Oscillations of functionally graded microbeams

International Journal of Engineering Science 110 (2017) 35–53

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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

Oscillations of functionally graded microbeams Mergen H. Ghayesh a,∗, Hamed Farokhi b, Alireza Gholipour a a b

School of Mechanical Engineering, University of Adelaide, South Australia 5005, Australia Department of Mechanical Engineering, McGill University, Montreal, Quebec H3A 0C3, Canada

a r t i c l e

i n f o

Article history: Received 7 September 2016 Revised 10 September 2016 Accepted 18 September 2016

Keywords: Microbeam Functionally graded material Shear-deformable Third-order shear-deformation Motion characteristics

a b s t r a c t The size-dependent oscillations of a third-order shear-deformable functionally graded microbeam are investigated taking into account all the longitudinal and transverse displacements and inertia as well as the rotation and rotary inertia. The modified couple stress theory along with the Mori–Tanaka homogenisation technique is employed to develop formulations for the elastic potential energy as well as the kinetic energy of the system. The energy of the system is balanced by the work of a harmonic excitation force via an energy method based on Hamilton’s principle, yielding the size-dependent coupled nonlinear continuous models of the functionally graded system for the longitudinal and transverse displacements as well as the rotational motion. A model reduction procedure, on the basis of a weighted-residual method, is applied without any simplifications on the displacement/inertia/rotation. This operation yields three sets of second-order reduced-order coupled model of the functionally graded system for the longitudinal, transverse, and rotational motions. These reduced-order models are solved via use of a continuation method in order to construct the nonlinear frequency-response and force-response curves of the functionally graded system. A linear analysis is also performed by means of an eigenvalue extraction method in order to determine the linear natural frequencies of the system. It is shown that the material gradient index as well as the length-scale parameter of the functionally graded system affects the system dynamics substantially. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Continuous microelements such as microbeams and microplates are the core elements of many microelectromechanical systems (MEMS) (Asghari, Kahrobaiyan, & Ahmadian, 2010; Baghani, 2012; Dehrouyeh-Semnani & Bahrami, 2016; Farokhi & Ghayesh, 2016a; Farokhi & Ghayesh, 2016b; Ghayesh & Farokhi, 2016a; Ghayesh, Farokhi, & Amabili, 2013a; Li and Hu, 2016; Rahaeifard, 2016). Functionally graded microscale continuous elements (Lü, Lim, & Chen, 2009) are a new class of electromechanical machine components which are gaining high interest mainly due to this fact that they are resistant to thermal and mechanical loadings at the same time; functionally graded continuous elements are a composite systems where ceramic and metal are combined using powder metallurgy technique – the ceramic is resistant to high temperature loads while the metal component is resistant to thermally induced fractures. There are many experimental investigations in the literature which show that the dynamical behaviour of continuous microelements is highly size-dependent (Akgöz & Civalek, 2013; Akgöz & Civalek, 2011; Dehrouyeh-Semnani, 2014; Dehrouyeh-Semnani, BehboodiJouybari, & Dehrouyeh, 2016; Ghayesh, Amabili, & Farokhi, 2013b; Hosseini & Bahaadini, 2016; Kahrobaiyan, Rahaeifard, Tajalli, & Ahmadian, 2012; ∗

Corresponding author. E-mail address: [email protected] (M.H. Ghayesh).

http://dx.doi.org/10.1016/j.ijengsci.2016.09.011 0020-7225/© 2016 Elsevier Ltd. All rights reserved.

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M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

Karparvarfard, Asghari, & Vatankhah, 2015; Kong, Zhou, Nie, & Wang, 2008; Mojahedi & Rahaeifard, 2016; Shafiei Kazemi, & Ghadiri , 2016a; Shafiei, Kazemi, & Ghadiri, 2016b; Taati, 2016); a phenomenon which cannot be predicted theoretically by the classical continuum mechanics – this paper employs the modified couple stress theory (Dai, Wang, & Wang, 2015; Dehrouyeh-Semnani, Dehrouyeh, Torabi-Kafshgari, & Nikkhah-Bahrami, 2015; Farokhi & Ghayesh, 2015a; Farokhi & Ghayesh, 2015b; Farokhi, Ghayesh, & Amabili, 2013a; Farokhi, Ghayesh, & Amabili, 2013b; Ghayesh & Amabili, 2014; Ghayesh & Farokhi, 2015a; Ghayesh, Farokhi, & Amabili, 2013c; Ghayesh, Farokhi, & Amabili, 2014; Gholipour, Farokhi, & Ghayesh, 2015; Li & Pan, 2015; S¸ ims¸ ek, 2010; Tang, Ni, Wang, Luo, & Wang, 2014) in order to take into account small-size influences (Farokhi, Ghayesh, Kosasih, & Akaber, 201a5) on the coupled dynamical response of the functionally graded system. The literature on the dynamical/static analyses of functionally graded microbeams can mainly be divided into two main groups in terms of models being considered. In the first group, the system motion is analysed using either the Euler-Bernoulli or Timoshenko beam theories; however, in the second class, higher-order shear-deformation beam theories have been employed. The literature on the first class is quite large. For example, Thai, Vo, Nguyen, and Lee (2015) employed the modified couple stress theory in order to examine the size-dependent buckling, bending, and free dynamics of functionally graded microbeams. Tajalli et al. (2013) employed a strain gradient elasticity theory in order to develop a formulation for a functionally graded Timoshenko microbeam. Nateghi and Salamat-talab (2013) analysed the effect of a thermal loading on the size-dependent buckling and free dynamics of a functionally graded microbeam via use of the modified couple stress theory. Kahrobaiyan, Rahaeifard, Tajalli, & Ahmadian, (2012) employed the framework of a strain gradient elasticity in order to develop a size-dependent model of a functionally graded Euler–Bernoulli microbeam. Arbind and Reddy (2013) contributed to the field by developing nonlinear finite element models for the buckling analysis of both the Timoshenko and Euler–Bernoulli microscale beams via use of the modified couple stress theory. Rahaeifard, Kahrobaiyan, Ahmadian, and Firoozbakhsh (2013) developed a size-dependent model of an Euler–Bernoulli microbeam on the basis of a strain gradient elasticity theory; they examined the bending response as well as the free dynamics of the system. Ke, Wang, Yang, and Kitipornchai (2012) examined the free nonlinear dynamics of a functionally graded microbeam by means of the modified couple stress theory. These studies were extended to higher-order shear-deformations (grouped in the second class); the literature on this group is not large. For example, Sahmani and Ansari (2013) employed a higher-order shear-deformation theory in order to obtain the size-dependent buckling behaviour of a functionally graded microbeam in the presence of a thermal loading by means of the generalised differential quadrature method. On the basis of the modified couple stress theory, S¸ ims¸ ek & Reddy (2013) developed a mathematical model for the buckling analysis of an elastically constrained functionally graded microbeam via use of a unified higher-order beam theory. Zhang, He, Liu, Gan, and Shen (2014) examined the size-dependent buckling response, bending, and free dynamical behaviour of a functionally graded microbeam on the basis of a strain gradient elasticity theory together with an improved version of a third-order shear-deformation theory. Sahmani, Bahrami, and Ansari (2014) employed a modified version of the strain gradient elasticity theory in order to examine the nonlinear free dynamics of the system. Ansari, Shojaei, and Gholami (2016) analysed the nonlinear size-dependent dynamical behaviour of a thirdorder shear-deformable functionally graded microbeam by means of the variational differential quadrature method and a strain gradient elasticity theory. Contributions of this paper to the field: This paper, for the first time, analyses the nonlinear size-dependent coupled rotational-transverse-longitudinal motion characteristics of a third-order shear-deformable functionally graded microbeam based on the modified couple stress theory by means of a continuation method by providing a stability analysis together with bifurcation types for a high-dimensional system. More specifically, the works of damping and external excitation, the size-dependent potential energy, and the kinetic energy of the system are developed in terms of the system parameters and the displacement field. An energy balance on the basis of Hamilton’s principle is employed in order to obtain the coupled rotational-transverse-longitudinal continuous model of the system. Based on a weighted-residual method, the model is reduced and then solved by means of the pseudo-arclength continuation method; the Floquet theory is employed for the stability analysis of the system in motion. The size-dependent frequency-response and force-response curves of this functionally graded system are constructed for all the rotational, transverse, and longitudinal motions. 2. Model development and solution method The schematic of a two-phase (ceramic-metal) functionally graded shear-deformable microbeam is depicted in Fig. 1. The geometry of the microbeam is shown by the following parameters: length L, thickness h, width b, and cross-sectional area A which is assumed to be uniform along its length. The microbeam is considered to be hinged at both ends and subjected to a transverse harmonic load per unit length of F(x) cos (ω t) (in the z direction), where t is time and ω denotes the excitation frequency. The Mori-Tanaka homogenisation scheme is used in order to obtain the effective material properties on the basis of the ceramic and metal properties; the Mori-Tanaka scheme is almost the most applicable method in describing the local effective properties of functionally graded materials (S¸ ims¸ ek & Reddy, 2013). The mixture of ceramic and metal changes continuously from ceramic at the bottom surface (z = h/2) to metal at the top surface (z = −h/2). Using the powder metallurgy, different distributions can be obtained along the microbeam thickness (i.e. the z direction). As a result, different material properties of the mixture such as mass density ρ , Young’s modulus E, Poisson’s ratio υ , and shear modulus μ vary continuously along the z coordinate.

