Nonlinear flexural behavior of temperature-dependent FG-CNTRC laminated beams with negative Poisson’s ratio resting on the Pasternak foundation

Engineering Structures 207 (2020) 110250

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Nonlinear flexural behavior of temperature-dependent FG-CNTRC laminated beams with negative Poisson’s ratio resting on the Pasternak foundation Jian Yang, Xu-Hao Huang, Hui-Shen Shen

T

⁎

School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China

ARTICLE INFO

ABSTRACT

Keywords: Beam Functionally graded materials Nanocomposites Temperature-dependent properties Negative Poisson’s ratio Pasternak elastic foundation

Nanocomposite materials, such as carbon nanotube-reinforced composites (CNTRCs), have emerged as a novel engineering material. They have received growing attentions in various engineering sectors. The fabrication process has also offered the possibility to design and make this type of material to have desired features, such as being functionally graded (FG) or/and having negative Poison’s ratio. This paper reports an investigation on the nonlinear flexural behavior of auxetic laminated beams with each layer is made of CNTRC. Each layer may have different CNT volume fractions and the functional grading occurs in the thickness direction of the beam in the piece-wise pattern. The extended rule of mixture model is used to evaluate the temperature-dependent material properties of CNTRCs. The governing equations for the nonlinear bending of FG-CNTRC laminated beams are derived based on the high order shear deformation beam theory. These equations include the geometrical nonlinearity in the von Kármán sense and take into account the thermal effect and the beam-foundation interaction. The nonlinear bending solutions can be obtained by employing a two-step perturbation approach. The nonlinear flexural responses of FG-CNTRC laminated beams under a uniform pressure in thermal environments are revealed and examined in details through a parametric study. Results showed that the negative Poisson’s ratio has a significant impact on the nonlinear flexural behavior of CNTRC laminated beams.

1. Introduction Fiber reinforced composite (FRC) beams are widely used in the airplane and automobile industry due to their light-weight nature and excellent mechanical performance. Different from isotropic materials, composites often show complex mechanical behaviors under external loadings. In their service life, those beams are often encountered with various loading scenarios, which may lead to large deflection. In those cases, geometrically nonlinear theory should be employed to allow for the large deflection effect. Many researchers have proposed a variety of methods for the nonlinear flexural analysis of laminated beams with or without considering the support from elastic foundation [1–6]. Auxetic materials, characterized as a negative Poisson’s ratio (NPR), are known to exhibit enhanced performance from conventional composite materials, such as higher shear modulus and fracture toughness, increased indentation resistance, superb vibration absorption ability, as well as lower fatigue crack propagation rate. The auxetic materials have a wide variety of multifunctional applications, for example, in energy storage,

⁎

biomedical, acoustics, photonics, and thermal management [7]. Potentially such materials could be used to produce sandwich beams with auxetic honeycomb cores [8–10]. Recently, Li et al. [11–14] conducted extensive studies on the nonlinear bending, vibration and thermal postbuckling responses of an auxetic honeycomb sandwich beam under various loading and environmental conditions. The auxetic concept has also been adopted in the fiber reinforced composite laminates by several researchers [15–20]. They showed that to change the stacking sequence and orientation angles may obtain auxetic laminates. It was found that a more severely anisotropic form of carbon fiber reinforced laminate can result into a NPR of higher magnitude [21]. However, there are few studies dealt with the analysis of the mechanical behaviors of the auxetic laminated beam/plate structures. Among those, Azoti et al. [22] analyzed the linear free vibration of a sandwich beam embedded with auxetic layers. The effect of NPR on the natural frequency of thick plates of arbitrary shape was investigated by Lim [23]. Chen and Feng [24] studied the nonlinear amplitudefrequency responses of a thin laminated plate embedded with auxetic

Corresponding author. E-mail address: [email protected] (H.-S. Shen).

https://doi.org/10.1016/j.engstruct.2020.110250 Received 14 November 2019; Received in revised form 15 January 2020; Accepted 15 January 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.

Engineering Structures 207 (2020) 110250

J. Yang, et al.

ends. Each ply is made of a mixture of CNT reinforcement and uniform matrix, and may have different fractions of CNT by volume. By adjusting the volume fraction of CNT in different plies, the piece-wise functionally graded (FG) CNTRC laminated beam can be produced. The beam is placed in a coordinate system (X, Y, Z) in which X and Y are in the length and width directions of the beam, respectively, and Z is in the downward direction perpendicular to the middle surface. The beam has length L, width b, and thickness h, and rests on a Pasternak elastic foundation. The force p0 imparted by the foundation when the beam experiences deflection can be expressed as

