Original Article

Nonlinear dynamicresponse and vibrationof functionally gradednanocompositecylindrical panelreinforced by carbonnanotubes in thermalenvironment

Nguyen Dinh Khoa1, Vu Minh Anh1,2

and Nguyen Dinh Duc1,2,3

Abstract

This paper investigated the nonlinear dynamic response and vibration of functionally

graded carbon nanotubes-reinforced composite cylindrical panels with the support

elastic foundations subjected to mechanical, thermal, and damping loads based on

Reddy’s higher order shear deformation shell theory. The cylindrical panel is reinforced

by single-walled carbon nanotubes which are graded through the panel thickness

according to the different linear functions. The effective material properties of the

panel are assumed to depend on temperature and estimated through the rule of mix-

ture. The nonlinear dynamic response and natural frequency for functionally graded

carbon nanotubes-reinforced composite cylindrical panel are determined by applying

the Galerkin method and fourth-order Runge–Kutta method. In numerical results, the

effects of geometrical parameters, temperature increment, nanotube volume fraction,

1Advanced Materials and Structures Laboratory, VNU-Hanoi, University of Engineering and Technology, Cau

Giay, Hanoi, Vietnam2InfrastructureEngineering Program, VNU-Hanoi, Vietnam-Japan University, TuLiem, Hanoi, Vietnam3National Research Laboratory, Department of Civil and Environmental Engineering, Sejong University,

Gwangjin-gu, Seoul, Korea

Corresponding author:

Nguyen Dinh Duc, Vietnam National University, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam.

Email: ducnd@vnu.edu.vn

Journal of Sandwich Structures & Materials

0(0) 1–32

! The Author(s) 2019

Article reuse guidelines:

sagepub.com/journals-permissions

DOI: 10.1177/1099636219847191

journals.sagepub.com/home/jsm

elastic foundations, and types of carbon nanotubes distributions on the nonlinear vibra-

tion of functionally graded carbon nanotubes-reinforced composite cylindrical panel are

studied and discussed in detail. The present theory and approach are validated by

comparing with those in the literature.

Keywords

Nonlinear dynamic and vibration, functionally graded carbon nanotubes-reinforced

composite cylindrical panel, Reddy’s higher order shear deformation shell theory,

elastic foundations, thermal environment

Introduction

Future applications require materials that are capable of performing well in avariety of environmental conditions, in particular, mechanical, thermal, and chem-ical properties. Besides, these materials are available in reasonable prices.Functionally graded carbon nanotubes-reinforced composites (FG-CNTRCs) areexpected to be a new generation of material that has potential applications invarious fields of science and technology such as defense, aerospace, energy, chem-ical, automotive, and petrochemical industries. They can be gas absorbers, samplesor actuators, support catalysts, oil probes, chemical sensors, and nano-reactors.Therefore, FG-CNTRCs are being received considerable attention of researchersand engineering communities for mechanical behaviors. Thermal postbucklinganalysis for nanocomposite cylindrical shells reinforced by single-walled carbonnanotubes (SWCNTs) subjected to a uniform temperature rise has been investi-gated by Shen [1]. Griebel and Hamaekers [2] examined the elastic moduli ofpolymer–carbon nanotube composites by molecular dynamics simulations of aSWCNT embedded in polyethylene. Zhang and Liew [3] studied viscous fluid-particle mixture induced vibration and instability of FG-CNTRC cylindricalshells integrated with piezoelectric layers. The free vibration characteristics ofcomposite plates reinforced with single-walled carbon nanotubes has been pre-sented by Mirzaei and Kiani [4]. Wave propagation of embedded viscoelasticFG-CNT-reinforced sandwich plates integrated with sensor and actuator basedon refined zigzag theory carried out by Kolahchi et al. [5]. Zhang et al. [6] studied afirst known investigation on the geometrically nonlinear large deformation behav-ior of triangular FG-CNTRC plates under transversely distributed loads. Shen andXiang [7] investigated nonlinear vibration of nanotube-reinforced composite cylin-drical panels resting on elastic foundations in thermal environments. Shen andXiang [8] introduced investigations on the nonlinear bending analysis panels rest-ing on elastic foundations in thermal environments; Kolahchi et al. [9] investigateddynamic stability analysis of temperature-dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elastomeric medium. The linear ther-mal buckling of a composite conical shell made from a polymeric matrix and

2 Journal of Sandwich Structures & Materials 0(0)

reinforced with carbon nanotube of CNTRC cylindrical fibers has been solved byMirzaei and Kiani [10]. Mirzaei and Kiani [11] investigated thermal buckling oftemperature-dependent FG-CNTRC conical shells. Furthermore, postbuckling ofnanotube-reinforced composite cylindrical shells in thermal environments is intro-duced by Shen [12]. Shen [13] investigated thermal buckling and postbucklingbehavior of FG-CNTRC cylindrical shells. The postbuckling analysis of FG-CNTRC plates resting on Pasternak foundations has been presented by Bidgoliet al. [14]. Shen [15] used SWCNTs with the five types of the configurations ofcarbon nanotubes-reinforced composites in cylindrical panel, FG-V, FG-A, FG-O,FG-X, and UD, to investigate the nonlinear bending of FG-CNTRC plates inthermal environments. Han and Elliott [16] have used molecular dynamic simu-lations to investigate the effect of volume fraction of single-walled carbon nano-tubes on the mechanical of nanocomposites. They indicated that Young’s modulusclearly increases when the CNT volume fraction increase.

Recently, many researchers focused on the vibration and dynamic responseproblem of composite structures because of the optimal purposes and structuralsafety. Seismic response of functionally graded-carbon nanotubes-reinforced sub-merged viscoelastic cylindrical shell in hygrothermal environment was studied byHosseini and Kolahchi [17]. The dynamic stability analysis of moderately thickcylindrical panels made from functionally graded material (FGM) by employingfinite strip formulations based on a Reddy-type third-order shear deformationtheory has been solved by Ovesy and Fazilati [18]. Viola et al. [19] solved staticanalysis of functionally graded conical shells and panels using the generalizedunconstrained third order theory coupled with the stress recovery. The nonlineardynamic response and vibration of nanotubes-reinforced composite (FG-CNTRC)shear deformable plates with temperature dependence material properties andsurrounded on elastic foundations were considered by Thanh et al. [20]. Ouet al. [21] researches nonlinear dynamic response analysis of cylindrical compositestiffened laminates based on the weak form quadrature element method. Duc andQuan [22] conducted nonlinear response of imperfect eccentrically stiffened FGMcylindrical panels on elastic foundation subjected to mechanical loads.Furthermore, nonlinear mechanical, thermal, and thermo-mechanical postbuck-ling of imperfect eccentrically stiffened thin FGM cylindrical panels on elasticfoundations has been presented by Duc et al. [23]. Thu and Duc [24] studiednonlinear stability analysis of imperfect three-phase sandwich laminated polymernanocomposite panels resting on elastic foundations in thermal environments.Shen and Wang [25] investigated nonlinear bending of FGM cylindrical panelsresting on elastic foundations in thermal environments. Kiani et al. [26] focused ondynamic analysis and active control of smart doubly curved FGM panels.Nonlinear dynamic response and vibration of nanocomposite multilayer organicsolar cell were developed by Duc et al. [27].

Furthermore, composite structures are usually subjected to high temperature.The geometrical parameters of structures are deformed and the material propertiesof structures are changed due to the presence of temperature. Therefore, it is

Khoa et al. 3

necessary to take into account the effect of temperature for a better understandingof the mechanical behavior composite plates and shells. Cong et al. [28] focused onnonlinear dynamic response of eccentrically stiffened FGM plate using Reddy’shigher order shear deformation shell theory (HSDT) in thermal environment.Najafi et al. [29] evaluated nonlinear dynamic response of FGM beams withWinkler–Pasternak foundation subject to noncentral low velocity impact in ther-mal field. The nonlinear thermal dynamic analysis of eccentrically stiffenedS-FGM circular cylindrical shells surrounded on elastic foundations usingHSDT in thermal environment focused is considered by Duc [30]. A finite elementformulation for thermally induced vibrations of FGM sandwich plates and shellpanels are investigated by Panday and Pradyumna [31]. An analytical approach toinvestigate the nonlinear dynamic response and vibration of imperfect eccentricallystiffened FGM double curved thin shallow shells on elastic foundation using asimple power-law distribution in thermal environment is presented by Duc andQuan [32]. Yang and Shen [33] developed vibration characteristics and transientresponse of shear-deformable functionally graded plates in thermal environments.Duc et al. [34] researched nonlinear vibration and dynamic response of imperfecteccentrically stiffened shear deformable sandwich plate with FGM in thermalenvironment. Yang et al. [35] introduced solution thermo-mechanical postbucklingof FGM cylindrical panels with temperature-dependent properties.