M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

37

Fig. 1. Schematic representation of an extensible functionally graded shear-deformable microbeam subject to a distributed harmonic excitation load in the transverse direction.

The nonlinear distribution of the ceramic and metal can be obtained using the following formulations

vc (z ) = (0.5 + z/h)n ,

(1)

vc + vm = 1,

(2)

in which vc and vm are the ceramic volume fraction and the metal one, respectively; n is the material gradient index. By changing the n value, the volume fraction of the ceramic and metal vary with respect to the thickness of the microbeam. According to the Mori-Tanaka scheme, the effective bulk modulus Ke and shear modulus μe of functionally graded materials can be formulated as (Fares, Elmarghany, & Atta, 2009; S¸ ims¸ ek & Reddy, 2013)

μe − μm vc = , μc − μm 1 + vm (μc − μm )/[μm + μm (9Km + 8μm )/(6(Km + 2μm ) )] Ke − Km vc = . Kc − Km 1 + vm (Kc − Km )/(Km + 4μm /3 )

(3)

Based on continuum mechanics, the effective values of Young’s modulus E, Poisson’s ratio υ , and mass density ρ can be expressed as

E (z ) =

9Ke μe , 3Ke + μe

(4)

υ (z ) =

3Ke − 2μe , 6Ke + 2μe

(5)

ρ ( z ) = ρm v m + ρc v c .

(6)

The displacement field is denoted by u(x,t), w(x,t), and φ (x,t) which are the longitudinal displacement, transverse displacement, and the rotation of the transverse normal, respectively. Based on the modified couple stress theory, the variation of the elastic strain energy δ U of a continuum can be expressed as

δU =

 V

(δ ε : σ + δ χ : m)dv,

(7)

where σ and m denote the stress tensor and deviatoric part of the symmetric couple stress tensor, respectively, and δ ε and δ χ are the variations of the strain tensor and the symmetric curvature tensor, respectively. For an isotropic linear elastic material, σ and m are given by (Ghayesh & Farokhi, 2013)

σ = λ tr (ε ) I + 2μe ε,

(8)

m = 2 l 2 μe χ,

(9)

in which λ and μe are the Lamé constants, and l is the material length-scale parameter in the modified couple stress theory.

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M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

The components of the displacement field in the x, y and z directions are presented by ux , uy , and uz , respectively, to be the displacement of a point located at a distance z from the mid-plane of the functionally graded third-order sheardeformable microbeam; these components are related to the displacement field of the mid-plane by (Ghayesh & Farokhi, 2016b)

ux (x, z, t ) = u(x, t ) + zφ (x, t ) −

4z 3 3 h2



φ (x, t ) +

 ∂ w(x, t ) , ∂x

uy (x, z, t ) = 0, uz (x, z, t ) = w(x, t ).

(10)

The rotation vector θ and the symmetric curvature tensor χ are defined as

1 2

θ = ∇ × u, χ=

1 2



(11)

 T  ∇θ + ∇θ ,

(12)

with u being the displacement vector. Inserting the displacement field (Eq. (10)), into the rotation and the symmetric curvature tensors (Eqs. (11) and (12), respectively) results in

θy =

1 2



φ−

 

∂ w 4z 2 ∂w − 2 φ+ , ∂x ∂x h

θx = θz = 0 ,

(13)



 2

∂φ ∂ 2 w 4z − − 2 ∂ x ∂ x2 h   2z ∂w χyz = χzy = − 2 φ + , ∂x h χxx = χyy = χzz = χxz = χyz = 0.

χxy = χyx =

1 4

∂φ ∂ 2 w + ∂ x ∂ x2



,

(14, 15)

The components of the strain tensor are given by

 2   ∂u 1 ∂w ∂φ 4z3 ∂φ ∂ 2 w εxx = + +z − + , ∂x 2 ∂x ∂ x 3 h2 ∂ x ∂ x2    4z 2 1 ∂w εxz = εzx = 1− 2 , φ+ 2 ∂x h εyy = εzz = εxy = εyz = 0.

(16)

Consequently, the variation of the strain energy stored in the functionally graded extensible shear-deformable microbeam can be formulated as

 

  4z 3 ∂ ∂w ∂ ∂ ∂ ∂w ∂ δU = δu + δ w + z δφ − 2 δφ + δ σxx ∂ x ∂ x ∂ x ∂ x ∂ x ∂ x ∂x 3 h V  

    4z 2 4z 2 ∂ ∂ 1 ∂w ∂ ∂ ∂w ∂ + σxz 1 − 2 δφ − δw − 2 δφ + δ δφ + δ w + mxy ∂x 2 ∂x ∂x ∂x ∂x ∂x ∂x h h    4z ∂ + myz − 2 δφ + δw dv ∂x h    

  L  c ∂w ∂ ∂ ∂ ∂ ∂ ∂ ∂w = N δu + δ w + M δφ − 1 P δφ + δ +Q δ w + δφ ∂x ∂x ∂x ∂x 3 ∂x ∂x ∂x ∂x 0   

 

 1 1 ∂ ∂ ∂w ∂w ∂ ∂ ∂ − c1 R δ w + δφ + Y δφ − δ − c1 H δφ + δ ∂x 2 ∂x ∂x ∂x 2 ∂x ∂x ∂x   ∂ − c1V δφ + δ w dx, ∂x 

(17)

where

c1 =

4 h2

(18a)

M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

 2    ∂u 1 ∂w ∂φ E11 c1 ∂φ ∂ 2 w N = σxx dA = A11 + + B11 − + , ∂x 2 ∂x ∂x 3 ∂ x ∂ x2 A