layers subject to in-plane excitation. Functionally graded materials (FGMs) are a novel type of composite materials adopted in a wide range of engineering applications [25]. The effects of auxeticity of the materials on the compressive postbuckling and thermal buckling behaviors of the FGM plates resting on elastic foundations were studied by Shariyat and his co-authors [26,27]. In the aforementioned studies, however, the value of NPRs (ranging from −0.3 to −0.9) is only a virtual value, which does not correspond to the real composite materials or the metamaterials. Since the discovery of carbon nanotube (CNT) by Iijima in 1991[28], extensive studies on CNT have been conducted by many researchers and the extraordinary material properties of CNT have been widely reported [29–31]. Their results showed that the material properties of CNT are anisotropic [29], size-dependent [30] and temperature dependent [31]. Carbon nanotube reinforced composites (CNTRCs) are a new emerging composite material in which the ratio of the two inplane Young’s moduli is more than 40 and the in-plane shear modulus is very small in comparison with the longitudinal Young’s modulus. With such material properties, the auxetic CNTRC can be produced [32]. The FGM concept was first applied to carbon nanotube reinforced composites by Shen [33], and the graded distributions of CNT within an isotropic matrix were designed specifically for the purpose of improved structural mechanical performances. Shen and Xiang [34] conducted the analyses for the large deflection bending, nonlinear vibration and thermal postbuckling of single layer FG-CNTRC beams resting on the Pasternak foundation based on a higher order shear deformation beam theory. Following the work of Shen [33], the nonlinear bending analysis of FG-CNTRC structures have been further studied by other research teams [35–38] using different methods. Recently, Shen and his co-authors presented the nonlinear free vibration analysis of CNTRC laminated plates and cylindrical shells with NPRs [39,40]. The nonlinear bending behaviors of beams or plates or cylindrical shells are different nonlinear problems. For nonlinear bending problem the superposition principle is no longer valid. Hence, the present nonlinear bending problem of FG-CNTRC laminated beams should be re-solved. Like in [41,42], CNT reinforcements are assumed to be uniformly distributed and aligned [43] in each layer. Unlike in [41,42] where each CNTRC layer is assumed to be linear functionally graded, in the current study the CNTRC layers are arranged in a piecewise functionally graded (FG) pattern in the thickness direction of the beam. The out-of-plane NPR of the beam is first determined by choosing specific stacking sequence and lamination angle. The temperature dependent material properties of CNTRC layers are determined through the extended rule of mixture model in a way that the CNT’s efficiency parameters are evaluated by matching the rule of mixture model results against the molecular dynamics (MD) simulation results. The governing equations are derived based on the high order shear deformation beam theory taking into account the von Kármán-type of kinematic nonlinearity as well as the beam-foundation interaction effects. The effect of thermal environmental condition is also considered in this work. A two-step perturbation approach is employed to solve the governing equations to examine the nonlinear flexural behaviors and the finite element method (FEM) is employed to calculate the effective Poisson’s ratio (EPR)-deflection curves of CNTRC laminated beams in the large deflection range.

¯ p0 = K¯1 W

¯ / dX 2) K¯2 (d 2W

(1)

¯ is the beam deflection in the Z direction, K¯1 denotes the where W Winkler foundation stiffness and K¯2 denotes the shear layer stiffness of the foundation. In the current study, 6-plies CNTRC laminated beams are considered. In order to conduct piece-wise functionally graded CNTRC laminated beams with FG- , FG-V, FG-O and FG-X patterns, an arithmetic series of 0.11, 0.14 and 0.17 of CNT volume fractions, as reported in Han and Elliott [44], is selected. For FG- , the CNT volume fractions are arranged as [(0.11)2/(0.14)2/(0.17)2] for six plies. For FG-V, the distribution of CNT reinforcements is inversed as [(0.17)2/(0.14)2/ (0.11)2]. For FG-O, a mid-plane symmetric graded distribution of CNT reinforcements is achieved, i.e. [0.11/0.14/0.17]S. For FG-X, the arrangement is inversed as [0.17/0.14/0.11]S. A uniformly distributed (UD) CNT reinforced beam with CNT volume fraction of each layer VCN = 0.14 is also considered for comparison purpose. It is worth noting that the two cases of UD and FG-CNTRC laminated beams will have the same total volume fraction of CNT. 2.1. Effective material properties of CNTRC plies It is known that the material properties of nano-materials may be changed when the small scale effects are taken into account [45]. When a matrix is reinforced with CNTs, its elastic properties cannot be determined accurately using the conventional micromechanical model. In order to estimate the effective material properties of the CNTRC laminated beam, the extended rule of mixture model is employed. In this model, the CNT efficiency parameters j (j = 1,2,3) are introduced to account for the effects relating to the interaction and the load transfer between the polymer matrix and CNT. According to this rule, the effective Young’s moduli and shear modulus of each ply can be estimated by [33]

E11 = 2

E22 3

G12

CN 1 VCN E11

+ Vm E m

(2a)

=

VCN V + mm CN E E22

(2b)

