Material structure and size effects on the nonlinear dynamics of electrostatically-actuated nano-beams

Author’s Accepted Manuscript Material structure and size effects on the nonlinear dynamics of electrostatically-actuated nano-beams M. Shaat, A. Abdelkefi

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S0020-7462(16)30326-2 http://dx.doi.org/10.1016/j.ijnonlinmec.2016.11.006 NLM2736

To appear in: International Journal of Non-Linear Mechanics Received date: 3 June 2016 Revised date: 19 August 2016 Accepted date: 18 November 2016 Cite this article as: M. Shaat and A. Abdelkefi, Material structure and size effects on the nonlinear dynamics of electrostatically-actuated nano-beams, International Journal of Non-Linear Mechanics, http://dx.doi.org/10.1016/j.ijnonlinmec.2016.11.006 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 galley proof before it is published in its final citable 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.

Material structure and size effects on the nonlinear dynamics of electrostatically-actuated nano-beams M. Shaat, A. Abdelkefi* Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA. [email protected] (M. Shaat) [email protected] (A. Abdelkefi) *Corresponding author. Tel.: +15756466546; fax:+15756466111.

Abstract An accurate nonlinear model for electrostatically actuated beams made of nanocrystalline materials is proposed accounting for the beam material structure and the beam size effects. Two sets of measures are incorporated in the context of the proposed model to account for the inherent properties (the material structure related properties) and the acquired properties (the size dependent properties) of the beam. The inherent properties of the beam are modeled via a micromechanical model while the acquired properties modeled via a non-classical continuum beam theory. The micromechanical model for nanocrystalline materials is proposed where the necessary measures to account for the effects of the grain size, the voids percent and size, and the interface (grain boundary) are incorporated. All the measures presented in the micromechanical model are related to the material structure to correctly model the structure of nanocrystalline materials. According to the classical couple stress and Gurtin-Murdoch surface elasticity theories, a size-dependent Euler-Bernoulli beam model is developed to model the mechanics of electrostatically actuated nano-beams. For the first time, the impacts of the beam material structure along with the beam size on the nonlinear dynamics and pull-in instability behaviors of electrostatically actuated nano-beams are intensively studied. The performed analyses through the present effort reflect the great impacts of the beam material structure and the beam size on the static pull-in, the natural frequencies, the dynamic pull-in, and nonlinear dynamics of electrostatically actuated nano-beams.

Keywords: nanocrystalline materials, nonlinear dynamics, electrostatically actuated beams, surface effects, couple stress.

1. Introduction

--------------------------------------------------

With the dramatic growth of nanotechnology, the production and use of engineered nanoparticles, nano-components, and nano-devices have been rapidly increasing in the past few years. Nano-devices have received significant interests in Micro-Electro-Mechanical Systems (MEMS) and Nano-ElectroMechanical Systems (NEMS). Designed devices for micro-/nano-scale applications are usually composed of elastic micro-/nano-sized structures made of nanomaterials. Nanomaterials have unique behaviors compared to their conventional counterparts.

Electromechanical systems are usually composed of an elastic conductive micro-/nano-resonator suspended above a rigid conductive plate and an electric voltage is applied in the gap between them that enforces the elastic resonator to deform changing the gap and hence the electrostatic energy of the system. The instant at which the resonator touches the fixed plate and it cannot restore its original position is known as pull-in instability. Detecting exactly the instant at which the pull-in of the system can happen is essential for designing these systems. Recently, behaviors of electromechanical systems are studied in different efforts [1-8]. The behavior of electrostatic actuated beams subjected to a static potential has been investigated in [1-4] where a closed form for the static pull-in is derived [1, 2]. The nonlinear dynamics of electrostatically actuated beams have been studied in different works [5-7]. In addition, the dynamic pullin phenomenon has been intensively studied in [7]. The applicability of actuated beams for bio-mass sensing has been discussed in [8]. In the aforementioned studies, the mechanics of the electromechanical systems has been investigated on the bases of the classical theories of continuum mechanics. However, the mechanics of micro-/nano-solids in micro-/nano-scale applications are highly influenced by their sizes which exceeds the limit of applicability of the classical theories of continuum mechanics [9-11]. Moreover, these systems are usually made of nanomaterials which have unique properties compared to their conventional counterparts. Therefore, new models that have measures to capture both (1) the material structure and (2) the entire material size effects are needed. The mechanics of elastic structures in different MEMS and NEMS applications have been studied using one of the non-classical theories, such as the nonlocal theory [12-14], couple stress theory [15-17], and strain gradient theory [18-20]. Furthermore, based on the surface elasticity theory, the pull-in instability of electrostatically actuated nanobeams [18, 20] and nanoplates [21-23] have been investigated. Although these theories and models incorporate new measures to capture the material size effects, they still lack the essential measures to model the material behavior as a function of its internal structure. To demonstrate this fact, the effects of the grain size on the effective Young’s modulus of nanocrystalline materials can be considered. As an example, the Young’s modulus of the crystalline diamond decreased from its conventional value 1150 GPa to

GPa and

GPa for two distinct average grain

sizes, 20 nm and 6 nm, respectively [24]. This example demonstrates that the properties of nanomaterials

2

are strongly dependents on the nature of their material’s structure. Thus, to model nanomaterials, extrameasures are needed to capture the mechanics of the materials as functions of their structures. Recently, Shaat and Abdelkefi [25, 26] proposed a novel model that has the merit to capture effects of the material structure along with the material size on the mechanics of nano-beams made of nanocrystalline materials. In their model, the couple stress theory was used to capture the material size effects and a size-dependent micromechanical model was developed to model the heterogeneous material structure of the nanocrystalline beam. In the mentioned studies, the vibrations and the static pull-in instability behaviors of nano-beam resonators for mass-sensing were investigated. To study the vibrations and the static pull-in instability of actuated resonators, the constant part of the electrostatic potential is considered. However, to study the nonlinear dynamics and the dynamic pull-in of these resonators, the beam should be subjected to a variable electrostatic actuation. In the present study, we investigate the material structure and the size effects on the nonlinear dynamics, stability, instability, and the static and the dynamic pull-in behaviors of electrostatically actuated beams. A size-dependent micromechanical model for a three-phase heterogeneous material structure is proposed and used to model the beam material structure. In the context of the micromechanical model, some measures are incorporated to relate the beam performance to the size of the inhomogeneities inside its material structure. To account for the beam size effects, including the grain rigid rotations and the beam free-surface energies, a new beam model is developed according to the classical couple stress theory and Gurtin-Murdoch surface elasticity theory. In the framework of the continuum model, measures are incorporated to relate the beam performance to its size. The paper is organized as follows: in section 2, the effective elastic moduli of beams made of nanocrystalline materials accounting for the porosity and the grain size effects are estimated via a size-dependent micromechanical model. Then, the essential concepts and equations for couple stress-based elastic continua with surface elasticity are presented in section 3. In section 4, a nanocrystalline-electrostatically-actuated-nano-beam is modeled according to the classical couple stress theory and Gurtin-Murdoch surface elasticity theory. In addition, the solutions for the static pull-in and the natural frequency of the actuated beams are derived. To investigate the influences of the beam material structure and the beam size on its nonlinear dynamics, a reduced-order model based on Galerkin method and Euler-Lagrange equations is derived. A nanocrystalline silicon beam is considered as a case study in section 5 to determine the impacts of the beam material structure and size on its static deflection, natural frequency, nonlinear dynamics, and pullin instabilities. In section 6, this work is summarized highlighting the main contributions and conclusions.

