Nonlinear buckling of eccentrically stiffened nanocomposite cylindrical panels in thermal environments

Thin–Walled Structures 146 (2020) 106428

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Nonlinear buckling of eccentrically stiffened nanocomposite cylindrical panels in thermal environments Nguyen Dinh Duc a, b, c, *, Seung-Eock Kim c, Tran Quoc Quan a, Duong Tuan Manh a, Nguyen Huy Cuong a a b c

Department of Construction and Transportation Engineering, VNU, Hanoi - University of Engineering and Technology, 144 Xuan Thuy, CauGiay, Hanoi, Viet Nam NTT Institute of High Technology, Nguyen Tat Thanh University, Disctric 4, Ho Chi Minh City, Viet Nam National Research Laboratory, Department of Civil and Environmental Engineering, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul 05006, South Korea

A R T I C L E I N F O

A B S T R A C T

Keywords: Analytical solutions Nonlinear buckling Imperfect nanocomposite cylindrical panels Reddy’s third order shear deformation shell theory Eccentrically stiffeners Thermal environment

Based on Reddy’s third order shear deformation shell theory and Galerkin method, this paper introduces analytical solutions to study nonlinear buckling behaviors of imperfect carbon nanotube reinforced composite cylindrical panels on elastic foundations in thermal environments. The panels are reinforced by single-walled carbon nanotubes and the eccentrically longitudinal and transversal stiffeners. The effects of geometrical pa­ rameters, eccentrically stiffeners, elastic foundations, initial imperfection, temperature increment and nanotube volume fraction on the mechanical behaviors of the nanocomposite cylindrical panels are also examined in numerical results. Some comparisons with results of other authors show the accuracy of the present theory and approach.

1. Introduction Nanocomposites are advanced materials in which at least one phase with nanoscale geometrical parameters is added in a polymer, metal or ceramic matrix to enhance mechanical properties of the material. Recently, nanocomposites have attracted the attention of scientists around the world due to their widely applications. Gleich et al. [1] investigated the thermal stability of nanocomposite Mo2BC hard coat­ ings deposited by magnetron sputtering. Hassanzadeh-Aghdam et al. [2] studied the role of carbon nanotube coating on the carbon fiber surfaces in the effective thermal conductivities of the unidirectional polymer hybrid nanocomposites. Hassanzadeh-Aghdam et al. [3,4] considered the influences of silica nanoparticle aggregation on the creep resistance of nanocomposites which consists of polymer matrix and nanoparticle, and elastic-plastic and thermo-elastic behaviors of nanocomposites with aluminum matrix and silicon carbide nanoparticles. Further, Kong et al. [5] established an assembled method to obtain nanocomposites with silica particles and multi-walled carbon nanotubes, and the resulting nanocomposites were incorporated to fabricate rubber composites. Dong et al. [6,7] presented analytical studies on linear and nonlinear free vibration characteristics and dynamic responses of spinning

functionally graded graphene reinforced thin cylindrical shells with various boundary conditions and subjected to a static axial load and free vibration characteristics of the functionally graded graphene reinforced porous nanocomposite cylindrical shell with spinning motion, respec­ tively. In 2018, Duc et al. [8] presented the first analytical approach to investigate the nonlinear vibration and dynamic response of nano­ composite organic solar cell plate subjected to external pressure uni­ formly distributed on the surface of the plate based on the classical plate theory. The mechanical and tribological properties of C–SiC polymer nanocomposites were studied by Li et al. [9]. Oktem and Adali [10] considered the buckling of shear deformable polymer/clay nano­ composite columns with uncertain material properties by multiscale modeling via a micromechanical approach. Li et al. [11] focused on the parametric instability of a functionally graded cylindrical thin shell under both axial disturbance and thermal environment. Carbon nanotubes are molecular-scale tubes of graphene sheet. Due to their outstanding thermal conductivity, mechanical, and electrical properties; carbon nanotubes are used in various practical fields such as energy storage, field emission, transistors, sensors, composite materials, conductive adhesives and connectors and so on. Iakovlev et al. [12] reported an approach to enhance the transmittance of single-walled

* Corresponding author. Department of Construction and Transportation Engineering, VNU, Hanoi - University of Engineering and Technology, 144 Xuan Thuy, CauGiay, Hanoi, Viet Nam E-mail address: [email protected] (N.D. Duc). https://doi.org/10.1016/j.tws.2019.106428 Received 5 February 2019; Received in revised form 8 June 2019; Accepted 25 September 2019 Available online 18 October 2019 0263-8231/© 2019 Elsevier Ltd. All rights reserved.

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carbon nanotube films placed onto polymeric substrates without losses in the film conductivity by a low-power laser treatment. García-Macías et al. [13] provided some insight into the linear buckling analysis of functionally graded carbon nanotube reinforced cylindrical curved panels under compressive and shear loading. Nagaraj et al. [14] researched the mechanical behavior of carbon nanotubes reinforced AA 4032 bimodal alloys using microhardness test and compression test; Russello et al. [15] investigated experimentally the transverse electrical conductivity of thin-ply carbon fibre laminates, enhanced with carbon nanotubes. Shen and Xiang [16,17] presented studies on the postbuck­ ling of functionally graded carbon nanotube reinforced composite cy­ lindrical panels resting on elastic foundations subjected to thermal load and postbuckling of carbon nanotube reinforced composite cylindrical panels on Pasternalk type elastic foundations subjected to the combi­ nation of mechanical and thermal loads. Based on the first order shear deformation theory, Lei et al. [18] investigated the stability of func­ tionally graded composite laminated plate reinforced carbon nanotubes. In 2018, Ninh [19] focused on thermal torsional post buckling of func­ tionally graded carbon nanotube reinforced composite cylindrical shells with sur-bonding piezoelectric layers and embedded in an elastic me­ dium. Recently, Duc et al. [20] introduced an analytical approach to study the nonlinear vibration and dynamic response of imperfect poly­ mer nanocomposites double curved shallow shells reinforced carbon nanotube subjected to blast loads. By considering the pre-buckling effect and in-plane constraint, Zhou et al. [21] performed an accurate nonlinear buckling analysis of a functionally graded porous graphene platelet reinforced composite cylindrical shells under axial compressive load. Hosseini and Kolahchi [22] presented a mathematical model for the behaviors of cylindrical shells reinforced by carbon nanotubes and carbon fibers subjected to hygrothermal load based on Kelvin-Voigt model; Lei et al. [23] presented a first-known dynamic stability anal­ ysis of carbon nanotube-reinforced functionally graded cylindrical panels under static and periodic axial force by using the mesh-free kp-Ritz method. Moreover, Zhang and Liew [24] focused on postbuck­ ling analysis of nanocomposite plates reinforced by carbon nanotubes and subjected to mechanical load. Hassanzadeh-Aghdam et al. [25] developed a multiscale micromechanical modeling approach to predict elastic properties of carbon fiber reinforced polymer hybrid composites. Analytical solution for nonlinear postbuckling of functionally graded carbon nanotube-reinforced composite shells with piezoelectric layers and Molecular dynamics simulations of the polymer/amine functional­ ized single-walled carbon nanotubes interactions are studied in the work [26,27] of Ansari et al. Stiffeners are secondary plates or sections which are reinforced to structures to strengthen their stiffness and durability. As a result, the mechanical behaviors of composite structures reinforced by stiffeners have received considerable interest. Yu et al. [28] presented a numerical investigation of the dynamic performance of steel plate shear walls stiffened by non-welded multi-rib stiffeners. Tartaglia et al. [29] carried out the influence of rib stiffeners on the response of extended end-plate joints. Shahandeh and Showkati [30] investigated the buckling and post-buckling behaviors of ring-stiffened pipelines were investigated at a small scale through experiments and the finite element method. Liu et al. [31] dealt with the influence of stiffeners on plate vibration and noise radiation induced by turbulent boundary layers by wind tunnel mea­ surements. Li and Chen developed [32] a phenomenological-based mechanical finite element model for the prediction of ultimate compression loads and failure modes of wing relevant composite panels stiffened with I-shaped stiffeners, of which the edge is subjected to impact loading. Duc and Quan carried out [33] the nonlinear response of eccentrically stiffened FGM cylindrical panels on elastic foundation subjected to mechanical loads. Kumar et al. [34] focused on the static and dynamic analysis of pressure vessels with various stiffeners. Chen et al. [35] numerically investigated the shear buckling behavior of welded stainless steel plate girders with transverse stiffeners. The novelty of this paper is using analytical approach to investigate

