Nonlinear primary resonance of third-order shear deformable functionally graded nanocomposite rectangular plates reinforced by carbon nanotubes

Accepted Manuscript Nonlinear primary resonance of third-order shear deformable functionally graded nanocomposite rectangular plates reinforced by carbon nanotubes R. Ansari, R. Gholami PII: DOI: Reference:

S0263-8223(16)31247-8 http://dx.doi.org/10.1016/j.compstruct.2016.07.023 COST 7623

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

13 January 2016 7 June 2016 19 July 2016

Please cite this article as: Ansari, R., Gholami, R., Nonlinear primary resonance of third-order shear deformable functionally graded nanocomposite rectangular plates reinforced by carbon nanotubes, Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct.2016.07.023

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Nonlinear primary resonance of third-order shear deformable functionally graded nanocomposite rectangular plates reinforced by carbon nanotubes R. Ansari*,a, R. Gholami*,b a b

Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran

Abstract. The aim of this paper is to numerically investigate the geometrically nonlinear primary resonance of third-order shear deformable functionally graded carbon nanotube reinforced composite (FG-CNTRC) rectangular plates with various edge supports subjected to a harmonic excitation transverse force. The nanocomposite plates are composed of a mixture of matrix and single-walled carbon nanotubes (SWCNTs), and the effective mechanical properties are obtained by means of the rule of mixture. Employing the von Kármán hypotheses and Reddy’s third-order shear deformation plate theory, the nonlinear equations of motion for the in-plane and out-of-plane directions as well as the corresponding boundary conditions are derived using Hamilton’s principle. In the numerical solution procedure, the generalized differential quadrature (GDQ) method is used for the discretization, and a numerical Galerkin scheme is then employed to convert the discretized nonlinear partial differential equations (PDEs) into a Duffing-type nonlinear time-varying set of ordinary differential equations (ODEs). Afterwards, a time periodic discretization and the pseudo-arc length continuation method are utilized to determine the frequency- and force-responses of FG-CNTRC plates. Moreover, the curves corresponding to the nonlinear free vibration are provided. The influences of important parameters on the frequency-response and force-response curves of the FGCNTRC plates are studied in the numerical results. Keywords: Nonlinear primary resonance; FG nanocomposite plates; Reddy’s third-order shear deformation theory; Numerical solution procedure.

1. Introduction The discovery of carbon nanotubes (CNTs) [1, 2] in 1990s has led to introducing novel areas in the nanotechnology because of their widespread applications in many nanodevices. CNTs have remarkable characteristics such as low density, high aspect ratio, high strength and high stiffness [3, 4]. Especially, their superior mechanical, thermal and electrical properties make them so promising for using in novel nano- and micro-electromechanical systems (NEMS and MEMS). In order to design such systems, it is essential to understand the mechanical behaviors of CNTs accurately. Due to the difficulties of experimental methods and computational cost of molecular dynamics simulations, the continuum mechanics modelling has received wide attention from researchers in order to study the mechanical characteristics of nano-systems [5-16]. *

Corresponding authors. [email protected] (R. Ansari) and [email protected] (R. Gholami).

1

It has been shown that, instead of traditional fibers, single- and multi-walled carbon nanotubes (SWCNTs and MWCNTs) can be utilized for reinforcement in composites to considerably enhance their mechanical, electrical and thermal properties [17-19]. Incorporating CNTs into various matrices such as metal, ceramic and polymer ones makes different nanocomposite materials. For instance, using the conventional manufacturing schemes and without damaging CNTs, the CNT-based polymer composites can be fabricated. It should be remarked that with regard to various application purposes, the CNT/polymer composites are categorized into two main classes, namely the structural composites and functional composites. The properties as well as the performance of these composites depend on various parameters, such as the type of CNTs, morphology and structural geometry of CNTs, type of matrix, the CNT dispersion, existing interfacial reactions between CNTs and matrix and processing strategy. More details about this issue can be found in [20]. The use of CNTs instead of the conventional carbon fibers results in a novel class of materials known as the carbon nanotubereinforced composites (CNTRC) which possessing widespread range of usages in NEMs, MEMs, electronic devices and so on [21, 22]. Furthermore, by following the concept of functionally graded materials (FGMs) and varying the volume fraction of aligned CNTs along the thickness direction of structures, a novel class of promising materials known as FG-CNTRC has been proposed by Shen [23]. Although a wide range of investigations on the constitutive and material properties of CNTRC materials have carried out by the researchers [24-28], due to the use of this sophisticated class of materials in the actual structural applications, it is necessary to examine the mechanical behaviors of the elementary building blocks in FG-CNTRC structures such as beams, plates and shells. Therefore, various studies have been carried out to analyze the influence of different factors on the design and fabrication parameters of FG-CNTRC structures [11, 12, 29-36]. For instance, using an element-free approach [37-40], the buckling of FG-CNTRC skew plates was investigated by Zhang et al.[41]. By development of a third-order shear deformable plate model and using the state-space Levy method, the free vibration of FG-CNTRC plates subjected to the in-plane loads was examined by Zhang et al.[42]. Dynamic stability of FG-CNTRC cylindrical panels subjected to the periodic compressive axial loads was numerically studied by Lie et al.[43]. The element-free kp-Ritz method was implemented by Lei et al.[44] to simulate the buckling of FG-CNTRC plates under different in-plane mechanical loads. An overview of the recent investigations on the linear and nonlinear mechanical behaviors of FG-CNTRC beams, plates and shells is mentioned below. Employing the finite element method as well as the first-order shear deformation plate theory, the bending and free vibration of the moderately thick composite rectangular plates reinforced by SWCNTs were investigated by Zhu et al. [45]. Shen [46] analytically investigated the thermal buckling and postbuckling of FG-CNTRC cylindrical shells according to the higher-order shear deformation shell theory with a von Kármán-type of kinematic nonlinearity. A numerical solution scheme including the weak formulation of nonlinear governing equations and element-free IMLS-Ritz 2

method was presented to investigate the large deflection of FG-CNTRC skew, thick rectangular and straight-sided quadrilateral plates resting on Pasternak foundations [29, 47, 48]. The large amplitude free vibration of FG-CNTRC plates embedded on an elastic foundation in thermal environment analytically was examined by Wang and Shen [49]. Ke et al.[8] studied the nonlinear free vibration of FG-CNTRC Timoshenko beams with various edge supports using the Ritz method as well as a direct iterative approach. In the context of Timoshenko beam theory and by use of an efficient numerical procedure, Ansari et al.[50] analyzed the periodic forced vibration of FG-CNTRC beams subjected to the periodic transverse load. The free vibration of FG-CNTRC thick plates with elastically restrained edges and moderately thick rectangular plates were analyzed by Zhang et al.[51] and Lei et al.[52]. Based on the three-dimensional theory of elasticity, Yas et al.[53] investigated the vibrational characteristics of FG-CNTRC cylindrical panels. The elastodynamic analysis of FG-CNTRC plate was numerically performed using the first-order shear deformation plate theory and element-free kpRitz method [54]. A numerical analysis was provided by Lin and Xiang [55] to study the nonlinear free vibration of nanocomposite beams reinforced by SWCNTs. Zhang et al.[56] presented a weak form of theoretical formulation based upon the first-order shear deformation shell theory, von Kármán hypotheses and the kp-Ritz method to study the influence of volume fraction of carbon nanotubes, span angle, edge-to-radius ratio, thickness and boundary conditions on the geometrically nonlinear bending of FG-CNTRC cylindrical panels. Recently, Guo and Zhang [57] investigated different kinds of the periodic and chaotic motions of a reinforced composite plate with CNTs under the combination of parametric and forcing excitations. Moreover, the postbuckling of FG-CNTRC rectangular plates with edges elastically restrained against translation and rotation under axial compression was investigated based on the first-order shear deformation theory and von Kármán assumption using the numerical element-free IMLS method [58]. The first-order shear deformation plate theory is widely used for the modeling of moderately thick plates. However, to obtain accurate results in this theory, the value of shear correction factor should be appropriately determined. This value depends on several factors such as geometrical parameters, boundary and loading conditions. On the other hand, higher-order shear deformation plate theories can be used for the modeling of thick and moderately thick plates without using the shear correction factor. To the best of authors’ knowledge, in the context of the third-order shear deformation theory, the coupled in-plane and out-of-plane geometrically nonlinear primary resonance of FG-CNTRC rectangular plates subjected to the distributed transverse periodic load has not been investigated yet. Therefore, in the present paper, the coupled in-plane and out-of-plane nonlinear primary resonance of third-order shear deformable FG-CNTRC rectangular plates with various edge supports subjected to a harmonic excitation transverse force is studied using a numerical approach. By utilizing the von Kármán hypothesis, Reddy’s third-order shear deformation plate theory and Hamilton’s principle, the geometrically nonlinear equations of motion for the in-plane and out-of-plane directions and the associated boundary conditions are obtained. After the discretization of the nonlinear governing 3

equations in the space domain by employing the generalized differential quadrature (GDQ) method, a numerical-based Galerkin method is used to obtain a Duffing-type nonlinear time-varying set of ordinary differential equations (ODEs). Then, using the time periodic discretization rule results in the discretized form of Duffing equation in the time domain which gives a set of nonlinear algebraic equations. Finally, the pseudo-arc length continuation method is utilized to obtain the geometrically nonlinear resonant dynamics of third-order shear deformable FG-CNTRC rectangular plates with different edge supports via plotting the frequency-response and force-response curves. The influences of parameters such as the nanotube volume fraction, CNT distributions, length-to-thickness ratio and boundary conditions on the nonlinear dynamical characteristics of the FG-CNTRC plates are illustrated and discussed.

2. Theoretical formulation In this section, at first, the material properties of FG-CNTRC are discussed. Then, using the thirdorder shear deformation plate theory, the von Kármán-type geometrical nonlinearity and Hamilton’s principle, the nonlinear governing equations of FG-CNTRC plate as well as the corresponding boundary conditions are obtained.