Different from publications above, the novelty of this work is studying aboutthe nonlinear dynamic response and vibration of FG-CNTRC cylindrical panel inthermal environment using HSDT. The cylindrical panels are reinforced bySWCNTs which are graded through the panel thickness according to the differentlinear functions. The volume fraction and type distribution of carbon nanotubeswill be considered. The effective material properties of the panel are assumed todepend on temperature and estimated through the rule of mixture. Numericalresults for nonlinear dynamic response and natural frequencies of the FG-CNTRC cylindrical panel are determined by applying the Galerkin method andfourth-order Runge–Kutta method.

The modeling of functionally graded carbonnanotubes reinforced

Assumption that this study considers a FG-CNTRC cylindrical panel is placed incoordinate system ðx; y; zÞ and in which ðx; yÞ plane on the middle surface of thepanel and z on thickness direction ð�h=2 � z � h=2Þ as shown in Figure 1.Besides, the panel is supported by elastic foundation with the thickness h, arclength b, axial length a, and radius of curvature R of the panel.

Elastic modules of CNTRCs and FG-CNTRC

In this study, the FG-CNTRC material consists of poly(methyl methacrylate),PMMA, which is strengthened by (7,7) SWCNTs.

4 Journal of Sandwich Structures & Materials 0(0)

According to mixture rule, the effective shear and Young’s modulus of the FG-CNTRC material can be given as [1,6–8,10,12]

E11 ¼ g1VCNTECNT11 þ VmEm;

g2E22

¼ VCNT

ECNT22

þ Vm

Em;

g3G12

¼ VCNT

GCNT12

þ Vm

Gm

(1)

where Vm and VCNT are the volume fractions of the matrix and the carbon nano-tubes, respectively. GCNT

12 ;ECNT11 ; ECNT

22 are shear and Young’s modulus of the FG-CNTRCs layer; Em and Gm are corresponding properties of the matrix and gi ði ¼1; 3Þ are the carbon nanotubes efficiency parameters.

In this study, we consider five types of FG-CNTRC: FG-O, FG-V, FG-X, UD,and FG-A as Figure 2. The distribution of the volume fractions of the matrix andthe carbon nanotubes change according to the panel thickness are the linear func-tions. The volume fractions of the FG-CNTRC take the following forms [4,6,10]

VCNTðzÞ ¼

2V�VCT 2

zj jh

� �FG� Xð Þ

V�VCT 1� 2

zj jh

� �FG� Vð Þ

V�VCT 1þ 2

z

h

� �FG� Að Þ

2V�VCT 1� 2

zj jh

� �FG�Oð Þ

V�VCT UDð Þ

; VmðzÞ¼ 1� VCNTðzÞ

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

(2)

Figure 1. Geometry and coordinate system of functionally graded carbon nanotubes-reinforcedcomposite cylindrical panels on elastic foundations.

Khoa et al. 5

where

V�CNT ¼ wCNT

wCNT þ qCNT=qmð Þ � qCNT=qmð ÞwCNT(3)

where qm, qCNT and wCNT are the densities of matrix and carbon nanotubes, themass fraction of carbon nanotubes, respectively. The effective Poisson’s ratio isfunction depends on position and temperature change. Moreover, the thermalexpansion coefficients in the transverse and longitudinal directions of the FG-CNTRC and is written as [1,6–8, 10,12]

�12 ¼ V�CNTv

CNT12 þ Vm�m (4)

where �CNT12 and �m are Poisson’s ratio of the carbon nanotubes and the matrix,respectively.

a11 ¼ VCNTECNT11 aCNT11 þ VmEmam

VCNTECNT11 þ VmEm

;

a22 ¼ ð1þ �CNT12 ÞVCNTaCNT22 þ ð1þ �mÞVmam � �12a11

(5)

in which am and aCNT11 ; aCNT22 are the thermal expansion coefficients of the matrixand the carbon nanotubes, respectively.

The materials properties of matrix are given as [1,6–8, 10,12]

�m ¼ 0:34; am ¼ 45 1þ 0:0005DTð Þ � 10�6=K;

Em ¼ ð3:52� 0:0034TÞGPa(6)

with T ¼ T0 þ DT; T0 ¼ 300K (room temperature) and DT is the tempera-ture increment.

For the carbon nanotubes, the material properties of (7,7) SWCNTs are highlydependent to temperature. According to Shen and Xiang [8], based on thePoisson’s ratio of SWCNTs with conditions are �CNT12 ¼ 0:175 and thicknessobtained for h ¼ 0:067m. These properties of SWCNTs at five certain temperaturelevels, i.e., T ¼ 300; 400; 500; 700; 1000 are determined. Specifically, thermalexpansion coefficient, shear modulus, and Young’s modulus of SWCNTs arelisted in Table 1.

gi ði ¼ 1; 3Þ, which are carbon nanotubes efficiency parameters, are introducedin equation (1). These values of gi ði ¼ 1; 3Þ are estimated by matching the effectiveproperties of FG-CNTRC material: the shear modulus G12 and Young’s modulusE11 and E22. The carbon nanotubes efficiency parameters utilized in this paper areas given in Table 2 [1,6–8,10,12].

6 Journal of Sandwich Structures & Materials 0(0)

Basic equations

The geometrical compatibility equation for a FG-CNTRC cylindrical panel iswritten as [37–39]

@2e0x@y2

þ @2e0y@x2

� @2c0xy@x@y

¼ @2w

@x@y

!2

� @2w

@x2@2w

@y2� 1

R

@2w

@x2(7)

Table 1. Temperature-dependent material properties for (7, 7) SWCNTs (�CNT12 ¼ 0:175and h ¼ 0:067m).

T ðKÞ ECNT11 ðTPaÞ ECNT22 ðTPaÞ GCNT12 ðTPaÞ aCNT11 ð�10�6=KÞ aCNT22 ð�10�6=KÞ

300 5.6466 7.0800 1.9445 3.4584 5.1682

400 5.5679 6.9814 1.9703 4.1496 5.0905

500 5.5308 6.9348 1.9643 4.5361 5.0189

700 5.4744 6.8641 1.9644 4.6677 4.8943

1000 5.2814 6.6220 1.9451 4.2800 4.7532

Figure 2. The configurations of five types of functionally graded carbon nanotubes -reinforcedcomposites in cylindrical panel.

Khoa et al. 7

where e0x, e0y and c0xy are normal strains and the shear strain at the middle surface of

the panel, respectively.The nonlinear equilibrium equation of a FG-CNTRC cylindrical panel on the

elastic foundations is written as [37–39]

@Nx

@xþ @Nxy

@y¼ J0

@2u

@t2þ I1

@2ux

@t2� c1J3

@3w

@x@t2(8a)

@Nxy

@xþ @Ny

@y¼ J0

@2v

@t2þ I1

@2uy

@t2� c1J3

@3w

@y@t2(8b)

@Qx

@xþ @Qy

@y� c2

@Rx

@xþ @Ry

@y

� �þNx

@2w

@x2þ 2Nxy

@2w

@x@y

þNy@2w

@y2þ c1

@2Px

@x2þ 2

@2Pxy

@x@yþ @2Py

@y2

!þNy

R� k1wþ k2

@2w

@x2þ @2w

@y2

!þ q

¼ J0@2w

@t2þ 2eJ0

@w

@t� c21J6

@4w

@x2@t2þ @4w

@y2@t2

!

þ c1 J3@3u

@x@t2þ @3v

@y@t2

!þ I4

@3u1

@x@t2þ @3u2

@y@t2

!" #

(8c)

@Mx

@xþ @Mxy

@y� c1

@Px

@xþ @Pxy

@y

� ��Qx þ c2Rx ¼ I1

@2u

@t2þ V2

@2ux

@t2� c1I4

@3w

@x@t2

(8d)

@Mxy

@xþ @My

@y� c1

@Pxy

@xþ @Py

@y

� ��Qy þ c2Ry ¼ I1

@2v

@t2þ V2

@2uy

@t2� c1I4

@3w

@y@t2

(8e)

Table 2. The CNT efficiency parameters for (7, 7).

Rule of mixture

V�CNT g1 g2 g3

0.12 0.137 1.022 0.715

0.17 0.142 1.626 1.138

0.28 0.141 1.585 1.109

8 Journal of Sandwich Structures & Materials 0(0)

where

Ji ¼Z h=2

�h=2

q zð Þzidz; ði ¼ 0; 1; 2; 3; 4; 6Þ;

Ii ¼ Ji � c1Jiþ2; V2 ¼ J2 � 2c1J4 þ c21J6; c2 ¼ 3c1;

qðzÞ ¼ VCNTqCNT þ Vmqm

(9)

and u; v;w are displacement components corresponding to the coordinates x; y; zð Þ;k1 is Winkler foundation modulus; k2 is the shear layer foundation stiffness ofPasternak model; e is the viscous damping coefficient; coefficients Iiði ¼ 0� 4; 6Þare given in Appendix 1.