the nonlinear buckling behaviors of imperfect eccentrically stiffened carbon nanotube reinforced nanocomposite sandwich cylindrical panels subjected to the combination of thermal load and mechanical load. The effective material properties of the nanocomposite cylindrical panels are assumed to depend on temperature and estimated through the rule of mixture. Basic equations are derived based on Reddy’s third order shear deformation shell theory taking into account the effects of the thermal stress in both the panels and the stiffeners then solved by the Galerkin method. Numerical results show the effects of geometrical parameters, eccentrically stiffeners, elastic foundations, initial imperfection, tem­ perature increment and nanotube volume fraction on the load – deflection curves of the eccentrically stiffened carbon nanotube rein­ forced nanocomposite cylindrical panels. 2. Modeling of eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panels Consider an eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panels on elastic foundations in coordinate system ðx; y; zÞ as shown in Fig. 1. The radii of curvature, thickness, axial length and arc length of the panel are R, h, a and b, respectively. The panel is reinforced by eccentrically longitudinal and transversal stiff­ eners with the width and thickness of longitudinal and transversal stiffeners dx ; hx and dy ; hy respectively; the spacing of the longitudinal and transversal stiffeners sx ; sy . The quantities Ax ; Ay are the crosssection areas of stiffeners and Ix ; Iy ; zx ; zy are the second moments of cross-section areas and the eccentricities of stiffeners with respect to the middle surface of panel, respectively. In order to provide continuity between the panel and stiffeners, suppose that stiffeners are made of full Methyl methacrylate. The carbon nanotube reinforced nanocomposite cylindrical panel is made of Methyl methacrylate, referred to as PMMA, reinforced by (10,10) single-walled carbon nanotubes. The effective shear and Young’s modulus of the carbon nanotube reinforced nanocomposite cylindrical panel are determined as [16–18,24]. E11 ¼ η1 VCNT ECNT 11 þ Vm Em ;

η2 E22

η3 G12

¼

VCNT Vm þ ; Em ECNT 22

¼

VCNT Vm þ ; Gm GCNT 12

(1)

CNT CNT where ECNT 11 ; E11 ; G12 and Em; Gm are Young’s and shear modulus of the carbon nanotube and the matrix, respectively. VCNT and Vm are the volume fractions of the carbon nanotube and the matrix and ηi ði ¼ 1; 3Þ are the carbon nanotube efficiency parameters. In this study, we consider three types of the carbon nanotube rein­ forced nanocomposite cylindrical panel, i.e. FG-V, FG-O and FG-X. The volume fractions of the carbon nanotube and the matrix are expressed as linear functions of the panel thickness [16–18,24]. 8 * � z� ðFG VÞ > V CNT 1 þ 2 > h > > > > > > > � � > < jzj * ðFG OÞ ; VCNT ðzÞ ¼ 2V CNT 1 2 h > > (2) > > > > > > > > : * jzj ðFG XÞ 4V VCT h

Vm ðzÞ ¼ 1

VCNT ðzÞ:

where V *CNT ¼

2

wCNT ; wCNT þ ðρCNT =ρm Þ ðρCNT =ρm ÞwCNT

(3)

N.D. Duc et al.

Thin-Walled Structures 146 (2020) 106428

Fig. 1. Configuration and the coordinate system of an eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panels on elastic foundations.

with wCNT is the mass fraction of carbon nanotube, ρCNT and ρm are the densities of carbon nanotube and matrix, respectively. The Poisson’s ratio, the Young’s modulus and the thermal expansion coefficient of the matrix are determined to depend on temperature as [16–18,24].

where νCNT 12 and νm are Poisson’s ratio of the carbon nanotube and the matrix, respectively. The thermal expansion coefficients in the longitudinal and transverse directions of the carbon nanotube reinforced nanocomposite cylindrical panel are defined as follows [16–18,24].

νm ¼ 0:34; αm ¼ 45ð1 þ 0:0005ΔTÞ � 10

α11 ¼

Em ¼ ð3:52

6

� K;

CNT VCNT ECNT 11 α11 þ Vm Em αm ; VCNT ECNT 11 þ Vm Em � VCNT αCNT α22 ¼ 1 þ νCNT 12 22 þ ð1 þ νm ÞVm αm

(4)

0:0034TÞ GPa;

where T ¼ T0 þ ΔT; ΔT is the temperature increment in the environment containing the panel and T0 ¼ 300K (room temperature). The mechanical properties of (10,10) single-walled carbon nano­ tubes are assumed to depend on temperature in which five levels of temperature are considered in Table 1 [16,17]. The Poisson’s ratio of single-walled carbon nanotubes is chosen to be constant. νCNT ¼ 0:175: 12 The carbon nanotubes efficiency parameters ηi ði ¼ 1; 3Þ depend on volume fraction of carbon nanotube and are calculated by the extended rule of mixture to molecular simulation results [16,17]. Specifically, three case of volume fraction of carbon nanotube are considered: η1 ¼ 0:137; η2 ¼ 1:022; η3 ¼ 0:715for the case of V *CNT ¼ 0:12 ð12%Þ; η1 ¼

ν12 α11 ;

CNT with αCNT 11 ; α22 and αm are the thermal expansion coefficients of the carbon nanotube and the matrix, respectively.

3. Basic equations In this study, Reddy’s third order shear deformation shell theory is used to establish basic equations and investigate the mechanical be­ haviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to thermal and mechanical loads. �rma �n The strain – displacement relations taking into account von Ka nonlinear terms are [36,37]. 0 1 0 1 0 1 1 3 0 0 1 0 1 ! k ε x x εx B C B C B kx C � γ � k2 γ 0xz 1 C 3 C B B 0 C xz 3 2 @ εy A ¼ B @ xz A; þ z ¼ B εy C þ zB ky C þ z B ky C; γ @ 0 A @ 1 A @ 3 A γ 0yz yz k2yz γ xy γ k k

0:142; η2 ¼ 1:626; η3 ¼ 1:138for the case of V *CNT ¼ 0:17 ð17%Þ and

η1 ¼ 0:141; η2 ¼ 1:585; η3 ¼ 1:109for the case of V *CNT ¼ 0:28 ð28%Þ.

The Poisson’s ratio of the carbon nanotube reinforced nano­ composite cylindrical panel could be expressed as a function of tem­ perature and position as [16–18,24].

xy

(5)

ν12 ¼ V *CNT vCNT 12 þ Vm νm ;

(6)

xy

xy

(7)

Table 1 Temperature-dependent material properties for (10, 10) single-walled carbon nanotubes [16,17]. Temperature (K)

ECNT 11 ðTPaÞ

ECNT 22 ðTPaÞ

GCNT 12 ðTPaÞ

αCNT 11 ð � 10

300 400 500 700 1000

5.6466 5.5679 5.5308 5.4744 5.2814

7.0800 6.9814 6.9348 6.8641 6.6220

1.9445 1.9703 1.9643 1.9644 1.9451

3.4584 4.1496 4.5361 4.6677 4.2800

3

6

=KÞ

αCNT 22 ð � 10 5.1682 5.0905 5.0189 4.8943 4.7532

6

=KÞ

N.D. Duc et al.

Thin-Walled Structures 146 (2020) 106428

where

0

1

. 1 u;x þ w2;x 2 0 1 � � 0 φx þ w;x . . C C @ γ xz A ¼ ; 2 C w R þ w;y 2 A γ 0 φy þ w;y

0

0 B εx C B B 0C B B εy C ¼ B B C

@

γ0xy

@ v;y

A

yz

u;y þ v;x þ w;x w;y

0

1 1 0 1 0 1 φx;x B kx C 2 B 1C B C @ kxz A B ky C ¼ @ φy;y ¼ A; B C k2yz @ 1 A φ þ φ x;y y;x kxy

0

1

0

3

� � B kx C φx þ w;x B 3 C 3c1 ; B ky C C¼ φy þ w;y B @ 3 A kxy

B c1 @

φx;y þ φy;x þ 2w;xy

Introduction of Eq. (7) into Eqs. 9 and 11 and the results into Eq. (12) yields the constitutive relations as Nx ¼ B11 ε0x þ B12 ε0y þ B13 k1x þ B14 k1y þ B15 k3x þ B16 k3y ​ Ny ¼

B12 0x

B24 k1y

B22 0y

ε þ

ε þ

B14 k1x

þ

B16 k3x

þ

þ

B26 k3y ​

Φ1

B17 Φs1x ;

Φ2

B27 Φs1y ;