2.1. Material properties of the CNTRC periodic transverse force
. It is assumed that the plate is made of a mixture of the isotropic matrix As illustrated in Fig. 1, consider a CNTRC plate of length a, width b and thickness h subjected to the

and SWCNTs. Moreover, the CNT distribution is considered as the uniform distribution (UD) or

distribution of CNTs namely, FGA, FGO and FGX are considered. The volume fraction of CNT

functionally graded distribution in the thickness direction. Three types of functionally graded

∗ UD:  = 

corresponding to the each type of CNT distribution is calculated as

FGA:  = 1 −

2| | ∗   ℎ

FGO:  = 2 1 − FGX:  =

where

∗

 =

(1a) (1b)

2| | ∗   , ℎ

(1c)

4|| ∗

ℎ 



"

(1d)

"

(2)

 + ! " # $ − ! " # $ 

where Λ &'( stands for the mass fraction of CNT. Also, ρ&'( and ρ* respectively denote the densities of CNT and matrix.

4

The rule of mixture can be utilized to calculate the effective material properties of CNTRC nanocomposites [59, 60]. According to the aforementioned rule, the effective material properties of  +,, = -,  +,, + # + #

CNTRC plates are obtained as follows [23]

-.

 # =  + # +.. +.. +

-

 # =  + # /,. /,. /

(3a) (3b) (3c)

0 =  0  + # 0# " =  "



+ # "

(3d)

#

(3e)

&'( &'( &'( where the Young’s modulus and shear modulus of CNT are represented by E,, , E.. and G,. . For

isotropic matrix, the aforementioned properties are signified by E* and G*. The CNT efficiency parameters η3 j = 1,2,3 denote the load transfer between the nanotubes and polymeric phases and any scale effect. Furthermore, volume fraction of CNT is indicated by V*. Also, one can write

 + # = 1. Poisson’s ratios of CNT and matrix are also represented by ν&'( and ν* , respectively.

The considered CNTRC plate is defined in the Cartesian coordinate system 0 ≤ , ≤ :, 0 ≤ . ≤ 2.2 Nonlinear Governing equations and associated boundary conditions

;, − ℎ⁄2 ≤  ≤ ℎ⁄2. Introducing the components of displacement of an arbitrary point of plate along the axes = ; ? = 1,2,3 as @A= , the displacement field of CNTRC plate based on the third-order shear deformation plate theory can be defined as

B, ,, . ,   = @A= C= = D@E +  FE − G, HFE + I,E JK CE + IC .

(4)

where G, = 4⁄3ℎ .; uN and I respectively symbolize the in-plane displacement and the vibration amplitude of an arbitrary

point located at the mid-plane of the CNTRC plate. Moreover, FE

represents the rotation of the middle surface normal. It is remarked that in the present study, the coordinates. Moreover, ?, O = 1,2,3 and P, Q = 1,2. Also, the capital Latin indices, e.g., I and J can symbol comma appeared in the equations stands for the partial differentiation with respect to the

take the specified values. Using the defined displacement field and applying the von Kármán-type of

1  G, HFE,S + FS,E + 2I,ES J RES = H@E,S + @S,E + I,E I,S J + HFE,S + FS,E J − 2 2 2

kinematic nonlinearity, the non-zero components of the strain tensor are obtained as T ,  = RES +  RES +  RES ,

T UE = 2RE = PHFE + I,E J = PUE ,

in which G. = 4⁄ℎ. .

P = 1 − G. . .

(5)

Also, based upon the linear elasticity, the non-zero constituents of the stress tensor can be written through the kinematic parameters as follows 5

Z,, a,, YZ ] `a W .. W _ ,. Z,. = _ 0 XZ, \ _ 0 W W VZ. [ ^ 0

where

a,, =

a,. a.. 0 0 0

+,, , 1 − 0,. 0.,

0 0 abb 0 0

R,, 0 g Y R 0 W .. ] W f 0 f R,. R \ 0 fX W , W R add e V . [

0 0 0 acc 0

a.. =

+.. , 1 − 0,. 0.,

(6)

a,. =

0., +,, 1 − 0,. 0.,

(7)

Hamilton’s principle can be used to obtain the governing equations and corresponding boundary conditions. Based on this principle, one can write (q

δ i Πk − Πl + mn dt = 0 (r

(8)

where Πk and ms denote the Kinetic energy and potential strain energy of system, respectively. Moreover, mn stands for the external potential energy. The symbol δ signifies the delta Kronecker.

Introducing the in-plane force resultants tES , bending moment resultants uES , higher-order bending moments vES , transverse forces aE and higher-order transverse forces wE are related to the stress components ZES and ZE as follows

xtES , uES , vES , aE , wE y = i

{ ⁄.

|{⁄.

xZES ,  ZES ,  ZES , ZE , . ZE yz

(9)

The variation of potential strain energy is expressed as

{ .



T ,  T yz~, δms = i i Z=} δR=} z z~ = i xtES δRES + uES δRES + vES δRES + aE − G.wE δUE {  | .



(10)

Substituting Eq. (6) into Eq. (9) results in the following relations

~,, t,, Y t ] `~ ,. .. Wt W _ 0 ,. W W _ Wu,, W _€,, u.. = _€,. Xu,. \ _ 0 Wv W W ,, W _ ,, W v.. W _ ,. V v,. [ ^ 0 ~cc a, 0 a. … †=‡ w, ‚cc w. 0

~,. ~.. 0 €,. €.. 0 ,. .. 0

0 0 ~bb 0 0 €bb 0 0 bb

0 T ~dd U, ˆ‰ T Š 0 U. ‚dd

€,, €,. 0 ‚,, ‚,. 0 ƒ,, ƒ,. 0

€,. €.. 0 ‚,. ‚.. 0 ƒ,. ƒ.. 0

0 0 €bb 0 0 ‚bb 0 0 ƒbb

,, ,. 0 ƒ,, ƒ,. 0 „,, „,. 0

,. .. 0 ƒ,. ƒ.. 0 „,. „.. 0

RT ] 0 Y ,, T g R .. W W 0 T W f WU,. bb W W f , W 0 f W R,, , 0 f R.. , \ ƒbb f XU,. W 0 f WR  W ,, W 0 fW  W R W „bb e W ..  VU,. [

(11a)

(11b)

= =  where U,. = 2R,. ; ? = 1,2,3 and

6

~‹Œ 1 Y ] Y ] €‹Œ  W W W W W W  W W { ⁄. { ⁄. ‹Œ ~‹Œ P  =i a‹Œ . z ; , Ž = 1,2,6,  ‘ = i a‹Œ ’. P“ z ; , Ž = 4,5. ‚‹Œ  X \ X‚‹Œ \ |{ ⁄. |{ ⁄. d ƒ W W W  W W ‹Œ W W W V„‹Œ [ Vb [

(12)

Moreover, the kinetic energy of CNTRC plate can be calculated as {

1 . m• = i i "@A–= @A–= z z~ 2  |{ .

1 = i xT @– E. + I– .  + 2, @– E F–E − 2G,  H@– E F–E + @– E I–,E J 2 

(13)

. yz~, + . − 2G, d + G,. b F–E. + G, G, b − d I–,E F–E + G,. b I–,E

in which dot denotes the differentiation with respect to the time and ~ stands for the surface area of the CNTRC plate. Moreover, Œ ; Ž = 0, … ,6 are defined as { ⁄.

(14)



(15)

"  Œ z˜ ; Ž = 0, … ,6,

Œ = i

|{ ⁄.

Furthermore, the potential energy due to the transverse load
is expressed as

m™ = i
Iz~, 

One can obtain the nonlinear governing equations of motion and corresponding boundary conditions variations. After integrating by parts and computing the coefficients of δuN , δw and δψN and by substituting Eqs. (10), (13) and (15) into (8) and using the fundamental lemma of calculus of

equaling the corresponding coefficients to zero, one can obtain the following mathematical

tES,E = T @œ S + , FœS −  G, HFœS + Iœ,S J,

formulations

(16a)

aE,E − G. wE,E + G,vES,ES + HtES I,S J,E +


= T Iœ + G,d FœE,E + G,H @œ E,E − G,b Iœ,EE − G,b FœE,E J,

uES,E − G,vES,E − aS + G. wS = , − G,  @œ S + . − 2G, d + G,. b FœS − G, d − G, b Iœ,S , @E = 0 žŸ tES  S = 0

(16b) (16c)

and the boundary conditions can be written as

I = 0 žŸ xaE − G.vE + G, vES,S + tES I,S y E = 0, FE = 0 žŸ HuES − G, vES J S = 0, I,E = 0 žŸ vES  S = 0,

(17a) (17b) (17c) (17d)

Inserting Eq. (11) into Eq. (16) gives the following governing equations expressed in terms of the components of displacement

7

~,, H@,,,, + I,,I,,, J + ~,. + ~bbH@.,,. + I,.I,,. J + ~bb H@,,.. + I,, I,.. J + €,, F,,,, + €,. + €bb F.,,. + €bbF,,.. − G, ,, HF,,,, + I,,,, J

− G, ,. + bb HF.,,. + I,,..J − G,bbHF,,.. + I,,.. J

(18a)

= T @œ , + , Fœ, −  G, HFœ, + Iœ,, J,

~bb H@.,,, + I,,, I,. J + ~,. + ~bb H@,,,. + I,,I,,. J + ~.. H@.,.. + I,. I,..J + €bb F.,,, + €,. + €bb F,,,. + €..F.,.. − G, bb HF.,,, + I,,,. J

− G, ,. + bb HF,,,. + I,,,.J − G,..HF.,.. + I,... J

(18b)

= T @œ . + , Fœ. −  G, HFœ. + Iœ,.J,

~cc − G. ‚cc HF,,, + I,,, J + ~dd − G. ‚dd HF.,. + I,.. J . + G, D,,H@,,,,, + I,,, + I,, I,,,, J

. + ,. + 2bb H@.,,,. + I,,. + I,. I,,,. J + ƒ,,F,,,,,

+ ƒ,. + 2ƒbb HF.,,,. + F,,,.. J − „,, G, HF,,,,, + I,,,,, J

− „,. G, HF,,,.. + F.,,,. + 2I,,,.. J + 2bbH@,,,.. + I,,, I,.. + I,, I,,.. J . − 2„bb G, HF.,,,. + F,,,.. + 2I,,,.. J + ,.H@,,,.. + I,,. + I,,I,,.. J

(18c)

. + .. H@.,... + I,.. + I,. I,... J + ƒ..F.,... − „.. G,HF.,... + I,.... JK

+ t I  +


= T Iœ + G,d HFœ,,, + Fœ.,. J

+ G, D H@œ ,,, + @œ .,. J − G, b HIœ,,, + Iœ,.. J − G, b HFœ,,, + Fœ.,. JK,

€,, − G, ,,H@,,,, + I,, I,,, J + €,. + €bb − G, ,. − G,bb H@.,,. + I,. I,,. J + €bb − G,bb H@,,.. + I,, I,.. J + ‚,, − G, ƒ,, F,,,,

+ ‚,. + ‚bb − G, ƒ,. − G, ƒbb F.,,. + ‚bb − G,ƒbb F,,..

