Nonlinear behavior of a nano-scale beam considering length scale-parameter

Applied Mathematical Modelling xxx (2013) xxx–xxx

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Nonlinear behavior of a nano-scale beam considering length scale-parameter Hamed Mobki a, Morteza H. Sadeghi a, Ghader Rezazadeh b,⇑ Mohammad Fathalilou a, Ali-asghar keyvani-janbahan a a b

Department of Mechanical Engineering, University of Tabriz, Iran Department of Mechanical Engineering, Urmia University, Iran

a r t i c l e

i n f o

Article history: Received 26 September 2011 Received in revised form 4 August 2013 Accepted 8 October 2013 Available online xxxx Keywords: NEMS Casimir van der Waals Couple stress theory Length-scale parameter Pull-in voltage

a b s t r a c t Size dependent behavior of materials arises for a structure when the characteristic size such as thickness or diameter is close to its internal length-scale parameter. In these cases, ignoring this behavior in modeling may leads to incorrect results. In this paper, strong effects of size dependence on static and dynamic behavior of electro-statically actuated nano-beams have been studied. The fixed points of the Aluminum nano-beams have been determined and shown that for a given DC voltage, there is a considerable difference between the calculated fixed points using classic beam theory and modified couple stress theory. In addition, it has been also shown that ignoring couple stress theory results in an order of magnitude error in calculated static and dynamic pull-in voltages. Some previous studies have applied the classic beam theory in their models and introduced a considerable hypothetical value of residual stress to justify the discrepancies between experimental and theoretical results. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction With the fast growth of nano scale technology, the possibility of substituting this new technology with micro technology, due to high speed and low energy consuming, has been increased. Nano Electro Mechanical Systems (NEMS) devices such as, nano-tweezers [1–3], super sensitive sensor [4,5], resonators [6], electrostatic switches [7,8], random access memory [9], actuators and sensors [10,11] are widely designed, analyzed, fabricated and used. Electro-statically actuated devices form a broad class of Micro Electro Mechanical Systems (MEMS) and NEMS devices due to their simplicity, as they require few mechanical components and small voltage levels for actuation [12]. In such devices, a conductive flexible beam/plate is suspended over a stationary ground plate and a potential difference is applied between them. As the microstructure/nanostructure is balanced between electrostatic attractive force and mechanical (elastic) restoring force, both electrostatic and elastic restoring force are increased when the electrostatic voltage increases. When the voltage reaches the critical value, pull-in instability occurs. Pull-in is a situation at which the elastic restoring force can no longer balance the electrostatic force. Further increasing the voltage will cause the structure to have dramatic displacement jump causing structural collapse and failure. Pull-in instability is a snap-through like behavior and it is saddle-node bifurcation type of instability [13]. In NEMS systems, by decreasing the geometric dimension, Casimir and van der Waals (vdW) effects play major role especially in terms of the mechanical behavior of these systems. The Casimir force represents the attraction of two uncharged ⇑ Corresponding author. Tel.: +98 914 145 1407. E-mail addresses: [email protected], [email protected] (G. Rezazadeh). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.10.001

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H. Mobki et al. / Applied Mathematical Modelling xxx (2013) xxx–xxx