Nxy ¼ B31 γ 0xy þ B32 k1xy þ B33 k3xy ; Mx ¼ B13 ε0x þ B14 ε0y þ B43 k1x þ B44 k1y þ B45 k3x þ B46 k3y My ¼

B14 0x

ε þ

B24 0y

ε þ

B44 k1x

þ

B54 k1y

þ

B46 k3x

þ

B56 k3y

Φ3

B17 Φs2x ;

Φ4

B27 Φs2y ;

Mxy ¼ B32 γ0xy þ B62 k1xy þ B63 k3xy ; Px ¼ B71 ε0x þ B16 ε0y þ B73 k1x þ B46 k1y þ B75 k3x þ B76 k3y

Φ5 B17 Φs4x ; 3 Py ¼ þ þ þ þ þ B86 ky Φ6 B27 Φs4y ; Pxy ¼ B33 γ0xy þ B63 k1xy þ B93 k3xy ; Qx ¼ B94 γ 0xz þ B95 k2xz ; Qy ¼ B96 γ0yz þ B97 k2yz ; Kx ¼ B98 γ 0xz þ B99 k2xz ; Ky ¼ B100 γ 0yz þ B101 k2yz ; B16 0x

ε

where 1

C A:

φy;y þ w;yy

where c1 ¼ 4=3h2 ; u; v are the displacement components along the x; y directions, respectively, and φx ; φy are the rotations of normals to the midsurface with respect to y and x axes, respectively. Hooke’s law for cylindrical panel taking into account the tempera­ ture – dependent properties is defined as �28 9 8 9 � 8 93 � Q11 Q12 εx > σx > 0 0 0 �� > α11 > > > > > > > > > >ε > > �� > > >7 � > > > > > 6 0 0 0 �6< y = < α22 > < σ y = � Q12 Q22 =7 7: γ σyz ¼ �� 0 ΔT 0 Q55 0 0 ��6 0 (9) 6> yz > > > > >7 > > 0 0 Q44 0 ��4> > �� 0 > > 0 > > σxz > >5 > γ xz > > > > > > > : ; ; � : : ; γxy σxy 0 0 0 0 0 Q66 �

Q11 ¼

(8)

1

φx;x þ w;xx

E11 v21 E11 E22 ; ​ Q12 ¼ ; ​ Q22 ¼ ; Q44 ¼ G23 ; ​ Q55 v12 v21 1 v12 v21 1 v12 v21

B82 0y

ε

B46 k1x

B84 k1y

B76 k3x

¼ G13 ; ​ Q66 ¼ G12 : (10) (

and the detail of coefficients Bij ; Φi ði ¼ 1; 6Þ; Φs1x ; Φs1y ; Φs2x ; Φs2y may be

found in Appendix A. Based on Reddy’s third order shear deformation shell theory, the nonlinear equilibrium equations of a perfect carbon nanotube reinforced nanocomposite cylindrical panel are [36,37].

The stress – strain relations of the stiffeners is defined as follows ) � �� � σ sx 1 εx E0 0 ¼ E0 α0 ΔT; (11) εy 0 E0 σ sy 1 2v0

with E0 ; α0 are the Young’s modulus and thermal expansion coefficient of the stiffeners, respectively. The force and moment resultants of a carbon nanotube reinforced nanocomposite cylindrical panel reinforced by stiffeners may be calcu­ lated as follows Zh=2

h=2þh Z i

�

σi 1; z; z3 dz þ

ðNi ; Mi ; Pi Þ ¼

Zh=2

�

si

Zh=2

σxy 1; z; z3 dz; ðQi ; Ri Þ ¼ h=2

Zh=2

� di

(14a)

Nxy;x þ Ny;y ¼ 0;

(14b)

� � Ny 3c1 Kx;x þ Ky;y þ c1 Px;xx þ 2Pxy;xy þ Py;yy þ R

þ ​ Nx w;xx þ 2Nxy w;xy þ Ny w;yy þ q dz; ði ¼ x; yÞ; Mx;x þ Mxy;y Mxy;x þ My;y

�

σ iz 1; z2 dz; ði ¼ x; yÞ;

Qx þ 3c1 Kx Qy þ 3c1 Ky

(14c)

k1 w þ k2 r2 w ¼ 0;

� c1 Px;x þ Pxy;y ¼ 0; �

c1 Pxy;x þ Py;y ¼ 0;

(14d) (14e)

in which k1 is Winkler foundation modulus, k2 is the shear layer foun­ dation stiffness of Pasternak model, q is an external pressure uniformly distributed on the surface of the panel. The strain components in Eq. (13) are represented in terms of stress function f and deflection function w as follows

h=2

�

σj 1; z2 dz; ði ¼ x; y; j ¼ xz; yzÞ:

ðQi ; Ri Þ ¼

Nx;x þ Nxy;y ¼ 0;

Qx;x þ Qy;y

h=2

h=2

� Nxy ; Mxy ; Pxy ¼

σsi 1; z; z3

(13)

h=2

(12)

4

N.D. Duc et al.

ε0x ¼

Thin-Walled Structures 146 (2020) 106428

B22 B12 B13 B22 B14 B12 1 B14 B22 B24 B12 1 B15 B22 B16 B12 3 f;yy f;xx kx ky kx Δ Δ Δ Δ Δ

� L11 ðφx Þ ¼ I11 φx ;x þ I12 φx;xxx þ I13 φy ;xyy ; L12 φy ¼ I21 φy ;y þ I22 φy ;xxy þ I23 φy ;yyy ;

B16 B22 B26 B12 3 B22 B12 B27 B12 s B17 B22 s ky þ Φ1 Φ2 Φ1y þ Φ1x ; Δ Δ Δ Δ Δ

ε0y ¼

L13 ðwÞ ¼ I31 w;xx þ I32 w;yy þ I33 w;xxxx þ I34 w;yyyy þ I35 w;xxyy

� 1 L14 ðf Þ ¼ I *11 f;xxxx þ I *12 f;xxyy þ I *13 f;yyyy þ f;xx ; L22 φy ¼ I51 φy ;xy ; R

B11 B12 B11 B14 B13 B12 1 B11 B24 B14 B12 1 B11 B16 B15 B12 3 f;xx f;yy kx ky kx Δ Δ Δ Δ Δ

Pðw; f Þ ¼ f;yy w;xx

B11 B26 B16 B12 3 B12 B11 B17 B12 s B11 B27 s ky Φ1 þ Φ2 Φ1x þ Φ1y ; Δ Δ Δ Δ Δ γ 0xy ¼

1 B32 1 B33 3 f;xy k k ; B31 B31 xy B31 xy

(15)

(16)

B212 :

γ 0xy;xy ¼ w2;xy

w;xx w;yy þ 2w;xy w*;xy

w;xx w*;yy

B22 B11 ; J2 ¼ ; J3 ¼ Δ Δ � B15 B22 B16 B12 J5 ¼ c1 Δ � B11 B16 B15 B12 J6 ¼ c1 Δ

�

1 B31

w;yy w*;xx

� � B12 B16 B22 B26 B12 ; J4 ¼ c1 Δ Δ � B14 B12 B33 B32 þ ; c1 Δ B31 B31 � B13 B12 ; Δ

2

B13 B22 B11 B14

L13 ðw* Þ ¼ I31 w*;xx þ I32 w*;yy ; P1 ðw* ; f Þ ¼ f;yy w*;xx

2f;xy w*;xy þ f;yy w*;yy ;

and the details of Iij ; I*i may be found in Appendix B. Inserting Eq. (15) into Eq. (18), we have J1 f;yyyy þ J2 f;xxxx þ J3 f;xxyy þ J4 φy;yyy þ J5 φx;xyy þ J6 φx;xxx þ J7 φy;xxy þ J8 w;xxxx

w;xx : R (18)

þJ9 w;yyyy þ J10 w;xxyy ¼ w2;xy

with w* ðx; yÞ is imperfection function and denotes initial small imper­

J1 ¼

∂3 f ∂3 f þ I *32 ; 3 ∂y ∂y∂x2

(20)

(17)

f;xy ;

The geometrical compatibility equation for a carbon nanotube reinforced nanocomposite cylindrical panel taking into account the ef­ fects of initial geometric imperfection is written as

ε0x;yy þ ε0y;xx

L33 ðwÞ ¼ I91 w;y þ I92 w;xxy þ I93 w;yyy ; L34 ðf Þ ¼ I *31

L23 ðw* Þ ¼ I61 w*;x þ I62 w;xxx ; L33 ðw* Þ ¼ I91 w*;y ;

and the Airy’s stress function fðx; y; tÞ is defined as Nx ¼ f;yy ; ​ Ny ¼ f;xx ; ​ Nxy ¼

