On the vibration of postbuckled functionally graded-carbon ...scientiairanica.sharif.edu/article_21383_c8e94bbf...posite annular plate with inner radius, a, outer radius, b, and thickness,

Scientia Iranica F (2019) 26(6), 3857{3874

Sharif University of TechnologyScientia Iranica

Transactions F: Nanotechnologyhttp://scientiairanica.sharif.edu

On the vibration of postbuckled functionallygraded-carbon nanotube reinforced composite annularplates

R. Gholamia;� and R. Ansarib

a. Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, Lahijan, P.O. Box 1616, Iran.b. Faculty of Mechanical Engineering, University of Guilan, Rasht, P.O. Box 3756, Iran.

Received 30 May 2018; received in revised form 4 January 2019; accepted 22 April 2019

KEYWORDSFree vibrationof postbucklednanocompositeannular plate;Postbucklingbehavior;Carbon nanotube-reinforced composites;Numerical approach;GDQ method.

Abstract. This paper studies the free vibration characteristics of post-buckled Func-tionally Graded (FG) carbon nanotube (CNT) reinforced annular plates. The analysiswas performed by employing a Generalized Di�erential Quadrature (GDQ)-type numericaltechnique and pseudo-arc length scheme. The material properties of FG-carbon nanotubereinforced composite (CNTRC) plates were evaluated by an equivalent continuum approachbased on the modi�ed rule of mixture. The vibration problem was formulated based on theFirst-order Shear Deformation Theory (FSDT) for moderately thick laminated plates andvon K�arm�an nonlinearity. By employing Hamilton's principle and a variational approach,the nonlinear equations and associated Boundary Conditions (BCs) were derived, whichwere then discretized by the GDQ method. The postbuckling behavior was investigated byplotting the secondary equilibrium path as the deection-load curves. Thereafter, the freevibration behaviors of pre- and post-buckled FG-CNTRC annular plates were examined.E�ects of di�erent parameters including types of BCs, CNT volume fraction, an outerradius-to-thickness ratio, and an inner-to-outer radius ratio were investigated in detail.

© 2019 Sharif University of Technology. All rights reserved.

1. Introduction

Since the discovery of carbon nanotubes (CNTs) byIijima in 1991 [1], considerable advances have beenmade in the realm of nanotechnology. CNTs are themost extraordinary materials that have been discoveredby mankind over the past thirty years. Characterizedby extraordinary properties, they have attracted agreat deal of attention from the scienti�c communityand beyond [2-5]. These materials have the potential

*. Corresponding author. Tel./Fax: +98 1342222906Email address: gholami [email protected] (R. Gholami)

doi: 10.24200/sci.2019.51145.2029

to revolutionize di�erent �elds such as medicine, elec-tronics, material science, energy storage, etc. CNTsare reported to enjoy many desired properties suchas high tensile strength and Young's modulus. Thehigh strength of CNTs makes them the sti�est known�ber discovered so far. Further, CNTs enjoy excellentthermal and electrical conducting properties and caneither show metallic or semi-conducting behavior basedon their size, chirality, and purity. Thus, CNTs canbe used as reinforcements to enhance the physical andmechanical properties and electrical conductivity ofthe polymeric structures. Because of some charac-teristics such as wear and corrosion resistance, lowdensity, light weight, and low cost, polymer-basedcomposites are extensively utilized in the industrial and

3858 R. Gholami and R. Ansari/Scientia Iranica, Transactions F: Nanotechnology 26 (2019) 3857{3874

engineering usages including marine and automotivetechnologies, military, and the agricultural industry [6].The addition of CNTs to polymers may result inthe enhancement of many mechanical, electrical, andoptical properties of polymer-based nanocomposites.These superior properties make them the best can-didate for use in various usages such as actuators,biomedical devices, chemical sensors, and smart mem-ory devices [7,8]. Ajayan et al. [9] fabricated carbonnanotube reinforced composites (CNTRCs) for the �rsttime in 1994. Since then, a number of studies have beenperformed to utilize CNTs as reinforcement for variousmaterials such as polymer, ceramic, and metals. Forinstance, Hassanzadeh-Aghdam and Mahmoodi [10]conducted a comprehensive analysis of the mechanicalproperties of CNT-reinforced metallic nanocompositesby proposing an analytical approach. The e�ects ofCNT volume fraction, interphase, and geometry on thethermal expansion behavior of CNT-reinforced metalliccomposites were studied by Hassanzadeh-Aghdam etal. [11,12]. Foroughi et al. [13] experimentally exam-ined the inuence of CNTs on the mechanical andbioactive properties of bioglass-ionomer cement. More-over, AfzaliTabar et al. [14] investigated the CNTs andnano-porous graphene on the silica nanohybrid Picker-ing emulsion. Recently, Ra�ee et al. [15] experimen-tally studied the vibrational and damping behaviorsof functionalized multi-walled CNT-reinforced epoxynanocomposites as the passive damping components.

