Thermal pre- and post-snap-through buckling of a geometrically imperfect doubly-clamped microbeam made of temperature-dependent functionally graded materials

Accepted Manuscript Thermal pre- and post-snap-through buckling of a geometrically imperfect doubly-clamped microbeam made of temperature-dependent functionally graded materials Amir Mehdi Dehrouyeh-Semnani, Hasan Mostafaei, Mohammad Dehrouyeh, Mansour Nikkhah-Bahrami PII: DOI: Reference:

S0263-8223(16)32728-3 http://dx.doi.org/10.1016/j.compstruct.2017.03.003 COST 8321

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

2 December 2016 18 January 2017 1 March 2017

Please cite this article as: Dehrouyeh-Semnani, A.M., Mostafaei, H., Dehrouyeh, M., Nikkhah-Bahrami, M., Thermal pre- and post-snap-through buckling of a geometrically imperfect doubly-clamped microbeam made of temperature-dependent functionally graded materials, Composite Structures (2017), doi: http://dx.doi.org/10.1016/ j.compstruct.2017.03.003

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Thermal pre- and post-snap-through buckling of a geometrically imperfect doubly-clamped microbeam made of temperature-dependent functionally graded materials Amir Mehdi Dehrouyeh- Semnania*), Hasan Mostafaei b), Mohammad Dehrouyeh c) , Mansour Nikkhah-Bahrami a) a) School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran b) Department of Civil Engineering, Isfahan University of Technology, Isfahan, Iran c) Department of Civil Engineering, Babol University of Technology, Babol, Iran

Abstract Owing to inappropriate fabrication processing it is likely to fabricate a geometrically imperfect functionally graded (FG) microbeam. In addition, the experimental tests show that the classical continuum theory is incapable of interpreting the mechanical behavior of microstructures when the role of size-dependency is significant. Therefore, this investigation aims to examine the influences of geometric imperfection on the nonlinear stability behavior as a prominent characteristic of microstructures for a thermally loaded doubly-clamped microbeam made of temperature-dependent FGMs by taking into account the size effect phenomenon. The geometrically nonlinear size-dependent governing equation of system is derived in the framework of modified couple stress theory in conjunction with Euler-Bernoulli beam theory and the classical rule of mixture. Based on the static form of governing equation, a closed-form solution for the nonlinear critical snap-through buckling temperature rise as well as the nonlinear thermal stability behavior of system in pre- and post-snap-through buckling domains is proposed and analytical study is then carried out by consideration of different effective parameters i.e., dimensionless imperfection amplitude, size-dependency, temperature-dependency, and power index. To verify the closed-form solution, the dynamic response of system is numerically evaluated by implementation of Galerkin scheme in conjunction with Runge-kutta finite difference method.

Keywords Geometrically imperfect doubly-clamped microbeam; Temperature-dependent FGMs; Critical snap-through buckling temperature rise; Nonlinear size-dependent thermal stability; Modified couple stress theory.

*

Corresponding author: Email address: [email protected], [email protected] (A.M. Dehrouyeh-Semnani)

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1. Introduction Functionally graded materials are inhomogeneous composite materials formed of two or more constituent phases with smoothly and continuously variable composition. Functionally graded materials have reduced thermal stresses and stress concentrations and have the capability of resisting high temperature gradient environments without losing structural reliability in comparison with isotropic and laminated materials. These advantages, together with their high strength and light weight, have made functionally graded materials an appropriate replacement for conventional materials. Hence, a lot of investigations have been carried out to examine the mechanical characteristics of functionally graded structures from macro to nano scale [1-18]. The experimental tests [19-23] proved that the mechanical behavior of microstructure may not obey the classical continuum theory due to the size effect phenomenon. Therefore, various higher order continuum theories have been developed to interpret the size-dependent behavior of structure in such scale. Yang et al. [24] introduced a modified couple stress continuum theory which is capable of explaining the size effect phenomenon in microstructures by employing one material length scale parameter in addition to the conventional Lame’s constants. Soon after that, this higher-order continuum theory became a popular non-classical continuum theory. Modified couple stress continuum theory has been employed by many researchers to establish the sizedependent mathematical models of mechanical elements as well as to examine the influence of size-dependency on mechanical behavior of microsystems [25-47]. Microbeams as continuous elements are widely used in many micro-electro-mechanical systems (MEMS) such as in microswitches, micro energy harvesters, atomic force microscopes, electro-statically excited micro-actuators, vibration shock sensors, and biosensors. Therefore, a lot of studies performed to explore the size-dependent mechanical characteristics of microbeams based on modified couple stress theory. The works associated with the size-dependent mechanical behavior of functionally graded microbeams based on modified couple stress theory are reviewed. Asghari et al. [48, 49] developed FG Euler-Bernoulli and Timoshenko beam models for static and free vibration analyses. Reddy [50] derive nonlinear governing equations of FG microbeams based on Euler-Bernoulli and Timoshenko beam theories for analysis of statics, free and forced vibrations as well as stability. Ke and Wang [51] investigated dynamic stability of FG Timoshenko microbeams under time-dependent axial force. Abbasnejad, et al. [52] studied nonlinear static and dynamic stability characteristics of a FG Euler-Bernoulli microbeam under electrostatic force. Akgöz and Civalek [53] explored free vibration characteristics of nonuniform FG Bernoulli–Euler microbeam. Nateghi et al. [54] developed a third-order shear deformable FG microbeam for static and dynamic analyses. Şimşek & Reddy [55] carried out an analytical study on static bending and free vibration behaviors of FG microbeams using various higher order beam theories. Zamanzadeh et al. [56] investigated stability behavior of a FG microbeam subjected to nonlinear electrostatic pressure and thermal changes regarding convection and radiation. Arbind and Reddy [57] developed nonlinear finite element formulations based on FG Euler-Bernoulli and Timoshenko beam models for static and dynamic

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analyses. Komijani et al. [58] examined nonlinear thermal stability and vibration of pre/postbuckled of extensible FG microbeams under in-plane thermal loading and resting on elastic foundation. Thai et al. [59] established governing equations of a three layered functionally graded microbeams based on Timoshenko beam theory for static bending, buckling and free vibration analyses. Akgöz and Civalek [60] conducted an analytical study on thermo-mechanical buckling behavior of an embedded FG simply supported microbeam based on sinusoidal shear deformation beam theory. Şimşek [61] performed a nonlinear study on free vibration of an axially functionally graded microbeam based on Euler-Bernoulli beam theory. DehrouyehSemnani et al. [62] developed a mathematical model for a three layered damped microbeam with functionally graded faces and viscoelastic core for vibration analysis. Shafiei et al. [63] investigated nonlinear free vibration of axially functionally graded tapered microbeams based on Euler-Bernoulli beam theory and von Kármán’s geometric nonlinearity. Ilkhani & HosseiniHashemi [64] analyzed free vibrations and stability behavior of spinning microbeam based on both Euler–Bernoulli and Timoshenko beam theories considering effects of tangential load and Coriolis force. Shafiei et al. [65] explored nonlinear free vibration of extensible tapered microbeams made of porous and imperfect FGMs. Shojaeian and Zeighampour [66] studied pullin behavior of FG sandwich microbridges subjected to electrostatic actuation effect and intermolecular Casimir forces based on Euler–Bernoulli, Timoshenko and Reddy beam theories. Akbarzadeh Khorshidi et al. [67] analyzed shear deformable FG microbeams in postbuckling domain using the first-order and higher-order beam theories. Ghorbani Shenas et al. [68] conducted a numerical study on free vibrational analysis of rotating pre-twisted FG Timoshenko cantilevered microbeams in thermal environment. Trinh et al. [69] examined mechanical behavior of FG microbeams using classical first-order, third-order, sinusoidal, and quasi-3D beam theories. Ghayesh et al. [70] investigated nonlinear forced oscillation of a third-order shear-deformable FG microbeam. Finally, it should be pointed out that all the reviewed works are related to the size-dependent mechanical analysis of geometrically perfect FG microbeams. In reality, due to improper manufacturing process, the geometrically imperfection in microbeams could be caused which may significantly affect the mechanical behavior of microbeams. Hence, some investigations have been made to study the influence of geometric imperfection on the size-dependent mechanical behavior of microbeams based on modified couple stress theory. Most of these investigations are associated with the size-dependent mechanical behavior of geometrically imperfect microbeams made of homogeneous isotropic materials [71-82]. The studies related to the stability behavior of geometrically imperfect homogeneous isotropic microbeams are reviewed. Farokhi and Ghayesh [74] investigated the size-dependent stability characteristics of a thermally loaded first-order shear deformable microbeam with geometric imperfection and under simply supported boundary conditions. They indicated the trend of natural frequency as well as the bifurcation diagram with respect to the temperature rise of an imperfect case is totally different from a perfect one. However, the snapthrough buckling didn’t investigate in this work. Farokhi and Ghayesh [73] and Farokhi et al. [78] analyzed size-dependent steady state response of Euler-Bernoulli and Timoshenko pinned-

