Buckling modelling of ring and stringer stiffened cylindrical shells aggregated by graded CNTs

Accepted Manuscript Buckling modelling of ring and stringer stiffened cylindrical shells aggregated by graded CNTs B. Sobhaniaragh, M. Nejati, W.J. Mansur PII:

S1359-8368(17)30881-8

DOI:

10.1016/j.compositesb.2017.05.045

Reference:

JCOMB 5068

To appear in:

Composites Part B

Received Date: 10 March 2017 Revised Date:

20 April 2017

Accepted Date: 13 May 2017

Please cite this article as: Sobhaniaragh B, Nejati M, Mansur WJ, Buckling modelling of ring and stringer stiffened cylindrical shells aggregated by graded CNTs, Composites Part B (2017), doi: 10.1016/ j.compositesb.2017.05.045. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Buckling Modelling of Ring and Stringer Stiffened Cylindrical Shells Aggregated by Graded CNTs B. Sobhaniaragh a,b,c∗ , M. Nejati c , W.J. Mansur b

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a. Department of Mechanical Construction and Production, Faculty of Engineering, Ghent University, Gent, Belgium. b. Laboratório de Métodos de Modelagem e GeofísicaComputacional, Department of Civil Engineering, Federal University of Rio de Janeiro, RJ, Brazil. c. Young Researchers Club, Islamic Azad University, Arak Branch, Arak, Iran

Abstract:

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In this paper, based on Third-order Shear Deformation Theory (TSDT), the mechanical buckling behavior of the continuously graded Carbon Nano-Tube (CNT)-reinforced shells stiffened by stringer and rings is studied. A two-parameter Eshelby-Mori-Tanaka (EMT) approach, which is capable of dominating the aggregation degree of CNTs within ceramic matrix phase, is proposed to estimate the effective material properties of the nanocomposite. In the present work, the equilibrium equations are obtained using variational approach based on TSDT of Reddy. The stability equations of the continuously graded CNRTC Stiffened Cylindrical Shell (SCS) are extracted via the concept of the adjacent equilibrium criterion.

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Opposed to most available works in which the stiffener effects were smeared out over the respective stiffener spacing, in the present paper, the stiffeners are modeled as Euler-Bernoulli beams. A semianalytical solution employing the Generalized Differential Quadrature Method (GDQM) along with the trigonometric expansion is implemented to solve the stability equations. A detailed parametric study is

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conducted to highlight the impacts of various geometrical and material parameters involved on the critical mechanical buckling of the SCS aggregated by graded CNTs. The results obtained showed that the graded

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distribution of the CNTs close to the surface, where contains higher amount of CNT volume fraction, results in increasing the critical mechanical buckling load. Furthermore, it has been inferred that the aggregation tendency of the CNTs has a negative effect on the mechanical buckling behavior of the continuously graded SCS.

Keywords: Computational modeling; Stringer and ring stiffeners, Mechanical buckling, Two-parameter Eshelby-Mori-Tanaka; Adjacent equilibrium criterion.

∗

Correspondent author: phone: (+55) 21996350721, fax: (+55) 2139387393, E-mail: [email protected]

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ACCEPTED MANUSCRIPT 1. Introduction Shell structures exhibit complicated coupling phenomena such as a strong influence of transverse shear and transverse normal deformation, membrane-bending coupling owing to non-zero curvature, which makes them enough stiff to endure both in-plane and out-plane loads. Among different kinds of shell structures,

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those with axisymmetric geometry, in particular Cylindrical Shells (CSs), are frequently used in the vast majority of engineering fields, especially civil, mechanical, marine, and aerospace engineering. Classical shell theories are predominantly based on Love-Kirchhoff assumptions, which ignore transverse shear strains

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as reported by Naghdi [1]. Therefore, those theories are applicable to the elastic thin shells with the ratio of the thickness to the radius of curvature of the middle surface of the shell being smaller than 1/20 [2].

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Reissner [3] presented a comprehensive study on the importance of the transverse shear and normal stresses. The shear deformation has been included in the shell theory for the first time by Reissner [4] and Mindlin [5], which is known as the First-order Shear Deformation Theory (FSDT). The FSDT not only assumes constant transverse shear stress, but does it also require a shear correction factor to satisfy the geometrical boundary conditions on the lower and upper surfaces. The limitations of the classical theories and the FSDT

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motivated researchers to develop several Higher-order Shear Deformation Theories (HSDT). Among various types of the HSDT such as the second-order shear deformation formulation of Whitney and Sun [6] and the Third-order Shear Deformation Theory (TSDT) of Lo et al. [7] with 11 unknowns, Kant [8] with six

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unknowns, Bhimaraddi and Stevens [9] with five unknowns, Hanna and Leissa [10] with four unknowns, the TSDT of Reddy [11,12] with five unknowns is the most widely adopted theory in the study of shells owing

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to appropriate efficiency and simplicity without any shear correction coefficient. On the other hand, the CSs are frequently reinforced by stiffening elements with the aim of enhancing load-carrying capability with a comparatively small extra weight. The most widely used stiffeners are the circumferential and meridional stiffeners, which are named rings and stringers, respectively. When stiffened CSs are subjected to thermomechanical loads, they might be buckled. Consequently, there is a need to study the stability of such structures for effective design and cost-optimized structures. In the last years, a large number of the works have been devoted to the buckling and post-buckling of the stiffened shells composed of Functionally Graded Materials (FGMs) [13], which are a new generation of advanced composite materials consisting of

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ACCEPTED MANUSCRIPT two or more constituent phases with a continuously variable distribution. Nevertheless, most of those studies have been established based on the classical shell theory, which does not take into account the important effects of shear deformation and rotatory inertia, resulting in significant inaccuracy, in particular for the larger values of shell’s thickness. For example, Bich et al [14] investigated the nonlinear static and dynamic

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buckling of imperfect eccentrically stiffened FG thin circular CSs subjected to axial compression based on classical thin shell theory. They concluded that stiffeners enhance the static and dynamic stability and loadcarrying capacity of FG circular cylindrical shells. Based on Kirchhoff-Love hypothesis, the buckling of stiffened thin-walled CSc by rings and stringers made of FG metal-ceramic subjected to axial compression

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loading was studied by Najafizadeh et al. [15]. Dung and Hoa [16] presented nonlinear buckling and postbuckling analysis of eccentrically stiffened FG CSs under external pressure. Based on the classical shell

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theory, the equilibrium equations were derived using the smeared stiffeners technique with the geometrical nonlinearity in von Karman sense. Galerkin method was used to determine closed-form expressions to obtain critical buckling loads. An analytical solution for the mechanical buckling of truncated conical shells made of FG metal-ceramic, subjected to axial compressive load and external uniform pressure based on

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FSDT was presented by Dung and Chan [17]. In this study, the closed-form expression for determining the critical buckling load of the shell was obtained. Duc and Thang [18] studied the nonlinear buckling of imperfect eccentrically stiffened FG thin circular CSs with temperature-dependent properties surrounded on

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elastic foundation in thermal environment. Their results showed that the addition of stiffeners inclines the mechanical and thermal loading capacity of the FG shell. Motivated by these considerations, the present

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work is established based on the TSDT of Reddy with five unknowns to derive the equilibrium and stability equations of the Stiffened Cylindrical Shell (SCS). Ever since the discovery of carbon nanotubes (CNTs) by S. Iijima [19] numerous attempts have been made to incorporate them as a fibrous reinforcing agent in polymer, ceramic or metal matrix composites. In the past years, the vast majority of CNT-Reinforced Composites (CNTRCs) research has been concentrated on polymer matrices [20] whereas comparatively few efforts [21] have been made in reinforcing ceramic matrices by CNTs. The combination of extraordinary characteristics of CNTs with intrinsic advantages of ceramic materials such as thermal stability, high corrosion resistance, light density and electrical insulation, generates CNT-Ceramic Matrix Composites (CNT-CMCs) with peculiar functional and structural materials.

