post-buckled multilayer FG-GPLRPC rectangular plates

Accepted Manuscript

Nonlinear stability and vibration of pre/post-buckled multilayer FG-GPLRPC rectangular plates Raheb Gholami , Reza Ansari PII: DOI: Reference:

S0307-904X(18)30440-2 https://doi.org/10.1016/j.apm.2018.08.038 APM 12452

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

25 July 2017 12 August 2018 29 August 2018

Please cite this article as: Raheb Gholami , Reza Ansari , Nonlinear stability and vibration of pre/postbuckled multilayer FG-GPLRPC rectangular plates, Applied Mathematical Modelling (2018), doi: https://doi.org/10.1016/j.apm.2018.08.038

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Research Highlights:

Proposing the nonlinear equations of parabolic shear deformable FG-GPLRPC plates



Prediction of nonlinear stability and vibration of pre- and post-buckled FG-GPLRPC plates



Investigating the impacts of GPL distribution scheme and weight fraction



Exploring the impacts of the GPLs’ and plate’s geometry and boundary conditions

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

Nonlinear stability and vibration of pre/post-buckled multilayer FG-GPLRPC rectangular plates Raheb Gholamia,*, Reza Ansarib

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

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

M

a

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Abstract

The present study examines the nonlinear stability and free vibration features of multilayer functionally graded graphene platelet-reinforced polymer composite (FG-GPLRPC) rectangular plates under

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compressive in-plane mechanical loads in pre/post buckling regimes. The GPL weight fractions layerwisely vary across the lateral direction. Furthermore, GPLs are uniformly dispersed in the polymer matrix

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of each layer. The effective Young’s modulus of GPL-reinforced nanocomposite is assessed via the modified Halpin-Tsai technique, while the effective mass density and Poisson’s ratio are attained by the rule of mixture. Taking the von Kármán-type nonlinearity into account for the large deflection of the FG-

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GPLRPC plate, as well as utilizing the variational differential quadrature (VDQ) method and Lagrange equation, the system of discretized coupled nonlinear equations of motions is directly achieved based upon a parabolic shear deformation plate theory; taking into account the impacts of geometric nonlinearity, in-plane loading, rotary inertia and transverse shear deformation. Afterwards, first, by neglecting the inertia terms, the pseudo-arc length approach is used in order to plot the equilibrium postbuckling path of FG-GPLRPC plates. Then, supposing a time-dependent disturbance about the postbuckling equilibrium status, the frequency responses of pre/post-buckled FG-GPLRC plate are *

Corresponding Author. Tel. /fax: +98 1342222906, E-mail address: [email protected] (R. Gholami). 1

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obtained in terms of the compressive in-plane load. The influences of various vital design parameters are discussed through various parametric studies. Keywords: Graphene nanoplatelets; Higher-order shear deformable FG-GPLRPC rectangular plates; Postbuckling path; Vibration in pre- and post-buckled regimes; Numerical approach.

1. Introduction

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In the past decades, the development of polymeric and laminated composite materials results in their widespread utilization in different engineering and industrial fields such as marine, automotive and oil industries as well as sport goods. The material properties of polymeric composite materials strongly depend on the material properties of fillers as well as intermolecular interaction and interface area existing between the matrix and fillers. Meanwhile, due to the superior mechanical, electrical and thermal

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properties of carbonaceous nanofillers [1-4], i.e., the graphene and carbon nanotubes (CNTs), compared to the conventional fillers including the carbon and glass fillers, the graphene, CNTs and their derivatives as a modern and promising alternative to conventional fillers have been used in the composite materials. Therefore, the nanocomposites reinforced by the CNTs and graphene as a subset of composite materials have received considerable attention from the research community and many researchers [5-9]. On the other hand, due to the static/dynamic forces, the plate-like structures as a fundamental element of many

M

mechanical systems are usually subjected to the deformation or oscillations. For instance, the external applied compressive in-plane loads may be result in the instability and subsequently destructing of the

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plate-like structures. Therefore, investigations of the mechanical behaviors such as the stability and vibrational characteristics of plate-like structures reinforced with the carbonaceous nanofillers are important steps in designing, fabricating and improving the performance of such structures.

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The literature concerning the mechanical behaviors of structures reinforced with the carbonaceous nanofillers on the basis of the linear or nonlinear models may be classified mainly into two general

CE

categories. The investigations performed on the CNT-reinforced structures were conducted in the first group; the second group is the extension of the analysis on the graphene-platelet reinforced beam-, plate-

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and shell-like structures. Some works existing in the literature are reviewed herein. The literature concerning the first group of investigations is quite numerous [10-18]. For example, the nonlinear vibration of first- and third-order shear deformable functionally graded carbon nanotubereinforced composite (FG-CNTRC) beams were examined by Lin and Xiang [19]. According to the Mori–Tanaka scheme and using an equivalent continuum model, the free vibration of CNTRC plates was examined by Formica et al.[20]. Shen and his associates investigated the postbuckling and nonlinear vibration of FG-CNTRC plates, panels and shells [21-24]. On the basis of their results, the critical buckling load, natural frequencies and postbuckling of the plate- and shell-like structures exposed to the 2

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external compressive mechanical loads will be increased by adding the CNTs to the polymer matrix. The vibration of FG-CNTRC plates under thermal environment was analyzed utilizing the element-free kpRitz technique [25]. Based upon the mixed variational theorem, a uniform formulation of finite prism techniques was developed by Wu and Li [26] to examine the three-dimensional vibration of FG-CNTRC plates with different edge conditions. Using the Ritz method, the vibrational response of FG-CNTRC conical panels was examined by Kiani et al.[27]. Tornabene et al.[28] carried out the free vibration of

