Postbuckling analysis of axially-loaded functionally graded GPL-reinforced composite conical shells

Thin–Walled Structures 148 (2020) 106594

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Postbuckling analysis of axially-loaded functionally graded GPL-reinforced composite conical shells R. Ansari *, J. Torabi , E. Hasrati Faculty of Mechanical Engineering, University of Guilan, P.O Box 3756, Rasht, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: FG-GPLRC Conical shell Postbuckling analysis Semi-analytical method VDQ method

In this study, a semi-analytical technique is employed to analyze the postbuckling of functionally graded gra­ phene platelet reinforced composite (FG-GPLRC) conical shells under compressive meridional loading. The nonuniform distribution of graphene platelets (GPLs) along the shell thickness is considered and the modified Halpin-Tsai micromechanical model is implemented to determine the overall material properties of nano­ composite shell. The mechanical behavior of FG-GPLRC conical shell is modeled on the basis of first-order shear deformation theory (FSDT) and von-K� arm� an’s nonlinear strain-displacement relations. The governing equations are formulated in the variational framework. The semi-analytical solution based on the variational differential quadrature method (VDQM) and Fourier series is developed. To trace the postbuckling path, the pseudo arclength continuation scheme in conjunction with the load disturbance approach was employed. In order to analyze the influences of geometrical factors, weight fractions and dispersion patterns of GPLs on the post­ buckling characteristics of FG-GPLRC conical shells, various numerical results are comparatively reported.

1. Introduction Circular shell structures are one of the fundamental structural ele­ ments in many different engineering applications. The investigation of the postbuckling behavior of axially-loaded circular cylindrical and conical shells is the serious instability problems of shell structures. Various research works on the initial [1–4] and advanced [5–12] post­ buckling analysis of circular shells can be found in the literature. For instance, the semi-finite element method was implemented by Hong and Teng [8] to analyze the imperfection sensitivity and postbuckling of isotropic circular shells. Mechanical postbuckling analysis of isotropic geometrically imperfect conical shells was reported by Jabareen and Sheinman [10] in accordance with the classical shell theory. The post­ buckling path was traced by the use of the Galerkin approach and arc-length procedures along with the Newton-Raphson and finite dif­ ference methods. Patel et al. [12] studied the postbuckling of laminated conical shells under combined torsion, external pressure and axial compression along with thermal loads. They employed the first-order shear deformation theory (FSDT) of shells and semi-analytical finite element method to model the problem. On the other hand, the improvement of material science technologies has resulted in advanced engineering materials with outstanding

mechanical specifications. Among these, owing to the remarkable properties, carbon-based nanostructures such as carbon nanotube (CNT) and graphene have received a great deal of attention. These materials have been recently introduced as the potential reinforcement for the high-performance nanocomposite structures. The excellent properties of graphene and its derivatives such as the high specified surface area, the relatively low cost and the high specific strength, introduce the graphene-based nanocomposites as the best candidates for the light­ weight structures. In the last couple of years, applying the concept of non-uniform distribution of reinforcements within the polymer matrix has led to the appearance of the functionally graded carbon nanotube reinforced composite (FG-CNTRC) [13] and functionally graded graphene platelets reinforced composite (FG-GPLRC) [14,15] materials. Accordingly, various investigations have been performed on the thermo-mechanical analysis of FG-CNTRC [16–23] and FG-GPLRC [24–35] structures. For the sake of brevity, only the brief review of buckling and postbuckling analysis is outlined in the following. Shen et al. [36–38] investigated the postbuckling of FG-CNTRC plate and cylindrical shell on the basis of the higher-order shear deformation �rma �n geometric nonlinearity. Ansari et al. [39] con­ theory and von-Ka ducted the numerical investigation based on the VDQM on the buckling of FG-CNTRC conical panels under axial loading. The further

* Corresponding author. E-mail address: [email protected] (R. Ansari). https://doi.org/10.1016/j.tws.2019.106594 Received 30 June 2019; Received in revised form 24 November 2019; Accepted 30 December 2019 Available online 28 January 2020 0263-8231/© 2020 Elsevier Ltd. All rights reserved.

