Rectangular and skew shear buckling of FG-CNT reinforced composite skew plates using Ritz method

Aerospace Science and Technology 77 (2018) 388–398

Contents lists available at ScienceDirect

Aerospace Science and Technology www.elsevier.com/locate/aescte

Rectangular and skew shear buckling of FG-CNT reinforced composite skew plates using Ritz method Y. Kiani a,∗ , M. Mirzaei b a b

Faculty of Engineering, Shahrekord University, Shahrekord, Iran Department of Mechanical Engineering, Faculty of Engineering, University of Qom, Qom, Iran

a r t i c l e

i n f o

Article history: Received 23 December 2017 Received in revised form 7 February 2018 Accepted 17 March 2018 Available online 20 March 2018 Keywords: Carbon nanotube reinforced composite Ritz method Gram–Schmidt process Skew plate Functionally graded Rectangular and skew shear

a b s t r a c t Present research deals with the shear buckling behaviour of composite skew plates reinforced with aligned single walled carbon nanotubes (CNTs). Distribution of CNTs across the thickness of the skew plate are assumed to be uniform or functionally graded. Two different types of shear loads are considered. The case of rectangular shear which produces pure shear and the case of skew shear which results in a combined uniform shear and uniaxial tension/compression. Suitable for moderately thick plates, first order shear deformation plate theory is used to estimate the displacement field of the plate. The equivalent properties of the composite media are obtained by means of the refined rule of mixtures approach which contains efficiency parameters to capture the size dependent properties of the CNTs. With the aid of the Hamilton principle, transformation of the orthogonal coordinate system to an oblique one and the conventional Ritz method whose shape functions are constructed according to the Gram–Schmidt process, the stability equations of the plate are established and solved for two different types of loading, namely rectangular and skew shear loads. As shown, through introduction of a proper functionally graded pattern, i.e., FG-X pattern, the buckling load of the plate may be increased, significantly. © 2018 Elsevier Masson SAS. All rights reserved.

1. Introduction Carbon nanotubes (CNTs) stand as a promising candidate for reinforcement of the composites due to their exceptional mechanical, thermal and electrical properties. Volume fraction of CNT in a polymeric/metal matrix is an important factor to improve the mechanical properties of a composite media reinforced with CNTs. It is shown that, through introduction of CNTs as reinforcements, stiffness of the structure may be enhanced significantly, more than other conventional reinforcements. Distribution type of CNTs through a specific direction of an structural element is another factor which may affect the global or local properties of an structural element reinforced with CNT. Consequently, CNTs as reinforcement may be distributed in a functionally graded pattern across the thickness of a structural element. This type of composite is known as functionally graded carbon nanotube reinforced composites (FG-CNTRC). An extensive overview on the modelling and analysis of FG-CNTRC beams, plates and shells is provided by Liew et al. [1].

*

Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Kiani), [email protected] (M. Mirzaei). https://doi.org/10.1016/j.ast.2018.03.022 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.

The fundamental research on the subject of FG-CNTRC structures belongs to Shen [2]. In this research, it is shown that with the introduction of CNTs, as reinforcements with nonuniform distribution across the plate thickness, the bending moments induced by lateral loading may be alleviated significantly. Shen proposed that due to higher elasticity modulus of CNTs in comparison to matrix, to reach higher flexural rigidity in plate, CNTs should be inserted in surfaces near the top and bottom of the plate. Motivated by the mentioned interesting work of Shen, various investigators analysed the influences of CNTs on the stability behaviour of CNTRC structures. An overview of the works on the stability of FG-CNTRC rectangular and skew plates is mentioned in the next. Thermal buckling response of FG-CNTRC rectangular plates with arbitrary combinations of boundary conditions is analysed by Mirzaei and Kiani [3]. In this research, both uni-axial and biaxial types of edge compression induced by uniform temperature rise loading are covered. Properties of the plate are considered to be temperature dependent. Lei et al. [4] investigated the buckling of rectangular plates with symmetric distribution of CNTs across the plate thickness using an element free kernel particle Ritz method. The equivalent properties of the media are estimated according to either the Eshelby–Mori–Tanaka approach or extended

Y. Kiani, M. Mirzaei / Aerospace Science and Technology 77 (2018) 388–398

rule of mixtures. The fact that enrichment of the matrix with more CNT enhances the critical buckling loads of plate is highlighted in this research. Kiani [5] analysed the shear buckling response of FG-CNTRC rectangular plates using an energy formulation. In this research, various combinations of free, simply supported and clamped boundary conditions are analysed. As shown, when surfaces near the top and bottom faces of the plate are enriched with more CNT, the buckling load of the plate may be increased significantly. Kiani [6] also obtained the buckling loads and buckled shapes of rectangular plates subjected to the case of nonuniform uni-axial compression. In this study the case of a plate subjected to parabolic edge compression is investigated. With the aid of Airy stress formulation, at first, the distribution of in-plane stresses due to the applied parabolic loads is obtained. The effect of Winkler elastic foundation, CNT volume fraction and CNT dispersion profile on the buckling loads of rectangular FG-CNTRC plates is studied by Zhang et al. [7] using an IMLS Ritz method. Lei et al. [8] investigated the buckling behaviour of composite laminated rectangular plates composed of FG-CNTRC layers using a meshless kp-Ritz method. Similar to linear buckling analysis which results in the buckling loads and buckled shapes of FG-CNTRC plates, nonlinear stability analysis which results in buckling load as well as postbuckling equilibrium path is also observed in the researches. Thermal postbuckling response of shear deformable thick rectangular plates made of FG-CNTRC is investigated by Shen and Zhang [9]. In this research, plates which are simply supported all around are analysed. Solution method of this research is based on the two step perturbation technique. It is shown that, critical buckling temperatures of the plate may be enhanced significantly through achieving a nonuniform distribution of CNTs across the thickness, however plates with intermediate volume fraction of CNTs does not have necessarily intermediate critical buckling temperatures. The Ritz method is used to obtain the eigenvalue problem associated with the buckling temperature and buckled shape of the plate. Based on a single term Galerkin formulation, Rafiee et al. [10] proposed a closed form solution for thermal postbuckling analysis of rectangular plates using the first order shear deformable plate model. Shen and Zhu [11] investigated the postbuckling of FG-CNTRC rectangular plates subjected to uniform compression using a two step perturbation technique. In this research, plates with all edges simply supported are considered and distribution of CNTs across the plate thickness is considered to be symmetric with respect to the mid-surface. Kiani [12] investigated the thermal postbuckling of FG-CNTRC rectangular plates using an energy method. Various combinations of simply supported and clamped boundary conditions are taken into consideration in this research. Results of this study show that, in the temperature-deflection curves of FG-CNTRC plates, secondary instability takes place which is designated as a snap-through phenomenon in the postbuckling range. Shen and Zhu [13] discussed the effects of FG-CNTRC face sheets on the postbuckling of sandwich rectangular plates subjected to uniform compression using a two step perturbation technique. In comparison to rectangular plate, less attention is devoted to skew plates which is due to the more complex geometry of the plate. In skew plates, in general, equations should be transformed into an oblique coordinate to apply the boundary conditions directly. Zhang et al. [14] applied an element free formulation to the stability equations of a skew plate to analyse the buckling of skew plates subjected to uniaxial compression, biaxial compression and combined compression/tension. The developed solution method is suitable for arbitrary combinations of free, simply supported and clamped types of boundary conditions. Lei et al. [15] analysed the effects of Pasternak elastic foundation on the buckling of FG-CNTRC skew plates using an element-free approach. Based on a two dimensional generalised differential quadrature method,