 2     ∂u 1 ∂w ∂φ F11 c1 ∂φ ∂ 2 w M = σxx zdA = B11 + + D11 − + , ∂x 2 ∂x ∂x 3 ∂ x ∂ x2 A

 2     ∂u 1 ∂w ∂φ G11 c1 ∂φ ∂ 2 w 3 P = σxx z dA = E11 + + F11 − + , ∂x 2 ∂x ∂x 3 ∂ x ∂ x2 A      ∂w ∂w Q = σxz dA = A55 φ + − B55 c1 φ + , ∂x ∂x A      ∂w ∂w 2 R = σxz z dA = B55 φ + − D55 c1 φ + , ∂x ∂x A      1 2 1 ∂φ ∂ 2 w ∂φ ∂ 2 w Y = mxy dA = l 2 A55 − − l B c + , 55 1 2 ∂ x ∂ x2 2 ∂ x ∂ x2 A      1 2 1 ∂φ ∂ 2 w ∂φ ∂ 2 w H = mxy z2 dA = l 2 B55 − − l D c + , 55 1 2 ∂ x ∂ x2 2 ∂ x ∂ x2 A    ∂w 2 V = myz zdA = −l B55 c1 φ + , ∂x A

39



in which

{A11 , B11 , D11 , E11 , F11 , G11 } = {A55 , B55 , D55 } =

 A





A

(18b)

(18c)

(18d)

(18e)

(18f)

(18g)

(18h)

(18i)



E (z ) 1, z, z2 , z3 , z4 , z6 dA,

 2 4 E (z ) 1, z , z d A . 2[1 + υ (z )]

(18j)

The kinetic energy of the functionally graded microbeam can be obtained via the following formulation



2  2  2       2c1 I4 ∂ u c1 2 I6 ∂φ ∂ 2w ∂u ∂φ ∂u ∂φ ∂φ ∂ 2 w I1 + I3 + + + 2I2 − + ∂t ∂t 9 ∂ t ∂ x∂ t ∂t ∂t 3 ∂t ∂ t ∂ x∂ t 0     2  2c1 I5 ∂φ ∂φ ∂ 2 w ∂w − + + I1 dx (19) 3 ∂t ∂ t ∂ x∂ t ∂t

1 T = 2



L

with

{I1 , I2 , I3 , I4 , I5 , I6 } =

 A

  ρ (z ) 1, z, z2 , z3 , z4 , z6 dA .

(20)

The only external force is the distributed harmonic transverse force, the variation of which can be expressed as

δWT =



0

L

F (x ) cos (ω t ) δ w dx.

(21)

The work due to a viscous damping can be formulated as (cd and cr are the damping coefficients)

δWD = −cd



0

L



   L ∂w ∂u ∂φ δu + δ w d x − cr δφ dx. ∂t ∂t ∂t 0

(22)

Generalised Hamilton’s principle given by the following is applied



t2

t1

(δ T − δU + δWT + δWD ) dt = 0,

(23)

by substituting, Eqs. (17), (19), (21), and (22), resulting in the following coupled size-dependent nonlinear equations of motion of the shear-deformable functionally graded microbeam for the longitudinal and transverse displacements as well as the rotation respectively

I1

 c1 I4 ∂ 2 φ c1 I4 ∂ 3 w ∂ 2u  ∂N ∂u + I − − − + cd = 0, 2 3 3 ∂ x∂ t 2 ∂x ∂t ∂t2 ∂t2

(24)

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M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

  3   c1 2 I6 c1 ∂ 2 P ∂ 2 w c1 2 I6 ∂ 4 w c1 I5 c1 I4 ∂ 3 u ∂ ∂w ∂ φ I1 2 − + − + − N − 2 2 2 2 9 ∂ x ∂t 3 9 3 ∂ x∂ t ∂x ∂x 3 ∂ x2 ∂t ∂ x∂ t −



∂Q ∂ R 1 ∂ 2 Y c1 ∂ 2 H ∂V ∂w + c1 − − + c1 − F (x ) cos (ω t ) + cd = 0, ∂x ∂ x 2 ∂ x2 2 ∂ x2 ∂x ∂t

 2  3 c1 I5 c1 I6 ∂ 2φ ∂ w + − 9 3 ∂t2 ∂t2 ∂ x∂ t 2 1 ∂Y ∂ M c1 ∂ P c1 ∂ H ∂φ − + + Q − c1 R − + − c1 V + cr = 0. ∂x 3 ∂x 2 ∂x 2 ∂x ∂t

I2 −

c1 I4 3

 ∂ 2u



(25)

+

I3 −

2c1 I5 c1 2 I6 + 3 9



(26)

Inserting Eq. (18) into Eqs. (24)–(26) gives the equations of motion of the functionally graded extensible shear-deformable microbeam in terms of the displacement field as follows

I1

 c1 I4 ∂ 2 φ c1 I4 ∂ 3 w ∂ 2u  ∂ 2u ∂ w ∂ 2w + I2 − − − A11 2 − A11 2 2 2 3 3 ∂ x∂ t ∂ x ∂ x2 ∂t ∂t ∂x   2 3 E11 c1 ∂ φ E11 c1 ∂ w ∂u − B11 − + + cd = 0, 3 3 ∂ x3 ∂t ∂ x2

(27)

  3   c1 2 I6 ∂ 2 w c1 2 I6 ∂ 4 w c1 I5 c1 I4 ∂ 3 u ∂ φ ∂ u ∂ 2w ∂ 2u ∂ w I1 2 − + − + − A11 + 9 ∂ x2 ∂ t 2 3 9 3 ∂ x∂ t 2 ∂ x ∂ x2 ∂ x2 ∂ x ∂t ∂ x∂ t 2   ∂φ ∂ 2 w ∂ 2 φ ∂ w  3A  ∂ w 2 ∂ 2 w E c ∂ 3 u E11 c1 11 11 1 − B11 − + − − 3 ∂ x ∂ x2 2 ∂x 3 ∂ x3 ∂ x2 ∂ x ∂ x2     F11 c1 A55 l 2 G11 c1 2 D55 l 2 c1 2 ∂ 3 φ G11 c1 2 A55 l 2 B55 l 2 c1 D55 l 2 c1 2 ∂ 4 w + − + − + + + + + 3 9 4 4 9 4 2 4 ∂ x3 ∂ x4     ∂φ ∂ 2 w ∂w + −A55 + 2B55 c1 − D55 c1 2 − B55 l 2 c1 2 + − F (x ) cos (ω t ) + cd = 0, ∂ x ∂ x2 ∂t 

 2  3  c1 I5 E11 c1 ∂ 2 u c1 I6 ∂ 2φ ∂ w  + − − B11 − 2 2 2 9 3 3 ∂t ∂t ∂ x∂ t ∂ x2     E11 c1 ∂ w ∂ 2 w G11 c1 2 A55 l 2 D55 l 2 c1 2 ∂ 2 φ 2F11 c1 B55 l 2 c1 − B11 − + −D11 + − − + − 2 3 ∂x ∂x 3 9 4 2 4 ∂ x2      G11 c1 2 D55 l 2 c1 2 ∂ 3 w  F11 c1 A55 l 2 ∂w 2 2 2 + − + − + A55 − 2B55 c1 + D55 c1 + B55 l c1 φ+ 3 9 4 4 ∂x ∂ x3 ∂φ + cr = 0. ∂t

I2 −

c1 I4 3

 ∂ 2u



+

I3 −

2c1 I5 c1 2 I6 + 3 9

(28)



(29)