2. Estimation of the effective elastic moduli of nanocrystalline materials

3

The real structure of polycrystalline materials is featured with many imperfections including interstitials, grain boundaries, twined boundaries, triple junctions, and dislocations, as presented in Figure 1. The role of these imperfections on the elastic properties of polycrystalline materials increases as the grain size reduces. Thus, for polycrystalline materials with coarse grains, it is acceptable to represent their structures as homogeneous ones. On the other hand, for nanocrystalline materials, the grain size is very small which increases the contribution of the other imperfections such as the interface and the porosities. Therefore, the structure of nanocrystalline materials should be molded as a highly heterogeneous structure. In addition, experimental investigations demonstrated that the elastic material properties of nanocrystalline materials strongly dependent on the grain sizes and the interface [24, 27, 28]. Therefore, to estimate the effective properties of nanocrystalline materials, a micromechanical model with the merit to represent the effective properties as a function of the grain size should be used.

Figure 1: A schematic of the real structure of nanocrystalline materials.

In the present study, the elastic properties of nanocrystalline materials are estimated using the sizedependent micromechanical model proposed in [25]. This model has the merit to capture the inhomogeneity size effects by incorporating the inhomogeneity surface energy effects into the conventional Mori-Tanaka micromechanical model. The model for the two-phase materials is first presented by Duan et al. [29] and modified in [25] for multi-phase materials. The micromechanical model for multi-phase materials, that is presented in [25], is developed using the decoupling method pioneered by Huang et al. [30] in which the multi-phase composite is decomposed into a set of two-phase composites. In this model, a two-phase composite for each inclusion type is assumed with the matrix material is the matrix of the multi-phase composite and then solved using the two-phase micromechanical model. After that, the effective properties of the multi-phase composite is obtained using the following formula:

4

∏

(1)

where

is an elastic property of the multi-phase composite (it can be

shear modulus and

is the bulk modulus),

( (

) , )

where

is the

is the effective elastic property of the two-phase

composite with phase-1 is the matrix and the inclusion type multi-phase composite where it is decoupled into property

or

.

is the number of phases of the

two-phase composites. The effective elastic

can be estimated using the two-phase model presented in [25, equations (19-24)].

In the present analyses, the nanocrystalline beam is modeled as a three-phase composite with nanograins and nano-voids as two inhomogeneities and the interface is the matrix phase. With the aid of the model presented in [25], the effective elastic moduli of the decoupled two-phase composites can be estimated as follows: (

) [ (

[

(

)

(

(

[ [

(3)

(

)]

(

[

(2)

)]

)] )

( [

)]

(

(4)

)] )] )

(

(5)

where (

)( (

)

)

[(

)(

(6) )

(

) (7)

( [ (

)]

) (

)

(

)(

)(

) (8)

(

)( (

)

(

5

))]

(

) (

)

)( ( (

)( )

(

(

( )(

(

(

( (

(

)( (

)

[

)

)))

)] )(

(9)

(10)

[ (

(

(11)

)] (

and

))

)

( where

))

(12) )

(

(

)))

(13)

are the effective elastic moduli of the decoupled two-phase composite with the nano-

grains are the inclusions and the interface is the matrix. Also,

and

are the effective elastic moduli

of the decoupled two-phase composite with the interface is the matrix and the nano-voids are the inhomogeneities. The surface parameters of the grains surfaces,

and

, and the voids surfaces,

and

, are incorporated into equation (2)-(13) to capture the inhomogeneities surface energy effects. ⁄

⁄

,

⁄

, and

⁄

,

surface parameters of the grains and the voids, respectively, where

and

are the non-dimensional are the average radii of the

grains and the voids, respectively. To estimate the effective elastic moduli of the nanocrystalline material, the grain material parameters, , and

, and the interface material parameters,

, and

, should be known where

is the

Poisson’s ratio. Usually, the nano-grains, nearly, have the same Young's modulus of their coarse counterparts [25-28, 31, 32]. On the other hand, the separate distance between atoms in the interface is larger than the distance between atoms inside the grain interior; hence, the interface has a lower Young's modulus [25-28, 31-34]. The atomic lattice model proposed in [25, 26, 31, 32] can be used to estimate the Young's modulus of the interface in terms of the grain Young’s modulus, the atomic density of the grain core,

, and the atomic density of the interface, [(

)(

)

(

)(

, as follows: )

]

6

(14)

where

are two material constants (see Table 1 in [25]). The interface Poisson’s ratio is assumed

and

equals the Poisson’s ratio of the grains. The volume fractions of the grains, interface, and voids are governed as follows: ( where

) ,

and

(

)

(15)

are the volume fractions of the interface, the grains, and the voids, respectively.

denotes the volume fraction of the grains with respect to the volume fraction of the interface matrix. depends on the interface thickness,

(

, and the grain radius,

, as follows: (16)

)

It should be mentioned that, in the context of the presented micromechanical model, the measures needed to relate the material properties to the inhomogeneities size are incorporated. The measures , and

, and

,

are introduced to capture the inhomogeneities surface energy effects. In addition,

are used to account for the interface and the porosity effects.

3. Couple stress-based elastic continua with surface elasticity In elastic continua with conventional macro-scale sizes, the grains inside the material structure are very small in comparison to the whole continuum size. Therefore, representing grains as mass points allowed to translate in the context of the continuum theory is acceptable. Thus, the infinitesimal strain tensor is defined as the fundamental measure for the deformation energy of linear-elastic continua. On the other hand, in elastic continua with micro-/nano-scale sizes, the ratio of the continuum size to the grain size intensively decreases and may approach unity (for the case of single-crystalline continua). Therefore, the grain has to be represented as a volume element that may deform and/or rotate. Hence, new deformation measures, in addition to the conventional infinitesimal strain, have to be introduced to reflect the effects of the new degrees of freedom of the grains on the deformation energy of the continuum. According to our definition of the acquired properties, the extra degrees of freedom, i.e. rotation and deformation, of the grains are acquired properties. Therefore, these non-classical fields should be modeled via a continuum theory such as the nonlocal theory, strain gradient theories, or couple stress theories. In the absence of external loading, the free surface tension of micro-/nano-bodies induces a surface pre-strain that changes the initial surface area of the body [35-38]. Furthermore, a compressive residual stress field in the interior bulk of materials produces because of the free surface tension (or more general the free surface residual stress) [38, 39]. Therefore, an essential feature of the surface elasticity theory is to distinguish the initial area from the deformed area to account for the surface pre-strain (pre-energy). Furthermore, the bulk constitutive equations should be adopted for the induced residual compressive stresses due to the free surface stress [38-40]. 7

In the present study, for linear elastic continua with surface elasticity, the material bulk is modeled according to the classical couple stress theory while the surface is modeled according to Gurtin-Murdoch surface elasticity theory. To capture the grains rigid rotation effects, the bulk strain energy density is formulated as a function of the infinitesimal strain and the rotation gradient tensor according to the classical couple stress theory. In addition, the surface strain energy density is represented by the infinitesimal surface strain and the difference between the trace of the Lagrangian surface finite strain and the trace of the infinitesimal surface strain. In the context of the classical couple stress theory [11, 41, 42], the material particles (grains) inside the material structure are modeled as rigid volumes that may rotate and translate. Therefore, to capture the particle rigid rotation effects, a rotation gradient tensor is introduced in the strain energy density function. Thus, the strain energy density is a function of the infinitesimal strain gradient

in addition to the rotation

, and it can be expressed, for isotropic continua, as follows [11, 43] (

where

)

represent the 3-D coordinates of the continuum.