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pinned microbeam with geometric imperfection under axial load, respectively. Likewise the previous work, a complete different trend for the stability behavior of imperfect case was obtained also the snap-through buckling wasn’t explored in this work. The snap-through and pull-in instabilities of a geometrically imperfect doubly-clamped Euler-Bernoulli microbeam under DC and AC voltages were studied by Ghayesh et al. [81]. Shojaeian et al. [82] examined the snap-through and pull-in instabilities of previous case under electro-elastic force and prestress considering Cassimere force. Some comparative studies with available experimental data in literature were performed. It should be pointed out that all the results in Refs. [71-82] were obtained based on numerical methods. The size-dependent free and forced vibrations of geometrically imperfect FG microbeams via modified couple stress theory have been numerically studied by Dehrouyeh-Semnani et al. [83] and Ghayesh et al. [84], respectively. To the best knowledge of authors, no single study exists which examines the size-dependent stability response of geometrically imperfect doubly-clamped microbeam made of FGMs under in-plane thermal loading based on modified couple stress continuum theory. Since the thermo-mechanical stability behavior of geometrically imperfect FG microbeam is a prominent characteristic of such microstructures, this paper aims to study the nonlinear size-dependent thermal stability characteristics of a geometrically imperfect temperature-dependent FG microbeam with clamped-clamped boundary conditions under uniform thermal loading. The outline of the paper is as follows: In Section 2, the mathematical formulations of modified couple stress theory and temperature-dependent functionally graded materials are reviewed. The coupled longitudinal-transverse governing equations and related boundary conditions of geometrically imperfect FG microbeams under in-plane uniform thermal loading are then developed by means of Hamilton's principle in conjunction with modified couple stress theory, Euler-Bernoulli beam theory, and von Kármán nonlinearity. Afterwards, the coupled governing equations and also the corresponding boundary conditions are rewritten in terms of transverse displacement. In Section 3, on the basis of governing equation in the static form and the clamped-clamped boundary conditions a closed form solution for thermal stability response is proposed. Afterwards, the governing equation in dynamic form is numerically solved by means of Galerkin discretization procedure in conjunction with an embedded Runge-Kutta method. In Section 4, by aid of the obtained closed-form solution, the critical temperature in which snap-through buckling occurs and the equilibrium path of system in pre- and post-snapthrough buckling are explored by regarding different effective parameters. Some results obtained based on the analytical solution are then verified with those achieved by implementation of the numerical method. The paper concludes with Section 5, where the work is summarized and the final remarks are presented.

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2. Mathematical formulation 2.1. Modified couple stress theory In view of modified couple stress theory, the strain energy
in a linear elastic isotropic material occupying region  can be written as [24]:

U =

1 ( σ : ε + m : χ ) dV 2 V∫

(1)

where , , , and  are the strain tensor, the Cauchy (classical) stress tensor (conjugated to ε ), the symmetric curvature tensor, and the deviatoric part of the couple stress tensor (conjugated to χ ), respectively. These tensors satisfy the geometrical and the constitutive equations [24]:

1 1 T T ∇u + ( ∇u ) , χ = ∇θ + ( ∇θ ) 2 2 (2a-d) 2 σ = λtr ( ε ) I + 2µ ε, m = 2µ
χ where  = ( ,  ,  ) stands for the displacement vector along x, y and z axes, respectively. The x, y, and z coordinates are considered across the length, the thickness, and the width of microbeam, respectively. In addition is the rotation vector,  and  are Lamé constants, and ℓ is a material length scale parameter. The rotation vector can be calculated by use of the following relationship [24]. ε=

(

)

(

)

1 1  ∂u ∂u   ∂u ∂u   ∂u ∂u   θ = curl ( u ) =  3 − 2  i +  1 − 3  j +  2 − 1  k  2 2  ∂y ∂z   ∂z ∂x   ∂x ∂y  

(3)

2.2. Functionally graded materials It is supposed that the geometrically imperfect functionally graded (FG) microbeam is made of two distinct materials (metal and ceramic) and the volume fraction of material phases are assumed to change smoothly across the thickness direction of microbeam. There exist various methods i.e., the classical rule of mixture (Voigt rule) [6, 12-14, 48-50, 57, 58, 62, 83], the exponential model [56], Mori-Tanaka scheme [35, 51, 55, 68, 70, 84], and the self-consistent estimate [18], to evaluate the effective physical and mechanical properties of functionally graded materials (FGMs) based on the above-mentioned assumptions. Among all the methods, the classical rule of mixture and Mori-Tanaka scheme are very popular to estimate the effective properties FGMs. The volume fraction of the ceramic and metal phases of the FG microbeam for both the mentioned models is assumed to be given by

z 1 −h h ≤z≤ Vc = ( + ) n , Vm = 1 − Vc , (4) h 2 2 2 where n and h denote the non-negative power law exponent and the thickness of geometrically imperfect FG microbeam, respectively. In addition, the subscripts m and c refer to the metal and ceramic materials, respectively.

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According to the classical rule of mixture, the effective material properties can be obtained by use of a simple relationship (Eq. (5)). In this method, each effective material property is only dependent on the corresponding material property of the metal and ceramic phases and the volume fractions. In Mori–Tanaka scheme, extracting the effective material properties is based on the distributed small spherical particles (metal phase) into matrix (ceramic phase) [18]. Based on Mori-Tanaka scheme, the effective Young’s modulus and Poisson’s ratio can be calculated as functions of the effective bulk and shear modulus which these modulus evaluated as functions of both the bulk and shear modulus of metal and ceramic phases in addition to the volume fraction. Moreover, the effective thermal expansion coefficient can be obtained as functions of abovementioned effective modulus and the thermal expansion coefficient of ceramic and metal phases. It should be pointed out that according to Mori-Tanaka scheme; there exists no particular formulation for the density. In a recent research, Kiani and Eslami [18] investigated temperature-dependent and – independent postbuckling of circular FG plates with geometric imperfection based on the classical rule of mixture, Mori-Tanaka scheme and the self-consistent estimate. They stated that the predicted critical buckling temperature/heat flux based on these approaches did not differ significantly with respect to each other. Based on the reported results, the maximum difference was estimated about 3%, which was observed at high temperature levels. Hence, in this study the classical rule of mixture as a simple method is employed to calculate the effective material properties of FGMs. Based on the classical rule of mixture (Voigt rule), the temperature-dependent properties of FG materials along the thickness of FG microbeam can be expressed as:

P (z,T ) = Pm (T ) V m (z ) + Pc (T ) V c (z ) (5) where  can be replaced by any material properties. Substituting Eq. (4) into Eq. (5) results in: n

z 1 (6) P (z,T ) = Pm (T ) + Pcm (T )  +  , Pcm (T ) = Pc (T ) − Pm (T ) h 2 It should be pointed out that the lower and upper surfaces of FG microbeam are metal-rich and ceramic-rich, respectively (see Fig. 1).

2.3. Governing equations Consider an extensible Euler–Bernoulli microbeam with a geometric imperfection in z direction (Fig. 1), undergoing moderately large transverse displacements. The engineering strain ( ) accounting for stretching of the midplane surface of the microbeam can be obtained by [85]

ds −1 (7) ds 0 where  and  stand for the lengths of a line element of the centerline of the imperfect microbeam in the undeformed and deformed configurations, respectively. In addition, , , and

ε0 =

6

 denote the longitudinal displacement, transverse displacement, and geometric imperfection respectively.  and  as functions of , , and  can be obtained by 1/2

1/2   ∂u 2  dw 0 ∂w  2   dw 0  ds 0 =  1 + +  dx , ds =   1 +  +   dx dx  ∂x      ∂x   dx In view of Eq. (8), Eq. (7) can be rewritten as

(8)

1/2

  ∂u  2  dw 0 ∂w  2  +   1 +  +   ∂ x dx ∂ x      ε0 =  1/2  dw 0  1 +  dx  

(9)

−1





Expanding Eq. (9) and retaining terms up to second order as well as assuming   to be 



small compared with    result in the following expression for the engineering strain ( ) 2

∂u ∂w dw 0 1  ∂w  (10) ε0 = + +   ∂x ∂x dx 2  dx  To take into account the bending effect, the bending strain field is considered based on EulerBernoulli beam theory as follows ∂ 2w ∂x 2 The total strain can be obtained by summing the strains given in Eqs. (10)-(17).