− G, ƒ,, − G,„,, HF,,,, + I,,,, J − G, ƒbb − G, „bb HF,,.. + I,,..J − G, ƒ,. + ƒbb − G,„,. − G, „bb HF.,,. + I,,.. J − ~cc − G.‚cc HF, + I,, J

= , − G,  @œ , + . − 2G,d + G,.b Fœ, − G,d − G,b Iœ,,,

8

(18d)

€bb − G, bb H@.,,, + I,,, I,. J + €,. + €bb − G, ,. − G,bbH@,,,. + I,, I,,. J + €.. − G,.. H@.,.. + I,. I,.. J + ‚bb − G,ƒbb F.,,,

+ ‚,. + ‚bb − G, ƒ,. − G, ƒbb F,,,. + ‚.. − G,ƒ.. F.,.. − G, ƒbb − G,„bb HF.,,, + I,,,. J

− G, ƒ,. + ƒbb − G,„,. − G, „bb HF,,,. + I,,,. J

(18e)

− G, ƒ.. − G,„.. HF.,.. + I,... J − ~dd − G. ‚dd HF. + I,.J

= , − G,  @œ . + . − 2G,d + G,.b Fœ. − G, d − G, b Iœ,. .

where t I = HtES I,S J,E . In the same way, the boundary conditions can be expressed in terms of the displacement. The mathematical formulation of various edge supports can be chosen using Eqs.

(17a)-(17d). In the present study, three types of boundary condition are chosen which are mathematically expressed as follows

@, = @. = I = F. = u,, − G, v,, = v,, = 0 : ¡z¢¡£ , = 0, : a. All edges simply supported (SSSS)

@, = @. = I = F, = u.. − G, v.. = v.. = 0 : ¡z¢¡£ . = 0, ; @, = @. = I = F, = F. = I,, = 0 : ¡z¢¡£ , = 0, :

(19)

b. All edges clamped (CCCC)

@, = @. = I = F, = F. = I,. = 0 : ¡z¢¡£ . = 0, ;

@, = @. = I = F. = u,, − G, v,, = v,, = 0 : ¡z¢¡£ , = 0, :

(20)

c. Two opposite edges simply-supported – the remaining edges clamped (SCSC)

@, = @. = I = F, = F. = I,. = 0 : ¡z¢¡£ . = 0, ;

(21)

2.3 Non-dimensional governing equations Now, introducing the following non-dimensional quantities

¤, ¥ = D

, . : , K , @E , I → ℎ@E , I, - = , : ; ℎ

§=

:  ~,,T
:. ,¨ = © ,
ª = , ; : TT ~,,T ℎ

€‹Œ ‚‹Œ ‹Œ ƒ‹Œ ƒ‹Œ H;‹Œ , z‹Œ , G‹Œ , «‹Œ , ℎ‹Œ J = ! , , , , $ ; , Ž = 1,2,6, .  d ~,,T ℎ ~,,T ℎ ~,,T ℎ ~,,Tℎ ~,,T ℎb

:‹Œ =

~‹Œ ‚‹Œ Œ ; , Ž = 1,2,4,5,6, z‹Œ = ; , Ž = 4,5, Œ̅ = ; Ž = 0,1, … ,6. ~,,T ~,,T ℎ. TT ℎ Œ

(22)

where ~,,T and TT stand for the values of ~,, and T for a homogeneous matrix plate, the nondimensional forms of governing equations can be expressed as follows

9

1 § 1 :,, !@,,,, + I,, I,,, $ + §:,. + :bb  !@.,,. + I,.I,,.$ + §. :bb !@,,.. + I,, I,.. $ -

1 + ;,, F,,,, + §;,. + ;bb F.,,. + §. ;bb F,,.. − G̅,G,, !F,,,, + I,,,, $ § 1 − §G̅,G,. + Gbb  !F.,,. + I,,.. $ − §. G̅, Gbb !F,,.. + I,,.. $ -

(23a)

1 = T̅ @œ , + ,̅ Fœ, − ̅ G̅, !Fœ, + Iœ,, $, -

§ 1 § :bb !@.,,, + I,,, I,. $ + § :,. + :bb  !@,,,. + I,, I,,.$ + §. :.. !@.,.. + I,. I,.. $ § + ;bb F.,,, + § ;,. + ;bb F,,,. + § . ;.. F.,.. − G̅, Gbb !F.,,, + I,,,. $ 1 § − §G̅,G,. + Gbb  !F,,,. + I,,,. $ − § . G̅, G.. !F.,.. + I,... $ § = T̅ @œ . + ,̅ Fœ. − ̅ G̅, !Fœ. + Iœ,. $, -

(23b)

:cc − G̅. zcc H-F,,, + I,,, J + §:dd − G̅. zdd H-F.,. + §I,..J

+

G̅, 1 . 1 § . § ­G,, !@,,,,, + I,,, + I,, I,,,, $ + §G,. + 2Gbb  !@.,,,. + I,,. + I,. I,,,. $® -

+

G̅, 1 ­+«,,F,,,,, + § «,. + 2«bb HF.,,,. + §F,,,..J − ℎ,, G̅, !F,,,,, + I,,,,, $® -

+

G̅, 2§ 1 . 1 ­−2§ℎbb G̅, !F.,,,. + §F,,,.. + I,,,.. $ + § . G,. !@,,,.. + I,,. + I,, I,,.. $® -

+

+

G̅, 2§ 1 1 ­−§ℎ,.G̅, !§F,,,.. + F.,,,. + I,,,.. $ + 2§ . Gbb !@,,,.. + I,,, I,.. + I,, I,,.. $® G̅, § . § § ­+§ G.. !@.,... + I,.. + I,. I,... $ + §  «.. F.,... − § ℎ.. G̅, !F.,... + I,.... $® -

¯ I +
ª = T̅ Iœ + +t

+

G̅, d̅ HFœ,,, + §Fœ.,. J -

G̅, G̅, b̅ HIœ,,, + §. Iœ,.. J − G̅,b̅ HFœ,,, + §Fœ.,. J®, ­̅ H@œ ,,, + §@œ .,. J − -

10

(23c)

1 § ;,, − G̅, G,,  !@,,,, + I,, I,,, $ + § ;,. + ;bb − G̅, G,. − G̅, Gbb !@.,,. + I,. I,,. $ 1 + §. ;bb − G̅, Gbb !@,,.. + I,, I,.. $ + z,, − G̅, «,,F,,,, -

+ §z,. + zbb − G̅, «,. − G̅, «bbF.,,. + §. zbb − G̅, «bb F,,..

1 1 − G̅, «,, − G̅, ℎ,,  !F,,,, + I,,,, $ − § . G̅, «bb − G̅, ℎbb  !F,,.. + I,,.. $ § − §G̅, «,. + «bb − G̅, ℎ,. − G̅, ℎbb  !F.,,. + I,,.. $ − - :cc − G̅. zcc H-F, + I,,J

= ,̅ − G̅, ̅ @œ , + .̅ − 2G̅, d̅ + G̅,. b̅ Fœ, −

(23d)

G̅,  ̅ − G̅, b̅ Iœ,, , - d

§ 1 ;bb − G̅,Gbb  !@.,,, + I,,, I,. $ + §;,. + ;bb − G̅, G,. − G,Gbb  !@,,,. + I,,I,,. $ § + § . ;.. − G̅, G.. !@.,.. + I,. I,.. $ + zbb − G̅,«bb F.,,, + § z,. + zbb − G̅, «,. − G̅, «bb F,,,. + §. z.. − G̅, «.. F.,.. § − G̅, «bb − G̅, ℎbb  !F.,,, + I,,,. $ -

1 − §G̅,«,. + «bb − G̅,ℎ,. − G̅, ℎbb  !F,,,. + I,,,. $ § − § . G̅, «.. − G̅, ℎ..  !F.,.. + I,... $ − - :dd − G̅. zdd H-F. + §I,. J -