Among the published papers on the mechani-cal characteristics of CNT-reinforced composites, themajority have been devoted to the reinforcement ofpolymers by CNTs [16-24]. This is because of therelative ease of polymer processing, which demandslower temperatures for consolidation compared to met-als and ceramic matrix composites. The fascinatingmechanical properties of CNTs over carbon �bers haveresulted in increasing use of CNT-reinforced compositestructures. The main di�erence between these twotypes of composites lies in the low quantity of CNTsused in the CNTRCs [25-27]. Meguid and Sun [28]stated that by increasing the CNT volume fractionbeyond a speci�ed limit, the mechanical properties ofCNTRCs will deteriorate. As a result, the concept ofFunctionally Graded (FG) materials has been incorpo-rated in the modeling of CNTRCs in order to use CNTsmore e�ciently in the reinforced composites. The localbuckling of CNTRC beams induced by the bendingwas studied by Vodenitcharova and Zhang [29]. Theimperfection sensitivity of the primary resonances ofFG-CNTRC beams under periodic transverse loadingwas examined by Gholami et al. [30]. A Mori-Tanaka-based equivalent model was utilized by Formica etal. [31] to study the free vibration of CNTRC plates.Their study showed that the maximum enhancementof the properties of �ber composites was obtainable by

uniformly aligning CNTs with the loading direction.Ansari et al. [32] examined the nonlinear forced vi-bration of Timoshenko beams made of FG-CNTRCs.Shen and He [33] studied the nonlinear vibration ofembedded FG-CNTRC curved panels under thermalloading. It was found that nonlinear vibration behaviorof CNTRC panels was considerably a�ected by the FG-CNT reinforcements. The nonlinear forced vibrationof FG-CNTRC rectangular plates based upon Mindlinand Reddy's plate theories was analyzed by Ansariet al. [34,35]. In addition, Ansari et al. [36] ana-lytically studied the postbuckling of piezoelectric FG-CNTRC shells. Lin and Xiang [37] investigated thefree vibrational characteristics of SWCNT-reinforcednanocomposite beams. The variational technique ofHamilton's principle and sense of von K�arm�an's non-linearity were used to derive the energies of the CNT-reinforced composite beams. Then, by employing thep-Ritz technique, the free vibration problem of thebeam was solved. The free vibration of FG-CNTRCcylindrical shells under the thermal loading was ana-lyzed by Song et al. [38] upon employing the assumedmodes approach. Mehrabadi et al. [39] examined linearbuckling of FG-CNTRC plates under uniaxial andbiaxial compression. In a study conducted by Lie etal. [40], free vibrations of SWCNT-reinforced nanocom-posite plates were analyzed by the kp-Ritz method.Shen et al. [41] presented a study on the vibrationalresponse of thermally postbuckled sandwich CNTRCplates resting on the elastic mediums. Ahmadi etal. [42] employed a multi-scale �nite element procedureto obtain the mechanical properties of carbon �ber-CNT-polyimide nanocomposites and, then, examinethe buckling of rods made of these nanocomposites.Recently, according to a variational approach, Gholamiand Ansari [43] provided a weak form of mathematicalmodeling to study the nonlinear resonant responses ofshear deformable FG-CNTRC annular sector plates.In addition, the resonance of multi-scale laminatednanocomposite rectangular plates was examined byGholami et al. [44]. According to the First-orderShear Deformation Theory (FSDT) and Rayleigh-Ritzscheme, the free vibration of nanocomposite sphericalpanels and shells of revolution was studied by Wang etal. [45].

In this work, upon employing the GeneralizedDi�erential Quadrature (GDQ) approach, the freevibration problem of postbuckled FG-CNTRC annularplates with Uniformly Distributed (UD) and FG rein-forcements is numerically formulated. It is assumedthat the material properties of FG-CNTRCs are ob-tained by employing a modi�ed rule of mixture-basedequivalent model. The postbuckling problem is for-mulated on the basis of the FSDT with a von K�arm�antype of kinematic nonlinearity. By applying Hamilton'sprinciple, the nonlinear equations and corresponding

R. Gholami and R. Ansari/Scientia Iranica, Transactions F: Nanotechnology 26 (2019) 3857{3874 3859

BCs are derived and, then, discretized by the GDQmethod. In addition, pseudo-arc length algorithm isemployed to �nd the secondary equilibrium paths ofCNTRCs plates. The free vibration of postbuckledCNTRC annular plates is formulated as a standard lin-ear eigenvalue problem. Impacts of design parametersincluding type of BCs, CNT volume fraction, inner-to-outer radius ratio, and outer radius-to-thickness ratioon the equilibrium postbuckling path and fundamentalfrequencies in the pre- and post-buckled con�gurationsare investigated.

2. Mathematical formulation

2.1. CNTRCs and material propertiesAs illustrated in Figure 1, an SWCNT-reinforced com-posite annular plate with inner radius, a, outer radius,b, and thickness, h, is assumed. It is considered thatthe SWCNT reinforcements are UD or FG throughthe thickness. The structure of the CNT signi�cantlya�ects the properties of the nanocomposites. Thusfar, di�erent micromechanical models such as the Mori-Tanaka [46,47] and Voigt models, as well as the rule ofthe mixture [26,48], have been proposed to obtain thematerial properties of CNTRCs. The former is usedfor micro-particles and the latter extensively for theCNTRCs. On a nanoscale, both of these approachesshould be extended to capture the small-scale e�ect. Ithas been demonstrated that both of Mori-Tanaka andVoigt techniques have an identical level of accuracyin treating the static and dynamic problems of FGceramic-metal beams [49], plates [50], and shells [51].Accordingly, applying the modi�ed version of the ruleof mixture, one can express the e�ective Young's and