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CNT/fiber-polymer multiscale laminated plates and shells employing the generalized differential quadrature (GDQ) technique. Several micromechanics approaches were implemented to assess the overall mechanical properties of composite layers. Then, a comparison was made between the numerical results corresponding to various micromechanical techniques. By employing the Airy stress function, Galerkin and fourth-order Runge-Kutta methods, Thanh et al.[29] inspected the nonlinear dynamics of embedded

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FG-CNTRC plates with considering the effect of temperature rise on the material properties. Setoodeh and Shojaee [30] inspected the large amplitude vibration of FG-CNTRC plates with quadrilateral shape. Recently, Gholami et al.[31] numerically examined the effects of initial imperfection on the nonlinear resonance of FG-CNTRC beams exposed to the periodic lateral loading. Large amplitude resonance of FG-CNTRC rectangular plates was studied by Ansari and Gholami [32]. The nonlinear dynamic stability of Kirchhoff FG-CNTRC plates was investigated by Fu et al.[33]. They utilized a two-step perturbation

used

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scheme and incremental harmonic balanced method to solve the problem. Recently, Shen and Wang [34] a two-step perturbation scheme to analyze the nonlinear vibration of simply-supported (SS)

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mechanically- and thermally-postbuckled FG-CNTRC rectangular plates. Unlike the studies performed on the mechanical features of FG-CNTRC structures, the literature regarding the second group, i.e., investigations on the mechanical responses of the functionally graded-

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graphene platelet-reinforced polymer composite (FG-GPLRPC) structures, is very limited. Applying the finite element (FE) package ABAQUS, the torsional buckling of FG-GPLRPC circular cylindrical shell

CE

with cutout was examined by Wang et al.[35]. Wu et al.[36] formulated the governing equations of FGGPLRPC beams in thermal environment and subsequently discretized the achieved equations via the

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GDQ method in order to study the dynamic stability of FG-GPLRPC beams. Feng et al.[37] carried out the nonlinear vibrations of Timoshenko beams made of FG-GPLRPCs. Feng et al. [38] analyzed the geometrically nonlinear bending of FG-GPLRPC beams. Furthermore, using the first-order shear deformation theory (FSDT), Song et al.[39] studied the vibrations of fully simply-supported FG-GPLRPC rectangular plates. Also, the FE method was utilized by Zhao et al.[40] to investigate the bending and vibration of FG-GPLRPC trapezoidal plates. Yang et al.[41] explored the thermo-elastic bending of plates reinforced with GPLs with uniform, linear and nonlinear distribution patterns. Recently, Shen et al.[42, 43] utilized a two-step perturbation scheme to analytically examine the postbuckling and vibration 3

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features of composite laminated cylindrical panels reinforced with graphenes in the thermal environment. Moreover, on the basis of the three-dimensional elasticity theory, Yang et al.[44] examined the thermomechanical bending of FG-GPLRPC circular and annular plates. Employing the three-dimensional theory of elasticity and generalized Mian and Spencer method, the thermo-mechanical bending of clamped FGGPLRPC elliptical plates was analytically examined by Yang et al. [45]. Song et al.[46] used an analytical solution method to explore the buckling and postbuckling characteristics of perfect and

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imperfect FG-GPLRPC rectangular plates with SS boundaries. Also, using the differential quadraturebased iteration method, thermal postbuckling behavior of shear deformable FG-GPLRPC plates with several boundary conditions were reported by Wu et al. [47]. Recently, Gholami and Ansari [48] applied the higher-order shear deformation plate theory (HSDPT) to describe the nonlinear resonance of thick FG-GPLRC rectangular plates subjected to the uniform periodic force. In that work, by providing a weak

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form of mathematical modeling and employing an efficient numerical solution process, the frequencyamplitude and force-amplitude response curves were achieved.

Considering the literature, the geometrically nonlinear stability and vibration features of the pre/postbuckled multilayer FG-GPLRPC rectangular plates with various edge conditions using numerical approaches have not yet been investigated. In this work, a numerical approach is established to inspect the geometrically nonlinear static stability and vibration of pre/post-buckled multilayer FG-GPLRPC

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rectangular plates subjected to compressive in-plane loadings. This study can be considered as the continuation of the works of Shen and Wang [34], Song et al.[46] and Gholami and Ansari [48]. Material

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properties of the multilayer FG-GPLRPC plate made of an isotropic polymer matrix reinforced with the GPLs are supposed to be distributed through the transverse axis according to several schemes of GPL nanofillers and volume fractions of GPLs in each layer. Since the coupled nonlinear equations can be

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hardly solved via analytical methods, a numerical solution scheme is implemented that does not need any simplifying assumption. Herein, the variational differential quadrature (VDQ) approach [49] is applied to

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directly attain the discretized equations of motions according to the parabolic shear deformation plate theory (PSDPT) with von Karman-type geometric nonlinearity. This theoretical attempt helps in

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bypassing the need for acquiring the strong form of the formulation and any shear correction factor. The energy functional of multilayer FG-GPLRPC rectangular plate is expressed in the matrix form and in terms of the displacement vector. Then, utilizing the VDQ method, the discretized displacement vector and energy functional are attained. Finally, the Lagrange equation is used to arrive the discretized nonlinear equations of motion. In the following, first, by dropping the inertia terms, the problem is reduced to the nonlinear static equilibrium of compressed FG-GPLRPC plates. By applying the pseudo arc-length method, the nonlinear equilibrium postbuckling path of system is obtained. Then, by imagining a time-dependent small perturbation and dropping the time-dependent nonlinear terms, an eigenvalue 4

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problem corresponding to the frequency responses of pre/post-buckled multilayer FG-GPLRPC rectangular plate is attained. A numerical attempts are conducted to explore the impacts of GPL distribution scheme and weight fraction as well as the GPLs’ and plate’s geometry on the nonlinear stability and vibration of pre/post-buckled multilayer FG-GPLRPC rectangular plates.