2f;xy w;xy þ f;yy w;yy ; L21 ðφx Þ ¼ I41 φx þ I42 φx;xx þ I43 φx ;yy ;

L23 ðwÞ ¼ I61 w;x þ I62 w;xxx þ I63 w;xyy ; L24 ðf Þ ¼ I *21 f;xyy þ I *22 f;xxx ; � L31 ðφx Þ ¼ I71 φx ;xy ; L32 φy ¼ I81 φy ;yy þ I82 φy ;xx þ I83 φy ;

in which Δ ¼ B11 B22

k1 w þ k2 r2 w;

w;xx w;yy þ 2w;xy w*;xy

w;xx w*;yy

w;yy w*;xx

w;xx ; R (21)

where

B14 B22

B24 B12 Δ

fection of the panel. Replacing Eq. (15) into Eq. (13), then results into Eq. (14c) – (14e), the equilibrium equations are rewritten as follows � L11 ðφx Þ þ L12 φy þ L13 ðwÞ þ L14 ðf Þ þ Pðw; f Þ þ Pðw* ; f Þ þ q ¼ 0; � (19) L21 ðφx Þ þ L22 φy þ L23 ðwÞ þ L*23 ðwÞ þ L24 ðf Þ ¼ 0; � L31 ðφx Þ þ L32 φy þ L33 ðwÞ þ L*33 ðwÞ þ L34 ðf Þ ¼ 0;

� ;

J7 ¼

� B11 B26 c1 B11 B16

J8 ¼ c1 Δ � B11 B26 J10 ¼ c1

where

B16 B12

B11 B24

Δ

Δ

B15 B12

; J9 ¼ c1

B16 B12 Δ

B14 B12

þ c1

B16 B22

B15 B22

c1

� B33 B32 þ ; B31 B31

B26 B12 ; Δ � B16 B12 B33 : c1 2 Δ B31

(22)

Eqs. (19) and (21) are nonlinear equations in terms of variables w and f and used to study the mechanical behaviors of eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panels

5

N.D. Duc et al.

Thin-Walled Structures 146 (2020) 106428

on elastic foundations subjected to thermal and thermo-mechanical loads.

4.1. Nonlinear thermal buckling analysis Consider a simply supported imperfect eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel. The panel is exposed to temperature environments uniformly raised from stress free initial state Ti to final value Tf and temperature increment ΔT ¼ Tf Ti is considered to be independent from the panel thickness. The in-plane condition on immovability at all edges, i.e. u ¼ 0 at x ¼ 0; a and v ¼ 0 at y ¼ 0; b is [20]. Z bZ a Z aZ b ∂u ∂v dxdy ¼ 0; ​ dydx ¼ 0: (29) 0 0 ∂x 0 0 ∂y

4. Analytical solutions The imperfect eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel is assumed to be simply supported in which four edges of the panel are immovable. The boundary conditions are [20]. w ¼ 0; u ¼ 0; φy ¼ 0; Mx ¼ 0; Px ¼ 0; ​ Nx ¼ Nx0 at x ¼ 0; a; w ¼ 0; v ¼ 0; φx ¼ 0; My ¼ 0; Py ¼ 0; ​ Ny ¼ Ny0 at y ¼ 0; b;

(23)

From Eqs. (8) and (15) we can obtain the following expressions in which initial imperfection has been included

where Nx0 ; Ny0 are fictitious compressive edge loads at immovable edges. The boundary conditions can be satisfied if approximate solutions are chosen as [8,20,33]. 2 3 2 3 wðx; yÞ W sinλm x sinδn y 6 φx ðx; yÞ 7 6 Φx cosλm x sinδn y 7 6 7 7 6 (24) 4 φy ðx; yÞ 5 ¼ 4 Φy sinλm x cosδn y 5; * μh sinλm x cosδn y w ðx; yÞ

∂u B22 ¼ f;xx ∂x Δ c1

A1 ¼ WðW þ 2μ A3 ¼

Δ ; A2 ¼ WðW þ 2μ 32B11

λ2 hÞ m2 δn

Δ ; 32B22

H2 H3 H4 Φx þ Φy þ W; H1 H1 H1

c1

(25)

H1 ¼ J1 δ4n þ J2 λ4m þ J3 λ2m δ2n ; H2 ¼

H3 ¼

6 4

þJ4 δ3n

3

2

7 5; H4 ¼

6 6 4

B16 B22

B26 B12 Δ

B27 B12 s B17 B22 s Φ1y þ Φ1x Δ Δ

c1 w;yy þ φy;y

� �2 1 ∂w 2 ∂x

�

∂w ∂w* ; ∂x ∂x

B15 B12 Δ

ðw;xx þ φx;x Þ

c1

B11 B26

B16 B12 Δ

B17 B12 s B11 B27 s w Φ1x þ Φ1y þ Ry Δ Δ

�

w;yy þ φy;y

∂w ∂y

�2

�

∂w ∂w* : ∂y ∂y (30)

Introduction of Eqs. (24) and (25) into Eq. (30) and then the results into Eq. (29) give fictitious edge compressive loads as. Nxo ¼ m1 W þm2 Φx þm3 Φy þm4 WðW þ2μhÞþm5 Φ1 þm6 Φ2 þm7 Φs1x þm8 Φs1y ;

(26)

Nyo ¼ m*1 W þm*2 Φx þm*3 Φy þm*4 WðW þ2μhÞþm*5 Φ1 þm*6 Φ2 þm*7 Φs1x þm*8 Φs1y : (31)

From the last two equations of Eq. (28), the amplitude of the rota­ tions of normals to the midsurface may be expressed into the amplitude of the deflection of the panel as

3

2

J7 δn λ2m

B11 B16

ðw;xx þ φx;x Þ

B12 B11 B14 B13 B12 B11 B24 B14 B12 f;xx þ φx;x þ φy;y Δ Δ Δ

B12 B11 Φ1 þ Φ2 Δ Δ

and

2

B12 Φ2 Δ

∂v B11 ¼ f;yy ∂y Δ

with δ2 hÞ 2n λm

B16 B12 Δ

B22 þ Φ1 Δ

in which λm ¼ mπ=aδn ¼ nπ=b, μ is imperfection parameter of the panel (0 � μ � 1) and m; n are odd natural numbers representing the number of half waves in the x and ydirections, respectively. Inserting Eq. (24) into the compatibility equation (21) and solving obtained equation for unknown f yields 1 1 f ¼ A1 cos 2λm x þ A2 cos 2δn y þ A3 sinλm x sinδn y þ Nxo y2 þ Nyo x2 ; 2 2

B15 B22

B12 B13 B22 B14 B12 B14 B22 B14 B12 f;yy þ φx;x þ φy;y Δ Δ Δ

6 4

J5 λm δ2n

7 5;

þJ6 λ3m 3

J8 α4 þ J9 δ4n þJ10 λ2m δ2n

λ2m R

(27)

7 7: 5

Setting Eqs. 24 and 25 into Eq. (19) and applying the Galerkin method for the resulting equation, we obtain equations to investigate the mechanical behaviors of eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel on elastic foundations in thermal environments as � � l11 W þ l12 Φx þ l13 Φy þ l14 W þ μh Φx þ l15 W þ μh Φy �� � þ n1 Ny0 δ2n þ Nxo λ2m ðW þ μhÞ þn2 WðW þ μhÞ þ n3 WðW þ 2μhÞ þ n4 WðW þ μhÞðW þ 2μhÞ þ n5 q þ n5

Ny0 ¼ 0; R

l21 W þ l22 Φx þ l23 Φy þ n6 ðW þ μhÞ þ n7 WðW þ 2μhÞ ¼ 0; l31 W þ l32 Φx þ l33 Φy þ n8 ðW þ μhÞ þ n9 WðW þ 2μhÞ ¼ 0;

(28) Fig. 2. Comparisons of the thermal buckling of the un-stiffened carbon nano­ tube reinforced nanocomposite cylindrical panel subjected to uniform temper­ ature rise.

with the details of coefficients lij ðj ¼ 1 � 3; k ¼ 1 � 5Þ; nq ðq ¼ 1 � 9Þ are shown in Appendix C. 6

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Thin-Walled Structures 146 (2020) 106428

Φy ¼

ðl22 l31 ðl32 l23

l32 l21 Þ ðl22 n8 Wþ ðl32 l23 l22 l33 Þ

l32 n6 Þ ðl22 n9 ðW þ μhÞ þ ðl32 l23 l22 l33 Þ

l32 n7 Þ WðW þ 2μhÞ; l22 l33 Þ

Φx ¼

ðl23 l31 ðl33 l22

l33 l21 Þ ðl23 n8 Wþ ðl33 l22 l23 l32 Þ

l33 n6 Þ ðl23 n9 ðW þ μhÞ þ ðl33 l22 l23 l32 Þ

l33 n7 Þ WðW þ 2μhÞ: l23 l32 Þ (32)

Imposing Eqs. (31) and (32) into Eq. (28), the relationship between the uniform temperature rise and the amplitude of the deflection of the eccentrically stiffened carbon nanotube reinforced nanocomposite cy­ lindrical panel is determined as follows ΔT ¼ a11 Wn þ a12 ðWn þ μÞ þ a13 Wn ðWn þ 2μÞ

(33)

2

þa14 Wn ðWn þ μÞ þ a15 ðWn þ μÞ þ a16 Wn ðWn þ μÞðWn þ 2μÞ;

in which the details of coefficients a1i ð i ¼ 1 � 6Þ are given in Appendix D.