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Nomenclature

Kinematic and kinetic quantities q displacement vector u displacement projection on the mid-surface along the x direction v displacement projection on the mid-surface along the θ direction w displacement projection on the mid-surface along the z direction ψ rotation of the transverse normal about θ axis φ rotation of the transverse normal about x axis

Matrix notation q Displacement vector Ap*q p*q matrix In n � n identity matrix Derivatives matrix of the linear strain terms E1 ; E2 G1 ; G2 Derivatives matrix of the non-linear strain terms C Elasticity matrix C1 ; C2 ; C3 ; C4 ; C5 Elasticity matrix components

Material EC EM EGPL W0GPL VGPL VM lGPL hGPL wGPL WGPL

Operators AT ¼ transðAÞ; Transpose of A A ¼ diagðAÞ Diagonal of vector A A Discretization of matrix A δðAÞ Variation of A � Kronecker-product Fundamental mechanical quantities Strain vector In-plane strain vector Transverse strain vector Stress vector Π Total energy U Elastic strain energy U~ Elastic strain energy density We Potential energy of external forces Shear correction factor κs

ε ε1 ε2 σ

νGPL νm

parameters Effective elastic modulus Elastic modulus of the matrix Elastic modulus of the GPL Total weight fraction of GPLs Volume fraction of GPLs Volume fraction of matrix Average length of GPLs Thickness of GPLs Width of GPLs Weight fraction of GPLs Possion’s ratio of GPLRC Possion’s ratio of matrix

Numerical quantities Partial derivative with respect to x Partial derivative with respect to θ x Vector of nodal values in meridional direction Sx Integral operators in meridional direction Integral operators in circumferential direction Sθ Dpq 2-D differential operator numbers of grid points in x direction n1 numbers of grid points in θ direction n2

∂x ∂θ

Geometric quantities x; θ; z Meridional, circumferential and transverse directions h Thickness of the shell L Length of the shell V Volume A Area

Fig. 1. Distribution patterns of GPLs along the thickness direction of the shell.

applications of VDQM on the mechanical analysis of FG nanocomposite structures can be found in Refs. [40–44]. In addition, Nguyen et al. [45] performed the NURBS-based postbuckling analysis of FG-CNTRC shells based on FSDT and using modified Riks method. Shen et al. [46,47] studied the thermal and mechanical postbuckling of laminated functionally graded graphene-reinforced composite (FG-GRC) plates according to the third-order shear deformation theory. The temperature dependency of the material properties was taken into account and the perturbation technique was applied to solve the nonlinear differential equations. Based on the Mindlin plate theory, Yang et al. [48] and Song et al. [49] performed different studies on the postbuckling of FG-GPLRC beams and plates. The layer-wise variation of GPLs through the thickness direction was considered while the uni­ formly distributed GPLs were regarded in each layer. Additionally, the mechanical postbuckling of FG-GRC cylindrical shells under axial

loading and external pressure was presented by Shen et al. [50,51]. Reddy’s shell theory and the modified Halpin–Tsai micromechanical approach were implemented to derive the governing equations. Employing the differential quadrature-based iteration technique, Wu et al. [52] studied the thermal postbuckling of FG-GPLRC plates based on FSDT and by the use of modified Halpin-Tsai micromechanics model. Wang et al. [53] highlighted the buckling analysis of GPLRC cylindrical shell with cutout. The impacts of the geometry, position and orientation of the cutout were extensively examined on the buckling behaviors of the nanocomposite cylinder. The thermo-mechanical linear buckling analysis of FG-GRC conical shells was also presented by Kinai [54] based on FSDT and Donnell kinematic relations. The nonlinear buckling behavior of functionally graded and composite conical shells was analyzed by Sofiyev and his coworker [55,56] through a parametric study. The superposition and Galerkin methods were employed to find 2

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along the thickness of shell are taken into account. In order to present the governing equations in the context of the variational approach, the �rma �n energy functional is derived on the basis of FSDT and von-Ka geometric nonlinearity. The semi-analytical solution based on the vari­ ational differential quadrature method (VDQM) is presented in which the numerical differential quadrature operators and analytical Fourier series are applied in meridional and circumferential directions, respec­ tively. Finally, the arc-length continuation technique and load distur­ bance method are employed to obtain the postbuckling path. The influences of geometrical and material factors on the postbuckling characteristics of FG-GPLRC conical shells are examined through the diverse comparative results. 2. Functionally graded graphene platelets reinforced composites The polymer and GPLs are considered as the constituents of nano­ composite respectively as the matrix phase and reinforcement. The modified Halpin-Tsai model is presented to account the impacts of GPLs geometry and dimension on the effective properties of FG-GPL rein­ forced composites. Hence, the effective Young’s modulus can be pre­ sented as [49]. � � � � 3 1 þ ξL ηL VGPL 5 1 þ ξW ηW VGPL (1) EM þ EM EC ¼ 8 1 ηL VGPL 8 1 ηW VGPL

Fig. 2. Schematic view and geometrical parameters of the conical shell.

