1 compares the pull-in voltages obtained using AEM (this study) with the results conducted in previous research. There is good agreement between the pull-in voltage obtained by the proposed method and DQM, FI and HAM. Table 1.Comparison of pull in voltage based on different methods Method
DQM(Kuang and Chen, 2004)
FI(Rokni et al., 2012)
V PI (V)
28.1
28.24
HAM(Moghimi Zand and Ahmadian, 2009)
28.86
7
Present study
28.12
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4-2Young’s modulus and length scale parameter estimation of poly silicon In this paper, experimental data reported in the literature are used to identify the Young’s modulus and length scale parameters of poly silicon based on strain gradient theory. The geometric parameters of 15 cases used for identification are given in Table 2. Table 2. Geometric parameters of used microbeams Boundary Condition
L (μm)
b (μm)
h (μm)
g (μm)
Ref.
1-4
Clamped-Clamped
210-510
100
1.5
1.18
(Tilmans and Legtenberg, 1994)
5-14
Clamped-Free
100-500
40
2.1
2.4
15
Clamped-Free
100
15
1.8
5
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Case
(Gupta, 1998)
cr
(Ballestra et al., 2008)
us
Cases 5-14 have a constant radius of curvature Rc 40mm (Gupta, 1998). Using the finite-difference script in (Gupta, 1998), it has been shown that the pull-in voltage of a curved cantilever can be calculated as a function of pull-in voltage of the ideal uniformly flat case using the following equation:
l2 l2 2 ) 0 0006347( ) ) gR c gR c
an
V Curved V Flat (1 0.5096(
(17)
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where V Curved and V Flat are the pull-in voltages of a curved and flat beam, respectively. By fitting the empirical and simulated results based on the classical Euler-Bernoulli beam model, effective Young’s modulus of poly silicon was estimated as E 166GPa in (Tilmans and Legtenberg, 1994)and E 153GPa in (Gupta, 1998) based on the empirical results of 1.5μm (cases 1-4) and 2.1μm (cases 5-14) microbeams respectively. Because of size effect, reducing the beam thickness leads to higher stiffness which cannot be modeled using classical continuum theory. Static pull-in voltages of cases 514 based on classical continuum theory ( E 166GPa and E 153GPa ) are compared with experimental ones (Gupta, 1998) in Fig. 3. As can been seen in Fig. 3, estimated value of Young’s modulus for a 1.5μm height microbeam ( E 166GPa ) leads to higher pull-in voltage for 2.1μm height microbeam because of size effect. For modeling the beam based on the strain gradient theory, Young’s modulus and length scale parameters of poly silicon should be calibrated using experimental results. In the present study, by varying the Young’s modulus and length scale parameter simultaneously and comparing the obtained pull-in voltages with the reported experimental data, it was found that the following parameters lead to the best agreement with the experimental results: E 156GPa and l 0 l1 l 2 Cl 90nm .
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80 Exp. E=166GPa E=153GPa
70
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60
cr
50 40
us
30
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20
0 100
150
200
250
M
10
300
350
400
450
500
d
Fig.3. Simulated pull-in voltage of cases 5-14 based on classical continuum theory in comparison with experimental results
te
Comparison of the average relative errors of pull-in voltages of different cases based on strain gradient (SG) theory with the classical (CL) ones is demonstrated in Table 3 .
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Table 3. Average relative errors of pull-in voltages of different cases based on classical and strain gradient continuum theories
h (μm)
1.5 (cases 1-4) 1.8 (cases 15) 2.1 (case 5-14)
CL( E 153GPa )
Error (%) CL( E 166GPa )
2.27 4.28 1.48
1.94 0.7 4.34
SG( E 156GPa, Cl =90nm ) 1.89 1.3 2.21
As can be seen in Table 3, predicted pull-in voltages based on strain gradient theory with calibrated parameters show the best agreement with the experimental ones. It should be noted that all of the cases investigated in this study are narrow beams ( b 5h ), thus estimated Young’s modulus for these cases is plate modulus ( E E / (1 ) ). By setting 0.23 (Tilmans and Legtenberg, *
2
1994), Young’s modulus of poly silicon can be found as E * 147.7GPa . 4-2-1 Natural frequencies of micro and nanobeams based on strain gradient theory In this section, the natural frequency of microbeams based on strain gradient theory using calibrated material parameters ( E * 147.7GPa and Cl 90nm ) is compared with experimental results. Using the least square fitting to experimental results, the following equation was presented for natural resonant frequency of cases 1-4 in (Tilmans and Legtenberg, 1994).
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(18)
( f 1l ) 2 7.5031 10 7 ( h l ) 2 814.94
where f 1 is the natural frequency of microbeams.