=

VCN V + mm CN G G12

(2c)

where in the above equation and the rest of this study, CN and m inCN CN dicate the properties of CNT and matrix, respectively. E11 , E22 and Em CN are the Young’s moduli and G12 and Gm are the shear moduli. VCN and Vm are the volume fractions of the CNT and the matrix, which satisfy the relationship of VCN + Vm = 1. The value of j in Eq. (2) may be obtained through matching the results from Eq. (2) with the ones from MD simulations [44]. The Poisson’s ratio of each layer is given by the conventional rule of

2. Modelling of FG-CNTRC laminated beams Consider a laminated beam consisting of N plies and having two pin

2

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J. Yang, et al.

mixture 12

B12 B13 0 0 B16 CN 12

= VCN

+ Vm

B22 B23 0 0 B26

(3)

m

CN where 12 and m are Poisson’s ratios of the CNT and the matrix. They are weakly dependent on temperature. The thermal expansion coefficients in the longitudinal and transverse directions of each layer can be expressed by [46] CN CN m VCN E11 11 + Vm E CN m VCN E11 + Vm E

11

=

22

= (1 +

CN 12 ) VCN

CN 22

CN CN where 11 , 22 and and the matrix.

1

B32 B33 0 0 B36

=

0 0 B44 B45 0 0 0 B45 B55 0 B11 B13 0 0 B16

(4a) m) V m m

12 11

B21 B23 0 0 B26 B5

(4b)

2

B31 B33 0 0 B36

=

0 0 B44 B45 0

are thermal expansion coefficients of the CNT

0 0 B45 B55 0

B11 B12 0 0 B16 B21 B22 0 0 B26

For an anisotropic laminated beam, the out-of-plane effective e e Poisson’s ratios 13 and 23 can be expressed as e 13

= -

e 23

=

A5 - 1 B6 - 1 B5 - 1 D

B5

3

B31 B32 0 0 B36

=

0 0 B44 B45 0 0 0 B45 B55 0

,

A5 - 2 B6 - 2 B5 - 2 D

=

B21 B31 0 0 B61

B22 B32 0 0 B62

B23 B33 0 0 B63

0 0 B44 B45 0

0 0 B45 B55 0

B26 B36 0 0 B66

B6

(5a)

1

,

=

B11 B31 0 0 B61

B12 B32 0 0 B62

B13 B33 0 0 B63

0 0 B44 B45 0

0 0 B45 B55 0

B16 B36 0 0 B66

,

D16 D26 D36 0 0 D66

,

in which

A21 A22 0 0 A26

B6

A31 A32 0 0 A36 0 0 A 44 A 45 0

2

,

0 0 A 45 A55 0 A61 A62 0 0 A66

D11 D12 D21 D22

A11 A12 0 0 A16

D=

A31 A32 0 0 A36 A23 =

0 0 A 44 A 45 0 0 0 A 45 A55 0 A61 A62 0 0 A66

A22 A23 0 0 A26

1

=

D31 D32 0 0 0 0 D61 D62

D13 D23 D33 0 0 D63

0 0 0 0 0 0 D44 D45 D45 D55 0 0

(5b)

In particular, when the beam is mid-plane geometrically symmetric with UD or FG-O and FG-X functionally graded patterns where the coupling stiffnesses Bij = 0 (ij = 1,2,3,6), the effective Poisson’s ratios e e 13 and 23 can be simply written as

A32 A33 0 0 A36 A5

,

B61 B62 0 0 B66

A23 B6 - 2 B5 - 3 D

A13 =

,

B61 B63 0 0 B66

2.2. Effective Poisson’s ratios of CNTRC laminated beams

A13 B6 - 1 B5 - 3 D

,

B62 B63 0 0 B66

m

+ (1 +

m

B5

e 13

0 0 A 44 A 45 0

=

0 0 A 45 A55 0

2 A16 (A22 A36 A23 A26 ) + A13 (A26 A22 A66 ) + A12 (A23 A66 A26 A36 ) 2 2 2 A26 A33 2A23 A26 A36 + A23 A66 + A22 (A36 A33 A66 )

(6a)

A62 A63 0 0 A66 e 23

A11 A13 0 0 A16

=

2 A16 A23

A16 (A12 A36 + A13 A26 ) + A12 A13 A66 A11 (A23 A66 A26 A36 ) 2 2 2 A16 A33 2A13 A16 A36 + A13 A66 + A11 (A36 A33 A66 )

A31 A33 0 0 A36 A5

2

=

0 0 A 44 A 45 0

(6b)

,

where Aij are the beam stretching stiffness as defined by

0 0 A 45 A55 0

N

Aij =

A61 A63 0 0 A66

b k=1

3

hk hk 1

(C¯ij )k dZ

(i , j = 1

6)

(7)

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J. Yang, et al.

and S¯T is defined by

in which C¯ij are the transformed stiffness coefficients, and

[C¯ij]

1

= [S¯ij]

(8)

where S¯ij are the transformed flexibility coefficients, and the details of which may be presented as follows:

Ax = Q¯ 11 (c 2

S66 ) c 2s 2 + S66 (s 4 + c 4 ),

(9)

where Sij are the flexibility coefficients, defined by

S11 = S22 = S44 =

1 , E11 1 , E22 1 , G 23

12

S12 =

E11 23

S23 =

E22 1 , G13

S55 =

13

, S13 =

E11

S66 =

1 G12

and (10b)

where E11, E22, G12, G13, G23, 12 , 13 and 23 are Young’s moduli, shear moduli, and Poisson’s ratios of the laminate, and is the lamination angle with respect to the beam X-axis.

