stability analysis of a capacitive fgm micro-beam using modified couple stress theory

Published by AMSS Press, Wuhan, ChinaActa Mechanica Solida Sinica, Vol. 26, No. 4, August, 2013 ISSN 0894-9166

STABILITY ANALYSIS OF A CAPACITIVE FGMMICRO-BEAM USING MODIFIED COUPLE STRESS

THEORY

Behrokh Abbasnejad Ghader Rezazadeh� Rasool Shabani

(Mechanical Engineering Department, Urmia University, Urmia, Iran)

Received 2 June 2011, revision received 12 September 2011

ABSTRACT Based on the Modified Couple Stress Theory, a functionally graded micro-beamunder electrostatic forces is studied. The FGM micro-beam is made of two materials and materialproperties vary continuously along the beam thickness according to a power-law. Dynamic andstatic pull-in voltages are obtained and it is shown that the static and dynamic pull-in voltages forsome materials cannot be obtained using classic theories and components of couple stress mustbe taken into account. In addition, it is shown that the values of pull-in voltages depend on thevariation through the thickness of the volume fractions of the two constituents.

KEY WORDS MEMS, FGM micro-beam, stability, pull-in voltage, electrostatic pressure, modifiedcouple stress theory

I. INTRODUCTIONFunctionally graded materials are usually a class of microscopically inhomogeneous composites made

of a mixture of a ceramic and a metal with spatially varying continuous material properties. Continuityprevents FGM from delamination due to large inter-laminar stresses, initiation and propagation of cracksbecause of large plastic deformation at the interfaces. So FGMs have more advantages in comparisonwith other composites. The concept of FGMs was first proposed in Japan in 1984 by the Sendai groupduring a work in a space plane project, thereafter FGMs are developed for a wide range of applications:such as reactor vessels, fusion energy devices, biomedical materials, aircrafts, space vehicles and militaryapplications. Therefore, it is very important to know and analyze the static and dynamic behavior ofthe FGM structures. Up until now, dynamic and static analysis of FGM beams has been shown in manyresearches on the macroscopic scale using classical continuum theory[1–3].

Nowadays, FGM beams are being used vastly in micro and nano structures such as thin films in theform of shape memory alloys[4,5], micro- and nano-electromechanical systems (MEMS and NEMS)[6–8]

and also atomic force microscopes (AFMs)[9].The thickness of the beams used in MEMS, NEMS and AFMs, is typically on the order of microns

and sub-microns. Size dependent behavior is an inherent property of materials which appears for abeam when the characteristic size such as thickness or diameter is close to the internal material lengthscale parameter[10]. The size-dependent static and vibration behavior in micro scale beams have beenexperimentally observed in metals[11–13], polymers[14–16] and poly-silicones[17]. Lacking intrinsic lengthscale parameters of the materials, the scale free classic theories of mechanics cannot give sufficientprediction of the behavior of the materials.

� Corresponding author. E-mail: [email protected]

· 428 · ACTA MECHANICA SOLIDA SINICA 2013

Some researchers introduced the classic couple stress elasticity theory based on Cosserat ContinuumMechanics[18–23]. The classic couple stress theory contained two classical and two additional materialconstants for isotropic elastic materials. Yang et al. resolved the arbitrary nature of couplings in theclassical couple stress theory without the use of rigid vector attachment conditions by introducing ahigher order equilibrium condition as used in the micro-polar theory and indicated that the couple stresstensor must be a symmetric tensor. Since the symmetric part of the curvature tensor is the additionalmeasure of deformation that conjugates to the couple stress, the anti-symmetric part of the curvaturetensor does not conjugate to the couple stress and it does not appear explicitly in the deformationenergy density function. They proposed the modified couple stress theory (MCST) in which the couplestress tensor is symmetric and involves only one internal material length scale parameter[24]. UtilizingMCST and the Hamilton principle, Park and Gao studied the static behavior of size-dependent Euler-Bernoulli micro-beams[25]. Kong et al. derived the governing equation, initial and boundary conditionsof an Euler-Bernoulli beam using the MCST and Hamilton principle[10]. As they reported, the stiffnessof beams is size-dependent. Also, the difference between the stiffness obtained by the Classic Theory(CT) and those predicted by the MCST is significant when the beam characteristic size is comparableto the internal material length scale parameter.