1
(xˆ )T (wˆ )dx 0
Mu K 1u K 3u 3 0
The parameters in Eq. (19) are
0
K 1 ( 0 ( xˆ ) 0
d (xˆ ) d 4 (xˆ ) d 2 (xˆ ) ˆ ˆ ( x ) N ( x ) )dxˆ dxˆ 6 dxˆ 4 dxˆ 2 6
1
K 3 1 ( xˆ )
d 2 ( xˆ ) d ( xˆ ) 2 ( ( ) dxˆ )dxˆ 2 dxˆ dxˆ 0 1
M
0
(20)
an
1
us
1
M ( xˆ ) 2 dxˆ ,
(19)
cr
0
ip t
To find the natural frequency of microbeam using strain gradient theory, we set V 0 in Eq. (4). By applying Galerkin’s projection method, the governing differential equation of beam midpoint, u (tˆ) , can be expressed as
te
d
where (xˆ ) is the first mode shape of the microbeam based on strain gradient theory. To find the resonant frequencies experimentally, a.c. actuation voltage was kept small so that the hard spring effect can be ignored (Tilmans and Legtenberg, 1994). Thus to compare the simulated results with experimental ones, the nonlinear term in Eq. (19) is neglected ( K 3 0 ) and the dimensional natural frequency of a microbeam is found as
1 2T
(21)
K1 M
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f sg
where T is defined in Eq. (5).
Table 4 compares the frequencies of microbeams using the proposed model (Eq. (19)) with experimental results (Tilmans and Legtenberg, 1994). As can be seen in this table, the predicted results using the proposed model for poly silicon microbeam show the excellent agreement with experimental results. Table 4. Comparison of microbeams frequencies (KHz) based on strain gradient theory with experimental ones Case 1 2 3 4
Exp.(Tilmans and Legtenberg, 1994) 324.48 163.58 104.03 75.02
SG 324.58 163.65 104.08 75.06
4-3Effect of size on the static response of micro- and nano-beams To show the effect of size on the static response of nanobeams, a clamped-clamped beam (case 1) is considered in this section and the effect of size is investigated on the deflection and pull-in voltage of the beam through the variation of the beam thickness by keeping the cross-sectional shape (b / h ) , aspect ratio (l / h ) and normalized voltage (V / h ) of the beam the same. The beam midpoint deflection is plotted versus the beam height in Fig.4 for three axial stresses based on strain gradient and classical theories which are presented in this figure as SG and CL.As can be
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seen, the effect of size increases when decreasing the beam height for all three axial stresses and diminishes when the beam height is far greater than the length scale parameter. This figure also indicates that applying axial tension stress leads to lower deflection in contrast with compressive stress. However, based on strain gradient theory, the axial stress has no effect when the thickness of the beam is approximately equal to the material length scale parameter.
ip t
0.25
cr
0.2
us
0.15 x a m n
CL =0
0.1
0
5
10
M
0.05
0
CL = 0
an
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15 h/Cl
20
CL =- 0 SG =- 0 SG =0 SG = 0 25
30
te
d
Fig.4. Normalized Beam midpoint deflection as a function of beam height for different axial stresses
2.6
Non-linear CL Non-linear SG Linear CL Linear SG
Ac ce p
2.5 2.4 2.3 2.2
)l C / h( /I P V
2.1 2
1.9 1.8 1.7 1.6
5
10
15
20
25
30
h/Cl
Fig.5. Pull-in voltage as a function of beam height for different axial stress
Pull-in voltage of a beam based on strain gradient and classical theories is plotted in Fig.5 with and without mid-plane stretching. Fig.5 indicates that when the nanobeam thickness is in the order of
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length scale parameters, pull-in voltage of NEMS predicted by strain gradient is about 1.6 times that from classical theory and the two theories converge to each other when the height of the beam becomes far greater than the length scale parameter. The figure also reveals that considering midplane stretching leads to higher pull-in voltage for both theories; however, it has no effect for beam thicknesses in the order of the length scale parameter as for the axial stress depicted in Fig. 4.
Conclusion
Ac ce p
References
te
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ip t
In this paper, based on the analogue equation method and Gauss-Seidel iteration technique, an efficient method is proposed to find the static response of micro- and nano-beams under electrical actuation. The Euler-Bernoulli beam assumption based on strain gradient theory has been used for micro and nano-beam modeling. The accuracy of the proposed method is verified by comparing the pull-in voltages with those available in the literature. Poly silicon Young’s modulus and length scale parameters are determined as E * 147.7GPa and l 0 l1 l 2 Cl 90nm by fitting the predicted pull-in voltages of different sets of MEMS reported experimental data. The Galerkin decomposition method is used to convert the partial differential equation to an ordinary one and a closed form expression is presented to find the natural frequency of micro and nano-beams based on strain gradient theory. Analytically obtained natural frequencies based on the proposed model for poly silicon showed the best agreement with the reported empirical data in the literature The effect of the size, beam thickness, axial stress and mid-plane stretching have been investigated and the following results have been deduced: Reducing the beam thickness decreases the normalized deflection of nano- and micro-beams resulting in higher pull-in voltages in comparison to classical continuum theory results. For both classical and non-classical theories, tension axial load leads to lower deflection and higher pull-in voltage in contrast with compression force, however, non-classical theory indicates that nano-beam sensitivity to different axial loads reduces when decreasing the beam thickness. Based on classical theory, mid-plane stretching has the same effect for different beam thicknesses; however, based on non-classical theory, its effect diminishes when the beam thickness decreases.
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