Figure 1. Schematics of an FG-CNTRC annular plate.

shear modules of CNTRCs as follows [48]:

E11 = �1VcntEcnt11 + VmEm; (1a)

�2E22

=VcntEcnt22

+VmEm

; (1b)

�3G12

=VcntGcnt12

+VmGm

: (1c)

By considering this point that, through theformula-tions, sub-/super-scripts \m" and \cnt" signify thematrix and CNT, respectively, in Eq. (1), G and E rep-resent shear and Young's modules, and �j (j = 1; 2; 3)identi�es the CNT e�ciency parameter, which isattributed to the scale-dependent properties. It isnotable that �j will be later obtained by matchingthe material properties achieved from the MolecularDynamics (MD) simulations with those obtained fromthe rule of mixture. In addition, Vcnt and Vm denotethe volume fractions of CNT and mixture, respectively,and have the following relationship as follows:

Vcnt + Vm = 1: (2)

The FG-CNTRCs are supposed to be in two di�erentcon�gurations, namely O and X types. For conve-nience, in the following, the two types of FG-CNTRCsare indicated by FGO and FGX. For the case, the FG-CNTRC is referred to as FGO, and the middle surfaceof the composite is CNT-rich, while, for the FGX, bothouter and inner faces are CNT-rich. In this study,the UD, FGO, and FGX distributions of CNTs are ofspecial concern and are expressed as below [48]:

UD : Vcnt = V �cnt; (3a)

FGO : Vcnt = 2�

1� 2 jzjh

�V �cnt; (3b)

FGX : Vcnt =4 jzjhV �cnt; (3d)

where:

V �cnt =�cnt

�cnt +��cnt�m

�� cnt�m ��cnt : (4)In the preceding equation, �cnt is the mass fractionof CNT, and � denotes the mass density. Similar tothe previous case, e�ective Poisson's ratios � and � areobtained by Wang and Shen [52]:

�12 = Vcnt�cnt12 + Vm�m; �21 = �12E22=E11; (5)

� = Vcnt�cnt + Vm�m: (6)


2.2. Nonlinear equations of motion andcorresponding BCs

Consider a cylindrical coordinate system (r; �; z) inwhich its origin is placed at the center of the mid-plane of the FG-CNTRC annular plate, and r, �,and z-axes denote radial, tangential, and thicknessdirections, respectively. Considering the axisymmetricdeformation, the displacement components, ur, u� anduz, along r; �, and z axes, respectively, are obtained byLiew et al. [53]:

ur = u (t; r) + z r (t; r) ; u� = 0; uz = w (t; r) ; (7)

where u(t; r) and w(t; r) are the radial and transversedisplacement components of middle-plane, respectively,and r(t; r) is the rotation about �-axis. In addition,t denotes time. Of note, the displacement �eld de�nedin Eq. (7) is based on the FSDT.

By applying Eq. (7), the strain-displacement re-lations are expressed by:

"r =@u@r

+12

�@w@r

�2+ z

@ r@r

; "� =ur

+ z rr;

"rz = "zr =12

� r +

@w@r

�: (8)

Additionally, according to the linear elasticity and thevon K�arm�an hypothesis, the nonlinear stress compo-nents can be de�ned by:8


�T =12

ZS

(I0

"�@u@t

�2+�@w@t

�2#

+ 2I1@u@t@w@t

+ I2�@ r@t

�2)dS; (15a)

�w =ZS

"12N0r

�@w@r

�2#dS: (15b)

Using Hamilton's principle [54]:t2Zt1

(��T � ��s + ��w) dt = 0; (16)

where � denotes the variation operator; one can achievethe governing equations and all possible BCs. Byinserting Eqs. (14) and (15) into (16), taking thevariation of u, w, and r through the integration byparts, and lastly by equating the coe�cients of �u, �w,and � r to zero, the following expressions are obtainedfor the governing equations of motion (Eqs. (17a)-(17c)) and the BCs (Eqs. (18a)-(18c))@Nr@r

+Nr �N�

r= I0

@2u@t2

+ I1@ r@t2

; (17a)

@Qr@r

+Qrr

+1r@@r

�rNr

@w@r

�+N0r

�@2w@r2

+1r@w@r

�= I0

@2w@t2

; (17b)

@Mr@r

+Mr �M�

r�Qr = I2 @ r@t2 + I1

@2u@t2

; (17c)

and:�u = 0 or Nr = 0; (18a)

�w = 0 or�Nr +N0r

� @w@r

+Qr = 0; (18b)

� r = 0 or Mr = 0: (18c)

By inserting Eq. (12) into Eqs. (17), the governingequations are determined in terms of displacementcomponents as follows:

A11�@2u@r2

+1r@u@r

+@w@r

@2w@r2

+12r

�@w@r

�2�+B11

�@2 r@r2

+1r@ r@r

��A22 ur2

�B22 rr2 �A122r

�@w@r

�2= I0

@2u@t2

+ I1@ r@t2

; (19a)

ksA55�@2w@r2

+1r@w@r

+@ r@r

+ rr

�+N (w)

+N0r

�@2w@r2

+1r@w@r

�= I0

@2w@t2

; (19b)