389

Fig. 1. Schematic, coordinate system and geometrical characteristics of FG-CNTRC skew plate.

Malekzadeh and Shojaee [16] investigated the buckling of arbitrary quadrilateral plates comprising of FG-CNTRC laminates. The stability equations of the plate are obtained according to the Trefftz criterion and the first order shear deformation plate theory. Using an isogeometric approach, Zhang et al. [17] obtained the optimal orientation of CNTs as reinforcements to reach the maximum buckling capacity of CNT-reinforced composite plates. This study is developed based on the third order shear deformation plate theory of Reddy. It is shown that, the efficiency of the skew plate can be significantly improved by simply placing the CNTs in the correct orientation. Kiani [18] investigated the thermal buckling of skew plates with different simply supported or clamped boundary conditions using a Ritz formulation. To the best of the present author knowledge and as the above literature survey reveals, the buckling response of skew plates subjected to shear stresses on the boundary is not reported yet. This research aims to fill this gap in the open literature. To this end, two different types of shear loads, namely, rectangular shear and skew shear loads are considered. Plate is made of a polymeric matrix reinforced by aligned single walled CNTs. Distribution of CNTs across the plate thickness may be uniform or functionally graded. Effective mechanical properties of the media are obtained by using a refined rule of mixtures approach. Prebuckling loads due to such applied stresses are obtained. The stability equations of the plate are obtained by transforming the orthogonal coordinate system to an oblique one which makes it possible to apply the boundary conditions directly. The discreted form of the governing equations is obtained using the Ritz method where shape functions are constructed by means of the Gram–Schmidt process. The eigenvalue problem is solved for different patterns of CNT, different CNT volume fractions, various skew angles, side to thickness ratios, aspect ratios and types of loading. 2. Basic formulation A skew plate with thickness h, edges a and b and skew angle α is considered. Orthogonal coordinate system is assigned to the corner of the mid-surface of the plate. The assigned coordinate system, geometrical characteristics and the schematic of the plate are shown in Fig. 1. The composite plate is made from a polymeric matrix which is reinforced with aligned single walled carbon nanotubes. As mentioned earlier, distribution of CNTs as reinforcements across the skew plate thickness may be uniform or functionally graded. When distribution of CNTs across the plate is functionally graded, it is usually referred to as functionally graded carbon nanotube reinforced composite (FG-CNTRC) skew plate. From the mathematical point of view, various dispersion profiles may be considered for the CNTs across the thickness of the plate, however, linearly graded

390

Y. Kiani, M. Mirzaei / Aerospace Science and Technology 77 (2018) 388–398

three efficiency parameters η j , j = 1, 2, 3 which are used to capture the size dependent properties of the CNTRC plate. In Eq. (2), CN CN E 11 , E 22 and G CN 12 are the Young modulus and shear modulus of SWCNTs, respectively. Furthermore, E m and G m indicate the corresponding properties of the isotropic matrix. In Eq. (2) volume fraction of CNT and volume fraction of matrix are denoted by V CN and V m , respectively which should satisfy the condition

Table 1 Volume fraction of CNTs as a function of thickness coordinate for various cases of CNTs distribution [20–22]. CNTs distribution

V CN

UD CNTRC

∗ V CN

FG-V CNTRC

∗ 1+2 V CN

FG-O CNTRC

∗ 1−2 2V CN

FG-X CNTRC

∗ 4V CN





| z|

z h

| z|  h

V CN + V m = 1

h

patterns of CNTs are more observed in the researches due to their consistency with the fabrication processes [19]. As a result, three types of FG-CNTRC plates may be achieved which are known as FG-V, FG-X and FG-O. These three types along with the uniformly distributed (UD)-CNTRC skew plate are considered in the present research. Table 1 presents the distribution of volume fraction of CNT as a function of thickness coordinate in various CNTRC plates. It is easy to check from Table 1 that, both uniform and functionally graded patterns of CNTRC plates will have the same total volume fraction of CNTs across the cross section which is denoted ∗ . Through such feature, the buckling characteristics of UDby V CN ∗ and FG-CNTRC may be compared with respect to each other. V CN may be obtained as a function of mass density of CNTs, ρ CN , mass density of matrix ρ m and mass fraction of CNTs w CN as ∗ V CN =

w CN w CN

+ρ

ρ − w CN ρ CN /ρ m

CN / m

(1)

A comparison among the distribution pattern of CNTs reveals that, in FG-X pattern, the top and bottom surfaces of the plate are enriched by the maximum volume fraction of CNTs whereas the mid-surface is free of CNTs. In FG-O, distribution pattern is inverse. The top and bottom surfaces are free of CNTs and the mid-surface is enriched with the maximum volume fraction of CNTs. In type FG-V, the bottom surface is free of CNT and the top one is enriched with the maximum volume fraction of CNT. In UD type, unlike the other three FG types, volume fraction of CNT is constant at each surface of the plate. Various methods are proposed to estimate the effective material properties of the CNTRC media. Among them, Mori–Tanaka scheme [23] and the rule of mixtures [24] approach are more observed through the researches. Rule of mixtures approach is a simple and efficient approach to obtain the properties of the fibre reinforced composite media. However, due to the severe differences between the properties of polymeric matrix and CNTs and size dependent material properties of the nanocomposites, this approach fails in accurate estimation of properties. A refined rule of mixtures which consists efficiency parameters is used extensively to accurately estimate the properties of the CNTRC beams, plates, panels and shells [25–28]. In this approach, auxiliary parameters are introduced into the rule of mixtures approach to match the data obtained by the rule of mixtures approach with those obtained by the molecular dynamics simulations [20]. Accordingly, Young’s modulus and shear modulus of the composite media may be written as [20] CN E 11 = η1 V CN E 11 + V m Em