= ,̅ − G̅,̅ @œ . + .̅ − 2G̅,d̅ + G̅,.b̅ Fœ. −

§G̅,  ̅ − G̅, b̅ Iœ,. . - d

1 §. § § ¯ I = Ht ¯,, I,, J + Ht ¯.. I,.J + Ht ¯ I J + Ht ¯ I J t ,, ,. - ,. ,. ,, - ,. ,, ,. 1 § ¯,, = :,, !@,,, + I,,. $ + :,.§ !@.,. + I,.. $ + ;,, F,,, + ;,. §F.,. t 221 § − G̅, G,, !F,,, + I,,, $ − §G̅, G,. !F.,. + I,.. $, § . 1 . ¯.. = :..§ !@.,. + I,. $ + :,. !@,,, + I,, $ + ;.. §F.,. + ;,. F,,, t 22§ 1 − §G̅,G.. !F.,. + I,.. $ − G̅, G,. !F,,, + I,,, $, § 2§ ¯,. = :bb !§@,,. + @.,, + I,, I,.$ + ;bb H§F,,. + F.,,J − G̅,Gbb !§F,,. + F.,, + t I $ , - ,,. 1 § ¯,, = ;,, !@,,, + I,,. $ + ;,. § !@.,. + I,.. $ + z,,F,,, + z,.§F.,. u 221 § − G̅, «,, !F,,, + I,,, $ − §G̅,«,. !F.,. + I,.. $, where

11

(23e)

(24a)

(24b)

(24c)

§ . 1 I,. $ + ;,.@,,, + I,,.  + z.. §F.,. + z,. F,,, 22§ 1 − §G̅,«.. !F.,. + I,.. $ − G̅, «,. !F,,, + I,,, $, § 2§ = ;bb !§@,,. + @.,, + I,, I,. $ + zbbH§F,,. + F.,, J − G̅, «bb !§F,,. + F.,, + I,,. $, -

¯.. = ;.. § !@.,. + u ¯,. u

3. Solution strategy The application of the numerical solution procedure including the GDQ method, numerical Galerkin approach and time periodic discretization scheme to the analysis of nonlinear vibrational characteristics of macro-, micro- and nano-structures [61-64] is provided an outstanding proficiency, accuracy and great potential in solving the nonlinear partial differential equations. For the subjected to the periodic transverse load (
ª = « cos Ω¨; f is the forcing amplitude and Ω signifies the geometrically nonlinear primary resonance characteristics of shear deformable FG-CNTRC plates

external excitation frequency) considered in the current study, firstly, to discretized the nonlinear governing equation of motion as well as the associated boundary conditions, the GDQ method [65] is

implemented. Thereafter, the numerical Galerkin scheme is employed to reduce the partial differential equations to a set of nonlinear time-varying Duffing-type ordinary differential equations. To numerically solve the obtained nonlinear Duffing-type equations, the time periodic differential operators are used. This method helps us to discretize the Duffing-type equations in the time domain and convert them to a set of nonlinear algebraic parameterized equations. Now, employing the pseudo arc-length continuation method enables us to examine the primary resonant characteristics of the FGCNTRC plates via plotting frequency-response and force-response curves.

3.1. Discretization on space domain The GDQ method is utilized to discretize the nonlinear governing equations (24a)-(24e). Applying the GDQ method results in the discretized governing equation of motion which can be expressed as the

´µœ + ¶µ + ¶ '· µ + ¸ cos Ω¨ = ¹

following form

in

which

overdot

k

stands

for

(25) the

differentiation

with

respect

to

time;

µ=

xB,k , Bk. , ºk , », k , ». k y represents the displacement vector with 5 ¼ elements containing the time-

discrete grid points along the ¤- and ¥-axes, respectively; the stiffness matrix ¶ and mass matrix ´ dependent generalized coordinates, to be numerically calculated. n and m are the number of total

identify two 5 ¼ × 5 ¼ matrices. Moreover, ¸ and ¶ ¾¿ µ are 5 ¼ × 1 vectors representing the

forcing amplitude and nonlinear stiffness vectors, respectively. It is remarked that the considered boundary conditions are discretized in the same way. The details of the discretization procedure are provided in Appendix.

12

3.2. Numerical Galerkin method Now, applying the numerical Galerkin method to Eq. (25) results in reducing the dimensions of purpose, firstly, ¶ '· µ and ¸ cos Ω¨ are assumed to be zero and the harmonic solution of system obtained equations and gives a Duffing-type set of ordinary differential equations of motion. For this

À e3ω in which ω associated with the free vibration problem is subsequently considered as µ = µ

denotes the linear fundamental frequency. Inserting the assumed solution in the linearized equation leads to

À = Ã. ´µ À, µ À = xB À ,k , » À k. yk ¶µ Ä,k , B Ä k. , º Ä k, »

(26)

To take the influence of the edge conditions into account, all the discretized boundary conditions are inserted into the discretized governing equations. Substituting the boundary conditions into the stiffness and inertia matrices and subsequently rearranging the discretized governing equations and

À . ¶ ÆÇ µ À È É Ê Ì = ÅÃ ´ÆÆ µ Ê È ¶ ÇÇ µ ÀË ¹

(27)

. À À H¶ ÆÆ − ¶ÊË ¶ |, ËË ¶ ËÊ Jµ Ê = à ´ÊÊ µ Ê Í |, À Ë = ¶ ËË ¶ ËÊ µ À Ê µ

(28)

corresponding edge conditions results in the following relation

¶ Å ÆÆ ¶ ÇÆ

in which subscripts ; and z signify the boundary and domain grid points, respectively. Eq. (27) can be written as an uncoupled equation as follows

‰

Above eigenvalue problem can be solved easily to give the fundamental frequency as well as the and considering the Î first linear mode shapes, the response of Eq. (28) can be written as

associated mode shapes of FG-CNTRC plates with different edge supports. By some manipulation

µ = ÏÐ

in which the Galerkin’s basis function Ï represents a 5 ¼ × 5k matrix and q cÓ×, stands for the (29)

reduced generalized coordinates vector.

Ô = ´ÏМ + ¶ÏÐ + ¶ '· ÏÐ − ¸ cos Ωτ

Accordingly, inserting Eq. (29) into (25) gives the residual as follows (30)

To simultaneously perform the procedure of multiplication of the residual by the base functions and integrating over the domain in the numerical Galerkin method, a matrix operator is defined as follows

ÖcÓ×c'* = Ïk ×,

(31)

in which × denotes the integral operator which are calculated according to the differential quadrature and Taylor series as defined in [66].

defined integral operator Ö as follows

The following time-varying Duffing-type set of ODEs can be obtained by multiplying Eq. (30) by the

À М + ¶ ÀÐ + ¶ À ¾¿ ÏÐ = ¸Ø cos Ω∗ τ ´ À cÓ×cÓ = Ö´Ï, ´

in which

À cÓ×cÓ = Ö¶Ï, ¶

(32)

À '· ÏÐ = Ö¶ '· ÏÐ, ¸ØcÓ×, = Ö¸ ¶ 13

(33)

generalized coordinates form 5 ¼ × 1 to 5Î.

It is obvious that using the numerical Galerkin method reduces the number of unknown quantities of

3.3. Time periodic discretization scheme À [67] (c and ωÛ are the non-dimensional damping coefficient and non-dimensional ÙØ = 2c/ωÛ ¶ To consider the energy dissipation due to a viscous damping, the damping matrix is assumed as

fundamental frequency) and the non-conservative damping forces of viscous type ÙØЖ is incorporated

À М + ¶ À Ð + ÙØЖ + ¶ À ¾¿ ÏÐ = ¸Ø cos Ωτ ´

into Eq.(32). Thus, one can write

Then, by introducing ¨ ∗ = ¨⁄Ü where T ∗ = 2π⁄Ω, Eq. (34) can be rewritten as

!

∗

Ω . Ω À М + ! $ ÙØЖ + ¶ ÀÐ + ¶ À ¾¿ ÏÐ = ¸Ø cos 2πτ∗ $ ´ 2π 2π

(34)

(35)

in which overdot represents the differentiation with respect to τ∗ . Now, considering a mesh distribution along the time domain, the time differentiation operators can be calculated according to the derivatives of periodic sin function as base function in spectral collocation approach [68] (for more details see Ref. [61]). Then, applying the aforementioned time differentiation operators, i.e., ßà∗ , to Eq. (35) gives the following relation

!

=

k Ω . Ω , k À âß. Àâ + ¶ À '· Ïâ = ¸Øã, $ ´ + ! $ ÙØâßà∗ + ¶ ∗ à 2á 2á

ã=

ä… , Gž£H2ᨌ∗ J, … å,

Ž = 1,2, … , t

(36)

where t is the even number of discrete points in the time space. Moreover, âcÓ×æç denotes the nodal values of qτ∗ at grid point τè∗.

Finally, Eq. (36) can be expressed in the vectorized form as follows

Ω . . Ω À K + ! $ Dß, Ø À Jì í¡G â + í¡G D¶ À '· ÏâK é! $ Dßà∗ ⊗ ´ Hë ∗ à∗ ⊗ ÙK + à ⊗ ¶ 2á 2á − Hëà∗ ⊗ ¸ØJãk = ¹

îí¡G â , Ω, «  = ¹

( 37)

Eq. (37) can be expressed as a set of the parameterized nonlinear equations as follows

î: ℝc#×ðñ ò. → ℝc#×ðñ ,

In Eq. (35), í¡G â represents the vectorization of the matrix â and óÂ∗ denotes an t × t identity (38)

tensor. The pseudo-arc length continuation method can be used to estimate the frequency-response and force-response characteristics of FG-CNTRC plate with various edge supports by solving the system of nonlinear algebraic equations (38). It is remarked that in the case of frequency-response a external excitation frequency ratio of system Ω⁄Ãô  is assumed to be given value. Moreover, the

constant forcing amplitude is considered. Similarly, to achieve the force-response curve of system, the

linear frequency as well as the linear solution of system are chosen as the initial guess.