B11

"@2u@r2

+1r@u@r

+@w@r

@2w@r2

+B112r

�@w@r

�2#�B22 ur2 �D22

rr2

+D11�@2 r@r2

+1r@ r@r

�� B12

2r

�@w@r

�2� ksA55

� r +

@w@r

�= I2

@ r@t2

+ I1@2u@t2

; (19c)

where:

N (w) =

(A11

"@u@r

+12

�@w@r

�2#+A12

ur

+B11@ r@r

+B12 rr

)�1r@w@r

+@2w@r2

�+

(A11

�@2u@r2

+@w@r

@2w@r2

�+A12

�1r@u@r� ur2

�+B11

@2 r@r2

+B12�

1r@ r@r� 1r2 r�)

@w@r

: (20)

The BCs provided in Eq. (18) show the possibleedge conditions for the FG-CNTRC annular plates.Consequently, the BCs are as follows:

For the simply supported CNTRC annular plates:

Nr = w = Mr = 0: (21a)

For clamped CNTRC annular plates:

Nr = w = r = 0: (21b)

The following dimensionless quantities are introducedso as to non-dimensionalize the governing equations ofmotion:

� =rb; � =

bh; fu;wg ! h fu;wg ;

r = r; �N0r =N0rA110

; � =tb

rA110I10

;

fa11; a22; a12; a55g = fA11; A22; A12; A55gA110 ;


fd11; d22; d12; d55g = fD11; D22; D12gA110h2 ;��I0; �I1; �I2 = � I0I00 ; I1I00h ; I2I00h2� ; (22)where A110 and I00 show the values of A11 and I0 fora homogeneous matrix plate. Thus, one obtains:

a11

"@2u@�2

+1�@u@�

+1�@w@�

@2w@�2

+1

2��

�@w@�

�2#+ b11

�@2 r@r2

+1�@ r@�

�� a22 u�2 � b22

r�2

� A122��

�@w@�

�2= �I0

@2u@�2

+ �I1@ r@�2

; (23a)

ksa55�@2w@�2

+1�@w@�

+ �@ r@�

+ � r�

�+ �N0r

�@2w@�2

+1�@w@�

�+

1�

�a11

"@u@�

+12�

�@w@�

�2#+ a12

u�

+ b11@ r@�

+ b12 r�

��1�@w@�

+@2w@�2

�+

1�

�a11�@2u@�2

+1�@w@�

@2w@�2

�+ a12

�1�@u@�� u�2

�+ b11

@2 r@�2

+ b12�

1�@ r@�� r�2

��@w@�

= �I0@2w@�2

; (23b)

b11

"@2u@�2

+1�@u@�

+1�@w@�

@2w@�2

+1

2��

�@w@�

�2#� b22 u�2 � d22

r�2

+ d11�@2 r@�2

+1�@ r@�

�� d12

2��

�@w@�

�2� ksa55�

�� r +

@w@r

�= �I2

@ r@�2

+ �I1@2u@�2

: (23c)

Further, by non-dimensionalizing the BCs, the follow-ing expressions are obtained for the simply supported

(Eq. (24a)) and clamped (Eq. (24b)) BCs:

a11

"@u@�

+12�

�@w@�

�2#+ a12

u�

+ b11@ r@�

+ b12 r�

= w = 0;

b11

"@u@�

+12�

�@w@�

�2#+ b12

u�

+ d11@ r@�

+ d12 r�

= 0; (24a)

a11

"@u@�

+12�

�@w@�

�2#+ a12

u�

+ b11@ r@�

+ b12 r�

= w = r = 0: (24b)

3. GDQ method

The GDQ method as an e�cient numerical approachcan be utilized for solving the boundary value problemsincluding the ordinary and partial di�erential equa-tions. Unlike the �nite element method that is usuallyemployed for solving the weak form of equations, theGDQ technique represents a powerful tool for solvingthe equations in the strong form with great e�ciencyand accuracy using a small number of discrete meshpoints [55].

3.1. IntroductionWith the aid of the GDQ technique [56-58], the pthorder derivative of g(r) is attained in the followingform:

@pg (p)@pp

��r=ri

=NXj=1

Arijg (rj) ; (25)

where N is the number of total discrete points. Byconsidering a column vector F:

F = [gj ] = [g (rj)] = [g (r1) ; g (r2) ; : : : ; g (rN )]T ;

(26)

where gj = g(rj) indicates the amount of g(r) at rj , andan operational matrix of di�erentiation on the basis ofEq. (25) is achieved as in the following form:

@p

@rp(F) = DprF = [D

pr ]i;j fFjg ; (27)

where:Dpr = [D

pr ]i;j = Apij ; i; j = 1 : N; (28)

where Aij gives the weighting coe�cients obtained asby Eq. (29) as shown in Box I, in which P(ri) =�Nj=1;j 6=i(ri � rj), and Ir denotes an N � N identitymatrix.