Recently, Tsiatas studied a new Kirchhoff plate model for the static analysis of isotropic micro-platesbased on the MCST[26]. Kahrobaiyan et al. investigated the size-dependent dynamic characteristics ofatomic force microscope micro-cantilevers[27]. Wang considered the size-dependent vibration charac-teristics of fluid-conveying micro-tubes[28]. And further, Xia and Wang discussed the size effect on thenonlinear bending, nonlinear vibration and post buckling of micro-beams[29]. All above papers studiedthe homogenous materials. However Asghari et al. investigated the size-dependent static and vibrationbehavior of micro-beams made of functionally graded materials (FGMs) analytically[30]. And mostrecently Ke and Wang studied the size effect on dynamic stability of functionally graded micro-beamssubjected to an axial excitation load based on MCST[31].

Microelectromechanical systems (MEMS) are generally classified according to their actuation mech-anisms. One of the most important actuation mechanisms is electrostatic[32]. Study of micro sensors andmicro actuators driven by an electrostatic force because of their small size, batch production, low energyconsumption, low cost and compatibility with the integrated circuits (ICs) is very important. These sys-tems are main components of many devices such as switches[33], micro-mirrors[34], micro-resonators[35],micro-actuators[36], accelerometers[37], and tunable capacitors[38]. Micro-beams under voltage drivingare widely used in many MEMS devices such as capacitive micro-switches and resonant micro-sensors.These devices are fabricated, to some extent, in a more mature stage than some other MEMS devices. Asthe microstructure is balanced between electrostatic attractive forces and mechanical (elastic) restor-ing forces, both electrostatic and elastic restoring forces are increased when the electrostatic voltageincreases. When the voltage reaches the critical value, pull-in instability happens. Pull-in is the pointat which the elastic restoring force can no longer balance the electrostatic force. Further increasing thevoltage will cause the structure to have dramatic displacement jump which causes structure collapseand failure. Pull-in instability is a snap-through like behavior and it is saddle-node bifurcation typeof instability[39]. In micro-mirrors[34] and micro-resonators[34] the designer avoids this instability toachieve stable motions, while in switching applications[33] the designer exploits this effect to optimizedevices’ performance. Hence it is important to pay attention to static and dynamic stability of the FGMmicro-beams.

In order for an MEMS structural layer to satisfy all material and economical requirements and sincea single layer can’t always meet the needs, Witvrouw and Mehta proposed the use of a non-homogenousfunctionally graded material (FGM) layer to achieve the desired electrical and mechanical propertiesand suggested that a polycrystalline-SiGe (poly-SiGe) layer can be an appropriate choice[40]. Hasanyanet al. studied the pull-in instabilities in a functionally graded MEMS caused by the heat producedby the electric current[41]. Jia et al.[42] studied the nonlinear pull-in characteristics of microswitchesconsisting of either homogeneous material or non-homogeneous functionally graded material (FGM)with two material phases under the simultaneous electrostatic and intermolecular forces using CT.

Recently Ballestra et al.[43] showed that there are significance differences between the experimentalresults and results of the numerical analysis based on the classical theory of elasticity for the pull-involtage analysis for gold micro-beams. Sadeghian et al.[44] illustrated that there is a strong size-dependent

Vol. 26, No. 4 Behrokh Abbasnejad et al.: Stability Analysis of Capacitive FGM Micro-beam · 429 ·

mechanical property as the characteristic dimensions (such as micro-beam or nano-beam thickness) ofthe structure approach the material length-scale parameter.