14

4. Results and discussion With a view to achieve a comprehensive set of geometrically nonlinear vibrational results, the numerical computations have been performed to clarify the nonlinear primary resonance of thirdorder shear deformable FG-CNTRC rectangular plates by changing the nanotube volume fraction, CNT distributions, length-to-thickness ratio and boundary conditions. The FG-CNTRC plates with CCCC, SSSS and CSCS edge conditions are considered in the present study. Furthermore, it is remarked that the distributions of CNTs in CNTRC plates are considered as UD, FGA, FGO and FGX. at ambient temperature are taken as [23]: + # = 2.5 /v:, "# = 1150 ΢⁄¼ ,0 # = 0.34. The The matrix is assumed to be made of Poly methyl methacrylate (PMMA) and the material properties

material properties of single-walled (10, 10) armchair CNTs at ambient temperature as a

   are assumed to be as [23]: +,, = 5.6466 Üv:, +.. = 7.0800 Üv:, /,. =

1.9445 Üv:, "  = 1400 ΢⁄¼ , 0  = 0.175. Moreover, to determine the CNT efficiency reinforcement

parameters presented in Eq. (1), it is needed to match Young’s moduli +,, , +..  and shear moduli

/,.  of CNTRC calculated by the extended rule of mixture with those estimated by the MD

simulations [69, 70]. For the three selected CNT volume fractions, the efficiency parameters are ∗

 = 0.12 ∶ -, = 0.137, -. = 1.022,

calculated as follows ∗



= 0.17 ∶ -, = 0.142, -. = 1.626,

∗

 = 0.28 ∶ -, = 0.141, -. = 1.585,

- = 0.715, - = 1.138, - = 1.109

8 in which the variations of maximum vibration amplitude I#ùú  of FG-CNTRC plate versus the A comprehensive parametric examination is carried out and numerical results are provided in Figs. 2-

excitation frequency ratio Ω/Ãô (excitation frequency to linear frequency ratio) and the forcing amplitude are plotted. Moreover, the response corresponding to the nonlinear free vibration of FG-

CNTRC plate as well as natural frequency are given in the figures. Since there is always some kind of damping, whether internal or external, in real physical systems that dissipates energy, a viscously damped system has been considered. A non-dimensional damping coefficient of 0.01 has been chosen to numerically investigate the primary resonance of FG-CNTRC plates. To verify the validity and accuracy of the developed model and solution methodology, the primary resonance results of a square aluminum plate with fully simply-supported edge conditions obtained from the present analysis are compared with the results given by Alijani et al. [71] in Fig. 2. Reasonable agreement has been found between the results of the present approach and those in [71].

Also, a comparison study of the frequency parameters (ω Ä =Ã ¯ ;. ü"# ⁄+#⁄ℎ; Ã ¯ denotes the dimensional linear frequency) of FG-CNTRC plates with SSSS and CCCC boundary conditions is

provided in Table 1 for various distributions of CNTs including UD, FGO and FGX. It is noted that in 15

this example, the matrix is assumed to be made of Poly ((m-phenylenevinylene)-co-[(2,5-dioctoxy-pphenylene) vinylene]), denoted as PmPV. It is seen that present numerical results are in good agreement with those given by Selim et al.[72]. The frequency-response curves of FG-CNTRC plates with various edge supports corresponding to different numbers of mode are provided in Fig. 3. This figure illustrates the converging trend of the present numerical-based Galerkin method. Moreover, it can be seen that considering three modes is sufficient to obtain the frequency-response of FG-CNTRC plates. The influence of CNT volume fraction on the frequency-response and force-response curves of the third-order shear deformable FGX-CNTRC plates with various boundary conditions is illustrated in Figs. 4 and 5, respectively. As can be seen, a hardening-type behavior is observed in the frequencyresponse and force-response curves, because of the consideration of the geometric nonlinearity and presence of cubic nonlinear terms resulting in occurring the resonance at excitation frequencies higher that the linear frequencies. The increase of the CNT volume fraction causes a reduction in the amplitude peak of FG-CNTRC and increases the fundamental linear and nonlinear frequencies. The nonlinear hardening-type behavior of FG-CNTRC plates with various edge supports is influenced by the contribution of the cubic nonlinear terms. Moreover, the influence of edge conditions is clearly demonstrated in Fig. 4. It can be found that as the edges become softer, the linear and nonlinear frequencies decrease, the vibration amplitude and nonlinear effects increase and the hardening-type response of the FG-CNTRC plates is intensified. This is due to the fact that the hardening effects become more pronounced for FG-CNTRC plates having the lower stiffness. It is notable that the truthful design and fabrication of sensors and resonators made of FG-CNTRC materials need to the accurate estimation of frequency-response curve of system. Also, according to Fig. 5, in lower forcing amplitudes, it can be seen that increasing the forcing amplitude leads to two limit point bifurcations and a jump phenomenon is occurred. For appropriately large values of forcing amplitude, increasing in the case of ∗ = 0.12, the lower response amplitude is predicted compared to other CNT volume the forcing amplitude increases the maximum vibration amplitude of FG-CNTRC plates. In general,

fractions.

Fig. 6 addresses to the nonlinear frequency-response of CNTRC plates as well as the linear frequency and nonlinear free vibration response for different types of CNT distribution. It can be observed that for all types of edge supports, the greatest linear fundamental frequency is achieved by FGX distribution which is followed by UD, FGA and FGO distributions, respectively. Thus, it can be deduced that the distribution of CNTs near the top and bottom surfaces are more effective in strengthening the plate stiffness than that near the mid-plane. Also, it can be concluded that the FGXtype distribution makes better use of CNTs as CNTs are more distributed near the top and bottom surfaces which are the regions experiencing higher bending compared to those close to the mid-plane. For the given geometric parameters, the plates with the FGA- and FGO-types of CNT distribution include the lowest and highest nonlinear hardening-type behavior, respectively. The influence of CNT 16

distribution is more considerable on the frequency-response of FG-CNTRC plate with SSSS boundary condition. Furthermore, due to the highest linear stiffness, FGX-CNTRC plate includes the highest natural frequency compared to the types of CNT distribution. Moreover, the force-response curves of FG-CNTRC plate with different boundary conditions are plotted in Fig. 7 for various CNT distributions. The lowest value of bifurcation point corresponds to FGO-CNTRC plates regardless to the type of boundary condition. Moreover, for approximately larger values of forcing amplitude, the Depicted in Figs. 8 and 9 are the influence of length-to-thickness ratio :⁄ℎ on the frequencyhighest maximum vibration amplitude belongs to FGA-CNTRC plates.

response and force-response curves of FGX-CNTRC plates with CCCC, CSCS and SSSS boundary conditions, respectively. Regardless of the type of boundary condition, it is observed that a decrease when :⁄ℎ increases. Moreover, the hardening-type behavior is more significant for SSSS plates. in the linear fundamental frequencies and typical hardening nonlinearity of FGX-CNTRC plate occurs

Furthermore, it is remarked that the applying the third-order shear deformation plate theory compared with small length-to-thickness ratios. Moreover, an increase in :⁄ℎ leads to increasing the vibration to the Kirchhoff plate theory enables us to estimate nonlinear vibrational response of CNTRC plates

The influence of the excitation frequency ratios Ω⁄Ãô on the force-response curves of the FGXamplitude of FG-CNTRC plates.

CNTRC plate with various edge conditions is demonstrated in Fig. 10. It can be seen that for the excitation frequency ratios less than unity, when the forcing amplitude gradually increases the However, increasing the forcing amplitude in the case of Ω⁄Ãô > 1 results in two limit bifurcation in maximum vibration amplitude of FGX-CNTRC plate increases without any jumps and bifurcations.

the force-response curve and a jump from lower response amplitude to the higher one. In the

approximately high forcing amplitudes, increasing the excitation frequency ratio leads to increasing Ω⁄Ãô > 1, increasing the excitation frequency ratio results in occurring the first bifurcation point in the response amplitude of FGX-CNTRC plates with various boundary conditions. Furthermore, for

the higher values of forcing amplitude.

5. Concluding remarks The geometrically nonlinear primary resonance of third-order shear deformable FG-CNTRC plates with various edge supports subjected to a harmonic excitation transverse force was examined in the present study. It was assumed that CNTs are graded in thickness direction of the plate and effective material properties are calculated using the rule of mixture. The third-order shear deformation plate theory, von Kármán hypotheses and Hamilton’s principle as well as the fundamental lemma of the calculus of variation were implemented to obtain the nonlinear governing equations of motion and corresponding boundary conditions. Afterwards, the discretized equations of motion of the FGCNTRC plates were obtained using the GDQ method and applying the numerical-based Galerkin method resulted in a set of Duffing-type second-order time-varying nonlinear ODEs. This time17

varying set was discretized in the time domain via a time periodic discretization to transform into a new set of nonlinear algebraic equations which was solved numerically using the pseudo-arc length continuation method. Then, the nonlinear primary resonance characteristics of FG-CNTRC plates was given in the form of frequency-response and force-response curves. Examining the nonlinear frequency-response of the third-order shear deformable FG-CNTRC plates revealed that: the increase of the CNT volume fraction results in decreasing the vibration amplitude peak of FG-CNTRC and increasing the fundamental frequency, but it may be lead to increasing or decreasing the hardening-type nonlinear behavior of CNTRC plates. Moreover, regardless of the type of boundary condition, an increase in the length-to-thickness ratio decreases the linear fundamental frequencies and typical hardening nonlinearity of FG-CNTRC plate. Moreover, examining the force-response curves of the FG-CNTRC plates illustrated that: for the excitation frequency ratios less than unity, increasing the forcing amplitude gradually leads to Ω⁄Ãô > 1, the FG-CNTRC plate displays two limit bifurcation points with a jump phenomenon. The increasing the amplitude peak of FG-CNTRC plate without any jumps and bifurcations. Moreover, for

CNT distribution has the highest effect on the force-response of FG-CNTRC plate with SSSS edge supports. Also, the lowest value of bifurcation point corresponds to FGO-CNTRC plates and for approximately larger values of forcing amplitudes, the highest maximum vibration amplitude belongs to FGA-CNTRC plates.