ε b = −z

(11)

2

∂u 1  ∂w  ∂w dw 0 ∂ 2w (12) +  + − z  ∂x 2  ∂x  ∂x dx ∂x 2 It should be noticed that the other components of strain tensor based on Euler-Bernoulli beam theory are zero. Consider the displacement field for a geometrically imperfect microbeam based on Euler-Bernoulli beam theory as follows:

ε xx =

∂w , u 3 = w ( x ,t ) + w 0 ( x ) (13) ∂x The non-zero components of curvature tensor (corresponding to zero initial curvature) for an extensible microbeam with geometric imperfection based on Euler-Bernoulli beam theory can be obtained by [72, 77, 81]: u1 = u − z

χ xy = χ yx

1 ∂ 2w =− 2 ∂x 2

(14)

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By substitution of Eqs. (12) and (14) into Eqs. (2c-d) and the assumption of uniaxial stressstrain relationship[86], one can obtain the non-zero components of the stress and the couple stress tensors as follows:

σ xx = E (T , z ) (ε xx − εT ) =  ∂u 1  ∂w  2 ∂w dw 0 d 2w E (T , z )  z +  + − − α (T , z )∆T  ∂x 2  ∂x  ∂x dx dx 2  ∂ 2w m xy = m yx = − µ (T , z )
2 (T , z ) 2 ∂x in which

  

(15)

(16)

εT = α (T , z )∆T

(17)

where α and ∆ are the thermal expansion coefficient of FGMs and the temperature variation from a reference temperature  , respectively. In view of Eq. (1) and considering the thermal effect, the size-dependent strain energy U can be achieved by U =

1 (σ xx ε xx + 2m xy χ xy − σ xx εT ) dV = 2 ∫V 2 1 L   ∂u 1  ∂w  ∂w dw 0   A11  +    + 2 ∫0   ∂x 2  ∂x  ∂x dx  

∂ 2w -2B 11 2 ∂x

2

 ∂u 1  ∂w  2 ∂w dw 0   ∂ 2w  + + D + S ( )  +   11 12   2  ∂x dx   ∂x   ∂x 2  ∂x 

(18)

2

2   ∂u 1  ∂w  2 ∂w dw 0  T ∂ w - 2N  +  + + 2 M + N  0  dx   ∂x dx  ∂x 2  ∂x 2  ∂x   in which T

{A11 , B11 , D11} = ∫ E (T , z ) {1, z, z 2 } dz , A

{N

T

T

,M

E (T , z )
2 (T , z ) dz 2(1 +ν (T , z )) A

S12 = ∫

} = ∫ E (T , z )α (T , z )∆T {1, z}d z ,

N 0 = ∫ E (T , z ) (α (T , z )∆T

A

A

)

2

(19)

dz

where ! , ! and "! are the thermal stress resultants and # , $ , % , and &  are the extensional, extensional-bending, bending, and in-plane shear stiffness coefficients, respectively. The kinetic energy of system can be expressed as:

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2 2 2  ∂u ∂ 2w   ∂ 2w   1 L    ∂u   ∂w   K = ∫ I 0    +   − 2I 1  +I2    dx 2 0    ∂t   ∂t   ∂t ∂t ∂x  ∂t ∂x       in which

{I 0 ,

I 1, I 2 } = b ∫

h /2

−h /2

ρ (z) {1, z, z 2 } dz

(20)

(21)

The variation of the work done by the viscous damping due to the transverse motion is given by L

∂w δw dx ∂t 0 where ' is the viscous damping coefficient of the transverse displacement.

δW d = −cw ∫

(22)

One can obtain the governing equations and related boundary conditions by means of Hamilton's principle as follows: -

∂ ∂x

  ∂u 1  ∂w 2 ∂w dw 0 +  −NT  A11   +  ∂x dx  ∂x 2  ∂x  

 ∂ 2w − B  11 ∂x 2 

 ∂ 2u ∂ 3w  + I 0 2 − I1 2 = 0  ∂t ∂t ∂x 

 ∂u 1  ∂w 2 dw 0 ∂w  ∂2  ∂ 2w - 2  B11  +  + − D + S −M T ( )  11 12  2 ∂x  ∂ x 2 ∂ x dx ∂ x ∂ x      ∂u 1  ∂w 2 dw 0 ∂w  ∂  ∂ 2w  A + + − B    11 11   ∂x  dx ∂x  ∂x 2  ∂x 2  ∂x    ∂ 2w d 2w 0 +N T  2 + dx 2  ∂x

   

  ∂w dw   0  +    ∂x dx   

(23a-b)

 ∂w ∂ 2w ∂ 3u ∂ 4w + c + I + I − I =0  w 0 1 2 ∂t ∂t 2 ∂t 2 ∂x ∂t 2 ∂x 2 

 ∂u 1  dw  2 dw 0 ∂w  ∂ 2w T A11  +   − B 11 2 − N = 0 or u =u s  + ∂ x 2 dx dx ∂ x ∂ x     ∂ 

 ∂u 1  dw  2 dw 0 ∂w  ∂ 2w T  B 11  +   − ( D11 + S 12 ) 2 − M  + ∂x  dx ∂x  ∂x  ∂x 2  dx 

  

(24a-c) 2 2 3   ∂u 1  dw  2 dw 0 ∂w    ∂w dw 0  ∂w ∂u ∂w T +  A11  +   +  − B 11 ∂x 2 − N   ∂x + dx  − I 1 ∂t 2 + I 2 ∂t 2 ∂x = 0 or w =w s    ∂x 2  dx  dx ∂x   2   ∂u 1  dw  dw 0 ∂w  ∂ 2w  ∂w ∂w s −  B 11  +  + − ( D11 + S 12 ) 2  + M T = 0 or =     dx ∂x  ∂x  ∂x ∂x   ∂x 2  dx  By removing the geometric imperfection  from the governing equations and boundary conditions, Eqs. (23a-b) and (24a-c) reduce to those derived for an Euler-Bernoulli FG extensible

microbeam based on modified couple stress theory by Reddy [50]. In addition, the nonlinear size-dependent governing equations of a homogeneous isotropic extensible microbeam with

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geometric imperfection established by Ghayesh and Amabili [72] can be recovered from the present model by neglecting the influences of functionally graded materials i.e., (=0 or ∞ and thermal loading i.e., ∆=0. It is assumed that the longitudinal inertia and the longitudinal-bending inertia are negligible. Besides, the derivation of the thermal force ! with respect to * is zero. Consequently, Eq. (23ab) can be written as follow:   ∂u 1  ∂w 2 ∂w dw 0  ∂ 2w   A11  +   + ∂x dx  − B 11 ∂x 2  = 0   ∂x 2  ∂x     Eq. (26) can be obtained by integrating Eq. (25). ∂ ∂x

(25)

 ∂u 1  ∂w  2 ∂w dw 0  ∂ 2w A11  B +  + − = C 1 (t )  11  ∂x dx  ∂x 2  ∂x 2  ∂x 

(26)

where + is a constant that represents the induced axial force due to the midplane stretching. Integrating Eq. (26) results in the following relation 2  x  1  ∂w  dw 0 ∂w   ∂w A11 u + ∫   + = C 1 (t ) x + C 2 (t )  dx  − B 11  0 dx ∂x   ∂x  2  ∂x  

(27)

where + is another integration constant. For the midplane stretching and thermal effect to be important, the microbeam ends must be restrained. Therefore, the related boundary conditions for the axial displacement are supposed as follows: u ( 0, t ) = 0, u ( L , t ) = 0

(28)

By applying the boundary conditions associated with the axial displacement, one can obtain + and + as follows: ∂w ( 0,t ) A L  1  ∂w  dw 0 ∂w  B  ∂w ( L , t ) ∂w ( 0, t )  C 1 = 11 ∫   + dx − 11  − , C 2 = − B 11 (29)    L 0  2  ∂x  dx ∂x  L  ∂x ∂x ∂x  Finally, the governing equations and related boundary conditions can be rewritten in terms of transverse displacement  as follows: 2

2   ∂ 4w  A11 L  1  ∂w  dw 0 ∂w  B 112 D − + S −    dx   11 12   + 4 ∫ 0 A ∂ x L 2 ∂ x dx ∂ x      11   

−

B 11  ∂w ( L , t ) ∂w ( 0, t )  T −  −N L  ∂x ∂x 

2 2   ∂ w d w 0 +  2 dx 2   ∂x

 ∂w ∂ 2w ∂ 4w +c + − =0 I I  w 0 2 ∂t ∂t 2 ∂t 2∂x 2 

10

(30)

  ∂3w  A B2 −  D11 − 11 + S 12  3 +  11 A11   ∂x  L −

∫

L

0

B 11  ∂w ( L ) ∂w ( 0 )  T −  −N L  ∂x ∂x 

 1  ∂w  2 dw 0 ∂w     + dx ∂x  dx 2 ∂ x    

 ∂w dw 0  ∂ 3w +   + I 2 2 = 0 or w =w s dx  ∂t ∂x   ∂x

2 B  A L  1  ∂w  dw 0 ∂w  B 11  ∂w ( L ) ∂w ( 0 )   − 11  11 ∫   + dx − −    A11  L 0  2  ∂x  dx ∂x  L  ∂x ∂x  

  ∂ 2w B2 +  D11 − 11 + S 12  2 + M T = 0 A11   ∂x

or

(31a-b)

