Thermal bifurcation buckling of piezoelectric carbon nanotube reinforced composite beams

Computers and Mathematics with Applications (

)

–

Contents lists available at ScienceDirect

Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa

Thermal bifurcation buckling of piezoelectric carbon nanotube reinforced composite beams M. Rafiee a,d , Jie Yang b,∗ , Siritiwat Kitipornchai c a

Department of Civil and Architectural Engineering, City University of Hong Kong, Kowloon, Hong Kong

b

School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, P.O. Box 71, Bundoora, VIC 3083, Australia

c

School of Civil Engineering, the University of Queensland, St. Lucia, Brisbane, QLD 4072, Australia

d

Department of Mechanical Engineering, Bu-Ali Sina University, Hamedan, Iran

article

info

Article history: Received 17 August 2012 Received in revised form 24 February 2013 Accepted 13 April 2013 Keywords: Thermal buckling Bifurcation Beam Carbon nanotube reinforced composite

abstract The nonlinear thermal bifurcation buckling behavior of carbon nanotube reinforced composite (CNTRC) beams with surface-bonded piezoelectric layers is studied in this paper. The governing equations of piezoelectric CNTRC beam are obtained based on the Euler– Bernoulli beam theory and von Kármán geometric nonlinearity. Two kinds of carbon nanotube-reinforced composite (CNTRC) beams, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FGCNTRC beam are assumed to be graded in the thickness direction. The SWCNTs are assumed aligned, straight and with a uniform layout. Exact solutions are presented to study the thermal buckling behavior of beams made of a symmetric single-walled carbon nanotube reinforced composite with surface-bonded piezoelectric layers. The critical temperature load is obtained for the nonlinear problem. The effects of the applied actuator voltage, temperature, beam geometry, boundary conditions, and volume fractions of carbon nanotubes on the buckling of piezoelectric CNTRC beams are investigated. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Due to their exceptional mechanical, thermal and electrical properties, carbon nanotubes (CNTs) are considered as one of the most promising reinforcement materials for high performance structural and multifunctional composites with tremendous application potentials [1–5]. There is a wide range of applications in which the vibrational characteristics of CNTs are significant. A review related to the importance and modeling of vibration behavior of various nanostructures can be found in Gibson et al.’s [2]. In these works it has been suggested that the nonlocal elasticity theory developed by Eringen [3,4] should be used in the continuum models for accurate prediction of vibration behaviors. This is due to the scale effect of the nanostructures. The importance of accurate prediction of nanostructures’ vibration characteristics have been discussed by Gibson et al. [2]. In the last few years a lot of research has been performed on composite structure reinforcement by nanotubes. Most of this research relates to carbon nanotubes and composite structure properties. A review and comparisons of the mechanical properties of single- and multi-walled carbon nanotube reinforced composites fabricated by various processes are given by Coleman et al. [6], in which the composites based on chemically modified nanotubes show the best results since functionalization should significantly enhance both dispersion and stress transfer. Tensile tests of multi-walled carbon nanotube

∗

Corresponding author. Tel.: +61 3 9925 6169; fax: +61 3 9925 6108. E-mail address: [email protected] (J. Yang).

0898-1221/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.camwa.2013.04.031