2. Theoretical formulation ⁄

under the in-plane loadings

and

individual layers with the same thickness

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A multilayer FG-GPLRPC rectangular plate composed of

is schematically illustrated in Fig. 1. The plate is

supposed to be made of polymer matrix reinforced with the GPLs. Moreover, the GPLs are considered to be uniformly distributed within each layer with the variable weight fraction form layer to layer through the transverse direction on the basis of various distribution patterns of GPL nanofillers. By choosing a ⁄

considered FG-GPLRPC plate are labeled.

⁄ ), the material points of

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Cartesian coordinate system (

This section is organized as follows. In subsection 2.1, the modified Halpin-Tsai model as well as the rule of mixture are employed to calculate the effective material properties of FG-GPLRPCs. In subsection 2.2 the matrix description of the strain-displacement and strain-stress relations are presented according to the higher-order PSDPT. A matrix representation is provided for the energy functional in subsection 2.3.

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Subsection 2.4 deals with using the VDQ technique to provide the discretized energy functional of FGGPLRPC rectangular plate. Finally, the discretized equations of FG-GPLRPC plates are directly obtained

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in subsection 2.5 using the Lagrange equations.

2.1. Micromechanics of the FG-GPLRPCs

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Before deriving the governing equations of motion, it is necessary to obtain the effective material properties of each layer. In this regard, in the current work, three kinds of distribution configuration are

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supposed for the GPL nanofillers through the thickness direction. These distribution patterns include the U-, X- and O-GPLRCs, as schematically shown in Fig. 2. It is remarked that for the considered GPL

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distribution patterns, the physical neutral plane is equal to the geometric mid-plane. Therefore, the geometric middle surface can be considered as the reference plane. It is noted that in this study, the geometries and mechanical properties associated with the matrix and GPLs are respectively indicated by subscripts “m” and “GPL”. Also, the quantities associated with the FG-GPLRPCs are illustrated by the subscript “eff”. For the considered GPL distribution schemes, the volume fraction of GPLs corresponding to the kth layer can be computed as follows

5

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( ) ( )

|

( )

(

. Also, (

|⁄

(1b)

|

|⁄

)

indicating the total volume fraction of GPLs can be arrived as

)(

⁄

(2)

)

According to Eq. (2), in addition to the GPL weight fraction both matrix

(1c)

and GPLs

,

.

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where

(1a)

depends on the mass density of

According to the Halpin–Tsai micromechanics approach, one can expressed the Young’s modulus of FG-

where

and

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GPLRPCs as following expression [50, 51]

can be predicted as follows [50]

where two parameters

and

M

⁄ ⁄ and

(4)

appeared in Eq. (4) are computed as

⁄ ⁄ In Eq. (5),

(3)

(5)

are Young’s modulus. Furthermore, the influences of the size and geometry of and

appeared in Eq. (5).

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the GPL reinforcements are included into computations via two parameters These parameters are defined as )

(

)

PT

(

(6)

where the average length, width and thickness of GPL nanofillers are denoted by

CE

respectively.

Moreover, to predict the effective mass density

and effective Poisson’s ratio

,

and

,

, the well-known rule

of mixture is employed. These quantities are estimated by the following relations

AC

(7a)

where

(7b)

is the Poisson's ratio.

2.2. Matrix description of the strain-displacement and constitutive relations Among various plate theories, the classical Kirchhoff plate theory cannot capture the impacts of transverse shear deformation and rotary inertias. Furthermore, FSDT is not enable to accurately model the

6

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distribution of the shear stresses. This problem can be resolved by introducing an adequate shear correction factor to be determined. The higher-order shear deformation theories considered herein do not need the shear correction factor and satisfy zero transverse shear stress status on the up and down surfaces. Therefore, they are used to mathematically formulate the motion of thick plate-like structures. In the current study, a higher-order PSDPT in conjugation with the von Kármán-type geometric nonlinearity is implemented.

)

(

)

(

)

(

)

(

)

( )(

(

)

( )(

(

)

in which ( )

(

where the

)

is the time and

-axis;

,

,

,

( (

) )

) )

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(

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According to the PSDPT, the displacement filed is given as follows [52-54]

(8)

(9)

are the displacement of a given point placed on the FG-GPLRPC plate along and

stand for the five unknown displacement functions of the mid-plane of

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FG-GPLRPC rectangular plate while ( ) illustrates the shape function indicating the distribution of the transverse shear stresses and strains across the FG-GPLRPC plate thickness.

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By collecting the above-mentioned five unknown displacement functions into a vector

, the

displacement field given by Eq. (8) can be expressed as follows ̃

]

[

CE

( )

( )

[

( )]

PT

[

( )

(10) ] [

]

AC

where the displacement vector is denoted by ̃ . Also,

[

]

stands for the augmented displacement vector.

On the basis of Eq. (8), the von Kármán type strain-displacement elements are of the form

7

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(

)

(

)

̃ ] ̃

[

[

̃

( ) [

]

]

[

( )

]

̃

( )

[

]

where ( )

( )

( )

( )

( )

( )

(11)

]

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[ ̃

(12)

(

) (

)

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The matrix description of the nonlinear kinematic relations defined in Eq. (11) is as follows

where denotes the Hadamard product [55]; and [

]

⁄ ⁄

⁄

,

(

) and

[

]

ED

⁄

⁄ ⁄

⁄

PT CE

AC (

) ( 〈

⁄

)

〉

where 〈 〉

〈 〈

(14a)

〉

] (14b)

]

⁄

⁄

⁄ ⁄

[

⁄ ⁄ ⁄

[

[

are expressed as follows

M

⁄

,

(13)

0

⁄ ⁄ ⁄

〉

(〈

1

] 〈

〉

〈

〉

)

(14c)

〉 ( ).