∂w ∂w s = ∂x ∂x

By ignoring the effect of functionally graded materials as well as the thermal loading, the current governing equation reduces to that obtained by Farokhi and Ghayesh [77] for a homogeneous isotropic extensible microbeam with geometric imperfection based on modified couple stress theory. In order to be able to write the governing equation in normalized form, the following dimensionless quantities are introduced: D − B 112 A11 x w w t , η = , η0 = 0 , τ = 11 (32) L L L I 0 L4 By substituting these parameters into Eqs. (30) and (31a-b), the dimensionless governing equation and corresponding boundary conditions are obtained as follow:

ζ =

2 ∂ 4η  1  1  ∂η  d η0 ∂η  dζ (1 + θ ) 4 − α1 ∫0    + ∂ζ 2 ∂ ζ d ζ ∂ ζ      

 ∂η (1,τ ) ∂η ( 0,τ )    d 2η d 2η0  ∂ 2η ∂ 4η −α 2  − + −δ 2 2 = 0  − γ  + ∂ζ   d ζ 2 d ζ 2  ∂τ 2 ∂τ ∂ζ  ∂ζ

(33)

2 ∂ 3η  1  1  ∂η  d η0 ∂η  d ζ − (1 + θ ) 3 + α1 ∫   + 0  2 ∂ζ   ∂ζ d ζ ∂ ζ     

 ∂η (1) ∂η ( 0 )    ∂η d η0 −α 2  − +  − γ  ∂ζ    ∂ζ d ζ  ∂ζ

 ∂ 3η δ + =0  ∂τ 2 ∂ζ 

or η =ηs

2  1  1  ∂η  d η0 ∂η  α 22  ∂η (1) ∂η ( 0 )  ζ − α 2 ∫   +  d − −   0  2 ∂ζ  α1  ∂ζ ∂ζ   d ζ ∂ζ     

∂ 2η +κ = 0 ∂ζ 2 in which + (1 + θ )

or

∂η ∂ζ

 ∂η  =   ∂ζ s

11

(34)

A11L 2 B11L S 12 α1 = , α2 = , θ= 2 2 D11 − B11 A11 D11 − B 11 A11 D11 − B 112 A11

(35)

I NT L2 MTL cw L2 γ= , κ = , c = δ = 22 d 2 2 D11 − B11 A11 D11 − B11 A11 I 0L I 0 (D11 − B112 A11 ) The boundary conditions of a doubly-clamped FG microbeam with geometric imperfection are as follows:

η ( 0,τ ) = η (1,τ ) =

∂η ( 0,τ )

∂η (1,τ )

(36) =0 ∂ζ ∂ζ Therefore, the governing equation for a FG doubly clamped microbeam can be rewritten as:

=

 1  1  ∂η  2 d η0 ∂η    d 2η d 2η 0  ∂η ∂ 2η ∂ 4η + −δ 2 2 = 0 (1 + θ ) 4 − α1 ∫0    +  d ζ − γ  2 +  + cd ∂ζ dζ 2  ∂τ ∂τ 2 ∂τ ∂ζ   d ζ  2  ∂ζ  d ζ ∂ζ   ∂ 4η

(37)

3. Solution methods 3.1. Statics-based closed form solution In this subsection, a closed form solution for evaluation of pre- and post-snap through behavior of system based on the governing equation Eq. (33) in static form is developed. The governing equation in the static form can be obtained by dropping the time-dependent terms in Eq. (33). 2   d 2η d 2η  d 4η  1  1  d η  d η0 d η  0 + =0  d ζ − γ  (1 + θ ) 4 − α1 ∫0    + 2 2  dζ 2 d d d d d ζ ζ ζ ζ ζ        

(38)

Eq. (38) can be rewritten as: 2 2 d 4η 2 d η 2 d η0 +λ = −λ dζ 4 dζ 2 dζ 2 in which

(39)

2 1  1  dη  d η0 d η   1  λ = γ α − +  dζ  (40)  (1 + θ )  1 ∫0  2  d ζ  ∂ζ d ζ   The homogeneous solution of Eq. (40) with the constant coefficient defined by Eq. (39) can be obtained by 2

η h (ζ ) = c 1 + c 2ζ + c 3 cos(λζ ) + c 4 sin( λζ ) where ' -', are constants to be determined by applying the boundary conditions.

12

(41)

Initial configuration of a geometrically imperfect doubly-clamped FG microbeam is supposed to have the following form [87, 88] 1 2

η 0 (ζ ) = A 0 (1 − cos(2πζ ))

(42)

where # is the imperfection amplitude. So, the particular solution of Eq. (39) can be written as follow

η p (ζ ) = c 5 cos(2πζ ) + c 6 sin(2πζ ) Substituting the particular solution Eq. (43) into the governing equation Eq. (39), yields

λ 2A0 , c6 = 0 c5 = − 2(4π 2 − λ 2 )

(43)

(44)

Consequently, the solution of Eq. (39) with the geometric imperfection configuration given in Eq. (42) can be obtained by

λ 2A0 η (ζ ) = ηh (ζ ) + η p (ζ ) = c1 + c 2ζ + c 3 cos( λζ ) + c 4 sin(λζ ) − 2 cos(2πζ ) (45) 4π ( 4π 2 − λ 2 ) Applying the boundary conditions given in Eq. (36), yields the following four algebraic equations

λ 2A0 =0 2(4π 2 − λ 2 ) c 2 + c 4λ = 0

c1 + c 3 −

(46a-d)

λ 2A0 =0 c1 + c 2 + c 3 cos λ + c 4 sin λ − 2 ( 4π 2 − λ 2 )

c 2 − c 3λ sin λ + c 4λ cos λ = 0 The first three equations are solved for the constants ' , ' and ', in terms of ' . One can obtain the following equations (λ 2 A 0 − 2c1 (4π 2 − λ 2 ))λ (1 − cos λ ) 2(4π 2 − λ 2 )(λ − sin λ ) λ 2 A 0 − 2c1 (4π 2 − λ 2 ) c3 = (47a-c) 2(4π 2 − λ 2 ) (λ 2 A 0 − 2c1 (4π 2 − λ 2 ))(1 − cos λ ) c4 = 2(4π 2 − λ 2 )(sin λ − λ ) Substitution of Eq. (47a-c) into Eq. (46a-d) and simplifying, the following characteristic equation can be derived c2 =

13

 λ 2 A 0 − 2c1 ( 4π 2 − λ 2 )   −2 + 2 cos λ − λ sin λ    (48) =0 2 2   sin λ π λ 2 4 −  ( )   In addition to the trivial solution, the following solutions can be achieved for the characteristic equation. c1 =

λ 2 A0 2 ( 4π 2 − λ 2 )

2 − 2 cos λ − λ sin λ = 0

or

(49a-b)

In the case of an imperfect microbeam (# ≠ 0) c1 =

λ 2A0 2 ( 4π 2 − λ 2 )

(50)

Substituting Eq. (50) into Eq. (47a-c) results in

c 2 =c 3 = c 4 = 0 (51) Substitution of Eq. (51) into Eq. (45) results in the following solution for a geometrically imperfect doubly-clamped FG microbeam. 1 2 in which

η (ζ ) = A (1 − cos 2πζ )

A=

(52)

λ 2b (4π 2 − λ 2 )

(53)

Substitution of Eq. (52) into Eq. (40) yields

 A 2π 2 A 0 A π 2   1  + λ = γ − α1   2   (1 + θ )   4 2

(54)

Eq. (54) can be rewritten as: A = −A 0 ± A 02 −

4

α1π 2

( (1 + θ ) λ

2

−γ )

(55)

 can be determined by substitution of Eq. (53) into Eq. (54). ∆ 3 + a2 ∆ 2 + a1∆ + a0 = 0 in which

(56)

 γ   16γπ 4 8π 2γ 2α A 2  α A 2   , a1 = + π 4 16 + 1 0  , a2 = −  + π 2  8 + 1 0   , ∆ =λ 2 (57) 1+θ 1+θ 1+θ  4(1 + θ )   1 + θ   Eq. (56) is a cubic polynomial with respect to ∆ and always has at least one real root. As two roots of Eq. (56) coalesce, the snap-through buckling takes place. To that end, we have a0 = −

14

( 4γ + π (α A 2

1

2 0

)

3

− 16(1 + θ ) ) − 1728α1A 02π 6 (1 + θ ) 2 = 0

(58)

The critical dimensionless thermal force /01 in which the snap-through buckling happens can be obtained by solving Eq. (58) with respect to / . 