In spite of existence of many research concerning the mechanical behavior of MEMS structures, thereis not enough study in the static and dynamic behaviors of electrostatically actuated FGM micro-beamsbased on MCST. Therefore in this paper an FGM micro-beam suspended over a substrate is considered.The micro-beam and substrate are subjected to an electrostatic force by applying a voltage to the beamand substrate. Static and dynamic stability of the FGM micro-beam is studied when the voltage isapplied statically and when it is implemented as a step DC voltage. And the results based on MCSTare compared with those of CT.

II. MODEL DESCRIPTION AND MATHEMATICAL MODELINGAn Euler-Bernoulli FGM micro-beam subjected to a distributed electrostatic load fe (x, t) with

length L, width b, thickness h is shown in Fig.1. Material properties of the beam i.e., Young and shearmodulus, Poisson’s ratio and mass density vary continuously along the beam thickness are function ofz based on a power law and can be given by[3]

P (z) = (Pl − Pu)

(1

2−

z

h

)k

+ Pu (1)

where subscripts u and l refer to material properties of the upper and lower surfaces respectively. k isthe non-negative power-law exponent, which show the material variation profile through the thicknessof the FGM micro-beam. It is worth noting that k = 0 corresponds to a classic beam made of purelower material, k = ∞ corresponds to a classic beam made of pure upper material, and 0 < k < ∞

corresponds to an FGM micro-beam made of two materials; the lower surface made of the first materialand the upper surface made of the second material. And k shows the variation rate of the ceramicconstituent percent along the micro-beam thickness. According to the MCST the strain energy can bewritten as[22]

U =1

2

∮

Ω

(σijεij + mijχij)dΩ (2)

where the stress tensor, σij , strain tensor, εij , the deviatoric couple stress tensor, mij , and symmetriccurvature tensor, χij are respectively defined as

σij = λεkkδij + 2Gεij (3)

εij =1

2(ui,j + uj,i) (4)

mij = 2l2Gχij (5)

χij =1

2(θi,j + θj,i) (6)

where λ and G are the Lame constants and l is the material length scale parameter. ui and θi are thecomponents of the displacement and rotation vectors. The components of rotation vector are related

Fig. 1. Schematic view of an FGM micro-beam.


to the components of the displacement vector field as following[10]:

θi =1

2curl (u)i (7)

As shown in Fig.1 the displacement field, based on Euler-Bernoulli beam theory, is given by

u = u0 (x, t)− zϕ (x, t) , v = 0, w = w (x, t) (8)

where u, v, w are the x, y and z directions of the displacement vector, respectively, and u0 is the axialdisplacement of the mid-plane of the micro-beam in the x direction. The rotation angle ϕ (x, t) is relatedto the deflection and can be defined as

ϕ ≈∂w (x, t)

∂x(9)

Considering small deformations, longitudinal strain of the FGM micro-beam in the x direction is

εxx =∂u (x, t)

∂x=

∂u0 (x, t)

∂x− z

∂2w (x, t)

∂x2(10)

From Eqs.(7), (9) and (10) it follows that:

θy = −∂w (x, t)

∂x, θx = θz = 0 (11)

Substituting Eq.(11) into Eq.(6), components of the curvature tensor can be expressed as

χxy = −1

2

∂2w (x, t)

∂x2, χxx = χyy = χzz = χxz = χyz = 0 (12)

Substituting Eq.(10) into Eq.(3), the axial stress in x direction can be expressed as

σxx = E (z)

(∂u0 (x, t)

∂x− z

∂2w

∂x2

)(13)

In Eq.(13), E for the plane stress condition is equal to E(z) and for the plane strain condition (wide

beam) is equal to E (z) /[1− (ν (z))

2]

where ν (z) is the Poisson’s ratio. Submitting Eq.(12) into Eq.(5)

one can get:

mxy = −G (z) l2 (z)

(∂2w (x, t)

∂x2

), mxx = myy = mzz = myz = mzx = 0 (14)

Under no axial force, equilibrium equations of forces and couples in a given section of the cantileverFGM micro-beam can be written as∫

A

σxxdA = 0 (15)

M = Mσ + Mm =

∫

A

σxxzdA +

∫

A

mxydA (16)

where Mσ and Mm are components of the bending moment due to the classic stress and couplestress tensors respectively and M is the external moment applied to the given section.