In the solution process, it is preferable to use dimensionless equations; as such, the following quantities are proposed to make the equations of motion dimensionless

x∗ =

x (u, w ) , ( u∗ , w∗ ) = , L h L

l

I  I I I I I (I1 ∗ , I2 ∗ , I3 ∗ , I4 ∗ , I5 ∗ , I6 ∗ ) = 1 , 2 , 3 2 , 4 3 , 5 4 , 6 6 , I10 I10 h I10 h I10 h I10 h I10 h   F L4

D

I

L4

110 10 φ ∗ = φ , η = , ls = , f = , t∗ = t , =ω h h hD110 D110 I10 L4   A11 h2 B11 h D11 E11 F11 G11 (a11 , b11 , d11 , e11 , f11 , g11 ) = , , , , , , D110 D110 D110 D110 h D110 h2 D110 h4    

(a55 , b55 , d55 ) =

A55 h2 B55 D55 , , D110 D110 D110 h2

, cd ∗ = cd

L4 , cr ∗ = cr I10 D110

L4 , I10 D110 h4

(30)

in which D110 and I10 are defined as the values of D11 and I1 of a homogeneous metal microbeam (i.e. vm = 1). Substituting these dimensionless quantities into Eqs. (27)–(29) and dropping the asterisk notation for brevity results in the following

M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

41

coupled nonlinear partial differential equations

I1

I1



  4I4 ∂ 2 φ 4I4 ∂ 3 w 4e11 2 ∂ 2 φ ∂ 2u  ∂ 2u ∂ w ∂ 2w  η + I2 − − − a11 η2 2 − a11 η − b11 − 2 2 2 2 3 ∂t 3 η ∂ x∂ t ∂x ∂x 3 ∂t ∂x ∂ x2 4e11 η ∂ 3 w ∂u + + cd = 0, 3 ∂ x3 ∂t

(31)

    4I 16I6 ∂ 3 φ ∂ 2 w 16I6 ∂ 4 w 4I4 ∂ 3 u ∂ u ∂ 2w ∂ 2u ∂ w 5 − + − + − a η + 11 3η 9 η ∂ x∂ t 2 3 η ∂ x∂ t 2 ∂ x ∂ x2 ∂ x2 ∂ x ∂t2 9η 2 ∂ x2 ∂ t 2    ∂φ ∂ 2 w ∂ 2 φ ∂ w  3a  ∂ w 2 ∂ 2 w 4e η ∂ 3 u 4e11 11 11 η − b11 − + − − 3 ∂ x ∂ x2 2 ∂x 3 ∂ x3 ∂ x2 ∂ x ∂ x2   3   4 2 2 4 f11 a55 ls 16g11 16g11 a55 ls ∂ φ 2 2 2 ∂ w + − + − + 4d55 ls η 3 + + + 2b55 ls + 4d55 ls 3 9 4 9 4 ∂x ∂ x4     3 ∂φ ∂ 2w ∂w 2 + −a55 + 8b55 − 16d55 − 16b55 ls + η2 2 − f (x ) cos (t ) + cd = 0, η ∂x ∂t ∂x  ∂ 3w   4e11 2 ∂ 2 u η − b − 11 9η 3 ∂t2 ∂t2 ∂ x∂ t 2 ∂ x2     2 2 4e11 16g11 a55 ls 8 f11 ∂ 2φ ∂w ∂ w 2 2 η − b11 − + −d11 + − − + 2b55 ls − 4d55 ls η2 2 2 3 ∂x ∂x 3 9 4 ∂x   3   2  4 16g11 4 f11 a55 ls ∂ w  2 2 3 ∂w + − + − 4d55 ls η 3 + a55 − 8b55 + 16d55 + 16b55 ls η φ+η 3 9 4 ∂x ∂x ∂φ + cr = 0. ∂t

I2 −

4I4 3

 ∂ 2u



+ I3 −

16I6 8I5 + 3 9

 ∂ 2φ

+

 16I

6

−

(32)

4I5 3η

(33)

The Galerkin technique is employed as a discretisation tool in order to reduce the nonlinear partial differential equations of coupled motion into sets of nonlinear ordinary differential equations. In this regard, the displacement field can be approximated using the following series expansions

w(x, t ) =

ˆ M 

ϕk (x )qk (t ),

(34)

ψk (x ) pk (t ),

(35)

ϕk (x )rk (t ),

(36)

k=1

φ (x, t ) =

Nˆ  k=1

u(x, t ) =

Qˆ  k=1

in which the parameters rk (t),qk (t), and pk (t) denote the kth generalised coordinates of the longitudinal, transverse, and rotational motions, respectively; ϕ k shows the kth eigenfunction for the transverse motion of a linear hinged-hinged beam and ψ k = ϕ ’k /(kπ ). In these equations F(x) = f1 (giving f1 cos ( t)) is considered for the transverse external force. By applying the Galerkin scheme on Eqs. (31)–(33), three sets of a second-order nonlinear ordinary differential equations are obtained as follows

I1

Qˆ  

   ˆ  1   Nˆ  1 M 4I  4I  ϕi ϕ j dx r¨ j + I2 − 4 ϕi ψ j dx p¨ j − 4 ϕi ϕ j dx q¨ j 3 3η 0 0 0 j=1 j=1 j=1

  1    1  ˆ ˆ  1 Qˆ    M M Nˆ    4e − a11 η2 ϕi ϕ j dx r j + η ϕi ϕ j ϕk dx q j qk − b11 − 11 η2 ϕi ψ j dx p j 1

j=1

+

ˆ M 4e11 η  3

0



j=1

i = 1, 2,...,Qˆ ,

1 0

 ϕi ϕ  j d x q j + cd

Qˆ  j=1

3

0

j=1 k=1



1 0



j=1

0

ϕi ϕ j dx r˙ j = 0 (37)

42

M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

I1

ˆ  M 

  1  ˆ  1 Qˆ M 16I6  4I4    ϕi ϕ j dx q¨ j − 2 ϕi ϕ j dx q¨ j + ϕi ϕ j dx r¨ j 3η 9η 0 0

0

j=1

+



1

 4I

5

−

3η

η

−a11

16I6 9η



ˆ Qˆ  M 

Nˆ  j=1



− b11 −

 4e11 3

1 0



j=1 k=1

j=1



1

0

ϕi ψ j dx p¨ j 

ϕi

j

+

k



1 0



ϕi

ψ  ϕ  dx j

 ˆ ˆ ˆ  1 M M M 3     r j qk + ϕi ϕ j ϕk ϕl dx q j qk ql 2 0

ϕi ϕ j ϕ  k dx 

1 0

j=1 k=1





ϕ  ϕ  dx

ˆ Nˆ  M 

η

j=1



j=1 k=1 l=1

 +

k

1

0





ϕi ψ j ϕ  k dx

p j qk

 1  Qˆ 4e11 η   − ϕi ϕ j dx r j 3 0 j=1

 Nˆ  1  2  4 f11 a55 ls 16g11 2 + − + − + 4d55 ls η ϕi ψ j dx p j 3



9

4

2

16g11 a55 ls 2 2 + + 2b55 ls + 4d55 ls 9 4

+

+ −a55 + 8b55 − 16d55 − 16b55 ls 



1 0

f1 ϕi dx cos( t ) + cd



η2

1 0

1

0

ˆ  M 

1 0

j=1

ˆ  M  j=1

ˆ M 



 ϕi

ϕ  dx

qj

j

 ϕi ϕ j dx q j + η3

 Nˆ 

ϕi ψ  j dx p j

0

j=1

 ϕi ϕ j dx q˙ j = 0,

 