γ c = 4π 2 (1 + θ ) + π 2  3α11/3A 02/3 (1 + θ )2/3 −

α1A 02 

(59)  4   Eventually, the response of a geometrically imperfect doubly-clamped FG microbeam under in-plane thermal loading can be obtained by the following equation.

1 2

 (1 + θ ) ∆ − γ   (1 − cos 2πζ ) α1π  in which ∆ can be obtained from Eq. (56).

η (ζ ) =  −A 0 ± A 02 −

4

2

(60)

In the case of a geometrically perfect doubly-clamped FG microbeam (# = 0)

4sin

λ

(sin

λ λ

λ

(61) cos ) = 0 2 2 2 2 The lowest root of the characteristics equation Eq. (61) is 24. Substituting the values of 5=0 and  = 24 into Eq. (47a-c) results in:

−

c 2 = c 4 = 0, c3 = − c1 (62) Substitution of Eq. (62) into Eq. (41) the following expression may be obtained for a perfect FG microbeam. 1 (63) A (1 − cos 2πζ ) 2 where # is the dimensionless amplitude of buckling. Substituting Eq. (63) into Eq. (40) results in

η (ζ ) =

A =±

2

π

1

γ − 4π 2 (1 + θ )  α1

(64)

Eq. (64) verifies that the critical dimensionless thermal force is 44  (1 + :) which is in agreement with linear analysis. Therefore, the dimensionless amplitude c is zero for the dimensionless thermal forces which are lower than the critical one.

3.2. Dynamics-based numerical solution In this subsection, the stability behavior of system is numerically determined based on the governing equation Eq. (37). To that end, the integro-partial differential equation of motion is discretized by aid of Galerkin method. The following series expansion is introduced to discretize the governing equation.

15

N

η (ξ ,τ ) = ∑φn (ξ ) p n (τ )

(65)

n =1

in which <= represent the nth generalized coordinate of transverse motion, and >= denotes the nth eigenfunction for the transverse motion of a linear doubly-clamped microbeam. The eigenfunctions can be obtained by:

φn(4) = ωn2φn (66) φn (0) = φn (1) = φn′ (0) = φn′ (1) = 0 It is obvious that the eigenfunctions satisfy the boundary conditions listed in Eq. (36). Application of Galerkin method to Eq. (37), by inserting Eq. (65) into Eq. (37), multiplying the resultant equations by the corresponding eigenfunctions, and integrating with respect to ? from 0 to 1, results in the following nonlinear ordinary differential equations. Mij pj + Kij p j + Cij p j + Aijk p j pk + Bijkl p j pk pl = Fi , i,j,k,l = 1,2,..N in which 1

1

∂ 2φ j

0

0

∂ζ 2

M ij = ∫ φi φ j d ζ −δ ∫ φi

(67)

dζ

1

Cij = c d ∫ φi φ j d ζ 0

∂ 4φ j

   1 d η0 ∂φ j d 2η 0 d d ζ ζ    4 2 2 ∫ ∫ 0 0 ∂ζ ∂ζ   0 dζ   0 d ζ ∂ζ (68)  1 ∂φ j ∂φk    1 ∂ 2φk   1 d η 0 ∂φ j α1  1 d 2η0 A ijk = −  ∫ φi dζ  ∫ d ζ  − α1  ∫ φi dζ  ∫ dζ  2 2  0 dζ 2    0 ∂ζ ∂ζ   0 ∂ζ   0 d ζ ∂ζ 2  1 ∂ φl   1 ∂φ j ∂φk  B ijkl = −α1  ∫ φi d dζ  ζ   ∫0 2 0 ∂ζ    ∂ζ ∂ζ  2 1 d η0 Fi = −γ ∫ φi dζ 0 dζ 2 The embedded Runge–Kutta’s finite difference method RK5(4) established by Dormand and Prince [89] is utilized to solve Eq. (67). 1

K ij = (1 + θ ) ∫ φi

1

d ζ + γ ∫ φi

∂ 2φ j



d ζ − α1 

1

φi

4. Thermal snap-through buckling In this section, the nonlinear thermal stability behavior of a temperature-dependent functionally graded doubly-clamped microbeam with geometric imperfection under uniform temperature rise is investigated via the statics-based closed form solution proposed in Eqs. (59) and (60). Moreover, some analytical results are verified with the dynamics-based numerical solution presented in subsection 3.2.

16

In this study, the metal and ceramic phases of functionally graded material are supposed to be made of SUS304 and Si3N4, respectively. The material properties of SUS304 and Si3N4 are considered to be temperature-dependent according to Touloukian model [90]. 2 2 P (T ) = P0 ( P−1T −1 + 1 + PT ) 1 + P2T + PT 3

(69)

where  denotes a typical material property of the metal and ceramic phases. The constants @ ,  ,  ,  , and  for each property of the metal and ceramic phases are given in Table 1 [90]. For the temperature-independent case (TID) the material properties are calculated at the reference temperature  =300. Furthermore, for the temperature dependent case (TD) the material properties of the metal and ceramic phases at an arbitrary temperature are evaluated based on Eq. (69). In this work, it is assumed that the constant  for the material length scale parameter of both the metal and ceramic phases is 10µm and the constants @ ,  ,  , and  for both the material length scale parameters are zero. In other words, the material length scale parameters like the densities and Poisson’s ratios are temperature-independent. In addition, in the analysis the aspect ratio A/ℎ is considered to be 60.

4.1. Critical snap-through buckling temperature rise In this subsection, an investigation is conducted to show influences of the dimensionless imperfection amplitude A , the power index n, and the ratio of the thickness to the material length scale parameter ℎ/ℓ on the critical temperature rise Δ01 at which the system loses its stability via a snap-through buckling. The size-dependent and –independent critical snap-through buckling temperatures rise Δ01 as a function of A for several values of ℎ/ℓ and n are depicted in Fig. 2. The results plotted in the figure illustrate that the general trend of Δ01 with respect to A is an initially ascending followed by a descending for both the modified couple stress theoryand classical theory-based models. It can be seen from the figures that the system becomes unstable at the reference temperature i.e., Δ01 =0 at a particular dimensionless imperfection F F amplitude A . The plots reveal that A has a descending trend with respect to ℎ/ℓ. The results show that the assumption of temperature-independency leads to increase of Δ01 . In addition, it can be deduced that when the role of size-dependency is increases (i.e., ℎ/ℓ is decreased) the difference between Δ01 predicted based on TID model with that obtained via the TD model rises. In order to better illustrate the effect of power index n on the critical temperature rise Δ01 , Fig. 3 is constructed by use of the temperature-dependent model. As seen from the first two plots, the critical temperature rise Δ01 has a descending trend with respect to the power index n based on both the classical and non-classical models. However, the third plot indicates that the F trend around A at which the system buckles at the temperature reference, is different from those depicted in the first two plots.

17

4.2. Pre- and post-snap-through buckling characteristics In this subsection, the nonlinear size-dependent stable and unstable equilibrium paths of a geometrically imperfect doubly-clamped FG microbeam under in-plane uniform thermal loading are addressed via use of Eq. (64) in conjunction with Eq. (60). Besides, some comparative studies between the analytical and numerical solutions are performed. Figure 4 illustrates the size-dependent and-independent stable and unstable equilibrium paths of temperature-dependent-based model of thermally loaded geometrically imperfect doublyclamped FG microbeam for A =0.015 and power indexes n=0, 0.5, 5, ∞. The upper and lower branches (solid lines) are associated with the stable responses and the middle branches (dash lines) related to the unstable responses. As seen from the plots, in the pre-snap-through buckling domain (∆<Δ01 ) there is only one stable response. Unlike the pre-buckling response of a perfect doubly-clamped FG microbeam under thermal loading [58], this response is non-zero. In addition, it can be seen that this response continues in the post-snap-through domain (∆>Δ01 ). In other words, this response is independent of the snap-through buckling. It can be easily deduced from the figure that the amplitude of this response has an ascending trend with respect to the temperature rise ∆. The plots indicate that the dimensionless amplitude of this response has a descending trend with respect to the power index n. In other words, the maximum and minimum amplitudes of the upper branch at a prescribed temperature rise ∆ are corresponding to the metal-rich (n=∞) and ceramic-rich (n=0) cases, respectively. Moreover, as ∆ is increased the role of power index in the dimensionless amplitude becomes more remarkable. Fig. 4 shows that due to the snap-through buckling one stable response (solid line) and one unstable response appear in the solution. Unlike, the post-buckling response of a perfect doublyclamped FG microbeam under thermal loading [58], the upper branch doesn’t coalesce into the middle (the unstable response) and the lower (the stable response) branches at the critical temperature rise Δ01 . This is the reason for calling this type of buckling as snap-through buckling. It should be pointed out this type of bifurcation is called perturbed pitch fork bifurcation. The depicted results demonstrate that by increasing ∆ the absolute values of # + # for the unstable responses decrease and tend to zero. However, it has an ascending trend with respect to ∆ for the lower branch. The results indicate that when the influence of sizedependency is increased i.e., ℎ/ℓ is decreased, the values of # + # for the upper branches decrease. Due to the remarkable influence of size-dependency on the onset of snap-through buckling, the effect of size-dependency on the middle and lower branches is more significant. In the post-snap-through buckling domain, based on both the classical and non-classical models at a prescribed ∆ the absolute value of # + # for the lower and middle branches decreases and increases, respectively, by increasing the role of size-dependency i.e., decreasing ℎ/ℓ. The influence of dimensionless imperfection amplitude A on the temperature-dependent stable and unstable responses of system in pre- and post-snap-through buckling domains based on the classical and non-classical models is shown in Fig. 5. It can be seen that the dimensionless