18

Appendix. using the GDQ method, first of all, the grid points along the non-dimensional ¤- and ¥-directions To discretize the governing equations of motion as well as the corresponding boundary conditions

(0 ≤ ¤ ≤ 1 and 0 ≤ ¥ ≤ 1) can be considered based on the shifted Chebyshev–Gauss–Lobatto grid

1 ? −1 ¤= = !1 − Gž£ á$ , ? = 1,2,3, … ,   2  −1 1 O−1 ¥} = !1 − Gž£ á$ , O = 1,2,3, … , ¼ 2 ¼−1

distribution as follows

(A-1)

a¤, ¥ can be approximately evaluated as a weighted linear sum of the function values at all grid

Now, utilizing the Kronecker tensor product, the derivatives of an arbitrary two-dimensional function

þ ™ò௤ a ௤ ™ = Dßక ⨂ß఍ K â. þ¤ ௤ þ¥ ௤

points as follows

(A-2)

in which the pth- and qth-order derivatives of function a with respect to the ¤ and ¥ are denoted by p and q, respectively. Moreover, the weighting coefficients matrices of the pth- and qth-order derivatives along the ¤- and ¥-directions are signified by ßక

™

and ß఍ , respectively; the symbol ⊗ ௤

represents the Kronecker product and â denotes a column vector expressed as

â = ሾa ¤, , ¥, , … , a¤ , ¥, , a¤, , ¥. , … , a¤ , ¥. , … , a¤, , ¥# , … , a¤ , ¥# ሿk.

(A-3)

Moreover, using a recursive scheme, the weighting coefficients of rth-order derivative ß௥ = ‚=} is ௥

calculated as ‚=}

௥

=} , where =} is a   ×   identity matrix Ÿ=0 Y  ࣪ =  W , where ࣪ =  = ෑ = − ௞  ?, O = 1, … ,   : z ? ≠ O : z Ÿ = 1 W W H= − } J࣪H} J ௞ୀ,; =ஷ௞ W ௥|, Í. = YŸ ൥ࣱ , ࣱ ௥|, − ࣱ=} ൩, ?≠O =} == XW = − } W  Í ?, O = 1, … ,   : z Ÿ ≥ 2 W ௥ X W − ෍ ࣱ=௞ , ? = O WW ௞ୀ, VV ௞ஷ=

(A-3)

Now, using the described scheme, the discretized components of ¶, ´ and ¸ as well as the nonlinear part ¶ '· can be expressed as follows

¶,, = :,, ë఍ ⨂ßక + :bbκ. ß఍ ⨂ëక , ¶,. = :,. + :bb §ß఍ ⨂ßక , .

.

¶, = − DG̅, G,, ë఍ ⨂ßక + § . G̅, G,. + 2Gbb ß఍ ⨂ßక Kൗ-, 

.

,

¶,d = ;,, − G̅,G,, ë఍ ⨂ßక + ;bb − G̅, Gbb κ. ß఍ ⨂ëక , .

¶,c = H;,. + ;bb − G̅, G,. + Gbb J§ß఍ ⨂ßక , ,

,

.

19

,

,

(A-4)

¶ ., = :,. + :bb §ß఍ ⨂ßక , ¶ .. = :..κ. ß఍ ⨂ëక + :bb ë఍ ⨂ßక , ,

,

.

.

¶ . = − D§G̅, G,. + 2Gbb ß఍ ⨂ßక + §  G̅, G..ß఍ ⨂ëక Kൗ-, ,

.



¶ .d = H;,. + ;bb − G̅, G,. + Gbb J§ß఍ ⨂ßక , ,

,

¶ .c = ;.. − G̅,G.. κ. ß఍ ⨂ëక + ;bb − G̅,Gbb ë఍ ⨂ßక , ¶ , =

.

.

G̅, G,, G̅,§ . G̅, G,. + 2Gbb § ,  . 2Gbb + G,. ß఍. ⨂ß, ë఍ ⨂ßక + , ¶ . = ß఍ ⨂ßక , క -

¶  = :cc − G̅. zcc ë఍ ⨂ßక + :dd − G̅.zddκ.ß఍ ⨂ëక − .

.

ℎ,, G̅,. d ë఍ ⨂ßక . -

−

2§ℎ,.G̅,. . 4§ .ℎbbG̅,. . §d ℎ.. G̅,. d . . ß ⨂ß − ß ⨂ß − ß఍ ⨂ëక , ఍ ఍ క క -. -. -.

−

§ . G̅,. ℎ,. + 2ℎbb ß఍. ⨂ß, క , -

+

G̅, § «.. − ℎ.. G̅,ß఍ ⨂ëక , -

¶ d = :cc − G̅. zcc -ë఍ ⨂ßక + ,

G̅,  . , !«,, − ℎ,, G̅, ë఍ ⨂ßక + § . «,. + 2«bbß఍ ⨂ßక $ -

¶ c = §:dd − G̅. zdd -ß఍ ⨂ëక + ,

G̅, § , . H«,. + 2«bb − G̅, ℎ,. + 2ℎbb Jß఍ ⨂ßక -

¶ d, = ;,, − G̅,G,, ë఍ ⨂ßక + §. ;bb − G̅,Gbb ß఍ ⨂ëక , ¶ d. .

.

= § ;,. + ;bb − G̅, G,. − G̅, Gbb ß఍ ⨂ßక , ,

,

¶ d = −- :cc − G̅. zcc ë఍ ⨂ßక − G̅, «,, − G̅,ℎ,,⁄- ë఍ ⨂ßక ,



− § . G̅, «,. + 2«bb − G̅,ℎ,. − 2G̅,ℎbb⁄- ß఍ ⨂ßక , .

,

¶ dd = Hz,, − G̅, «,,  − G̅, «,, − G̅,ℎ,, Jë఍ ⨂ßక − :cc − G̅.zcc- . ë఍ ⨂ëక .

+ § . Hzbb − G̅,«bb  − G̅, «bb − G̅, ℎbb Jß఍ ⨂ëక , .

¶ dc = H§z,. + zbb − G̅, «,. − G̅, «bb − §G̅, «,. + «bb − G̅, ℎ,. − G̅,ℎbbJß఍ ⨂ßక , ¶ c, = §;,. + ;bb − G̅, G,. − G,Gbb ß఍ ⨂ßక , ¶ c. ,

,

= § . ;.. − G̅, G.. ß఍ ⨂ëక + ;bb − G̅,Gbb ë఍ ⨂ßక , .

.

,

,

¶ c = −- :dd − G̅. zdd §ß఍ ⨂ëక − §G̅, «,. + 2«bb − G̅,ℎ,. − 2G̅,ℎbb⁄- ß఍ ⨂ßక ,

− §  G̅, «.. − G̅, ℎ.. /-ß఍ ⨂ëక , 

,

.

¶ cd = H§z,. + zbb − G̅, «,. − G̅, «bb − §G̅, «,. + «bb − G̅, ℎ,. − G̅,ℎbbJß఍ ⨂ßక , ¶ cc = §. Hz.. − G̅, «..  − G̅, «.. − G̅,ℎ..Jß఍ ⨂ëక .

,

,

+ Hzbb − G̅,«bb  − G̅, «bb − G̅, ℎbb Jë఍ ⨂ßక − - . :dd − G̅. zdd ë఍ ⨂ëక . 20

.

´,, ` ¹ _ ´ = − _´, _´d, ^ ¹

¹ ´.. ´. ¹ ´c.

´, ´. ´ ´d ´c

´,, = ´.. = T̅ ë఍ ⨂ëక ,

´,d ¹ ´d ´dd ¹

¹ ´.c g f ´c f, ¹ f ´cc e

´ = T̅ ë఍ ⨂ëక +

G̅, G̅, b̅ . . ­− Dë఍ ⨂ßక + § . ß఍ ⨂ëక K®, -

´,d = ´d, = ´.c = ´c. = ,̅ − ̅ G̅, ë఍ ⨂ëక ,

(A-5)

, , ´, = −´, = − ̅ G̅,⁄- ë఍ ⨂ßక , ´. = −´. = − ̅ G̅, § ⁄- ß఍ ⨂ëక ,

´d = −´d = 

G̅, d̅ G̅,. b̅ G̅, d̅ G̅,.b̅ , ,  ë఍ ⨂ßక , ´c = −´c = §   ß఍ ⨂ëక , − − -

´dd = ´cc = .̅ − 2G̅, d̅ + G̅,. b̅ ë఍ ⨂ëక .

¸ = ሾ¹, ¹, ሾ… , «, … ሿ,×# , ¹, ¹ ሿk

¶ ௟ µ = ä¶ ௨ µk, ¶ ௩ µk , ¶ ௪ µk, ¶ ట r µk, ¶ ట q

where

¶ ௨r µ = ¶ ࢛૛ µ =

(A-6)

k µk

å .

(A-7)

:,. + :bb Î . :,, , . , , , !Dë఍ ⨂ßక Kº$ ∘ !Dë఍ ⨂ßక Kº$ + DHß఍ ⨂ëక JºK ∘ !Dß఍ ⨂ࡰక Kº$ +

:bb Î . , . !Dë఍ ⨂ßక Kº$ ∘ !Dë఍ ⨂ßక Kº$, -

:,. + :bb Î :.. Î  , . , , , DHß఍ ⨂ëక JºK ∘ DHß఍ ⨂ëక JºK + !Dë఍ ⨂ßక Kº$ ∘ !Dß఍ ⨂ßక Kº$ :bb § . , !Dë఍ ⨂ßక Kº$ ∘ DHß఍ ⨂ëక JºK, + -

1 Î. , ¯૚૚ ∘ !Dë఍ ⨂ß, Kº$ + Hß, ⨂ëక J !‫ۼ‬ ¯ .. ∘ DHß, ⨂ëక JºK$ ¶ ୵ µ = Dë఍ ⨂ßక K ‫ۼ‬ క ఍ ఍ Î Î , ¯,. ∘ DHß఍, ⨂ëక JºK$ + Hß఍, ⨂ëక J ‫ۼ‬ ¯,. ∘ !Dë఍ ⨂ß, + Dë఍ ⨂ßక K !‫ۼ‬ క Kº$ +

G̅, . . ,  ­G !Dë఍ ⨂ßక Kº$ ∘ !Dë఍ ⨂ßక Kº$ + !Dë఍ ⨂ßక Kº$ ∘ !Dë఍ ⨂ßక Kº$ -. ,,

+

2§ . Gbb G̅, . . , . , !Dë఍ ⨂ßక Kº$ ∘ DHß఍ ⨂ëక JºK + !Dë఍ ⨂ßక Kº$ ∘ !Dß఍ ⨂ßక Kº$ -.