1

ˆ, i = 1, 2,...,M

(38)

    Mˆ  1   Nˆ  1  16I 8I 4I5  16I  6 ψi ϕ j dx r¨ j + I3 − 5 + 6 ψi ψ j dx p¨ j + − ψi ϕ  j dx q¨ j 3 9 9η 3η 0 0 0 j=1 j=1 j=1 

 1   ˆ ˆ  1 Qˆ   M M    4e11 − b11 − η2 ψi ϕ j dx r j + η ψi ϕ  j ϕk dx q j qk

I2 −

4I4 3

 Qˆ 

2

0

j=1

j=1



−





3



1

0

j=1

j=1 k=1 2

16g11 a55 ls 8 f11 2 2 + −d11 + − − + 2b55 ls − 4d55 ls 3 9 4

 +

2

4 f11 a55 ls 16g11 2 − + − 4d55 ls 3 9 4

η

+ a55 − 8b55 + 16d55 + 16b55 ls ˆ  N  j=1

0

1

 ψi ψ j dx p˙ j = 0

2



η

ˆ  M  0

j=1



+ cr



3

ˆ M  j=1



1 0

1

0

 η

2

 Nˆ  j=1

1 0

 ψi

ψ  dx j

pj

 ψi ϕ  d x qj j 

ψi ϕ  j dx

qj + η

4

 Nˆ  j=1

ˆ, i = 1, 2,...,N

1 0

  ψi ψ j dx p j (39)

where the dot shows differentiations with respect to dimensionless time and prime notations denote the differentiations with respect to the dimensionless axial coordinate. Considering a change of variables (xi = q˙ i , yi = p˙ i , and zi = r˙ i ) in Eqs. (37)–(39) transforms these second order nonlinear ˆ +N ˆ +Qˆ ) into doubled dimensional first-order ordinary differential equations (2(M ˆ +N ˆ +Qˆ )). ordinary differential equations (M ˆ = Nˆ = Qˆ = 8 for mode contributions, a high-dimensional functionally graded third-order shear-deformable By assuming M microbeam is modelled and analysed. In order to obtain linear natural frequencies, an eigenvalue analysis (Farokhi, Ghayesh, & Hussain, 2015b; Ghayesh & Amabili, 2013; Ghayesh and Farokhi, 2015b; Ghayesh, Kazemirad, & Reid, 2012) is applied. For the nonlinear analysis, frequency-response and force-response of the system are obtained by means of the pseudo-arclength continuation technique. Stability analysis is conducted via the Floquet theory. 3. Verifications of coupled size-dependent nonlinear equations of motion For the validation of the equations of motion for the shear-deformable functionally graded microbeam developed in Section 2, these equations are simplified for a microbeam made only from epoxy, by assuming constant E and υ and σ = Eε

M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

Fig. 2. Poisson’s ratio variations with respect to the thickness of the FG microbeam for different values of n.

Fig. 3. Young’s modulus variations with respect to the thickness of the FG microbeam for different values of n.

43

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M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

Fig. 4. Frequency-response curves of the functionally graded shear-deformable extensible microbeam: (a, b) the maximum amplitudes of the first generalised coordinate of the transverse and the rotational motions, respectively and (c) the minimum amplitude of first generalised coordinate of the longitudinal motion. n = 2, ls = 0.2, f1 = 2.0, and ζ = 0.01. Solid and dashed lines represent the stable and unstable solutions, respectively.

and μ = E/2(1 + ʋ). As a result, Eqs. (18j) and (20) can be expressed as

 bh3 bh5 bh7 , 0, , , {A11 , B11 , D11 , E11 , F11 , G11 } = E bh, 0, 12 80 448  bh3 bh5 , , {A55 , B55 , D55 } = μ bh, 12

 {I1 , I2 , I3 , I4 , I5 , I6 } = ρ

80

bh3 bh5 bh7 bh, 0, , 0, , 12 80 448

(40)

.

(41)

By inserting Eqs. (40) and (41) into Eqs. (18b)–(18i) one can get

c1 =

4 h2

(42a)

 2  ∂u 1 ∂w N = σxx dA = bh + , ∂x 2 ∂x A 

(42b)

M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

45

Fig. 5. Force-response curves of the functionally graded shear-deformable extensible microbeam: (a, b) the maximum amplitudes of the first generalised coordinate of the transverse and the rotational motions, respectively and (c) the minimum amplitude of first generalised coordinate of the longitudinal motion. n = 2, ls = 0.2, /ω1 = 1.2, and ζ = 0.01. Solid and dashed lines represent the stable and unstable solutions, respectively.

 M=

A

 P=

A

σxx zdA =

σxx z3 dA =

bh3 15

∂φ bh3 ∂ 2 w − , ∂x 60 ∂ x2

bh5 ∂ 2 w bh5 ∂φ − , 105 ∂ x 336 ∂ x2

 ∂w , ∂x A    bh3 ∂w 2 R = σxz z dA = , φ+ 30 ∂x A 

Q=

σxz dA =

2bh 3

 Y =

A

mxy dA =

A

mxy z2 dA =

(42d)



φ+

1 4bh ∂ 2 w 1 2 2bh ∂φ l − l2 , 2 3 ∂x 2 3 ∂ x2

 H=

(42c)

1 2 bh3 l 2 30

∂φ 1 2 2bh3 ∂ 2 w − l , ∂ x 2 15 ∂ x2

(42e)

(42f) (42g) (42h)

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M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

Fig. 6. Frequency-response curves of the functionally graded shear-deformable extensible microbeam: (a, b) the maximum amplitudes of the first generalised coordinate of the transverse and the rotational motions, respectively and (c) the minimum amplitude of first generalised coordinate of the longitudinal motion. n = 2, f1 = 2.0, and ζ = 0.01; (ls = 0.0, 0.2, 0.4, 0.6).

 V =

A

myz zdA = −l 2

bh 3

  ∂w . φ+ ∂x

(42i)

Eqs. (42a)–(42i) and (41) are substituted into Eqs. (24)–(26) which results in three equations for longitudinal, transverse, and rotational motions, respectively. These equations are the same as those given in Eqs. (14)–(16) in Farokhi & Ghayesh (2015c) given by

 2  ∂ 2u ∂u ∂ u ∂ 2w ∂ w − EA + + cd = 0, ∂t ∂t2 ∂ x2 ∂ x2 ∂ x     2  1 ∂ 4w ∂ 2w 16 ∂ 3 φ ∂ 2u ∂ w ∂ u ∂ 2w 3 ∂ 2w ∂ w ρA 2 + ρI − − EA + + 105 ∂ x∂ t 2 21 ∂ x2 ∂ t 2 ∂t ∂ x2 ∂ x ∂ x ∂ x2 2 ∂ x2 ∂ x      

16 ∂ 3 φ 8μA ∂φ 1 ∂ 3φ 4 1 ∂ 4w ∂ 2w 7 ∂ 4w ∂φ ∂ 2 w 2 + EI − − + + μ A l − − + 21 ∂ x4 105 ∂ x3 15 ∂ x ∂ x2 15 ∂ x4 5 ∂ x3 3 h2 ∂ x ∂ x2 ∂w − F (x ) cos (ω t ) + cd = 0, ∂t ρA

(43)

(44)

M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

47

Fig. 7. Frequency-response curves of the functionally graded shear-deformable extensible microbeam: (a, b) the maximum amplitudes of the first generalised coordinate of the transverse and the rotational motions, respectively and (c) the minimum amplitude of first generalised coordinate of the longitudinal motion. ls = 0.2, f1 = 2.0, and ζ = 0.01; (ceramic(n = 0), n = 2, n = 5, and metal).