4W 11

S11

+ S12

X4

¯ K¯1 W

S21

3W ¯

1 N¯x = L

L 0

+ 2W ¯

K¯2

+ S22

X3

x

X3

X2

2¯

x X2

A¯11 2

B¯11 + A¯11 X 2

2M ¯T

+ N¯x

X2

2W ¯

X2

4 ¯ E11 3h2

S26 ¯x

¯ T P¯T ] = [ N¯ T M

X

+

X2

N¯ T dX

(12)

[Ax ]k (1, Z , Z 3) T dZ

k=1

hk 1

2 22

T1

=

x2 x

x2

T , C2 =

x=

X , L

Nx =

L2N¯ x 2D¯ , 11

(

(13)

11, 23

( (

=

14 , 16, T1

=

q

=

12,

¯ W , L

W=

(K1 W

q

+

23

T3

T , C3 = (

x

(

21,

L4 4D¯

26)

L

= =

L2AxT 2D¯ , 11 qL3 4D¯ , 11

=

1

1 A¯11 L

T 3,

L4

,

(15)

+

14

2W

x

15

x

x2

dx

+

K2

W x

26

T3

T 6)

2W

x2

)

NT x

(16)

C3

ST =0 x

(17) (18)

T,

E0 I

¯x, L2 2hD¯ 11

=

L2A¯11 2D¯ , 11

T 6)

2

L2

2D¯ , 11

L2 E0 I

),

S21, S22),

),

4 ¯ E 3h2 11

),

4 ¯ E 3h2 11

B¯11 =

),

2

S11, S12,

4 ¯ E , 3h2 11

11

4 ¯ P 3h2 x

x

2

=

11

(M¯ ,

), (K , k ) = K¯ (

1 ( D¯ 11

(B¯ (B¯ ,

2D¯ 11

(

11

22 )

L2 2D¯ S23, 13 11 15)

x

(Mx , Px ) =

(K1, k1) = K¯1

hk

b

22 )

11

and the non-dimensional parameters are defined by

N¯ T S¯T + =0 X X 2W ¯

2

W x

13

2

0

2M T

C2

x2

x3

C1 =

where ¯ x is the rotation of the normal of the mid-plane about the Y axis, and all the coefficients Sij (i = 1,2; j = 1,2,3,6) are defined in Appendix A. The effects of the elastic foundation and the differential temperature are also included in Eqs. (11)–(13), where the thermal forces N¯ T , mo¯ T and the higher order moments P¯T caused by the temperature ments M rise are defined by N

+ Q¯ 16 2cs (

where

(11)

¯ ¯x + W X

¯ 2 ¯ W + B¯11 x X X

22 )

x2

3W 21

+q

=0

S23

+ c2

11

2W

2N T 16

x

x3

+ C1

x2

The beam is exposed to a differential temperature field and is subjected to a transverse uniform distributed load (UDL) q. Based on the high order shear deformation beam theory [47,48] and the von Kármán-type nonlinear strain–displacement relationship, the governing equations of a FG-CNTRC laminated beam, which includes the reaction from supporting foundation, can be written as [34] 2N ¯T

3 12

x4 2W

2.3. Governing equations of CNTRC laminated beam

3¯

+ Q¯ 12 (s 2

22 )

A two-step perturbation approach was developed by Shen [50] and was successfully to solve different kinds of nonlinear problems of beams by many research teams [51–57]. To utilize this method for solving the nonlinear bending problem of FG-CNTRC laminated beams, the governing equations (11)–(13) are first re-written in the dimensionless forms as

(10a)

c = cos , s = sin

4W ¯

+ s2

3. Solving process

,

1 , E33

, S33 =

11

where 11 and 22 are the thermal expansion coefficients for the kth ply as given in Eq. (4). Q¯ ij are the transformed elastic constants as defined in Appendix A. It is worth noting that Eq. (13) only holds valid when the beam has constrained ends in the longitudinal direction, namely, the displacement in the longitudinal direction U¯ (X = 0) = U¯ (X = L) = 0 . As pointed out by Shen [49], for the nonlinear bending or the nonlinear vibration problem, the longitudinal constraints can be either way, but for the postbuckling, at least one end should be freely moveable in the longitudinal direction, regardless of the rotational condition.