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ACCEPTED MANUSCRIPT The potential of developing high-performance CNT-CMCs, which are capable of withstanding conditions of high temperature, chemical attack, and wear, is very appealing with applications as diverse as gas turbines, aerospace materials, and automobiles. According to the authors’ literature survey, either Extended Rule of Mixture (ERM) [22-24] or

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Eshelby-Mori-Tanaka (EMT) [25-27] has been used to determine the effective material properties of the CNTRCs. With regard to the former micromechanics model, in the pioneering efforts Shen [22] developed the conventional rule of mixture by introducing CNT efficiency parameters. The efficiency parameters introduced were computed by matching the elastic moduli of CNTRCs estimated by Molecular Dynamics

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(MD) and the rule of mixture. In the past years, the ERM has played a striking role in estimating the global material properties of structures reinforced by CNTs and has been used in the large number of studies

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investigating mathematical modeling and mechanical responses of such nanocomposites structures. Nevertheless, the ERM confronts grave limitations inasmuch as it disregards several intrinsic features of CNTs dispersed within the matrix of composites, such as aggregation or waviness. Accordingly, equivalent continuum model based on the EMT, which was firstly applied and validated to the CNTRCs by Odegard et

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al. [25], has received a significant degree of attention in predicting the effective mechanical properties. The EMT is predominantly established based on the equivalent elastic inclusion concept of Eshelby [28] in combination with the idea of the average stress within the matrix phase proposed by Mori and Tanaka [29].

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Particularly, notable enhancements and adjustments of the EMT approach were carried out by [30-33] to contemplate the CNT aggregation degree, orientation, curviness and length in combination with

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experimental study. As reported in various Scanning Electron Microscopy (SEM) image [34], the striking aspects of CNTs such as high stiffness, a peculiar aspect ratio, and strong Van der Waals bonds [35] make them to aggregate in the nanocomposite, thereby forming some local regions with higher concentration of CNTs than the average volume fraction in the nanocomposite. Taking the CNT aggregation effects with making use of EMT model, Sobhaniaragh et al. [27,36] investigated the natural frequencies and bending behavior of nanocomposites shells reinforced by agglomerated CNTs. In addition, the interested reader shall refer to Sobhaniaragh’s PhD thesis [37] to find comprehensive details on both ERM and EMT. Furthermore, this methodology has been followed by Fantuzzi et al. [38], Kamarian et al. [39], Tornabene et al. [40],

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ACCEPTED MANUSCRIPT Heshmati [41] to study the impacts of agglomeration and distribution of CNTs on the free vibration characteristics of nanocomposites using EMT. Experimental results conducted by Meguid and Sun [42] demonstrated that there was a limit to volume fraction of CNTs scattered within the matrix phase beyond which a drop in the mechanical properties was

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observed. Accordingly, the designer should take the advantage of low limit of CNT volume fraction to fulfill multi-functional requirements. To this end, in the computational modeling of CNTRCs, the fundamental idea of FGMs might be integrated to adequately enhance the mechanical function of structures reinforced by CNTs. The FGMs make use of certain desirable features of each constituent phases and tailor the

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distribution of material properties so that the desired responses to given mechanical and thermal loadings are achieved or natural frequencies are modified to a required manner. In recent years, a number of researchers

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implemented the idea of smooth gradation of spatial CNT volume fraction in the modeling of nanocomposites structures to achieve desired structural response. Among all available literature, a large number of papers [26,36,38-41,43-57] have been dedicated to the bending and vibrational analyses of FG CNTRCs. In addition, the buckling and postbuckling behavior of FG CNTRCs plates can be found in [58-

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70]. Few studies [71-76], however, have been devoted to buckling analysis of the FG CNTRCs shell structures. It should be noted that the shell studied in all literature [71-76] has not been reinforced by stiffeners. Recently, the linear buckling analysis of the FG CNTRCs conical shells subjected to lateral

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pressure was studied by Jam and Kiani [75]. The governing equilibrium equations of the shell were obtained based on the Donnell shell theory assumptions consistent with the FSDT. The ERM was used to estimate

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effective material properties of CNTRC conical shell. It was concluded that increasing the CNTs volume fraction results in higher critical buckling loads of the shell. García-Macías et al. [76] investigated the critical buckling analysis of FG-CNTRC cylindrical curved panels under compression and shear forces by finite elements analyses. The CNTs were assumed to be randomly oriented, distributed in the polymer matrix phase. The results obtained showed that CNTs distributed close to the top and bottom of the panel induce higher stiffness values. To the best of the authors’ knowledge, there is no study on the mechanical buckling of the stiffened CSs reinforced by agglomerated carbon nanotubes. Accordingly, in order to fill in this research gap, the present paper is devoted to studying the mechanical buckling behavior of continuously graded SCS

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ACCEPTED MANUSCRIPT aggregated by graded CNTs under axial and lateral loads. Two types of stiffeners, i.e. exterior and interior, are contemplated based on the eccentricity of the stiffeners with regard to the mid-surface of the CS. Taking contribution of stringer and rings into account, the smeared stiffener technique is employed with the assumption of closely spaced stiffener system.In the present study, to solve the stability equations the

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Generalized Differential Quadrature Method (GDQM) [77] along with the trigonometric expansion is employed. Comparing with distinguished numerical techniques such as viz. Finite Element Method (FEM), the Boundary Element Method (BEM), and the meshless method, the GDQM has proven its highly

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efficiency and accuracy in various computational mechanics problems [78].

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2. Theoretical Formulations

Consider a CNT-reinforced SCS of finite length L , mean radius R using a global cylindrical coordinate system ( x , y , z ) to label the material point in the unstressed reference configuration, as depicted in Fig. 1. In this paper, the CS is stiffened not only by closely spaced circular rings, but also by longitudinal stringers attached to the shell’s skin. Note that the Z r and Z s denote the eccentricity of the stiffeners with

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stringers, respectively.

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regard to the mid-surface of the CS. Also, the hr , br and hs , bs assign height and width of the ring and

2-1. Micromechanics of Agglomerated Carbon Nanotubes

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In this section, a two-parameter EMT model is presented to evaluate effective mechanical properties of the CNTRC SCS.The equivalent continuum model is based on the theory of Eshelby for elastic inclusions. Originally, one single inclusion in a semi-infinite elastic, isotropic, and homogeneous medium forms the limiting assumption of the Eshelby theory. The model is extended by Mori-Tanaka method to capture multiple in-homogeneities embedded into a finite domain. The EMT is mainly established based on the equivalent elastic inclusion concept of Eshelby [28] along with the idea of the average stress within the matrix phase proposed by Mori-Tanaka [29].Owing to low bending stiffness and strong Van der Waals bonds [35], CNTs dispersed in the CNT tend to aggregate in some regions. That is why such regions contain higher accumulation of CNTs in comparison with average volume fraction in the whole CNTRC. The areas

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ACCEPTED MANUSCRIPT with CNTs accumulated are assumed to have spherical shape and contemplated as inclusions, which have different material properties from outside. Hence, CNTs dispersed in the matrix are sorted according to their positions, i.e., embedded inside or outside the inclusions, as demonstrated in Fig. 1. According to twoparameter EMT model, the total volume Vr of the reinforcing phase is separated into two different parts

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[30]:

Vr =Vr inc +Vr m

(1)

in which Vr inc and Vr m designate the volume of CNTs embedded in the inclusions (accumulated areas) and