+ §. G,. + 2Gbb  !Dß఍ ⨂ßక Kº$ ∘ !Dß఍ ⨂ßక Kº$ + DHß఍ ⨂ëక JºK ∘ !Dß఍ ⨂ßక Kº$® ,

+ +

,

,

,

,

,

.

§. G,. G̅, , , , , , . , !Dß఍ ⨂ßక Kº$ ∘ !Dß఍ ⨂ßక Kº$ + !Dë఍ ⨂ßక Kº$ ∘ !Dß఍ ⨂ßక Kº$ -. G̅, § d G.. . . ,  !DHß఍ ⨂ëక JºK ∘ DHß఍ ⨂ëక JºK + DHß఍ ⨂ëక JºK ∘ DHß఍ ⨂ëక JºK$, -.

21

¶ ψr µ =

;,. + ;bb Î . ;,, , . , , , !Dë఍ ⨂ßక Kº$ ∘ !Dë఍ ⨂ßక Kº$ + DHß఍ ⨂ëక JºK ∘ !Dß఍ ⨂ßక Kº$ -

¶ ψq µ =

;,. + ;bb Î ;.. Î  , . , , , DHß఍ ⨂ëక JºK ∘ DHß఍ ⨂ëక JºK + !Dë఍ ⨂ßక Kº$ ∘ !Dß఍ ⨂ßక Kº$ -

with

+

;bb Î . , . !Dë఍ ⨂ßక Kº$ ∘ !Dë఍ ⨂ßక Kº$, -

+

;bb § . , !Dë఍ ⨂ßక Kº$ ∘ DHß఍ ⨂ëక JºK. -

¯ ,, = :,, ÅDë఍ ⨂ß, K B, + ‫ۼ‬ క +:,. Î ÅDß఍ ⨂ëక K B. + ,

1 , , !Dë఍ ⨂ßక K º$ ∘ !Dë఍ ⨂ßక K º$È 2-

§ , , , !Dß఍ ⨂ëక K º$ ∘ !Dß఍ ⨂ëక K º$È + ;,, − G̅, G,, Dë఍ ⨂ßక K », 2-

+ ;,. − G̅, G,.Î Dß఍ ⨂ëక K ». −

¯ .. = :.. Î ÅDß, ⨂ëక K B. + ‫ۼ‬ ఍

,

G̅, G,, §. G̅, G,. . . Dë఍ ⨂ßక K º − Dß఍ ⨂ëక K º, -

§ , , !Dß఍ ⨂ëక K º$ ∘ !Dß఍ ⨂ëక K º$È 2-

+ :,. ÅDë఍ ⨂ßక K B, + ,

1 , , !Dë఍ ⨂ßక K º$ ∘ !Dë఍ ⨂ßక K º$È 2-

+ ;.. − G̅, G.. Î Dß఍ ⨂ëక K ». + ;,. − G̅, G,.  Dë఍ ⨂ßక K »,

−

,

§ . G̅, G.. . G̅,G,. . Dß఍ ⨂ëక K º − Dë఍ ⨂ßక K º, -

¯ ,. = :bb ÅÎ Dß, ⨂ëక K B, + Dë఍ ⨂ß, K B. + ‫ۼ‬ ఍ క

1 , , !Dë఍ ⨂ßక K º$ ∘ !Dß఍ ⨂ëక K º$È -.

+ ;bb ቂÎ Dß఍ ⨂ëక K », + Dë఍ ⨂ßక K ». ቃ ,

,

− G̅, Gbb !Î Dß఍ ⨂ëక K », + Dë఍ ⨂ßక K ». + ,

,

,

2§ , , Dß఍ ⨂ßక K º$. -

where ë఍ and ëక signify ¼ × ¼ and   ×   identity tensors, respectively and ∘ denotes the Hadamard product. It is remarked that the boundary conditions can be discretized in the same manner.

22

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26

Table 1: Comparison study of frequency parameters of FG-CNTRC plates ℎ⁄; = 0.1, : ⁄; = 1 List of Captions:

Fig. 1: Schematic of CNTRC plates. (a) UD plate, (b) FGA plate, (c) FGO plate (d) FGX plate

Fig. 2: Comparison of the frequency response curves for the fully simply-supported isotropic square plate made of pure aluminum (h/a = 0.01) ∗ (: ⁄ℎ = 12, :⁄; = 1,  = 0.12, G = 0.01, « = 0.06)

Fig. 3: Frequency-response curve of FGX-CNTRC plate corresponding to different numbers of mode subjected to various edge supports (:⁄ℎ = 10, : ⁄; = 1, G = 0.01, « = 0.06)

Fig. 4: Influence of CNT volume fraction on the frequency-response curves of FGX-CNTRC plate subjected to various edge supports (:⁄ℎ = 10, : ⁄; = 1, G = 0.01, Ω⁄Ãô = 1.4)

Fig. 5: Influence of CNT volume fraction on the force-response curves of FGX-CNTRC plate to various edge supports (:⁄ℎ = 10, : ⁄; = 1, ∗ = 0.28, G = 0.01, « = 0.06

Fig. 6: Influence of CNT distribution on the frequency-response curves of FG CNTRC plate subjected various edge supports (: ⁄ℎ = 10, :⁄; = 1, ∗ = 0.28, G = 0.01, Ω⁄Ãô = 1.4)

Fig. 7: Influence of CNT distribution on the force-response curves of FG CNTRC plate subjected to Fig. 8: Influence of aspect ratio :⁄ℎ on the frequency-response curves of FGX-CNTRC plate subjected to various edge supports (:⁄; = 1, ∗ = 0.28, G = 0.01, « = 0.06)

Fig. 9: Influence of aspect ratio :⁄ℎ on the force-response curves of FGX-CNTRC plate subjected to various edge supports (:⁄; = 1, ∗ = 0.28, G = 0.01, , Ω⁄Ãô = 1.4)

Fig. 10: Influence of the excitation frequency ratio Ω⁄Ãô  on the force-response curves of FGXCNTRC plates subjected to various edge supports (:⁄ℎ = 10, : ⁄; = 1, ∗ = 0.28, G = 0.01)

27

ࢂ∗࡯ࡺࢀ

Table 1: Comparison study of frequency parameters of FG-CNTRC plates ℎ⁄; = 0.1, :⁄; = 1 UD

FGO

FGX

mode Present

Selim et al. [72]

Present

Selim et al. [72]

Present

Selim et al. [72]

Fully simply-supported FG-CNTRC plate 0.11

1

13.5597

13.557

11.3467

11.319

14.6983

14.683

2

17.6849

17.732

16.1008

16.137

18.6494

18.687

1

14.3662

14.360

12.1404

12.110

15.4199

15.402

2

18.3792

18.421

16.6971

16.728

19.3626

19.398

1

16.8431

16.841

14.1015

14.074

18.1959

18.191

2

22.0383

22.098

19.9593

20.002

23.3708

23.435

0.14

0.17

Fully clamped FG-CNTRC plate 0.11

0.14

0.17

1

18.2690

18.091

16.8486

16.473

18.9804

18.636

2

23.8323

23.482

22.4768

22.243

25.3259

24.063

1

18.9063

18.671

17.5049

17.166

19.6469

19.156

2

24.3557

24.079

23.0342

22.849

24.8804

24.697

1

22.9007

22.578

20.8177

20.694

23.7812

23.228

2

29.7815

29.339

27.9509

27.774

30.6720

30.239

28

ℎൗ 2 ℎൗ 2



,

;

:

ℎൗ 2 ℎൗ 2

.

,

,



;

;

:

.

(b) FGA

(a) UD

ℎൗ 2 ℎൗ 2



:

ℎൗ 2 ℎൗ 2

.

,



;

(d) FGX

(c) FGO

Fig. 1: Schematic of CNTRC plates. (a) UD plate, (b) FGA plate, (c) FGO plate (d) FGX plate

29

:

.