(17) and

are the conventional Lame’s constants.

is a material length parameter introduced in the couple stress theory where

is a modulus of bending

and twisting have the dimension of force [41]; consequently, must have the dimension of length.

is a

dimensionless material parameter. For the positive definiteness of the strain energy density ,

,

(18)

If the material parameter

, the classical couple stress theory recovers the modified couple stress

theory [11] proposed by Yang et al. [44]. On the other hand, if

, the classical couple stress theory

reduces to the consistent couple stress theory [11] which is proposed by Hadjesfandiari and Dargush [45]. It should be mentioned that, for continua made of nanocrystalline materials, the effective Lame constants can be estimated using the proposed micromechanical model in section 2. An essential feature of Gurtin-Murdoch surface elasticity theory is that the bulk of the elastic material is treated as geometrically linear and the surface is formulated up to the second-order finite deformation of the surface strain [35]. Therefore, the surface strain energy density function depends on the infinitesimal surface strain, ( where

)

(

, and the squares of the surface displacement gradients as follows: )

(

)

(

represent the in-plane coordinates of the surface, and

)

(19) is a 3-D dummy index. The

displacement gradient term in equation (19) represents the difference between the trace of Lagrangian strain and the trace of the Eulerian strain. This term is introduced to capture the surface pre-strain effects [45].

8

From equation (17) the constitutive equations according to the classical couple stress theory, for isotropic continua, can be obtained as follows: (20)

( )

(

)

(21)

while the surface constitutive equations according to Gurtin-Murdoch surface elasticity theory can be obtained from equation (19) as follows: (

)

(

)

(22) (23)

where

and

are the surface Lame constants and

A couple stress tensor, tensor,

( ),

is the surface tension.

, is introduced in equation (21) in addition to the classical force stress

to capture the grains rigid rotation effects. Moreover, two surface stress tensors are

introduced to capture the surface effects where of-plane stress tensor, i.e.

is the in-plane surface stress tensor and

is the out-

represents the out-of-plane unit direction.

It should be noted that the introduced measures (material parameters), ,

,

, and

, are used to

relate the beam behavior to its size. These measures are related to the entire material size where their role decreases as the beam size increases. These measures are independent from the measures introduced in the framework of the micromechanical model. Thus, in the presented study, two independent sets of measures are used to model the mechanics of nanobeams made of nanocrystalline materials.

4. Nonlinear dynamics of electrostatically actuated nano-beams made of nanocrystalline materials The system under investigation consists of an actuated elastic beam, made of a nanocrystalline material, integrated with a platform at its free end, as shown in Figure 2. The beam resonator is suspended over a fixed electrode and an electrical field is applied in the gap between the fixed electrode and the attached platform. The beam is considered with a length , a width , and a thickness . The material structure of the nanocrystalline beam is represented as a three-phase composite with the interface is the matrix and the grains and the voids as nano-inhomogeneities, as shown in Figure 2. To model the material structure effects on the nonlinear dynamics of the considered actuated beam, the proposed micromechanical model (in section 2) is used to model the heterogeneity nature of the beam’s material structure. Moreover, to capture the beam size effects on its nonlinear dynamics, the actuated

9

beam is modeled according to the classical couple stress theory and Gurtin-Murdoch surface elasticity theory. The classical couple stress theory is used to reflect the effects of the grains rigid rotation on the dynamics of the actuated beam. In addition, Gurtin-Murdoch surface elasticity theory is used to capture the surface energy and surface tension effects of the beam free-surfaces on its dynamics. Next, the conventional Euler-Bernoulli beam theory is adopted for the incorporation of the grain rotation effects and the beam’s free surfaces energy and tension effects. Then, using Hamilton’s principle, the governing equations of motion are derived considering the electrostatic field affects the actuated beam. After that, the derived equations of motions are analytically solved to perform the static and the frequency analyses. Finally, a nonlinear reduced-order model is derived and utilized to study the nonlinear dynamics of the actuated beam.

Figure 2: A schematic of an electrostatically-actuated beam made of a nanocrystalline material.

4.1 . Derivation of the equation of motion and boundary conditions In this study, the beam shown in Figure 2 is modeled according to Euler-Bernoulli beam assumptions where the displacement fields are defined as follows: ( ) where

( )

(24)

are the displacements of a point (

) located inside the beam domain and

transverse displacement of a point on the centeriodal axis of the beam in the

plane.

The kinematical variables to be employed are the bulk infinitesimal strain tensor, (

), the rotation gradient tensor, ) i.e.

stands for the upper surface and

(

)

( ) is the

, and the surface strains,

(

)

(

stands for the lower surface. According to the defined

displacement fields, the non-zero kinematical components can be expressed as follows:

10

;

;

;

;

(25) (

To satisfy the non-classical surface balance conditions, field,

should be adopted for the out-of-plane stress field

beams. However, following Lu et al. [40],

)

. Usually,

, the force stress for thin plates and

is assumed varying linearly through the beam thickness

and satisfies the surface conditions: (

)

(

)

(26)

Consequently, for couple stress-based nanocrystalline beams with surface elasticity, the constitutive equations (20)-(23) can be rewritten as follows: [( (

)( (

)

(

))

]

)

(27)

(

)

(

)

where a homogenous surface is assumed to bound the beam; therefore,

and

are considered the

same at all beam surfaces. The term in the square brackets in the force-stress constitutive equations is added to correct the stress field for the out-of-plane stress field

. This term depends on

and

which are the transverse shear surface stresses at the upper and the lower surfaces of the beam, respectively, and it depends on

which is the Poisson’s ratio.

According to equations (24) and (25), the non-zero components of the bulk and surface stresses are derived as follows: (

(

)

)

(

)(

)

(

) (28)

; (

)

;

(

)

;

To derive the governing equation of the beam under consideration, Hamilton’s principle is utilized which states that: ∫( where

)

(29)

and , respectively, denote the kinetic energy and the potential energy of the actuated beam.