material bodies due to the modification of the zero-point energy associated with the electromagnetic modes in the gap between them [14]. The well known example is two conductive parallel plates that are pulled towards each other with a force proportional to d4, where d is the distance between two plates. The vdW force is related to the electrostatic interaction between dipoles at the atomic scale [15]. One of the most important differences between these two forces is that the Casimir force between plates depends only on the spatial properties of the objects, whereas the vdW force depends on the material properties as well as the geometric properties. Hendrik Casimir predicted Casimir effects in 1948 for perfectly conductive parallel plates [16] and Kenneth et al. have extended these phenomena to real-world materials [17]. In micro and nano structures, many researches show that the materials have strong size dependence in deformation behavior [18–20]. Size-dependent behavior is an inherent property of materials, which appears for a beam when the characteristic size such as thickness or diameter is close to the internal material length scale parameter [21]. As the conventional theories of mechanics are not capable of describing such size effects, due to the lack of material length scale parameter, couple stress theory [22] and nonlocal theories [23,24] have been employed to deal with the size dependence behavior. The classical couple stress elasticity theory is a higher order continuum theory that contains two higher-order material length scale parameters appear in addition to the two classical Lame constants [25]. To reduce the difficulties of determining length scale parameters of materials by experiments, Yang et al. introduced the modified couple stress theory, in which the couple stress tensor is symmetric and only one internal material length scale parameter is involved, unlike those in the classical couple stress theory [22]. Utilizing the modified couple stress theory, Park and Gao studied the static response of an Euler–Bernoulli beam and interpreted the outcomes of an epoxy polymeric beam bending test [19]. Recently, Shengli et al. derived the governing equation, initial and boundary conditions of an Euler–Bernoulli beam using the modified coupled stress theory and the Hamilton principle [21]. As they reported, the stiffness of beams is size-dependent. Also, the difference between the stiffness obtained by the classical beam theory and those predicted by the modified couple stress theory is significant when the beam characteristic size is comparable to the internal material length scale parameter. In spite of many research about the size dependent behavior of materials in material literature, there is no comprehensive study about this behavior in NEMS world. In NEMS technology, the nano-beams can be made of various materials such as aluminum and gold [26], and silicon [27]. These materials have a significant length scale parameter and the difference between classic beam and couple stress theories, especially in lower beam characteristic size, is expected to be considerable [28]. In this paper, size dependent behavior of electrostatically-actuated nano-beam considering vdW and Casimir forces is investigated. Also, size dependent behavior of electro-statically actuated nano-beams is modeled, using classic beam and modified couple stress theories. Governing static and dynamic equations are solved using Step by Step Linearization Method (SSLM) and reduced order model, respectively. The pull-in voltage, detachment length and natural frequency are compared in both classic and modified couple stress theories for aluminum nano-beams. 2. Model description and theoretical development 2.1. Model description In this part the mathematical model of the nano-beam is developed and the static and dynamic behaviors of the model are studied. The effect of different parameters such as pull-in, detachment length, on behavior of the system is studied. The nano-beam is modeled as a distributed system except for the bifurcation analysis in which it is assumed as a single-DOF mass-spring system. Fig. 1a and b shows schematic views of a NEM switch and its cross-section, respectively. It consists of a nano-beam suspended over a stationary conductor plate, with length L, thickness h, width b and initial gap of G0. Attractive electrostatic force due to an applied voltage V as well as vdW and Casimir forces pulls the nano-beam down towards the substrate. The nano-beam is considered isotropic with Young modulus E, density q cross section area of S and and cross section inertia moment of I. Also, Fig. 2 shows simplified one-DOF equivalent mass-spring model of the nano-beam, for bifurcation analysis. In this figure m and K represent the equivalent mass and the spring constant of the nano-beam, respectively. In the next section the mathematical model of the nano-beam based on the equivalent mass-spring model and its adjustments with the Euler–Bernoulli beam model will be presented. 2.2. Mathematical modeling According to the modified couple stress theory of Yang et al., the strain energy U in a deformed isotropic linear elastic material occupying volume — V is given by [22]:

U¼

1 2

Z V —

ðrij eij þ mij vij Þd— V;

ð1Þ

where r, e, m, and v are the volume of the material, stress tensor, strain tensor, deviatoric part of the couple stress tensor, and symmetric curvature tensor, respectively, which are defined by:

Please cite this article in press as: H. Mobki et al., Nonlinear behavior of a nano-scale beam considering length scale-parameter, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.10.001

H. Mobki et al. / Applied Mathematical Modelling xxx (2013) xxx–xxx

3

Fig. 1. Schematic view of a beam-based NEM switch.

Fig. 2. Mass-spring model of nano-beam.

rij ¼ kekk dij þ 2Geij

ð2Þ

1 2

eij ¼ ðui;j þ uj;i Þ

ð3Þ

mij ¼ 2l Gvij

2

ð4Þ

cij ¼ ðhi;j þ hj;i Þ

1 2

ð5Þ

´ e’s constants; and l, u, and h are material length scale parameter, displacement vector, For Eqs. (2)–(5), k and G are Lam and rotation vector, respectively.

h¼

1 curlðuÞ: 2

ð6Þ

The square of the length scale parameter l introduced in Eq. (4) is proportional to the ratio of the modulus of curvature to the modulus of shear [29], and l is therefore regarded as a material property measuring the effect of couple stress [30]. In order to determine parameter l for a specific material, some typical experiments such as micro-bend test, micro-torsion test and specially micro/nano indentation test can be carried out [31]. Referring to the coordinate system (x, z) shown in Fig. 1, in which x-axis coincides with the centroidal axis of the un-deformed beam and z-axis is the symmetry axis, the displacement components in the Euler–Bernoulli beam can be represented by [19]:

u ¼ zwðx; tÞ;

v ¼ 0;

w ¼ wðx; tÞ;