 ρI

16 ∂ 3 w 68 ∂ 2 φ − 2 105 ∂ t 105 ∂ x∂ t 2



+ μA l 2



 + EI

2 ∂ 2φ 1 ∂ 3w 4 − + 3 5 ∂x 15 ∂ x2 3 h2

68 ∂ 2 φ 16 ∂ 3 w − 3 105 ∂ x 105 ∂ x2

 +

8 μA 15

 

∂w ∂φ + cr = 0. φ+ ∂x ∂t

  ∂w φ+ ∂x (45)

4. Mechanical properties of shear-deformable functionally graded microbeam The mechanical properties of the functionally graded microbeam along the normalised thickness (z/h) are depicted in this section. Illustrated in Fig. 2 is Poisson’s ratio variation of the shear-deformable functionally graded microbeam in terms of the normalised thickness (z/h). For a fixed distance from the mid-plane, a larger material gradient index results in a larger Poisson’s ratio; as the material gradient index n is considered fixed, Poisson’s ratio decreases with the normalised thickness (z/h). Young’s modulus variations of the shear-deformable functionally graded microbeam are shown in Fig. 3 in terms of the normalised thickness (z/h). As shown, for a constant value of the material gradient index, a larger Young’s modulus corresponds to a larger z/h; for a constant value of z/h, Young’s modulus becomes smaller when n is larger.

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M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

Fig. 8. Back-bone curves of the functionally graded shear-deformable extensible microbeam: (a, b) the maximum amplitudes of the first generalised coordinate of the transverse and the rotational motions, respectively and (c) the minimum amplitude of first generalised coordinate of the longitudinal motion. n = 2, ls = 0.2, and ζ = 0.01.

5. Frequency-responses and force-responses of size-dependent coupled nonlinear motion The frequency-responses and force-responses are plotted for the functionally graded system with properties: h = 15 μm, L/h = 80, and b/h = 4; the microbeam consists of Aluminium with EAl = 70 GPa, ρ Al = 2702 kg/m3 , and υAl = 0.3 and SiC (ceramic) with ESiC = 427 GPa, ρ Al = 3100 kg/m3 , and υAl = 0.17. The frequency-response curves of the functionally graded shear-deformable microbeam are shown in Fig. 4, for the transverse, rotational, and longitudinal motions in sub-figures (a), (b), and (c), respectively. The following system parameters have been chosen for these plots: n = 2, ls = 0.2, f1 = 2.0, and ζ = 0.01 (ζ defines as the modal damping ratio); for this set of parameters, the fundamental natural frequency for transverse motion can be obtained as ω1 = 14.6580. It can be noted that the type of motion is hardening, due to the geometric nonlinearity of stretching-type. There are two saddle-node bifurcations present at points /ω1 = 1.1774 and 1.0450, where the system response for all the three types of the motion is unstable between these two points. Theoretically speaking, as the normalised excitation for frequency (i.e. the excitation frequency divided by the fundamental natural frequency of the transverse motion) is increased, the motion amplitude for the three types of the motion increase gradually from /ω1 = 0.9, until the first saddle-node bifurcation at /ω1 = 1.1774 is reached, where the response ultimately becomes unstable. Decreasing the normalised excitation frequency, this unstable solution branch becomes stable again at /ω1 = 1.0450 via a secondary saddle-node bifurcation. The response amplitude for all of the types of the motion decreases afterwards by increasing the normalised excitation frequency.

M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

49

Fig. 9. Force-response curves of the functionally graded shear-deformable extensible microbeam: (a, b) the maximum amplitudes of the first generalised coordinate of the transverse and the rotational motions, respectively and (c) the minimum amplitude of first generalised coordinate of the longitudinal motion. ls = 0.2, /ω1 = 1.2, and ζ = 0.01; (ceramic(n = 0), n = 2, n = 5, and metal).

As shown in Fig. 5, it has been found that as the forcing function amplitude in the transverse direction is increased from f1 = 0.0, the response amplitude for all the three types of the motion (shown in sub-figures (a), (b), and (c)) increases accordingly until it hits the first saddle-node bifurcation at f1 = 17.4517 where the motion becomes unstable; the corresponding motion amplitude for the first generalised coordinate of the transverse and rotational motions as well as the second generalised coordinate of longitudinal motion have been obtained as 0.2887, 0.0136, and −0.0010, respectively. The following system parameters have been chosen for the plots of Fig. 5: n = 2, ls = 0.2, /ω1 = 1.2, and ζ = 0.01. As the forcing amplitude is decreased, this now unstable solution branch becomes stable once again at point B via the second saddle-node bifurcation (f1 = 2.1559); the response amplitude for all the three types of the motions increases afterwards by increasing the forcing amplitude in the transverse direction. Illustrated in Fig. 6 is the effect of the length-scale parameter, as the representative of size effects, on the frequencyresponse curves of the functionally graded shear-deformable microbeam. These curves have been obtained for a system with the following parameters: n = 2, f1 = 2.0, and ζ = 0.01; (ls = 0.0, 0.2, 0.4, 0.6). As is well-known when the length-scale parameter is set to zero, the case of the classical continuum mechanics is obtained; this case possesses the largest peakamplitude between all the cases studied here. For all the values of the length-scale parameter, a hardening type nonlinearity is observed; therefore, the length-scale parameter does not have qualitative effect on the system response for all the three motion types. As the length-scale parameter is increased, the nonlinearity decreases slightly, hence a reduction in the peakamplitude occurs.

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M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

Fig. 10. Force-response curves of the functionally graded shear-deformable extensible microbeam: (a, b) the maximum amplitudes of the first generalised coordinate of the transverse and the rotational motions, respectively and (c) the minimum amplitude of first generalised coordinate of the longitudinal motion. n = 2, /ω1 = 1.2, and ζ = 0.01, (ls = 0.0, 0.2, 0.4, 0.6).

For the functionally graded system with the dimensionless parameters of ls = 0.2, f1 = 2.0, and ζ = 0.01, the influence of the gradient index of the material property on the frequency-response curves of the size-dependent coupled system is illustrated in Fig. 7. It can be seen in this figure that the lowest fundamental natural frequency for the transverse motion belongs to the metal case. The smallest peak-amplitude belongs to the ceramic case; as the gradient index increases the peak-amplitude of motions increases accordingly. However, it seems that the effect of the gradient index on the nonlinearity of the system is not significant. If we compare the effect of the gradient index on the response of the system from qualitative perspective, the value of the gradient index does not change the type of the nonlinearity as well as the number of saddlenode bifurcations. Fig. 8 shows the back-bone curves of the system for the transverse, rotational, and longitudinal motions, in sub-figures (a), (b), and (c), respectively. As seen in this figure, the back-bone curves, for all the three motion types, tend to the right, implying a hardening-type nonlinear behaviour. For several values of the material gradient index, the coupled force-response curves of the shear-deformable functionally graded microbeam, for the transverse, rotational, and longitudinal motions, are plotted in sub-figures of (a), (b), and (c), of Fig. 9, respectively. It can be seen that, between the cases studied, the metal microbeam undergoes both the jumps at larger forcing amplitudes. It is also expected that the ceramic microbeam behaves in an opposite way to the metal microbeam in this sense, however, it is not the case; instead, the case with n = 5 experiences a jump to the upper branch at a relatively