2

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

composites have demonstrated that an addition of small amounts of the nano-scaled reinforcement results in considerable increases in the elastic modulus and break stress [7,8]. Wuite and Adali firstly used the classical laminated beam theory to analyze symmetric cross-ply and angle-ply laminated beams stacked with multiple transversely isotropic layers reinforced by CNTs in different aligned directions, and isotropic beams composed of randomly oriented CNTs dispersed in a polymer matrix, based on micromechanical constitutive models developed according to the Mori–Tanaka method [9]. A type of novel actuator comprising a piezoceramic matrix reinforced with SWCNTs was proposed to apply for active control of smart structures by Ray and Batra [10]. The effective properties of the resulting piezoelectric composites were determined by a micromechanical analysis and the active control system exhibited better performance in the controllability of vibrations. The same authors [11] then proposed another new hybrid piezoelectric composite containing SWCNTs and piezoelectric fibers as reinforcements embedded in a conventional polymer matrix. The value of the effective piezoelectric coefficient related to the in-plane actuation increases significantly due to the addition of CNTs. Functionally graded materials (FGMs) are inhomogeneous composites characterized by smooth and continuous variations in both compositional profile and material properties and have found a wide range of applications in many industries. Emerging intelligent structures, including these FGM members bonded with piezoelectric sensors or actuators, are practically very important since they are related to structural vibration control, structural health monitoring, transportation engineering and thermal stress control. Therefore, it is of great importance to analyze the behavior of FG structures with surface-bonded piezoelectric layers. Due to the above-mentioned favorable features, many studies have been conducted on the static and dynamic behavior of FG structures with surface-bonded piezoelectric layers. Vodenitcharova and Zhang [12] studied the shear bending and shear with local buckling of nanocomposite beams reinforced by SWCNTs. For the analysis of problem they use the Airy stress function. Special attention has been given on the radial flexibility of SWCN in stress–strain and buckling. Seidel and Lagoud [13] carried out a research on micromechanical elastic properties of reinforced structures by CNTs for determination of elastic properties of nanotubes. In the above-mentioned paper, the graphite plate properties were used to determine the nanotubes’ elastic features. Zhu et al. [14] presented bending and free vibration analyses of thin-tomoderately thick composite plates reinforced by single-walled carbon nanotubes with the finite element method based on the first order shear deformation plate theory. In another paper, Wang and Shen [15] studied the nonlinear vibration of a reinforced composite plate by SWCNTs in a thermal environment. Also Shen and Zhang [16] investigated the thermal buckling of a composite plate reinforced by CNT (FG distribution). Shen [17] investigated the buckling and postbuckling of nanocomposite plates with functionally graded nanotube reinforcements subjected to uniaxial compression in thermal environments. Jafari Mehrabadi and his coworkers [18] investigated the mechanical buckling of a functionally graded nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes (SWCNTs) based on Mindlin plate theory subjected to uniaxial and biaxial in-plane loadings. Shen and Zhu [19] studied the buckling and postbuckling behavior of functionally graded nanotube-reinforced composite plates in thermal environments. Shen and Zhu [20] investigated the postbuckling of sandwich plates with nanotube-reinforced composite face sheets resting on elastic foundations. Alibeigloo [21] reported an analytical solution for thermo-elasticity analysis of FGM beams integrated with piezoelectric layers. By assuming simply supported edge conditions, he used the state space method in conjunction with the Fourier series in the longitudinal direction to obtain exact solutions. An analytical method for deflection control of FGM beams containing two piezoelectric layers is reported by Gharib et al. [22]. Zhu and Sankar [23] solved the two-dimensional elasticity equations for an FG beam subjected to transverse loads using a mixed Fourier series–Galerkin method. Shi and Chen [24] studied the problem of a functionally graded piezoelectric cantilever beam subjected to different loadings. A pair of stress and induction functions was proposed in the form of polynomials and a set of analytical solutions for the beam subjected to different loadings was obtained. Nonlinear analysis of buckling, free vibration and active control and dynamic stability for piezoelectric functionally graded beams in a thermal environment was investigated by Fu and his coworkers [25]. They derived a nonlinear governing equation for the functionally graded material beam with surface-bonded piezoelectric actuators by Hamilton’s principle. Using the finite difference method and the Newmark method synthetically, they obtained numerical solutions for the nonlinear dynamic equations of functionally graded beams with piezoelectric patches. Kiani et al. [26] investigated the thermal buckling of piezoelectric functionally graded material beams ith surface-bonded piezoelectric layers which are subjected to both thermal loading and constant voltage. The free vibration and dynamic response of a simply supported functionally graded piezoelectric cylindrical panel impacted by time-dependent blast pulses are analytically investigated by Bodaghi and Shakeri [27]. Ke and his coworkers [28] investigated the nonlinear vibration of piezoelectric nanobeams subjected to an applied voltage and a uniform temperature change based on the nonlocal theory and Timoshenko beam theory. Yang and Chen [29] presented a theoretical investigation in free vibration and elastic buckling of beams made of functionally graded materials (FGMs) containing open edge cracks by using the Bernoulli–Euler beam theory and the rotational spring model. Rafiee and Kalhori [30] and Shooshtari and Rafiee [31] studied the nonlinear free and forced vibration of a beam made of functionally graded (FG) materials based on the Euler–Bernoulli beam theory and von Kármán geometric nonlinearity under different boundary conditions. Free vibration and elastic buckling of beams made of functionally graded materials (FGMs) containing open edge cracks were studied based on the Timoshenko beam theory [32]. Based on the Timoshenko beam theory and von Kármán nonlinear kinematics and using the Ritz method, Ke et al. [33] studied the postbuckling response of beams made of functionally graded materials (FGMs) containing an open edge crack. Kong et al. [34] developed modified couple stress theory for a third-order shear deformation functionally graded (FG) micro beam. Ke et al. [35] presents

M. Rafiee et al. / Computers and Mathematics with Applications (

(a) FGO-CNTRC.

)

–

3

(b) FGX-CNTRC.

(c) UD-CNTRC. Fig. 1. Configurations of the carbon nanotube reinforced composite piezoelectric beams cross section. (a) FGO CNTRC piezoelectric beam; (b) FGX CNTRC piezoelectric beam; and (c) UD CNTRC piezoelectric beam.

a dynamic stability analysis of functionally graded nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on the Timoshenko beam theory. Rafiee et al. [36] investigated the large amplitude free vibration of functionally graded carbon nanotube reinforced composite beams with surface-bonded piezoelectric layers subjected to a temperature change and an applied voltage. Buckling analysis of CNTRC piezoelectric beams, which to the best of the authors’ knowledge do not exist in the literature, is conducted in this study. The nonlinear bifurcation problem of single-walled carbon nanotube reinforced composite piezoelectric beams which are subjected to thermal loading and applied constant voltage are considered here. Three types of boundary conditions are assumed for the beam. Based on the Euler–Bernoulli beam theory, the equilibrium equations for the beam are derived and the eigenvalue solution is carried out to obtain the critical temperature load. The influences of the geometry, temperature change, external electric voltage, boundary conditions and carbon nanotube volume fraction on the postbuckling behavior of the CNTRC piezoelectric beams are discussed in detail. 2. Theoretical formulations A CNTRC beam of length L and height h with surface-bonded piezoelectric layers with a thickness hp is shown in Fig. 1 such that total height is H. The SWCNT reinforcement is either uniformly distributed (referred to as UD) or functionally graded in the thickness direction (referred to as FG). FG-O and FG-X CNTRC are the functionally graded distribution of carbon nanotubes over the thickness direction of the composite beam (see Fig. 2). Piezoelectric actuators are symmetrically and perfectly bonded (neglecting the adhesive thickness) on the top and bottom surface of the substrate CNTRC host. The length of the beam is L, and the thicknesses of the piezoelectric CNTRC beam, the substrate CNTRC host and piezoelectric actuator are H , h and hp , respectively. 2.1. CNTRC and piezoelectric properties CNTRCs can be employed in high-temperature environments and many of the constituent materials may possess temperature-dependent properties. It is supposed that the CNTRC beam is linear elastic throughout the deformation, and that the beam is initially stress free at T0 (in Kelvin) and is subjected to a uniform temperature variation 1T = T − T0 .

4

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

Fig. 2. Effect of distributions of carbon nanotubes on thermal buckling temperature T cr of CNTRC piezoelectric beams under boundary conditions ∗ (hp /h = 1/12, VCN = 0.12, V0 = 0, h = 10 mm).