In the middle surface, the normal and shear strains of FG-CNTRC cylindricalpanels are defined as

e0x ¼ @u0

@x� w0

Rþ 1

2

@w0

@x

� �2

; e0y ¼@v0

@yþ 1

2

@w0

@y

!2

c0xy ¼@u0

@yþ @v0

@xþ @w0

@x

@w0

@y

(10)

In which u0; v0;w0 are displacement component of a genetic point on the refer-ence surface.

The relationships between strain and displacements at a distance “z” usingHSDT and von Karman nonlinear terms are [36,37]

ex ¼ e0x � z@2w0

@x2þ f zð Þ @/

01

@x

ey ¼ e0y � z@2w0

@y2þ f zð Þ @/

02

@y

cxy ¼ c0xx � 2z@2w0

@x@yþ f zð Þ @/

01

@yþ f zð Þ @/

02

@x

cxz ¼@u

@zþ @w

@x¼ f 0 zð Þ/0

1

cyz ¼@v

@zþ @w

@y¼ f 0 zð Þ/0

2

(11)

with

/01 ¼ ux þ

@w0

@x; /0

2 ¼ uy þ@w0

@y

Khoa et al. 9

f zð Þ ¼ z 1� 4=3ð Þ z=hð Þ2h i

f0 zð Þ ¼ d

dzf zð Þ

(12)

where ux and uy are rotation angles around the x and y axes, respectively.Hooke’s law for a FG-CNTRC cylindrical panel is defined as [7]

rxxryyrxyrxzryz

26666664

37777775¼

Q11 Q12 0 0 0

Q12 Q22 0 0 0

0 0 Q66 0 0

0 0 0 Q44 0

0 0 0 0 Q55

26666664

37777775

exxeyyexyexzeyz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

� DT

a11a220

0

0

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

(13)

where

Q11 ¼ E11

1� �12�21; Q22 ¼ E22

1� �12�21; Q12 ¼ �21E11

1� �12�21; Q44 ¼ G23;

Q55 ¼ G13; Q44 ¼ G12

(14)

According to Shen [1], we assume that G13 ¼ G12 and G23 ¼ 1:2G12

The force and moment resultants of FG-CNTRC cylindrical panel are deter-mined by

Ni;Mi;Pið Þ ¼Z h=2

�h=2

rið1; z; z3Þdz; i ¼ x; y; xy;

Qi;Kið Þ ¼Z h=2

�h=2

rizð1; z2Þdz; i ¼ x; y

(15)

Inserting equations (11) and (13) into equations (15), the constitutive relationsare obtained as

Nx ¼ C11e0x þ C12e0y þ C13k1x þ C14k

1y þ C15k

3x þ C16k

3y � U1

x;

Ny ¼ C12e0x þ C22e0y þ C14k1x þ C24k

1y þ C16k

3x þ C26k

3y � U1

y;

Nxy ¼ C31c0xy þ C32k1xy þ C33k

3xy;

Mx ¼ C13e0x þ C14e0y þ C43k1x þ C44k

1y þ C45k

3x þ C46k

3y � U2

x;

My ¼ C14e0x þ C24e0y þ C44k1x þ C54k

1y þ C46k

3x þ C56k

3y � U2

y;

10 Journal of Sandwich Structures & Materials 0(0)

Mxy ¼ C32c0xy þ C62k1xy þ C63k

3xy;

Px ¼ C15e0x þ C16e0y þ C45k1x þ C46k

1y þ C75k

3x þ C76k

3y � U3

x;

Py ¼ C16e0x þ C26e0y þ C46k1x þ C56k

1y þ C76k

3x þ C86k

3y � U3

y;

Pxy ¼ C33c0xy þ C63k1xy þ C93k

3xy;

Qx ¼ G11c0xz þ G12k2xz;

Qy ¼ G21c0yz þ G22k2yz;

Kx ¼ G12c0xz þ G32k2xz;

Ky ¼ G22c0yz þ G42k2yz

(16)

and the linear parameters C1j j ¼ 1; 6� �

;C2j j ¼ 2; 4; 6ð Þ;C3j j ¼ 1; 3� �

;C4j j ¼ 3; 6� �

;C5j j ¼ 4; 6ð Þ; C6j j ¼ 2; 3ð Þ;C7j j ¼ 5; 6ð Þ;Ci6 i ¼ 8; 9ð Þ; Gij i; j ¼ 4; 2� �

;Uxi i ¼ 1; 3� �

;Uyi i ¼ 1; 3� �

are given in Appendix 1.Calculating from equations (12) and equations (16), one can write

e0x ¼ H11Nx �H12Ny þH13k1x þH14k

1y þH15k

3x þH16k

3y þH11U

1x �H12U

1y;

e0y ¼ �H12Nx þH22Ny þH23k1x þH24k

1y þH25k

3x þH26k

3y �H12U

1x þH22U

1y;

c0xy ¼1

C31Nxy � C32

C31k1xy �

C33

C31k3xy

(17)

with

H11 ¼ C22

C11C22 � C212

; H12 ¼ C12

C11C22 � C212

; H13 ¼ C14C12 � C13C22

C11C22 � C212

;

H14 ¼ C24C12 � C14C22

C11C22 � C212

; H15 ¼ C16C12 � C15C22

C11C22 � C212

; H16 ¼ C26C12 � C16C22

C11C22 � C212

;

H22 ¼ C11

C11C22 � C212

; H23 ¼ C13C12 � C14C11

C11C22 � C212

; H24 ¼ C14C12 � C24C11

C11C22 � C212

;

H25 ¼ C15C12 � C16C11

C11C22 � C212

; H26 ¼ C16C12 � C26C11

C11C22 � C212

(18)

The Airy’s stress function f x; y; tð Þ is defined as

Nx ¼ @2f

@y2; Ny ¼ @2f

@x2; Nxy ¼ � @2f

@x@y(19)

Khoa et al. 11

Replacing equation (19) into equations (8a) and (8b) yields

@2u

@t2¼ � I1

J0

@2u1

@t2þ c1J3

J0

@3w

@x@t2(20a)

@2v

@t2¼ � I1

J0

@2u2

@t2þ c1J3

J0

@3w

@y@t2(20b)

Replacing equations (20a) and (20b) into equations (8c) to (8e) leads to

@Qx

@xþ @Qy

@y� c2

@Rx

@xþ @Ry

@y

� �þNx

@2w

@x2þ 2Nxy

@2w

@x@y

þNy@2w

@y2þ c1

@2Px

@x2þ 2

@2Pxy

@x@yþ @2Py

@y2

!� k1wþ k2

@2w

@x2þ @2w

@y2

!þNy

Rþ q

¼ J0@2w

@t2þ 2eJ0

@w

@tþ c21J

23

J0� c21J6

� �@4w

@x2@t2þ @4w

@y2@t2

!

þ I4c1 � I1J3c1J0

� �@3u1

@x@t2þ I4c1 � I1J3c1

J0

� �@3u2

@y@t2

(21a)

@Mx

@xþ @Mxy

@y� c1

@Px

@xþ @Pxy

@y

� ��Qx þ c2Rx

¼ V2 � I21J0

� �@2u1

@t2þ c1J3I1

J0� c1I4

� �@3w

@x@t2

(21b)

@Mxy

@xþ @My

@y� c1

@Pxy

@xþ @Py

@y

� ��Qy þ c2Ry

¼ V2 � I21J0

� �@2u2

@t2þ c1J3I1

J0� c1I4

� �@3w

@y@t2

(21c)

By substituting equations (10) and (12) into equation (16) and then into equa-tions (21) gives

R11 wð Þ þ R12 u1ð Þ þ R13 u2ð Þ þ R14 fð Þ þ P x; fð Þ þ q ¼ J0@2w

@t2þ 2eJ0

@w

@t

þ c21J23

J0� c21J6

� �@4w

@x2@t2þ @4w

@y2@t2

!þ I4c1 � I1J3c1

J0

� �@3u1

@x@t2þ I4c1 � I1J3c1

J0

� �@3u2

@y@t2;

R21 wð Þ þ R22 /xð Þ þ R23 /y

� �þ R23 fð Þ ¼ V2 � I21J0

� �@2u1

@t2þ c1J3I1

J0� c1I4

� �@3w

@x@t2;