Also, the stress-strain equation for kth layer of FG-GPLRPC plate is stated as ( )

( ) ( )

(15) 8

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in which

indicating the augmented stress vector and

signifying the elastic stiffness matrix are

expressed as ̃

[

̃ ]

̃ -

(16a)

[ ( )

Furthermore, ( )

̃

]

[

]

for the kth layer are as

( )

( )

( )

(16b)

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,̃

( )

(

2.3. Matrix description of the energy functional

(17)

)

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The strain and stress relations defined in Eqs. (13) and (15) in terms of augmented displacement vector can be employed to calculate the strain energy. Therefore, one can express the following relation ∫∫

( )

∑∫ ∫

(

( )

( ) ( )

( )

( ) ( )

(18)

)

M

∑∫ ∫

( )

in which the surface area is denoted by .

∑∫ ∫

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Inserting Eq. (13) into (8) provides the potential strain energy of FG-GPLRPC plate as follows ( )

)

PT

,(

( ) ( )

(

)

( )

( )

(

) (19)

)( ) -

(

CE

Eq. (18) can be rewritten as follows ∫

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where

(

̅

̅

∑∫

∑∫

̅

( )

( )

( )

̅

̅

[

̅

] ̅

∑∫

̅

)

( )

( ) ( )

(20)

[

]

(21a) ( )

( )

[ ] ̅

( )

∑∫

9

( )

̅

̅

∑∫

( )

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[

]

[

]

[

] (21b)

]

Moreover,

,

,

[

,

,

{

]

and

]

appeared in Eq. (21b) are calculated as follows ()

∑∫

}

[

*

+

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[

(

)

(22)

()

∑∫

(

)

∫∫

̃̇ ̃̇

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Furthermore, using displacement field (10), the kinetic energy of FG-GPLRPC can be computed as ̃̇

∑∫ ∫

∫ ̇ (

( ) ̃̇

(23)

) ̇

∑∫

M

In Eq. (23), the symbol dot symbolizes the differentiation with respect to the time. Also, we have ( )

(

)

)(

(

)

PT

∑∫

ED

[

[

+

CE

*

∑∫

]

AC where

[

(

)

(

)(

(

∑∫

)

[

]

] ( )*

+

Additionally, the work due to external in-plane loadings ∫(

(24a)

) )

∫

, (

(24b) is expressed as 〈 〉

〈 〉

)

(25)

] is the load vector; including the components of applied in-plane loads.

2.4. Discretizing the energy functional of FG-GPLRPC plate Now, to define the mesh grid points through the

- and

distribution with desirable accuracy and stability is utilized 10

- axes, the Chebyshev-Gauss-Lobatto grid

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(

) (26)

(

)

where the number of grid points through the

- and

-directions are denoted by

and

, respectively.

By employing the VDQ procedure [49] and using the differential and integral processors as discussed in

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the Appendices A and B, respectively, the discretized energy functional of FG-GPLRPC plates is attained. The matrix operators and aforementioned approach are given in Appendices A and B, respectively. It is noticed that the utilized differential and integral operators are not discussed in this work. More explanations about the VDQ method and mathematic process are given in [49, 56].

By applying the VDQ method on Eq. (20), the discretized form of strain energy of system is stated as )

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( where ̅ 〈

̅ 〉

〈

̅

̅

〉

(27)

̅

(28)

Moreover, the kinetic energy of FG-GPLRPC plates can be written as ) ̇

in which )

(

)

(30)

ED

(

(29)

M

̇ (

Finally, the discretized form of external work done on the system can be written as 〈 〉

where

.

〈 〉

)

(31)

PT

(

operator;

CE

In the relations corresponding to the discretized form of energy functional, the denoting the displacement vector and

and

signifying the integral

representing the discretized form of

and

are computed as

AC

(32a)

11

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( )

[ [

]

( )

] ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

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( )

]

( )

[

]

( ) ( )

( )

( )

ED

]

M

( )

( )

[

(32b)

,

( )

( )

[

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( )

[

where the differential processors with respect to respectively, the integral matrix operators in

and

- and

( )

PT

] ( )

are signified by -directions and

the column vectors including the displacement variables

(32c)

,

,

, ,

and

, and

,

( )

;

and

and

are,

represent

in the used two-

CE

dimensional grid points.

2.5. Weak form of nonlinear equations Now, the Lagrange equations can be implemented in order to directly arrive discretized nonlinear

AC

governing equation of motion of FG-GPLRPC plates. The Lagrange equations can be expressed as [57] (

̇

(33)

)

Inserting relations (27), (29) and (31) into Eq. (33) gives the following relation ̈

(34)

where (35a) 12

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(35b) 〈

〉

〈

〉

(35c) (35d) (35e)

〈 〉

(35g)

3. Solution method

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〈 〉

The utilized solution procedure in this section contains two main steps: (1): The nonlinear stability characteristics of system as the postbuckling load-deflection curve will be achieved for various boundary conditions via solving a system of nonlinear algebraic equations; (2) Using the postbuckling path achieved for the buckled FG-GPLRPC rectangular plate, the free vibration behavior in the pre- and post-

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buckled states will be obtained via solving a eigenvalue problem.