where

Table 1 Comparison of the dimensionless buckling load of FG-GPLRC cylindrical shells (α ¼ 0; R1 ¼ 200 mm; h ¼ 4 mm; L ¼ 1000 mm; W 0GPL ¼ 0:5%. Type UD FGA FGO FGX

Linear bifurcation buckling load

Nonlinear buckling load

Ref. [63]

Present study

Present study

2.7409 2.5099 2.1792 3.1961

2.7371 2.5543 2.2677 3.1283

2.3635 2.4272 1.9219 2.5973

ηL ¼

ξL ¼ 2ðlGPL = hGPL Þ; ξW ¼ 2ðwGPL = hGPL Þ

0.12

0.28

UD FGA FGO FGX UD FGA FGO FGX

CC

(3)

are the geometrical factors of GPLs. The volume fraction of GPLs are also presented as

Present study

Ref. [64]

Present study

72.39 61.55 58.08 86.05 115.68 99.63 93.16 137.84

72.41 61.58 58.11 86.07 115.71 99.66 93.18 137.87

57.96 51.02 50.23 63.73 114.93 106.66 100.73 135.43

57.98 51.04 50.24 63.75 114.97 106.69 100.76 135.48

WGPL Þ

(4)

where ρGPL and WGPL stand for GPLs’ mass density and weight fraction and ρm is the density of the polymer. Additionally, Poisson’s ratio of nanocomposite is given as

SS

Ref. [64]

WGPL WGPL þ ρρGPL ð1 M

Table 2 Comparison of the linear buckling load of FG-CNTRC conical shell (h ¼ 2 mm; � β ¼ 30 ; L=R1 ¼ 3; R1 =h ¼ 30). Type

(2)

in which EGPL and Em are Young’s modulus of GPLs and matrix phase and VGPL is the volume fraction of GPLs. Furthermore

VGPL ¼

V�cn

1 ðEGPL =EM Þ 1 ;η ¼ ξL W ðEGPL =EM Þ ξW

ðEGPL =EM Þ ðEGPL =EM Þ

νc ¼ VGPL νGPL þ VM νM

(5)

where VM is the matrix volume fraction with VGPL þ VM ¼ 1. In accordance with the non-uniform distribution of GPLs along the thickness direction, the material properties of FG-GPLRC vary through this specific direction. In this study, besides the uniform distribution (UD), three different functionally graded patterns known as FGA, FGO and FGX are considered, as shown in Fig. 1. In order to model the FG distribution of GPLs, both continuous and layer-wise models were considered in the literature. The continuous variation of GPLs is prac­ tically troublesome, and a layerwise approach should be used in prac­ tical applications. However, in order to model the mechanical behavior of FG-GPLRC, both layerwise and continuous approaches can be employed in the governing equations. In the present work, the contin­ uous dispersion of GPLs is assumed. So, the mathematical expression of the GPLs weight fraction for each type can be given as � � 2z UD : WGPL ¼ W 0GPL FGA : WGPL ¼ 1 W 0GPL FGO h � � � � 2jzj 2jzj W 0GPL FGX : WGPL ¼ 2 W 0GPL : WGPL ¼ 2 1 (6) h h

the upper and lower critical axial loads. In addition, Heydarpour et al. [57] presented the thermoelastic analysis of FG-GPLRC spherical shells subjected to the thermo-mechanical loadings based on the Lord-Shulman theory using the layerwise differential quadrature method. Furthermore, an accurate nonlinear buckling study was con­ ducted for the functionally graded porous GPLRC cylindrical shells by Zhou et al. [58] based on Donnell’s theory and higher-order shear deformation shell theory. In this study, a semi-analytical investigation is carried out on the postbuckling of FG-GPLRC conical shells subjected to compressive meridional loading. The overall material properties of GPL-reinforced composites are obtained with the aid of modified Halpin-Tsai model. Besides the uniform pattern, three different FG distributions of GPLs

in which, W0GPL is the total weight fraction of GPLs, z is the coordinate along the thickness direction and h is the thickness of the shell. 3

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Table 3 Comparison of linear and nonlinear dimensionless buckling loads (~ σcr ¼ W0GPL

) of clamped FG-GPLRC cylindrical shell (L=R1 ¼ 2).