ð7Þ

where u, v and w are, respectively, the x-, y-, and z-components of the displacement vector and the rotation angle w of the centroidal axis of the beam based on Euler–Bernoulli assumptions can be approximated by: Please cite this article in press as: H. Mobki et al., Nonlinear behavior of a nano-scale beam considering length scale-parameter, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.10.001

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H. Mobki et al. / Applied Mathematical Modelling xxx (2013) xxx–xxx

w¼

@wðx; tÞ @x

ð8Þ

for small deformations considered here. From Eqs. (3), (7), and (8) for the plane stress (narrow beam) and plane strain conditions (wide beam), components of the strain tensor are as follow, respectively:

exx ¼

@u @2w z 2 @x @x

Plane stress :

eyy ¼ v z

Plane strain :

ezz ¼

@2w ; @x2

m 1m

z

ezz ¼ v z

@2w ; @x2

@2w ; @x2

exy ¼ eyz ¼ eyz ¼ ezx ¼ 0;

ð9Þ

eyy ¼ 0; exy ¼ eyz ¼ eyz ¼ ezx ¼ 0;

and from Eqs. (6)–(8) that

hy ¼ 

@wðx; tÞ ; @x

hx ¼ hz ¼ 0:

ð10Þ

Substituting Eq. (10) in Eq. (5) results: 2

wðx;tÞ cxy ¼  12 @ @x 2 cxx ¼ cyy ¼ czz ¼ cyz ¼ czx ¼ 0

ð11Þ

and inserting Eq. (9) into Eq. (2) yields:

@2w ; ryy ¼ rzz ¼ rxy ¼ ryz ¼ rzx ¼ 0 @x2 E @2w E @2w ¼ z ; r ¼ z ; ryy ¼ rzz ¼ rxy ¼ ryz ¼ rzx ¼ 0; yy ð1  m2 Þ @x2 ð1  m2 Þ @x2

Plane stress :rxx ¼ Ez Plane strain :rxx

ð12Þ

´ e’s constants k and G (shear modulus) are defined in terms of where m is the Poisson’s ratio of the beam material, and Lam Poisson’s ratio and modulus of elasticity as following [32]:

Em ð1 þ mÞð1  2mÞ E : G¼ 2ð1 þ mÞ k¼

ð13Þ

As indicated in Eq. (12) rxx for the both cases; plane stress (narrow beam) and plane strain (wide beam) conditions can be written as:

rxx ¼ E z

! @2w ; @x2

ð14Þ

 represents modulus of elasticity (E) and plate modulus (E/(1  m2)), for the plane stress and plane strain conditions, where E respectively. Similarly, the use of Eq. (12) in Eq. (4) gives:

mxy ¼ Gl

2 @2 w ; @x2

ð15Þ

mxx ¼ myy ¼ mzz ¼ myz ¼ mzx ¼ 0: Substituting Eqs. (9), (11), (14), and (15) into Eq. (1) leads to:

U¼

1 2

Z

L

Mx

0

@ 2 wðx; tÞ 1 dx  @x2 2

Z 0

L

Y xy

@ 2 wðx; tÞ dx; @x2

ð16Þ

where the resultant moment Mx and couple moment Yxy are defined, respectively, as:

Mx ¼ Y xy ¼

R

R

rxx zdA; mxy dA:

ð17Þ

Also, the kinetic energy of the nano-beam can be written as following:

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H. Mobki et al. / Applied Mathematical Modelling xxx (2013) xxx–xxx

T¼

1 2

Z

L

qS

0

 2 @wðx; tÞ dx; @t

ð18Þ

where q and S are material density and the cross section area of the nano-beam. The variation of work done by the lateral traction force q(x, t) can be written as:

dW ¼

Z

L

qðx; tÞdwðx; tÞdx:

ð19Þ

0

Employing Hamilton principle:

Z

t

ðT  U þ WÞdt ¼ 0

d

ð20Þ

0

and using Eqs. (16), (18), (19), and (20), the governing equation of dynamic motion of the capacitive nano-beam will be: 4 2  þ GSl2 Þ @ w þ qS @ w ¼ q : ðEI ext 4 @x @t 2

ð21Þ

The lateral external force q(x, t) for the case is the sum of the electrostatic, vdW, and Casimir forces, i.e.:

qext ¼ qelect þ qv dW þ qCasimir

ð22Þ

 as It can clearly be seen in Eq. (21) the bending rigidity of the beam concerns with two parts, a part associated with (EI) the bending rigidity of the classical theory, another part associated with GAl2 relates to the modified couple stress theory. The presence of l enables the incorporation of the material micro-structural features in the new model and renders it possible to explain the size effect. Clearly, when the micro-structural effect is suppressed by letting l = 0, the new model defined by Eq. (21) will reduce to the classical Euler–Bernoulli beam model [19]: 4 2  @ w þ qS @ w ¼ q : EI ext 4 @x @t 2