Fig. 3. Comparisons of the nonlinear response of the un-stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to combination of uniform temperature rise and axial compressive loads.

4.2. Nonlinear buckling thermo-mechanical analysis A simply supported imperfect eccentrically stiffened carbon nano­ tube reinforced nanocomposite cylindrical panel on elastic foundations with all immovable edges is considered. The panel is subjected to axial compressive loads Fx (Pascals) uniformly distributed at two curved edges x ¼ 0; a and simultaneously exposed to temperature environ­ ments. In this case, Nx0 ¼ Fx h and the in-plane condition on immov­ ability at edges y ¼ 0; b is Z aZ b ∂v dxdy ¼ 0: (34) 0 0 ∂y The fictitious compressive edge loads at immovable edges y ¼ 0; b are determined by setting Eq. (30) into Eq. (34) as. Nyo ¼ B12 þ Φ1 B11

Fig. 4. Effects of the stiffeners on the buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel sub­ jected to uniform temperature rise.

Δ I21 φx B11 Φ2 þ

Δ I22 φy B11 B17 B12 s Φ B11 1x

Δ Δ δ2n I23 W þ WðW þ 2μhÞ B11 B11 8

B12 Fx h B11

B27 Φs1y :

Subsequently, introduction of Eq. (35) into Eq. (28) gives. � � � Fx ¼ a21 Wn þa22 Wn þ μ þ a23 Wn Wn þ 2μ þ a24 Wn ðWn þ μ 2

þa25 ðWn þ μÞ þa26 Wn ðWn þ μÞðWn þ 2μÞþa27 ;

(35)

(36)

Fig. 5. Effects of the stiffeners on the buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel sub­ jected to combination of uniform temperature rise and axial compressive loads. Fig. 6. Effects of the Winkler foundation k1 on the buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to uniform temperature rise. 7

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Thin-Walled Structures 146 (2020) 106428

Fig. 7. Effects of the Pasternak foundation k2 on the buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to uniform temperature rise.

Fig. 9. Effects of the Pasternak foundation k2 on the buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to the combination of uniform temperature rise and axial compressive loads.

Fig. 8. Effects of the Winkler foundation k1 on the buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to the combination of uniform temperature rise and axial compressive loads.

Fig. 10. Effects of temperature increment ΔT on the buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to the combination of uniform temperature rise and axial compressive loads.

where the details of coefficients a2i ð i ¼ 1 � 6Þ may be found in Appendix E. Eq. (36) is basic equation which is used to investigate the relation of axial compressive loads – deflection amplitude of imperfect eccentri­ cally stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to the combination of uniformly raised temperature field and uniform axial compressive loads.

based on a higher order shear deformation theory with different values of elastic foundations stiffness k1 and k2 . The input data are chosen as a= b ¼ 1:2; b=h ¼ 100; a=R ¼ 0:5; V *CNT ¼ 0:28. As can be seen, the re­ sults in this paper are so close with the ones obtained in work of Shen and Xiang [17]. Next, Fig. 3 compares the nonlinear response of an un-stiffened carbon nanotube reinforced nanocomposite cylindrical panel in case of type X of carbon nanotube reinforcements subjected to the combi­ nation of uniform temperature rise and axial compressive loads with a= b ¼ 0:96; a=R ¼ 0:5; b=h ¼ 20; ΔT ¼ 400 K with the numerical results given by Shen and Xiang [16] based on a higher-order shear deformation �rma �n-type of kinematic nonlinearity. It is also theory with a von Ka found that there is a very close agreement observed between the present results and existing predictions.

5. Numerical results and discussion 5.1. Validation To validate the accuracy of present study, Fig. 2 shows the com­ parisons of the load – deflection curves of an un-stiffened carbon nanotube reinforced nanocomposite cylindrical panel with UD carbon nanotube reinforcements subjected to uniform temperature rise between the results in this paper with the ones presented by Shen and Xiang [17] 8

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Thin-Walled Structures 146 (2020) 106428

Fig. 11. Effects of the carbon nanotube volume fraction V *CNT on the buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nano­ composite cylindrical panel subjected to uniform temperature rise. Fig. 14. Effects of ratio a=R on the buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel sub­ jected to combination of uniform temperature rise and axial compressive loads.

Fig. 12. Effects of the carbon nanotube volume fraction V *CNT on the buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nano­ composite cylindrical panel subjected to combination of uniform temperature rise and axial compressive loads. Fig. 15. The buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to uniform temperature rise with different types of carbon nanotube reinforcements.

5.2. Load – deflection amplitude relations 5.2.1. Effect of stiffeners The influences of stiffeners on the mechanical behaviors of the car­ bon nanotube reinforced nanocomposite cylindrical panel subjected to uniform temperature rise and the combination of uniform temperature rise and axial compressive loads are described in Figs. 4 and 5, respec­ tively. The results show that the load – carrying capacity of the eccen­ trically stiffened panel is higher than one of un-stiffened panel. In other words, the stiffeners can enhance the load – carrying capacity for the carbon nanotube reinforced nanocomposite cylindrical panel. Figs. 4 and 5 also analyze the effect of initial imperfection with coefficient μ on the mechanical behaviors of the carbon nanotube reinforced nano­ composite cylindrical panel. In case of small deflection amplitude, initial imperfection has a negative influence on the load – carrying capacity of the panel. Specifically, the load – carrying capacity of the panel will increases when the coefficient μ increases. However, an opposite trend will occur when the deflection amplitude is high enough. When the

Fig. 13. Effects of ratio a=b on the buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel sub­ jected to uniform temperature rise. 9

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Thin-Walled Structures 146 (2020) 106428

5.2.5. Effect of geometrical parameters Figs. 13 and 14 indicate the effects of geometrical parameter a=b on the mechanical behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to uniform tem­ perature rise and ratio a=R on the mechanical behaviors of the eccen­ trically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to the combination of uniform temperature rise and axial compressive loads, respectively. It can be seen that the load – carrying capacity of the eccentrically stiffened carbon nanotube rein­ forced nanocomposite cylindrical panel increases when increasing the ratios a=b and a=R: The effects of initial imperfection on the mechanical behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel also shown in Figs. 13 and 14. 5.2.6. Effect of types of carbon nanotube reinforcements Figs. 15 and 16 compare the mechanical behaviors of the eccentri­ cally stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to uniform temperature rise and the combination of uniform temperature rise and axial compressive loads with types V and O of carbon nanotube reinforcements. It is easy to see that the load – carrying capacity of the panel with types V of carbon nanotube rein­ forcement is higher than the one of the panel with types O of carbon nanotube reinforcement.

Fig. 16. The buckling behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to the combination of uniform temperature rise and axial compressive loads with different types of carbon nanotube reinforcements.

6. Concluding remarks

deflection is high enough, the load – carrying capacity of the panel de­ creases with increase of factor coefficient μ.