Before proceeding the solution methodology, it is declared that in VDQ method, attending the essential boundary conditions is sufficient to achieve the reliable and precise numerical results [49]. In the current work, the following essential boundary conditions will be used for various edge supports: a. Fully simply-supported edge supports (SSSS)

M

(36)

PT

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b. Fully clamped edge supports (CCCC)

-axis and simply-supported edges through the

CE

c. Clamped edges along with the

(37)

-axis (CSCS) (38)

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In order to attain the static equilibrium postbuckling path of FG-GPLRPC plate, the following steps will be followed: (1) the inertia terms of Eq. (34) are removed; (2) the external applied in-plane compressive forces are assumed as

̃ and

̃ (it is noticed that in the present study, the biaxial

postbuckling of FG-GPLRPC rectangular plate is investigated i.e.,

) and the displacement

variables vector associated with the static postbuckling problem is assumed to be following relation is obtained

13

. Therefore,

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(

̃) (

where

̃

(

(

)

)

(

)

) (

(39) ). ̃

(3) by considering the linear part of Eq. (39) (i.e., (

)

) and inserting the discretized

boundaries into the stiffness and geometric stiffness matrices, one can achieve the solution of buckling problem including the critical buckling loads and corresponding buckling mode shapes.; (4) by inserting

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the residual of discretized boundaries into the residual vector of domain as well as using the first buckling load and its mode shape as an initial guess, the pseudo arc-length algorithm together with the modified Newton–Raphson scheme [58] is utilized to the numerically solve the set of parameterized nonlinear static equilibrium equations; achieving the nonlinear equilibrium postbuckling path as the postbuckling load-deflection curve.

Now, to examine the vibrational behavior of FG-GPLRC rectangular plate around the buckled

total displacement vector

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configuration, a time-dependent small perturbation about the buckled state is assumed. To this end, the is considered as follows [59-61]

in which includes the static postbuckling equilibrium configuration

(40)

and the dynamic disturbance

.

Substituting the obtained static displacement vector via the postbuckling analysis into Eq. (34) and

M

omitting the nonlinear elements of the dynamic elements, the following set of linear equations for the small amplitude vibration of FG-GPLRPC plates around the buckled state is arrived ̈

ED

̅

(41)

where ̅ signifying the stiffness matrix associated with the pre- and post-buckled FG-GPLRPC plate

̅

̃

̃

̃

̃

(

)

(

)

(42)

and subsequently inserting it into Eq. (41), the following eigenvalue

CE

Then, assuming

PT

which can be obtained as follows

problem is achieved (̅

̃

)̃

(43)

AC

in which ̃ signifies the natural fundamental frequency. By employing the equilibrium situation of each load rise in the pre- and post-buckling configurations as well as the use of Eq. (43), the natural fundamental frequencies of FG-GPLRPC plates corresponding to the pre- and postbuckling states can be obtained as the fundamental frequency versus the pre- and postbuckling loads.

4. Results and discussion 14

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The results are generated for the geometrically nonlinear stability and free vibrational behaviors of multilayer FG-GPLRPC rectangular plates with different edge conditions namely SSSS, CCCC and CSCS in the pre- and post-buckled configurations employing the proposed formulation and utilized solution procedure. Herein, the mechanical properties of isotropic polymer matrix are: ⁄

and

[62]. Also, unless otherwise specified, the GPLs with following

properties and geometries are selected:

,

,

,

, as indicated in [63, 64]. Moreover, it is supposed that the multilayer FG-GPLRPC

rectangular plate includes

layers with total thickness

the following nondimensional parameters are used ̃ √

(

)⁄

)⁄

̃(

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and

,

. Through the computations,

(44) , GPLs’

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It is intended to explore the impacts of vital factors including the GPL distribution scheme,

and plate geometries and type of edge supports on the nonlinear stability and natural frequency of multilayer FG-GPLRPC rectangular plates in the pre- and post-buckled configurations. In this regard, the postbuckling load-deflection curve as the nondimensional maximum deflection (i.e., Non. Dim. Max. Deflection) (

) is drawn versus the nondimensional applied load (i.e., Non. Dim. Applied Load) ( ).

The values of nondimensional critical buckling loads (

) are also illustrated in the figures. Moreover, to

M

show the free vibration features, in the pre- and post-buckled configurations, the nondimensional frequency (i.e., Non. Dim. Frequency) ( ) versus the nondimensional applied load ( ) are provided. It is

ED

noticed that unless otherwise stated, the X-GPLRC distribution pattern is used in the computations. Before proceeding the numerical investigations, the accuracy of the proposed mathematical formulations and solution technique is validated. In this regard, the nondimensional critical buckling loads and

and

PT

fundamental frequencies of FG-GPLRPC plates with various edge conditions, GPL distribution schemes are provided in Tables 1 and 2, respectively. These results are compared with those provided by

CE

Wu et al.[65]; indicating a good agreement. Depicted in Figs. 3 and 4 is the effect of GPL weight fraction on the nonlinear stability/postbuckling

AC

equilibrium path and free vibrational characteristics of multilayer FG-GPLRPC plates in the pre- and post-buckled states. As demonstrated in Fig. 3, increasing

results in enhancing the critical buckling

load, shifting the static equilibrium path to the right side and consequently increasing the strength and load-carrying ability of multilayer FG-GPLRPC plates. Also, as illustrated in Fig. 4, in the pre-buckled regime, increasing the applied compressive in-plane force results in decreasing the natural frequency of FG-GPLRPC plate. More increasing the compressive loading leads to tending to the natural frequency to zero at the buckling load of system. This behavior is due to reduction of the total stiffness of considered FG-GPLRPC plate and consequently losing the its stability. In the pre-buckled state, an increase in 15

ACCEPTED MANUSCRIPT

leads to increasing the natural frequency of system. In contrast to the pre-buckled state, at a specified value of compressive in-plane load, increasing the value of

results in lower natural fundamental

frequencies. Figs. 5 and 6 highlight the impact of GPL distribution scheme on the nonlinear stability as well as the free vibration characteristics of FG-GPLRPC plates in the pre- and post-buckled configurations. The curve corresponding the plate made of the net epoxy is also plotted for a direct comparison. Regardless to the