18

imperfection amplitude # decreases for the upper branch when A is increased. In addition, the results show that the upper branch of the imperfect model crosses the upper branch of the perfect model. The crossing occurs at a higher temperature rise for an imperfect microbeam with a smaller A . It can be deduced that the amplitude # based on the imperfect model is greater than that based on the perfect model before crossing, but after crossing a reverse scenario occurs. As shown in the figure, the absolute dimensionless amplitude of the middle and lower branches increases as A is increased. In other words, the trend of the absolute dimensionless amplitude with respect to A based on the upper branch has an exact opposite of those based on the middle and lower branches. As seen, a crossing happens between the responses of the imperfect microbeam and the lower branch of the perfect microbeam for both the size-dependent and – independent models. In addition, it can be inferred that the crossing between the lower branch of the perfect microbeam can be took place with the middle branch or the lower branch of the imperfect microbeam depending on the values of A and ℎ/ℓ. The effect of temperature-dependency on the size-dependent and –independent responses of system in the pre- and post-snap-through buckling domains for n=0.5, 5 and A =0.015 is depicted in Fig. 6. On the basis of the classical and non-classical models, considering the temperature-independent (TID) case for material properties results in reduction of the absolute dimensionless amplitude of system for the whole branches. In other words, the temperatureindependent model considers additional stiffness for the system. Moreover, it can be inferred that the role of temperature-dependency in the stable responses of system becomes more significant at the higher temperature rise ∆. Before initiating to compare the statics-based analytical solution with the dynamics-based numerical one, it should be pointed out that the numerical solution proposed in this study is capable of capturing only the stable equilibrium paths. Moreover, in the post-snap-through buckling domain with two stable equilibrium paths (the upper and lower branches), the final numerical result is dependent on the initial conditions. In Fig. 7 the stable statics-based analytical results depicted in Fig. 4b are compared to those obtained via the dynamics-based numerical results. As expected (the snap-through buckling is a static instability), there is a good agreement between the analytical and numerical results in prediction of pre- and post-snap-through buckling behavior of system. Finally, it should be pointed out that the value of 'G has no influence on the results, but by increasing it the required time to obtain the final solution decreases.

5. CONCLUSION A size-dependent model based on modified couple stress continuum theory is developed to examine nonlinear thermal stability behavior of a geometrically imperfect FG Euler-Bernoulli microbeam under clamped-clamped boundary conditions and in-plane uniform thermal loading. Statics-based analytical solution is established to explore the influence of different dimensionless parameters on the nonlinear critical snap-through buckling temperature rise Δ01 as well as the nonlinear stable and unstable equilibrium paths of system. Furthermore, a dynamics-based

19

numerical solution is proposed to verify the statics-based analytical solution. After performing some parametric studies, the following results are concluded: It is found that Δ01 has an initially ascending trend followed by a descending trend with respect to the dimensionless imperfection amplitude A . In addition, it is revealed that when the power index n is increased Δ01 decreases except around the A in which the system buckles at the reference temperature (Δ01 =0). It is revealed that the system under consideration has one non-zero equilibrium path (the upper branch) in the pre-snap-through buckling region which continues in the post-snap-through buckling region. In addition to this path in the post-snap-through buckling region, the system has one stable non-zero equilibrium path (the lower branch) and one non-zero unstable equilibrium path (the middle branch) in this region. The absolute dimensionless amplitude # of stable equilibrium paths (the upper and lower branches) increases when the temperature rise is increased or when the power index is decreased, but for the unstable equilibrium path a reverse scenario takes place (see Figs. 3 and 4). Moreover, the obtained results indicate that the sizedependency yields the absolute dimensionless amplitude of stable equilibrium paths reduces at a prescribed ∆. The results indicate that the assumption of temperature-independency for the properties of functionally graded materials results in the critical snap-through buckling temperature rise (Δ01 ) increases and also the dimensionless absolute amplitude of stable equilibrium paths decreases. The role of temperature-dependency in the amplitude of system becomes more significant at the higher temperature rise. A good agreement is observed between the results of statics-based analytical solution with those obtained via use of the dynamics-based numerical solution.

References [1] [2] [3] [4] [5]

R. Kandasamy, R. Dimitri, and F. Tornabene, "Numerical study on the free vibration and thermal buckling behavior of moderately thick functionally graded structures in thermal environments," Composite Structures, vol. 157, pp. 207-221, 2016. F. Tornabene, N. Fantuzzi, E. Viola, and R. C. Batra, "Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory," Composite Structures, vol. 119, pp. 67-89, 2015. N. Fantuzzi, F. Tornabene, M. Bacciocchi, and R. Dimitri, "Free vibration analysis of arbitrarily shaped Functionally Graded Carbon Nanotube-reinforced plates," Composites Part B: Engineering, http://dx.doi.org/10.1016/j.compositesb.2016.09.021. F. Alijani, M. Amabili, and F. Bakhtiari-Nejad, "Thermal effects on nonlinear vibrations of functionally graded doubly curved shells using higher order shear deformation theory," Composite Structures, vol. 93, pp. 2541-2553, 2011. F. Alijani and M. Amabili, "Effect of thickness deformation on large-amplitude vibrations of functionally graded rectangular plates," Composite Structures, vol. 113, pp. 89-107, 2014.

20

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

[19] [20] [21]

M. Komijani, Y. Kiani, S. E. Esfahani, and M. R. Eslami, "Vibration of thermoelectrically post-buckled rectangular functionally graded piezoelectric beams," Composite Structures, vol. 98, pp. 143-152, 2013. M. Akbari, Y. Kiani, M. M. Aghdam, and M. R. Eslami, "Free vibration of FGM Lévy conical panels," Composite Structures, vol. 116, pp. 732-746, 2014. Y. Kiani and M. R. Eslami, "Thermal buckling and post-buckling response of imperfect temperature-dependent sandwich FGM plates resting on elastic foundation," Archive of Applied Mechanics, vol. 82, pp. 891-905, 2012. S. Sahmani, M. M. Aghdam, and M. Bahrami, "On the postbuckling behavior of geometrically imperfect cylindrical nanoshells subjected to radial compression including surface stress effects," Composite Structures, vol. 131, pp. 414-424, 2015. M. Şimşek, "Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions," Composite Structures, vol. 133, pp. 968-978, 2015. M. Şimşek, "Buckling of Timoshenko beams composed of two-dimensional functionally graded material (2D-FGM) having different boundary conditions," Composite Structures, vol. 149, pp. 304-314, 2016. L. Li, Y. Hu, and L. Ling, "Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory," Composite Structures, vol. 133, pp. 1079-1092, 2015. L. Li, X. Li, and Y. Hu, "Free vibration analysis of nonlocal strain gradient beams made of functionally graded material," International Journal of Engineering Science, vol. 102, pp. 77-92, 2016. L. Li and Y. Hu, "Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material," International Journal of Engineering Science, vol. 107, pp. 77-97, 2016. Ö. Civalek, "Free vibration of carbon nanotubes reinforced (CNTR) and functionally graded shells and plates based on FSDT via discrete singular convolution method," Composites Part B: Engineering, vol. 111, pp. 45-59, 2017. H. Ersoy, K. Mercan, and Ö. Civalek, "Frequencies of FGM shells and annular plates by the methods of discrete singular convolution and differential quadrature methods," Composite Structures, http://dx.doi.org/10.1016/j.compstruct.2016.11.051. Ç. Demir, K. Mercan, and Ö. Civalek, "Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel," Composites Part B: Engineering, vol. 94, pp. 1-10, 2016. Y. Kiani and M. R. Eslami, "Thermal Postbuckling of Imperfect Circular Functionally Graded Material Plates: Examination of Voigt, Mori–Tanaka, and Self-Consistent Schemes," Journal of Pressure Vessel Technology, vol. 137, pp. 021201-021201-11, 2014. C. Liebold and W. H. Müller, "Comparison of gradient elasticity models for the bending of micromaterials," Computational Materials Science, vol. 116, pp. 52–61, 2016. J. Lei, Y. He, S. Guo, Z. Li, and D. Liu, "Size-dependent vibration of nickel cantilever microbeams: Experiment and gradient elasticity," AIP Advances, vol. 6, p. 105202, 2016. D. Liu, Y. He, D. J. Dunstan, B. Zhang, Z. Gan, P. Hu, and H. Ding, "Toward a further understanding of size effects in the torsion of thin metal wires: an experimental and theoretical assessment," International Journal of Plasticity, vol. 41, pp. 30-52, 2013.