Fig. 2: Comparison of the frequency response curves for the fully simply-supported isotropic square plate made of pure aluminum (h/a = 0.01)

30

(b) SSSS

(a) CCCC 1.4

Amplitude ( wmax )

0.8

k= 1 k= 2 k= 3 k= 4

k=1 k=2 k=3 k=4

1.2 1 Amplitude ( wmax )

1

0.6

0.4

0.8 0.6 0.4

0.2

0 0.4

0.2

0.6

0.8 1 1.2 1.4 Frequency ratio ( Ω/ωL )

1.6

1.8

0 0.4

0.6

0.8

1

1.2 1.4 1.6 1.8 2 Frequency ratio ( Ω /ωL )

2.2

Fig. 3: Frequency-response curve of FGX-CNTRC plate corresponding to different numbers of mode ∗ (: ⁄ℎ = 12, :⁄; = 1,  = 0.12, G = 0.01, « = 0.06)

31

2.4

2.6

(b) SSSS

(a) CCCC

1.4

1 Vcnt = 0.12, ωL = 1.697

0.8

Vcnt = 0.17, ωL = 2.157

0.7

Vcnt = 0.28, ωL = 2.318

*

*

1.2

*

1

Vcnt = 0.12, ωL = 1.244 *

Amplitude ( wmax )

Vcnt = 0.17, ωL = 1.563

0.6 0.5 0.4 0.3

*

Vcnt = 0.28, ωL = 1.719

0.8 0.6 0.4

0.2

0.2

0.1 0 0.6

0.8

1 1.2 1.4 1.6 Frequency ratio ( Ω/ωL )

1.8

0 0.4 0.6 0.8

2

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Frequency ratio ( Ω /ωL )

(c) CSCS *

Vcnt = 0.12, ωL = 1.603

1

*

Vcnt = 0.17, ωL = 2.028 Amplitude ( wmax )

Amplitude ( wmax )

*

0.9

0.8

*

Vcnt = 0.28, ωL = 2.172

0.6 0.4 0.2 0 0.6

0.8

1

1.2 1.4 1.6 1.8 Frequency ratio ( Ω /ωL )

2

2.2

Fig. 4: Influence of CNT volume fraction on the frequency-response curves of FGX-CNTRC plate subjected to various edge supports (:⁄ℎ = 10, : ⁄; = 1, G = 0.01, « = 0.06)

32

(b) SSSS

0.8

0.7

0.7

0.6 Amplitude ( wmax )

0.9

0.8

0.6 0.5 0.4 0.3

*

Vcnt = 0.12, ωL = 1.697

0.2

0.4 0.3 *

Vcnt = 0.12, ωL = 1.244 *

*

0.1 0

0.5

0.2

Vcnt = 0.17, ωL = 2.157

Vcnt = 0.17, ωL = 1.563

0.1

*

*

Vcnt = 0.28, ωL = 1.719

Vcnt = 0.28, ωL = 2.318 0

0.2

0.4

0.6 0.8 1 1.2 Forcing Amplitude ( f )

1.4

0

1.6

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Forcing Amplitude ( f )

0.8

(c) CSCS 0.9 0.8 0.7 Amplitude ( wmax )

Amplitude ( wmax )

(a) CCCC

0.6 0.5 0.4 0.3

*

Vcnt = 0.12, ωL = 1.603

0.2

*

Vcnt = 0.17, ωL = 2.028

0.1 0

*

Vcnt = 0.28, ωL = 2.172 0

0.2

0.4 0.6 0.8 1 Forcing Amplitude ( f )

1.2

1.4

Fig. 5: Influence of CNT volume fraction on the force-response curves of FGX-CNTRC plate subjected to various edge supports (: ⁄ℎ = 10, :⁄; = 1, G = 0.01, Ω⁄Ãô = 1.4)

33

0.9

1

(b) SSSS

(a) CCCC 1

0.8

0.9

FGA, ωL = 2.224

0.8

0.6

FGO, ωL = 2.095

0.7

0.5

FGX, ωL = 2.318

0.4

0.6 0.5 0.4

UD, ωL = 1.627

0.3

FGA, ωL = 1.619

0.2

0.2

FGO, ωL = 1.414

0.1

0.1

FGX, ωL = 1.719

0.3

0 0.6

0.8

1 1.2 1.4 Frequency ratio ( Ω /ωL )

1.6

0 0.6 0.8

1.8

1

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Frequency ratio ( Ω /ωL )

(c) CSCS 0.9 0.8 0.7 Amplitude ( wmax )

Amplitude ( wmax )

Amplitude ( wmax )

UD, ωL = 2.257

0.7

0.6

UD, ωL = 2.134 FGA, ωL = 2.093 FGO, ωL = 1.975 FGX, ωL = 2.172

0.5 0.4 0.3 0.2 0.1 0 0.6

0.8

1 1.2 1.4 1.6 Frequency ratio ( Ω /ωL )

1.8

2

Fig. 6: Influence of CNT distribution on the frequency-response curves of FG CNTRC plate subjected ∗ to various edge supports (: ⁄ℎ = 10, :⁄; = 1,  = 0.28, G = 0.01, « = 0.06)

34

(b) SSSS

0.7

0.7

0.6

0.6 Amplitude ( wmax )

0.8

0.5 0.4 0.3

UD, ωL = 2.257

0.2

FGA, ωL = 2.224 FGO, ωL = 2.095

0.1 0

0.2

0.4

0.6 0.8 1 1.2 Forcing Amplitude ( f )

1.4

0.5 0.4 0.3

UD, ωL = 1.627

0.2

FGA, ωL = 1.619 FGO, ωL = 1.414

0.1

FGX, ωL = 2.318 0

FGX, ωL = 1.719 0

1.6

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Forcing Amplitude ( f )

0.8

(c) CSCS 0.7 0.6

Amplitude ( wmax )

Amplitude ( wmax )

(a) CCCC 0.8

0.5 0.4 0.3 UD, ωL = 2.134 0.2

FGA, ωL = 2.093

0.1

FGO, ωL = 1.975 FGX, ωL = 2.172

0 0

0.2

0.4 0.6 0.8 1 Forcing Amplitude ( f )

1.2

1.4

Fig. 7: Influence of CNT distribution on the force-response curves of FG CNTRC plate subjected to various edge supports (:⁄ℎ = 10, : ⁄; = 1, ∗ = 0.28, G = 0.01, Ω⁄Ãô = 1.4)

35

0.9

1

(a) CCCC

(b) SSSS

a/h = 6, ωL = 2.749

1

1.4 1.2

a/h = 10, ωL = 2.318

0.8

Amplitude ( wmax )

a/h = 12, ωL = 2.195 a/h = 14, ωL = 2.099

0.6

a/h = 16, ωL = 2.017 a/h = 20, ωL = 1.878

0.4

1

a/h = 8, ωL = 1.848 a/h = 10, ωL = 1.719 a/h = 12, ωL = 1.603 a/h = 14, ωL = 1.497

0.8 0.6

a/h = 16, ωL = 1.399 a/h = 20, ωL = 1.228

0.4 0.2

0.2 0 0.7

0.8

0.9

1 1.1 1.2 1.3 1.4 Frequency ratio ( Ω /ωL )

1.5

0 0.4 0.6 0.8

1.6

1 1.2 1.4 1.6 1.8 2 Frequency ratio ( Ω /ωL )

2.2 2.4 2.6

(c) CSCS 1.2 a/h = 6, ωL = 2.498

1

a/h = 8, ωL = 2.296 a/h = 10, ωL = 2.172

Amplitude ( wmax )

Amplitude ( wmax )

a/h = 8, ωL = 2.487

a/h = 6, ωL = 2.004

0.8

a/h = 12, ωL = 2.082 a/h = 14, ωL = 2.008

0.6

a/h = 16, ωL = 1.943 a/h = 20, ωL = 1.826

0.4 0.2 0 0.6

0.8

1 1.2 1.4 Frequency ratio ( Ω /ωL )

1.6

1.8

Fig. 8: Influence of aspect ratio :⁄ℎ on the frequency-response curves of FGX-CNTRC plate subjected to various edge supports (:⁄; = 1, ∗ = 0.28, G = 0.01, « = 0.06)

36

(a) CCCC

(b) SSSS 1

1.2

0.9 0.8 Amplitude ( wmax )

0.8 a/h = 6, ωL = 2.749

0.6

a/h = 8, ωL = 2.487 a/h = 10, ωL = 2.318

0.4

0

0

0.2

0.4

0.6 a/h = 6, ωL = 2.004

0.5

a/h = 8, ωL = 1.848 a/h = 10, ωL = 1.719

0.3

a/h = 12, ωL = 1.603

a/h = 16, ωL = 2.017

0.2

a/h = 14, ωL = 1.497

a/h = 20, ωL = 1.878

0.1

a/h = 14, ωL = 2.099

0.2

0.7

0.4

a/h = 12, ωL = 2.195

0.6 0.8 1 1.2 1.4 Forcing Amplitude ( f )

1.6

1.8

0

2

a/h = 16, ωL = 1.399 a/h = 20, ωL = 1.228

0

0.2

0.4 0.6 0.8 1 Forcing Amplitude ( f )

1.2

(c) CSCS 1.2 1 Amplitude ( wmax )

Amplitude ( wmax )

1

0.8 a/h = 6, ωL = 2.498

0.6

a/h = 8, ωL = 2.296 a/h = 10, ωL = 2.172

0.4

a/h = 12, ωL = 2.082 a/h = 14, ωL = 2.008

0.2

a/h = 16, ωL = 1.943 a/h = 20, ωL = 1.826

0

0

0.2

0.4

0.6 0.8 1 1.2 Forcing Amplitude ( f )

1.4

1.6

1.8

Fig. 9: Influence of aspect ratio : ⁄ℎ on the force-response curves of FGX-CNTRC plate subjected to various edge supports (:⁄; = 1, ∗ = 0.28, G = 0.01, , Ω⁄Ãô = 1.4)

37

1.4

(b) SSSS

(a) CCCC 0.8

0.7

0.7

0.6

Amplitude ( wmax )

0.5 0.4 Ω/ωL = 0.8

0.3

Ω/ωL = 0.9

0.2

Ω/ωL = 1.1

0.1

Ω/ωL = 1.2

0.5 0.4 0.3 0.2 0.1

Ω /ωL = 1.4

0

0

0.2

0.4

0.6 0.8 1 1.2 1.4 Forcing Amplitude ( f )

1.6

1.8

0 0

2

0.2

0.4 0.6 0.8 1 Forcing Amplitude ( f )

1.2

(c) CSCS 0.8 0.7 0.6 Amplitude ( wmax )

Amplitude ( wmax )

0.6

0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6 0.8 1 1.2 1.4 Forcing Amplitude ( f )

1.6

1.8

2

Fig. 10: Influence of the excitation frequency ratio Ω⁄Ãô  on the force-response curves of FGXCNTRC plates subjected to various edge supports (:⁄ℎ = 10, : ⁄; = 1, ∗ = 0.28, G = 0.01)

38

1.4