Apij =

8>>>>>>>>>>>>>:Ir; r = 0P(ri)

(ri�rj)P(rj) ; i 6= j and i; j = 1; : : : ; N and p = 1p�A1ijAp�1ii � A

p�1ij

ri�rj�; i 6= j and i; j = 1; : : : ; N and p = 2; 3; : : : N � 1

� NPj=1;j 6=i

Apij ; i = j and i; j = 1; : : : ; N and p = 1; 2; 3; : : : N � 1(29)

Box I

3.2. Postbuckling analysisWith the aid of Chebyshev-Gauss-Lobatto points asthe grid points, the mesh generation can be obtainedby:

�i = �+1� �

2

�1� cos i� 1

N � 1��; i = 1 : N; (30)

where � = a=b. The discretized form of displacementcomponents is de�ned as the following vectors:

UT = [U1; : : : ; UN ] ;

WT = [W1; : : : ;WN ] ;

T = [1; : : : ;N; ] ; (31)

where Ui = u (�i) ;Wi = w (�i) ;i = (�i). Byassuming �N0r = �P , utilizing the GDQ scheme, anddiscarding the inertia terms, the equilibrium equationsare discretized by:

a11�D2�U + D

1�U �A1 + 1�

�D2�W

� � �D1�W�+

12��D1�W

� � �D1�W� �A1�+ b11

�D2� + D

1� �A1�� a22U �A2

�b22�A2� a122��D1�W

��D1�W��A1 =0; (32a)�sa55

�D2�W + D

1�W �A1�

+ �sa55��D1� + �A1�

� P �D2�W + �D1�W� �A1�+

1�

�a11�D1�U +

12��D1�W

� � �D1�W��+ b11D1� + a12U �A1 + b12 �A1

�� D2�W + �D1�W� �A1�

+1�

�a11�D2�U +

1��D2�W

� � �D1�W��+ a12

�D1�U �A1 �U �A2�+ b11D2�

+ a12�D1�U �A1 �U �A2�

+ b12�D1� �A1 � �A2�� D1�W� = 0;

(32b)

b11�D2�U + D

1�U �A1 + 1�

�D2�W

� � �D1�W�+

12��D1�W

� � �D1�W� �A1�+ d11

�D2� + D

1� �A1�� b22U �A2

� d22 �A2 � b122��D1�W

� � �D1�W� �A1� �sa55� �� + D1�W� = 0; (32c)

where � shows the Hadamard product [59] and ATi =�1=�i1; 1=�i2; : : : ; 1=�iN

�. Following the same procedure

used for the discretization of the equilibrium equation,one can discretize the BCs (Eqs. (18)) similarly. Theset of nonlinear equations of the domain can be de�nedby:

G : R3N+1 ! R3N ;G (P;X) = 0;

XT =�UT;WT;T

�; (33)

where parameter P is the axial load.The previous equation due to the presence of

P is a parameterized equation. Here, by employingthe pseudo-arc length continuation technique, thisequation will be solved. To this end, by substitutingthe residual of equations relevant to boundaries into theresidual of the domain G(P;X), the edge conditions aresatis�ed. This assumption implies that the elements of


G related to the grid points must be substituted withthose of discretized BCs.

3.3. Vibration study in the postbuckled regionHerein, the aim is to examine the linear free vibrationof a buckled CNTRC plate. To accomplish this goal,by introducing small disturbances ud, wd, and d,respectively, around the buckled con�gurations us, ws,and s, the time evolution of that disturbance will beobtained as follows:

u (�; �) = us (�) + ud (�; �) ; w (�; �)

= ws (�) + wd (�; �) ; r (�; �)

= s (�) + d (�; �) : (34)

Thereafter, by inserting the previous equation into thegoverning equation and ignoring the nonlinear time-dependent terms, the linear free vibration problem isobtained by:

mx + kx = 0; (35)

where dot indicates the derivative with respect to � , xdenotes the generalized coordinate, and m and k arethe inertia and sti�ness matrices, respectively, whichcan be determined by:

xT = [ud; wd; d] ;

m =

24 �I0 0 �I10 �I0 0�I1 0 �I2

35 ;k =

24 kuu kuw ku kwu kww kw k u k w k

35 : (36)The elements of k are introduced as follows:

kuu = a11�@2

@�2+

1�@@�

�� a22 1�2 ;

kuw =a11�

�@2ws@�2

@@�

+@ws@�

@2

@�2+

1�@ws@�

@@�

�� a12��

@ws@�

@@�;

ku = b11�@2

@�2+

1�@@�

�� b22 1�2 : (37a)

kwu =1�

�a11

@@�

+a12�

��1�@ws@�

+@2ws@�2

�+

1�

�a11

@2

@�2+ a12

�1�@@�� 1�2

��@ws@�

;

kww =ksa55�@2

@�2+

1�@@�

�+a11�2

�1�@ws@�

+@2ws@�2

�@ws@�

@@�

+a11�2

�@ws@�

@2

@�2+@2ws@�2

@@�

�@ws@�

+1�

(a11

"@us@�

+12�

�@ws@�

�2#+ a12

us�

+ b11@ s@�

+ b12 s�

)�1�@@�

+@2

@�2

�� P

�@2

@r2+

1r@@r

�+

1�

�a11�@2us@�2

+1�@ws@�

@2ws@�2

�+ a12

�1�@us@�� us�2

�+ b11

@2 s@�2

+ b12�

1�@ s@�� s�2

��@@�;

kw =1�

�b11

@@�

+ b121�

��1�@ws@�

+@2ws@�2

�+

1�

�b11

@2

@�2+ b12

�1�@@�� 1�2

��@ws@�

+ ksa55��@@�

+1�

�; (37b)

k u = b11�@2

@�2+

1�@@�

�� b22�2;

k w =b11�

�@ws@�

@2

@�2+@2ws@�2

@@�

+1�@ws@�

@@�

�� d12��

@ws@�

@@�� ksa55� @@� ;

k = �d22�2 + d11�@2

@�2+

1�@@�

�� ksa55�2: (37c)