S¯16 = (2(S11 S12) S66 ) c 3s + (2(S12 S22) + S66 ) cs 3, S¯22 = S22 c 4 + (2S12 + S66 ) c 2s 2 + S11 s 4, S¯23 = S23 c 2 + S13 s 2 S¯26 = (2(S11 S12) S66) cs3 + (2(S12 S22) + S66) c 3s S¯36 = 2(S13 S23 ) cs , S¯44 = S44 c 2 + S55 s 2, S¯45 = (S55 S44 ) cs , S¯55 = S55 c 2 + S44 s 2, 2S12)

(14b)

where T = T-T0 is the temperature rise from the reference temperature T0 at which there are no thermal-induced strains. In Eq. (14a)

S¯11 = S11 c 4 + (2S12 + S66 ) c 2s 2 + S22 s 4 , S¯12 = (S11 + S22 S66 ) c 2s 2 + S12 (c 4 + s 4 ), S¯13 = S13 c 2 + S23 s 2,

S¯66 = 2(2(S11 + S22

4 ¯T P 3h2

¯T S¯T = M

L2 2hD¯ 11

(D , T x

4 T F 3h2 x

), (19)

where the dimensionless foundation stiffnesses k1 and k2 will only appear in the following numerical examples, E0 is the reference value of

(14a)

4

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J. Yang, et al.

Em at the room temperature (T0 = 300 K), and AxT , DxT , FxT are defined by

Table 2 Efficiency parameters for CNT/PmPV nanocomposites [33]. VCN

hk

N

(AxT , DxT , FxT ) =

[Ax ]k (1, Z , Z 3) dZ

b k=1

(20)

hk 1

0.11 0.14 0.17

where Ax is given in detail in Eq. (15). Eqs. (16) and (17) are the governing equations for FG-CNTRC laminated beams with immovable end conditions. These equations are adopted in the following analysis. It is worth noting that Eq. (16) along is not sufficient for analyzing cantilever beams. To attain the load–deflection relation, we assume that

2

1

0.149 0.150 0.149

3

0.934 0.941 1.381

0.934 0.941 1.381

jw (x ), j

W (x , ) = j =1 x (x ,

j

)=

xj (x ),

j=1 q

j

=

j

(21)

j=1

where is a small perturbation parameter that has no exact physical meaning in the first step. In the current study the theoretical solution for the simply supported FG-CNTRC laminated beam is derived. For this purpose, the first term of wj (x) is assumed to have the form as (22)

(1) w1 (x ) = A10 sin mx

Meanwhile, the thermal bending moments in Eqs. (16) and (17) are expanded in the Fourier sine series to satisfy the simply supported boundary conditions, namely,

(MxT , SxT ) =

4 k = 1,3,...

1 sin kx k

(23)

Substituting Eq. (21) into Eqs. (16) and (17), collecting the terms of the same order of , a set of perturbation equations is obtained. By using Eqs. (22) and (23) to solve these equations step by step, the asymptotic solutions are obtained as (24)

(1) W (x , ) = A10 sin mx + O ( 4 )

x (x ,

)=

(1) B10

(25)

cos mx + O ( 4)

and q

=

(1) q

(1) (A10 )+

(2) q

(1) 2 (A10 ) +

(3) q

(1) 3 (A10 ) + O(

4

(26)

)

It is worth noting that the perturbation series is a divergent series. Which order solution is closer to the real solution needs to be determined by experimental verification or by comparing with the theoretical exact solution. Contrary to Zhang’s conclusion [58], the higher order perturbation solution is not necessarily more correct than the lower order solution. (1) All coefficients in Eqs. (24)–(26) can be written in terms of A10 , in (1) Eqs. (24) and (26), ( A10 ) is considered as the alternative perturbation parameter that is related to the maximum deflection Wm. In the case of the simply supported beam, m is usually taken to be 1. From Eqs. (24)

Fig. 1. Poisson’s ratio

13

for ( 1 / -

1/ 2 )S

laminates.

Table 1 Temperature-dependent material properties for (10,10) SWCNT (tube length = 9.26 nm, mean tube radius = 0.68 nm, tube thickness = 0.067 nm, [59]. Temperature (K)

CN E11 (TPa)

CN E22 (TPa)

CN G12 (TPa)

300 400 500

5.6466 5.5679 5.5308

7.0800 6.9814 6.9348

1.9445 1.9703 1.9643

5

-6 CN 11 (× 10 /K)

3.4584 4.1496 4.5361

CN 12 =0.175)

-6 CN 22 (× 10 /K)

5.1682 5.0905 5.0189

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J. Yang, et al.

Fig. 3. Comparisons of load–deflection curves for the (0/90/0) beam subjected to UDL.