It is assumed that the CNTRC beam is made from a mixture of SWCNT, graded distribution in the thickness direction, and a matrix which is assumed to be isotropic. The effective material properties of the two-phase nanocomposites, mixture of CNTs and an isotropic polymer, can be estimated according to the Mori–Tanaka scheme [37] or the rule of mixtures [38,39]. Due to simplicity and convenience, in the present study the rule of mixtures was employed by introducing CNT efficiency parameters and the effective material properties of CNTRC beams can thus be written as [40] CN E11 = η1 VCN E11 + Vm E m

η2 E22

η3 G12

= =

VCN

+

CN E22

VCN GCN 12

+

(1a)

Vm

(1b)

Em Vm

(1c)

Gm

CN CN m m where E11 , E22 and GCN 12 indicate the Young’s moduli and shear modulus of SWCNTs, respectively, and E and G represent the corresponding properties of the isotropic matrix. To account for the scale-dependent material properties, ηj (j = 1, 2, 3), the CNT efficiency parameters, are introduced and can be calculated by matching the effective properties of CNTRC obtained from the MD simulations with those from the rule of mixtures. VCN and Vm are the volume fractions of the carbon nanotubes and matrix, respectively, and it is noticeable that the sum of the volume fractions of the two constituents equals unity. The uniform and two types of functionally graded distributions of the carbon nanotubes along the thickness direction of the nanocomposite beams depicted in Fig. 1 are assumed to be as follows,

 ∗ VCN = VCN (UD − CNTRC )       V = 2 1 − 2 |z | V ∗ FGO − CNTRC ) CN CN ( h      | |  ∗ VCN = 2 2 z VCN (FGX − CNTRC )

(2)

h

where ∗ VCN =

wCN wCN + (ρCN /ρm ) − (ρCN /ρm )wCN

(3)

where wCN is the mass fraction of the nanotube, and ρCN and ρm are the densities of the carbon nanotube and matrix, respectively. The thermal expansion coefficient can be expressed as CN αh = VCN α11 + Vm α m

(4)

where α and α are thermal expansion coefficients of the carbon nanotube and matrix. Poisson’s ratio and mass density ρ can be calculated by CN 11

m

CN νh = VCN ν12 + Vm ν m ,

ρh = VCN ρ CN + Vm ρ m

(5)

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

5

CN where ν12 and ν m are Poisson’s ratios of the carbon nanotube and matrix, respectively. It is assumed that the material property of the nanotube and matrix is a function of temperature, so that the effective material properties of CNTRCs, like Young’s modulus, shear modulus and thermal expansion coefficients, are functions of temperature and position. The piezoelectric material properties are assumed to be temperature-independent, and C11 , αp denoting the reduced elastic constant and thermal expansion coefficient, respectively. The reduced elastic modulus E, thermal expansion coefficient α and density ρ are piecewise functions with respect to variable z as

C11 Q11 C11

 

E

α

ρ

ν = 

  ρp ρh , ρp

  αp αh , αp

 ,

  νp νh νp

−H /2 ≤ z < −h/2 −h/2 ≤ z < h/2 h/2 ≤ z ≤ H /2

(6)

where E , α, ρ , and υ are the Young’s modulus, thermal expansion coefficient, density and Poisson’s ratios. Here, the density ρh and Poisson’s ratios νh are considered temperature-independent (TID). 2.2. Governing equations Based on the Kirchhoff–Love hypothesis, the displacement of the piezoelectric CNTRC beam could be denoted by u¯ (x, z , t ) and w ¯ (x, z , t ) respectively, taking the form [41] u¯ (x, z , t ) = u (x, t ) − z

∂w (x, t ) ∂x

(7)

w ¯ (x, z , t ) = w (x, t )

(8)

where t is time, u(x, t ) and w(x, t ) are displacement components of a point in the neutral surface. The von Kármán type nonlinear strain–displacement relationship gives

εx =

∂u ∂ 2w 1 −z 2 + ∂x ∂x 2



∂w ∂x

2

− α11 (z , T ) 1T .

(9)

The normal stress σxx including thermal effect is given by the linear elastic constitutive law as

 σxx = E11 (z , T )

∂u ∂ 2w 1 −z 2 + ∂x ∂x 2



∂w ∂x

2 

 − α11 (z , T ) 1T .

(10)

Assuming the piezoelectric material with temperature-independent material properties, the linear thermo-piezoelectric–elastic constitutive relationship of the piezoelectric material is given as

 σ = C11 p xx

∂ 2w 1 ∂u −z 2 + ∂x ∂x 2



∂w ∂x

2 

 − αp (z , T ) 1T − d31 Ez

(11)

where C11 , ap and d31 denote the reduced elastic constant, thermal expansion coefficient and piezoelectric strain constant, respectively. In light of the fact that the thickness of the piezoelectric layer is very thin and self-induced electric potential is much smaller than the applied voltage, we could describe the relationship between applied voltage V (t ) and electric field intensity [42] within a piezoelectric actuator as [43,44]: Ez =

V (t ) hp

.

(12)

By using Hamilton’s principle, the equations of motion can be derived as

∂ Nx =0 ∂x   ∂ ∂ 2 Mx ∂w + N + Q (x, t ) = 0. x ∂ x2 ∂x ∂x

(13a) (13b)

The force and moment per unit length of the beam expressed in terms of the stress through the thickness, according to the Euler–Bernoulli beam theory, are h/2

 Nx =

−h/2

 Mx =

h/2

−h/2

σxx dz +



z σxx dz +

h/2+hp h/2



σxxp dz +

h/2+hp

h/2

−h/2



−(h/2+hp )

p z σxx dz +



σxxp dz

−h/2

−(h/2+hp )

p z σxx dz

(14)

(15)

6

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

Nx and Mx are obtained as



   ∂ u 1 ∂w 2 ∂ 2w Nx = A11 + − B11 2 − N T − N p ∂x 2 ∂x ∂x     ∂ u 1 ∂w 2 ∂ 2w Mx = B11 + − D11 2 − M T − M p ∂x 2 ∂x ∂x

(16a)

(16b)

where the superscripts ‘‘T ’’ and ‘‘P’’ in Eqs. (16) represent the thermal and electric loads, respectively, and N T , M T as well as N P , M P are defined as



NT NP

MT MP



H /2



E α 1T Ed31 Ez



= −H / 2





z dz ,



1

(17)

A11 , B11 and D11 in Eqs. (16) are the stretching stiffness, stretching–bending coupling stiffness and bending stiffness coefficients, respectively, which are defined as

{A11 , B11 , D11 } =

H /2



−H / 2

E 1, z , z 2 dz .