η2 E 22

η3 G 12

= =

V CN CN E 22

V CN G CN 12

+ +

Em Gm

As claimed by Shen [20], the effective Poisson ratio depends weakly on position and therefore takes the form ∗ CN ν12 = V CN ν12 + V m ν m

(4)

Different shear deformation plate theories may be used to estimate the plate kinematics, see e.g. [29–33]. As an efficient theory to analyse the global characteristics of the plates and shells, first order shear deformation plate theory is used in this research to estimate the plate kinematics as functions of the mid-surface characteristics and cross section rotations. It is known that, through usage of first order shear deformation plate theory, buckling characteristic of thin, moderately thick and even thick plates may be achieved accurately. According to the first order shear deformation plate theory, through the length, width and thickness displacement components of the plate may be written in terms of those belong to the mid-surface and cross section rotations as [34–36]

u (x, y , z) = u 0 (x, y ) + zϕx (x, y ) v (x, y , z) = v 0 (x, y ) + zϕ y (x, y ) w (x, y , z) = w 0 (x, y )

(5)

where in the above equation, displacement components u, v and w are associated to displacements along x, y and z directions, respectively. Besides, a subscript 0 indicates the characteristics of the mid-surface. Transverse normal rotations about the x and y axes are denoted by ϕ y and ϕx are, respectively. According to the first order theory, in-plane strain components are linear functions of thickness coordinate whereas out-of-plane shear strain components are constant across the thickness. Therefore one may write

⎧ ⎫ εxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ εyy ⎪ ⎬

γxy

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ γxz ⎪ ⎭

=

γ yz

⎫ ⎧ εxx0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ε y y0 ⎪

γxy0

⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ γxz0 ⎪

+z

γ yz0

⎧ ⎫ κxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ κyy ⎪ ⎬

κxy

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ κxz ⎪ ⎭

(6)

κ yz

where, again the subscript 0 indicates the features of the midsurface. The components of the strain on the mid-surface of the plate may be written as

⎧ ⎫ εxx0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ε y y0 ⎪ ⎬

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

u 0, x v 0, y

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

γxy0 = u 0, y + v 0,x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ x + w 0, x ⎪ ⎩ γxz0 ⎪ ⎭ ⎪ ⎪ ⎪ ⎭ ⎩ γ yz0 ϕ y + w 0, y

(7)

and the components of change in curvature compatible with the first order shear deformation theory are

Vm Vm

(3)

(2)

It is seen from Eq. (2) that, the refined rule of mixtures differs essentially with the conventional rule of mixtures approach in

⎧ ⎫ κxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ κyy ⎪ ⎬

κxy

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ κxz ⎪ ⎭

κ yz

=

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

ϕx,x ϕ y, y

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

ϕx, y + ϕ y,x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎭ ⎩ 0

(8)

Y. Kiani, M. Mirzaei / Aerospace Science and Technology 77 (2018) 388–398

391

where in the above equations and in the rest of this manuscript (),x and (), y denote the derivatives with respect to the x and y directions, respectively. Under linear elastic deformations of the FG-CNTRC skew plates, constitutive law for the plate takes the form

⎧ ⎫ σxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σyy ⎪ ⎬

τ yz

⎪ ⎪ ⎪ ⎩ τxz

τxy

⎡

Q 11 ⎢ Q 12 ⎢ =⎢ 0 ⎪ ⎣ 0 ⎪ ⎪ ⎭ 0

Q 12 Q 22 0 0 0

0 0 Q 44 0 0

0 0 0 Q 55 0

⎤⎧

0 ε ⎪ ⎪ xx 0 ⎥⎪ ⎨ εyy ⎥ 0 ⎥ γ yz ⎦⎪ ⎪ 0 ⎪ ⎩ γxz Q 66 γxy

⎫ ⎪ ⎪ ⎪ ⎬ (9)

⎪ ⎪ ⎪ ⎭

In Eq. (9), Q i j (i , j = 1, 2, 4, 5, 6) are the reduced material stiffness coefficients compatible with the lumped conditions and are obtained in terms of the elasticity and shear modulus as well as the Poisson ratios as [21]

Q 11 =

E 11 1 − ν12 ν21

, Q 22 =

E 22 1 − ν12 ν21

, Q 12 =

ν21 E 11 1 − ν12 ν21

Q 44 = G 23 , Q 55 = G 13 , Q 66 = G 12

(10) Fig. 2. Two different kinds of shear loads applied to a skew plate.

Stress resultants of the first order plate theory may be obtained upon integration of stress field through the thickness. Stress resultant components in this case become [37]

⎧ N xx ⎪ ⎪ ⎪ N ⎪ yy ⎪ ⎪ ⎪ N xy ⎪ ⎪ ⎨

M xx M ⎪ yy ⎪ ⎪ ⎪ M xy ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Q xz Q yz

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

=

⎧ σxx ⎪ ⎪ ⎪ σ ⎪ yy ⎪ ⎪ ⎪ τxy +0.5h⎪ ⎪ ⎨

−0.5h

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

+0.5h

( Ai j , B i j , D i j ) =

zσxx dz z σyy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ zτxy ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ κτxz ⎪ ⎭

κτ yz

M xx

M yy ⎪ ⎪ ⎪ ⎪ M xy ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Q yz Q xz

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎡

A 11 ⎢ A 12 ⎢ ⎢ 0 ⎢ ⎢B = ⎢ 11 ⎪ ⎢ B 12 ⎪ ⎪ ⎢ ⎪ ⎪ ⎢ 0 ⎪ ⎪ ⎣ 0 ⎪ ⎪ ⎭ 0

A 12 A 22 0 B 12 B 22 0 0 0

⎫ ⎧ εxx0 ⎪ ⎪ ⎪ ⎪ ⎪ ε y y0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γ ⎪ ⎪ xy0 ⎪ ⎪ ⎬ ⎨

κxx × κyy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ κxy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γ ⎪ ⎪ yz0 ⎭ ⎩ γxz0

0 0 A 66 0 0 B 66 0 0

B 11 B 12 0 D 11 D 12 0 0 0

B 12 B 22 0 D 12 D 22 0 0 0

0 0 B 66 0 0 D 66 0 0

0 0 0 0 0 0 κ A 44 0

⎤

0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎦ κ A 55





Q i j , z Q i j , z2 Q i j dz

(13)