This paper presents analytical solutions for the nonlinear buckling of the imperfect eccentrically stiffened carbon nanotube reinforced nano­ composite cylindrical panel on elastic foundations subjected to the combination of uniform temperature rise and axial compressive loads. Elastic modules of the polymer matrix and carbon nanotubes are assumed to be temperature dependent. Fundamental equations are derived based on the Airy’s stress function and Reddy’s third order shear deformation shell theory taking into account the effects of initial geo­ metric imperfection, von Karman-Donnell nonlinear terms and the thermal stress in both the panels and the stiffeners. The approximate analytical solutions for simply supported boundary conditions are sug­ gested and the Galerkin’s procedure is applied to determine the explicit relations between load and deflection amplitude. Some conclusions can be obtained from the numerical analysis:

5.2.2. Effect of elastic foundations Figs. 6–9 illustrate the effect of elastic foundations with two co­ efficients k1 ; k2 on the mechanical behaviors of the eccentrically stiff­ ened carbon nanotube reinforced nanocomposite cylindrical panel subjected to uniform temperature rise and the combination of uniform temperature rise and axial compressive loads, respectively. Obviously, the load – carrying capacity of the eccentrically stiffened carbon nano­ tube reinforced nanocomposite cylindrical panel increases when two coefficients k1 ; k2 of elastic foundations increase. The results from these figures also show that the beneficial effect of Pasternalk foundation with coefficients k2 is better than Winkler foundation with modulus k1 . Furthermore, the initial imperfection considerably impact on the me­ chanical behaviors of the eccentrically stiffened carbon nanotube rein­ forced nanocomposite cylindrical panel.

(i) Carbon nanotube and stiffeners enhance the stiffness of the car­ bon nanotube reinforced nanocomposite cylindrical panel. (ii) The temperature increment strongly negative effect on the load – carrying capacity of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel. (iii) Elastic foundations have beneficial effect on the load-carrying capacity of carbon nanotube reinforced nanocomposite cylindri­ cal panel. Furthermore, the influence of Pasternalk foundation is stronger than Winkler foundation. (iv) The nonlinear buckling of the carbon nanotube reinforced nanocomposite cylindrical panel is very sensitive with the change of initial imperfection. The effect of initial imperfection on the nonlinear buckling of the panel in case of small deflection amplitude is opposite of the effect of initial imperfection on the nonlinear buckling of the panel in case of large deflection amplitude. (v) The load-carrying capacity of carbon nanotube reinforced nano­ composite cylindrical panel with type V is higher than the loadcarrying capacity of carbon nanotube reinforced nanocomposite cylindrical panel with type O. (vi) The geometrical parameters have significant influences on the nonlinear buckling of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel.

5.2.3. Effect of temperature increment The influence of temperature increment and initial imperfection on the mechanical behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to the combina­ tion of uniform temperature rise and axial compressive loads is considered in Fig. 10. As expected, an increase of temperature increment leads to a decrease of the load – carrying capacity of the cylindrical panel. Moreover, from these figures, the mechanical behaviors of the cylindrical panel is very sensitive with the change of initial imperfection. 5.2.4. Effect of the carbon nanotube volume fraction Figs. 11 and 12 show the effects of the carbon nanotube volume fraction V *CNT on the mechanical behaviors of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel subjected to uniform temperature rise and the combination of uniform tempera­ ture rise and axial compressive loads, respectively. As can be observed, the higher the carbon nanotube volume fraction V *CNT is, the higher the load – carrying capacity of the eccentrically stiffened carbon nanotube reinforced nanocomposite cylindrical panel is.

The numerical results are also compared with other analytical results 10

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Thin-Walled Structures 146 (2020) 106428

to validate the accuracy of the present approach and theory.

Acknowledgement

Declaration of competing interest

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2018.04. The authors are grateful for this support.

The authors declare no conflict of interest.

Appendix A Z

h=2

B11 ¼

Q11 dz þ h=2

Z

E0 AT1 ; sT1

h=2

B12 ¼

Q12 dz; h=2

Z

h=2

B13 ¼

Q11 zdz þ h=2

Z

E0 AT1 zT1 ; sT1

h=2

B14 ¼

Q12 zdz; h=2

Z

h=2

Q11 z3 dz þ

B15 ¼ h=2

Z

E0 AT1 zT1 sT1

�3 þ

�3 dT1 hT1 E0 zT1 ; 4sT1

þ

�3 dT2 hT2 E0 zT2 ; 4sT2

þ

�3 E0 hT1 dT1 ; 12sT1

þ

�3 �2 �5 E0 dT1 hT1 zT1 E0 dT1 hT1 þ ; T T 2s1 80s1

h=2

Q12 z3 dz;

B16 ¼ h=2

B17 ¼

1 dT1 ; 1 2ν0 sT1 Z

h=2

B22 ¼

Q22 dz þ h=2

Z

E0 AT2 ; sT2

h=2

B24 ¼

Q22 zdz þ h=2

Z

E0 AT2 zT2 ; sT2

h=2

Q22 z3 dz þ

B26 ¼ h=2

B27 ¼

1

1 dT2 ; 2ν0 sT2

Z

h=2

B31 ¼

E0 AT2 zT2 sT2

�3

Q66 dz; h=2

Z

h=2

B32 ¼

Q66 zdz; h=2

Z

h=2

Q12 z3 dz;

B33 ¼ h=2

Z

h=2

Q11 z2 dz þ

B43 ¼ h=2

Z

E0 AT1 zT1 sT1

�2

h=2

Q12 z2 dz;

B44 ¼ h=2

Z

h=2

Q11 z4 dz þ

B45 ¼ h=2

Z

E0 AT1 zT1 sT1

�4

h=2

Q12 z4 dz;

B46 ¼ h=2

11

N.D. Duc et al.

Z

Thin-Walled Structures 146 (2020) 106428

h=2

Q22 z2 dz þ

B54 ¼ h=2

Z

E0 AT2 zT2 sT2

�2 þ

�3 Z E0 hT2 dT2 ; B ¼ 56 12sT2

h=2

Q22 z4 dz þ h=2

E0 AT2 zT2 sT2

�4 þ

�3 �2 �5 E0 dT2 hT2 zT2 E0 dT2 hT2 þ ; 2sT2 80sT2

h=2

Q66 z2 dz;

B62 ¼ h=2

Z

h=2

Q66 z4 dz;

B63 ¼ h=2

�2 �3 E0 dT1 zT1 hT1 2 h=2 h=2 � � � T �3 � �5 Z h=2 Z T T 5 T T 7 T T 4 T T 2 � E0 d1 h1 E0 d1 h1 15E0 d1 z1 h1 15E0 d1 z1 hT1 T T 6 6 þ Q11 z dz þ E0 A1 z1 þ ; B75 ¼ þ þ ; B76 ¼ 80 448 12 80 h=2 Z

h=2

Q11 z3 dz þ E0 AT1 zT1

B71 ¼

Z

h=2

B82 ¼ h=2

Z

Q22 z3 dz þ E0 AT2 zT2

�3

�3

þ

þ

�3 Z E0 zT1 dT1 hT1 ; B73 ¼ 4

�3 Z E0 zT2 dT2 hT2 ; B84 ¼ 4

h=2

Q11 z4 dz þ E0 AT1 zT1

h=2 h=2

Q22 z4 dz þ E0 AT2 zT2

�4

�4

þ

þ

�2 �3 �5 E0 dT2 zT2 hT2 E0 dT2 hT2 þ ; 2 80

h=2

Q66 z6 dz;

B93 ¼ h=2

Z

h=2

B86 ¼ h=2

Q22 z6 dz þ E0 AT2 zT2

Zh=2 B94 ¼ h=2

E0 dT2 hT2 448

�7

15E0 dT2 zT2 12

þ

h=2 h1

þ

15E0 dT2 zT2 80

�2

hT2

�5 ;

Zh=2 Q55 z2 dz; h=2

Zh=2 Q44 z2 dz; B99 ¼

Zh=2 Q44 z4 dz; B101 ¼

h=2

Zh=2

�3

h=2

Zh=2

1 ¼ 1 2ν0

hT2

Q55 dz; B100 ¼

h=2

h=2

�4

Zh=2 Q44 z2 dz; B96 ¼

Q55 z2 dz; B98 ¼

B97 ¼

þ

Zh=2 Q44 dz; B95 ¼

Zh=2

Φs1x

�6

Q55 z4 dz;

h=2

dT 1 E0 α0 ΔT Tx dz; Φs1y ¼ 1 2ν0 sx

h=2

Zh=2 E0 α0 ΔT

dTy

h=2 h1

sTy

dz;

Φs1x ¼ E0 α0 h1 ΔT; Φs1y ¼ E0 α0 h2 ΔT; Φ1 ¼ ðQ11 α11 þ Q12 α22 ÞhΔT; Φ2 ¼ ðQ12 α11 þ Q22 α22 ÞhΔT;