(18)

If the axial inertia and rotational inertia terms are neglected, (13) gives

 Nx = A11

∂u 1 + ∂x 2



∂w ∂x

2 

− N T − N p = Nx0 .

(19)

In the present work, CNTRC piezoelectric beams with axially immovable ends are considered. For beams with immovable ends (i.e. u = 0, at x = 0 and L), integrating (19) with respect to x leads to 0=

L [u]xx= =0

 L = 0

 1 1  Nx0 − N T − N p − A11 2



∂w ∂x

2 

dx.

(20)

Hence,

 L  1

A11

Nx = Nx0 =

L

2

0

∂w ∂x

2 

dx − N T − N p .

(21)

From (16b), bending moments can be re-expressed in terms of deflection as

 Mx = −D11

∂w ∂x

2

− MT − MP .

(22)

With (21) and (22) in mind and axial inertia being neglected, (13) can be reduced to



A11

 L  1

L

0

2

∂w ∂x

2 

 dx

 ∂ 2w ∂ 2w ∂ 4w  − D 4 − N T + N0P =0 2 ∂x ∂x ∂ x2

(23)

where N0P = 2V0 C11 d31 , which could be obtained from Eq. (17) and it is valid while the upper and lower piezoelectric actuators are applied by the same constant excitation voltage V (t ) = V0 . It should be noted that the above equation is nonlinear due to the fact that Nx0 is nonlinear in w . 3. Thermal postbuckling analysis In order to determine the solution to the postbuckling problem of CNTRC piezoelectric beams, the following dimensionless quantities are introduced x

ξ= , L

˜ = D

D11 r 2 A110

w ˜ = ,

w r

,

A˜ =



H /2



−H /2

A11 A110

,

H /2

z 2 dz

r =

θ˜ = T

−H / 2

N

T

A110

,

dz ,

θ˜ = P

L

η= , r

N0P A110

(24)

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

7

where A110 is taken as the value of A11 of a homogeneous beam and r is the radius of gyration of the cross section. Neglecting the over-tilt sign for w , Eq. (23) can be rewritten in dimensionless form as

   2   1  ˜T ˜P 2 2 ∂ 4w  θ + θ η 1 ∂w A˜  ∂ w = 0. + d ξ − ˜ ˜ 0 2 ∂ξ ∂ξ 4 ∂ξ 2 D D

(25)

The boundary conditions are given by

w(0) = w(1) = w′′ (0) = w ′′ (1) = 0,

(26a)

w(0) = w (0) = w(1) = w (1) = 0,

(26b)

w(0) = w ′ (0) = w(1) = w′′ (1) = 0

(26c)

′

′

for hinged–hinged, clamped–clampedand clamped–hinged beams respectively. It is recalled that Eq. (25) may be reduced to  ˜

1

D

0

the linear problem if the term − A˜

1 2



∂w ∂ξ

2

for a specific buckled configuration, θ˜ T + θ˜ P >

dξ is dropped. It is easily verified that the integral in Eq. (25) is a constant

A˜

η2

1

  1 2

0

∂w ∂ξ

2 

dξ , and the general solutions of Eq. (25) are [32,45]

w (ξ ) = m1 sin (Λξ ) + m2 cos (Λξ ) + m3 ξ + m4 , (27)      (θ˜ T +θ˜ P )η2 − A˜  1 1 ∂w 2 dξ and unknown constants m (j = 1, . . . , 4) are to be determined from boundwhere Λ = j ˜ ˜ 0 2 ∂ξ D D ary conditions at each end. Inserting Eq. (27) into boundary conditions in Eqs. (26a)–(26c), one can obtain a homogeneous matrix equation whose determinant should be zero to guarantee non-trivial solutions det [H (Λ)] = 0.

(28)

The characteristic equation for Λ can be obtained for different boundary conditions as Hinged–Hinged: sin(Λ) = 0.

(29)

Clamped–Clamped: 2 − 2 cos(Λ) − Λ sin(Λ) = 0.

(30)

Clamped–Hinged: tan(Λ) − Λ = 0.

(31)

Obviously, the lowest positive solution of Eqs. (29)–(31) is the critical buckling load. Solving Eq. (29) for Λ yields Λ = nπ , where n is an integer. Then the corresponding mode shapes w (ξ ) for the hinged–hinged boundary condition are given by

w (ξ ) = δ sin (nπ ξ )

(32)

where δ is a constant to be determined. Now, it is recalled that

 Λ = 2

 θ˜ T + θ˜ P η2 ˜ D

−

A˜

˜ D

1

 0

  1

2

∂w ∂ξ

2 

dξ .

(33)

By using Eq. (30), the above equation leads to

δ=±

2η



  Λ2 θ˜ T + θ˜ P − D˜ 2 . ˜A2 Λ η

(34)

For clamped–clamped boundary conditions, solving Eq. (30) for Λ yields 2π , 8.9868, 4π , and 15.4505 as the first four eigenvalues and the corresponding mode shapes w (ξ ) are given by

w (ξ ) = δ



cos Λ − 1 sin Λ − Λ

sin (Λξ ) − cos (Λξ ) −

 Λ (cos Λ − 1) ξ +1 sin Λ − Λ

(35)

where δ is a constant to be determined. The expression w (ξ ) governs both symmetric and antisymmetric buckling shapes. Substituting (35) into (33) yields the same postbuckling relation given by (34).