Now, by employing the GDQ technique, one candiscretize Eq. (35) as follows:

MX + KX = 0; (38)

in which:

XT =�UTd ;w

Td ;

Td�;


M =

24 �I0D0� 0 �I1D0�0 �I0D0� 0�I1D0� 0 �I1D0�

35 ;K =

24 Kuu Kuw Ku Kwu Kww Kw K u K w K

35 : (39)The components of K are determined by:

kuu = a11�D2� + A1}D1�� a22A2}D0� ;

kuw =a11�

��D2�Ws

�}D1� + �D1�Ws�}D2�+ A1 � �D1�Ws�}D1�� a12

��A1 � �D1�Ws�}D1�� ;

ku = b11�D2� + A1}D1�� b22A2}D0� ; (40a)

kwu =1��a11D1� + a12A1

�} �A1 � �D1�Ws�+ D2�Ws�+

1��a11D2� + a12

�A1}D1� �A2}D0��

} �D1�Ws� ;kww =ksa55

�D2� + A1}D1��

+a11�2�D2�Ws + A1 � �D1�Ws�� D1�Ws�

}D1� + a11�2��

D1�Ws�}D2� + �D2�Ws�}D1��

� �D1�Ws�+ 1��a11��D1�Us�+

12��D1�Ws

� � �D1�Ws��+ a12A1� �D0�Us�+ b11D1�s + b12A1 � �D0�s��} �D2� + A1}D1��+

1�

�a11�

D2�Us +1��D1�Ws

� � �D2�Ws��+ a12

�A1 � �D1�Us��A2 � �D0�Us��

+ b11D2�s + b12�

A1 � �D1�s��A2��D0�s��}D1��P �D2�+A1}D1�� ;

(40b)

kw =ksa55��D1� + A1

�+

1��b11D1� + b12A1

} �D2�Ws + A1 �D1�Ws�+

1�

�b11D2� + b12

�A1

}D1� �A2}D0��} �D1�Ws� ;

k u = b11�D2� + A1}D1�� b22A2}D0� ;

k w =b11�

�D1�Ws}D2� + D2�Ws}D1� + A1

� �D1�Ws�}D1�� d12� A1 � �D1�Ws�}D1� � ksa55�D1� ;

k = �d22A2 + d11 �D2� + A1}D1�� ksa55�2D0� ;(40c)

where } represents the SJT product (SJT is theabbreviation of Shanghai Jiao Tong Univ.) [59,60]. Byconsidering the harmonic solution of the form X =~Xei!� , Eq. (39) turns into:

�!2M~X + K~X = �K� !2M� ~X = 0; (41)where ! denotes the non-dimensional frequency.

Substitution of the BCs into K and M and therearrangement of the discretized equations and theassociated BCs yield the following eigenvalue problem:�

Kdd KdbKbd Kbb

� �~Xd~Xb

�=�!2Mdd ~Xdf0g

�; (42)

where subscripts d and b are the domain and boundarymesh grid points, respectively.

Eq. (42) can be uncoupled through the followingexpressions:(�

Kdd �Kdb(Kbb)�1Kbd�

~Xd = !2Mdd ~Xd~Xb = (Kbb)

�1Kbd ~Xd:(43)

Now, !i (i = 1; 2; 3; : : :) and their corresponding modeshapes ~XT =

h~XTd ; ~XTb

ican be achieved by �nding a

solution to Eq. (43).


4. Numerical results and discussion

The formulation and solution procedure developedin the previous sections are utilized to present thenumerical results for the postbuckling behavior andthe free vibration of the FG-CNTRC annular plates.E�ects of various factors on the static equilibrium post-buckling path and frequencies are shown by conductinga non-dimensional study. Poly methyl methacrylate(PMMA) with �m = 0:34; �m = 1150 kg=m3; Em =2:5 GPa [26] and armchair (10,10) SWCNTswith �cnt = 0:175;Gcnt12 = 1:9445 TPa;Ecnt11 =5:6466 TPa;Ecnt22 = 7:08 TPa; �cnt = 1400 kg=m3 atroom temperature (300 K) [61] are selected as matrixand reinforcements, respectively.

Furthermore, the values of CNT e�ciency param-eters �1, �2, and �3 for three di�erent CNT volumefractions are considered as follows [26]:

V �cnt = 0:12 : �1 =0:137; �2 =1:022 ; �3 = 0:715;

V �cnt = 0:17 : �1 =0:142; �2 =1:626 ; �3 = 1:138;

V �cnt = 0:28 : �1 =0:141; �2 =1:585 ; �3 = 1:109:

In what follows, a sequence of letters including \SS"and \C" is used to represent the simply supported andclamped BCs, respectively.