M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

51

Fig. 11. Comparison of frequency-responses of the FGM extensible shear-deformable microbeam (current study) and epoxy microbeam (from Ref. H. Farokhi & Ghayesh, 2015c): (a, b) the maximum amplitudes of the first generalised coordinate of the transverse and the rotational motions, respectively and (c) the minimum amplitude of first generalised coordinate of the longitudinal motion. ls = 3.5, f1 = 0.832, and ζ = 0.01. Solid and dashed lines represent the stable and unstable solutions, respectively.

small forcing amplitude. For all the values of the material gradient index, there are two saddle-node bifurcations present in the coupled force-response curves of the shear-deformable functionally graded microbeam. Illstrated in Fig. 10 are the force-responses of the coupled motion of the functionally graded microbeam for several values of the length-scale parameter in order to analyse the influence of small-size effects on the force-response motion characteristics of the functionally graded system; the other parameters of the system are selected as n = 2, /ω 1 = 1.2, and ζ = 0.01. As the length-scale parameter is increased, both the first and second saddle-node bifurcations take place at larger forcing amplitudes; this is the same for all the three motion types – physical meaning of this effect is that a microscale functionally graded beam requires a larger value of the forcing function to experience a jump, compared to a macroscale functionally graded beam. For all of the three motion types, the first post-jump motion (at the first saddle-node bifurcation) is associated with larger amplitudes for larger values of the length-scale parameter; this seems to contradict the effect of additional stiffness imposed by the length-scale parameter. 6. Conclusions The aim of this paper was to conduct an investigation into the size-dependent coupled transverse-rotational-longitudinal motion characteristics of a third-order shear-deformable functionally graded extensible microbeam in the framework of the modified couple stress theory; none of the displacements and the rotation as well as translational and rotary inertia was

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M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

neglected in the theoretical modelling and numerical simulations. Based on the Mori-Tanaka homogenisation scheme, the modified couple stress theory, constitutive relations, and the relation between the symmetric curvature and the deviatoric part of the symmetric couple stress tensors, the energies of the extensible shear-deformable functionally graded system were dynamically balanced by the work of an external harmonically-varying transverse force. This operation yielded the coupled nonlinear equations of motion for the axial and transverse directions as well as rotation. These continuous motion models for the shear-deformable functionally graded extensible microbeam were discretised using the Galerkin method, retaining all the translational and rotational inertia and considering a high-dimensional nonlinearly coupled terms. Numerical simulations were conducted by means of the pseudo-arclength continuation. The nonlinear results for the motion characteristics of the shear-deformable functionally graded extensible microbeam shows that: (1) the resonant response of the functionally graded system is of a hardening-type; (2) two saddle-node bifurcations were observed with unstable solution in between; (3) there is no other types of bifurcation (rather than the saddle-node ones) in the frequency and force-response curves of all the transverse, rotational, and longitudinal motions; (4) the number of the bifurcation points in the force-response curves is two; (5) as the length-scale parameter is increased, the peak-amplitude for the nonlinear resonance decreases and the resonance region is shifted to larger values of the excitation frequency for all the three motion types; (6) investigating the effect of the material gradient index on the frequencyresponses showed that the smallest peak-amplitude belongs to the ceramic microbeam – as the material gradient index increases, the peak-amplitude of the resonance in all of the three types of the motion increases; (7) influences of the material gradient index on the coupled force-responses showed that the metal microbeam experiences both the jumps at largest forcing amplitude between the cases studied; however, the ceramic case does not undergo at the smallest one; (8) the forceresponses for different value of the length-scale parameter revealed that, for all the three motion types, for larger values of the length-scale parameter, both the first and second saddle-node bifurcations are delayed to larger forcing amplitudes – the physical meaning of which is that the microscale functionally graded beam experiences the jumps at larger forcing amplitudes compared to the case of a macroscale functionally graded beam. Acknowledgement The financial support to this research by the start-up grant of the University of Adelaide is gratefully acknowledged. Appendix. Validation of results with literature for a pure epoxy microbeam The theoretical model and numerical simulations developed in this paper for a functionally graded shear-deformable extensible microbeam have been verified by obtaining the corresponding epoxy microbeam frequency-responses by simplifying our equations of motion for an epoxy microbeam and conducting the simulations by means of our own computer codes; the comparison with Ref. Farokhi & Ghayesh (2015c) is illustrated in Fig. 11, showing excellent agreement. Material properties of epoxy microbeam are given as: l = 17.6 μm, h/l = 3.5; L/l = 560; b/l = 14; E = 1.44 GPa; ʋ = 0; ρ = 1220. Reference Akgöz, B., & Civalek, Ö. (2011). Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. International Journal of Engineering Science, 49, 1268–1280. Akgöz, B., & Civalek, Ö. (2013). A size-dependent shear deformation beam model based on the strain gradient elasticity theory. International Journal of Engineering Science, 70, 1–14. Ansari, R., Shojaei, M. F., & Gholami, R. (2016). Size-dependent nonlinear mechanical behavior of third-order shear deformable functionally graded microbeams using the variational differential quadrature method. Composite Structure, 136, 669–683. Arbind, A., & Reddy, J. (2013). Nonlinear analysis of functionally graded microstructure-dependent beams. Composite Structure, 98, 272–281. Asghari, M., Kahrobaiyan, M. H., & Ahmadian, M. T. (2010). A nonlinear Timoshenko beam formulation based on the modified couple stress theory. International Journal of Engineering Science, 48, 1749–1761. Baghani, M. (2012). Analytical study on size-dependent static pull-in voltage of microcantilevers using the modified couple stress theory. International Journal of Engineering Science, 54, 99–105. Dai, H. L., Wang, Y. K., & Wang, L. (2015). Nonlinear dynamics of cantilevered microbeams based on modified couple stress theory. International Journal of Engineering Science, 94, 103–112. Dehrouyeh-Semnani, A. M. (2014). A discussion on different non-classical constitutive models of microbeam. International Journal of Engineering Science, 85, 66–73. Dehrouyeh-Semnani, A. M., & Bahrami, A. (2016). On size-dependent Timoshenko beam element based on modified couple stress theory. International Journal of Engineering Science, 107, 134–148. Dehrouyeh-Semnani, A. M., BehboodiJouybari, M., & Dehrouyeh, M. (2016). On size-dependent lead-lag vibration of rotating microcantilevers. International Journal of Engineering Science, 101, 50–63. Dehrouyeh-Semnani, A. M., Dehrouyeh, M., Torabi-Kafshgari, M., & Nikkhah-Bahrami, M. (2015). An investigation into size-dependent vibration damping characteristics of functionally graded viscoelastically damped sandwich microbeams. International Journal of Engineering Science, 96, 68–85. Fares, M., Elmarghany, M. K., & Atta, D. (2009). An efficient and simple refined theory for bending and vibration of functionally graded plates. Composite Structure, 91, 296–305. Farokhi, H., & Ghayesh, M. H. (2015c). Nonlinear coupled dynamics of shear deformable microbeams. International Journal of Dynamics and Control, 1–12. Farokhi, H., & Ghayesh, M. H. (2015b). Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams. International Journal of Engineering Science, 91, 12–33. Farokhi, H., & Ghayesh, M. H. (2015a). Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory. International Journal of Mechanical Sciences, 90, 133–144. Farokhi, H., & Ghayesh, M. H. (2016b). Nonlinear size-dependent dynamics of an imperfect shear deformable microplate. Journal of Sound and Vibration, 361, 226–242.