8

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

Table 1 Comparison of critical buckling temperature difference (1T cr ) for clamped–clamped isotropic beams (L/h = 25, h = 10 mm, hp = 1 mm, E = 70 × 109 , υ = 0.3, α = 23 × 10−6 ). Source

V0 (V)

Present Kiani and Eslami [26] Error (%)

−500

−200

0

200

500

377.147 379.804 0.704

376.555 379.21 0.704

376.16 378.81 0.704

375.765 378.41 0.704

375.173 377.816 0.704

Table 2 Temperature-dependent material properties for (10, 10) SWCNT (L = m 9.26 nm, R = 0.68 nm, h = 0.067 nm, ν12 , ρ CN = 1400 kg/m3 ). Source: From Wang and Shen [15]. Temperature (K)

cnt E11 (TPa)

cnt α11 (×10−6 /K)

300 500 700

5.6466 5.5308 5.4744

3.4584 4.5361 4.6677

In the case of clamped–hinged CNTRC piezoelectric beams, one may obtain the following mode shapes w (ξ ) as the same procedure we had for the prior boundary condition as

w (ξ ) = δ



1 − cos (Λ)

Λ − sin (Λ)

sin (Λξ ) − cos (Λξ ) − Λ

1 − cos (Λ)

Λ − sin (Λ)

ξ +1

 (36)

where δ is the same relation given by (34). The critical temperature of CNTRC piezoelectric beams based on the Euler–Bernoulli beam theory can be obtained as

     H /2 Λ 2 P M ˜ − θ˜ − θ˜ E α dz + T0 . T = A110 D η −H / 2 cr

(37a)

4. Numerical results and discussions Since no thermal buckling results for CNTRC piezoelectric beams are available in open literature, the present theoretical formulations are validated in Table 1 by comparing our results for clamped FGM beam with piezoelectric layers with the ones provided by Kiani and Eslami [26]. The geometrical and material parameters are L/h = 25, h = 10 mm, hp = 1 mm, E = 70 × 109 , υ = 0.3, α = 23 × 10−6 . The material properties are assumed to be temperature independent. A good agreement is observed for different applied voltages. The material properties and effective thickness of SWCNTs used for analysis are properly chosen in the present paper by MD simulations [15]. Typical results are listed in Table 2. We assume, η1 = 0.137, η2 = 1.022 and η3 = 0.715 for the ∗ ∗ case of VCN = 0.12, and η1 = 0.142, η2 = 1.626 and η3 = 1.138 for the case of VCN = 0.17, and η1 = 0.141, η2 = 1.585 ∗ and η3 = 1.109 for the case of VCN = 0.28 [28]. Here PmPV material is considered. The material properties of the matrix in this example are temperature dependent ρ m = 1150 kg/m3 , ν m = 0.34, α m = 45(1 + 0.00051T ) × 10−6 /K and E m = (3.52 − 0.0034T ) GPa, in which T = T0 + 1T and T0 = 300 K (room temperature). The (10, 10) SWCNTs are selected as the reinforcements. The material parameters of the piezoelectric material is independent of temperature with Ep = 63.0 GPa, ρp = 7600 kg/m3 , αp = 0.9 × 10−6 /K, υp = 0.3 and d31 = 2.54 × 10−1 0 m/V. ∗ Table 3 shows the effects of CNT volume fraction VCN (=0.12, 0.17, 0.28) on dimensionless thermal buckling temperacr m0 2 m0 ture λ = 1T α η (where α refers to the thermal expansion coefficient of matrix α m at room temperature) for CNTRC piezoelectric beams under different boundary conditions (V0 = 0, L/H = 100, h = 10 mm, hp /h = 1/12). It can be ∗ found that the buckling temperatures of FGX CNTRC beams are increased with increase in nanotube volume fraction VCN but the beam with intermediate nanotube volume fraction will not have an intermediate buckling load. In contrast, for the ∗ ∗ cases of FGO CNTRC and UD-CNTRC the beam with VCN = 0.12 and VCN = 0.28 will have highest buckling temperature among the three, respectively. It can be also found that the buckling temperature of FG-CNTRC beams is higher than that of the UD-CNTRC beams and the buckling temperature of UD-CNTRC beams is higher than that of the FGO-CNTRC for the ∗ same nanotube volume fraction VCN . It is also obvious that dimensionless thermal buckling temperature λ = 1T cr α m0 η2 for clamped–clamped beams is higher than the clamped–hinged and hinged–hinged ones, respectively. Table 4 shows the effects of applied voltage (V0 ) and piezoelectric-to-CNTRC host thickness ratio hp /h on dimensionless ∗ thermal buckling temperature λ = 1T cr α m0 η2 of CNTRC piezoelectric beams (L/H = 100, h = 10 mm, VCN = 0.12). The positive voltage decreases the thermal buckling temperature of the CNTRC piezoelectric beams (λ), and the negative voltage increases the dimensionless thermal buckling temperature of beams, which is coincident with the fact that an axial compressive and tensile force would be generated in the beam by the positive and negative voltage, respectively. Compared

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

9

Table 3 ∗ Effects of CNT volume fraction VCN and boundary conditions on dimensionless thermal buckling temperature λ = ∆T cr α m0 η2 for CNTRC piezoelectric beams (V0 = 0, L/H = 100, h = 10 mm, hp /h = 1/12). B.C.