Substituting Eq.(13) into Eq.(15) one can get:

∂u0 (x, t)

∂x=

(∫A zE (z) dA∫A

E (z) dA

)(∂2w (x, t)

∂x2

)(17)

Considering Eqs.(12) and (13), the components of the bending moment; Mσ and Mm can be obtainedas

Mσ =

∫

A

zE (z)

(∂u0 (x, t)

∂x− z

∂2w (x, t)

∂x2

)dA = (EI)eq

∂2w (x, t)

∂x2(18)

Mm =

∫

A

−G (z) l2 (z)

(∂2w (x, t)

∂x2

)dA =

(GAl2

)eq

∂2w (x, t)

∂x2(19)


where

(EI)eq =

∫

A

[zE (z)

(∫A

zE (z) dA∫A E (z) dA

)− z2E (z)

]dA (20)

(μAl2

)eq

=

∫

A

G (z) (l (z))2dA (21)

From Eqs.(18) and (19) the bending moment at a given section in the FGM micro-beam in termsof the transversal deflection w (x, t) can be expressed as

M = Mσ + Mm =((EI)eq +

(GAl2

)eq

)(∂2w (x, t)

∂x2

)(22)

In case of fixed-fixed boundary conditions, the sum of the longitudinal stresses is not equal to zeroand its value depends on the beam deflection. This means that the transversal deflection of the beamis coupled with the longitudinal displacement of the beam through the nonlinear terms of the straintensor.

εxx =∂u (x, t)

∂x+

1

2

(∂w (x, t)

∂x

)2

(23)

For convenience the nonlinear term of the strain tensor can be averaged along the beam length andconsequently a mean value of the generated axial force can be given as[36]

Ta (w) =1

L

∫ L

0

∫

A

E (z)1

2

(∂w (x, t)

∂x

)2

dAdx (24)

Therefore, the equation of static deflection of the FGM micro-beam considering axial forces due to thefixed-fixed boundary conditions is given by

((EI)eq +

(GAl2

)eq

) ∂4w (x, t)

∂x4− Ta (w)

∂2w (x, t)

∂x2= fe (x, t) (25)

The distributed external load fe (x, t) in the proposed case study is a nonlinear displacement de-pendent electrostatic force, which is introduced as[45]

fe (x, t) =εbV 2

2 (g0 − w)2 (26)

where, ε is the permittivity of the air within the gap, b is the width of the FGM micro-beam, g0 is theinitial gap between the micro-beam and the substrate and V is the applied DC voltage. Consideringinertial terms, the governing equation for dynamic motion of the micro-beam is obtained as

(ρA)eq∂2w (x, t)

∂t2+

((EI)eq +

(GAl2

)eq

) ∂4w (x, t)

∂x4− Ta (w)

∂2w (x, t)

∂x2=

εbV 2

2 (g0 − w)2 (27)

where:

(ρA)eq =

∫

A

ρ (z) dA (28)

For convenience, the following non-dimensional parameters are utilized:

w =w

g0, x =

x

L, t =

t

t∗, z =

z

h, t∗ =

[(ρA)eq L4

ElI

]1/2

s1 =(EI)eq +

(GAl2

)eq

ElI, s2 =

εbL4

2ElIg30

Ta (w) =Ta (w) g2

0h

2ElI, Ta (w) =

∫ 1

0

∫ 1/2

−1/2

E (z)