−0.5h

(11)

In the above equation, κ is the shear correction factor of the first order plate theory. As known, adoption of a shear correction factor results in more accurate critical buckling loads and somehow compensate the errors due to the assumption of uniform transverse strains. Evaluation of accurate shear correction factor for FG-CNTRC plates is not straightforward since this factor depends on the boundary conditions, material properties, geometry and loading type. In this research, κ is set equal to κ = CN 5/(6 − ν12 V m − ν m V CN ) which is used extensively by Liew and his co-authors in structural examination of FG-CNTRC beams, plates and shells. Substitution of Eq. (9) into Eq. (11) with the simultaneous aid of Eqs. (6)–(8), and (10) generates the stress resultants in terms of the mid-surface characteristics of the plate as

⎧ N xx ⎪ ⎪ ⎪ N ⎪ yy ⎪ ⎪ ⎪ N xy ⎪ ⎪ ⎨

In the above equation, the stiffness components A i j , B i j , and D i j indicate the stretching, bending-stretching, and bending stiffnesses, respectively, which are calculated by

As mentioned earlier, there are two kinds of shearing loads and both of them are considered in the present research. The case of rectangular shear load (R-Shear) and the case of skew shear load (S-Shear). These two different types of loading are shown in Fig. 2. As a first step in buckling analysis, the prebuckling loads should be evaluated accurately. Considering a small element in the prebuckling state, as shown in Fig. 2, the induced stresses due to the applied shear stress S in prebuckling state may be evaluated easily as 0 R-Shear : N xx = 0,

N 0y y = 0,

0 N xy =S

0 = 2S tan(α ), S-Shear : N xx

N 0y y = 0,

0 N xy =S

(14)

As expected for the especial case of rectangular plates subjected to shear, i.e. α = 0◦ , the prebuckling state of stresses are similar. The rectangular shear loading results in pure uniform shear resultants, while skew shear type of loading yields shear stresses combined with compression/tension stresses. In order to satisfy the boundary conditions on the plate, it is more appropriate to use the oblique coordinate system (x, y ) instead of the previously defined orthogonal coordinate system (x, y ). From simple geometry the oblique coordinate is given by

x = x − y tan(α ) y = y sec(α )

(15)

The components of rotations in the oblique coordinates are obtained as

ϕx (x, y , t ) = ϕx (x, y , t ) cos(α ) ϕ y (x, y , t ) = −ϕx (x, y , t ) sin(α ) + ϕ y (x, y , t ) (12)

(16)

And the partial derivatives are related by

(),x = (),x (), y = −(),x tan(α ) + (), y sec(α )

(17)

392

Y. Kiani, M. Mirzaei / Aerospace Science and Technology 77 (2018) 388–398

In the oblique coordinate system the components of the midsurface strain field from Eq. (7) change to

⎧ ⎫ εxx0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ε y y0 ⎪ ⎬

γxy0

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ γxz0 ⎪ ⎭

=

γ yz0

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

u 0, x

ϕy

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

v 0, y sec(α ) − v 0,x tan(α ) u 0, y sec(α ) − u 0,x tan(α ) + v 0,x ⎪ ⎪ ⎪ ϕx cos(α ) + w 0,x ⎪ ⎭ − ϕx sin(α ) + w 0, y sec(α ) − w 0,x tan(α ) (18)

And the components of change of curvature from Eq. (8) with the aid of Eqs. (16) and (17) change to

⎧ ⎫ κxx ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ κyy ⎪ ⎬

κxy

3. Ritz approximation Recalling Eqs. (21) and (22) and applying the Green–Gauss theorem to the integrand (20) may result in the five coupled partial differential equations and the associated boundary conditions. On the other hand, energy based techniques also may be used to solve the shear buckling problem of the skew plate. Conventional Ritz method as a powerful tool is used in this study to obtain the matrix representation of the governing equations associated to the onset of shear buckling in a skew plate subjected to different loading types. Beforehand, using the general idea of separation of variables technique, each of the essential variables of the problem, i.e. u 0 , v 0 , w 0 , ϕx and ϕ y may be written as

u 0 (x, y ) =

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ κxz ⎪ ⎭ ⎫ ⎧ ϕx,x cos(α ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ϕ y, y sec(α ) + ϕx,x sec(α ) tan(α ) − ϕx, y tan(α ) − ϕ y,x tan(α ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ϕx, y + ϕ y,x − 2ϕx,x sin(α ) 0 0

⎪ ⎪ ⎪ ⎪ ⎭

v 0 (x, y ) =

δU + δ V = 0

where in Eq. (20), δ U is the virtual strain energy of the skew plate in the oblique coordinate system which may be calculated as

a b +0.5h δU = (σxx δ εxx + σ y y δ ε y y + τxy δ γxy 0 −0.5h

(21)

and δ V is the virtual potential energy of the prebuckling loads due to the induced prebuckling loads which are given in Eqs. (14). Recalling the classical definition of boundary conditions, it is known that, clamped and simply supported edges, unlike the free edge, apply the additional twisting moments when is necessary. Therefore, for any type of FG-CNTRC plates, with symmetric or nonsymmetric distribution of CNTs across the mid-plane of the plate, bifurcation type of buckling happens when edges are combinations of clamped and simply supported. On the other hand, for plates with at least one edge free, when CNTs are distributed nonsymmetrically with respect to the mid-surface, bifurcation does not occur. Therefore in the present study, when at least one edge of the plate is free, only FG-O, FG-X and UD patterns of CNT dispersion profiles are considered. For the case when boundary conditions are combinations of simply supported and clamped, in addition to FG-X, FG-O and UD profiles, FG-V type of CNT also results in bifurcation. Considering that the above conditions are met for the occurrence of bifurcation buckling, the potential energy of the prebuckling forces at the onset of buckling is equal to

a b δV = − 0



Ny Nx  

0

 0 + N xy w 0, y δ w 0,x d ydx

(22)

W i j N iw (x) N w j ( y)

i =0 j =0

ϕx (x, y ) =

Ny Nx  

X i j N ix (x) N xj ( y )

i =0 j =0

ϕ y (x, y ) =

Ny Nx  

y

y

Y i j N i (x) N j ( y )