To form the strain energy of a couple stress-based beam with surface elasticity, the bulk strain energy density function (equation (17)) is integrated over the beam volume

, and the surface strain energy

density function (equation (19)) is integrated over the beam surfaces. The total potential energy of the 11

beam combines the strain energy (restoring energy) of the beam and the electrostatic energy (applied work) affecting the beam. The first variation of the total potential energy,

, of the electrostatically

actuated beam according to the classical couple stress and surface elasticity theories is defined as follows: ∫(

∮(

)

) (30)

(

( ))

(

(

)) ( )

The last term in equation (30) represents the electrostatic force that affects the beam at its free-end. denotes the permittivity of the dielectric vacuum between the two conductive electrodes, and

is the surface area of the platform, as presented in Figure 2. The

beam deflects because of the affected electrostatic energy which depends on the value of the applied voltage, , and the gap between the two electrodes,

. It should be mentioned that the electrostatic force

defined in equation (29) accounts for the fringing field effects. As shown in equation (30), the electrostatic force depends on the induced deflection of the beam’s free-end,

( ), where the increase in

the beam deflection is accompanied with an increase in the contribution of the electrostatic energy on the beam deflection. There is a limiting voltage value, known as the pull-in voltage, at which the electrostatic energy outweighs the restoring elastic energy of the beam and the beam deflection suddenly increases until it touches the fixed electrode. The value of the pull-in voltage is a critical parameter which should be accurately detected. The variation of the kinetic energy of the beam, neglecting the beam’s rotatory inertia, can be defined as follows: ∫

̇

̇

̈( )

(31)

The beam under consideration is integrated with a platform of mass,

. The function of the platform is

specified according to the application. For example, for biomass sensing and gas detection applications, the platform behaves as an attractor for the particles whose mass will be detected. This platform can be covered with antibodies to attract the matching biological cells, or it can be covered with a chemical compound that matches with a certain gas to attract. The variation of the potential energy in terms of the stress resultants can be obtained by substituting equations (24), (25), and (28) into equation (30):

12

∫ (

) (32) [(

where

)

(

)

]

is the electrostatic force at the beam free-end which is defined as: (

(

( ))

(

)) ( )

(33)

The stress resultants in equation (32) can be defined as follows (refer to the Appendix for the detailed derivations of the stress resultants): ∫

{

(

(

) )

[

(

]} )

∫ (34) ∮

(

)(

)

∮ Substituting equation (34) into equation (32), the variation of the total potential energy in terms of the beam deflection can be obtained. After that, by substituting the obtained equation and equation (31) into equation (29), the governing equation of the beam can be written in the following form: ̈

(35)

The considered beam is a cantilever beam as presented in Figure 2. Therefore, the following boundary conditions are defined: (

) (

) (

) (

where

(

(36)

)

)

is the platform inertia force that affects the beam at its free end. This force is defined as: ̈( )

(37)

13

The stiffnesses presented in equations (35) and (36) are defined as: ;

(38)

with (

(

) ) (39) )(

(

)

[

(

] )

It should be mentioned that the same beam resonator has been considered in [8, 26]. However, the beam was modeled according to the classical beam theory in [8] and according to the couple stress theory in [26]. Thus, dropping the surface terms

and

from equations (35) and (36) reduces the proposed

beam model to the couple stress-based beam model [26]. Also, dropping the couple stress term,

,

along with the surface terms reduces the model to the classical Euler-Bernoulli beam model [8]. In the next analyses, the governing equations (35)-(39) are normalized by using the following nondimensional parameters: ( )

( )

;

;

(40)

√

Consequently, the normalized governing equation and associated boundary conditions of the beam are given by: ̈ (

(41)

) (

)

(

)

(

)

(

(42)

)

where (

(

( ))

(

)) ( )

(43)

̈( ) where ;

(44)

;

14

4.2 . Relationship between static equilibrium and applied DC voltage In most MEMS and NEMS, the elastic resonator is forced to deflect by a DC bias and then it is actuated to vibrate about its static equilibrium by an AC harmonic voltage. Thus, the applied voltage, ( )

, is a combination of

amplitude of the applied AC voltage while

which is a constant DC voltage and

which is the

is the excitation frequency. Usually, the amplitude of the

applied DC voltage is much higher than the amplitude of the AC voltage; therefore, it is acceptable to neglect the higher-order term of the AC voltage [5], i.e. ( )

.

Because of the applied DC voltage, it is essential to detect the static equilibrium position of the elastic beam which will directly affect its dynamic behavior. By dropping the time dependent terms from equations (40) and (41), the governing equation and the associated boundary conditions of the static behavior of the beam can be written as: (45) ( ) ( ) ( ) (

(46)

( ) )

where (

(

( ))

(

( )

)) with

(47)

where the electrostatic force only depends on the constant DC voltage

.

The solution for equation (45) for the static deflection is expressed as: ( )

(48)

By applying the given boundary conditions in equation (46), the constants

,

,

, and

are obtained.

Substituting the obtained constants into equation (48) gives the static deflection distribution along the beam length as follows: ( )

[

]

(

)[

]

It should be mentioned that the electrostatic force, ( ). Thus, to determine the static deflection

(49)

depends on the beam’s free-end deflection

( ), the free-end beam deflection

( ) is needed to

form the electrostatic force. Therefore, substituting equation (49) into equation (47) and putting the static deflection of the beam can be related to the applied DC voltage through the following relation:

15

,

( ))

(

√

(

( ) ( )))

(

(50)

with To recover the solutions for the classical model, with

is simply set to zero (or

) and

is replaced

in equations (49) and (50). It should be mentioned that similar equations are previously derived

by Aboelkassem et al. [8] where the beam is modeled based on the classical Euler-Bernoulli beam theory neglecting the effects of the couple stress, the surface, and fringing field. In addition, in a previous work, we considered a similar system and derived the solutions considering only the couple stress effects for a nanocrystalline beam [26] neglecting surface and fringing field effects. The merit of the present model when compared to the previous models is the inclusion of both the couple stress and the surface effects on the performance of the beam resonator considering fringing field effects.

4.3 . Relationship between natural frequencies and applied DC voltage To determine the natural frequencies and associated mode shapes of the electrostatically-actuated beam, the beam’s deflection is decomposed in the following form: (

)

( )

( )

(51)

where ( ) denotes the mode shape and

is its corresponding natural frequency.

Substituting equation (51) into equations (49) and (50) and considering only the linear part of the electrostatic force leads to the following equation of motion and boundary conditions which describe the dynamics of the beam about its static position ( )

( )

( ):

( )

(52)

( ) ( ) ( ) ( )

(53) ( )

{

( )

where [

(

( ))

(

(

( ))

)]

(54)

The general solution of equation (52) can be expressed as follows: ( )

(̅̅̅ )

(̅̅̅ )

(̅̅̅ )

where

16

(̅̅̅ )

(55)

̅̅̅

√ √ (56)

̅̅̅

√ √ The coefficients

,

,

, and

in equation (55) are then obtained by applying the boundary

conditions given in equation (53). By utilizing the first two equations of the boundary conditions (53), equation (55) can be rewritten as: ( )

(

(̅̅̅ )

(̅̅̅ ))

̅̅̅ ̅̅̅

(̅̅̅ )

(

(̅̅̅ ))

(57)

Then, using the last two equations of the boundary conditions (53), the following characteristic equation is obtained: ̅̅̅

{(

(̅̅̅

̅̅̅)

̅̅̅ ̅̅̅

̅̅̅

(̅̅̅ ̅̅̅

̅̅̅)

̅̅̅

̅̅̅) (̅̅̅

̅̅̅

̅̅̅

̅̅̅)} (58)

{((̅̅̅

̅̅̅) ̅̅̅) (̅̅̅

̅̅̅ ̅̅̅

(̅̅̅̅

̅̅̅ ̅̅̅ ̅̅̅

̅̅̅)

̅̅̅

̅̅̅)}

Solving equation (58) gives an infinite number of natural frequencies

where stands for the mode

number. After that, the mode shape function for each mode is determined as: ( )

((

(̅̅̅ )

(̅̅̅ )) (59)

(

̅̅̅ ̅̅̅

̅̅̅ ̅̅̅

̅̅̅

̅̅̅

̅̅̅ ̅̅̅

̅̅̅

)(

where ̅̅̅ and ̅̅̅ can be obtained for each mode

(̅̅̅ )

̅̅̅ ̅̅̅

(̅̅̅ )))

from equation (56).

denotes the amplitude of the

shape function. To recover the solutions for the couple stress-based beam model,

is simply set to zero. This makes

√ and, hence, equation (58) reduces to the one presented in [26].