∗ VCN

UD

FGO

FGX

CC

0.12 0.17 0.28

45.0465 43.4610 45.5308

30.6093 28.5485 29.9518

63.2856 63.9512 72.6249

CH

0.12 0.17 0.28

23.0385 22.2276 23.2862

15.6547 14.6008 15.3185

32.3666 32.707 37.1431

HH

0.12 0.17 0.28

11.2616 10.8653 11.3827

7.65233 7.13713 7.48796

15.8214 15.9878 18.1562

CNT distribution

Table 4 Effects of applied voltage V0 and piezoelectric-to-CNTRC host thickness ratio hp /h on dimensionless thermal buckling temperature λ = ∆T cr α m0 η2 for CNTRC piezoelectric ∗ beams with different boundary conditions (L/H = 100, h = 10 mm, VCN = 0.12). B.C.

CC

V0 (V)

−500 0 500

CH

−500 0 500

HH

−500 0 500

hp /h = 1/8

hp /h = 1/12

hp /h = 1/20

UD

FGX

UD

FGX

UD

FGX

48.1845 45.9516 43.7187

64.4648 62.1289 59.7931

47.2822 45.0465 42.8109

65.6244 63.2856 60.9467

46.8042 44.5664 42.3286

67.1633 64.8220 62.4808

25.7342 23.5013 21.2684

34.1109 31.775 29.4392

25.2741 23.0385 20.8028

34.7054 32.3666 30.0277

25.0307 22.7929 20.5551

35.4936 33.1524 30.8112

13.7208 11.4879 9.25499

17.8681 15.5322 13.1964

13.4973 11.2616 9.026

18.1602 15.8214 13.4825

13.3794 11.1416 8.90378

18.5468 16.2055 13.8643

Table 5 ∗ Effects of length-to-thickness L/H and carbon nanotube volume fraction VCN on thermal buckling temperature T cr for CNTRC piezoelectric beams (hp /h = 1/12, V0 = 0, h = 10 mm). B.C.

CC

CH

HH

L/H

25 50 100 25 50 100 25 50 100

∗ VCN = 0.12

∗ VCN = 0.17

∗ VCN = 0.28

UD

FGO

FGX

UD

FGO

FGX

UD

FGO

FGX

433.471 333.368 308.342 368.262 317.066 304.266 333.368 308.342 302.085

390.694 322.674 305.668 346.384 311.596 302.899 322.674 305.668 301.417

487.513 346.878 311.720 395.901 323.975 305.994 346.878 311.720 302.930

428.773 332.193 308.048 365.860 316.465 304.116 332.193 308.048 302.012

384.588 321.147 305.287 343.262 310.815 302.704 321.147 305.287 301.322

489.485 347.371 311.843 396.910 324.227 306.057 347.371 311.843 302.961

434.906 333.727 308.432 368.996 317.249 304.312 333.727 308.432 302.108

388.746 322.187 305.547 345.388 311.347 302.837 322.187 305.547 301.387

515.185 353.796 313.449 410.054 327.513 306.878 353.796 313.449 303.362

with the thermal load (Table 5), electric load has no pronounced effect on the fundamental frequency, which could be predicted from the formulation by the truth that the value of the piezoelectric strain constant d31 is much smaller than the thermal expansion coefficient α . At the same time, the thickness difference between the piezoelectric layers and FGM layer is another factor. Another interesting result that can be found in this table is the effect of piezoelectric-to-CNTRC host thickness ratio hp /h on dimensionless thermal buckling temperature λ = 1T cr α m0 η2 of CNTRC piezoelectric beams. It can be seen that thermal buckling temperature of the UD CNTRC piezoelectric beams decreases with a decrease in piezoelectricto-CNTRC host thickness ratio hp /h, but thermal buckling temperature of the FGX CNTRC piezoelectric beams increases. Table 5 gives the effects of L/H on thermal buckling temperature T cr for CNTRC piezoelectric beams (hp /h = 1/12, V0 = 0, h = 10 mm) of the CNTRC piezoelectric beam with different distributions of nanotubes i.e. UD, FGO and FGX. It can be seen that the thermal buckling temperature of the CNTRC piezoelectric beams decreased with an increase in the L/H ratio. This is due to the fact that any increase in the L/H ratio will decrease the global stiffness of the structure. Fig. 2 plots the different distribution of nanotubes (UD, FGO and FGX) on thermal buckling temperature T cr for CNTRC piezoelectric beams for different values of L/H. It can be found that the buckling temperatures of FGX CNTRC beams are higher than UD CNTRC. As previously mentioned, higher stiffness of the FGX distribution rather than the UD one is the reason. Also it can be observed that the thermal buckling temperature of the CNTRC piezoelectric beams decreased with an increase in the L/H ratio.

10

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

∗ Fig. 3. Effect of carbon nanotube volume fraction VCN on dimensionless thermal buckling temperature T cr of FGX-CNTRC piezoelectric beams (hp /h = 1/12, V0 = 0, N0 = 0, h = 10 mm).

∗ Fig. 4. Effect of hp /h on dimensionless thermal buckling temperature T cr of FGX-CNTRC piezoelectric beams (V0 = 0, h = 10 mm, VCN = 0.12).

∗ Fig. 3 depicts the effect of CNT volume fraction VCN (=0.12, 0.17, 0.28) on thermal buckling temperature T cr for CNTRC piezoelectric beams with an FGX Distribution of carbon nanotubes (hp /h = 1/12, V0 = 0, h = 10 mm). It can be seen ∗ that the increase of the CNT volume fraction VCN yields an increase of thermal buckling temperature but the beam with intermediate nanotube volume fraction will not have intermediate buckling. Fig. 4 demonstrate the effects of piezoelectric-to-CNTRC thickness ratio (hp /h) on thermal buckling temperature T cr for ∗ CNTRC piezoelectric beams with FGX Distribution of carbon nanotubes (V0 = 0, h = 10 mm, VCN = 0.12). In this example, the thickness of the host CNTRC kept constant and piezoelectric thickness changed accordingly. It can be seen that the thermal buckling temperature is decreased as the piezoelectric-to-CNTRC host thickness ratio (hp /h) is increased. The static bifurcation diagram for the first buckled configuration of FGX-CNTRC piezoelectric beams under clamped– clamped, hinged–hinged, and clamped–hinged boundary conditions at ξ = 0.5 against the axial load is illustrated in Fig. 5 ∗ (hp /h = 1/12, VCN = 0.12, V0 = 0, h = 10 mm, L/H = 25). As the axial thermal load exceeds the critical buckling load, the straight position loses stability and the beam buckles. Fig. 6 shows the effect of different distributions of carbon nanotubes (UD, FGO and FGX CNTRC) on the bifurcation diagram for the static deflection of CNTRC piezoelectric beams