(23)

i =0 j =0

β

β

where in the above equations N i (x), i = 0, 1, 2, . . . , N x , N j ( y ), j = 0, 1, 2, . . . , N y and β = u , v , w , x, y are the shape functions which have to be chosen according to the essential boundary conditions. Various types of shape functions are used extensively by researchers. Polynomial shape functions, trigonometric and those developed based on Chebyshev polynomials are used extensively. Also another type of shape functions which are of interest in the open literature are those developed using the Gram–Schmidt process. This process develops a set of orthogonal shape functions of various type. In the rest an overview of this process is provided. 4. Gram–Schmidt process Given a function (polynomial) ξ0 (s), an orthogonal set of functions (polynomials) may be developed in an arbitrary interval c ≤ s ≤ d according to Gram–Schmidt process as follows

ξ1 (s) = (s − ζ1 )ξ0 (s) ξk (s) = (s − ζk )ξk−1 (s) − ηk ξk−2 (s), where in the above equation, ζk and

d ζk = c d

,

q(s)ξk2−1 (s)ds

c

sq(s)ξk−1 (s)ξk−2 (s)ds

d

q(s)ξk2−2 (s)ds

k≥2

(24)

ηk are obtained as

k≥1

c

d

ηk =

sq(s)ξk2−1 (s)ds

c 0 0 N xx w 0,x δ w 0,x + N 0y y w 0, y δ w 0, y + N xy w 0, x δ w 0, y

V i j N iv (x) N vj ( y )

i =0 j =0

w 0 (x, y ) =

(20)

+ κτxz δ γxz + κτ yz δ γ yz )dzd ydx

Ny Nx  

(19)

To obtain the stability equations of the skew plate associated to the onset of buckling, static version of virtual displacements may be used [37]. For the case of a skew plate when prebuckling loads act as external forces, virtual displacement principle takes the form [37]

0

U i j N iu (x) N uj ( y )

i =0 j =0

κ yz

=

Ny Nx  

,

k≥2

(25)

where in the above equations q(s) is the weight function. It is easy to check that, the developed set of functions ξk (s), k = 0, 1, 2, . . . according to the above process are orthogonal with respect to the weight function q(s) in the interval c ≤ s ≤ d. Equivalently

Y. Kiani, M. Mirzaei / Aerospace Science and Technology 77 (2018) 388–398

the following property is established between the functions ξk (s), k = 0, 1, 2, . . .

d q(s)ξn (s)ξm (s)ds = amn

(26)

393

Table 2 Mechanical properties of (10, 10) armchair SWCNT at room temperature [9] (tube length = 9.26 nm, tube mean radius = 0.68 nm, tube thickness = 0.067 nm). T [K]

CN E 11 [TPa]

CN E 22 [TPa]

G CN 12 [TPa]

CN ν12

300

5.6466

7.0800

1.9445

0.175

c

where amn is nonzero when m and n are identical. Otherwise, amn is equal to zero. 5. Fundamental shape function

7. Numerical results and discussion

The process mentioned in the previous section may be used to generate the shape functions in both x and y directions for each of the essential variables u 0 , v 0 , w 0 , ϕx and ϕ y . In the present research, the weight function is chosen as unity, i.e. q(x) = q( y ) = 1 and the domain problem is 0 ≤ x ≤ a and 0 ≤ y ≤ b. As seen from the recursive Eq. (24), the complete set of orthogonal shape functions depends of the choice of the first one, ξ0 (s). This function should be chosen according to the essential boundary conditions of the problem. For a skew plate subjected to shear loads, the in-plane boundary conditions are of movable type. Therefore both normal to edge and tangent to edge displacements are not restrained at the support. Three different out-of-plane boundary conditions are also used in the present research. A clamped (C) edge in which both of the rotations and lateral displacement are equal to zero at the edge. A simply supported (S) one, in which lateral displacement and the tangential slope are restrained at the support. And finally the free edge when none of the rotations and lateral displacement are restrained at the edges. As an example of choosing the fundamental shape function for each of the essential variables consider a plate which is free at x = 0, clamped at x = a, simply supported at y = b and clamped at y = 0. Therefore the fundamental shape functions for such plate may be considered as

N 0u (x) = 1,

N 0u ( y ) = 1

N 0v (x) = 1,

N 0v ( y ) = 1

N 0w (x)

N 0w ( y )

=1− ,

N 0x (x) = 1 − y

x

N 0 (x) = 1 −

a

x a x a

=

N 0x ( y ) =

,

y

N0 ( y) =

,

y b y b

 1−

 1−

y



b y



In the present research, the buckling behaviour of skew plates subjected to uniform shear stresses on the boundary is analysed. In the subsequent results, the following convention is used for the boundary conditions. For instance, in an SFSC plate, the first letter is associated to x = 0, the second letter is the boundary condition at y = 0, the third letter denotes the boundary conditions at x = a and finally the last letter is associated with the boundary at y = b. Unless otherwise stated, poly(methyl methacrylate), referred to as PMMA, is selected for the matrix with material properties E m = 2.5 GPa and ν m = 0.34. (10,10) armchair SWCNT is chosen as the reinforcements. Elasticity modulus, shear modulus and Poisson’s ratio of SWCNT are highly dependent to temperature. However in the present research, only plates which are operating at room temperature are considered. Material properties of (10,10) armchair SWCNT at room temperature are given in Table 2. Han and Elliott [38] performed a molecular dynamics simulation to obtain the mechanical properties of nanocomposites reinforced with SWCNT. However in their analysis, the effective thickness of CNT is assumed to be at least 0.34 nm. The thickness of CNT as reported should be at most 0.142 nm [39]. Therefore molecular dynamics simulation of Han and Elliott [38] is re-examined [20]. The so-called efficiency parameters, as stated earlier, are chosen to match the data obtained by the modified rule of mixture of the present study and the molecular dynamics simulation results [20]. For three different volume fractions of CNTs, these parameters are as: η1 = 0.137 and η2 = 1.022 for ∗ = 0.12. η = 0.142 and η = 1.626 for V ∗ = 0.17. η = 0.141 V CN 1 2 1 CN ∗ = 0.28. For each case, the efficiency paand η2 = 1.585 for V CN rameter η3 is equal to 0.7η2 . The shear modulus G 13 is taken equal to G 12 whereas G 23 is taken equal to 1.2G 12 [20,27].

b 7.1. Comparison studies

y

(27)

b

6. Solution procedure With the introduction of the above functions and the simultaneous aid of Eqs. (24) and (25), the complete set of orthogonal shape functions may be achieved. Afterwards the stability equations describing the onset of buckling of a skew plate under shear loads takes the form

Ke X − Kg X = 0

(28) e

eigenvalues are critical buckling load and eigenvectors are associated with buckled pattern of the plate.