Table 4 ¯ max / h Nonlinear relationships between pressure q0L4/E0I and beam deflection W for three kinds of UD and FG-X CNTRC laminated beams subjected to UDL at T = 300 K (L/h = 20, h = 0.006 m). Lay-up

¯ max /h W

(20/-20/20)S

UD FG-X UD FG-X UD FG-X

(70/-70/70)S (90/-90/90)S

0.5

1.0

1.5

2.0

2.5

12.6286 14.3357 0.3937 0.4988 0.4378 0.5583

49.2449 54.0646 1.8622 2.2443 2.0057 2.4273

133.8368 144.5798 5.4800 6.4833 5.8336 6.9179

290.3918 311.2745 12.3219 14.4625 13.0516 15.3408

542.8978 579.5418 23.4627 27.4287 24.7898 29.0068

and (26) the load-central deflection relationship can be written as

¯m qL3 W = AW(0) + AW(1) 4D ¯ 11 L Fig. 2. Poisson’s ratio

23

for ( 1 / -

1/ 2 )S

Table 3 e Negative Poisson’s ratios 13 for (20/-20/20)S and ¯ /h = 0 (h = 6 mm). laminated beams at W T(K)

UD

(20/-20/20)S 300 −0.62 400 −0.76 500 −0.95 (70/-70/70)S 300 −0.62 400 −0.76 500 −0.95

e 23

laminates.

2

¯m W L

+ AW(2)

+ AW(3)

¯m W L

3

+ ...

(27)

in which

for (70/-70/70)S CNTRC

FG-V

FG-

FG-X

FG-O

−0.53 −0.67 −0.86

−0.53 −0.67 −0.86

−0.54 −0.67 −0.86

−0.54 −0.67 −0.86

−0.53 −0.67 −0.86

−0.53 −0.67 −0.86

−0.54 −0.67 −0.86

−0.54 −0.67 −0.86

AW(0) =

Aw(1) =

AW(2) =

AW(3) =

6

T3

11

4

(

12

2

2

15

T 6)

T3

14

21 22

23

+

21 22

12 22

23

+

T 23

+ (K1 + K2 )

T1

T

23

+

23

3

16

13

(28)

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J. Yang, et al.

Fig. 5. Effect of foundation stiffness on the flexural load–deflection curves of CNTRC laminated beams with negative Poisson’s ratios subjected to UDL: (a) (20/-20/20)S beams; (b) (70/-70/70)S beams.

Fig. 4. Flexural load–deflection curves of CNTRC laminated beams with negative Poisson’s ratios subjected to UDL: (a) (20/-20/20)S beams; (b) (70/-70/ 70)S beams.

4. Numerical results and discussion

e and that with the configof (20/-20/20)S has the maximum NPR 13 e . Hence, (20/-20/ uration of (70/-70/70)S has the maximum NPR 23 20)S and (70/-70/70)S CNTRC laminated beams are considered in the e e following examples. The negative Poisson’s ratios 13 and 23 for these two CNTRC laminated beams under different thermal environmental conditions T = 300, 400 and 500 K are listed in Table 3. The study on the nonlinear bending behavior of FG-CNTRC laminated beams has not been attempted previously. To partly validate the present method, a laminated beam of the (0/90/0) configuration and made from bimodular composite material is considered. The beam is subjected to UDL only. The dimensionless load–deflection curves are plotted in Fig. 3. It can be seen that the present analytical results agree well with those of Ghazavi and Gordaninejad [1] based on the first order shear deformation beam theory. The beam has L/h = 10. The Young’s moduli and Poisson’s ratios for tension and compression of t t graphite/epoxy are, respectively, E11 = 165.5 GPa, E22 = 8.276 GPa, c c c t E E = 0.32, = 151.7 GPa, = 7.586 GPa, = 0.3, and shear 11 22 12 12 moduli for both tension and compression are G12 = G13 = G23 = 2.586 GPa. Fig. 3 shows that the present FEM results are slightly higher than the present analytical solutions, whereas the FEM results of Ghazavi and Gordaninejad [1] are slightly lower than the present analytical solutions.

In this section, numerical illustrations are presented in order to clarify the effect of NPR on the nonlinear flexural behavior of CNTRC laminated beams supported by a Pasternak elastic foundation under thermal environmental conditions. The effective material properties of CNTRCs are assumed to be temperature dependent. (10,10) SWCNTs with hCN = 0.067 nm are selected as reinforcement. The temperaturedependent material properties of (10,10) SWCNTs are listed in Table 1 [59]. The CNT efficiency parameters j (j = 1,2,3) are obtained by matching the Young’s moduli E11 and E22 of CNTRCs predicted from the rule of mixture model to those from the MD simulations, as previously reported in Shen [33]. These parameters are listed in Table 2, and we assume that 3 = 2 and G23 = G13 = G12. Poly{(m-phenylenevinylene)–co-[(2,5-dioctoxy-p-phenylene) vinylene]}, referred to as PmPV, is chosen for matrix, and the material properties of PmPV are assumed to be m = 0.34, m =45(1 + 0.0005 T) × 10-6/K and E m = (3.51–0.0047 T) GPa, in which T = T0 + T and T0 = 300 K (room temperature). Hence, m = 45.0 × 10-6/K andE m = 2.1 GPa when T = 300 K. From Eq. (5), the relationship between the effective Poisson’s ratios (EPRs) and the lamination angles are calculated and plotted in Figs. 1 and 2. It is observed that the laminate with the configuration

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Fig. 6. Effect of temperature variation on the flexural load–deflection curves of CNTRC laminated beams with negative Poisson’s ratio subjected to UDL: (a) (20/-20/20)S beams; (b) (70/-70/70)S beams.