21

[22] [23] [24] [25] [26] [27] [28]

[29] [30] [31] [32] [33] [34] [35] [36]

D. C. C. Lam, F. Yang, A. C. M. Chong, J. Wang, and P. Tong, "Experiments and theory in strain gradient elasticity," Journal of the Mechanics and Physics of Solids, vol. 51, pp. 1477-1508, 2003. C. Motz, T. Schöberl, and R. Pippan, "Mechanical properties of micro-sized copper bending beams machined by the focused ion beam technique," Acta Materialia, vol. 53, pp. 4269-4279, 2005. F. Yang, A. C. M. Chong, D. C. C. Lam, and P. Tong, "Couple stress based strain gradient theory for elasticity," International Journal of Solids and Structures, vol. 39, pp. 2731-2743, 2002. B. Akgöz and Ö. Civalek, "Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams," International Journal of Engineering Science, vol. 49, pp. 1268-1280, 2011. M. Şimşek, M. Aydın, H. H. Yurtcu, and J. N. Reddy, "Size-dependent vibration of a microplate under the action of a moving load based on the modified couple stress theory," Acta Mechanica, vol. 226, pp. 3807-3822, 2015/11/01 2015. A. M. Dehrouyeh-Semnani, M. Dehrouyeh, M. Torabi-Kafshgari, and M. NikkhahBahrami, "A damped sandwich beam model based on symmetric–deviatoric couple stress theory," International Journal of Engineering Science, vol. 92, pp. 83-94, 2015. R. Ansari, R. Gholami, M. Faghih Shojaei, V. Mohammadi, and M. A. Darabi, "Sizedependent nonlinear bending and postbuckling of functionally graded Mindlin rectangular microplates considering the physical neutral plane position," Composite Structures, vol. 127, pp. 87-98, 2015. M. Mohammadabadi, A. R. Daneshmehr, and M. Homayounfard, "Size-dependent thermal buckling analysis of micro composite laminated beams using modified couple stress theory," International Journal of Engineering Science, vol. 92, pp. 47-62, 2015. N. Shafiei, A. Mousavi, and M. Ghadiri, "Vibration behavior of a rotating non-uniform FG microbeam based on the modified couple stress theory and GDQEM," Composite Structures, vol. 149, pp. 157-169, 2016. A. M. Dehrouyeh-Semnani and M. Nikkhah-Bahrami, "The influence of size-dependent shear deformation on mechanical behavior of microstructures-dependent beam based on modified couple stress theory," Composite Structures, vol. 123, pp. 325-336, 2015. S. Yang and W. Chen, "On hypotheses of composite laminated plates based on new modified couple stress theory," Composite Structures, vol. 133, pp. 46-53, 2015. M. H. Ghayesh, H. Farokhi, and G. Alici, "Size-dependent performance of microgyroscopes," International Journal of Engineering Science, vol. 100, pp. 99-111, 2016. M. Mojahedi and M. Rahaeifard, "A size-dependent model for coupled 3D deformations of nonlinear microbridges," International Journal of Engineering Science, vol. 100, pp. 171-182, 2016. I. Eshraghi, S. Dag, and N. Soltani, "Bending and free vibrations of functionally graded annular and circular micro-plates under thermal loading," Composite Structures vol. 137, pp. 196-207, 2016. A. M. Dehrouyeh-Semnani, M. Dehrouyeh, H. Zafari-Koloukhi, and M. Ghamami, "Sizedependent frequency and stability characteristics of axially moving microbeams based on modified couple stress theory," International Journal of Engineering Science, vol. 97, pp. 98-112, 2015.

22

[37] [38] [39]

[40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52]

H. Farokhi, M. H. Ghayesh, and S. Hussain, "Large-amplitude dynamical behaviour of microcantilevers," International Journal of Engineering Science, vol. 106, pp. 29-41, 2016. A. M. Dehrouyeh-Semnani, "The influence of size effect on flapwise vibration of rotating microbeams," International Journal of Engineering Science, vol. 94, pp. 150-163, 2015. M. Tang, Q. Ni, L. Wang, Y. Luo, and Y. Wang, "Nonlinear modeling and sizedependent vibration analysis of curved microtubes conveying fluid based on modified couple stress theory," International Journal of Engineering Science, vol. 84, pp. 1-10, 2014. S. Hosseini-Hashemi, F. Sharifpour, and M. R. Ilkhani, "On the free vibrations of sizedependent closed micro/nano-spherical shell based on the modified couple stress theory," International Journal of Mechanical Sciences, vol. 115–116, pp. 501-515, 2016. M. SoltanRezaee and M. Afrashi, "Modeling the nonlinear pull-in behavior of tunable nano-switches," International Journal of Engineering Science, vol. 109, pp. 73-87, 2016. A. M. Dehrouyeh-Semnani, M. BehboodiJouybari, and M. Dehrouyeh, "On sizedependent lead-lag vibration of rotating microcantilevers," International Journal of Engineering Science, vol. 101, pp. 50-63, 2016. H. M. Sedighi and A. Bozorgmehri, "Dynamic instability analysis of doubly clamped cylindrical nanowires in the presence of Casimir attraction and surface effects using modified couple stress theory," Acta Mechanica, vol. 227, pp. 1575-1591, 2016. M. Şimşek, "Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He’s variational method," Composite Structures, vol. 112, pp. 264-272, 2014. Y.-R. Kwon and B.-C. Lee, "A mixed element based on Lagrange multiplier method for modified couple stress theory," Computational Mechanics, DOI: 10.1007/s00466-0161338-3 2016. A. M. Dehrouyeh-Semnani and A. Bahrami, "On size-dependent Timoshenko beam element based on modified couple stress theory," International Journal of Engineering Science, vol. 107, pp. 134-148, 2016. R. Ansari, R. Gholami, A. Norouzzadeh, and S. Sahmani, "Size-dependent vibration and instability of fluid-conveying functionally graded microshells based on the modified couple stress theory," Microfluidics and Nanofluidics, vol. 19, pp. 509-522, 2015. M. Asghari, M. T. Ahmadian, M. H. Kahrobaiyan, and M. Rahaeifard, "On the sizedependent behavior of functionally graded micro-beams," Materials & Design, vol. 31, pp. 2324-2329, 2010. M. Asghari, M. Rahaeifard, M. H. Kahrobaiyan, and M. T. Ahmadian, "The modified couple stress functionally graded Timoshenko beam formulation," Materials & Design, vol. 32, pp. 1435-1443, 2011. J. N. Reddy, "Microstructure-dependent couple stress theories of functionally graded beams," Journal of the Mechanics and Physics of Solids, vol. 59, pp. 2382-2399, 2011. L.-L. Ke and Y.-S. Wang, "Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory," Composite Structures, vol. 93, pp. 342-350, 2011. B. Abbasnejad, G. Rezazadeh, and R. Shabani, "Stability analysis of a capacitive fgm micro-beam using modified couple stress theory," Acta Mechanica Solida Sinica, vol. 26, pp. 427-440, 2013.