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

11

static deflection at ξ=0.5

a

0

50

100

150

200

250

300

350

400

250

300

350

400

250

300

350

400

ΔT (K)

static deflection at ξ=0.5

b

0

50

100

150

200 ΔT (K)

static deflection at ξ=0.5

c

0

50

100

150

200 ΔT (K)

Fig. 5. Bifurcation diagram for the dimensionless static deflection of FGX-CNTRC piezoelectric beams at ξ = 0.5 with the axial thermal load showing ∗ the first buckled configuration at different volume fraction of carbon nanotubes VCN : (a) clamped–clamped, (b) hinged–hinged, and (c) clamped–hinged (hp /h = 1/12, V0 = 0, h = 10 mm, L/H = 25).

at ξ = 0.5 with the axial thermal load showing the first buckled configurations under different boundary conditions ∗ (hp /h = 1/12, V0 = 0, h = 10 mm, L/H = 25, VCN = 0.12). 5. Conclusions An exact solution for nonlinear buckling analysis of CNTRC piezoelectric beams based on the Euler–Bernoulli beam theory under different boundary conditions which are subjected to in-plane temperature variation, and electrical loading has been

12

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

static deflection at ξ=0.5

a

0

50

100

150

200

250

150

200

250

150

200

250

ΔT (K)

static deflection at ξ=0.5

b

0

50

100 ΔT (K)

static deflection at ξ=0.5

c

0

50

100 ΔT (K)

Fig. 6. Effect of different distribution of carbon nanotubes (UD, FGO and FGX CNTRC) on the bifurcation diagram for the dimensionless static deflection of CNTRC piezoelectric beams at ξ = 0.5 with the axial thermal load showing the first buckled configuration: (a) clamped–clamped, (b) hinged–hinged, and ∗ (c) clamped–hinged (hp /h = 1/12, V0 = 0, h = 10 mm, L/H = 25, VCN = 0.12).

presented. The analysis is performed within the context of von Kármán type displacement–strain relationship. Two kinds of carbon nanotube-reinforced composite (CNTRC) beams, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRC beam are assumed to be graded in the thickness direction. The SWCNTs are assumed aligned, straight and a uniform layout. A parametric study for CNTRC piezoelectric beams with different values of CNT volume fraction and under different sets of thermal environmental conditions has been carried out. It is found that in some cases the beam with intermediate nanotube volume fraction does not have intermediate buckling temperature. The results show that the buckling load of the CNTRC piezoelectric beam can be increased as a result of a functionally graded reinforcement.