Fig. 7. EPR-deflection curves of CNTRC laminated beams subjected to UDL: (a) (20/-20/20)S beams; (b) (70/-70/70)S beams.

After validating the presented formulation and solution method, a parametric study has been carried out and key results are shown in Table 4 and Figs. 4–8. For these examples, the beam has L/h = 5, 10 and 20, and the thickness of each CNTRC ply is identical and the total thickness of the beam h = 0.006 m. Four types of FG-CNTRC laminated beams, i.e. FG- , FG-V, FG-O and FG-X, are considered. An UD-CNTRC laminated beam with the same thickness is also considered as a reference case. The boundary conditions are pin-pin and two ends are constrained in the longitudinal direction. It should be mentioned that in ¯ /h and q0L4/E0I represent the dimensionless for the beam’s Figs. 4–8 W central deflection and the applied UDL, respectively, where E0 is the Young’s modulus of Em at T = 300 K. Fig. 4 presents the load–deflection curves of (20/-20/20)S and (70/70/70)S CNTRC laminated beams with L/h = 20 influenced by four types of different CNT reinforcement design arrangements, i.e. FG- , FG-V, FG-O and FG-X. Also plotted are the results of the same beam of UD type for the comparison purpose. One can see that the FG- (20/20/20)S beam has the lowest, while the FG-V (20/-20/20)S beam has the greatest deflection amongst the four FG cases in Fig. 4(a). In contrast, the UD (70/-70/70)S beam has the highest, while the FG- (70/-

70/70)S beam has the lowest deflections in Fig. 4(b). It can also be seen that the load–deflection curve of FG-X type falls lower than that of the UD type for both (20/-20/20)S and (70/-70/70)S beams. This observation is also valid for the (90/-90/90)S CNTRC laminated beam as e e shown in Table 4, in which the Poisson’s ratios 13 and 23 are both positive. These numerical results are useful for the benchmarking purpose by others. Therefore, in the following parametric analysis cases, only UD and FG- CNTRC laminated beams are considered. Fig. 5 shows the impact of the foundation stiffness on the flexural behavior of UD and FG- (20/-20/20)S and (70/-70/70)S CNTRC laminated beams with L/h = 10 subjected to UDL and resting on elastic foundations at T = 300 K. Two foundation models are considered. The stiffnesses are (k1, k2)=(1000, 100) for the Pasternak elastic foundation, (k1, k2)=(1000, 0) for the Winkler elastic foundation and (k1, k2) =(0, 0) for the beam without any elastic foundation. Like the cases of single layer CNTRC beam [34], the beam load–deflection curves decrease with increase in foundation stiffness for both (20/-20/20)S and (70/-70/70)S CNTRC laminated beams. Fig. 6 illustrates the effect of temperature variation on the nonlinear flexural behavior of UD and FG- (20/-20/20)S and (70/-70/70)S

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figures we observe that the EPR-deflection curves first appear to move downward and then upward and the variation of EPR becomes small ¯ /h is sufficiently large. From Fig. 7, it is found that the negative when W e value of the Poisson’s ratio 13 reaches up to −1.18 for the (20/-20/20)S e beam with L/h = 20, and that for 23 is up to −1.25 for the (70/-70/ 70)S beam with L/h = 20. Unlike what Fig. 4(b) shows, the FG-X (70/70/70)S beam’s EPR-deflection curve falls to the lowest and the FG-O (70/-70/70)S beam has the highest EPR-deflection curve, as shown in Fig. 7(b). Similar to the trend shown in Fig. 5(a) and (b), the EPRdeflection curves increased with the increase in temperature for both (20/-20/20)S and (70/-70/70)S beams, as shown in Fig. 8(a) and (b). 5. Concluding remarks The large-deflection nonlinear flexural responses of functionally graded CNTRC laminated beams with negative Poisson’s ratios, which are resting on Pasternak elastic foundations and placed in thermal environments, have been presented. The novelty of this study can be reflected by the identification of the negative Poisson’s ratio of CNTRC laminated beams with the functionally graded configurations by performing the nonlinear bending analysis. The specification of the volume fraction of CNT in each ply will effectively lead to various piece-wise functionally graded CNTRC laminated beams. Based on the CNT volume fraction in each ply, those beams are characterized as UD, FG- , FG-V, FG-O and FG-X types showing the different variation along the thickness direction. The extended rule of mixture model is used for obtaining the temperature-dependent material properties of FG-CNTRC beams. The two-step perturbation approach is employed to solve the geometrically nonlinear governing equations. We found that the (20/-20/20)S e CNT/PmPV laminated beam has the maximum NPR 13 , while the (70/e 70/70)S CNT/PmPV laminated beam has the maximum NPR 23 . Numerical results reveal that the nonlinear flexural responses are significantly affected by the value of EPR. Conflict of interest

Fig. 8. Effect of temperature variation on the EPR-deflection curves of CNTRC laminated beams subjected to UDL: (a) (20/-20/20)S beams; (b) (70/-70/70)S beams.