ð23Þ

Referring to Eq. (22), to the electrostatic force (per unit length) will be [33]:

qelect ¼

e0 bV 2 2ðG0  wÞ2

ð24Þ

;

where e0 = 8.854  1012 C2N1m2 is the permittivity of vacuum within the gap, V is the electrical potential difference within the beam (movable electrode) and the ground electrode. Also, Casimir force between nano-beam and substrate is given by [16]:

qcasimir ¼

b p2 — hc 240ðG0  wÞ4

ð25Þ

;

where — h ¼ 1:055  1034 Js is the Planck’s constant, c = 2.998  108 ms1 is the speed of light. And the vdW force per unit length of the nano-beam is given by [34]:

qv dW ¼

Ab 6pðG0  wÞ3

ð26Þ

;

where A ¼ p2 C q21 is the Hamaker constant which lies in the range of (0.4  4)  1019 J [35]. q1 is the volume density of graphite, and C is a constant character in the interaction between the two atoms [36]. Although Eq. (21) is a complete model for analysis, however, as it is a distributed model with an infinite degree of freedom, it is not easy to use it for bifurcation study. So in order to simplify the analysis of the bifurcation behavior of the nano-beam, as mentioned before, an equivalent lumped mass-spring model is employed for this aim [36]: 2

m

d y dt

2

þ Ky ¼

e0 bLV 2 2

2ðG0  yÞ

þ

AbL 3

6pðG0  yÞ

þ

bp2 — hcL 240ðG0  yÞ4

;

ð27Þ

where m = qSL is the mass of the nano-beam, and K is the spring constant for the beam, defined as the ratio of the applied uniform force ‘q’ to the maximum beam deflection ‘ymax’. Thus, the spring constant depend on the cross-section shape as well as the boundary conditions. Considering mass-spring model, the springs constant for cantilever, and fixed fixed beam are 2 8ðEIþGSl2 Þ Þ and 384ðEIþGSl , respectively [36]. L3 L3 In order to increase the accuracy of the mass-spring model and adjusting it with the distributed model, corrective coefficients terms of a0, b0, and c0 are applied as: Please cite this article in press as: H. Mobki et al., Nonlinear behavior of a nano-scale beam considering length scale-parameter, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.10.001

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H. Mobki et al. / Applied Mathematical Modelling xxx (2013) xxx–xxx

"

2

meq

d y dt

2

þ Ky ¼ a0

#

e0 bLV 2

" þ b0

2ðG0  yÞ2

AbL

6pðG0  yÞ3

#

" þ c0

b p2 — hcL

240ðG0  yÞ4

# ;

ð28Þ

where meq is the equivalent mass of the nano-beam, a0 is the electrostatic corrective coefficient, b0 is the vdW corrective coefficient, and c0 is the Casimir corrective coefficient which will be explained later. For convenience, Eqs. (21) and (28) can be rewritten in a non-dimensional form using following non-dimensional parameters:

^ ¼ w

w ; G0

^¼ y

y ; G0

x ^x ¼ ; L

^t ¼ t ; t

ð29Þ

where variables with hat are the non-dimensional values of the related parameters. t⁄ is a characteristic time and is equal to qffiffiffiffiffiffiffi qffiffiffiffiffiffi meq SL4 for the mass-spring model and is equal to qEI for the distributed Euler–Bernoulli beam model. Using these non-diK mensional parameters Eqs. (21) and (28) can be changed as following:

^ @4w ^ @2w a0 V 2 b0 c0 þ 4 ¼ þ þ ; 2 3 2 ^ ^ @x ^ ^ ^ 4 @t ð1  wÞ ð1  wÞ ð1  wÞ

ð30Þ

2

^ d y aV 2 b c ^¼ þy þ þ ; 2 ^Þ2 ð1  y ^Þ3 ð1  y ^Þ4 d^t ð1  y

ð31Þ

where the primed non-dimensional parameters b0 (vdW force), c0 (Casimir force), and a0 (electrostatic force) are associated with the distributed model and non-primed ones related to the mass-spring model. These parameters are: 4