g

where as usual K is the elastic stiffness matrix and K is the geometrical stiffness matrix. Besides, X is the unknown displacement vector comprising the unknowns U i j , V i j , W i j , X i j and Y i j where i = 0, 1, 2, . . . , N x and j = 0, 1, 2, . . . , N y . The elements of the geometrical stiffness matrix are dependent to the prebuckling loads. Therefore, the strain energy and the potential energy of the applied prebuckling loads are associated to, respectively, elastic and geometric stiffness matrices. The elements of the geometric stiffness matrix contain the unknown shear load parameter. Therefore the above problem should be treated as an eigenvalue problem whose

As mentioned earlier, the problem of shear buckling of skew plates made of FG-CNTRC is not reported so far. Therefore, comparison of this study is limited to the case of isotropic homogeneous plate subjected to shear. This problem is solved by Xiang et al. [40] for shear deformable plates and different types of shear loads (see Table 3). Therefore to assure the validity of our solution method with those developed by Xiang et al. [40] the present formulation is reduced to that of an isotropic homogeneous plate. This should be done easily when volume fraction of CNT is set equal to zero and the three efficiency parameters are set equal to unity. It should be mentioned that, in all of the numerical results of this study the number of shape functions is set equal to N x = N y = 14 after examination of convergence up to three digits. Comparison of our results with those of Xiang et al. [40] reveals that, the solution method and the developed formulation in this study are correct. 7.2. Parametric studies After comparing the results of this study with the available data in the open literature for the simple case of a plate made of an

394

Y. Kiani, M. Mirzaei / Aerospace Science and Technology 77 (2018) 388–398

Table 3 Comparison of shear buckling load parameter, λcr = S cr b2 /(π 2 D ) for skew plates with [40].

(a/b, h/b)

CCCC

CFFF

R-Shear

(1, 0.001) (2, 0.001) (1, 0.150) (2, 0.150)

α = 45◦ and ν = 0.3. Results of this study are compared with those of Xiang et al.

S-Shear

R-Shear

S-Shear

Present

[40]

Present

[40]

Pesent

[40]

Present

[40]

24.047 19.176 10.331 9.244

24.039 19.168 10.331 9.244

89.231 61.418 28.859 24.835

89.161 61.377 28.860 24.836

0.365 0.083 0.324 0.075

0.368 0.083 0.325 0.075

1.631 0.283 1.306 0.238

1.645 0.285 1.307 0.238

Table 4 ∗ = 0.17. Buckling load factor λcr of SSSS FG-CNTRC skew plates with a/b = 1, b/h = 20 and V CN Type

Pattern

α = −45◦

α = −30◦

α = −15◦

α = 0◦

α = 15◦

α = 30◦

α = 45◦

S-Shear

UD FG-X FG-O FG-V

41.826 47.615 33.373 37.832

44.054 51.162 34.260 39.227

47.504 55.598 36.681 42.116

53.141 62.112 41.600 47.566

63.214 73.206 51.150 57.895

83.436 96.185 66.381 75.652

124.372 141.949 106.230 118.200

R-Shear

UD FG-X FG-O FG-V

87.533 97.116 75.316 83.139

72.080 81.233 58.848 66.470

61.137 70.746 48.095 54.866

53.141 62.112 41.600 47.566

47.487 57.835 33.183 39.589

37.521 46.841 26.121 31.191

35.168 43.281 26.163 30.590

Table 5 ∗ = 0.17. Buckling load factor λcr of CSCS FG-CNTRC skew plates with a/b = 1, b/h = 20 and V CN Type

Pattern

α = −45◦

α = −30◦

α = −15◦

α = 0◦

α = 15◦

α = 30◦

α = 45◦

S-Shear

UD FG-X FG-O FG-V

44.606 49.714 36.824 40.993

50.322 56.457 41.089 45.981

56.497 63.488 46.216 51.738

64.142 72.067 52.937 59.138

75.536 84.681 63.487 70.577

97.297 108.426 84.397 92.996

149.691 165.586 133.016 145.479

R-Shear

UD FG-X FG-O FG-V

89.086 98.220 77.665 85.079

75.866 84.415 64.757 71.531

68.665 76.744 58.055 64.371

64.142 72.067 52.938 59.138

61.047 68.706 50.608 56.509

60.482 68.536 48.565 54.938

60.279 68.834 48.574 55.007

Table 6 ∗ = 0.17. Buckling load factor of λcr CCCC FG-CNTRC skew plates with a/b = 1, b/h = 20 and V CN Type

Pattern

α = −45◦

α = −30◦

α = −15◦

α = 0◦

α = 15◦

α = 30◦

α = 45◦

S-Shear

UD FG-X FG-O FG-V

46.925 51.893 39.647 43.774

51.940 58.025 43.221 48.074

57.907 64.890 48.123 53.612

65.973 73.915 55.305 61.496

78.768 87.987 67.294 74.442

103.203 114.816 89.174 98.332

155.743 171.861 139.696 152.297

R-Shear

UD FG-X FG-O FG-V

92.814 101.966 81.206 88.762

78.940 87.638 67.391 74.386

70.811 79.088 59.602 66.141

65.973 73.915 55.305 61.496

64.493 72.237 53.520 59.845

62.997 71.174 51.588 57.985

68.156 77.045 56.990 63.745

isotropic homogeneous material, parametric studies are conducted in this section for FG-CNTRC skew plates with different types of shear loads. It is of worth-noting that in numerical results of this section, number of shape functions in each direction is set equal to N x = N y = 14. Furthermore, the buckling load factor in all of the subsequent results is defined as

λcr =

S cr b2

π 2 Dm

(29)

where D m is the flexural rigidity of a plate with thickness h made from the isotropic polymeric matrix with elasticity modulus E m and the Poisson ratio νm . Tables 4 to 8 provide the critical buckling load factor of skew plates with aspect ratio a/b = 1 and side to thickness ratio a/h = 20. Tables 4 to 8 are associated to, in order, SSSS, CSCS, CCCC, CFFF and CCFF plates. For each case, two types of shear loading, namely rectangular and skew shear are considered. Furthermore, volume ∗ = 0.17. Different skew angles fraction of CNT is set equal to V CN

of the plate and different dispersion profiles are examined. It is seen that for all combinations of boundary conditions, FG-X pattern of CNT yields the maximum critical buckling shear load and FG-O pattern results in the minimum shear buckling load. The critical buckling load of S-shear and R-shear types of loading are the same for the especial case of a rectangular plate. When the shear load is applied as the S-shear, critical buckling load increases permanently as the skew angle increases. For the rectangular shear type of loading, however, variation of buckling load factor with respect to the skew angle is not monotonic. Fig. 3 gives the critical buckling load factor as a function of aspect ratio for skew plates with UD pattern of CNTs across the thickness. In this example, plates with SSSS boundary condition are considered and parameters of the plate are set equal to b/h = 20 and α = 45◦ . Both S-shear and R-shear are considered and three different volume fractions of CNT are compared with respect to each other. It is observed that, with increasing the volume fraction of CNT, buckling load of the plate enhances significantly. Such