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

13

Acknowledgments The work described in this paper was supported by a research grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 114911). The authors are grateful for the financial support. References [1] E.T. Thostenson, Z.F. Ren, T.W. Chou, Advances in the science and technology of carbon nanotubes and their composites: a review, Composites Science and Technology 61 (13) (2001) 1899–1912. [2] R.F. Gibson, O.E. Ayorinde, Y.F. Wen, Vibration of carbon nanotubes and their composites: a review, Composites Science and Technology 67 (1) (2007) 1–28. [3] A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54 (9) (1983) 4703–4710. [4] A.C. Eringen, Nonlocal Continuum Field Theories, Springer, New York, 2002. [5] M. Rafiee, S. Mareishi, M. Mohammadi, An investigation on primary resonance phenomena of elastic medium based carbon nanotubes, Mechanics Research Communications 44 (1) (2012) 51–56. [6] J.N. Coleman, U. Khan, W.J. Blau, Y.K. Gun’ko, Small but strong: a review of the mechanical properties of carbon nanotube–polymer composites, Carbon 44 (9) (2006) 1624–1652. [7] H.D. Wagner, O. Lourie, Y. Feldman, R. Tenne, Stress-induced fragmentation of multiwall carbon nanotubes in a polymer matrix, Applied Physics Letters 72 (2) (1998) 188–190. [8] D. Qian, E.C. Dickey, R. Andrews, T. Rantell, Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites, Applied Physics Letters 76 (20) (2000) 2868–2870. [9] J. Wuite, S. Adali, Deflection and stress behaviour of nanocomposite reinforced beams using a multiscale analysis, Composite Structures 71 (3–4) (2005) 388–396. [10] M.C. Ray, R.C. Batra, A single-walled carbon nanotube reinforced 1–3 piezoelectric composite for active control of smart structures, Smart Materials and Structures 16 (5) (2007) 1936–1947. [11] M.C. Ray, R.C. Batra, Effective properties of carbon nanotube and piezoelectric fiber reinforced hybrid smart composites, Journal of Applied Mechanics (ASME) 76 (3) (2009) 34503. [12] T. Vodenitcharova, L.C. Zhang, Bending and local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube, International Journal of Solids and Structures 43 (10) (2006) 3006–3024. [13] G.D. Seidel, D.C. Lagoud, Micromechanical analysis of the effective elastic properties of carbon nanotube reinforced composite, Mechanics of Materials 38 (8–10) (2006) 884–907. [14] P. Zhu, Z.X. Lei, K.M. Liew, Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory, Composite Structures 94 (4) (2012) 1450–1460. [15] Z.X. Wang, H.S. Shen, Nonlinear vibration of nanotube-reinforced composite plates in thermal environments, Computational Materials Science 50 (8) (2011) 2319–2330. [16] H.S. Shen, C.L. Zhang, Thermal buckling and post buckling behavior of functionally graded carbon nanotube-reinforced composite plates, Materials and Design 31 (7) (2010) 3403–3411. [17] H.S. Shen, Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments. Part II: pressure-loaded shells, Composite Structures 93 (10) (2011) 2496–2503. [18] S.J. Mehrabadi, B.S. Aragh, V. Khoshkhahesh, A. Taherpour, Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes, Composites Part B (Engineering) 43 (4) (2012) 2031–2040. [19] H.S. Shen, Z.H. Zhu, Buckling and postbuckling behavior of functionally graded nanotube-reinforced composite plates in thermal environments, CMC: Computers, Materials & Continua 18 (2) (2010) 155–182. [20] H.S. Shen, Z.H. Zhu, Postbuckling of sandwich plates with nanotube-reinforced composite face sheets resting on elastic foundations, European Journal of Mechanics. A. Solids 35 (2012) 10–21. [21] A. Alibeigloo, Thermoelasticity analysis of functionally graded beam with integrated surface piezoelectric layers, Composite Structures 92 (6) (2010) 1535–1543. [22] A. Gharib, M. Salehi, S. Fazeli, Deflection control of functionally graded material beams with bonded piezoelectric sensors and actuators, Materials Science and Engineering A 498 (1–2) (2008) 110–114. [23] H. Zhu, B.V. Sankar, A combined Fourier Series–Galerkin method for the analysis of functionally graded beams, Journal of Applied Mechanics (ASME) 71 (3) (2004) 421–424. [24] Z.F. Shi, Y. Chen, Functionally graded piezoelectric cantilever beam under load, Archive of Applied Mechanics 74 (3–4) (2004) 237–247. [25] Y.M. Fu, J.Z. Wang, Y.Q. Mao, Nonlinear vibration and active control of functionally graded beams with piezoelectric sensors and actuators, Journal of Intelligent Material Systems and Structures 22 (18) (2011) 2093–2102. [26] Y. Kiani, S. Taheri, M.R. Eslami, Thermal buckling of piezoelectric functionally graded material beams, Journal of Thermal Stresses 34 (8) (2011) 835–850. [27] M. Bodaghi, M. Shakeri, An analytical approach for free vibration and transient response of functionally graded piezoelectric cylindrical panels subjected to impulsive loads, Composite Structures 94 (5) (2012) 1721–1735. [28] L.L. Ke, Y.S. Wang, Z.D. Wang, Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory, Composite Structures 94 (6) (2012) 2038–2047. [29] J. Yang, Y. Chen, Free vibration and buckling analyses of functionally graded beams with edge cracks, Composite Structures 83 (1) (2008) 48–60. [30] M. Rafiee, H. Kalhori, Large amplitude free and forced oscillations of functionally graded beams, Mechanics of Advanced Materials and Structures (2012). http://dx.doi.org/10.1080/15376494.2012.680670. [31] A. Shooshtari, M. Rafiee, Nonlinear forced vibration analysis of clamped functionally graded beams, Acta Mechanica 221 (1–2) (2011) 23–38. [32] L.L. Ke, J. Yang, S. Kitipornchai, Y. Xiang, Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials, Mechanics of Advanced Materials and Structures 16 (6) (2009) 488–502. [33] L. Ke, J. Yang, S. Kitipornchai, Postbuckling analysis of edge cracked functionally graded Timoshenko beams under end shortening, Composite Structures 90 (2) (2009) 152–160. [34] S.L. Kong, S.J. Zhou, Z.F. Nie, K. Wang, Static and dynamic analysis of micro beams based on strain gradient elasticity theory, International Journal of Engineering Science 47 (4) (2009) 487–498. [35] L.L. Ke, J. Yang, S. Kitipornchai, Dynamic stability of functionally graded carbon nanotube-reinforced composite beams, Mechanics of Advanced Materials and Structures 20 (1) (2013) 28–37. [36] M. Rafiee, J. Yang, S. Kitipornchai, Large amplitude vibration of carbon nanotube reinforced functionally graded composite beams with piezoelectric layers, Composite Structures 96 (2013) 716–725. [37] D.L. Shi, X.Q. Feng, Y.Y. Huang, K.C. Hwang, H.J. Gao, The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotubereinforced composites, Journal of Engineering Materials and Technology 126 (3) (2004) 250–257. [38] A.M.K. Esawi, M.M. Farag, Carbon nanotube reinforced composites: potential and current challenges, Materials & Design 28 (9) (2007) 2394–2401.

14

M. Rafiee et al. / Computers and Mathematics with Applications (

)

–

[39] J.D. Fidelus, E. Wiesel, F.H. Gojny, K. Schulte, H.D. Wagner, Thermo-mechanical properties of randomly oriented carbon/epoxy nanocomposites, Composites Part A: Applied Science and Manufacturing 36 (11) (2005) 1555–1561. [40] H.S. Shen, Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments, Composite Structures 91 (1) (2009) 9–19. [41] S. Mareishi, M. Mohammadi, M. Rafiee, An analytical study on thermally induced vibration analysis of fg beams using different HSDTs, Applied Mechanics and Materials 249 (2013) 784–791. [42] K.M. Liew, J. Yang, S. Kitipornchai, Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading, International Journal of Solids and Structures 40 (15) (2003) 3869–3892. [43] M. Rafiee, M. Mohammadi, S. Sobhani Aragh, H. Yaghoobi, Nonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectric functionally graded laminated composite shells, Part I: theory and analytical solutions, Composite Structures 103 (2013) 179–187. [44] M. Rafiee, M. Mohammadi, S. Sobhani Aragh, H. Yaghoobi, Nonlinear free and forced thermo-electro-aero-elastic vibration and dynamic response of piezoelectric functionally graded laminated composite shells, Part II: numerical results, Composite Structures 103 (2013) 188–196. [45] S.A. Emam, A theoretical and experimental study of nonlinear dynamics of buckled beams, Ph.D. Dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA, 2002.