UD FGA FGO FGX UD FGA FGO FGX

Nonlinear buckling load

α ¼ 30

�

1.5

3

Linear bifurcation buckling load

​ ð%Þ

α ¼ 15

0.5

2πR1 ð1 ν2m Þ σcr � 10 Em h

α ¼ 15

�

α ¼ 30

�

�

h ¼ 0:01 R1

h ¼ 0:03 R1

h ¼ 0:01 R1

h ¼ 0:03 R1

h ¼ 0:01 R1

h ¼ 0:03 R1

h ¼ 0:01 R1

h ¼ 0:03 R1

5.2913 4.9459 4.3932 6.0591 11.890 10.468 9.1096 14.136

1.7692 1.6431 1.4534 2.0005 3.9760 3.4640 3.0238 4.6681

4.2617 3.9857 3.5388 4.8910 9.5771 8.4148 7.3252 11.398

1.4257 1.3232 1.1733 1.6255 3.2035 2.7926 2.4442 3.7943

4.4753 4.1752 3.6845 5.0678 10.129 8.688 7.669 12.003

1.4725 1.3909 1.2037 1.6921 3.3085 2.9322 2.5153 4.0185

3.9214 3.7098 3.2270 4.4338 8.8112 7.7090 6.7934 10.396

1.3488 1.2516 1.0937 1.4732 3.0319 2.6129 2.2222 3.4455

Fig. 3. Effects of thickness-radius ratios (h=R1 ) on the postbuckling path of clamped FG-GPLRC conical shells for different weight fractions (α ¼ 30 ; L= R1 ¼ 3; FGO). �

3. Fundamental relations

where u; v; w are the displacements of mid-surface, and ψ ; φ introduce the rotations about θ- and x - axis, respectively. Based on Eq. (7), the strain-displacement relations are expressed as [12].

We consider a truncated conical shell with the following geometrical parameters: semi-vertex angle α, radius of the cone at its small and large edges R1 and R2 , cone length along its generator L and constant thick­ ness h, which are defined in the conical coordinate system ðL1 � x � L2 ; 0 � θ � 2π; h=2 � z � h=2Þ and subjected to the compressive meridional mechanical load, as shown in Fig. 2. According to the FSDT, the displacement field is given by Ref. [12]. ux ðx; θ; zÞ ¼ uðx; θÞ þ zψ ðx; θÞ; uθ ðx; θ; zÞ ¼ vðx; θÞ þ zφðx; θÞ; uz ðx; θ; zÞ ¼ wðx; θÞ

ε ¼ ½ ε1 ε2 �T ; ε1 ¼ ½ εxx εθθ

γ xθ �T ; ε2 ¼ ½ γxz

γθz �T

(8)

in which 8 9 0 > > > � � ( 0 ) < κxx > = γ γ NL 0 0 ε1 ¼ εθθ ¼ εθθ þ εθθ þ z κθθ ε2 ¼ xz ¼ 0xz γ : ; > > > > > > γ θz > θz γ xθ ; > : γNL > ; > : κ0 > ; : γ0 > 8 9 < εxx =

(7)

4

8 9 0 > > > = < εxx >

8 9 NL > > > < εxx > =

xθ

xθ

xθ

(9)

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Fig. 4. Effects of thickness-radius ratios (h=R1 ) on the primary buckling mode shape of clamped FG-GPLRC conical shells (α ¼ 30 ; L=R1 ¼ 3; W 0GPL ¼ 1 %; FGO). �

with

9 8 u;x > > > > > > > < v;θ þ us w > = 0 þ εθθ ¼ xs xt >; > > > > > : γ0 > ; > > > > : v;x þ u;θ vs > ; xθ xs 8 9 0 > > > < εxx > =

8 w2 9 ;x > > > > > 2 > > > > > > > > < 2 > = w;θ NL ; εθθ ¼ 2 > > > > > 2ðxsÞ > ; > > : γ NL > > > > > xθ > > >w w > > : ; 9 8 NL > > > = < εxx >

;x

;θ

9 8 ψ ;x 8 9 > > 9 > > 0 > > > ( ) 8 > κ > > > w;x þ ψ < < xx = < φ;θ þ ψ s > = = γ 0xz 0 ¼ ; κθθ ¼ v w;θ xs 0 : > > > > γθz þ þ φ; > > ; > > : κ0 > > xt xs > ; : φ þ ψ ;θ φs > xθ ;x xs

(10)

xs

in which 2 1 P1 ¼ 4 0 0

NL NL where s ¼ sin α and t ¼ tan α. Note that εNL xx , εθθ and γ xθ are the nonlinear �n’s strain-displacement relations. The components of part of von-K� arma strain-displacement can be represented in a vector form as

1 2

ε1 ¼ P1 E1 q þ En q; ε2 ¼ E2 q

(11)