Y. Kiani, M. Mirzaei / Aerospace Science and Technology 77 (2018) 388–398

395

Table 7 ∗ = 0.17. Buckling load factor of λcr CFFF FG-CNTRC skew plates with a/b = 1, b/h = 20 and V CN Type

Pattern

α = −45◦

α = −30◦

α = −15◦

α = 0◦

α = 15◦

α = 30◦

α = 45◦

S-Shear

UD FG-X FG-O

3.043 4.090 1.878

3.722 4.985 2.307

4.416 5.872 2.779

5.189 6.837 3.330

6.151 8.022 4.027

7.462 9.656 4.950

9.204 11.852 6.216

R-Shear

UD FG-X FG-O

4.734 6.198 3.104

4.971 6.525 3.232

5.110 6.720 3.305

5.189 6.837 3.330

5.197 6.883 3.277

5.003 6.720 3.059

4.446 6.028 2.706

Table 8 ∗ = 0.17. Buckling load factor of λcr CCFF FG-CNTRC skew plates with a/b = 1, b/h = 20 and V CN Type

Pattern

α = −45◦

α = −30◦

α = −15◦

α = 0◦

α = 15◦

α = 30◦

α = 45◦

S-Shear

UD FG-X FG-O

16.631 20.405 11.918

17.409 21.534 12.304

18.386 22.817 12.977

19.573 24.255 13.967

21.302 26.203 15.605

24.590 29.800 18.735

31.444 37.427 24.675

R-Shear

UD FG-X FG-O

33.912 40.203 26.066

27.709 33.334 20.598

23.385 28.552 16.947

19.573 24.255 13.967

16.220 20.336 11.515

13.696 17.290 9.682

11.882 15.054 8.244

Fig. 3. Effects of aspect ratio and volume fraction of CNT on the buckling load factor λcr of SSSS CNTRC skew plate for two different types of applied loads.

Fig. 4. Effects of aspect ratio and volume fraction of CNT on the buckling load factor λcr of CCCC CNTRC skew plate for two different types of applied loads.

396

Y. Kiani, M. Mirzaei / Aerospace Science and Technology 77 (2018) 388–398

Fig. 5. Effects of aspect ratio and graded pattern of CNT on the buckling load factor λcr of CCCC CNTRC skew plate for two different types of applied loads.

Fig. 6. Effect of skew angle and type of shear loading on the buckling load factor λcr of CNTRC skew plates for different types of boundary conditions.

Y. Kiani, M. Mirzaei / Aerospace Science and Technology 77 (2018) 388–398

trend is expected due to the higher elasticity modulus of CNT in comparison to polymeric matrix. Fig. 4 provides the same results of Fig. 3 for the case of skew plates with all edges clamped. Same conclusions are observed where the buckling load of the plate enhances with increasing the CNT volume fraction. Fig. 5 gives the critical buckling load factor of skew plates with respect to aspect ratio and different patterns of CNT. In this exam∗ = 0.28 and geople, volume fraction of CNT is set equal to V CN ◦ metric characteristic of the plate are α = 45 and b/h = 25. Again it is verified that, FG-X pattern of CNT results in the maximum buckling load of the plate and FG-O pattern yields the minimum buckling load of the plate. For different types of boundary conditions, Fig. 6 gives the buckling load factor of the skew plate as a function of the skew angle. For each case of boundary conditions, S-Shear and R-Shear types of loading are considered. FG-X pattern of CNT dispersion ∗ = 0.17, a/b = 1 and b /h = 25 are considered. Buckling profile, V CN load factor is given for CFFF, CCCC, SSSS and CSCS plates. It is verified that by increasing the skew angle of the plate, buckling load of the plate increases when shear load is applied in an S-shear type. For the especial case of rectangular plate which is designated with the skew angle α = 0◦ , the critical buckling load for both cases of loading are the same which is expected. 8. Conclusion In the present research, the shear buckling behaviour of skew plates made of FG-CNTRC is investigated. For this purpose, the fundamental equations of the plate are transformed into an oblique coordinate system. Properties of the composite media are obtained using a refined rule of mixtures approach which contains efficiency parameters to capture the size dependency of the properties of the CNTs. Two different kinds of shear loading, namely, rectangular and skew shear are taken into account. First order shear deformation plate theory and the static version of the Hamilton principle are used to construct the equations of the plate. The governing equations are obtained using the conventional two-dimensional Ritz method where shape functions are constructed by means of the Gram–Schmidt process. The developed solution method is used to obtain the shear buckling loads of FG-CNTRC for various graded patterns of CNTs, different volume fractions of CNT, shear loading type, aspect ratio, thickness ratio and skew angle. It is verified that, the buckling load of the plate increases significantly with enrichment of the matrix with more CNT. Furthermore, to enhance the buckling capacity of the skew plate, CNTs should be dispersed according to the FG-X pattern. In this type of distribution pattern for CNTs, the CNTs are more dispersed near the free surfaces of the plate which results in the enhancement of the flexural rigidities of the plate. Conflict of interest statement We have no conflict of interest. Acknowledgements This work has been financially supported by the research deputy of Shahrekord University. The Grant Number was 95GRN1M40504. References [1] K.M. Liew, Z.X. Lei, L.W. Zhang, Mechanical analysis of functionally graded carbon nanotube reinforced composites: a review, Compos. Struct. 120 (2015) 90–97.