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volume of nanotubes dispersed in the matrix phase, respectively. Aggregation of the CNTs in the polymer matrix results in the mechanical properties of the material to degrade compared with a condition with no

aggregation parameters,

V inc , V

ζ=

ζ and ξ , given as

Vr inc Vr

ξ=

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CNTs aggregated. This peculiar aspect of the CNTRCs is described qualitatively by introducing the

(2)

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where V inc is the effective volume of the inclusions, and the parameter ζ elaborates the volume fraction of inclusions with regard to the total volume V of the material. If parameter ζ is equal unity, there is no aggregation in matrix phase, and by increasing the parameter ζ , the aggregation degree of CNTs drops. On

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the other hand, the parameter ξ defines the ratio of nanotubes volume embedded in the inclusions to the total volume of the CNTs. It is obvious that all CNTs are accumulated in the inclusions if ξ equals unity. It

ξ >ζ

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is worth noting that the following limitation should be satisfied so that the aggregation of CNTs occurs (3)

To predict the effective elastic moduli of the inclusions and the matrix phase, it is assumed that the CNTs are transversely isotropic and randomly oriented in the spherical inclusions. The effective bulk modulus

K in ( z ) and shear modulus Gin ( z ) of the inclusions are expressed by [30,52]

Kin ( z ) = K m +

Vr ( z )ξ (δ r − 3K mα r ) 3(ζ − Vr ( z )ξ + Vr ( z )ξα r )

(4)

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ACCEPTED MANUSCRIPT Gin ( z ) =Gm +

Vr ( z )ξ (ηr − 2Gm β r ) 2(ζ −Vr ( z )ξ +Vr ( z )ξα r )

(5)

In a similar manner, the effective bulk modulus K out ( z ) and shear modulus Gout ( z ) of the matrix are given

Gout ( z ) =Gm +

Vr ( z )(1−ξ )(δ r −3K mα r ) 3(1−ζ −Vr ( z )(1−ξ ) +Vr ( z )(1−ξ )α r )

(6)

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K out ( z ) = K m +

Vr ( z )(1−ξ )(ηr − 2Gm β r ) 2(1−ζ −Vr ( z )(1−ξ ) +Vr ( z )(1−ξ ) β r )

(7)

where K m and Gm denote the bulk and shear moduli of the matrix, respectively, and α r , β r , δ r , and η r

3( K m + Gm ) + kr + lr 3(Gm + kr )

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αr =

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are defined by

2 [ Gm (3K m + Gm ) + Gm (3K m + 7Gm ) ]  4Gm 1  4Gm + 2k r + lr + +  Gm + pr Gm (3K m + Gm ) + mr (3K m + 7Gm )  5  3(Gm + k r )

βr = 



1 2

(2k r + lr )(3K m +2Gm − lr )   Gm + k r  8G p

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1

δ r =  nr + 2lr + 3

m r η r =  (nr − lr ) + + 5 3 Gm + pr

 2( kr − lr )(2Gm + lr ) 8mr Gm (3K m + 4Gm ) +  3(Gm + kr ) 3K m ( mr + Gm ) + Gm (7 mr + Gm ) 

(8)

(9)

(10)

(11)

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in which k r , mr , nr and lr are the Hill’s elastic moduli for the CNTs phase [79,80]. The relation between the Hill’s elastic moduli and the material properties of CNTs can be found in Ref. [40]. Eventually, the

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effective bulk modulus K ( z ) and the effective shear modulus G ( z ) of the CNTRC are determined as

   Km  ζ −1   K out ( z )     K ( z ) = K out ( z ) 1+     Km 0 −1   1+ d ( z )(1−ζ )    K out ( z )  

(12)

   Gm  ζ −1   Gout ( z )     G ( z ) =Gout ( z ) 1+   Gm  −1   1+ β ( z )(1−ζ )  G ( z )   out  

(13)

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ACCEPTED MANUSCRIPT in which vout ( z ) designates the Poisson's ratio of the matrix phase given by

vout ( z ) =

3K out ( z ) − 2Gout ( z ) 2 [ 3K out ( z ) +Gout ( z ) ]

(14)

where

β ( z )=

1+ vout ( z ) 3(1−vout ( z ))

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d 0 ( z )=

2(4−5vout ( z )) 3(1−vout ( z ))

(15)

(16)

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Since the CNTRC turns out to be isotropic [30], the effective Young’s modulus E ( z ) and Poisson’s ratio

E ( z) =

9 K ( z )G ( z ) 3K ( z ) + G ( z )

v( z ) =

3K ( z ) − 2G ( z ) 6 K ( z ) + 2G( z )

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v ( z ) of the nanocomposite are denoted by

(17)

(18)

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In the aforementioned relations, Vr ( z ) and Vm ( z ) are the volume fraction of the reinforcing phase and the matrix, which satisfy the relation of Vr ( z ) + Vm ( z ) = 1 . The volume fraction of CNTs Vr ( z ) is characterized * by a smooth variation from the inner to the outer surface of the shell assuming that Vr ( z ) = VCNT VCNT ( z ) ,

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* where VCNT is the volume fraction of CNTs [81] and depends on the mass fraction of nanotubes, w CNT , and

* CNT

V

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density of both CNT ρCNT and the matrix ρm

 ρ  ρ =  CNT − CNT + 1 ρm  wCNT ρm 

−1

(19)

while VCNT ( z ) denotes the CNT distribution through the shell’s thickness. In this study, it is proposed that the CNT volume fraction follows sigmoidal power-law distribution

e sr% − 1 VCNT ( z ) = s 2 ( e −1)( es(r%−0.5) + 1)

(20)

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ACCEPTED MANUSCRIPT in which r% = ( z + h 2 ) h , the sigmoid exponent

s

controlling the CNT variation profile through the radial

direction. This profile exhibits sigmoidal variation for different values of sigmoid exponent, as depicted in Fig. 2. The CNT volume fraction shows a linear variation from 0 to 1 in radial direction if sigmoid exponent

s = 1. As the

s inclines, the continuously graded shell turns to be a laminated shell with two lamina,

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* whose CNT volume fraction of inner and out surfaces are 0 and VCNT . In the present work, the widely used

CNT profiles in the available literature [9,25,41], i.e., Profile-X, Profile- ∧ , and Profile-O also take into account. Note that sigmoidal profile with

s =1

and Profile- ∧ are symmetric with respect to the mid-

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surface of the shell. As reported in [27], CNT volume fractions with symmetric distributions through the thickness of the structure have higher capabilities to reduce/increase the natural frequency than those of

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uniformly and asymmetric distributions.