AbL b0 ¼ 6pðEIþGSl 2 4 ;  ÞG0

b ¼ 6bp0 AbL ; KG4 0

2

4

hcL p2 — hcL 0 bp — c0 ¼ 240ðbEIþGSl c ¼ c240KG 5 ; 2 5 ;  ÞG 0

0

ð32Þ

4

e0 bL 0 e0 bL a0 ¼ 2ðEIþGSl a ¼ a2KG 2 3 ; 3 :  ÞG 0

0

The equation of the static deflection of the nano-beam can be obtained from the dynamic motion equation eliminating inertial terms. Therefore using Eq. (30), the governing equation of the static deflection of the nano-beam can be written as: 4

^ d w a0 V 2 b0 c0 ¼ þ þ : 2 3 d^x4 ^ ^ ^ 4 ð1  wÞ ð1  wÞ ð1  wÞ

ð33Þ

3. Numerical approach Due to the presence of nonlinear terms in Eq. (33), the analytical solution methods may not be employed to obtain the pull-in voltage ‘Vpull-in’ and the detachment length of the nano-beam. Hence, the SSLM method together with Galerkin based reduced integration method are implemented to solve this equation. By using SSLM method, the smooth and continuous behavior of the beam as well as the magnitude of the nonlinear forces can be approximated in each iteration step [37]. It should be mentioned that there is a slight difference between static and dynamic application of the numerical method which will be discussed later. 3.1. Static deflection and pull-in voltage calculation The use of static SSLM demands gradually application of force. In the case of electrostatic force, the voltage can be increased from zero to the pull-in voltage gradually, so that it satisfies the quasi-equilibrium condition. However, vdW and Casimir forces depend only upon the non-dimensional gap size, so their values cannot be controlled manually. In order to solve this problem, it has been assumed that the mentioned forces have been applied gradually in a ‘virtual’ manner. For this purpose, a virtual variable ‘‘ k’’ has been introduced which is changed from zero to one. Indeed by multiplying this virtual variable to vdW and Casimir forces, application of SSLM to solve the static deflection equation will be capable [38]. ^ i is the non-dimensional displacement of the nanostructure, subjected to Vi Denoting superscript ‘i’ as the counting step, w under the virtual nano scaled of vdW and Casimir forces. By increasing the applied DC voltage and consequent change of the virtual variable ‘ ki’ the deflection at (i + 1)th step can be obtained as: iþ1 i ^ iþ1 ¼ w ^ i þ dw ^ ¼w ^ i þ ui ; V iþ1 ¼ V i þ dV&k  ¼k  þ dk )w

ð34Þ

where dV and d k are the voltage and virtual force variable changes between two successive steps, respectively. By considering a small value of dV and d k, the deflection variation in ith step ‘ui’ will be small enough that, in each step, we can take into account only the first two terms of the external force function in the Taylor expansion series. Eq. (33) for the ith step is: Please cite this article in press as: H. Mobki et al., Nonlinear behavior of a nano-scale beam considering length scale-parameter, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.10.001

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H. Mobki et al. / Applied Mathematical Modelling xxx (2013) xxx–xxx

! 4 i ^ d w a0 ðV i Þ2 b0 c0 i : ¼ þ k þ  d^x4 ^ i Þ2 ^ i Þ3 ð1  w ^ i Þ4 ð1  w ð1  w

ð35Þ

iþ1 ^ iþ1 , Vi+1 and k Substituting w in Eq. (33) from Eq. (34), and using the first two terms of Taylor expansion, results in: 

^i d4 w d^x4

4

i

0

i 2

a ðV Þ þ dd^xu4 ¼ ð1 þ ki ^ i Þ2 w



b0 ^ i Þ3 ð1w

  0 i2  0   0  0 a ðV Þ c0 c0 2a0 V i þ ð1cw^ i Þ4 þ 2ð1 þ ki ð13bw^ i Þ4 þ ð14w ui þ dV ð1 þ dk  ð1bw^ i Þ3 þ ð1w^ i Þ4 : ^ i Þ5 ^ i Þ3 ^ i Þ2 w w

ð36Þ

Subtracting Eq. (35) in Eq. (36) yields: d4 ui d^x4





2a0 ðV i Þ2 ^ i Þ3 ð1w

þ ki



3b0 ^ i Þ4 ð1w

0

c þ ð14w ^ i Þ5



i

2a0 V ui ¼ dV ð1 þ dk  ^ i Þ2 w



b0 ^ i Þ3 ð1w

 0 þ ð1cw^ i Þ4 :

ð37Þ

Eq. (37) is a linear ordinary differential equation that represents the variation of deflection along the nano-beam. This linear differential equation can be solved using Galerkin weighted residual Method in which u(x) can be approximated as:

uðxÞ ffi uN ðxÞ ¼

N X qj wj ðxÞ;