The authors declare that there are no conflicts of interests with publication of this work.

CNTRC laminated beams with L/h = 5 subjected to a uniform pressure exposed to temperature field T = 300, 400 and 500 K. It can be seen that the deflections are increased with increase in temperature for both (20/-20/20)S and (70/-70/70)S beams of UD type and FG- type. The negative (upward) initial deflection is clearly observed when temperature variations are under consideration. A finite elements commercial software has been used in order to perform the EPR-deflection curves. The EPR-deflection curves of (20/20/20)S and (70/-70/70)S CNTRC laminated beams in the large deflection range are calculated and are plotted in Figs. 7 and 8. From these

Acknowledgments This study was supported by the National Natural Science Foundation of China under Grant 51779138. The third author is grateful for this financial support. The first and second authors are also grateful for the supports from the Science Research Plan of Shanghai Municipal Science and Technology Committee under Grant 18DZ1205603, and the Innovation Program of Shanghai Municipal Education Commission under Grant 14ZZ027.

Appendix A In Eqs. (11)–(13), the coefficients Sij (i = 1,2; j = 1–6) are defined by

S11 =

4 ¯ F11 3h2

S12 = D¯ 11 S21 =

B¯11 ¯ B11 A¯11

4 ¯ F11 3h2

4 ¯ F11 3h2

S22 = D¯ 11

B¯11 E¯11 A¯11

4 ¯ H11 3h2

4 ¯ F11 3h2

4 ¯ E11 3h2

E¯11 ¯ B11 A¯11

4 ¯ F11 3h2

4 ¯ E11 3h2

4 ¯ H11 3h2

1 ¯ B11 A¯11

4 ¯ E11 3h2

2

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J. Yang, et al.

4 ¯ D55 h2

S23 = A¯ 55

S26 =

B¯11 A¯11

4 ¯ D55 h2

4 ¯ F55 h2

4 E¯11 3h2 A¯11

(A.1)

where A¯11, B¯11, D¯ 11, etc., are the beam reduced stiffness constants and can be written as [60]

A¯11 B¯11 E¯11 B¯11 D¯ 11 F¯11 = E¯11 F¯11 H¯11

A11 B11 E11 B11 D11 F11 E11 F11 H11 A22 A26 B22 B26 E22 E26

A12 A16 B12 B16 E12 E16 B12 B16 D12 D16 F12 F16 E12 E16 F12 F16 H12 H16

A26 A66 B26 B66 E26 E66

B22 B26 B26 B66 D22 D26 D26 D66 F22 F26 F26 F66

E22 E26 F22 F26 H22 H26

E26 E66 F26 F66 H26 H66

1

A12 A16 B12 B16 E12 E16

B12 B16 D12 D16 F12 F16

E12 E16 F12 F16 H12 H16

(A.2)

where Aij, Bij, Dij, etc., are the beam stiffness constants, defined as hk

N

(Aij , Bij , Dij , Eij, Fij , Hij ) =

b k=1

(A¯ 55 , D¯55, F¯55) =

(i, j = 1, 2, 6)

(A.3a)

hk

N

(Q55 )k (1, Z 2, Z 4 ) dZ

b k=1

(Q¯ ij )(1, Z , Z 2, Z 3, Z 4, Z 6) dZ

hk 1

(A.3b)

hk 1

in which

(Q¯45)2 Q¯44

Q55 = Q¯55

(A.4)

and Q¯ ij are the transformed elastic constants, defined by

Q¯ 11 c4 2c 2s 2 Q¯ 12 c 2s 2 c 4 + s 4 s4 2c 2s 2 Q¯ 22 = 3s cs 3 c c 3s ¯ Q16 3 3 cs c s cs 3 Q¯ 26 c 2s 2 2c 2s 2 Q¯ 66 Q¯44 Q¯45 = Q¯55

c2 s2 cs cs s2 c2

s4 c 2s 2 c4 cs 3 c 3s 2 c s2

4c 2s 2 4c 2s 2 4c 2s 2 2cs (c 2 s 2 ) 2cs (c 2 s 2) (c 2 s 2 ) 2

Q11 Q12 Q22 Q66 (A.5a)

Q44 Q55

(A.5b)

where

Q11 = Q22 = Q12 =

E11 (1

12 21)

E22 (1

12 21)

, ,

21 E11

(1

12 21)

(A.5c)

Q44 = G23, Q55 = G13, Q66 = G12 where E11, E22, G12, are the effective Young’s and shear moduli and

12

and

21

are Poisson’s ratios of the CNTRC layer, respectively.

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