2.3. Fundamental Equations

On the basis of the TSDT of Reddy [11], the displacement components of the mid-surface of the shell

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along the x, y, and z axes, designated by u , v , and w may be expressed as

u ( x, y , z ) = u0 ( x, y ) + zψ ( x, y ) + z 2θ x + z 3 λx

v( x, y, z ) = v0 ( x, y) + zφ ( x, y ) + z 2θ y + z 3λy

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w( x, y, z ) = w0 ( x, y )

(21)

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where u 0 , v0 , and w0 are the displacement on the mid-surface of the shell, ψ and φ represent mid-surface rotations of transverse normal about x and y axes, respectively. The above relations are reduced by satisfying the

stress-free

conditions

on

the

inner

and

outer

surfaces

of

the

shell,

i.e

σ xz ( x, y, ± h 2) = σ yz ( x, y, ± h 2) = 0 . As a result, one gets u ( x, y, z ) = u0 ( x, y) + zψ ( x, y ) − C1 z 3 (ψ + w0, x ) v( x, y, z ) = v0 ( x, y) + zφ ( x, y ) − C1 z 3 (φ + w0, y )

(22)

w( x, y, z ) = w0 ( x, y )

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ACCEPTED MANUSCRIPT in which C1 = − 4 3h 2 . The nonlinear strain-displacement expression are given by

ε xx(3)  ε xx  ε xx(0)  ε xx(1)     (0)   (1)  3  (3)  ε yy  = ε yy  + z ε yy  + z ε yy     (0)   (1)   (3)  γ xy  γ xy  γ xy  γ xy 

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(0) (2) γ xz  γ xz  2 γ xz    =  (0)  + z  (2)  γ yz  γ yz  γ yz 

and

ε xx(1)  ψ , x   (1)    ε yy  = φ, y   (1)    γ xy  ψ , y + φ, x 

ε xx(3)  ψ , x + w0, xx     (3)  ε yy  = −C1 φ, y + w0, yy ,  (3)    ψ , y + φ, x + 2w0, xy  γ xy 

γ xz(0)  ψ + w0, x  γ xz(2)  ψ + w0, x   (0)  =   ,  (2)  = −3C1   γ yz  φ + w0, y  γ yz  φ + w0, y 

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ε xx(0)  u0, x + w0,2 x 2   (0)    2 ε yy  = v0, y + w R + w0, y 2  ,  (0)    u + v + w0, x w0, y  γ xy   0, y 0, x

(23)

(24)

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where ε xx and ε yy are the normal strains and γ xy , γ xz and γ yz are the shear strains. Here, a comma denotes the partial derivative. According to the 2-D Hooke’s law, the constitutive equation of the continuously graded CNTRC SCS is elaborated by

0

0

0

Q44

0

0

Q55

0  ε xx    0  ε yy    0  γ yz   0  γ xz    Q66  γ xy   

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0

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σ xx   Q11 Q12    σ yy  Q21 Q22   0 τ yz  =  0    0 0 τ xz   0 τ xy   0  

0

0

(25)

where [ Q ] is the material stiffness matrix. Opposed to most available works [15-17] in which the stiffener effects were smeared out over the respective stiffener spacing, or in other words, no discrete stiffener effects were treated, in the present paper, the stiffeners are modeled as Euler-Bernoulli beams, and we assume that the normal to the CS and stiffener prior to bending remains straight after deformation. Accordingly, the displacement fields of the Euler-Bernoulli beam are specified as [12]

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∂wI ( x , y )  u ( x , y ) = u I ( x , y ) − z ∂x   w( x , y ) = wI ( x , y )

For stringer:

∂wI ( x , y )   u ( x , y ) = vI ( x , y ) − z ∂y   w( x , y ) = w ( x , y )  I

(27)

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For ring:

(26)

where u I , vI , and wI denote the displacements components at I-th nodal location through the CS thickness.

SC

The axial and circumferential strains, ε a and ε c , associated with rings and stringers are given, respectively,

εa =

∂vI 1 ∂wI 2 wI ∂2w + ( ) + − z ( 2I ) R ∂y 2 ∂y ∂y

εc =

∂uI 1 ∂wI 2 ∂2w + ( ) − z ( 2I ) ∂x 2 ∂x ∂x

M AN U

by

(28)

(29)

The Hook’s law for rings and stringers is defined as

TE D

σ xx = Es ε s σ yy = Er ε r

(30) (31)

where the subscripts “r” and “s” refer to ring and stringer stiffeners, respectively, and Ei is the

1 2S S

Ur =

1 2Sr

L

∫∫

α

 ∂ 2 wI   ∫ σ S ε S dA + GS J S (  dy dx A  ∂ x ∂ y  S 

(32)

AC C

US =

EP

Young's modulus. The strain energy of the stringer and ring stiffeners are expressed, respectively, by

0

0

L

α

0

0

∫ ∫

 ∂ 2 wI 2  σ ε dA + G J ( ) dy dx  ∫Ar r r r r ∂x∂y  

(33)

Substituting Eqs. (28) and (29) into Eqs. (32) and (33), one gets

  ∂u I  1 U = E A   s 2 S ∫0 ∫0  s s  ∂x    s  L 2π

 1  ∂wI  +   2  ∂x

  

2

 ∂2w I +E I  s s   ∂x 2

12

   

2

    

2

ACCEPTED MANUSCRIPT

   ∂vI 1 U = E A   r 2 S ∫0 ∫0  r r  ∂y   r  L 2π

 ∂2w I  −2 Z E A r r r  2 ∂y 

 wI 1  ∂wI  +   +    R 2  ∂y 

2   ∂v ∂w w   I + 1  I  + I   ∂y 2  ∂y  R   

2

   

2

   dydx  

 ∂2w I +E I  r r  ∂y 2 

 ∂2w  I  +G J r r    ∂x∂y  

where

h + hs , 2

Zr =

h + hr , 2

JS =

hs bs 3 , 3

Jr =

hr br 3 3

IS =

(36)

bs hs 3 + Z S2 AS , 12

2

2

    

2

   dydx  

(35)

Ir =

br hr 3 + Z r2 Ar 12

M AN U

ZS =

   

    

(34)

RI PT

2   ∂u  ∂2w  ∂w I   I + 1  I   + G J  s s  ∂x∂y   ∂x 2  ∂x       

SC

 ∂2w I −2 Z E A  s s s  2 ∂x 

TE D

in which S = 2π R N s , S = L N r , N r , and N s are the spacing of the ring and stringer stiffeners, r s respectively. In addition, the I r and I s represent the moment of inertia of the ring and stringer with respect

EP

to the mid-surface, respectively.

The forces and moments of the CS can be expressed by

AC C

 N xx  h 2 σ xx       N yy  = ∫ σ yy  dz ,  N  −h 2 τ   xy   xy 

 Pxx  h 2 σ xx      3  Pyy  = ∫ σ yy  z dz ,  P  − h 2 σ   xy   xy 

 M xx  h 2 σ xx       M yy  = ∫ σ yy  zdz ,  M  − h 2 σ   xy   xy 

Qxz  h 2 σ xz    = ∫  dz , Qyz  − h 2 σ yz 

 Rxz  h 2 σ xz  2   = ∫  z dz  Ryz  − h 2 σ yz 

(37)

13

ACCEPTED MANUSCRIPT Substituting Eqs. (23) and (25) into Eq. (37), we obtain the following expressions for force and moment resultants of the continuously graded CNTRC SCS as

N xx = A11ε xx(0) + B11ε yy(0) + A22ε xx(1) + B22ε yy(1) + A44ε xx(3)

N xy = D11γ xy(0) + D22γ xy(1) + D44γ xy(3) (0) M xx = ( A22ε xx(0 ) + B22ε yy ) + ( A33ε xx(1) + B33ε yy(1) ) + ( A55ε xx(3) + B55ε yy(3) )

SC

( 0) M yy = ( B22ε xx(0 ) + A22ε yy ) + ( B33ε xx(1) + A33ε yy(0) ) + ( B55ε xx(3) + A55ε yy(3) )

RI PT

( 0) N yy = ( B11ε xx(0 ) + A11ε yy ) + ( B22ε xx(1) + A22ε yy( 0) ) + ( B44ε xx(3) + A44ε yy(3) )

M xy = D22γ xy(0) + D33γ xy(1) + D55γ xy(3)

M AN U

( 0) Pxx = ( A44ε xx(0 ) + B44ε yy ) + ( A55ε xx(1) + B55ε yy(1) ) + ( A77ε xx(3) + B77ε yy(3) ) (0) Pyy = ( B44ε xx( 0) + A44ε yy ) + ( B55ε xx(1) + A55ε yy(0 ) ) + ( B77ε xx(3) + A77ε yy(3) )