12 Journal of Sandwich Structures & Materials 0(0)

R31 wð Þ þ R32 /xð Þ þ R33 /y

� �þ R34 fð Þ ¼ V2 � I21J0

� �@2u2

@t2þ c1J3I1

J0� c1I4

� �@3w

@y@t2

(22)

in which

R11 ¼ G11 þ c2G12 � 4G12

h2� 4c2G32

h2

� �@2w

@x2þ G21 þ c2G22 � 4G22

h2� 4c2G42

h2

� �

� @2w

@y2� c21A45

@4w

@x4� c21A46 þ 4c21A63 þ c21A55

� � @4w

@x2@y2� c21A56

@4w

@y4;

R12 ¼ G11 þ c2G12 � 4G12

h2� 4c2G32

h2

� �@u1

@xþ c1A43 � c21A45

� � @3u1

@x3

þ 2c1A62 � 2c21A63 þ A53c1 � c21A55

� � @3u1

@x@y2;

R13 ¼ G21 þ c2G22 � 4G22

h2� 4c2G42

h2

� �@u2

@yþ A54c1 � c21A56

� � @3u2

@y3

þ 2c1A62 � 2c21A63 þ c1A44 � c21A46

� � @3u2

@y@x2;

R14 ¼ c1A42@4f

@x4þ c1A41 � 2c1A61 þ A52c1ð Þ @4f

@x2@y2þ A51c1

@4f

@y4� 1

R

@2f

@y2;

P x; fð Þ ¼ @2f

@y2@2w

@x2� 2

@2f

@x@y

@2w

@x@yþ @2f

@y2@2w

@y2;

R21 ¼ c21A45 � c1A15

� � @3w

@x3þ 2c21A63 þ c21A46 � c1A16 � 2c1A33

� � @3w

@x@y2

� G11 � c22G32

� � @w@x

;

R22 ¼ A13 � c1A15 � c1A43 þ c21A45

� � @2/x

@x2

þ A32 � c1A33 � c1A62 þ c21A63

� � @2u1

@y2� G11 � c22G32

� �u1;

R23 ¼ A14 þ A32 � c1A16 � c1A44 � c1A33 � c1A62 þ c21A46 þ c21A63

� � @2u2

@x@y;

R24 ¼ A11 � A31 � c1A41 þ c1A61ð Þ @3f

@x@y2þ A12 � c1A42ð Þ @

3f

@x3;

R31 ¼ c21A55 þ 2c21A63 � 2c1A33 � c1A25

� � @3w

@x2@y

þ c21A56 � c1A26

� � @3w

@y3� B21 � c22B42

� � @w@y

;

Khoa et al. 13

R32 ¼ A32 � c1A33 þ A23 � c1A25 � A62c1 þ c21A63 � A53c1 þ c21A55

� � @2u1

@x@y;

R33 ¼ A32 � c1A33 � A62c1 þ c21A63

� � @2/y

@x2

þ A24 � c1A26 � A54c1 þ c21A56

� � @2u2

@y2� G21 � c22G42

� �u2;

R34 ¼ A22 � A31 þ A61c1 � A52c1ð Þ @3f

@x2@yþ A21 � A51c1ð Þ @

3f

@y3

(23)

Introduction of equations (17) into equation (7), the compatibility equation of

the FG-CNTRC cylindrical panel is rewritten as

H11@4f

@y4�H22

@4f

@x4� H12 þH12 � 1

C31

� �@4f

@x2@y2

þ H13 � c1H15ð Þ @3u1

@x@y2þ H23 � c1H25ð Þ @

3u1

@x3þ C32

C31� c1

C33

C31

� �@3u1

@y2@x

þ H14 � c1H16ð Þ @3u2

@y3þ H24 � c1H26ð Þ @3u2

@x2@yþ C32

C31� c1

C33

C31

� �@3u2

@x2@y

� c1H16@4w

@y4� c1H25

@4w

@x4� c1H26 þ 2c1

C33

C31þ c1H15

� �@4w

@x2@y2

¼ @2w@x@y

� �2 � @2w

@x2@2w

@y2þ @2w

@x2R

(24)

Solution procedure

In this paper, we assume that four edges of the FG-CNTRC cylindrical panel are

immovable and simply supported. The boundary conditions are

w ¼ u ¼ u2 ¼ Mx ¼ Px ¼ 0; Nx ¼ Nx0 at x ¼ 0; a;

w ¼ v¼u1 ¼ My ¼ Py ¼ 0; Ny ¼ Ny0 at y ¼ 0; b(25)

in which Nx0 and Ny0 are fictitious compressive edge loads at immovable edges.The mentioned conditions (25) can be satisfied identically if the approximate

solutions are chosen in the following form

w x; y; tð Þu1 x; y; tð Þu2 x; y; tð Þ

264

375¼

W tð ÞsinkmxsindnyUx tð ÞcoskmxsindnyUy tð Þsinkmxcosdny

264

375 (26)

14 Journal of Sandwich Structures & Materials 0(0)

where km ¼ mp=a, dn ¼ np=b; m; n are odd natural numbers representing thenumber of half waves in the x and y directions, respectively; and WðtÞ; Ux; Uy

are the amplitudes of the approximate solutions with condition is dependenton time.

Introduction of equations (26) into the compatibility equation (24), we obtainthe stress function f as

f x; y; tð Þ ¼ A1 tð Þcos2axþ A2 tð Þcos2byþ A3 tð Þsinaxsinbyþ 1

2Nx0y

2 þ 1

2Ny0x

2

(27)

in which

A1 ¼ W2b2

32F22a2; A2 ¼ W2a2

32F11b2; A3 ¼ F1Ux þ F2Uy � F3W (28)

and

F1 ¼H13 � c1H15ð Þab2 þ H23 � c1H25 þ C32

C31� c1

C33

C31

� �a3

�

�H11b4 þH22a4 þ 2H12 � 1

C31

� �a2b2

�

F2 ¼H14 � c1H16ð Þb3 þ H24 � c1H26 þ C32

C31� c1

C33

C31

� �ba2

�

�H11b4 þH22a4 þ 2H12 � 1

C31

� �a2b2

�

F3 ¼c1H16b

4 þ c1H25a4 þ c1H26 þ 2c1C33

C31þ c1H15

� �a2b2

�

�H11b4 þH22a4 þ 2H12 � 1

C31

� �a2b2

�

(29)

Replacing equations (27) to (29) into equations of motion (24) and then apply-ing Galerkin method yields

r11 � Nx0a2 þNy0b2

� �h iWþ r12Ux þ r13Uy þ r14W

2 þ r15W3

� 1

RNx0

16

ababþ F1b

2a2WUx32

3mnp2þ F2b

2a2WUy32

3mnp2þ 16

mnp2q

¼ 2eJ0@W

@tþ n2

@2W

@t2þ q2a

@2Ux

@t2þ q2b

@2Uy

@t2;

Khoa et al. 15

r21Wþ r22Ux þ r23Uy þ r24W2 ¼ q1

@2Ux

@t2þ q2b

@2W

@t2;

r31Wþ r32Ux þ r33Uy þ r34W2 ¼ q1

@2Uy

@t2þ q2b

@2W

@t2

(30)

where

q1 ¼ V2 � I21J0

; q2 ¼c1J3I1J0

� c1I4 (31)

and specific expressions of coefficients rij i ¼ 1� 3; j ¼ 1� 3ð Þ; n1; n2 are given inAppendix 1.