(∂w

(x, t

)∂x

)2

bdzdx

(29)


where El is the Young’s modulus of elasticity of the material of the beam lower surface. Inserting theseparameters into Eqs.(25) and (27), the following non-dimensional equations for the static and dynamicdeflection of the micro-beam are obtained:

s1

∂4w(x, t

)∂x4

− Ta (w)∂2w

(x, t

)∂x2

=s2V

2

(1− w

(x, t

))2 (30)

∂2w(x, t

)

∂t2+ s1

∂4w(x, t

)∂x4

− Ta (w)∂2w

(x, t

)∂x2

=s2V

2

(1− w

(x, t

))2 (31)

III. NUMERICAL SOLUTION3.1. Static Analysis

Because of nonlinearity of the governing equation, a step by step linearization (SSLM) method is usedto linearize it[36]. Afterwards, the obtained linearized differential equation is solved using a Galerkinbased weighted residual method. Using SSLM, the voltage applied to the micro-beam and substrateare increased from zero to its final value gradually. It’s supposed that wk

s is the displacement of theFGM micro-beam due to applied voltage V k. In the next step by increasing voltage, the displacementat the (k + 1)

thstep can be obtained as

V k+1 = V k + δV , wk+1s = wk

s + δws, δws = ψ(x) (32)

The equation of static deflection of the FGM micro-beam at (k + 1)th

step can be expressed as

s1

∂4wk+1s

(x, t

)∂x4

− Ta

(wk+1

s

) ∂2wk+1s

(x, t

)∂x2

= s2

(V k+1

1− wk+1s

(x, t

))2

(33)

Using Calculus of Variation Theory and keeping the first two terms of Taylor’s expansion in eachstep, it’s possible to rewrite Eq.(30) in terms of ψ as

s1∂4ψ

∂x4− Ta

(wk

s

) ∂2ψ

∂x2− 2s2

(V k

)2

(1− wks )

3 ψ − 2s2V kdV

(1− wks )

2 = 0 (34)

Considering a small value of δV , the value of ψ (x) will be expected to be small enough to obtain adesired accuracy.

It is worth pointing out that Ta

(wk+1

s

)in Eq.(33) is approximated by Ta

(wk

s

). This value can be

corrected using an iteration procedure. The obtained linear Eq.(34) can be solved by expressing ψ asfollows:

ψ (x) =

∞∑j=1

ajϕj (x) (35)

where ϕk (x) are the proposed shape functions for the FGM micro-beam, satisfying the accompanyingboundary conditions. Using the Galerkin weighted residual method, the unknown ψ (x) is approximatedby truncating the summation series to a finite number, n:

ψn (x) ∼=

n∑j=1

ajϕj (x) (36)

Substituting Eq.(36) into Eq.(34) and multiplying it by ϕi as a weight function in the Galerkin method,and integrating the outcomes from x = 0 to 1, a set of algebraic equations will be obtained. By solutionof these algebraic equations, deflection of the FGM micro-beam can be determined at any given appliedvoltage.


3.2. Dynamic Analysis

In the dynamic analysis of the FGM micro-beam due to nonlinear nature of the electrostatic force,direct usage of the Galerkin method is very complicated. Therefore, to prevent the complexity in thesolution, the nonlinear term is considered as a forcing term and its integration over the x domain isrepeated at each time step. By selecting time steps to be small enough, more accurate results willbe obtained. Based on the Galerkin reduced order model, an approximate solution of Eq.(31) can beexpressed as follows:

w(x, t

)∼=

n∑j=1

qj

(t)ϕj (x) (37)

Substituting Eq.(37) into Eq.(31) and multiplying the resultant by ϕi and integrating outcome fromx = 0 to 1, the following system of linear ordinary differential equations can be obtained:

n∑j=1

Mij qj

(t)

+

n∑j=1

(Km

ij + Kaij

)qj

(t)

= Fi (38)

where qj is the time dependent generalized coordinate of the system. Mij , Kmij and Ka

ij are the elementsof the effective mass, mechanical and axial stiffness matrices, respectively, which are given by

Mij =

∫ 1

0

ϕiϕjdx, Kmij = s1

∫ 1

0

ϕiϕ(iv)j dx, Ka

ij =

∫ 1

0

Ta (w) ϕiϕ′′

j dx

Fi =

∫ 1

0

s2V2

(1− w

(x, t

))2 ϕidx(39)

Equation (38) can be integrated over time by any integration method such as Rung-Kutta method. Itmust be noted that in the integration procedure of Ka

ij , integrating over the x domain must be repeatedat each time step due to the displacement dependency of the axial force.