ð38Þ

j¼1

where uN(x) is an approximation of u(x) and wj(x) is the jth shape function satisfying the entire boundary conditions of the ^ have identical boundary condinano-beam and q’s are the generalized deflections. It should be mentioned that u(x) and w tions. Substitution of Eq. (38) into Eq. (37) and multiplying by wr(x) (where r = 1,. . .,N) as the weight function in Galerkin method and calculating the integrand, a set of algebraic equations will be obtained. Solving these set of equations, the deflection at any given applied voltage and consequently the pull-in voltage of the nano-beam can be determined. 3.2. Dynamic analysis For obtaining the response of the system excited by a step DC voltage, dynamic analysis of the nano-beam has also been performed. Applying a minor modification of Eq. (21), by assuming the generalized deflections are function of time i.e.:

wðx; tÞ ¼

N X qi ðtÞwi ðxÞ;

ð39Þ

i¼1

and using Galerkin approximation method, the equation of dynamic response will be:

€ þ ½K½q ¼ ½F; ½M½q

ð40Þ

where:

½Mij ¼ qS

RL

w ðxÞwj ðxÞdx; 0 i 2 RL 4  ½Kij ¼ ðEI þ GSl Þ o ddxw4i wj dx; RL ½Fi ¼ 0 qext wi ðxÞdx;

ð41Þ

are the effective mass, spring, and actuating force matrixes respectively. It must be noted that the obtained dynamic equation is nonlinear due to nonlinear displacement dependent electrostatic, vdW, and Casimir forces. q(t) can be obtained from above set of nonlinear ordinary differential equations using an integration method over the time. It is worth mentioning that the integration of the forcing term through the beam length, at the each step of integration over the time, must be repeated. 4. Results and discussion In this section the results of the numerical simulation for different cases are presented and compared to those obtained by other investigators for validation. The results obtained by the proposed numerical method are compared with those of vakili et al. [37]. The considered sample is a fixed-fixed nano switch, where it’s geometrical and material properties are listed in Table 1. The results of this comparison are presented in Table 2 and as can be seen in this table, the results have good agreement.At this stage the convergence of the proposed method to the results of Ref. [37] is investigated. The physical properties and assumptions of the nano-beam are identical as before (presence of Casimir and vdW forces and absence of electrostatic force). In Table 3 Dimensionless Peak Deflection (DPD) is obtained for differentdk  ’s, and as can be seen DPD converges to the exact value by decreasing of d k . Similarly, Tables 4 shows the convergence of the pull-in voltage to the results of Ref. [37] with refinement of number of the shape modes. Please cite this article in press as: H. Mobki et al., Nonlinear behavior of a nano-scale beam considering length scale-parameter, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.10.001

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H. Mobki et al. / Applied Mathematical Modelling xxx (2013) xxx–xxx Table 1 Geometrical and material properties of micro switch. Properties

Value

Length Width Height Young’s modulus Go

70 nm 6 nm 6 nm 169 GPa 10 nm

Table 2 The obtained pull-in voltage with 100 grid points and dk ¼ 0:01 for differentdV. dV

1

0.5

0.2

0.1

0.05

0.02

0.01

0.005

Vpull-in [37] Vpull-in

7 7

6.5 6.5

6.4 6.4

6.4 6.4

6.3 6.4

6.28 6.3

6.28 6.29

6.28 6.29

Table 3 The values of d k for calculated initial beam deflection without electrostatic excitation. d k

0.5

0.25

0.2

0.1

0.02

0.01

0.005

Dimensionless peak deflection

0.008

0.01

0.012

0.013

0.013

0.014

0.014

Table 4 The obtained pull-in voltage with dV = 0.01 and dk ¼ 0:01 for different number of shape modes. Number of shape modes

N=1

N=2

N=3

N=4

Vpull-in

6.32

6.3

6.29

6.29

4.1. Determination of the corrective coefficients For obtaining the equivalent masses of the nano-beam for the fixed-fixed and cantilever beam, the natural frequency of the corrected mass-spring and the distributed model (in lack of the electrostatic force without considering vdW and Casimir forces are computed and equated to the each other. These obtained equivalent masses are: cantilev er meq ¼ 0:65m;

mfixedfixed ¼ 0:74m: eq

ð42Þ

The values of the pull-in voltage and detachment parameters for the mass-spring and the distributed model are obtained by the similar approach used for obtaining equivalent mass. i.e., by using SSLM method [38] and equating the corresponding parameters of the distributed and lumped system, b0, c0, and a0 are obtained as follow: cantilev er

b0

¼ 0:68;

fixed—fixed

b0

¼ 0:79;