En ¼ 〈G1 q〉G2 þ 〈G2 q〉G1

(12)

5

<■> ¼ diag(■), P1 , E1 and E2 are defined as 3 0 0 z 0 0 1 0 0 z 05 0 1 0 0 z

(13)

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Fig. 5. Effects of length-radius ratios (L=R1 ) on the postbuckling path of FG-GPLRC conical shells for different boundary conditions (α ¼ 20 ; h= R1 ¼ 0:03; W 0GPL ¼ 1%; UD). �

2

∂x

6 1=x 6 6 ∂θ =xs E1 ¼ 6 6 0 6 4 0 0

0 ∂θ =xs ∂x 1=x 0 0 0

0 1=xt 0 0 0 0

0 0 0

∂x 1=x ∂θ =xs

3 0 7 0 7 � 7 0 7; E2 ¼ 0 7 0 0 7 ∂θ =xs 5 ∂x 1=x

0 1=xt

∂x ∂θ =xs

1 0

and q ¼ ½u v w ψ φ�T is the vector of displacement components. In the same manner, the constitutive relation is determined as

�

C1 C¼ 0

(15)

σ ¼ Cε

0 0 0

3

2

3

∂x =2 0 0 ∂x 0 0 0 0 ∂θ =2xs 0 0 5; G2 ¼ 4 0 0 ∂θ =xs 0 0 5 0 0 ∂θ =xs 0 0 ∂x 0 0

2 � Q11 0 ; C1 ¼ 4 Q21 C2 0

Q12 Q22 0

3 � 0 Q55 0 5; C2 ¼ 0 Q66

0 Q44

(14)

� (17)

in which

where

σ ¼ ½ σxx σθθ

Q11 ðzÞ ¼

2 � 0 0 ; G1 ¼ 4 0 1 0

1

σxθ

σ xz σ θz �T

(16)

Ec ðzÞ Ec ðzÞνc ðzÞ Ec ðzÞ ; Q22 ¼ Q11 ; Q12 ðzÞ ¼ ; Q44 ðzÞ ¼ ; Q55 ¼ Q44 ; Q66 ¼ Q44 1 ν2c ðzÞ 2ð1 þ νc ðzÞÞ ν2c ðzÞ

6

(18)

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Fig. 6. Postbuckling path of FG-GPLRC conical shells for different distribution types of GPLs associated with various boundary conditions (α ¼ 30 ; h= R1 ¼ 0:03; L=R1 ¼ 3; W 0GPL ¼ 1%). �

4. Minimum total potential energy principle

Z δU ¼

In order to derive the governing equations in the variational framework, the minimum total potential energy principle is employed as follow δΠ ¼ δU þ δWe ¼ 0

A

� � 1 1 δqT ET1 C1 E1 þ ET2 C2 E2 þ ET1 C3 En þ ETn C4 E1 þ ETn C5 En qdA; 2 2 (23)

in which dA ¼ xs dx dθ and � � Z h=2 Z h=2 A B C1 ¼ PT1 C1 P1 dz ¼ ; C2 ¼ κ s C2 dz ¼ As ; C3 B D h=2 h=2 � � Z h=2 Z h=2 T A ¼ PT1 C1 dz ¼ ; C4 ¼ C1 P1 dz ¼ ½ A B � ¼ C3 ; B h=2 h=2 Z h=2 ¼ C1 dz ¼ A

(19)

where Π is the total potential energy, U is the elastic strain energy and We is the potential energy of external forces. In the elasticity theory, the strain energy is Z U~ dV (20) U ¼

h=2

V

(24)

where dV ¼ xs dx dθ dz and the strain energy density U~ is defined in a quadratic matrix representation as

with

1 1 1 1 U~ ðεÞ ¼ εT σ ¼ εT Cε ¼ εT1 C1 ε1 þ εT2 C2 ε2 2 2 2 2

ðA3�3 ; B3�3 ; D3�3 Þ ¼

Using Eqs. (20) and (21) the δU can be achieved as Z Z � δU ¼ δ U~ dV ¼ δεT1 C1 ε1 þ δεT2 C2 ε2 dV V

C5

Z

(21)

h=2 h=2

� C1 1; z; z2 dz; As2�2 ¼

Z

h=2

C2 dz h=2

(25)

where κs ¼ 5=6 is the shear correction factor. Note that due to the nonuniform FG distribution of GPLs along the thickness direction, the value of B is not zero and the bending-stretching coupling is observed. The variation of potential energy of external forces are also given as Z δWe ¼ δqT PdA (26)

(22)