CR IP T

type of edge conditions, the plates with X- and O-GPLRC schemes display the highest and lowest stabilities and load-carrying capacities, respectively. Moreover, the highest critical buckling load and also natural fundamental frequency in the pre-buckled state belongs to the X pattern which follow by U- and O-GPLRC distribution schemes, respectively. The reason leading to this behavior is due to more strengthening the FG-GPLRPC plates when more GPLs are placed near to upper and lower surfaces. ⁄

ratio on the nonlinear stability as well as the vibration

AN US

Figs. 7 and 8 explore the influence of

of FG-GPLRPC plates in the pre- and post-buckling situations. Regardless to the kind of the boundary conditions, it is revealed that an increase in

⁄

ratio results in increasing the critical buckling

load, natural frequency in the pre-buckled state and static load-carrying ability of the FG-GPLRPC rectangular plates. Moreover, for a given value of applied compressive load, a reduction in the natural ⁄

frequency in the postbuckled state occurs by increasing the ⁄

leads to reducing its effect. On the basis of the results, it can be

M

that more increasing the

. However, it should be remarked

deuced that the above-mentioned consequences is due to intensifying the bending resistance and stiffness

ED

of structures reinforced by GPLs by increasing the

⁄

.

Presented in Figs. 9 and 10 is the nonlinear equilibrium postbuckling path as well as the vibration ⁄

PT

response of FG-GPLRPC plates in the pre- and post-buckled configurations associated with various ratio. The width of GPL nanofillers is supposed to be constant. For a given amount of GPLs,

expanding the

⁄

causes the increase of the surface contact area and consequently providing a

CE

better load transfer between the polymer matrix and GPL nanofillers. In addition to increasing the static load-carrying capability and critical buckling load, it results in intensifying the bending rigidity and

AC

increasing the fundamental frequency in pre-buckled region. To better comprehend the impact of GPL geometry on the vibration and buckling responses of FG-GPLRPC rectangular plates, the frequency and buckling load of FG-GPLRPC plates are, respectively, plotted versus various

⁄

⁄

corresponding to

as displayed in Figs. 11 and 12. Although increasing the size of GPL nanofillers

results in increasing the frequencies and buckling loads, but this effects are negligible for the higher values of length-to-width and width-to-thickness ratios. Finally, Figs. 13 and 14 illustrate the nonlinear equilibrium postbuckling path as well as he free vibration of FG-GPLRPC plates with various edge supports and length-to-thickness ratios ( ⁄ ). For all 16

ACCEPTED MANUSCRIPT

considered edge supports, decreasing the ⁄ results in increasing the critical buckling load and carrying load-carrying capacity of degree of typical of FG-GPLRPC plates. Furthermore, for a specified value of applied compressive in-plane loading, increasing ⁄ leads to the lower and higher frequencies in the pre- and post-buckled configurations, respectively. Moreover, a comparison study is performed to show difference between results predicted by models on the basis of the FSDT and HSDPT. The results corresponding to the FSDT are illustrated with black dash lines. For the FG-GPLRPC plates with small

CR IP T

⁄ , it can be seen that the differences between the results estimated by the FSDT and HSDPT are considerable. However, for higher values of length-to-thickness ratios, these differences are insignificant.

5. Conclusion

The nonlinear stability and vibration of multilayer FG-GPLRPC rectangular plates under the compressive

AN US

in-plane loads in pre/post buckling regimes were investigated numerically. In the context of a variational approach, the PSDPT, von Kármán type geometric nonlinearity and Hamilton’s principle in conjugation with the VDQ technique were used to achieve the weak form of nonlinear equations of motions. The material properties of FG-GPLRPCs were described by means of the modified Halpin-Tsai model in conjugation with the rule of mixture. In the following, by ignoring the inertia portion, the nonlinear equilibrium path of multilayer FG-GPLRPC rectangular plate was obtained via solving a time-

M

independent set of discretized coupled nonlinear equation using the pseudo arc-length scheme. Then, by applying a time-varying disturbance about the buckled state, the frequency responses of pre- and post-

ED

buckled FG-GPLRPC plate were achieved via solving an eigenvalue problem. The impacts of weight fraction, distribution configuration and geometry of GPLs were discussed by plotting the postbuckling load-deflection curves as well as the nondimensional natural frequency versus the pre- and post-buckling

PT

loads associated with various boundary conditions. It was concluded that increasing the GPL weight fraction and nanofillers’ length-to-width and length-to-

CE

thickness ratios result in increasing the total stiffness of FG-GPLRPC rectangular plate and consequently lead to the higher natural frequencies in the pre-buckled state, higher critical buckling loads and higher

AC

stability and load-carrying capacity, while in the postbuckled state, the total stiffness and subsequently the stability and natural frequencies reduce. Moreover, among the various GPL distribution patterns, the FGGPLRPC rectangular plate with X-GPLRC distribution configuration has the highest critical buckling load, highest natural frequencies in prebuckling regime as well as the highest load-carrying capacity and lowest natural frequencies in the postbuckling zone.

Appendix A: Definition of the differential and integral processors a) Differential processor 17

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By considering a vector as following form , ( )

( )

(

)-

(A1)

where ( ) stands for the value of ( ) at each grid point,

, and

signifies the number of mesh grid

points, -th derivative of ( ) at each grid point can be calculated as follows ( )

( )

|

*

|

(A2)

| +

CR IP T

Now, based upon the differential quadrature (DQ) method [66], one can express Eq. (A2) as following form

where

( ) ( )

(

) (

( )

.