ð43Þ

v er ¼ 0:69; ccantile 0

cfixed—fixed ¼ 0:8; 0

ð44Þ

a0cantilev er ¼ 0:71;

afixed—fixed ¼ 0:82: 0

ð45Þ

4.2. Static pull-in parameter This section discusses the equilibrium points of the nano-beam considering vdW and Casimir forces. Referring to Eq. (31), physically equilibrium or fixed points exist in the range of 0 < y < 1, However, mathematically these points may also exist in _ Eq. (31) may be transformed into the following form: the range of 1 < y. Setting w ¼ y, dy dt dz dt

2

¼ z;

c b ¼ ð1yÞ2 þ ð1yÞ  y: 3 þ ð1yÞ4 aV

ð46Þ

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Fig. 3. Variation of the dimensionless gap of the nano-beams with applied voltage based on classic beam theory.

Fig. 4. Variation of the dimensionless gap of the nano-beams with applied DC voltage based on couple stress theory.

Fig. 5. Phase diagram for the nano-beam based on couple stress theory and given voltage 0 V.

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Fig. 6. Phase diagram for the nano-beam based on couple stress theory and given voltage 4 V.

Fig. 7. Phase diagram for the nano-beam based on couple stress theory and given voltage 8 V.

Fig. 8. Phase diagram for the nano-beam based on couple stress theory and given voltage 10.04 V.

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Fig. 9. Pull-in voltage VS length for nano-beam with h = 150 nm (l = 0.5 lm), G0 = 100 nm, b = 200 nm.

Fig. 10. Pull-in voltage VS initial gap for nano-beam with h = 150 nm (l = 0.5 lm), L = 8 lm, b = 200 nm.

At the equilibrium points, and in the presence of the electrostatic force the nano-beam is at rest, hence considering Eq. (46), equilibrium points are obtained as:

f ða; b; c; V; yÞ ¼ aV 2 ð1  yÞ2 þ bð1  yÞ þ c  yð1  yÞ4 ¼ 0:

ð47Þ

Figs. 3 and 4 depict equilibrium points for the Aluminum nano-beam versus applied voltage as a control parameter based on classic beam and modified couple stress theories respectively. The cantilever aluminum nano-beams properties are b = 200 nm, h = 150 nm, G0 = 100 nm, and L = 8 lm, and l = 0.5 lm [39]. As shown in these figures for given 0 < V < Vpull-in there exist two physically possible fixed points. In capacitive MEMS structures usually there are two types of fixed points [40]. The first ones are the physically fixed points, which are located between the beam and the ground conductive plate. These fixed points are physically possible. And the second are mathematically fixed points which are taken place beneath of the ground plate and these points physically are impossible. Physically and mathematically fixed point can be obtained from Eq. (47). In this paper only physically fixed points are extracted. As illustrated, motion trajectories in phase portraits (which are shown in Figs. 5–8), the first fixed point is a stable center and the second one is an unstable saddle node. Therefore in Figs. 3 and 4 continues and dashed curves represent the stable and unstable branches, respectively. In the aforementioned nano-beam, by increasing the control parameter V the physically possible fixed points are closing to each other and in the pull-in voltage, they coincide in the saddle node bifurcation point. As shown in Figs. 3 and 4, the calculated pull-in voltage based on classical beam theory and modified couple stress theory are 1.36 V and 10.02 V, respectively. This significant difference between the calculated pull-in voltages via the two methods, Please cite this article in press as: H. Mobki et al., Nonlinear behavior of a nano-scale beam considering length scale-parameter, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.10.001

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Fig. 11. Detachment length VS initial gap for nano-beam with h = 150 nm (l = 0.5 lm), b = 200 nm.

Fig. 12. Dimensionless fundamental frequency of the nano-beam versus applied DC voltage.

Fig. 13. Time history of aluminum nano-switch subjected to step-wise VDC = 1.24 v actuating based on classic beam theory.