Consider the FG-CNTRC panel which is subjected temperature environmentsand simultaneously exposed to uniform external pressure q. The condition onimmovability at all edges, i.e., u ¼ 0 at x ¼ 0; a and v ¼ 0 at y ¼ 0; b, is satisfiedin an average sense [23,32]

Z b

0

Z a

0

@u

@xdxdy ¼ 0;

Z a

0

Z b

0

@v

@ydydx ¼ 0 (32)

From equations (10) and (17), we obtain the following expressions

@u

@x¼ H11

@2f

@y2�H12

@2f

@x2þ H13 � c1H15ð Þ @u1

@xþ H14 � c1H16ð Þ @u2

@y

� 1

2

@w

@x

� �2

� c1H15@2w

@x2� c1H16

@2w

@y2þH11U

1x �H12U

1y;

@v

@y¼ �H12

@2f

@y2þH22

@2f

@x2þ H23 � c1H25ð Þ @u1

@xþ H24 � c1H26ð Þ @u2

@y

þ w

R� 1

2

@w

@y

� �2

� c1H25@2w

@x2� c1H26

@2w

@y2�H12U

1x �H22U

1y

(33)

Putting equations (27)–(29) into equation (33) and then the result into equation(32), we have

Nx0 ¼ T11Wþ T12Ux þ T13Uy þ T14W2 þ T15;

Ny0 ¼ T21Wþ T22Ux þ T23Uy þ T24W2 þ T25

(34)

in which the details of Tij i; j ¼ 1; 5� �

are found in Appendix 1.Replacing equations (34) into equation (32) yields

S11Wþ S12Ux þ S13Uy þ S14WUx

16 Journal of Sandwich Structures & Materials 0(0)

þ S15WUy þ S16W2 þ S17W

3 � T15

R

16

ababþ 16

mnp2q

¼ 2eJ0@W

@tþ n2

@2W

@t2þ q2a

@2Ux

@t2þ q2b

@2Uy

@t2;

r21Wþ r22Ux þ r23Uy þ r24W2 ¼ q1

@2Ux

@t2þ q2a

@2W

@t2;

r31Wþ r32Ux þ r33Uy þ r34W2 ¼ q1

@2Uy

@t2þ q2b

@2W

@t2

(35)

with

S11 ¼ r11 � T15a2 � b2T25 � T21

R

16

abab

� �;

S12 ¼ r12 � T12

R

16

abab; S13 ¼ r13 � T13

R

16

abab;

S16 ¼ r14 � T11a2 þ T21b2 þ T14

R

16

abab

� �; S17 ¼ r15 þ b2T24 � T14a2

� �;

S14 ¼ � T12a2 þ T22b2 þ F1b

2a232

3mnp2

� �; S15 ¼ � T13a2 þ T23b

2 þ F2b2a2

32

3mnp2

� �(36)

If q ¼ 0 and talking linear parts of the set of equation (35), by solving the

determinant equation (37), the natural frequencies of the perfect FG-CNTRC

cylindrical panel can be determined directly:

S11 þ n2x2 S12 þ q2mpa

x2 S13 þ q2npbx2

r21 þ q2mpa

x2 r22 þ q1x2 r23

r31 þ q2npbx2 r32 r33 þ q1x

2

¼ 0 (37)

By solving equation (37), three values of frequencies of the FG-CNTRC cylin-

drical panel in the axial, circumferential and radial direction are obtained and the

smallest one is considered.

Numerical results and discussion

Validation of the present study

To verify the reliability of the present approach and theory, the natural frequencies

f ¼ x=2p ðHzÞ of Al2O3/Ti-6Al-4V FGM cylindrical panel in thermal

Khoa et al. 17

environments in this paper are compared with results presented by Duc et al. [23]

based on the HSDT, which is shown in Table 3. The mechanical properties are

chosen as a=b ¼ 1; b=h ¼ 20. It is clear that a good agreement is obtained in

this comparison.

Natural frequency

In this section, we consider influences of many factors which can affect to natural

oscillation frequency of the FG-CNTRC cylindrical panels. Specifically, Table 4

shows the effect of types of FG-CNTRC, carbon nanotubes volume fraction

V�CNT and temperature increment DT on the natural frequency of FG-CNTRC

cylindrical panels with b=a ¼ 1; b=h ¼ 20; R=h ¼ 300. It is easy to see that the

value of the natural oscillation frequency of the FG-CNTRC cylindrical panel

Table 3. Comparison of natural frequencies for FGM cylindrical panels a ¼ b ¼ 0:1 mð Þ.

ðk1 ðGPa=mÞ;k2 ðGPa�mÞÞ b=h

N ¼ 0 N ¼ 1 N ¼ 5

Present

Duc

et al. [23] Present

Duc

et al. [23] Present

Duc

et al. [23]

(0,0) 20 2.815e3 2.971e3 1.654e3 1.722e3 1.422e3 1.465e3

30 1.926e3 2.001e3 1.056e3 1.161e3 913.4 986.9

40 1.421e3 1.528e3 816.6 891.7 712.4 753.9

(0.1,0) 20 2.955e3 3.058e3 1.812e3 1.892e3 1.623e3 1.685e3

30 2.115e3 2.190e3 1.456e3 1.507e3 1.356e3 1.419e3

40 1.795e3 1.841e3 1.384e3 1.423e3 1.355e3 1.398e3

(0.1,0.02) 20 3.315e3 3.381e3 2.589e3 2.624e3 2.703e3 2.726e3

30 2.756e3 2.812e3 2.405e3 2.452e3 2.416e3 2.474e3

40 2.713e3 2.747e3 2.410e3 2.432e3 2.314e3 2.362e3

Table 4. Effect of types of functionally graded carbon nanotubes-reinforced composite, carbonnanotubes volume fraction V�

CNT , and temperature increment DT on natural frequencies s�1ð Þ ofthe functionally graded carbon nanotubes-reinforced cylindrical panel.

DT ðKÞV�CNT ¼ 0:12 V�

CNT ¼ 0:17 V�CNT ¼ 0:28

FG-A UD FG-A UD FG-A UD

0 8976 9171 11541 11730 12487 12640

100 8403 8610 9369 11000 11706 11895

200 7791 8016 8600 10226 9158 11108

400 5250 6675 6807 8472 8989 9344

600 3335 4953 5774 6203 6544 7105

18 Journal of Sandwich Structures & Materials 0(0)

increases when the value of temperature increment is fixed and the value of

carbon nanotubes volume fraction increases. In contrast, the value of the natural

oscillation frequency of the FG-CNTRC cylindrical panel decreases when the

value of temperature increment increases and the value of carbon nanotubes

Table 5. Effects of carbon nanotubes volume fraction, ratio a=h, elastic foundations k1; k2, andtypes of functionally graded carbon nanotubes-reinforced composite on natural frequencies s�1ð Þof the functionally graded carbon nanotubes-reinforced composite cylindrical panels.

ðk1 ðGPa=mÞ;k2 ðGPa�mÞÞ a=h

V�CNT ¼ 0:12 V�

CNT ¼ 0:17

FG-O FG-A FG-V FG-O FG-A FG-V

(0,0) 20 1519 1143 704 2822 1744 1277

(0,0) 25 904 817 280 1752 1228 652

(0,0) 30 518 651 252 1131 938 150

(0.1,0.02) 20 2005 640 1489 3110 1151 1833

(0.1,0.02) 25 1457 586 1197 2082 965 1338

(0.1,0.02) 30 1190 456 1062 1545 652 1107

0 0.01 0.02 0.03 0.04 0.05-4

-2

0

2

4

6x 10

-5

t(s)

W(m

)

(m,n)=(1,1), R/h=400, a/b =1,k2 = 0, b/h=50, q=8000sin(500t)

k1 = 0

k1 = 0.25 GPa/m

k1 = 0.55 GPa/m

Figure 3. Effects of the Winkler foundation on the nonlinear dynamic response of thefunctionally graded carbon nanotubes-reinforced composite cylindrical panel.

Khoa et al. 19

0 0.01 0.02 0.03 0.04 0.05

-4

-2

0

2

4

6x 10

-5

t(s)

W(m

)

(m,n)=(1,1), R/h=400, k1 =0,

�T=0 , b/h=50, a/b =1, q=8000sin(500t)

k2 = 0

k2 = 0.02 GPa.m

k2 = 0.04 GPa.m

Figure 4. Effects of the Pasternak foundation on the nonlinear dynamic response of thefunctionally graded carbon nanotubes-reinforced composite cylindrical panel.

0 0.01 0.02 0.03 0.04 0.05-1

-0.5

0

0.5

1

1.5

2

2.5x 10

-4

t (s)

W(m

)

(m,n)=(1,1), R/h=400, b/h=20, a/b=1, q=8000sin(500t)

��T= 0

��T= 100 K

��T= 150 K

Figure 5. Effects of temperature increment DT on the nonlinear dynamic response of thefunctionally graded carbon nanotubes-reinforced composite cylindrical panel.

20 Journal of Sandwich Structures & Materials 0(0)

volume fraction is fixed. Additionally, this table also highlights that naturaloscillation frequency of FG-A cylindrical panel is lower than one of FG-UDcylindrical panel.

Besides, the elastic foundations and geometrical parameter a=h have significantinfluence on the natural frequencies of the functionally graded carbon nanotubes-reinforced, which is indicated in Table 5. Specifically, the value of the naturalfrequencies of the functionally graded carbon nanotubes-reinforced cylindricalpanel also increases when two parameters of elastic foundations k1; k2 or thevalue of carbon nanotubes volume fraction increases. The change of the value ofnatural frequency also depends on types of FG-CNTRC. Comparing three types ofFG-CNTRC: FG-A, FG-V, and FG-O, the natural frequency of FG-O the highestand the natural frequency of FG-V is the lowest of all. Geometrical parameter isalso illustrated in Table 5. It can be seen that the natural frequency of functionallygraded carbon nanotubes-reinforced cylindrical panel decreases when the ratioa=h increases.