To provide a convergence study and illustratethe accuracy of mathematical modeling, solution pro-cedure, and numerical results, the natural frequencyparameters of isotropic annular plates are provided inTable 1. It can be seen that the results are convergedby increasing N and are in excellent agreement withthose of Liew et al. [53]. In this study, N = 21 is usedin all computational e�orts. In addition, in Table 2,the critical buckling load parameters associated withvarious inner-to-outer radius ratios are compared withthose given in [62], illustrating very well agreement.

Depicted in Figure 2 are the static equilibriumpostbuckling paths as the maximum non-dimensional

Table 2. Comparison of critical buckling load parameter( ~Pcr = �Pcrb2=D110) of C-C isotropic annular plates(� = 0:3; b=h = 20).

a=b Present Ref. [62]

0.2 59.922 60.010.3 76.633 76.770.4 101.812 102.05

deection, wmax, versus the non-dimensional compres-sive radial load (Non. dim. radial load; P = �P=A110where �P is the dimensional compressive radial load)for the three di�erent prescribed distributions of CNTsin the CNTRC annular plates corresponding to fourvarious combinations of simply supported and clampededge supports. According to this �gure, it is revealedthat for a �xed value of the radial load, the maximumdeection, wmax, corresponding to an FGO-CNTRCannular plate is larger than those of the other two typesof CNTRC plates; the FGO-CNTRC annular has thelowest critical buckling load and the highest criticalbuckling load, and the maximum load-carrying capac-ity belongs to the FGX distribution pattern. It can beconcluded from this �gure that the addition of moreCNTs to the upper and lower surfaces of FG-CNTRCannular plates results in a considerable increase in thetotal sti�ness of system and induces more resistanceagainst bending. In addition, it is observed that theFG-CNTRC plates with C-C edge conditions have min-imum values of maximum deection and, subsequently,maximum values of critical buckling load, whereas, forthe case of the FG-CNTRC annular plates with fullysimply supported edges, an opposite trend is seen. Thedependence of the non-dimensional frequency (Non.dim. frequency: ! = �!b

pI00=A110 where �! denotes

the natural frequency) upon the non-dimensional com-pressive radial load in the pre- and post-buckled statesis exhibited in Figure 3. Based on the results exhibitedin Figure 3, it can be concluded that the fundamentalfrequencies of FG-CNTRC annular plates in the pre-

Table 1. Convergence of the frequency parameters�

~! = �!b2pI00=D110

�of isotropic annular plates with di�erent BCs

(� = 0:3; b=h = 5; a=b = 0:5).

BCs Mode N Ref. [53]5 7 9 11 13 21

SS-SS 1 31.3583 31.7365 31.7292 31.7292 31.7292 31.7292 31.872 104.3532 89.9127 89.8677 89.8647 89.8647 89.8647 90.64

SS-C 1 41.2354 41.2906 41.2883 41.2883 41.2883 41.2883 41.622 109.4586 94.3543 94.2975 94.2913 94.2914 94.2914 95.27

C-SS 1 37.5501 38.0643 38.0538 38.0538 38.0538 38.0538 38.362 107.2712 92.9592 92.8525 92.8512 92.8512 92.8512 93.78

C-C 1 47.7424 47.8078 47.8099 47.8099 47.8099 47.8099 48.312 110.4921 96.4347 96.2918 96.2885 96.2886 96.2886 97.39


Figure 2. Equilibrium postbuckling path of FG-CNTRC annular plates for three di�erent distributions of the CNTs(b=h = 40; V �cnt = 0:17; a=b = 0:2).

Figure 3. Vibration behavior of pre- and post-buckled FG-CNTRC annular plates for three di�erent distributions of theCNTs (b=h = 40; V �cnt = 0:17; a=b = 0:2).


buckled state decrease with an increase in the radialload. This is because the sti�ness of FG-CNTRCannular plates decreases by increasing the compressiveradial loading. By increasing the compressive radialload to a new high, the sti�ness matrix becomes azero matrix at a certain point, called the bucklingpoint. In this point, the FG-CNTRC annular platesdo not experience any vibration and, consequently, thefundamental frequency is zero. It can be interpretedthat the buckling point is a bifurcation point throughwhich the FG-CNTRC annular plate meets its sec-ondary equilibrium state known as the postbucklingregion. Prior to the bifurcation point, the frequenciescorrespond to the pre-buckling con�guration, and thoseafter the critical buckling load are concerned with thevibration in the post-buckled state. For a buckledplate, it is seen that by increasing the compressiveradial load, the fundamental frequencies increase. Thisimplies that a buckled plate can withstand additionalload without failure. In addition, it is deduced that thedimensionless frequency-load curves are continuous, yetnot di�erentiable at the buckling point. According tothis �gure, it is observed that at a �xed value of radialload, the fundamental frequencies associated with theFGO-CNTRC plates have lower values than CNTRCannular plates with FGX and UD patterns in the pre-buckled region. While, in the postbuckled region, it isseen that the fundamental frequencies corresponding to

FGO-CNTRC plates have larger values than the othertwo cases. It is due to this fact that the deection inthe postbuckled region intensi�es the nonlinear sti�nessmatrices. In addition, it is implied that the e�ect ofdeection is more signi�cant than the CNT distributionpattern. Hence, at a certain point in the postbuckledregion, since the deection of FGO-CNTRC annularplate is more than two other distribution patterns, itsfrequency is greater than the annular plate with UDand FGX patterns.