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Fig. 14. Time history of aluminum nano-switch subjected to step-wise VDC = 1.25 v actuating based on classic beam theory.

demonstrates that an aluminum nano-beam cannot be modeled through classic theories and must be modeled through couple stress theory, which captures its size dependent behavior. Figs. 5–8 present motion trajectories of the aluminum nano-beam based on couple stress theory for different values of applied voltage with different initial conditions. As shown in Figs. 5–7 there are basins of attraction of stable centers and repulsion of unstable saddle nodes. It is evident from figures that the substrate position acts as a singular point and velocity of the system near this singular point tends to infinity. The basin of attraction of the first stable center is bounded by a bold closed orbit. Depending upon the location of the initial condition, the system can be stable or unstable. Figs. 5–8 show that with increasing the applied voltage the basin of attraction of stable center is contracted and by imposing a voltage equal to the pull-in one, as shown in Fig. 8, the basin of attraction vanishes and the system becomes unstable for every initial condition. In these figures bold-black, continues-blue and dashed one curves represent Homoclinic, periodic and unstable orbits, respectively. Figs. 9 and 10 show pull-in voltage versus the length and initial gap, respectively. As shown in these figures, with decreasing the length and increasing the initial gap, pull-in voltage increases and the size effect of the nano-beam is also increased. Fig. 11 illustrates the variation of detachment length with respect to the initial gap, obtained based on the couple stress, and classical beam theories. As shown in this figure, the calculated initial gap based on classic beam theory is higher than that of couple stress theory for the same detachment length; furthermore, variation of detachment length with initial gap based on couple stress theory is steeper in comparison of classic beam theory. Fig. 12 shows the variation of the non-dimensional fundamental frequency of the aluminum nano-beam versus applied DC value from zero to pull-in voltage. As shown using the classic beam theory in these cases leads to erroneous results and modified couple stress theory must be utilized.

Fig. 15. Time history of aluminum nano-switch subjected to step-wise VDC = 9.22 v actuating based on couple stress theory.

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Fig. 16. Time history of aluminum nano-switch subjected to step-wise VDC = 9.23 v actuating based on couple stress theory.

Fig. 17. Phase portrait of the aluminum nano-beam.

The dynamic response of the nano-beam (such as a nano switch) to a step DC voltage is studied by using Galerkin based reduced order model. For this nano-beam with L = 8 lm, G0 = 100 nm, b = 200 nm, and h = 150 nm, the calculated dynamic pull-in voltage based on classic beam and couple stress theories are 1.25 and 9.23, respectively. The dynamic pull-in voltages are about 92% of the static one, which shows that the results are in good agreement with the reported works [41]. Figs. 13 and 15 show the time history of dimensionless gap for this nano-beam with actuating step voltage of 1.24 v– 9.22 v, respectively. We note that the nano-beam oscillates with these voltages, and dose not collapse. As shown in Figs. 14 and 16 by a small increase of the actuating step voltage by only about 0.01 V the nano-beam collapses. Fig. 14 shows the dynamic pull-in phenomenon (collapse point) with step DC voltages of 1.25 v (based on classical beam theory), and Fig. 16 shows the same phenomenon with 9.23 v (based on modified coupled stress theory). In Fig. 17, the phase portrait of the mentioned nano-beam with zero initial conditions and various step DC voltages is plotted. As shown in this figure, response of the nano-beam to small step DC voltages is periodic. By increasing the amount of the applied voltage, which is due to the displacement dependent of the nonlinear electrostatic force and decreasing the equivalent stiffness, the period of the oscillations is increased and a symmetric breaking is occurred in motion trajectories. However, for large enough applied step DC voltages, owing to the saddle type fixed point and singular point, the motion exhibit an unbounded non-periodic behavior through a homoclinic bifurcation. Also, as shown, for neighboring curves, there is a considerable difference between two theories.

5. Conclusion In the present study, size dependent behavior of electro-statically actuated nano-beams was studied. Using the energy method, a mathematical modeling was presented for the beam dynamic motion. Equation of static deflection was solved using Step-by-Step Linearization method (SSLM) and the equation of dynamic motion was solved using the Galerkin-based reduced order model. As results showed, in the aluminum nano-beams, which have a considerable length-scale parameter in Please cite this article in press as: H. Mobki et al., Nonlinear behavior of a nano-scale beam considering length scale-parameter, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.10.001

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comparison with their thickness, applying the classic beam theory leads to incorrect results and couple stress theory must be applied. In addition, it was illustrated that the effect of the size dependent behavior considerably grows by decreasing the thickness of the nano-beam. Furthermore bifurcation behavior of aluminum nano-beam was studied and it was shown that, the nano-beam has two physically fixed points in range of 0 < V < Vpull-in. Based on motion trajectory, it was proved that the first fixed point of nano-beam is stable and the second one is unstable. And also it was shown for V = Vpull-in two fixed point meet together and saddle-node bifurcation occurred. In addition was shown that in the case of application of step DC voltage the nano-beam can be dynamically unstable through a homoclinic bifurcation for voltages equal or greater than dynamic pull-in voltage. 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