V

Substituting Eq. (11) into (22) and considering dV ¼ dzdA, the variation of elastic strain energy is written as

A

Accordingly, substituting Eqs. (23) and (26) into Eq. (19) yields to

7

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Fig. 7. Effects of weight fractions (W 0GPL ) on the postbuckling path of clamped FG-GPLRC conical shells for various distribution types of GPLs (α ¼ 20 ; h= R1 ¼ 0:03; L=R1 ¼ 2). �

Z δqT A

�� � � 1 1 ET1 C1 E1 þET2 C2 E2 þ ET1 C3 En þETn C4 E1 þ ETn C5 En qþP dA¼0 2 2 (27)

S ¼ Sθ � sXT Sx

�

(30)

where Sx and Sθ are the integral operators in meridional and circum­ ferential directions, respectively. In addition, X is the vector of nodal points in the meridional direction

5. VDQM Based on the discretization approach of VDQM which was clearly explained in Refs. [59,60], the displacement vector and related strain vector are discretized over the space domain. In accordance with the VDQM, the quadratic representation of energy functional given in Eq. (27) is directly discretized as follows � � 1 1 δqT ET1 C1 E1 þ ET2 C2 E2 þ ET1 C3 En þ ETn C4 E1 þ ETn C5 En q þ δqT IP ¼ 0 2 2 (28)

X ¼ ½ x1

where Ci ¼ Ci � S ði ¼ 1; 2Þ, I ¼ I5�5 � S and the symbol ’’� ’’ denotes the Kronecker product. Note that q is the discretized form of displace­ ment vector defined as 2 11 3 2 11 3 2 11 3 2 11 3 2 11 3 v w φ ψ u 2u3 6 v21 7 6 w21 7 6 φ21 7 6 u21 7 6 ψ 21 7 7 6 7 6 7 7 6 7 6 6 v 6 7 6 ⋮ 7 6 ⋮ 7 6 ⋮ 7 6 ⋮ 7 6 ⋮ 7 7 6 7 6 7 7 6 7 7 6 6 q¼ 6 4 w 5; u¼ 6 un1 1 7; v¼ 6 vn1 1 7; w¼ 6 wn1 1 7; ψ ¼ 6 ψ n1 1 7; φ¼ 6 φn1 1 7 7 6 7 6 7 7 6 7 6 6 ψ 4 ⋮ 5 4 ⋮ 5 4 ⋮ 5 4 ⋮ 5 4 ⋮ 5 φ ψ n1 n2 un1 n2 vn1 n2 wn 1 n 2 φn1 n2 (29)

K ¼ ET1 C1 E1 þ ET2 C2 E2

x2

⋯

xn1 �T

and defined based on the Chebyshev-Gauss-Lobatto points as � � 1 i 1 1 cos xðiÞ ¼ π L 2 n1 1

(31)

(32)

Now, by defining the following matrices 1 1 Ks ¼ K þ N1 þ N2 2 3

N1 ¼ ET1 C3 En þ 2ETn C4 E1

(33)

3 N2 ¼ ETn C5 En 2 F ¼ IP Eq. (28) can be simplified as T

δq ðKs q þ FÞ ¼ 0 which results in the following governing equations

In addition, considering dA ¼ sxdxdθ, the two-dimensional integral operator S is introduced as 8

(34)

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Fig. 8. Effects of semi-vertex angle (α) on the postbuckling path of FG-GPLRC conical shells for different boundary conditions(h=R1 ¼ 0:02; L= R1 ¼ 2; W 0GPL ¼ 1%; FGA).

(35)

Ks q þ F ¼ 0 Moreover, the matrix operators E1 , E2 , G1 and G2 are 2 3 0 0 0 0 D10 6 X 7 XD01 =s X=t 0 0 6 7 6 XD01 =s D10 X 7 0 0 0 7; E2 E1 ¼ 6 6 0 7 0 0 D10 0 6 7 4 0 0 0 X XD01 =s 5 0 0 0 XD01 =s D10 X � � 0 0 D10 D00 0 ¼ 0 X=t XD01 =s 0 D00

ð1Þ

Dθ ¼ ½ 0

2

0 G1 ¼ 4 0 0

m sin mθ m cos mθ

0 0 0

D10 =2 0 XD01 =2s 0 0 D10

2m sin mθ

3 2 0 0 5 0 ; G2 ¼ 4 0 0 0

0 0 0

2m cos mθ

0 D10 XD01 =s 0 XD01 =s 0

T¼½1

ðqÞ

ð0Þ

X ¼ diag Iθ � ½ 1=x1

1=x2

⋯

T� 1=xn1 � ;

cos 2 mθ sin 2 mθ

… cosn2 mθ sinn2 mθ � (40)

in which m stands for the circumferential wave number and θ is defined as θ ¼ ½ θ1