IV. NUMERICAL RESULTSTo compare the obtained results with those existing in literature, a classic fixed-fixed wide micro-beam

used in Ref.[36] with the following geometrical and material properties is considered here.

E = 169 GPa, b = 50 μm, h = 3 μm, L = 350 μm, g0 = 1 μm

The calculated static pull-in voltage of the micro-beam is 20.1 V, which is in good agreement withthose published in Ref.[36].

Sadeghian et al.[44] experimentally showed that the size-dependent mechanical properties for a siliconcantilever are significant when the cantilever thickness approaches nano-meter scale. But for a Gold orNickel beam, the size dependent behavior is important even for the micro-scale beams[43,46,47]. Ballestraet al.[43] showed the experimentally obtained pull-in voltage for a gold micro-beam is about 2 timesthe theoretical results and P. Pacheco et al.[46] reported that experimentally obtained pull-in voltagesfor nickel beams are about 9 times the theoretical results. They supposed that the major difference isdue to the high intrinsic residual axial tensile stress developed in the nickel film during the fabricationprocess (on the order of 150 MPa). But today we can show that these differences not only are becauseof residual stress but also are due to the material internal length-scale parameter.

The micro-beam investigated here is made of Gold and Nickel as the first and second materialconstituents, which have considerable micro-scale length scale parameters. It is considered that thelower surface of the micro-beam is made of pure Gold and the upper surface is made of pure Nickel.Geometrical and material properties of the micro-beam are listed in Tables 1 and 2, respectively.

Table 1. Geometrical properties of the FGM micro-beam

Parameter Length L Width b Thickness h Initial gap g0 Permittivity of air ε0Value 541.8 μm 32.2μm 2.68μm 2.83μm 8.8541 × 10−12 F/m


Table 2. Material properties of the FGM micro-beam

Parameter Value

Material type Gold Nickel

Young’s modulus E 98.5 GPa 200 GPaPoisson’s ratio ν 0.44 0.31Mass density ρ 19300 kg/m3 8900 kg/m3

Shear modulus G 27 GPa 76 GPaLength scale parameter l 1.12 μm 5.0 μm

Fig. 2. Variation of the Young’s modulus (a) and the mass density through the thickness of the FGM micro-beam (b).

Based on parameters listed in Tables 1 and 2, mechanical behaviors of the micro-beam for differentvalues of the power law exponent (different are obtained and shown in Figs.2(a) and 2(b). These figuresshow the variation of Young’s modulus and density of the micro-beam along the thickness as stated inEq.(1).

4.1. Stability of Equilibrium Positions

Figure 3 illustrates the equilibrium positions or fixed points of the fixed-fixed micro-beams for k = 5versus applied voltage as a control parameter using the classic elasticity theory (dashed lines) andmodified couple stress theory (solid lines). As shown in Figs.3(a) and 3(b), for a given applied voltagebased on the lumped model analysis, the micro-beam has two fixed points or equilibrium positionsabove the substrate, but for the distributed model analysis for low applied voltages, the number ofequilibrium positions of the system is one and for high applied voltages is two. In order to study thestability of the fixed points, phase portraits are given for the micro-beam motion with k = 5 for differentapplied voltages and different initial conditions. As shown in Fig.4(a) when there is no applied voltage

Fig. 3. The center gap of the FGM fixed-fixed micro-beam versus applied voltage: (a) based on lumped model analysis;(b) based on distributed model analysis.