Pxy = D44γ xy(0) + D55γ xy(1) + D77γ xy(3)

Qyz = D11γ yz(0) + D33γ yz(2) Rxz = D33γ xz(0) + D55γ xz(2)

AC C

in which

(38)

EP

Ryz = D33γ yz(0) + D55γ yz(2)

TE D

Qxz = D11γ xz(0) + D33γ xz(2)

h2

{ A11 , A22 , A33 , A44 , A55 , A77 } = ∫

Q11 ( z )(1, z , z 2 , z 3 , z 4 , z 6 ) dz

−h 2

h 2

{B11 , B22 , B33 , B44 , B55 , B77 } = ∫

Q12 ( z )(1, z , z 2 , z 3 , z 4 , z 6 ) dz

−h 2

h 2

{D11 , D22 , D33 , D44 , D55 , D77 } = ∫

Q44 ( z )(1, z , z 2 , z 3 , z 4 , z 6 ) dz

−h 2

14

(39)

ACCEPTED MANUSCRIPT The equilibrium equations of the continuously graded CNTRC SCS under mechanical loadings shall be derived on the basis of the stationary potential energy criterion. The total potential energy of a CS stiffened by ring and stringer stiffeners is defined as [12]

V =U +Ω

(40)

RI PT

where Ω designate the potential energy of the applied loads, and U is the total strain energy of the SCS, which is given by

U = U Sh + U r + U S

(41)

SC

where U Sh represents the strain energy of the CS. The total strain energy U Sh for the CS based on the TSDT may be written

U sh = ∫

∫ ∫

(σ xxε xx + σ yyε yy + τ xy γ xy + τ xz γ xz + τ yz γ yz ) dz dy dx

M AN U

L 2π R h 2

0 −h 2

0

(42)

The potential energy of the loads, Ω , for a conservative system is defined as the negative of work done by applied loads as the shell is deformed. Accordingly, for the lateral pressure, N 0 y and axial compressive edge

L 2π R

∫ (N

1 Ω= ∫ 20

0

0x

TE D

load, N 0x , it is defined by [29]

+ N 0 y ) ( ( w, x )2 + (v, x )2 ) dydx

(43)

EP

2 It is worth noting that the contribution of the (v, x ) term in the buckling of the CS with short or moderate

length is negligible. In other words, the shortening owing to circumferential displacement of the CS under

AC C

mechanical buckling can be omitted for the CSs with length-to-radius ratio between 0.7 and 3 [82]. Substituting Eqs. (34-35), (42-43) into Eq. (41), the total potential energy of the SCS is given by L 2π R

U = R∫

∫

0

0

(

(

1 (0)2 (0) 2 (0) (0) (0) 2 (0)2 (0) 2  A11ε xx + A11ε yy + 2 B11ε xx ε yy + D11 γ xy + γ xz + γ yz 2

(

))

1 +  A22ε xx(1) ε xx(0) + A22ε yy(1) ε yy(0) + B22ε yy(1) ε xx(0) + B22ε xx(1) ε yy(0) + D22 γ xy(0) γ xy(1) 2

(

1 +  A33ε xx(1) 2 + A33ε yy(1) 2 + 2 B33ε xx(1) ε yy(1) + D33 γ xy(1) 2

2

) + 2D

15

33

)

  

 (γ yz(1) γ yz(0) + γ xz(1) γ xz(0) )  

ACCEPTED MANUSCRIPT

(

(

(3) (0) + A44ε xx(3)ε xx(0) + A44ε yy ε yy + B44ε yy(3)ε xx(0) + B44ε xx(3)ε yy(0) + D44 γ xy(3) γ xy(0)

))

(

 +  A55ε xx(3)ε xx(1) + A55ε yy(3)ε yy(1) + B55ε yy(3)ε xx(1) + B55ε xx(3)ε yy(1) + D55 γ xy(3) + γ xy(1) 

(

D55 (1) 2 γ xz + γ yz(1) 2 2

)

(

 1 D77 (3) (3)2 (3)2 (3) (3) γ xy  +  A77ε xx + A77ε yy + 2 B77ε xx ε yy + 2  2 2

 ∂u1 1  ∂wI  2   ∂ 2 wI   ∂ 2 wI 1  + E A + + E I − 2 Z A E   s s S S  S S S  2  2  ∂x 2  ∂x   2 S S   ∂x   ∂x   2

2

) 

RI PT

+

)

  ∂uI 1  ∂wI  +      ∂x 2  ∂x 

2

  

 ∂ 2 w   ∂vI wI 1  ∂wI  −2Zr Er Ar  2I   + +    ∂y   ∂y R 2  ∂y 

2

M AN U

SC

2 2 2  ∂v w 1  ∂w 2   ∂2wI   1    ∂ w I I I I +GS JS    +  Er Ar  + + +    + Er Ir  2  ∂ x ∂ y 2 S ∂ y R 2 ∂ y ∂ y        r  

2

2

 ∂ 2 wI     + G J    dydx  r r   ∂x∂y   

(44)

The equilibrium equation of continuously graded CNTRC SCS can be obtained by the variational approach. Employing the Euler equation [83], the general equilibrium equations of the SCS based on TSDT are

N xx , x + N xy , y

TE D

resulted as

2  ∂u I 1  ∂u I  2  ∂  1   ∂wI  +  + − Z S AS ES  Es As   =0  ∂x 2  ∂x   ∂x  S x   ∂x   

AC C

EP

 ∂v w 1  ∂w 2   ∂ 2WI    ∂  1  I I I   N xy , x + N yy , y +  E A + + − Z A E =0 r r  ∂y R 2  ∂y   r r r  ∂y 2    ∂y  Sr     ∂v w 1  ∂w 2   ∂ 2 wI   1 1  I I I   − N yy − E A + + − Z A E + N xx w0, xx r r  ∂y R 2  ∂y   r r r  ∂y 2   R RS r   

∂ 2 wI ∂2w 1 + N yy w0, yy + 2 N xy w0, xy + N0 y ( 2 ) + N xx − ∂y SS ∂x 2 + N0 x (

∂ 2 wI ∂2 w 1 ) + N + Qxz , x + Qyz , y − 3C1 ( Rxz , x + Ryz , y ) yy ∂x 2 Sr ∂y 2

∂2  1  +C1 ( Pxx , xx + 2 Pxy , xy + Pyy , yy ) − 2   ∂x  Ss 

3 2   h    ∂ wI   ES I S + Z S ES AS C1      2  2    ∂x  

16

(45)

(46)

ACCEPTED MANUSCRIPT 3 2 3 2   h    ∂u 1  ∂w     ∂  1   h  −  Z S AS ES + ES AS C1     I +  I     − 2  + E I Z E A C   r r r r r 1     2    ∂x 2  ∂x     ∂y  S r    2   

2 3 2  ∂ 2 wI   ∂ 2 wI  h    ∂vI 1  ∂wI  wI    ∂  1   Z A E E A C G J − + + + − ( )     r r r r r 1   S S   2 R    ∂x∂y  S S ∂x∂y  2    ∂y 2  ∂y   ∂y  

 h ∂  1  h − C1    ES AS   ∂x  S s  2  2

M AN U

3 2  ∂u I 1  ∂wI  2  h  h    ∂ wI    +  − Z A E − c     S S S  1   2    = 0   x x ∂ 2 ∂ 2 2          ∂x   

−Qyz + 3C1 Ryz + M xy , X + M yy , y − C1 ( Pxy , x + Pyy , y ) +

2

 h ∂  1  h − C1    Er Ar   ∂y  S r  2  2

(48)

3

3 2  ∂v w 1  ∂w 2  h  h    ∂ wI    I I I  −Z A E  −c + + =0   ∂y R 2  ∂y   r r r  2 1  2    ∂y 2       