0 0.01 0.02 0.03 0.04 0.05

-4

-3

-2

-1

0

1

2

3

4

x 10-5

W(m

)

(m,n)=(1,1) , R/h=400,b/h=10,�T=0, q=8000sin(500t)

a/b = 0.5

a/b = 2/3

a/b = 1

t(s)

Figure 6. Effects of ratio a=b on the nonlinear dynamic response of the functionally gradedcarbon nanotubes-reinforced composite cylindrical panel.

Khoa et al. 21

Nonlinear dynamic response

Figures 3 and 4 consider the effects of elastic foundations with parameter

Pasternak k1, Winkler k2 on the nonlinear dynamic response of the FG-

CNTRC cylindrical panel in thermal environments with a=b ¼ 1; R=b ¼ 20. It is

clear that the panel fluctuation amplitude decreases when the modulus k1 and the

modulus k2 increases. On other words, the panel fluctuation amplitude decreases

when panel is rested on the elastic foundations. Moreover, from comparison

between Figures 3 and 4, we can see that the beneficial effect of the Pasternak

foundation with modulus k1 is better than the effect of Winkler one with modu-

lus k2.The influence of temperature increment DT on the nonlinear dynamic response

of the FG-CNTRC cylindrical panel is shown in Figure 5. When temperature

increment DT increases, the panel fluctuation amplitude increases.The effects of geometrical parameters b=a; R=b and b=h on the nonlinear

dynamic response of the FG-CNTRC cylindrical panel in thermal environments

0 0.01 0.02 0.03 0.04 0.05

-6

-4

-2

0

2

4

6

8

x 10-6

t(s)

W(m

)

(m,n)=(1,1) , R/h=400, b/h=10, �T=0, q=8000sin(500t)

a/h = 10

a/h = 20

a/h = 30

Figure 7. Effects of ratio a=h on the nonlinear dynamic response of the functionally gradedcarbon nanotubes-reinforced composite cylindrical panel.

22 Journal of Sandwich Structures & Materials 0(0)

are given in Figures 6 to 8, respectively. An increase of the ratio b=a will lead to adecrease of the amplitude of the FG-CNTRC cylindrical panels. The effects ofratios b=h; R=b have similar behaviors.

The comparison panel fluctuation amplitude from the obtained results withdifferent carbon nanotube volume fraction V�

CNT ¼ 0:12; 0:17; 0:28 is consideredin Figure 9, respectively. We can see that the carbon nanotube volume fractionV�

CNT is chosen as 0.28 the amplitude of FG-CNTRC cylindrical panel is the lowest.And the carbon nanotube volume fraction V�

CNT is chosen 0.12 the amplitude ofFG-CNTRC cylindrical panel is the highest. To put it differently, the stiffness ofFG-CNTRC cylindrical panel is improved when the FG-CNTRC cylindrical panelis strengthened by the carbon nanotube. It is clear that carbon nanotubes have animportant role to increase the stability of functionally graded cylindrical panel.

Figure 10 gives the effect of the type of FG-CNTRC on the nonlinear dynamicresponse of the FG-CNTRC cylindrical panel in thermal environments. Threetypes of FG-CNTRC: FG-O, FG-V, and FG-A with geometrical dimensionsa=b ¼ 1; b=h ¼ 50; R=h ¼ 400 are considered. Obviously, the amplitude fluctua-tion of FG-V FG-CNTRC is lower than the amplitude fluctuation of

0 0.01 0.02 0.03 0.04 0.05-1

-0.5

0

0.5

1x 10

-4

t(s)

W(m

)

(m,n)=(1,1), V*

cn = 0.12,

�T=0 , b/h=10, a/b=1, q=8000sin(500t)

R/b = 5

R/b = 10

R/b = 30

Figure 8. Effects of ratio R=b on the nonlinear dynamic response of the functionally gradedcarbon nanotubes-reinforced composite cylindrical panel.

Khoa et al. 23

0 0.01 0.02 0.03 0.04 0.05-1

-0.5

0

0.5

1

1.5x 10

-4

t(s)

W(m

)

(m,n)=(1,1), �T=0 , b/h=20,

R/h=400, a/b=1, q=8000sin(500t)

V*

CNT =0.12

V*

CNT =0.17

V*

CNT =0.28

Figure 9. Effects of CNT volume fraction on the nonlinear dynamic response of the functionallygraded carbon nanotubes-reinforced composite cylindrical panel.

0 0.01 0.02 0.03 0.04 0.05-2

-1

0

1

2

3x 10

-4

t(s)

W(m

)

(m,n)=(1,1), R/h=400, �T=0 , b/h=50, a/b=1, q=8000sin(500t)

FG-O CNTRC

FG-V CNTRC

FG-A CNTRC

Figure 10. The nonlinear dynamic response of the functionally graded carbon nanotubes-reinforced composite cylindrical panel with different types of carbon nanotubes reinforcements.

24 Journal of Sandwich Structures & Materials 0(0)

FG-AFG-CNTRC and higher than the amplitude fluctuation of FG-O FG-CNTRC. Therefore, FG-O FG-CNTRC should be used when manufac-tures model.

Conclusions

This work investigated the nonlinear dynamic response and vibration of FG-CNTRC cylindrical panel in thermal environment using both of the HSDT. Thepanel is supported by elastic foundations and subjected to thermal load. Usingstress function, Galerkin method, and fourth-order Runge–Kutta method, thenonlinear dynamic response and natural frequencies of FG-CNTRC cylindricalpanels are determined. The reliability of numerical results is evaluated by compar-ing with other results of the literature. From the obtained results, we can give someconclusions:

1. The stiffness of FG-CNTRC cylindrical panels is improved by carbon nano-tube. When the volume fraction of carbon nanotubes increases, the stiffness ofpanel will increase.

2. The amplitude fluctuation of the FG-CNTRC cylindrical panel stronglychanges according to the temperature increment.

3. The beneficial effect of the elastic foundations on the dynamic response of theFG-CNTRC cylindrical panel is clear and the beneficial effect of the Pasternakfoundation with modulus k1 is better than the effect of Winkler one with mod-ulus k2.

4. The effect of volume fraction of carbon nanotube and type of FG-CNTRCcylindrical panel on the nonlinear dynamic response and natural frequencyare considerable. The amplitude fluctuation of FG-V FG-CNTRC is lowerthan the amplitude fluctuation of FG-A FG-CNTRC and higher than theamplitude fluctuation of FG-O FG-CNTRC.

5. The nonlinear dynamic response of the FG-CNTRC cylindrical panel also isstrongly affected by the geometrical parameters.

The advantage of this work is using analytical solution, so the obtained results areexpressed explicitly in terms of the input parameters—we may change these param-eters and actively control the behaviors of the structures.

Highlights

• Nonlinear dynamic response and vibration• Functionally graded nanocomposite cylindrical panel reinforced by

carbon nanotubes• The shell resting on elastic foundations in thermal environments• Based on Reddy’s HSDT• Using analytical solution

Khoa et al. 25

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, author-

ship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, author-

ship, and/or publication of this article: This research is funded by National Foundation for

Science and Technology Development (NAFOSTED) of Vietnam, under grant number

107.02–2018.04. The authors are grateful for this support.

ORCID iD

Nguyen Dinh Duc http://orcid.org/0000-0003-2656-7497

References

1. Shen SH. Thermal buckling and postbuckling behavior of functionally graded carbon

nanotube-reinforced composite cylindrical shells.ComposPart B:Eng 2012; 43: 1030–1038.2. Griebel M and Hamaekers J. Molecular dynamics simulations of the elastic moduli of

polymer-carbon nanotube composites. Comput Methods Appl Mech Eng 2004;

193: 1773–1788.3. Zhang LW and Liew KM. Postbuckling analysis of axially compressed CNT reinforced

functionally graded composite plates resting on Pasternak foundations using an

element-free approach. Compos Struct 2016; 38: 40–51.4. Mirzaei M and Kiani Y. Free vibration of functionally graded carbon nanotube rein-

forced composite cylindrical panels. Compos Struct 2016; 142: 45–56.5. Kolahchi R, Zarei MS, Hajmohammad MH, et al. Wave propagation of embedded

viscoelastic FG-CNT-reinforced sandwich plates integrated with sensor and actuator

based on refined zigzag theory. Int J Mech Sci 2017; 130: 534–545.6. Zhang LW, Liu WH and Liew KM. Geometrically nonlinear large deformation analysis

of triangular CNT-reinforced composite plates. Int J Non-Linear Mech 2016; 86: 122–132.7. Shen SH and Xiang Y. Nonlinear vibration of nanotube-reinforced composite cylindri-

cal panels resting on elastic foundations in thermal environments. Compos Struct 2014;

111: 291–300.8. Shen SH and Xiang Y. Nonlinear bending of nanotube-reinforced composite cylindrical

panels restingonelastic foundations in thermal environments.EngStruct 2014; 80: 163–172.9. Kolahchi R, Safari M and Esmailpour M. Dynamic stability analysis of temperature-

dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elas-

tomeric medium. Compos Struct 2016; 150: 255–265.10. Mirzaei M and Kiani Y. Thermal buckling of temperature dependent FG-CNT rein-

forced composite plates. Meccanica 2015; 47: 1–17.11. Mirzaei M and Kiani Y. Thermal buckling of temperature dependent FG-CNT rein-

forced composite conical shells. Aerosp Sci Technol 2015; 47: 42–53.12. Shen SH. Postbuckling of nanotube-reinforced composite cylindrical shells in thermal

environments. Part II: pressure-loaded shells. Compos Struct 2011; 93: 2496–2503.13. Shen SH. Thermal buckling and postbuckling behavior of functionally graded carbon

nanotube-reinforced composite cylindrical shells.ComposPart B:Eng 2012; 43: 1030–1038.