4.4 . Reduced-order modeling To derive the reduced-order model of the present system, the Galerkin method and Euler-Lagrange equations are used. First, the total potential energy of the system is derived by integrating the bulk strain

17

energy density function (equation (17)) and the electrostatic force over the beam volume and by integrating the surface strain energy density function (equation (19)) over the beam surfaces as follows: (60) where the beam material bulk strain energy, (

∫ {(

, is derived in the form:

) )

[(

)( (

)

(

))] (61)

}

∫{

(

[(

)

)

](

)

(

) }

The surface tension term in equation (61) is added to satisfy the non-classical surface balance conditions (refer to equation (28)). The surface strain energy, ∫ ∮{ (

)

(

[(

)

∫{

) (

, is also derived as follows:

(

)

) ]} (

)

(62) )(

[(

)] (

) }

Consequently, the total potential energy of the system is derived as follows: ∫{ (

)

(

)

} (63)

(

( )

(

)) ( )

Second, the total kinetic energy of the system is derived in the form: ∫

̇

( ̇ ( ))

(64)

Third, the Lagrangian can be expressed in the following form:

18

∫

( ̇ ( )) ̇

∫

(

∫

)

(

) (65)

∫

(

(

( )

)) ( )

Finally, the equation of motion can be developed in the normalized form by utilizing the nondimensional parameters given in equation (40) and by approximating the deflection according to the following form: (

)

( )

( ) ( )

∑

(66)

( ) denotes the ith linear undamped mode shape of the beam and

where

( ) represents the ith

generalized coordinate. Applying Euler-Lagrange equations, the following multi-mode equations of motion are obtained: ̈

∫

∫

[(

(

(

( ) (

( )

( ) ( )

(

( ) )

( ) )

)

(67)

))]

where stands for the equation number and is a dummy index stands for the mode number from

.

The mass matrix and the stiffness matrix in equation (67) are defined as: ∫

( )

( ) (68)

∫

∫

By selecting the amplitude

such that the equivalent mass

is unity and by expanding the last two

terms of equation (67) using Taylor’s expansion, equation (67) can be written as follows: ̈ where

̇

(69) denotes the nonlinear terms of the electrostatic force which can be defined as follows:

19

(

)

( )

[(

( )

(

( ) )

) (70)

( where

(

( ) (

( )

( ) )

̅

))]

is added in equation (69) as a nondimensional damping parameter, i.e.

where

is the

damping factor of the beam. The coefficients in equation (69) are defined as: (

̅

(

( ))

(

( )

) ( ))

(

( ))

(

( ))

(

)

( )

( ) (71)

̿

(

( ))

( (

(

( ))

(

) ( ))

(

( ))

)

( )(

( ))

( )

The derived equation of motion (equation (69)) is then can be solved numerically to study the nonlinear dynamics of the actuated beam.

5. Size and material structure effects on the static and dynamic performance of the mechanical resonator The effects of the beam material structure along with the beam size on the static behavior, natural frequencies, and the nonlinear dynamics of electrostatically-actuated beams made of nanocrystalline materials are studied. To this aim, the derived reduced-order model in equation (69) is numerically solved when considering only one mode in the Galerkin discretization. A nano-beam made of a nanocrystalline silicon with thickness

, length

, and width

is considered to investigate the

beam size and material structure effects on its static and dynamic behaviors. The beam is equipped with a platform with

and

. The nondimensional total mass of the platform is considered

. The initial gap between the beam and the fixed electrode is set at

. The three-phase

micromechanical model presented in section 2 is used to estimate the effective Lame constants of the beam bulk. The material parameters of the nano-grains and the interface are given in Table 1. The Sinano-grains are considered with the conventional material parameters of Si-crystals. The material parameters of the interface are estimated using equation (14).

20

Table 1: The beam geometry and material parameters of the considered actuated beam. Beam geometry Beam free surface parameters ( Length scale parameter (nm)

)

; Material parameters , , ,

Phase-1 (Interface) Phase-2 (Si-nano-grains) Phase-3 (nano-voids) Surface parameters of the grains and voids ( )

[46] [2]

;

,

[31] [31]

. .

[46]

;

To investigate the impacts of the grains size on the static behavior, natural frequencies, and dynamics of the electrostatically-actuated nanocrystalline silicon beam, different sizes and volume fractions of the grains are considered. Three distinct sizes for the Si-grains (

) are utilized. For

nanocrystalline materials, the interface thickness and volume fraction depend on the grain size [25-28, 31, 32]. Therefore, an interface thickness

is considered for Si-grain radius

for Si-grain radius

, and

for Si-grain radius

, .

The Si-grains volume fractions are calculated using equations (15) and (16). The beam surface energy and surface tension contribute to the beam static and dynamic behaviors through the additional stiffness rigidities in the continuum model (defined in equations (35)). On the other hand, the surface energy of the grains and the voids inside the beam material structure contributes to the beam static and dynamic behaviors through the effective elastic moduli of the beam bulk material. The grains and the voids surface parameters along with the beam free surface parameters are presented in Table 1. The nano-voids inside the material structure are assumed having average radii equal those of the nano-grains. Moreover, for the sake of simplicity, the voids are assumed having the same surface parameters of the grains. To capture the grain rigid rotation effects, the classical couple stress theory is used where two additional material parameters

and

are presented in the beam constitutive equations. However, the

deflection-based equilibrium equation of the beam (equation (35)) depends only on the length scale parameter . Thus, the parameter

has no contributions to the beam deflection. The value of the scaling

parameter, , is reported in [2] for polycrystalline silicon by

.

Table 2 shows the effective bulk modulus, , shear modulus, , and density of the nanocrystalline silicon beam for different grain sizes and various values of the porosity percent. An increase in the interface thickness and a decrease in the effective moduli and density of the nanocrystalline silicon are observed when decreasing the grain radius from 100 nm to 20 nm. On the other hand, considering the grains’ surface energy effects, an increase in the effective shear modulus is observed when intensively decreasing the grain radius to 0.5 nm and when

. The reason for this unique behavior can be

attributed to the increase in the inhomogeneities surface to bulk ratio [25]. The table reflects both 21

hardening and softening effects of the grains’ and the voids’ surface energy where the elastic moduli may decrease or increase depending on the inhomogeneities size and their volume fractions.

Table 2. The effective material properties of nanocrystalline silicon beam for different grain sizes. 1

(nm)

100 20 0.5 100 20 0.5

(nm) 0%

1.02

10 %

1.02

0.97 0.6 0.5 0.873 0.54 0.45

Effective bulk modulus (GPa) WS2 NS3 61.72 61.82 37.34 37.48 28.09 33.00 45.55 45.68 29.12 29.45 6.57 26.28

1

The grains and the voids have the same average radius, i.e. WS: considering the grains’ surface energy effects. 3 NS: neglecting the grains’ surface energy effects. 4 The effective density ( )

Effective shear modulus (GPa) WS NS 75.33 75.38 43.54 43.6 35.85 38.37 52.062 52.13 32.51 32.72 58.1 29.24

Effective density4 (kg/m3) 2291.1 2181.7 2152.2 2062 1963.55 1936.9

.