TE D

(47)

  

SC

−Qxz + 3C1 Rxz + M xx , x + M xy , y − C1 ( Pxx , x + Pxy , y ) +

RI PT

∂ 2 wI  1 + (Gr J r ) =0 Sr ∂x∂y 

  

(49)

in which N xx and N yy are the applied forces to the ring and stringer stiffeners given by

N xx = ∫ σ xx dA =

h 2 + hs

AS

∫

N yy = ∫ σ yy dA =

h 2 + hr

∫

h2

2

(50)

2

 ∂2w  ∂v 1  ∂w  w brσ r dA = Er Ar ( I +  I  + I ) − Z r Ar Er  2I  ∂y 2  ∂y  R  ∂y 

(51)

AC C

AS

EP

h2

 ∂ 2 wI  ∂uI 1  ∂wI  bsσ s dA = Es As ( +   ) − Z s As Es  2  ∂x 2  ∂x   ∂x 

Taking the variational approach into account, the stability equations of continuously graded CNTRC SCS can be obtained. Using Taylor series, the total strain energy of the shell can be expanded about the equilibrium state as

1 1 ∆V = δ V + δ 2V + δ 3V + ....... 2 6

(52)

in which the first variation, δ V ,is correlated to the equilibrium state. The stability condition of the configuration of the SCS in the vicinity of the state of the equilibrium shall be determined by the sign of

17

ACCEPTED MANUSCRIPT second variation, δ V . Additionally, the δ V = 0 leads to the stability equations of the buckling problem 2

2

[83]. To proceed with the stability equations of the CNTRC SCS, the displacement components are expressed by

u0 = u00 + u10 ,

v0 = v00 + v01 ,

φ = φ 0 + φ1

RI PT

ψ = ψ 0 +ψ 1 ,

w0 = w00 + w01 (53)

where u 00 , v00 , w00 , ψ 0 , and φ 0 represent the displacement components of the state of the equilibrium. The

SC

displacement fields of the vicinity of the stable equilibrium are expressed by u 10 , v01 , w10 , ψ 1 , and φ 1 with regard to the position of the equilibrium. Consequently, two items denoting the stable equilibrium and the neighboring state form the stress resultants. It is worth noting that the applied load imposing on the

M AN U

configuration is deemed to be the critical buckling load if the δ 2V = 0 is satisfied. Substituting Eqs. (23) and (25) into Eq. (52) and using Eq. (52) and keeping the second order terms, one can derive the second variation of the potential energy, which upon employing the Euler equations, leads to the stability equations of the SCS reinforced by aggregated graded CNTs as

∂  1  ∂u1I ∂ 2 w1I   E A ( ) − Z A E ( )  =0  s s s ∂x  S  s s ∂x ∂x 2   s

(54)

∂  1 ∂v1I w1I ∂ 2 w1I  ( E A ( + ) − Z A E ( ) =0 r r r r r ∂y R ∂y  S ∂y 2  r

(55)

EP

N 1xy , x + N 1yy , y +

TE D

N 1xx , x + N 1xy , y +

AC C

∂v1I w1I ∂ 2 w1I  1 1 1  N yy − E A ( + ) − ZE A ( ) +  2 R RSr  r r ∂y R r r r ∂y  N 0 x w1, xx + 2 N xy0 w1, xy +

1 0 ∂ 2 w1I 1 0 ∂ 2 w1I N xx + N yy + S ∂x 2 S ∂y 2 s r

N0 y w1, yy + Q1xz + Q1yz − 3C1 ( R1zx , x + R1yz ) + C1 ( Pxx1 , xx + 2 Pxy1 , xy + Pyy1 , yy ) − ∂2  1   h   Es I s + ZEs As C1   2  ∂x  S   2 s s

3

 ∂ 2 w1I  h  ( 2 ) −  ZEs As + Es As C1   2  ∂x  s

18

3

 ∂u1I   )  − (  ∂x  

ACCEPTED MANUSCRIPT ∂2  1   h   Er I r + ZEr Ar C1   2  ∂y  S   2 r r

3

1 1  ∂ 2 w1I   h  ∂vI WI   ( ) − ZE A + E A C ( + )  −   2 r r 1  2  ∂y R    ∂y  r r r 3

∂2 1 ∂ 2 w1I 1 ∂ 2 wI [ (G J )+ (G J )] = 0 s s ∂x∂y S r r ∂x∂y ∂x∂y S s r 1 1 1 1 − Q 1xz − 3C1 R xz1 + M xx , x M xy , y − C1 ( Pxx , x + Pxy , y ) + +

∂  1   h h − C1      ∂x  S s   2 2

3

  h ∂wI 1 h ( ) − Z S AS ES  − C1   E A  s s ∂x 2   2

3

  h ∂vI 1 wI 1 h E A ( + ) − Z r Er Ar  − C1    r r R ∂y 2   2

3

M AN U

∂  1   h h − C1      ∂y  S r   2 2

 ∂ 2 wI 1   ( 2 )   = 0  ∂x  

SC

− Q 1yz + − 3C1 R 1yz + M 1xy , x M 1yy . y − C1 ( Pxy1 , x + Pyy1 , y ) + +

3

RI PT

(56)

 ∂ 2 wI 1   ( 2 )   = 0  ∂y  

(57)

(58)

0

where the superscript “1” denotes the state of stability. In aforementioned expressions, N xx0 and N yy represent the pre-buckling force resultants of the SCS reinforced by aggregated graded CNTs which deemed

2.4. Solution Methodology

TE D

that there are no shear and axial forces along y-direction.

In this section, the stability equations, Eqs. (54)-(58), is numerically solved for the CNTCR SCS

EP

reinforced by aggregated graded CNTs subjected to axial and radial compressive loadings to determine the 0

critical buckling load. Herein, it has been assumed that the forces N xx0 and N yy in x and y directions,

N xx0 = −

χ=

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respectively, with following assumption are obtained as follows

F

2π R

0 , N yy = − PR

P 2π R 2 F

(59)

The two edges of the SCS are assumed to be simply supported and, as a result, the boundary conditionsare given by

v10 = w10 = M xx1 = N xx1 = φ 1 = 0 at x = 0, L

(60)

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ACCEPTED MANUSCRIPT According to the simply supported boundary conditions, the following approximate solution is proposed for the displacement fields as follows

ny ) R ny v1 = v0 n ( x) cos( ) R ny w1 = w0 n ( x) sin( ) R ny ψ 1 = ψ n ( x) sin( ) R ny ϕ1 = ϕ n ( x) cos( ) R

(61)

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u1 = u 0 n ( x) sin(

where n=1, 2,... is the number of half waves in y-direction. At this step, the partial derivatives obtained by

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substituting Eq. (61) into the Eqs. (54)-(58) have to be specified in their discrete form by means of the GDQM. The interested reader shall find the basic aspects and all explanations of GDQM in [37,77]. In compact vector notation, the discrete form of stability equations along with the associated boundary conditions can be conveniently written as

[ Abd ]  db  + F  0  [ Add ] d d    Ag 

0   db    =0  Ag2    d d  

(62)

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[ Abb ]  [ Adb ]

1

in which subscripts ‘d’ and ‘b’ denote domain and boundary, respectively. Afterwards, through the

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condensation of boundary degrees of freedom, the critical buckling load of a SCS reinforced by graded aggregated CNTs is acquired from

] − [ Add ][ Abb ] [ Abd ]){d d } + F (  Ag −1

dd

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([ A

2

 −  Ag1  [ Abb ]

−1

[ Abd ]) = 0

(63)

3. Results and Discussion In the present work, the mechanical buckling behavior of the CNTCR SCS reinforced by aggregated graded CNTs subjected to combined compressive loadings is studied, and the impacts of several parameters including material profiles, ring and stringer stiffeners on critical buckling loads of the shell are presented and analyzed in detail. In the following, firstly, the reliability and versatility of the present methodology are evaluated by conducting comparison studies.