Fig. 4. Phase portraits of the FGM cantilever micro-beam with different initial conditions.

(V = 0 V), there exists only one stable center equilibrium position at zero (w = 0 ). But if one paysattention to Fig.4(b), it can be found that for a given applied voltage the first fixed point is a stablecenter and the second is an unstable saddle node. As shown in Fig.4(b), there are a basin of attractionof the stable center and a basin of repulsion of the unstable saddle node.

The first basin of attraction of the stable center is bounded by a homoclinic orbit. Depending onthe location of the initial condition, the system can be stable or unstable. As shown in Figs.3 and 4(c),as the applied voltage approaches a critical value, the stable (S.B) and unstable branches (U.S.B) ofthe fixed points meet together at a saddle-node bifurcation point[48]. The voltage corresponding to thesaddle node bifurcation point is well-known as the static pull-in voltage (Vsp) in the MEMS literature.However in Fig.4(d) for an applied voltage bigger than the static pull-in voltage, there is one unboundedbasin of repulsion of the unstable saddle node. In other words, when the applied voltage is equal orgreater than the static pull-in voltage there is no basin of stable attractors on the upper side of thesubstrate and the micro-beam is unstable for any initial conditions.

In addition, Fig.3 represents a comparison between the modified couple stress and classic beamtheories. As shown, applying MCST shifts right the saddle-node bifurcation point and hence increasesthe calculated static pull-in voltage.

Figure 5 shows the variation of the ratio of the static pull-in voltage calculated by MCST (V MCSTsp )

to the static pull-in voltage of a gold micro-beam calculated by CT (V Au-CTsp ) versus beam thickness

(h) for different value of k.As illustrated in Fig.5 by increasing the h , the static pull-in voltage ratio (V MCST

sp /V Au-CTsp ) for

k = 0 approaches one, but in lower value of h the differences between the two theories is so considerable.These differences for high enough value of k will be greater as a result of the high value of Nickel lengthscale parameter. In addition as shown in Fig.5 increasing the h, for high enough value of k value ofV MCST

sp /V Au-CTsp converges to V Ni-CT

sp /V Au-CTsp .

Figure 6(a) shows the variation of the ratio of the non-dimensional natural frequency calculatedusing MCST (ΩMCST) to the non-dimensional natural frequency of a gold micro-beam calculated using


Fig. 5. Variation of the static pull-in voltage ratio versus h.

CT (ΩAu-CT) versus beam thickness; h for different value of k. It can be shown that the differencebetween two theories depends on the value of h. Figure 6(a) depicts that for k = 0 the increase in thevalue of h causes the non-dimensional natural frequency ratio (ΩMCST/ΩAu-CT) is closed one. Howeverat lower value of h, the differences between the two theories is so considerable. As shown in Fig.6(a)when the value of k is higher enough, these differences will be greater because of high length scaleparameter of Nickel.

Figure 6(b) illustrates the non-dimensional natural frequency of the FGM fixed-fixed micro-beamusing MCST for different values of power law exponent (k) versus applied voltage. As shown in Fig.6(b),increasing the value of k increases the value of natural frequency for a given applied voltage due toincreasing in equivalent micro-beam stiffness. In addition, it is shown that the value of natural frequencyis decreased by increasing the applied DC voltages until the natural frequency of the micro-beam becomeszero at the static pull-in voltage. In other words, Fig.6(b) emphasizes that the pull-in instability is akind of stationary instability.

Fig. 6. (a) Variation of non-dimensional frequency ratio versus beam thickness and (b) non-dimensional frequency of thefixed-fixed FGM micro-beam versus applied DC voltage with h = 2.68 μm using MCST.