Appendix B

I11 ¼ ðB94

3c1 B95 Þ

� B11 B16 þB16 c21

B15 B12 Δ

� B15 B22 I13 ¼ B16 c21

� B15 B22 3c1 B99 Þ; I12 ¼ B71 c21

3c1 ðB98 c1

B16 B12 Δ

B11 B14

B13 B12

B13 B22

Δ

c1

B13 B22

B14 B12 Δ

þ c1 B73

B14 B12

c21 B75 ;

�

Δ � B33 B33 c1 c1 B31 6 c21 B76 þ 26 4 2

� B11 B16 þB82 c21

B15 B12 Δ

c1

�

�

Δ c1

B16 B12

B11 B14

B13 B12 Δ

� þ c1 B46

þB63 c1

c21 B93

12

� B32 3 B31 7 7; 5

h=2

Q12 z6 dz; h=2

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I21 ¼ ðB96

Thin-Walled Structures 146 (2020) 106428

3c1 B97 Þ

� B11 B26 þB16 c21

3c1 ðB100

B16 B12

c1

Δ

� � B33 þ2 B33 c1 c1 B31

� B16 B22 3c1 B101 Þ; I22 ¼ B71 c21

B32 B31

B11 B24

B14 B12

c1

B14 B22

B24 B12 Δ

� þ c1 B46

Δ

�

B26 B12 Δ

�

c21 B76

� c21 B93 ;

þ B63 c1

0

� B16 B22 I23 ¼ B16 c21

B26 B12

I32 ¼ I31 ; I33 ¼ B71 c21 I34 ¼ B16 c21 I35 ¼ B71 c21 þB16 c21

B16 B22

B15 B22

B26 B12 B26 B12

B16 B12 Δ

B16 B11

I *12 ¼ c1

B71 B22

B16 B12

� B11 B26 þB14 c1

B16 B12

� c21 B93 ;

B13 B22

B14 B12

B13 B22

þ B62 B71 B22

B16 B12

B26 B12 Δ

B24 B12

7 7 7; 5

� c1 B75

þ B73

B32 B31

� þ B63

� c1 B93 ;

c1

B16 B11

B71 B12 Δ

;

�

Δ B11 B24

B14 B12

�

Δ B26 B12

B14 B22

Δ

B15 B22

B16 B12 Δ B26 B12 Δ B26 B12 Δ

�

Δ

B32 B33 B14 B11 B13 B12 þ c1 ; I *22 ¼ B31 B31 Δ

B14 B22

B13 B12

3

� � B33 c1 B33 c1 B31

Δ

� B33 c1 B76 þ B33 c1 B31

� B16 B22 c1 B71 c1

Δ

�

B13 B12

c1 B63

B11 B14

c1 B45

þ B43

B14 B12

B11 B14

Δ

B16 B22

B15 B12

�

Δ

B16 B12

� B15 B22 c1 B71 c1

B33 ; B31

� B11 B16 3c1 B95 Þ; I42 ¼ B14 c1

B15 B12

c1

I61 ¼ I41 ; I62 ¼ B41 c1

2c1

Δ

Δ

B32 B31

� B16 B22 B71 c1

I63 ¼ B13 c1

B82 B11

Δ �

B14 B12

þB46

c21 B86 ;

� 2 B 2c21 B76 þ 4 c21 33 B31

B15 B12 Δ

B16 B12

B11 B16

� B16 B22 I51 ¼ B13 c1

6 c1 6 4

c21 B75 ;

B16 B12

Δ

Δ

2

c21 B86 ;

Δ

B11 B16

Δ

� B33 I43 ¼ B32 c1 B31 B13 B22

B15 B12 Δ

B16 B12

B11 B26

Δ

�

B11 B16

Δ

þ B16 c21

ðB94

B16 B12

B15 B22

þB16 c1

I *21

B11 B26

þ B82 c21

þ c1

3c1 B99 Þ

� B15 B22 þB13 c1 B71 c1

þ B16 c21

1 B16 B12 2 B11 B26 B c1 C Δ B C C þ c1 B84 þ B82 B B C @ B11 B24 B14 B12 A c1 Δ

B71 B12 ∂4 f * B16 B22 B82 B12 ∂4 f 1 ∂2 f ; I ¼ c1 þ ; 4 13 Δ Δ ∂x ∂y4 R ∂x2

I41 ¼ 3c1 ðB98

2

B24 B12 Δ

B16 B12

þ B82 c21

Δ

�

B14 B22

Δ

Δ

I *11 ¼ c1

6 6 c1 6 4

B15 B22

Δ B16 B22

c1

Δ

�

B32 B31

B16 B12 Δ

þ B16 c1

B24 B12 Δ � þ B63

�

� B11 B26 þ B16 c1

B11 B16

B15 B12 Δ

B11 B26

B16 B12

B32 B31

� þ B62

B11 B24

Δ

c1 B93

c1 B45 þ B14 c1

B11 B16

B15 B12 Δ

� c1 B75 ;

� � B32 B33 c1 B63 þ 2 c1 Δ B31 �� � 2 B16 B12 B ; c1 B76 þ 2 c1 33 c1 B93 Δ B31

c1 B46 þ B14 c1 þ B16 c1

� B33 c1 B46 þ B32 c1 B31

þ B44

B11 B26

B16 B12

13

c1 B63

� B14 B12 3 Δ 7 7; 5

N.D. Duc et al.

Thin-Walled Structures 146 (2020) 106428

� B15 B22 I71 ¼ B14 c1

�

� B11 B16 þ B24 c1

�

� B11 B26 þ B24 c1

B16 B12 B13 B22 B14 B12 B15 B12 B11 B14 B13 B12 Δ Δ Δ Δ � � � � � � B33 B32 B33 B32 þ B62 c1 B63 c1 B33 c1 þ B63 c1 B93 þB44 c1 B46 þ B32 c1 B31 B31 B31 B31 2 � B15 B22 B16 B12 B13 B22 B14 B12 � 3 B16 c1 Δ Δ 6 7 6 7 c1 6 7; � � 4 5 B11 B16 B15 B12 B11 B14 B13 B12 þB82 c1 þ B46 c1 B76 Δ Δ

� B16 B22 I81 ¼ B14 c1 �

2 6 6 c1 6 4

B16 c1

B26 B12

B14 B22

B16 B12

B32 B31

B15 B22

B16 c1

þ B62

c1 B63

3c1 B101 Þ

ðB96

B16 B12

B15 B22

�

B14 B12

B16 B12 Δ

þ B24 c1

Δ 3

B14 B12

�

Δ

c1 B56 ;

B32 B31

� þ B63

� c1 B93 ;

3c1 B97 Þ; B15 B12 Δ

B11 B16 Δ

�

� 2 B c1 B76 þ 2 c1 33 B31

B11 B24

c1 B86

þ B84

� � B33 c1 B33 c1 B31

B11 B16

þ B82 c1

B16 B12

7 7 7 þ B54 5

�

Δ

Δ

0

B11 B24

Δ �

I91 ¼ I83 ¼ 3c1 ðB100

B24 B12 Δ

B11 B26

� B33 I82 ¼ B32 c1 B31

B24 B12 Δ

Δ

þB82 c1

B B c1 B B @

B14 B22

Δ

B16 B22

�

I92 ¼ B14 c1

B26 B12

�

� B32 B33 c1 B46 þ 2 c1 B31

� c1 B63

B15 B12 1 C C C; C A

c1 B93 0

I93 ¼ B14 c1

I *31 ¼ c1

B16 B22

B26 B12 Δ

B14 B22

B24 B12 Δ

B82 B11

B16 B12 Δ

c1

þ B24 c1

B11 B26

B16 B22

B16 B12 Δ

B82 B12 Δ

; I *32 ¼

c1 B56

B24 B11

1

B16 B22 B26 B12 B B16 c1 Δ B c1 B B @ B11 B26 B16 B12 þB82 c1 Δ

C C C; C A c1 B86

B14 B12 Δ

B32 B33 þ c1 ; B31 B31

Appendix C � � � ab H4 I11 λm 4 þ I34 δm 4 k1 k2 λ2 þ δm 2 þ I35 λn 2 δm 2 þ X1 ; 4 H1 � � ab H2 8 H2 ab 8 ab H3 2 2 l12 ¼ λm 2 δ2 n ; l15 ¼ λ mδ n; ; l14 ¼ I11 λm þ I12 λn 3 þ I13 λn δm 2 þ X1 4 3 H1 mnπ2 3 mnπ2 H1 H1 � � � � ab 1 2 H3 l13 ¼ I21 δm þ I22 δn λm 2 þ I23 δn 3 þ I *11 λ4 þ I *12 λm 2 δn 2 þ I *13 δm 4 λm ; 4 R H1

l11 ¼

0 l21 ¼

l22 ¼ l23 ¼ l33 ¼

� � ab H4 H4 I62 λ3m þ I63 λm δ2n þ I *21 λm δ2n þ I *22 λm 3 ; l31 ¼ 4 H1 H1

ab B B 4 @

I92 λ2m δn þ I93 δn 3 þ

1

C C; H4 H4 A I *31 δ3n þ I *32 λm 2 δn H1 H1

� � ab H2 H2 I42 λm 2 þ I43 δn 2 I41 þ I *21 λm δn 2 þ I *22 λm 3 ; 4 H1 H1 � � � � ab H3 H3 ab H2 H2 I51 δn λm þ I *21 λm δn 2 þ I *22 λm 3 I92 λn δm þ I *31 δn 3 þ I *32 λm 2 δn ; l32 ¼ ; 4 4 H1 H1 H1 H1 � � ab H3 H3 I81 Φy δm 2 þ I82 Φy λm 2 I83 Φy þ I *31 δn 3 þ I *32 λm 2 δn ; 4 H1 H1

14

N.D. Duc et al.