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3.1. Validation of the Present Study Owing to lack of relevant results required for mechanical buckling of the continuously graded CNTRC CS reinforced by ring and stringer stiffeners for direct comparison, validation of the presented formulation is

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conducted in such a way that the results are compared with those of isotropic stiffened and un-stiffened CS, and functionally graded SCS. The critical buckling load of an isotropic stiffened and un-stiffened CS composed by alumina in the work by Almroth and Brush [83] is taken into account for the first comparison

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case. The reference geometry for this comparison case is a CS with L R = 1 and R = 0.3 . As can be seen in Table 1, the results obtained from this comparison are in good agreement.

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In Table 2, the accuracy of the presented work using two-parameter EMT is compared with those of the ERM by Shen [74] with graded aligned, straight CNTs. In this table, Poly (methyl methacrylate), referred to as PMMA, is picked for the matrix, and the (10, 10) SWCNTs are chosen as reinforcements. What is more, the temperature is assumed to be T = 300 K (room temperature) at which there are no thermal strains. The un-stiffened CS with L = 160.9 mm is assumed subjected to an axial load. It can be found from Table 2 that

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there is well agreement between the results confirming the accuracy of the present methodology. According to Table 2, it is worth noting that critical buckling load determined by EMT for the two types of CNTs

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profiles and UD-profile are higher than those of the ERM for different CNT volume fractions.

3.2. Parametric Studies

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After evaluating the reliability of the present methodology, detailed parametric studies for mechanical buckling analysis of CNTCR SCS reinforced by aggregated graded CNTs subjected to axial and lateral loads for different types of CNT volume fraction profiles and various mid-radius to thickness ratio, length-to-mean radius ratio, and exterior and interior stiffeners are computed. For all results presented here, the buckling load intensity factor is expressed in terms of a parameter IFcr = Fcr L2 π 2 Dc in which Dc is a rigidity modulus of the CS composed of ceramic material which is equal to Dc = Ec h3 12(1 − vc 2 ) . It is assumed that the stiffeners are made by metal with E = 70 GPa and v = 0.3 .The (10,10) SWCNTs as the reinforcement phase and Alumina as the ceramic matrix [84,85] are chosen. Han and Elliott [86] picked for

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ACCEPTED MANUSCRIPT ECNT 11 = 600 GPa , ECNT 22 = 10 GPa , GCNT 12 = 17.2 GPa for (10,10) SWCNTs. Such a low value of Young’s modulus, as reported by [87], is due to the fact that the effective thickness of CNTs is assumed to be 0.34 nm or more. Notwithstanding, the effective thickness of SWCNTs should be smaller than 0.142 nm [88]. Accordingly, the material properties and effective thickness of SWCNTs used for the present work

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suitably are chosen based on the MD simulation results reported by Shen and Zhang [89]. The material properties of the (10,10) SWCNTs are ECNT 11 = 5.6466 TPa , ECNT 22 = 7.08 TPa , GCNT 12 = 1.9445 TPa and vCNT 12 = 0.175 at room temperature (300 K). Note that the effective wall thickness for (10,10)-tube is

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0.067 nm, which satisfies the Vodenitcharova-Zhang criterion [88], and the value of 0.34 nm for tube wall thickness is an unacceptable assumption for SWCNTs. The CNT volume fraction used in this section is

deriving the critical buckling load.

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* VCNT = 0.17 unless the otherwise mentioned. It should be noted that the n=1 and χ =1 have been used for

Fig. 3 demonstrates the effect of mid-radius to the thickness ratio, S , on the buckling load intensity factor of the SCS reinforced by agglomerated graded CNTs for various CNT profiles. Based on the achieved

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results, the critical buckling load increases with the increase of the S ratio, although the increase rate of the intensity factor is much more sensible for the thicker CS. Furthermore, the highest value of the intensity factor can be achieved by Profile-X. This implies that by continuously distributing of CNTs close to the

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inner and outer shell’s surface leads to enhancing the stiffness of the CS and, as a result, increasing the critical mechanical buckling load. It is interesting to note that the effect of the CNTs distribution through the

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shell’s thickness on the critical buckling load is substantially more evident for thinner shells. On the other hand, by comparison the results for Profile- ∧ and sigmoidal profile, it can be inferred that the graded distribution of the CNTs close to the surface, where contains higher amount of CNT volume fraction, results in increasing the critical mechanical buckling load. Fig. 4 presents the variation of buckling load intensity factor of the continuously graded CNTRC SCS with L R and S ratios. In this figure, it is assumed that the aggregation parameters are equal to each other and, as a result, this implies that there is no aggregation within the matrix phase and all the nano particles are scattered in the matrix. It can be inferred from Fig. 4that the variation of the L R ratio on the critical buckling load is more drastic for thinner SCSs. By

22

ACCEPTED MANUSCRIPT IF increasing L R ratio, the cr factor rises, representing higher discrepancies among intensity factors of SCSs with different values of S ratio. The influence of the aggregation parameter ζ on the buckling load intensity factor of the continuously graded CNTRC SCS for different values of sigmoidal parameter s is shown in Fig. 5. It can be seen in Fig. 5

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that by increasing the aggregation parameter ζ , which leads to the decrease of CNTs aggregation within the ceramic matrix, the intensity factor dramatically increases with higher rate at first, followed by lower

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increase rate for higher values of the ζ . From this figure, one can find the advantage of grading of CNTs through the thickness of the shell in increasing the critical buckling load compared with that of the discretely

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laminated composite. Increasing the sigmoidal parameter, s , the buckling response of the SCS approaches to that of the two-lamina composite such that the CNT volume fractions of the outer and inner laminasare 17 percent and 0 percent, respectively. In Fig. 5, the influences of the sigmoidal parameter, s , and the midradius to the thickness ratio, S , on the intensity factor of the SCS reinforced by agglomerated graded CNTs are demonstrated. As can be seen from Fig. 5, the consequence of the smoothly gradation of the CNTs on

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the critical buckling load can be found more appreciable in the SCS.

The effects of the different types of CNT profiles on the buckling load intensity factor of the SCS reinforced by agglomerated graded CNTs for several values of L R ratio are compared in Fig. 6. According

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to Fig. 6, the discrepancy among the intensity factors of various kinds of CNT profiles becomes more

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evident by increasing L R ratio. Table 3 illustrates the results obtained for the SCS reinforced by agglomerated graded CNTs for several values of the number of the ring and stringer stiffeners. In particular, the results for either exterior or interior stiffeners are compared. Certain important observations apparent from Table 3 are as follows: (i) with the increase in the number of the stiffeners, the buckling load intensity factor increase for various types of CNT profiles. (ii) The CS stiffened by exterior stiffeners has higher critical buckling load than that stiffened by interior stiffeners. Fig. 7 shows the variation of buckling load intensity factor with aggregation parameter ξ for different CNT profiles. According to this figure, it is interesting to note that the increase of the aggregation parameter ξ , or in other words the increase of the

23

ACCEPTED MANUSCRIPT CNTs tendency to be aggregated, has a negative effect on the mechanical buckling behavior of the continuously graded CNTRC SCS. In addition, it can be observed that the effect of the various types of CNT profiles on the critical buckling load is more significant for the SCS with lower aggregation degree of CNTs within the ceramic matrix phase.