23

[53] [54] [55] [56] [57] [58]

[59] [60] [61] [62]

[63] [64] [65] [66] [67]

B. Akgöz and Ö. Civalek, "Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory," Composite Structures, vol. 98, pp. 314-322, 2013. A. Nateghi and M. Salamat-talab, "Thermal effect on size dependent behavior of functionally graded microbeams based on modified couple stress theory," Composite Structures, vol. 96, pp. 97-110, 2013. M. Şimşek and J. N. Reddy, "Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory," International Journal of Engineering Science, vol. 64, pp. 37-53, 2013. M. Zamanzadeh, G. Rezazadeh, I. Jafarsadeghi-poornaki, and R. Shabani, "Static and dynamic stability modeling of a capacitive FGM micro-beam in presence of temperature changes," Applied Mathematical Modelling, vol. 37, pp. 6964-6978, 2013. A. Arbind and J. N. Reddy, "Nonlinear analysis of functionally graded microstructuredependent beams," Composite Structures, vol. 98, pp. 272-281, 2013. M. Komijani, S. E. Esfahani, J. N. Reddy, Y. P. Liu, and M. R. Eslami, "Nonlinear thermal stability and vibration of pre/post-buckled temperature- and microstructuredependent functionally graded beams resting on elastic foundation," Composite Structures, vol. 112, pp. 292-307, 2014. H.-T. Thai, T. P. Vo, T.-K. Nguyen, and J. Lee, "Size-dependent behavior of functionally graded sandwich microbeams based on the modified couple stress theory," Composite Structures, vol. 123, pp. 337-349, 2015. B. Akgöz and Ö. Civalek, "Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium," International Journal of Engineering Science, vol. 85, pp. 90-104, 2014. M. Şimşek, "Size dependent nonlinear free vibration of an axially functionally graded (AFG) microbeam using He’s variational method," Composite Structures, vol. 131, pp. 207-214, 2015. A. M. Dehrouyeh-Semnani, M. Dehrouyeh, M. Torabi-Kafshgari, and M. NikkhahBahrami, "An investigation into size-dependent vibration damping characteristics of functionally graded viscoelastically damped sandwich microbeams," International Journal of Engineering Science, vol. 96, pp. 68-85, 2015. N. Shafiei, M. Kazemi, and M. Ghadiri, "On size-dependent vibration of rotary axially functionally graded microbeam," International Journal of Engineering Science, vol. 101, pp. 29-44, 2016. M. Ilkhani and S. Hosseini-Hashemi, "Size dependent vibro-buckling of rotating beam based on modified couple stress theory," Composite Structures, vol. 143, pp. 75-83, 2016. N. Shafiei, A. Mousavi, and M. Ghadiri, "On size-dependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams," International Journal of Engineering Science, vol. 106, pp. 42-56, 2016. M. Shojaeian and H. Zeighampour, "Size dependent pull-in behavior of functionally graded sandwich nanobridges using higher order shear deformation theory," Composite Structures, vol. 143, pp. 117-129, 2016. M. Akbarzadeh Khorshidi, M. Shariati, and S. A. Emam, "Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory," International Journal of Mechanical Sciences, vol. 110, pp. 160-169, 2016.

24

[68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83]

A. Ghorbani Shenas, S. Ziaee, and P. Malekzadeh, "Vibrational behavior of rotating pretwisted functionally graded microbeams in thermal environment," Composite Structures, vol. 157, pp. 222-235, 2016. L. C. Trinh, H. X. Nguyen, T. P. Vo, and T.-K. Nguyen, "Size-dependent behaviour of functionally graded microbeams using various shear deformation theories based on the modified couple stress theory," Composite Structures, vol. 154, pp. 556-572, 2016. M. H. Ghayesh, H. Farokhi, and A. Gholipour, "Oscillations of functionally graded microbeams," International Journal of Engineering Science, vol. 110, pp. 35-53, 2017. H. Farokhi, M. H. Ghayesh, and M. Amabili, "Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory," International Journal of Engineering Science, vol. 68, pp. 11-23, 2013. M. H. Ghayesh and M. Amabili, "Coupled longitudinal-transverse behaviour of a geometrically imperfect microbeam," Composites Part B: Engineering, vol. 60, pp. 371377, 2014. H. Farokhi and M. H. Ghayesh, "Nonlinear motion characteristics of microarches under axial loads based on modified couple stress theory," Archives of Civil and Mechanical Engineering, vol. 15, pp. 401-411, 2015. H. Farokhi and M. H. Ghayesh, "Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams," International Journal of Engineering Science, vol. 91, pp. 1233, 2015. H. Farokhi and M. H. Ghayesh, "Nonlinear resonant response of imperfect extensible Timoshenko microbeams," International Journal of Mechanics and Materials in Design, pp. 1-13, 2015. H. Farokhi and M. H. Ghayesh, "Nonlinear size-dependent dynamics of microarches with modal interactions," Journal of Vibration and Control, http://dx.doi.org/10.1177/1077546314565439 2015. H. Farokhi and M. H. Ghayesh, "Size-dependent parametric dynamics of imperfect microbeams," International Journal of Engineering Science, vol. 99, pp. 39-55, 2016. H. Farokhi, M. H. Ghayesh, B. Kosasih, and P. Akaber, "On the nonlinear resonant dynamics of Timoshenko microbeams: effects of axial load and geometric imperfection," Meccanica, vol. 51, pp. 155-169, 2016. M. H. Ghayesh and H. Farokhi, "Internal energy transfer in dynamical behaviour of Timoshenko microarches," Mathematics and Computers in Simulation, vol. 112, pp. 2839, 2015. M. H. Ghayesh and H. Farokhi, "Coupled nonlinear dynamics of geometrically imperfect shear deformable extensible microbeams," Journal of Computational and Nonlinear Dynamics, vol. 11, p. 041001, 2016. M. H. Ghayesh, H. Farokhi, and G. Alici, "Size-dependent electro-elasto-mechanics of MEMS with initially curved deformable electrodes," International Journal of Mechanical Sciences, vol. 103, pp. 247-264, 2015. M. Shojaeian, Y. T. Beni, and H. Ataei, "Size-dependent snap-through and pull-in instabilities of initially curved pre-stressed electrostatic nano-bridges," Journal of Physics D: Applied Physics, vol. 49, p. 295303, 2016. A. M. Dehrouyeh-Semnani, H. Mostafaei, and M. Nikkhah-Bahrami, "Free flexural vibration of geometrically imperfect functionally graded microbeams," International Journal of Engineering Science, vol. 105, pp. 56-79, 2016.

25

[84]

[85] [86] [87] [88] [89] [90]

M. H. Ghayesh, H. Farokhi, A. Gholipour, S. Hussain, and M. Arjomandi, "Resonant responses of geometrically imperfect functionally graded extensible microbeams," ASME Journal of Computational and Nonlinear Dynamics, http://dx.doi.org/10.1115/1.4035214 2016. W. Lacarbonara, "A theoretical and experimental investigation of nonlinear vibrations of buckled beams," Citeseer, 1997. A. M. Dehrouyeh-Semnani and M. Nikkhah-Bahrami, "A discussion on incorporating the Poisson effect in microbeam models based on modified couple stress theory," International Journal of Engineering Science, vol. 86, pp. 20-25, 2015. M. Hafiz, L. Kosuru, A. Ramini, K. Chappanda, and M. Younis, "In-Plane MEMS Shallow Arch Beam for Mechanical Memory," Micromachines, vol. 7, p. 191, 2016. S. A. Emam, "A static and dynamic analysis of the postbuckling of geometrically imperfect composite beams," Composite Structures, vol. 90, pp. 247-253, 2009. J. R. Dormand and P. J. Prince, "A family of embedded Runge-Kutta formulae," Journal of Computational and Applied Mathematics, vol. 6, pp. 19-26, 1980. J. N. Reddy and C. D. Chin, "Thermomechanical analysis of functionally graded cylinders and plates," Journal of Thermal Stresses, vol. 21, pp. 593-626, 1998.

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Figure 1: The schematic representation of a geometrically imperfect functionally graded microbeam.

Figure 2: Critical snap-through buckling temperature rise HIJK as a function of dimensionless imperfection amplitude LM for different values of power indexN based on classical theory (CT) and modified couple stress theory (MCST).

Figure 3: Critical snap-through buckling temperature rise HIJK as a function of power index n for

different values of dimensionless imperfection amplitude LM based on classical theory (CT) and modified couple stress theory (MCST).

Figure4: Temperature-dependent (TD) stable (solid line) and unstable (dash line) equilibrium paths in pre- and post-snap-through buckling for LM =0.015 and different values of power index N based on classical theory (CT) and modified couple stress theory (MCST).

Figure 5: Influence of dimensionless imperfection amplitude LM on temperature-dependent (TD) stable (solid line) and unstable (dash line) equilibrium paths for power index N=5 based on classical theory (CT) and modified couple stress theory (MCST).

Figure 6: Influence of temperature-dependency on stable (solid line) and unstable (dash line) equilibrium paths for LM = 0.015 and power indexes N=0.5 & 5 based on classical theory (CT) and modified couple stress theory (MCST).

Figure 7: Comparative study between statics-based analytical solution (solid line) with dynamics-based numerical solution (circle) for temperature-dependent stable responses, LM = 0.015, O/P = 3.

Material SUS304 (Metal)

properties P-1 P0 P1 P2 P3 α (K-1) 0 12.33e-6 8.086e-4 0 0 E (Pa) 0 201.04e+9 3.079e-4 -6.534e-7 0 ρ (Kg/m3) 0 8166 0 0 0 ν 0 0.28 0 0 0 Si3N4 α (K-1) 0 5.87273e-6 9.095e-4 0 0 (Ceramic) E (Pa) 0 348.43e+9 -3.07e-4 2.16e-7 -8.946e-11 ρ (Kg/m3) 0 2170 0 0 0 ν 0 0.28 0 0 0 Table 1: Temperature-dependent coefficients for metal (SUS304) and ceramic (Si3N4) phases [90].