( )

(

)

)

/ ( )

∑

M

{ In Eq. (A4),

, and denotes )

identity matrix and

According to the Taylor series, the function ( ) in a neighborhood of )

CE

(

is obtained as (A5)

PT

b) Integral processor

∑

can be assessed as (A6)

|

Using the analytical approach, the integral of Eq. (A6) over the subinterval 0

AC

(A4)

ED

∏ (

( )

(A3)

stands for the differential matrix operator which can be computed using the following formula

AN US

( )

1 gives the

following relation ∫

( )

∑

(

)

( (

Moreover, the whole integral over , ,

) ) - is computed as

18

(A7) |

ACCEPTED MANUSCRIPT

( ∑(

( )

∫

) ( ∑

|

)

(

)

(

)

(

(

|

)

) (

| )

)

(A8)

By introducing a tow vector ̃ ( ) as follows ( *

)

)

(

(

)

(

)

where

(

)

(

)

(A9)

+

CR IP T

̃(

)

(

)

, one can represent the integration specified in Eq. (A8) as ( )

∫

∑ ̃(

(A10)

) ( )

( )

∫

(∑ ̃ (

)

( )

̃

)

in which ̃ is defined as the integral operator.

c) Generalization to -Dimensional space column vector with the size of (

where

)

-

/

and

(A12)

{

(A13)

PT

.

(A11)

-dimensional space by the following

ED

,

) in the

M

By introducing the nodal values of function (

AN US

Now, considering Eq. (A3), Eq. (A10) is displayed as follows

are, respectively, the number of mesh grid points in each dimension and number of

CE

dimensions, one can approximately calculate the partial derivative of function in

-dimensional space

using the matrix differential operator of Eq. (A4) as follows

AC

(

)

.

(

)

(

where the Kronecker tensor product is denoted by

)

( )

( )

(

/

)

(A14)

. Furthermore, one can express the integral operator

of -dimensional space as following form ∫

∫

(

∫

(̃

)

̃

̃

̃ ) (A15)

̃

19

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Appendix B: Discretization process of a functional and ( ) (

Assuming that

,

-) are, respectively, a constant square matrix and a column

vector, the scalar is defined as ∫

( )

(B1)

( )

( )

[

( ) ( )

CR IP T

in which ]

( ) Eq. (B1) can be represented as ( )

( ) and

Discretizing

∑ ∑ ∫

( )

AN US

∑∑∫

( ) over the domain (

It should be remarked that

and

components. Thus

M

(

(B4)

∫

signify the column vectors with

( ) [

] )

(B3)

gives the following relation

∑ ∑

)

(B2)

(B5)

ED

Using the integral operator defined in Appendix A, the integral of Eq. (B4) can be stated as ̃(

(B6)

)

PT

∑ ∑

Moreover, considering the relation

(

( )

)

in which

is a row vector,

and

CE

are column vectors, one can express Eq. (B6) as follows ∑ ∑

AC

where

(

(B7)

)

(̃). Furthermore, the Kronecker product of two matrices

and is shown by [

and

is defined as

as the following form (B8)

]

One can re-write Eq. (B7) as

20

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(

∑∑

(B9)

)

Eq. (B9) is equivalent to (B10)

)

where the discretized column vector [

and the transpose of it,

,

]

, are given by

CR IP T

(

-

It is noted that the derivative and variation of with recpect to column vector )

(

)

AN US

(

References

(B11)

are obtained as

(B12)

AC

CE

PT

ED

M

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24

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List of Captions: Table 1: Nondimensional critical buckling loads of FG-GPLRPC plates under a compressive uniaxial inplane loading in the

-axis ( ⁄

⁄

)

Table 2: Fundamental frequencies of FG-GPLRPC plates for various GPL volume fractions and ⁄

)

CR IP T

distribution configurations ( ⁄

Fig. 1: A schematic of FG-GPLRPC rectangular plate under transverse and in-plane loadings

Fig. 2: Schematic figure of three GPL’s distribution configurations through the lateral direction

Fig. 3: Nonlinear static equilibrium postbuckling path of FG-GPLRPC plates for various values of GPL weight fraction ( ⁄

⁄

)

Fig. 4: Free vibration response of pre- and post-buckled FG-GPLRPC plates for various values of GPL ⁄

)

AN US

weight fraction ( ⁄

Fig. 5: Nonlinear static equilibrium postbuckling path of FG-GPLRPC plates for various GPL distribution configurations ( ⁄

⁄

)

Fig. 6: Free vibration response of pre- and post-buckled FG-GPLRPC plates for various GPL distribution configurations ( ⁄

⁄

)

M

Fig. 7: Nonlinear static equilibrium postbuckling path of FG-GPLRPC plates associated with various GPL nanofillers’ length-to-thickness ratios ( ⁄

⁄

)

Fig. 8: Variation of first natural frequency of pre- and post-buckled FG-GPLRPC plates associated with

ED

various GPL nanofillers’ length-to-thickness ratios ( ⁄

⁄

)

Fig. 9: Nonlinear static equilibrium postbuckling path of FG-GPLRPC plates for different values of GPL ⁄

PT

nanofillers’ length-to-width ratio ( ⁄

⁄

)

Fig. 10: Variation of the first natural frequency of pre- and postbuckled FG-GPLRPC plates for different

CE

values of GPL nanofillers’ length-to-width ratio ( ⁄

⁄

⁄

)

Fig. 11: Influence of geometry of GPL nanofillers on the fundamental natural frequency of FG-GPLRPC plates ( ⁄

⁄

)

AC

Fig. 12: Influence of geometry of GPL nanofillers on the critical buckling load of FG-GPLRPC plates ( ⁄

⁄

)

Fig. 13: Nonlinear static equilibrium path of FG-GPLRPC square plates for various length-to-thickness ratio of plate (

)

Fig. 14: Free vibration response curve of pre- and post-buckled FG-GPLRPC square plates for different length-to-thickness ratio of plate (

)