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The influences of sigmoidal parameter, s , on the intensity factor of SCS reinforced by agglomerated graded CNTs for various values of the S ratio and the aggregation parameter ξ are presented in Figs. 8 and 9. According to these figures, by increasing the sigmoidal parameter and accordingly approaching to the

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laminated composites, the critical mechanical buckling of the SCS decreases. It is worth noting that this condition becomes more appreciable for higher values of the mid-radius to the thickness ratio, S .

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Fig. 10 presents the impact of volume fraction of the CNTs on the buckling load intensity factor of the SCS reinforced by agglomerated graded CNTs for various values of the aggregation parameter ζ for the sigmoidal profile. This figure evinces that by inclining the CNTs volume fraction, the buckling load

5. Conclusion

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intensity factor considerably increases.

This paper presented the mechanical buckling analysis of the continuously graded CNTRC SCS

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aggregated by CNTs subjected to axial and lateral loads.Taking the aggregation aspect of CNTs in nanoscale into account, a two-parameter EMT approach was adopted to predict effective material properties of

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the CNTRC. Notwithstanding most available studies in which the stiffener effects were smeared out over the respective stiffener spacing, in this work, the stiffeners have been modeled as Euler-Bernoulli beams. Based on TSDT of Reddy,the equilibrium equations of the CS stiffened by rings and stringers modelled as EulerBernoulli beams were obtained using variational approach. The stability equations derived by employing adjacent equilibrium criterion were solved with making use of the GDQM as well as trigonometric expansion. Based on the obtained simulation results, the following conclusions can be made: • By continuously distributing of CNTs close to the inner and outer shell’s surface leads to enhancing the stiffness of the CS and, as a result, increasing the critical mechanical buckling load.

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ACCEPTED MANUSCRIPT • Results indicate that the variation of the L R ratio on the critical buckling load is more drastic for thinner SCSs. • The numerical results reveal that, the effect of the various types of CNT profiles on the critical buckling load is more significant for the SCS with lower aggregation degree of CNTs within the ceramic matrix

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phase. • It has been inferred that the aggregation tendency of the CNTs has a negative effect on the mechanical buckling behavior of the continuously graded SCS.

capable of increasing the critical buckling load of the SCS.

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• Compared with the discretely laminated composites, smoothly grading of CNTs through the thickness is

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The authors encourage the extension of current work to study more features in the buckling modeling of ring and stringer stiffened cylindrical shells such as the various boundary conditions, and buckling mode

Acknowledgments

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shapes.

The authors would like to acknowledge the research support CNPq (ConselhoNacional de

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DesenvolvimentoCientífico e Tecnológico) with grant number of 164774/2014-9, and of FAPERJ

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(Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro) with grant number of E/26/201.453/2014.

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nanotube-reinforced composite plates. Materials & Design. 2010 31;31(7):3403-11.

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Table 1. Comparison of the critical buckling load for various values of mid-radius to thickness ratio, R h . ( L R = 1, R = 0.3m ).

R h

Ref. [83]

Stiffened Present

Ref. [83]

26.3023

26. 4529

26.2390

26.5272

50

9. 5116

9.4460

9.5533

100

2. 3891

2.3615

2.4076

200

0.5981

0.5904

0.6086

300

0.2663

0.2623

0.2730

9.4827

2.3659

0.5988 0.2672

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Un-stiffened Present

Table 2. Comparisons of critical axial buckling load (kN) for un-stiffened continuously graded CNTRC CS with aligned and straight CNTs ( R h = 100 , h = 1mm , Z =

VCNT = 0.12

VCNT = 0.17

*

*

Ref. [74] Present

18.75 19.82

Z = 300

Z = 500

18.72 19.91

AC C

Ref. [74] Present

19.35 20.00

Uniform profile

VCNT = 0.28 *

Profile-X

Uniform profile

Profile-X

21.81 22.93

30.43 31.57

35.53 36.45

37.77 39.01

47.18 48.80

22.06 23.33

31.11 32.09

37.06 38.13

39.60 40.55

46.52 48.87

30.57 32.11

35.14 36.46

37.31 38.29

45.99 46.80

EP

Ref. [74] Present

Profile-X

TE D

Uniform profile

Z = 100

L2 Rh , T = 300 K ).

21.37 22.14

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Table 3. Buckling load intensity factor of the SCS reinforced by agglomerated graded CNTs for various CNT profiles

and the number of the ring and stringer stiffeners, Ni ( L R = 1 , S

Ni

Interior stiffeners Profile-X Profile- ∧

Profile-O

Uniform profile

Sigmoidal profile

ξ = 1,

s = 1 ).

Exterior stiffeners Profile-X Profile- ∧

Profile-O

Uniform profile

Sigmoidal profile

0.9506 0.9519 0.9527 0.9533

0.8467 0.8480 0.8488 0.8494

0.9125 0.9139 0.9146 0.9152

0.8295 0.8308 0.8315 0.8321

1.7533 1.7693 1.7782 1.7853

1.5589 1.5749 1.5838 1.5909

1.6792 1.6952 1.7041 1.7112

1.5269 1.5424 1.5510 1.5578

6 15 20 24

0.8903 0.8906 0.8907 0.8909

0.9497 0.9498 0.9501 0.9503

0.8457 0.8458 0.8459 0.8461

0.9116 0.9119 0.9122 0.9125

0.8286 0.8288 0.8289 0.8291

0.8912 0.8926 0.8934 0.8941

6 15 20 24

1.6305 1.6308 1.6309 1.631

1.7428 1.7431 1.7433 1.7434

1.5484 1.5486 1.5488 1.5489

1.6687 1.6690 1.6692 1.6693

1.5167 1.5171 1.5173 1.5174

1.6416 1.6583 1.6676 1.6750

AC C

EP

TE D

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10

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Fig. 1. Schematic plot of the SCS aggregated by CNTs.

Fig. 2. Through-the-thickness sigmoidal distribution of CNT volume fraction for various sigmoid exponents.

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Fig. 3. Effect of mid-radius to the thickness ratio, S , on the buckling load intensity factor of the SCS reinforced by agglomerated graded CNTs; Green line: N r = N s = 7 , Red line: N r = N s = 24

AC C

EP

TE D

( L R = 2, s = 1, ξ = ζ = 0.5)

Fig. 4. Variation of buckling load intensity factor of the CNTRC SCS with L R and S ratios

( N r = N s = 10, s = 1, ζ = ξ = 0.5)

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Fig. 5. Effect of aggregation parameter ζ on the buckling load intensity factor of the continuously graded CNTRC SCS for different values of sigmoidal parameter s

AC C

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(ξ = 0.9, S = 20, L R = 2, N r = N s = 10)

Fig. 6. Effect of L R ratio on the buckling load intensity factor of the SCS reinforced by agglomerated graded CNTs for various CNT profiles (ξ = ζ = 0.7, S = 20, N r = N s = 20, s = 1)

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Fig. 7. Variation of buckling load intensity factor with aggregation parameter ξ for different CNT profiles: N r = N s = 7 , Red line: N r = N s = 24 (ζ = 0.1, S = 20, L R = 1, s = 1)

Fig. 8. Influences of sigmoidal parameter, s , and the mid-radius to the thickness ratio, S , on the intensity factor of SCS reinforced by agglomerated graded CNTs (ξ = ζ = 0.5, L R = 2, N r = N s = 15)

SC

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AC C

EP

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Fig. 9. Variation of buckling load intensity factor with aggregation parameter ξ and sigmoidal parameter (ζ = 0.1, L R = 1, S = 10, N r = N s = 10)

Fig. 10. Impact of the CNT volume fraction on the buckling load intensity factor for various aggregation parameter ζ (ξ = 0.9, L R = 2, S = 30, N r = N s = 8)