Thin-Walled Structures 146 (2020) 106428

� � � ab 8 H4 ab 2 2 ab Δ 4 ab Δ 4 I11 λ2m I21 δ2n ; n2 ¼ λ δ ; n4 ¼ λm þ δn ; 2 m n 4 3 H1 mnπ 64 B22 64 B11 � � � � 1 Δ ab 2 1 2 Δ ab * 2 2 n3 ¼ þ δ 4I *11 λ2m I λ δ ; 6 B11 mnπ2 n R 3 B22 mnπ2 12 m n

n1 ¼

4ab ab 2λ2 Δ ab 2δ2 Δ * ; n6 ¼ I61 λm ; n7 ¼ m I *22 ; n8 ¼ I83 δn ; n9 ¼ n I ; 2 4 4 3δn B11 3λm B22 31 mnπ

n5 ¼

Appendix D C1 C2 ; a12 ¼ ; G3 G3 þ ðG1 þ G2 ÞðWn þ μÞ þ ðG1 þ G2 ÞðWn þ μÞ h h

a11 ¼

C3 C4 ; a14 ¼ ; G3 1 G3 1 ðG ðG þ þ G ÞðW þ μ Þ þ þ G ÞðW þ μ Þ 1 2 n 1 2 n h2 h h2 h

a13 ¼

C5 Wn ðWn þ μÞðWn þ 2μÞ ; a16 ¼ C6 ; G3 1 G3 1 þ ðG1 þ G2 ÞðWn þ μÞ þ 2 ðG1 þ G2 ÞðWn þ μÞ 2 3 h h h h

a15 ¼

� G1 ¼ δ2n m*5 ððQ11 α11 þ Q12 α22 ÞhÞ þ m*6 ððQ12 α11 þ Q22 α22 ÞhÞ þ m*7 E0 α0 h1 þ m*8 E0 α0 h2 ; G2 ¼ λ2m ðm5 ððQ11 α11 þ Q12 α22 ÞhÞ þ m6 ððQ12 α11 þ Q22 α22 ÞhÞ þ m7 E0 α0 h1 þ m8 E0 α0 h2 Þ; � n5 * m ððQ11 α11 þ Q12 α22 ÞhÞ þ m*6 ððQ12 α11 þ Q22 α22 ÞhÞ þ m*7 E0 α0 h1 þ m*8 E0 α0 h2 ; R 5 � �� n5 m*1 � � n5 m*2 � ðl23 l31 l33 l21 Þ � n5 m*3 � ðl22 l31 l32 l21 Þ C1 ¼ l11 þ þ l12 þ þ l13 þ ; ðl33 l22 l23 l32 Þ ðl32 l23 l22 l33 Þ R R R

G3 ¼

�� C2 ¼

l12 þ

n5 m*2 � ðl23 n8 R ðl33 l22

l33 n6 Þ � n5 m*3 � ðl22 n8 þ l13 þ ðl32 l23 l23 l32 Þ R

l12 þ

n5 m*2 � ðl23 n9 R ðl33 l22

l33 n7 Þ � n5 m*3 � ðl22 n9 þ l13 þ ðl32 l23 l23 l32 Þ R

l14

δ2n m*2

�� C3 ¼ 2 6 C4 ¼ 6 4

þ n2

δ2n m*1

λ2m m2

λ2m m1

� C5 ¼

l14

δ2n m*2

λ2m m2

l14

δ2n m*2

λ2m m2

2 6 C6 ¼ 4

þ n4

δ2n m*4

� ðl23 l31 ðl33 l22

� l32 n6 Þ þ n1 ; l22 l33 Þ �� � l32 n7 Þ n5 m*4 ; þ n3 þ l22 l33 Þ Ry

l33 l21 Þ þ l15 l23 l32 Þ

δ2n m*3

l33 n6 Þ þ l15 l23 l32 Þ

δ2n m*3

λ2m m3

l33 n7 Þ þ l15 l23 l32 Þ

δ2n m*3

λ2m m3

λ2m m3

� ðl22 l31 ðl32 l23

l32 l21 Þ 3 l22 l33 Þ 7 7; 5

�

� ðl23 n8 ðl33 l22

� ðl23 n9 ðl33 l22 � 2 λm m4

ðl32 l23

� l32 n6 Þ ; l22 l33 Þ

� ðl22 n9 ðl32 l23

l32 n7 Þ 3 l22 l33 Þ 7 5:

� ðl22 n8

15

N.D. Duc et al.

Thin-Walled Structures 146 (2020) 106428

Appendix E � l11 þ l12

a21

a22

a23

a24

� � ðl23 l31 l33 l21 Þ Δ ðl22 l31 l32 l21 Þ n5 Δ I22 I23 þ l13 n5 ðl33 l22 l23 l32 Þ ðl l l22 l33 Þ RB11 Ry B11 � � 32 23 ¼ ; B12 B12 n5 ðWn þ μÞh λ2m δ2n Ry B11 B11 � � � � 0 Δ ðl23 n8 l33 n6 Þ Δ ðl22 n8 l32 n6 Þ 1 l12 n5 I21 I22 þ l13 n5 Ry B11 ðl33 l22 l23 l32 Þ Ry B11 ðl32 l23 l22 l33 Þ C B B C B C B �� C � � @ A B12 B17 B12 s Φ1 Φ2 þ Φ1x B27 Φs1y δ2n n1 B11 B11 � � � � ¼ ; B12 B12 n5 Wn þ μ h λ2m δ2n RB11 B11 �� � � � � Δ ðl23 n9 l33 n7 Þ Δ ðl22 n9 l32 n7 Þ n5 Δ δ2n l12 n5 I21 I22 þ l13 n5 þ n3 þ Ry B11 ðl33 l22 l23 l32 Þ RB ðl32 l23 l22 l33 Þ RB11 8 � � 11 ¼ h; B12 2 2 B12 n5 ðWn þ μÞh λm δn RB11 B11 �� � � � � Δ ðl23 l31 l33 l21 Þ Δ ðl22 l31 l32 l21 Þ Δ I23 δ2n l14 þ I21 δ2n þ l15 þ I22 δ2n þ n2 þ B11 ðl33 l22 l23 l32 Þ B ðl32 l23 l22 l33 Þ B11 �11 � h; ¼ B12 2 2 B12 n5 ðW þ μÞh λm δn RB11 B11 n5

Δ I21 RB11

�

� � � l33 n6 Þ Δ ðl22 n8 l32 n6 Þ þ l15 þ I22 δ2n B11 ðl l l23 l32 Þ l22 l33 Þ � 32 23 � h; 2 2 B12 ðWn þ μÞh λm δn B11 � � � l33 n7 Þ Δ ðl22 n9 l32 n7 Þ Δ δ2n 2 þ l15 þ I22 δ2n þ n4 δn B11 ðl32 l23 l22 l33 Þ B11 8 l23 l32 Þ � � h2 ; B12 2 2 B12 n5 ðWn þ μÞh λm δn RB11 B11 � B17 B12 s n5 � � ��; Φ2 þ Φ B27 Φs1y B12 B12 B11 1x R n5 h ðWn þ μÞh2 λ2m δ2n RB11 B11

�� � Δ ðl23 n8 l14 þ I21 δ2n B11 ðl33 l22 2 a5 ¼ B12 n5 RB11 �� � Δ ðl23 n9 l14 þ I21 δ2n B11 ðl33 l22 2 a6 ¼

a27 ¼

� B12 Φ1 B11

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