4.2. Application of a Step DC Voltage

Due to the dependency of the electrostatic force on both the voltage and deflection (w), pull-inoccurs at a voltage less than Vsp when the voltage is applied in the form of a step DC voltage. Figure7 shows the response of the fixed-fixed FGM micro-beam to a step DC voltage for k = 5. As shown inFig.7 the response of the FGM micro-beam to the application of small step DC voltages is a periodicresponse, and when increasing the value of the applied voltage, due to the displacement dependency ofthe nonlinear electrostatic force and decreased equivalent stiffness, period of the oscillations is increasedand symmetry breaking occurs in motion trajectories. It must be noted that the scenario of instabilityin the case of applying step DC voltage is different from its statically application. As depicted in Fig.3,when the applied DC voltage approaches the static pull-in voltage, the system tends to an unstable


Fig. 7. (a) Time history and (b) phase portrait of the micro-cantilever for k = 5.

equilibrium position by undergoing a saddle node bifurcation. A saddle node bifurcation is a locallystationary bifurcation and can be analyzed based on locally defined eigenvalues. In addition to localbifurcations, periodic orbits appear that cannot be analyzed based on locally defined eigenvalues. Suchphenomena are called global bifurcations[49].

In Fig.7(b), it is shown that how a periodic orbit approaches a homoclinic orbit at the dynamicpull-in voltage. Indeed, the periodic orbit ends at the dynamic pull-in voltage where a homoclinic orbitis formed. In another words, it can be said that a homoclinic bifurcation happens when the periodicorbit collides with a saddle point at the dynamic pull-in voltage. The results of the previously publishedreports show that the dynamic pull-in voltage is less than about 92% of the static one[50], which isin good agreement with the present results. Moreover, for a given micro-beam thickness the dynamicpull-in voltage predicted using MCST is greater than that predicted using CT.

Figure 8 shows the variation of the ratio of the dynamic pull-in voltage calculated using MCST(V MCST

dp ) to the dynamic pull-in voltage of a pure gold classic micro-beam calculated using CT (V Au-CTdp )

versus beam thickness (h) for different value of k. As illustrated in Fig.8, by increasing h the dynamicpull-in voltage ratio (V MCST

dp /V Au-CTdp ), like static pull-in voltage ratio for k = 0, approaches one, but

at lower value of h the differences between the two theories is so considerable. These differences forhigh enough value of k will be greater as a result of the high value of the Nickel length scale parameter.In addition, as shown in Fig.8 increasing h, for high enough value of k, the value of V MCST

dp /V Au-CTdp

converges to V Ni-CTdp /V Au-CT

dp .

Fig. 8. Variation of the dynamic pull-in ratio versus beam thickness.

V. CONCLUSIONIn the present work, the mechanical behavior of a fixed-fixed FGM micro-beam subjected to a

nonlinear electrostatic pressure using Modified Couple Stress Theory and Classic Theory was studied.It was assumed that the lower surface was made of pure Gold and the upper surface was made of pureNickel. Considering a power-law form to represent the continuous variation of material properties along


the beam thickness, the nonlinear differential equation of motion based on MCST was derived. Thestatic instability of the FGM micro-beam for a fixed-fixed micro-beam was studied through solving theequation of static deflection implemented with SSLM.

It was shown that for a given applied voltage based on the lumped model analysis, two equilibriumpositions or fixed points exist and in the distributed model analysis depending on the value of theapplied voltage, one or two equilibrium positions in the upper side of the substrate exist. Based onillustrated trajectories in phase portraits, the first of them is a stable center and the second one is anunstable saddle node.

Increasing the applied voltage, as a control parameter, the first and second fixed points in thestate-control space approach each other and in a specific voltage, called the pull-in voltage in MEMSliterature; they meet at a saddle node bifurcation point. Results showed that by increasing the power lawconstant (increasing the percent of the Nickel constituent) the position of the saddle node bifurcation,because of increased equivalent micro-beam stiffness, moves to the right in the state-space. In the caseof applying a step DC voltage, the system reach an unstable state by a global homoclinic bifurcation atthe critical voltage called the dynamic pull-in voltage, which is about 92% of the static one independentof values of k.

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