2

Next, the impacts of the grain size, the grain rigid rotation, and the beam surface energy and surface tension on (1) the static pull-in, (2) the natural frequencies, and (3) the nonlinear dynamics and the dynamic pull-in of nano-beams made of nanocrystalline materials are deeply investigated. The static pullin can be defined as the tendency of the nanobeam resonator to suddenly descend touching the fixed electrode when it is subjected to a constant electrostatic potential. The needed value of the constant electrostatic potential,

, that causes the beam’s pull-in is known as the static pull-in voltage. On the

other hand, when the beam is subjected to a variable electrostatic potential in addition to the constant electrostatic force, another force that depends on the forcing frequency affects the beam, and it can enforce the beam to pull-in. At this instant, the beam can pull-in faster, at electrostatic potential levels lower than the static pull-in potentials. The pull-in of the beam that depends on its dynamic force can be defined as the dynamic pull-in. To account for the beam material structure effects, four different cases are considered. In the first case, the beam is modeled using the conventional material parameters of silicon (

,

,

) where the effects of the heterogeneity of the material structure and the grain size are neglected. In the three other cases, the beam is modeled according to the proposed micromechanical 22

model accounting for the grain size effects where three different sizes of the grain are considered (

,

, and

). Moreover, to account for the beam size effects, each one

of the considered four case studies is solved according to the classical beam model (CM), i.e. couple stress and surface parameters are neglected, the surface elasticity model (SE), i.e. no couple stress, the couple stress model (CS), i.e. no free surface effects, and the proposed model (CS & SE), i.e. the model accounts for both couple stress and surface effects.

5.1 . Size and material structure effects on the static pull-in instability of the resonator Figure 3 shows the influences of the beam size and the beam material structure on the static deflection and the static pull-in of the considered nanocrystalline silicon beam. The beam free-end static deflection, ( ), as a function of the applied

voltage is plotted. The results of the electrostatically actuated

beam modeled using the conventional material parameters of silicon (Si) (

,

,

) are presented in Figure 3(a). In Figures 3(b)-(d), the beam is modeled considering the heterogeneity nature of the material structure and the grain size effects for different porosity percent (

and

) and when the grain and the void radii are (Figure 3(c)), and

(Figure 3(b)),

(Figure 3(d)).

The horizontal lines in the plotted figure are drawn to separate the stable static solutions from the unstable ones. The voltage values at the inflection points of the curves (i.e. the points at the ends of the stable branches and the start of the unstable ones) are the static pull-in voltages,

, of the beam. The

deflection corresponding to the pull-in voltage is known as the pull-in deflection which, as indicated in the figures, nearly equals one-third the initial gap,

, between the beam and the fixed electrode.

Inspecting the plotted curves in Figure 3, it clear that the beam material structure and the beam size have significant effects on the beam static deflection and static pull-in. In fact, when

, a decrease

in the inhomogeneities size is accompanied with a decrease in the beam stiffness and, hence, a decrease in the beam pull-in voltage. On the other hand, when

, an increase in the beam pull-in voltage is

shown when decreasing the average radius of the grains and the voids from

to

. To

illustrate the reasons for these behaviors, the hybrid contributions of the inhomogeneities inside the beam material structure should be explained. The interface inside the beam material structure always affects the beam stiffness with a softening mechanism. However, the surface energy of the inhomogeneities (grains and voids) may stiffen or harden the beam deepening on the surface material parameters, the inhomogeneities size, and their volume fractions [25, 26, 31, 32]. Also, decreasing the inhomogeneities size increases the contribution of their surface energy. The hardening effect that is observed when decreasing the grains and the voids size from

to

at

is due to the increase of the

inhomogeneities surface energy which, in this case, has a hardening effect. It can be concluded that to

23

accurately model electrostatically-actuated beam, the beam material structure should be modeled accounting for these phenomena. Neglecting one of these contributions can result in a wrong estimation of the static pull-in of the resonator. To figure out the beam size effects, the results according to the classical model (CM), the surface elasticity model (SE), and the couple stress model (CS) are compared to the proposed model (SE & CS). The plotted curves in Figure 3 show an increase in the limiting static pull-in voltage with the inclusion of the surface and the couple stress effects into the mathematical model. Thus, both the grain rotation and the beam surface energy stiffen the beam. Also, the figure shows the coupled effect of the beam material structure and beam size, as presented in Figure 3(c) and 3(d). The plots in Figure 3 address the importance of modeling the beam size and material structure effects. Neglecting any one of these essential phenomena during the design process of MEMS and NEMS nanodevices may result in under- or over-estimations of the beam behaviors which causes malfunctions of the designed device. To investigate the impacts of the fringing field on the beam static deflection and static pull-in, the static pull-in voltage values are presented in Table 3 for different grain sizes and according to the different models when considering/neglecting fringing field effects. It is clear that the fringing field decreases the beam’s static pull-in voltage values for all considered cases. Table 3: Fringing field effects on the nanocrystalline beam’s static pull-in. Static pull-in (nm) 100 20 0.5 1 2

CM WF1 NF2

SE WF NF

CS WF NF

SE & CS WF NF

CM WF NF

SE WF NF

CS WF NF

SE & CS WF NF

13.42

13.98

16.92

17.63

19.19

20.00

21.79

22.71

11.3

11.77

15.28

15.93

16.06

16.73

19.08

19.89

10.29

10.72

14.55

15.16

14.65

15.27

17.92

18.67

8.97

9.35

13.63

14.21

12.72

13.25

16.37

17.06

9.18

9.56

13.77

14.35

13.19

13.74

16.74

17.44

9.66

10.06

14.1

14.7

15.45

16.1

18.57

19.36

Considering fringing field effects. Neglecting fringing field effects.

24

(a)

(b)

(d)

(c)

25

Figure 3: The beam free-end nondimensional static deflection as a function of the applied fringing field effects). (a) The conventional silicon beam with ( nanocrystalline silicon beam with

,

voltage (considering

,

, (c) a nanocrystalline beam with nanocrystalline beam with

), (b) a , and (d) a

.