397

[2] H.S. Shen, Nonlinear bending of functionally graded carbon nanotube reinforced composite plates in thermal environments, Compos. Struct. 91 (2009) 9–19. [3] M. Mirzaei, Y. Kiani, Thermal buckling of temperature dependent FG-CNT reinforced composite plates, Meccanica 51 (2016) 2185–2201. [4] Z.X. Lei, K.M. Liew, J.L. Yu, Buckling analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method, Compos. Struct. 98 (2013) 160–168. [5] Y. Kiani, Shear buckling of FG-CNT reinforced composite plates using Chebyshev–Ritz method, Composites, Part B, Eng. 105 (2016) 176–187. [6] Y. Kiani, Buckling of FG-CNT-reinforced composite plates subjected to parabolic loading, Acta Mech. 228 (2017) 1303–1319. [7] L.W. Zhang, Z.X. Lei, K.M. Liew, An element-free IMLS-Ritz framework for buckling analysis of FG-CNT reinforced composite thick plates resting on Winkler foundations, Eng. Anal. Bound. Elem. 58 (2015) 7–17. [8] Z.X. Lei, L.W. Zhang, K.M. Liew, Buckling analysis of CNT reinforced functionally graded laminated composite plates, Compos. Struct. 152 (2016) 62–73. [9] H.S. Shen, C.L. Zhang, Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates, Mater. Des. 31 (2010) 3403–3411. [10] M. Rafiee, X.Q. He, S. Mareishi, K.M. Liew, Nonlinear response of piezoelectric nanocomposite plates: large deflection, post-buckling and large amplitude vibration, Int. J. Appl. Mech. 7 (2015) 1550074 (32 pages). [11] H.S. Shen, Z.H. Zhu, Buckling and postbuckling behavior of functionally graded nanotube-reinforced composite plates in thermal environments, Comput. Mater. Continua 18 (2010) 155–182. [12] Y. Kiani, Thermal post-buckling of FG-CNT reinforced composite plates, Compos. Struct. 159 (2017) 299–306. [13] H.S. Shen, Z.H. Zhu, Postbuckling of sandwich plates with nanotube-reinforced composite face sheets resting on elastic foundations, Eur. J. Mech. A, Solids 35 (2012) 10–21. [14] L.W. Zhang, Z.X. Lei, K.M. Liew, Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach, Composites, Part B, Eng. 75 (2015) 36–46. [15] Z.X. Lei, L.W. Zhang, K.M. Liew, Buckling of FG-CNT reinforced composite thick skew plates resting on Pasternak foundations based on an element-free approach, Appl. Math. Comput. 266 (2015) 773–791. [16] P. Malekzadeh, M. Shojaee, Buckling analysis of quadrilateral laminated plates with carbon nanotubes reinforced composite layers, Thin-Walled Struct. 71 (2013) 108–118. [17] L.W. Zhang, M. Memar Ardestani, K.M. Liew, Isogeometric approach for buckling analysis of CNT-reinforced composite skew plates under optimal CNTorientation, Compos. Struct. 163 (2017) 365–384. [18] Y. Kiani, Thermal buckling of temperature-dependent FG-CNT-reinforced composite skew plates, J. Therm. Stresses 40 (2017) 1442–1460. [19] H. Kwon, C.R. Bradbury, M. Leparoux, Fabrication of functionally graded carbon nanotube-reinforced aluminum matrix composite, Adv. Eng. Mater. 13 (2013) 325–329. [20] H.S. Shen, Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, part I: axially-loaded shells, Compos. Struct. 93 (2011) 2096–2108. [21] J.E. Jam, Y. Kiani, Buckling of pressurized functionally graded carbon nanotube reinforced conical shells, Compos. Struct. 125 (2015) 586–595. [22] K. Mehar, S.K. Panda, T.Q. Bui, T.R. Mahapatra, Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM, J. Therm. Stresses 40 (2017) 899–916. [23] 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 nanotube reinforced composites, J. Eng. Mater. Technol. 126 (2004) 250–257. [24] 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, Appl. Sci. Manuf. 36 (2005) 1555–1561. [25] Y. Kiani, Free vibration of functionally graded carbon nanotube reinforced composite plates integrated with piezoelectric layers, Comput. Math. Appl. 72 (2016) 2433–2449. [26] Y. Kiani, Free vibration of FG-CNT reinforced composite skew plates, Aerosp. Sci. Technol. 58 (2016) 178–188. [27] Y. Kiani, Free vibration of functionally graded carbon-nanotube-reinforced composite plates with cutout, Beilstein J. Nanotechnol. 7 (2016) 511–523. [28] M. Mirzaei, Y. Kiani, Thermal buckling of temperature dependent FG-CNT reinforced composite conical shells, Aerosp. Sci. Technol. 47 (2015) 42–53. [29] S. Yin, J.S. Hale, T. Yu, T.Q. Bui, S.P.A. Bordas, Isogeometric locking-free plate element: a simple first order shear deformation theory for functionally graded plates, Compos. Struct. 118 (2014) 121–138. [30] T. Yu, S. Yin, T.Q. Bui, S. Xia, S. Tanaka, S. Hirose, NURBS-based isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method, Thin-Walled Struct. 101 (2016) 141–156. [31] T.T. Yu, S. Yin, T.Q. Bui, S. Hirose, A simple FSDT-based isogeometric analysis for geometrically nonlinear analysis of functionally graded plates, Finite Elem. Anal. Des. 96 (2015) 1–10.

398

Y. Kiani, M. Mirzaei / Aerospace Science and Technology 77 (2018) 388–398

[32] T.V. Vu, N.H. Nguyen, A. Khosravifard, M.R. Hematiyan, S. Tanaka, T.Q. Bui, A simple FSDT-based meshfree method for analysis of functionally graded plates, Eng. Anal. Bound. Elem. 79 (2017) 1–12. [33] S. Liu, T. Yu, T.Q. Bui, Size effects of functionally graded moderately thick microplates: a novel non-classical simple-FSDT isogeometric analysis, Eur. J. Mech. A, Solids 66 (2017) 446–458. [34] T.V. Do, D.H. Doan, N.D. Duc, T.Q. Bui, Phase-field thermal buckling analysis for cracked functionally graded composite plates considering neutral surface, Compos. Struct. 182 (2017) 542–548. [35] S. Sadamoto, M. Ozdemir, S. Tanaka, K. Taniguchi, T.T. Yu, T.Q. Bui, An effective meshfree reproducing kernel method for buckling analysis of cylindrical shells with and without cutouts, Comput. Mech. 59 (2017) 919–932.

[36] S. Yin, T. Yu, T.Q. Bui, P. Liu, S. Hirose, Buckling and vibration extended isogeometric analysis of imperfect graded Reissner–Mindlin plates with internal defects using NURBS and level sets, Comput. Struct. 177 (2016) 23–38. [37] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells, Theory and Application, CRC Press, Boca Raton, 2003. [38] Y. Han, J. Elliott, Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites, Comput. Mater. Sci. 39 (2007) 315–323. [39] C.Y. Wang, L.C. Zhang, A critical assessment of the elastic properties and effective wall thickness of single-walled carbon nanotubes, Nanotechnology 19 (2008) 075705. [40] Y. Xiang, C.M. Wang, S. Kitipornchai, Buckling of skew Mindlin plates subjected to in-plane shear loading, Int. J. Mech. Sci. 37 (1995) 1089–1101.