θ2

… θn2 �T ; θ1 ¼ 0; θn ¼ 2π

(41)

In addition, Dθ is defined as ð1Þ

(36)

…

3 0 05 0

n2 m sin mθ n2 m cos mθ �

(42)

In these equations n1 and n2 are the numbers of grid points in x and θ directions, respectively. In addition, Dð1Þ x stands for the numerical dif­ ferential operators in the meridional direction. In this study, the dif­ ferential and integral operators in x direction are defined using the GDQ method whereas, the differential operator in θ direction is analytically calculated based on Fourier series as defined in Eq. (42). Furthermore, Simpson integration method is applied in the θ direction. The further details on GDQ method can be found in Ref. [59].

(37)

in which Dpq denotes the 2-D differential operator ð0Þ Dpq ¼ Dθ � DðpÞ x ; Dθ ¼ T; Dx ¼ Ix

cos mθ sin mθ

(38) (39)

where T is introduced as follows 9

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6. Results and discussion

are provided, the same prebuckling deformation is observed for different thickness-radius ratios. The effects of length-radius rations on the buckling load and post­ buckling path of FG-GPL reinforced composite conical shells are illus­ trated in Fig. 5 for CC, CS and SS boundary conditions. The uniform distribution and W0GPL ¼ 1% are considered. The numerical results reveal that the variations of length-radius ratio do not considerably affect the linear dimensionless buckling load. In addition, the same buckling stress ratios are observed for different L=R1 . The higher pre­ buckling axial deformation is also obtained by the increase of L=R1 . For instance, it is found that in the case of CC boundary condition, the prebuckling axial deformations for L=R1 ¼ 2; 4; 6 are u=h ¼ 0:72; 1:12; 1:45, respectively. Comparison of the results for different boundary conditions shows the lower buckling load ratios in the case of SS. Demonstrated in Fig. 6 are the variations of buckling load and postbuckling path of FG-GPLRC conical shells versus the GPLs distri­ bution patterns for different boundary conditions. The geometrical pa­ � rameters are α ¼ 30 ; h=R1 ¼ 0:03; L=R1 ¼ 3 and W0GPL ¼ 1%. It is apparent that the FGX and FGA distribution patterns result in the highest and lowest buckling capacity. It is also observed that the shell with FGX pattern has the highest dimensionless axial shortening in prebuckling state. This is due to the highest initial axial stiffness obtained due to the special distribution of GPLs through the FGX pattern. The postbuckling path of the clamped FG-GPLRC conical shells is depicted in Fig. 7 for different volume fractions and distribution patterns � of GPLs. the geometrical factors are: α ¼ 20 ; h=R1 ¼ 0:03; L=R1 ¼ 2. It is generally perceived that the GPLs volume fraction does not have significant effects on the postbuckling path, while some discrepancies can be seen in the case of non-uniform dispersion patterns. It is also concluded that increasing the GPLs volume fraction considerably en­ hances the stability of the shell and increases the dimensionless linear buckling load. The impacts of semi-vertex angle on the buckling load and post­ buckling analysis of GPL-reinforced composite conical shells are exhibited in Fig. 8 for CC, CS and SS boundary conditions. The FGA dispersion pattern with W0GPL ¼ 1% are considered where thicknessradius and length-radius ratios are h=R1 ¼ 0:02; L=R1 ¼ 2. The results indicate that the increment of semi-vertex angle reduces the dimen­ sionless linear buckling load. Furthermore, one can see the higher im­ pacts of nonlinear terms for the shells with the smaller values of α. The higher values of dimensionless axial shortening are observed in the case of larger semi-vertex angles.

The nonlinear governing equations for the postbuckling analysis of FG-GPLRC conical shells were derived in the previous section. The arclength continuation technique and load disturbance method were employed to find the postbuckling path. The basic idea of arc-length method is to consider the load parameter as an additional unknown governed by a constraint function [61,62] In this study, the epoxy with the material properties of EM ¼ 3 GPa, νM ¼ 0:34 is considered as the matrix and GPLs with the dimensions of lGPL ¼ 2:5 μm, wGPL ¼ 1:5 μm and hGPL ¼ 1:5 nm are chosen as the reinforcements with the following properties: EGPL ¼ 1:01 TPa and νGPL ¼ 0:186 [49]. In order to present the numerical results, the essential boundary conditions are considered for clamped (C) and simply-supported (S) edges as clamped: simply

supported:

At x ¼ L1 At x ¼ L2 At x ¼ L1 At x ¼ L2

: u ¼ v ¼ w ¼ ψ ¼ φ ¼ 0; : v ¼ w ¼ ψ ¼ φ ¼ 0; : u ¼ v ¼ w ¼ φ ¼ 0; : v ¼ w ¼ φ ¼ 0;

(43)

Different sets of boundary conditions including fully clamped (CC), clamped-simply supported (CS) and fully simply supported (SS) are considered. It should be noted that in the case of CS boundary condition, the edge at x ¼ L1 is clamped and the other edge is simply-supported. To verify the accuracy of the proposed model, the buckling load ratios (NGPL =Npolymer ) of FG-GPLRC cylindrical shells are compared in Table 1 with the results reported in Ref. [63] for variant distribution patterns of GPLs. The conformity of the results in the case of linear bifurcation buckling analysis shows the validity of this study. In the second com­ parison study, the linear buckling loads (Pcr ðKNÞ) of FG-CNTRC conical shells obtained by the present approach are compared with those re­ ported by Ansari and Torabi [64] in Table 2. As it is observed, the present results have excellent agreement with those of [64] which shows the accuracy of the proposed model. In order to analyze the postbuckling behavior of FG-GPLRC conical shell, two different analyses including the linear bifurcation analysis and the geometrically nonlinear analysis were performed in this study, and the variations of buckling ratio (~ σ = σ~cr ) versus dimensionless meridional displacement (u=h) are presented as the 2

νm Þ postbuckling path in which σ~cr ¼ 2πR1Eð1 σcr is the dimensionless linear mh

buckling load. Note that σcr is the linear bifurcation buckling load ob­ tained based on the common adjacent equilibrium buckling criterion [65]. To show the influences of geometrical nonlinear terms on the buckling behavior, the dimensionless linear bifurcation loads and nonlinear buckling loads of FG-GPLRC conical shell are compared in Table 3 for different distribution patterns and volume fractions of GPLs and semi-vertex angles. It is observed that the neglecting of the nonlinear effects leads to overestimating of buckling capacity. Com­ parison of the results implies that the impacts of nonlinear terms are more significant for the conical shells with smaller semi-vertex angle. For instance, in the case of h ¼ 0:01, W0GPL ¼ 0:5 % and UD distribution, it is observed that for smaller semi-vertex angle (α ¼ 15� ), the nonlinear buckling load is about 18% smaller than the linear bifurcation buckling load, while this value is about 8.5% for larger semi-vertex angle (α ¼ 30� ). The postbuckling path of the clamped FG-GPLRC conical shells is demonstrated in Fig. 3 for diverse thickness-radius ratios and GPLs volume fractions. The FGO pattern along with the geometrical param­ eters of α ¼ 30� , L=R1 ¼ 3 are regarded. The effects of thickness-radius ratios on the primary buckling mode shapes are also demonstrated in Fig. 4. It is found that the increase of the thickness-radius ratios de­ creases the linear dimensionless buckling load (~ σ cr ). It should be pointed out that the rise of thickness-radius ratio makes the shell more stable and increases the linear buckling load (σ cr ), however, the linear dimen­ sionless buckling load decreases. In addition, it is seen that the increase of h=R1 leads to less secondary buckling. Since the dimensionless results

7. Conclusion The nonlinear postbuckling analysis of FG-GPL reinforced composite conical shells was presented based on the analytically approach. The micromechanical model based on the modified Halpin-Tsai approach was employed to estimate the overall material properties of nano­ composite. Considering the von-K� arm� an nonlinear geometrical straindisplacement relations, the matrix representation of the total potential energy was derived based on the FSDT. The semi-analytical energybased solution procedure was presented in the context of the VDQM. Finally, the load disturbance method along with the arc-length contin­ uation scheme was applied to find the postbuckling path. Comparison of the results for linear and nonlinear buckling analysis indicated that omitting the nonlinear effects overestimates the buckling capacity. The results revealed that the volume fraction of GPLs does not significantly affect the postbuckling path while increases the linear buckling loads. It was also found that the FGX dispersion pattern of GPLs leads to the highest linear buckling load and dimensionless axial shortening. Furthermore, it was concluded that the increment of thickness-radius ratio increases the secondary buckling. Comparison of the postbuck­ ling path for different semi-vertex angles showed the more impacts of nonlinear terms on the buckling load ratio for the shells with the larger 10

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Thin-Walled Structures 148 (2020) 106594

semi-vertex angles.

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