26 Journal of Sandwich Structures & Materials 0(0)

14. Bidgoli MR, Karimi MS and Arani AG. Viscous fluid-particle mixture induced vibra-

tion and instability of FGCNT-reinforced cylindrical shells integrated with piezoelectric

layers. Steel Compos Struct 2015; 19: 713–733.15. Shen HS. Nonliear bending of functionally graded carbon nanotube reinforced com-

posite plate in thermal environments. Compos Struct 2009; 91: 3–19.

16. Han Y and Elliott J. Molecular dynamics simulations of the elastic properties of poly-

mer/carbon nanotube composites. Comput Mater Sci 2007; 39: 315–323.17. Hosseini H and Kolahchi R. Seismic response of functionally graded-carbon nanotubes-

reinforced submerged viscoelastic cylindrical shell in hygrothermal environment. Phys

E: Low-Dimens Syst Nanostruct 2018; 102: 101–109.

18. Ovesy HR and Fazilati J. Parametric instability analysis of moderately thick FGM

cylindrical panels using FSM. Compos Struct 2012; 108–109: 135–143.19. Viola E, Rossetti L, Fantuzzi N, et al. Static analysis of functionally graded conical

shells and panels using the generalized unconstrained third order theory coupled with

the stress recovery. Compos Struct 2014; 112: 44–65.

20. Thanh NV, Khoa ND, Tuan ND, et al. Nonlinear dynamic response and vibration of

functionally graded carbon nanotubes reinforced composite (FG-CNTRC) shear

deformable plates with temperature dependence material properties and surrounded

on elastic foundations. J Therm Stress 2017; 40: 1254–1274.21. Ou X, Yao X, Zhang R, et al. Nonlinear dynamic response analysis of cylindrical

composite stiffened laminates based on the weak form quadrature element method.

Compos Struct 2018; 203: 446–457.22. Duc ND and Quan TQ. Nonlinear response of imperfect eccentrically stiffened FGM

cylindrical panels on elastic foundation subjected to mechanical loads. Eur J Mech A

Solids 2014; 46: 60–71.23. Duc ND, Tuan ND, Quan TQ, Quyen NV, Anh TV. Nonlinear mechanical, thermal

and thermo-mechanical postbuckling of imperfect eccentrically stiffened thin FGM

cylindrical panels on elastic foundations. J Sandw Struct Mater 2015; 96: 155–168.24. Thu PV and Duc ND. Nonlinear stability analysis of imperfect three-phase sandwich

laminated polymer nanocomposite panels resting on elastic foundations in thermal

environments. VNU J Sci Math Phys 2016; 32: 20–36.25. Shen SH and Wang H. Nonlinear bending of FGM cylindrical panels resting on elastic

foundations in thermal environments. Eur J Mech A Solids 2015; 49: 49–59.26. Kiani Y, Sadighi M and Eslami MR. Dynamic analysis and active control of smart

doubly curved FGM panels. Compos Struct 2013; 102: 205–216.27. Duc ND, Seung-Eock K, Quan TQ, et al. Nonlinear dynamic response and vibra-

tion of nanocomposite multilayer organic solar cell. Compos Struct 2018;

184: 1137–1144.28. Cong PH, Anh VM and Duc ND. Nonlinear dynamic response of eccentrically stiffened

FGM plate using Reddy’s TSDT in thermal environment. J Therm Stress 2017;

40: 704–732.29. Najafi F, Shojaeefard MH and Googarchin HS. Nonlinear dynamic response of FGM

beams with Winkler–Pasternak foundation subject to noncentral low velocity impact in

thermal field. Compos Struct 2017; 167: 132–143.30. Duc ND. Nonlinear thermal dynamic analysis of eccentrically stiffened S-FGM circular

cylindrical shells surrounded on elastic foundations using the Reddy’s third-order shear

deformation shell theory. Eur J Mech A Solids 2016; 58: 10–30.

Khoa et al. 27

31. Panday S and Pradyumna S. A finite element formulation for thermally induced vibra-

tions of functionally graded material sandwich plates and shell panels. Compos Struct

2017; 160: 877–886.32. Duc ND and Quan TQ. Transient responses of functionally graded double curved

shallow shells with temperature-dependent material properties in thermal environment.

Eur J Mech A Solids 2014; 47: 101–123.33. Yang J and Shen SH. Vibration characteristics and transient response of shear-deformable

functionally graded plates in thermal environments. J Sound Vib 2002; 255: 579–602.34. Duc ND, Cong PH, Tuan ND, et al. Nonlinear vibration and dynamic response of

imperfect eccentrically stiffened shear deformable sandwich plate with functionally

graded material in thermal environment. J Sandw Struct Mater 2016; 18: 445–473.35. Yang J, Liew KM, Wu YF, et al. Thermo-mechanical post-buckling of FGM cylindrical

panels with temperature-dependent properties. Int J Solids Struct 2006; 43: 307–324.36. Matsumaga H. Free vibration and stability of functionally graded shallow shells accord-

ing to a 2-D higher-order deformation theory. Compos Struct 2008; 84: 132–146.37. Duc ND. Nonlinear static and dynamic stability of functionally graded plates and shells.

Hanoi: Vietnam National University Press, 2014.38. Brush DD and Almroth BO. Buckling of bars, plates and shells. New York: McGraw-

Hill, 1975.39. Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. Boca

Raton: CRC Press, 2004.

Appendix 1

G11 ¼XNk¼0

Q44 hk � hk�1ð Þ; G12 ¼XNk¼0

Q44h3k � h3k�1

3;

G21 ¼XNk¼0

Q55 hk � hk�1ð Þ; G22 ¼XNk¼0

Q55h3k � h3k�1

3;

G32 ¼XNk¼0

Q44h5k � h5k�1

5; G42 ¼

XNk¼0

Q55h5k � h5k�1

5;

C75 ¼XNk¼0

Q11h7k � h7k�1

7; C76 ¼

XNk¼0

Q12h7k � h7k�1

7;

C86 ¼XNk¼0

Q22h7k � h7k�1

7; C93 ¼

XNk¼0

Q66h7k � h7k�1

7;

C11 ¼XNk¼0

Q11 hk � hk�1ð Þ; C12 ¼XNk¼0

Q12 hk � hk�1ð Þ;

28 Journal of Sandwich Structures & Materials 0(0)

C13 ¼XNk¼0

Q11hk � hk�1ð Þ2

2; C14 ¼

XNk¼0

Q12h2k � h2k�1

2;

C15 ¼XNk¼0

Q11h4k � h4k�1

4; C16 ¼

XNk¼0

Q12h4k � h4k�1

4;

C22 ¼XNk¼0

Q22 hk � hk�1ð Þ; C24 ¼XNk¼0

Q22h2k � h2k�1

2;

C43 ¼XNk¼0

Q11h3k � h3k�1

3; C44 ¼

XNk¼0

Q12h3k � h3k�1

3;

C45 ¼XNk¼0

Q11h5k � h5k�1

5; C46 ¼

XNk¼0

Q12h5k � h5k�1

5;

C26 ¼XNk¼0

Q22h4k � h4k�1

4; C31 ¼

XNk¼0

Q66 hk � hk�1ð Þ;

C32 ¼XNk¼0

Q66h2k � h2k�1

2; C33 ¼

XNk¼0

Q66h4k � h4k�1

4;

C54 ¼XNk¼0

Q22h3k � h3k�1

3; C56 ¼

XNk¼0

Q22h5k � h5k�1

5;

C62 ¼XNk¼0

Q66h3k � h3k�1

3; C63 ¼

XNk¼0

Q66h5k � h5k�1

5;