5.2 . Size and material structure effects on the fundamental natural frequency of the resonator To investigate the effects of the beam material structure and the beam size on its natural frequencies, the nondimensional fundamental frequency of the beam as a function of the applied

is plotted in

Figure 4 for different grain sizes, different porosity percent, and according to the different models. The results of the beam model accounting for the couple stress and/or the free surface effects are compared to those of the classical beam model. The plotted curves show the high contributions of both the grains rigid rotation and the beam’s free surface effects. When the beam is only modeled according to the classical couple stress theory, it has a maximum nondimensional fundamental natural frequency as much as the one of the beam modeled via the classical theory. The reason is that the nondimensional frequencies, plotted in Figure 4, are normalized to the modified beam stiffness,

. However, the

incorporation of the surface elasticity increases this maximum limit. In fact, this can be attributed to the beam free-surface tension and the nondimensional surface stiffness

in equations (56) and (58). The

beam surface tension directly affects the induced nondimensional frequency of the beam via the nondimensional surface stiffness

. The contribution of this nondimensional surface stiffness decreases

with the increase in the beam’s modified stiffness,

. The contribution of

on the beam’s frequency

decreases with the increase in the beam size and/or the incorporation of the couple stress model. The couple stress and the surface parameters increases the value of the modified stiffness

. Thus, according

to the plotted curves, the nondimensional frequencies obtained according to the surface elasticity model are higher than the frequencies obtained according to the proposed model. The plotted curves in Figure 4 reflect the significant influences of the beam material structure on its natural frequencies. It is interesting to observe the effects of the grain size on the contributions of the couple stress and the surface energy. The obtained maximum nondimensional natural frequencies according to the couple stress are not affected with the grains size. However, decreasing the grain size may decrease (or increase) the beam modified stiffness

- depending on the grain size, the porosity

percent, and the grain volume fraction – and accordingly it increases (or decreases) the contribution of the nondimensional surface stiffness

and, hence, the induced frequencies of the beam.

According to the plotted curves in Figure 4 and the previous discussion, accounting for the coupled effects of the beam material structure and the beam size is important when modeling and designing MEMS/NEMS nano-devices.

26

(a)

(b)

(d)

(c)

27

Figure 4: The beam nondimensional fundamental frequency as a function of the applied nondimensional mass of the platform,

voltage with a

, (considering fringing field effects).

5.3 . Beam size and material structure effects on the nonlinear behavior and dynamic pullin instability of the resonator To investigate effects of the beam size and material structure on its nonlinear response and dynamic pull-in instability, the phase portraits and the frequency-response curves are plotted in Figures 5, 6, 7, and 8 for different material structure parameters and using different models. 5.3.1. Phase Portraits Effects of the initial conditions and the amplitude of the applied AC voltage,

, on the nonlinear

responses and the dynamic pull-in of the beam are presented in Figures 5, 6, and 7. The plotted curves in these figures show the phase portraits of the nano-beam actuated near its dynamic pull-in and modeled according to the proposed model for the damped case when the damping factor beam is deflected with a constant bias Table 3) and actuated with a

(where (in Figure 5),

is set equal to 0.01. The

is the static pull-in voltage as obtained in (in Figure 6), and

(in Figure 7).

Inspecting the plots in Figures 5, 6, and 7, it is clear that the beam reflects different behaviors depending on the AC voltage value. For zero time-dependent (variable) electrostatic force (

), the nano-beam shows a stable focus

and a saddle point, as shown in Figure 5. Various trajectories that describe the different motions of the actuated beam are obtained for different initial conditions. The red-dotted trajectories represent the periodic oscillations of the beam toward the stable focus (the static beam deflection,

( )). The actuated

beam shows a stable behavior for the initial conditions which leads to periodic trajectories. The solid-blue and the dashed-green trajectories describe the beam’s unbounded motion where the beam reflects unstable behavior for initial conditions which gives these trajectories.

28

Figure 5: Phase portraits describing the stable and the unstable behaviors of the nanocrystalline beam when ,

, and

. (Solved using the proposed model (SE & CS) where ,

,

,

, and considering fringing field).

Figure 6 shows different responses of the actuated nano-beam when it is subjected to AC actuation (

). For small initial conditions, the beam has double-periodic oscillations where the solid-red

stable limit cycle is shown which goes around twice before closing. It should be mentioned that there is a bifurcation point at which the simple limit cycle (with one-period oscillation) changes to the doubleperiod limit cycle. For higher initial conditions, the nano-beam may oscillate twice before its divergent/unbounded motion as presented with the black trajectories. Also, the beam may show unbounded motion which represented by the green trajectories. , no limit cycles are shown in the beam’s phase

When the applied AC voltage is set at

portrait, as presented in Figure 7. The red trajectories show many period-doubling before the divergent/unbounded motions. This phase portrait indicates the initiation of the dynamic pull-in of the actuated beam where the beam reflects no periodic oscillations through its time history for all initial conditions.

29

Figure 6: Phase portraits describing the stable and the unstable behaviors of the nanocrystalline beam when ,

, and

. (Solved using the proposed model (SE & CS) where ,

,

,

, and considering fringing field).

Figure 7: Phase portraits describing the stable and the instable behaviors of the nanocrystalline beam when ,

, and

. (Solved using the proposed model (SE & CS) where ,

,

,

, and considering fringing field).

To show the importance of considering the coupling between the couple stress and the surface effects when modeling the nonlinear dynamics of actuated nano-beams, we plot in Figure 8 the phase portraits which are obtained according to the proposed model and compared to those obtained accounting only for the couple stress or the surface effects. In these three cases, a beam with nano-inhomogeneities of an average radii,

and 10% porosity and subjected to

30

is

considered. The phase portrait represents a center (or a stable focus) and a saddle point when modeling the beam accounting for both the couple stress and the free-surface effects, as shown in Figure 8(a). The red solid trajectories passing through the saddle point are known as the separatices which separate regions of initial conditions that lead to periodic motions around the center or toward the stable focus (the solid blue trajectories) from regions of initial conditions that cause unbounded motions (the black dashed trajectories). The increase in the distance between the saddle point and the center (or stable focus) indicates the increase in the system stability. The distance between the center (or stable focus) and the saddle point decreases when considering only the free-surface energy effects, as shown in Figure 8(b). When modeling the beam accounting only for the couple stress effects, the saddle point and the center (or stable focus) merge in a single point converting the phase portrait to the one shown in Figures 8(c). The point of merge is an inflection point that indicates the initiation of the static pull-in. These results are expected because the static pull-in voltage is strongly dependent on the selected model, as shown in Figure 3(c). Indeed, the static pull-in voltage values when accounting for the couple stress and surface energy or only surface energy are less than the applied DC voltage (

). On the other hand, when

only considering the couple stress effects, the static pull-in voltage is exactly equal to

, as

presented in Table 3. The presented phase portraits reflect the importance of modeling the nonlinear dynamics of actuated nano-beams accounting for both the couple stress and the beam free-surface effects in order to design accurate nano-devices.

(a)

31

(b)

(c) Figure 8: Phase portraits for the beam molded according to (a) the proposed model, (b) the surface elasticity theory, and (c) the couple stress theory. (

,

,

, considering fringing

field effects).

The plotted curves in Figures 9 and 10 show the beam material structure and the beam size effects on its nonlinear dynamics. In Figure 9, the obtained separatices (the trajectory that separates the stable region from the unstable one in the phase portrait) when modeling the beam according to the proposed model are compared to those obtained accounting only for the couple stress or the free-surface effects. Because the beam is actuated with 85% of its static pull-in voltage all separatices in Figure 9 share the same center (or stable focus point) and the same saddle point. It follows from Figure 9 that the couple stress model gives the same separatrix as the one obtained using the classical model. However, because of the beam freesurface tension, the separatices obtained according to the proposed model or the surface elasticity theory

32

covers wider range of initial velocities that give stable periodic motions of the beam. According to Figure 9, the beam free-surface energy has a great contribution on its stable and the unstable behaviors. Figure 10 shows the separatices for different grain and voids radii when the beam is actuated with a constant value of

. Inspecting this figure, it is noted that different saddle points and

centers (stable focuses) are obtained for distinct grains and voids sizes. This is predicted because of the significant effect of the couple stress and surface elasticity on the static pull-in of the nano-beam, as presented in Figure 3. The results of the nanocrystalline silicon are compared to the conventional coarsegrained silicon. The coarse-grained silicon beam reflects a center (stable focus) and a saddle point. The distance between the center (stable focus) and the saddle point decreases when decreasing the grain size to nano-scale till an inflection point appears at

. On the other hand, a reverse trend is

observed when decreasing the grains and the voids average radius to 0.5 nm. According to Figure 9, the inhomogeneities surface energy has a great contribution on the stable and the instable behaviors of actuated beams.