M.H. Ghayesh et al. / International Journal of Engineering Science 110 (2017) 35–53

53

Farokhi, H., & Ghayesh, M. H. (2016a). Nonlinear size-dependent dynamics of microarches with modal interactions. Journal of Vibration and Control, 22, 3679–3689. Farokhi, H., Ghayesh, M. H., & Amabili, M. (2013a). Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory. International Journal of Engineering Science, 68, 11–23. Farokhi, H., Ghayesh, M. H., & Hussain, S. (2015b). Three-dimensional nonlinear global dynamics of axially moving viscoelastic beams. ASME Journal of Vibration and Acoustics, 138 011007-011007. Farokhi, H., Ghayesh, M. H., Kosasih, B., & Akaber, P. (2015a). On the nonlinear resonant dynamics of Timoshenko microbeams: Effects of axial load and geometric imperfection. Meccanica. Farokhi, H., Ghayesh, M., & Amabili, M. (2013b). Nonlinear resonant behavior of microbeams over the buckled state. Applied Physics A, 113, 297–307. Ghayesh, M. H., & Amabili, M. (2013). Steady-state transverse response of an axially moving beam with time-dependent axial speed. International Journal of Non-Linear Mechanics, 49, 40–49. Ghayesh, M. H., & Amabili, M. (2014). Coupled longitudinal-transverse behaviour of a geometrically imperfect microbeam. Composites Part B, 60, 371–377. Ghayesh, M. H., Amabili, M., & Farokhi, H. (2013b). Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. International Journal of Engineering Science, 63, 52–60. Ghayesh, M. H., & Farokhi, H. (2013). Nonlinear dynamics of a microscale beam based on the modified couple stress theory. Composites Part B, 50, 318–324. Ghayesh, M. H., & Farokhi, H. (2015b). Thermo-mechanical dynamics of three-dimensional axially moving beams. Nonlinear Dynamics, 80, 1643–1660. Ghayesh, M. H., & Farokhi, H. (2015a). Nonlinear dynamics of microplates. International Journal of Engineering Science, 86, 60–73. Ghayesh, M. H., & Farokhi, H. (2016b). Coupled nonlinear dynamics of geometrically imperfect shear deformable extensible microbeams. ASME Journal of Computational and Nonlinear Dynamics, 11 041001-041001. Ghayesh, M. H., & Farokhi, H. (2016a). Parametric instability of microbeams in supercritical regime. Nonlinear Dynamics, 83, 1171–1183. Ghayesh, M. H., Farokhi, H., & Amabili, M. (2013a). Nonlinear behaviour of electrically actuated MEMS resonators. International Journal of Engineering Science, 71, 137–155. Ghayesh, M. H., Farokhi, H., & Amabili, M. (2014). In-plane and out-of-plane motion characteristics of microbeams with modal interactions. Composites Part B, 60, 423–439. Ghayesh, M. H., Kazemirad, S., & Reid, T. (2012). Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: A general solution procedure. Applied Mathematical Modelling, 36, 3299–3311. Ghayesh, M., Farokhi, H., & Amabili, M. (2013c). Coupled nonlinear size-dependent behaviour of microbeams. Applied Physics A, 112, 329–338. Gholipour, A., Farokhi, H., & Ghayesh, M. H. (2015). In-plane and out-of-plane nonlinear size-dependent dynamics of microplates. Nonlinear Dynamics, 79, 1771–1785. Hosseini, M., & Bahaadini, R. (2016). Size dependent stability analysis of cantilever micro-pipes conveying fluid based on modified strain gradient theory. International Journal of Engineering Science, 101, 1–13. Kahrobaiyan, M. H., Rahaeifard, M., Tajalli, S. A., & Ahmadian, M. T. (2012). A strain gradient functionally graded Euler–Bernoulli beam formulation. International Journal of Engineering Science, 52, 65–76. Karparvarfard, S. M. H., Asghari, M., & Vatankhah, R. (2015). A geometrically nonlinear beam model based on the second strain gradient theory. International Journal of Engineering Science, 91, 63–75. Ke, L.-L., Wang, Y.-S., Yang, J., & Kitipornchai, S. (2012). Nonlinear free vibration of size-dependent functionally graded microbeams. International Journal of Engineering Science, 50, 256–267. Kong, S., Zhou, S., Nie, Z., & Wang, K. (2008). The size-dependent natural frequency of Bernoulli–Euler micro-beams. International Journal of Engineering Science, 46, 427–437. Li, L., & Hu, Y. (2016). Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material. International Journal of Engineering Science, 107, 77–97. Li, Y. S., & Pan, E. (2015). Static bending and free vibration of a functionally graded piezoelectric microplate based on the modified couple-stress theory. International Journal of Engineering Science, 97, 40–59. Lü, C., Lim, C. W., & Chen, W. (2009). Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. International Journal of Solids and Structures, 46, 1176–1185. Mojahedi, M., & Rahaeifard, M. (2016). A size-dependent model for coupled 3D deformations of nonlinear microbridges. International Journal of Engineering Science, 100, 171–182. Nateghi, A., & Salamat-talab, M. (2013). Thermal effect on size dependent behavior of functionally graded microbeams based on modified couple stress theory. Composite Structure, 96, 97–110. Rahaeifard, M. (2016). Static behavior of bilayer microcantilevers under thermal actuation. International Journal of Engineering Science, 107, 28–35. Rahaeifard, M., Kahrobaiyan, M., Ahmadian, M., & Firoozbakhsh, K. (2013). Strain gradient formulation of functionally graded nonlinear beams. International Journal of Engineering Science, 65, 49–63. Sahmani, S., & Ansari, R. (2013). Size-dependent buckling analysis of functionally graded third-order shear deformable microbeams including thermal environment effect. Applied Mathematical Modelling, 37, 9499–9515. Sahmani, S., Bahrami, M., & Ansari, R. (2014). Nonlinear free vibration analysis of functionally graded third-order shear deformable microbeams based on the modified strain gradient elasticity theory. Compos. Struct., 110, 219–230. Shafiei, N., Kazemi, M., & Ghadiri, M. (2016b). Nonlinear vibration of axially functionally graded tapered microbeams. International Journal of Engineering Science, 102, 12–26. Shafiei, N., Kazemi, M., & Ghadiri, M. (2016a). On size-dependent vibration of rotary axially functionally graded microbeam. International Journal of Engineering Science, 101, 29–44. S¸ ims¸ ek, M. (2010). Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory. International Journal of Engineering Science, 48, 1721–1732. S¸ ims¸ ek, M., & Reddy, J. (2013). A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory. Composite Structure, 101, 47–58. Taati, E. (2016). Analytical solutions for the size dependent buckling and postbuckling behavior of functionally graded micro-plates. International Journal of Engineering Science, 100, 45–60. Tajalli, S., Rahaeifard, M., Kahrobaiyan, M., Movahhedy, M., Akbari, J., & Ahmadian, M. (2013). Mechanical behavior analysis of size-dependent micro-scaled functionally graded Timoshenko beams by strain gradient elasticity theory. Composite Structure, 102, 72–80. Tang, M., Ni, Q., Wang, L., Luo, Y., & Wang, Y. (2014). Nonlinear modeling and size-dependent vibration analysis of curved microtubes conveying fluid based on modified couple stress theory. International Journal of Engineering Science, 84, 1–10. Thai, H.-T., Vo, T. P., Nguyen, T.-K., & Lee, J. (2015). Size-dependent behavior of functionally graded sandwich microbeams based on the modified couple stress theory. Composite Structure, 123, 337–349. Zhang, B., He, Y., Liu, D., Gan, Z., & Shen, L. (2014). Size-dependent functionally graded beam model based on an improved third-order shear deformation theory. European Journal of Mechanics-A/Solids, 47, 211–230.