U1x ¼

XNk¼0

Zhkhk�1

Q11a11zDTdzþXNk¼0

Zhkhk�1

Q12a22zDTdz;

U3x ¼

XNk¼0

Zhkhk�1

Q11a11z2DTdzþ

XNk¼0

Zhkhk�1

Q12a22z2DTdz;

U5x ¼

XNk¼0

Z hk

hk�1

Q11a11z3DTdzþ

XNk¼0

Z hk

hk�1

Q12a22z3DTdz;

U1y ¼

XNk¼0

Z hk

hk�1

Q12a11zDTdzþXNk¼0

Z hk

hk�1

Q22a22zDTdz;

U3y ¼

XNk¼0

Z hk

hk�1

Q12a11z2DTdzþ

XNk¼0

Z hk

hk�1

Q22a22z2DTdz;

U5y ¼

XNk¼0

Z hk

hk�1

Q12a11z3DTdzþ

XNk¼0

Z hk

hk�1

Q22a22z3DTdz;

Khoa et al. 29

A11 ¼ C13H11 � C14H12; A12 ¼ C13H12 þ C14H22; A13 ¼ C13H13 þ C14H23 þ C43;

A14 ¼ C13H14 þ C14H24 þ C44; A15 ¼ C13H15 þ C14H25 þ C45;

A16 ¼ C13H16 þ C14H26 þ C46;

A17 ¼ C13H11 � C14H12; A18 ¼ C13H12 þ C14H22; A21 ¼ C14H11 � C24H12;

A22 ¼ C14H12 þ C24H22; A23 ¼ C14H13 þ C24H23 þ C44;

A24 ¼ C14H14 þ C24H24 þ C54;

A25 ¼ C14H15 þ C24H25 þ C46; A26 ¼ C14F16 þ C24F26 þ C56;

A27 ¼ C14H11 � C24H12;

A28 ¼ C14H12 þ C24H22; A31 ¼ C32

C31; A32 ¼ C62 � C2

32

C31

� �;

A33 ¼ C63 � C33C32

C31

� �;

A41 ¼ C15H11 � C16H12; A42 ¼ C15H12 þ C16H22; A43 ¼ C15H13 þ C16H23 þ C45;

A44 ¼ C15H14 þ C16H24 þ C46; A45 ¼ C15H15 þ C16H25 þ C75;

A46 ¼ C15H16 þ C16H26 þ C76;

A47 ¼ C15H11 � C16H12; A48 ¼ C15H12 þ C16H22; A51 ¼ C16H11 � C26H12;

A52 ¼ C16H12 þ C26H22; A53 ¼ C16H13 þ C26H23 þ A46;

A54 ¼ C16H14 þ C26H24 þ C56;

A55 ¼ C16H15 þ C26H25 þ C76; A56 ¼ C16H16 þ C26H26 þ C86;

A57 ¼ C16H11 � C26H12;

A58 ¼ C16H12 þ C26H22; A61 ¼ C33

C31; A62 ¼ C63 � C32C33

C31

� �;

A63 ¼ C93 � C233

C31

� �;

r11 ¼ � G11 þ c2G12 � 4G12

h2� 4c2G32

h2

� �a2W� G21 þ c2G22 � 4G22

h2� 4c2G42

h2

� �b2W

� c21A45a4 � c21A46 þ 4c21A63 þ c21A55

� �a2b2 � c21A56b

4W� k1 � k2 a2 þ b2� �

� c1A42a4H3 � c1A41 � 2c1A61 þ A52c1ð ÞF3a2b2 � b4F3 � 1

RF3a

2;

r12 ¼ � G11 þ c2G12 � 4G12

h2� 4c2G32

h2

� �aþ c1A43 � c21A45

� �a3

þ 2c1A62 � 2c21A63 þ A53c1 � c21A55

� �ab2 þ c1A42a4H1

þ c1A41 � 2c1A61 þ A52c1ð ÞF1a2b2 þ b4F1 þ a2

RF1U;

30 Journal of Sandwich Structures & Materials 0(0)

r13 ¼ � G21 þ c2G22 � 4G22

h2� 4c2G42

h2

� �bþ A54c1 � c21A56

� �b3

þ 2c1A62 � 2c21A63 þ c1A44 � c21A46

� �a2b

þ c1A42a4F2 þ c1A41 � 2c1A61 þ A52c1ð Þa2b2F2 þ b4F2 þ a2

RF2;

r14 ¼ c1A42b2

4H22a2 þ A51c1

a2b2

4H11� 1

R

2a2a2b2

3H11mnp2��F3b

2a232

3mnp2;

r15 ¼ �2a4

32H11� 2

b4

32H22;

r23 ¼ � A14 þ A32 � c1A16 � c1A44 � c1A33 � c1A62 þ c21A46 þ c21A63

� �ab

� A11 � A31 � c1A41 þ c1A61ð Þb2aþ A12 � c1A42ð Þa3� �

F2;

r24 ¼ A12 � c1A42ð Þ 8b3H22ab

;

r31 ¼ � c21A55 þ 2c21A63 � 2c1A33 � c1A25

� �a2b

� c21A56 � c1A26

� �b3 � G21 � c22G42

� �b

þ A22 � A31 þ A61c1 � A52c1ð Þa2bþ b3 A21 � A51c1ð Þ� �

F3;

r21 ¼ � c21A45 � c1A15

� �a3 � G11 � c22G32

� �a� 2c21A63 þ c21A46 � c1A16 � 2c1A33

� �ab2

þ A11 � A31 � c1A41 þ c1A61ð Þb2aþ A12 � c1A42ð Þa3� �

H3;

r22 ¼ � A13 � c1A15 � c1A43 þ c21A45

� �a2 � A32 � c1A33 � c1A62 þ c21A63

� �b2

� G11 � c22G32

� �Ux � A11 � A31 � c1A41 þ c1A61ð Þb2aþ A12 � c1A42ð Þa3

� �F1;

r32 ¼ � A32 � c1A33 þ A23 � c1A25 � A62c1 þ c21A63 � A53c1 þ c21A55

� �ab

� A22 � A31 þ A61c1 � A52c1ð Þa2bþ b3 A21 � A51c1ð Þ� �

F1;

r33 ¼ � A32 � c1A33 � A62c1 þ c21A63

� �a2Uy � A24 � c1A26 � A54c1 þ c21A56

� �b2

� G21 � c22G42

� �Uy � A22 � A31 þ A61c1 � A52c1ð Þa2bþ b3 A21 � A51c1ð Þ

� �H2;

r34 ¼ A21 � A51c1ð Þ aH11

8

3ab;

n1 ¼ �E1

16

mpa

� �4

þ npb

� �4 !

;

n2 ¼ J0 þ c21 J6 � J23J0

� �mpa

� �2 þ npb

� �2� �;

Khoa et al. 31

T11 ¼ �F3b4

aabW�

H12 c1H25a2 þ c1H26b2 þ 1

R

� �þH22 c1H15a2þc1H16b

2� ��

H11H22 �H212

4

abab;

T12 ¼ F1b4

aabUx � H12 �F1H22aþ c1H25 �H23ð Þ½ � þH22 F1H12aþ c1H15 �H13ð Þ½ �½ �

H11H22 �H212

4

bab;

T13 ¼ H22 F2H11b� c1H16 �H14ð Þ½ � �H12 F2H12bþ c1H26 �H24ð Þ½ �½ �H11H22 �H2

12

4

aab;

T14 ¼ H22a2

8þH12b

2

8

� �1

H11H22 �H212

; S15 ¼ � H22

H11H22 �H212

H11U1x �H12U

1y

� �

� H12

H11H22 �H212

�H12U1x �H22U

1y

� �;

T21 ¼ �F3a4

babW�

H12 c1H15a2þc1H16b2

� �þH11 c1H25a2 þ c1H26b

2 þ 1

R

� �� H11H22 �H2

12

4

abab;

T22 ¼ � H12 F1H12aþ c1H15 �H13ð Þ½ � þH11 �F1H22aþ c1H25 �H23ð Þ½ �½ �H11H22 �H2

12

Ux4

bab;

T23 ¼ F2a4

babþ H12 F2H11b� c1H16 �H14ð Þ½ � �H11 F2H12bþ c1H26 �H24ð Þ½ �½ �

H11H22 �H212

4

aab;

T24 ¼ H12a2 þH11b2

� �8 H11H22 �H2

12

� � ; T25 ¼ � H12

H11H22 �H212

H11U1x �H12U

1y

� �

� H11

H11H22 �H212

�H12U1x þH22U

1y

� �:

32 Journal of Sandwich Structures & Materials 0(0)