E�ects of V �cnt on the equilibrium postbucklingpath and frequency-response curves are plotted inFigures 4 and 5 for the FGX-CNTRC annular plates.From these �gures, it is seen that as V �cnt increases,the maximum dimensionless deection decreases andcritical buckling load increases. It means that byincreasing V �cnt, the exibility of the CNTRC annularplate increases, too. This is due to the considerablesti�ness of CNTs. As it is expected, by increasing V �cnt,the frequencies of the CNTRC annular plates with FGXpattern increase in the pre-buckled region, whereasan opposite trend is observed for the postbucklingcon�guration.

Figures 6 and 7 show the e�ects of b=h on themaximum dimensionless deection and frequency ofthe FGO-CNTRC plates with C-C, SS-SS, C-SS, andSS-C BCs, respectively. It is observed that an increasein b=h leads to increasing and decreasing the maximum

Figure 4. Postbuckling path of FGX-CNTRC annular plates for di�erent amounts of V �cnt (b=h = 40; a=b = 0:3).


Figure 5. Free vibration of pre- and post-buckled FGX-CNTRC annular plates for various values of V �cnt(b=h = 40; a=b = 0:3).

Figure 6. Postbuckling characteristics of FGX-CNTRC annular plates for various values of b=h (V �cnt = 0:17; a=b = 0:2).


Figure 7. Free vibration response of pre- and post-buckled FGX-CNTRC annular plates for di�erent values of b=h(V �cnt = 0:17; a=b = 0:2).

dimensionless deection and the critical bucklingload, respectively. In other words, an increase in b=hresults in a decrease in the postbuckling load-carryingcapacity of FGannular plates. According to thedimensionless frequency-radial load curves, it is seenthat the frequency in the pre-buckled and post-buckledregions respectively decreases and increases as theplate aspect ratio rises.

Finally, the e�ects of a=b on the equilibriumpostbuckling path and vibration characteristics of theFGX-CNTRC annular plates with C-C, SS-SS, C-SS, and SS-C BCs are shown in Figures 8 and 9,respectively. It is deduced that an increase in a=bdecreases the maximum dimensionless deection andincreases the critical buckling load. In other words,the larger the di�erence between the outer and innerradii, the more stable the CNTRC plate. In addition,it is seen that increasing a=b causes the fundamentalfrequency to increase in the pre-buckled and deep post-buckled regions. In the post-buckled region, dependingon the geometry and compressive radial loading (and,consequently, the deection of plate), the fundamentalfrequency may decrease or increase.

5. Conclusion

In this study, a numerical methodology was adoptedto investigate the postbuckling and free vibration of

FG-CNTRC annular plates with di�erent BCs. To thisend, the FSDT along with the von K�arm�an geometricnonlinearity was utilized to formulate the underlyingproblem. The UD, FGO, and FGX distributions ofSWCNTs in the composite plates were considered.Upon employing an equivalent continuum model, thematerial properties of FG-CNTRCs were estimated.Governing equations were attained by Hamilton's prin-ciple and, then, discretized by a GDQ-based method.Prior to examining the vibration behavior of post-buckled CNTRC plates, postbuckling analysis wasperformed to obtain the buckling load and equilibriumpostbuckling path via the pseudo-arc length continua-tion scheme. Thereafter, the free vibration problem ofthe postbuckled CNTRC annular plates was solved as astandard linear eigenvalue equation. E�ects of variousparameters including types of BCs, CNT volume frac-tion, outer radius-to-thickness ratio, and inner-to-outerradius ratio on the postbuckling path and fundamentalfrequencies were investigated. Results showed that at a�xed value of the applied radial load, the fundamentalfrequencies of FGO-CNTRC plates are the smallest inthe pre-buckling region and the largest in the post-buckling region among the given cases. In addition,it was observed that by increasing the outer radius-to-thickness aspect ratio, the fundamental frequenciesincrease in the prebuckling and deep postbucklingregions.


Figure 8. Postbuckling behavior of FGX-CNTRC annular plates for various values of a=b (b=h = 40; V �cnt = 0:28).

Figure 9. Vibration characteristics of pre- and post-buckled FGX-CNTRC annular plates for various values of a=b(b=h = 40; V �cnt = 0:28).


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Biographies

Raheb Gholami received his BS, MS, and PhDdegrees in Mechanical Engineering from University ofGuilan, Rasht, Iran in 2008, 2010, and 2015, respec-tively. He is currently a faculty member at the De-partment of Mechanical Engineering, Lahijan Branch,Islamic Azad University. His research backgroundand interests include computational micro- and nano-mechanics, numerical techniques, nonlinear analyses,and prediction of mechanical behavior of beam, plateand shell-type structures.

Reza Ansari received his PhD degree from Universityof Guilan, Rasht, Iran in 2008, where he is currently afaculty member at the Faculty of Mechanical Engineer-ing. During his PhD program, he was also a visitingfellow at Wollongong University, Australia from 2006 to2007. He has authored more than 400 refereed journalpapers and 12 book chapters. His research backgroundand interests include computational mechanics, numer-ical analysis, continuum mechanics, nano-, micro-, andmacro-mechanics.

Documents

On the vibration of postbuckled functionally graded-carbon ...scientiairanica.sharif.edu/article_21383_c8e94bbf...posite annular plate with inner radius, a, outer radius, b, and thickness,