25

ACCEPTED MANUSCRIPT

Tables:

Table 1: Nondimensional critical buckling loads of FG-GPLRPC plates under a compressive uniaxial inplane loading in the

-axis ( ⁄

⁄

)

Type of boundary conditions

GPL

X-GPLRC

O-GPLRC

Present

Wu et al.[65]

Present

0.1

0.0414

0.0413

0.0906

0.3

0.0620

0.0619

0.1351

0.5

0.0826

0.0825

0.1801

0.1

0.0462

0.0460

0.0987

0.3

0.0759

0.0758

0.1599

0.5

0.1057

0.1055

0.1

0.0368

0.3 0.5

Present

Wu et al.[65]

0.0899

0.0697

0.0692

0.1346

0.1041

0.1037

0.1794

0.1386

0.1382

0.0984

0.0761

0.0760

0.1597

0.1238

0.1235

0.2211

0.2207

0.1721

0.1709

0.0366

0.0811

0.0809

0.0626

0.0622

0.0480

0.0478

0.1084

0.1072

0.0828

0.0823

0.0592

0.0588

0.1348

0.1331

0.1026

0.1021

0.0311

0.0680

0.0675

0.0523

0.0520

0.0310

AC

CE

PT

ED

Pure epoxy

SCSC

Wu et al.[65]

AN US

U-GPLRC

CCCC

M

pattern

SSSS

CR IP T

( )

distribution

26

ACCEPTED MANUSCRIPT

Table 2: Fundamental frequencies of FG-GPLRPC plates for various GPL volume fractions and distribution configurations ( ⁄

Present

Wu et al.[65]

Present

0.1

0.6348

0.6342

1.0844

0.3

0.7772

0.7764

1.3276

0.5

0.8973

0.8963

1.5328

0.1

0.6691

0.6689

1.1296

0.3

0.8594

0.8587

1.4454

0.5

1.0138

1.0135

1.6957

0.1

0.5985

0.5970

0.3

0.6839

0.5

0.7591

Wu et al.[65]

SCSC Present

Wu et al.[65]

1.0783

0.8889

0.8849

1.3200

1.0883

1.0833

1.5240

1.2545

1.2507

1.1295

0.9287

0.9286

1.4407

1.1874

1.1864

1.6949

1.3995

1.3969

1.0211

1.0222

0.8414

0.8374

0.6816

1.1766

1.1755

0.9627

0.9611

0.7564

1.3112

1.3095

1.0717

1.0696

AC

CE

PT

ED

M

O-GPLRC

CCCC

AN US

X-GPLRC

SSSS

CR IP T

( )

pattern

U-GPLRC

)

Type of boundary conditions

GPL distribution

⁄

27

ACCEPTED MANUSCRIPT

Figures:

𝑎

CR IP T

𝑥

𝑥 𝑤

𝑏

𝑁𝑥

M

𝑥

AN US

𝑁𝑥

AC

CE

PT

ED

Fig. 1: A schematic of FG-GPLRPC rectangular plate under transverse and in-plane loadings

28

(b) O-GPLRC

AN US

(a) U-GPLRC

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

Fig. 2: Schematic figure of three GPL’s distribution configurations through the lateral direction

29

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

30

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 3: Nonlinear static equilibrium postbuckling path of FG-GPLRPC plates for various values of GPL

AC

CE

PT

ED

weight fraction ( ⁄

31

⁄

)

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

32

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 4: Free vibration response of pre- and post-buckled FG-GPLRPC plates for various values of GPL

AC

CE

PT

ED

M

weight fraction ( ⁄

33

⁄

)

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

34

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 5: Nonlinear static equilibrium postbuckling path of FG-GPLRPC plates for various GPL

AC

CE

PT

ED

M

distribution configurations ( ⁄

35

⁄

)

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

36

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 6: Free vibration response of pre- and post-buckled FG-GPLRPC plates for various GPL distribution ⁄

AC

CE

PT

ED

M

configurations ( ⁄

37

)

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

38

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 7: Nonlinear static equilibrium postbuckling path of FG-GPLRPC plates associated with various GPL

AC

CE

PT

ED

M

nanofillers’ length-to-thickness ratios ( ⁄

39

⁄

)

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

40

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 8: Variation of first natural frequency of pre- and post-buckled FG-GPLRPC plates associated with

AC

CE

PT

ED

M

various GPL nanofillers’ length-to-thickness ratios ( ⁄

41

⁄

)

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

42

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 9: Nonlinear static equilibrium postbuckling path of FG-GPLRPC plates for different values of GPL ⁄

AC

CE

PT

ED

M

nanofillers’ length-to-width ratio ( ⁄

43

⁄

)

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

44

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 10: Variation of the first natural frequency of pre- and postbuckled FG-GPLRPC plates for different

AC

CE

PT

ED

M

values of GPL nanofillers’ length-to-width ratio ( ⁄

45

⁄

⁄

)

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

46

AN US

CR IP T

ACCEPTED MANUSCRIPT

M

Fig. 11: Influence of geometry of GPL nanofillers on the fundamental natural frequency of FG-GPLRPC ⁄

AC

CE

PT

ED

plates ( ⁄

47

)

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

48

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 12: Influence of geometry of GPL nanofillers on the critical buckling load of FG-GPLRPC plates ⁄

AC

CE

PT

ED

( ⁄

49

)

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

50

AN US

CR IP T

ACCEPTED MANUSCRIPT

M

Fig. 13: Nonlinear static equilibrium path of FG-GPLRPC square plates for various length-to-thickness

AC

CE

PT

ED

ratio of plate (

51

)

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

52

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 14: Free vibration response curve of FG-GPLRPC square plates in the pre- and post-buckling

AC

CE

PT

ED

configurations for different length-to-thickness ratio of plate (

53

)