(b) (a) Figure 9: Couple stress and surface effects on the nonlinear dynamics of the actuated beam for (a) undamped case, , and (b) the damped case,

, i.e.

.(

considering fringing field effects).

33

,

,

,

(b) (a) Figure 10: Grain size effects on the nonlinear dynamics of the actuated beam for (a) undamped case, the damped case,

i.e.

, and (b)

. (The beam is modeled according to the proposed model (SE & CS)

accounting for the fringing field effects.

,

).

5.3.2. Frequency response curves In Figure 11, the frequency-response curves obtained using the different models are presented. The plotted curves in this figure reflect the stable solutions for distinct forcing frequency, i.e.

.

The obtained curves are bent to the left because of the negative effective nonlinearity of the system (see equation (69)) indicating the softening behavior of the beam due to the nonlinear electrostatic force. It should be mentioned that, in the present study, there are many different parameters that influence the effective nonlinearity of the system including the couple stress, the surface parameters, and the beam material structure parameters. Inspecting Figure 11, horizontal gaps exist between the ends of the left and the right branches of the frequency-response curves. This indicates that the beam reaches the dynamic pull-in when actuated at the forcing frequencies cover these gaps. At these ranges of frequencies, no stable solutions exist for the beam dynamics. Thus, the left branches of the frequency-response curves end with bifurcation points beyond which the dynamic pull-in of the system occurs. On the other hand, other bifurcation points exist at the ends of the right branches which correspond to the stability limits defined by the unstable deflections (saddle points). The horizontal line in Figure 11 defines the stability limit that corresponds to the unstable static deflection (the saddle point when

). In addition to the frequency-

response curves, the limit cycles at different forcing frequencies are shown. Stable one-period limit cycles are found for the considered frequency range. Depending on the forcing frequency, various limit cycles are

obtained.

Moreover,

different

limit

cycles

are

obtained

when

modeling

the

beam

considering/disregarding the couple stress and/or the free-surface effects. As previously demonstrated, an underestimation or overestimation of the amplitude of the nano-beam can be obtained when considering

34

different models. This indicates the importance of the accurate modeling of nano-beams for sensing or actuation applications.

Figure 11: Frequency-response curves obtained using the different models for a nanocrystalline beam with and

(

,

,

, considering fringing field effect, i.e.

).

The plots in Figure 12 show the frequency-response curves for different grain and voids sizes of a nanocrystalline beam modeled according to the proposed model (SE & CS). Different behaviors are reflected in this figure where the beam may be featured with the dynamic pull-in, or it may have stable solutions for all the frequency range. Furthermore, various dynamic pull-in frequencies (band of frequencies) are obtained for distinct sizes of the inhomogeneities. In addition to the frequency-response curves, the limit cycles of the nanocrystalline beam for the distinct sizes of the inhomogeneities when are presented in Figure 13. The coarse-grain silicon-based beam reflects two limit cycles for two distinct sets of initial conditions. On the other hand, the nanocrystalline beams show only one limit cycle for all initial conditions. Inspecting the plots in Figure 12 and 13, it is concluded that the grain size significantly affects the strength of the softening behavior, the dynamic response or amplitude of the system, and the band range of the dynamic pull-in instability. The plotted curves in Figures 11, 12, and 13 demonstrate the great impacts of the beam material structure and the beam size on the nonlinear dynamics of electrostatically actuated beams in different micro/nano-scale applications. In addition, these figures demonstrate the importance of modeling nanobeams accounting for both the beam size and material structure effects.

35

Figure 12: Frequency-amplitude curves obtained using the proposed model (SE & CS) for a nanocrystalline beam with different grain and voids sizes and

(

,

fringing field effect, i.e.

,

, considering

).

Figure 13: The limit cycles for a nanocrystalline beam with different grain and voids sizes obtained using the proposed model (SE & CS) when

,

,

field effect, and

36

, ).

, considering fringing

6. Conclusions An accurate modeling has been carried out for electrostatically actuated nanobeams. In addition to account for the effects of the inhomogeneities surface energy and the interface through a micromechanical model, both the classical couple stress and Gurtin-Murdoch surface elasticity theories have been utilized to develop a size-dependent Euler-Bernoulli beam model. For the first time, effects of the beam material

structure along with the beam size on the nonlinear dynamics and pull-in instability behaviors of electrostatically actuated nano-beams were intensively studied. The results showed that an electrostatically actuated nanocrystalline silicon beam is strongly affected by the beam size and surface effects. It was demonstrated that to accurately model the mechanics of nano-beams in micro-/nano-scale applications, both the couple stress and the free-surface effects should be simultaneously considered. Moreover, the effects of the grain size, the interface, and the porosity were well-presented in order to determine the importance of considering the material structure effects when modeling nano-beams. By representing the beam’s deflection and the beam’s natural frequency as functions of the applied static voltage, it was revealed that the actuated beam can support different electrical potentials depending on its material structure and its size. Thus, neglecting one of the considered phenomena through the present study can result in wrong estimations of the static pull-in of the actuated nano-beams. In addition, the effects of the beam size and material structure on the beam’s nonlinear dynamics and dynamic pull-in instability were investigated. Different phase portraits trajectories that describe different dynamical behaviors were obtained depending on the initial conditions, the grain size, and the utilized beam model. Moreover, to study the effects of the forcing excitation frequency on the beam’s nonlinear dynamics, the frequency-response curves and the limit cycles were deeply investigated where different responses can be obtained depending on the beam material structure and the beam size. Based on this performed analysis, it was shown that it is necessary to model both the beam material structure along with its dynamics accounting for it size effects when designing nano-beams for micro-/nano-scale applications.

Appendix The surface stress components can be derived from equation (27) in the form: (

)

;

(A.1)

To derive the surface stress resultants presented in equation (34), the following line integrals should be defined with referring to Figure A.1:

37

∮

∫

∫

|

∫

∫

|

| (A.2)

∮

∫

∮

∫

∫

where

∫

|

∫

∫

∫

|

|

∫

|

|

is the -component of the unit vector.

With the aid of the defined line integrals, the surface stress resultants can be defined as follows: ∮

∮

(

∮

∮

)(

(

)

)

(A.3)

∮

Figure A.1: A schematic representations of the contour lines to derive the line integrals in equation (A.2).

38

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Highlight 

A nonlinear model for electrostatically actuated beams made of nanocrystalline materials is proposed.



Two sets of measures are incorporated in the context of the proposed model.



All the measures presented in the micromechanical model are related to the material structure.



A size-dependent Euler-Bernoulli beam model is developed to model the mechanics of the nano-beam.



The impacts of the material structure and size on the dynamics and pull-in instability are studied.

42