Mohammad Arefi.Ali Tabatabaeian.Masoud Mohammadi

Faculty of Mechanical Engineering.Department of Solid Mechanics.University of Kashan.Kashan 87317-51167.Iran

Keywords: Reinforced composite plate Graphene platelet Sinusoidal shear deformation theory Pasternak’s foundation Stress and deformation analysis

ABSTRACT The bending and stress analysis of a functionally graded polymer composite plate reinforced with graphene platelets are studied in this paper.The governing equations are derived by using principle of virtual work for a plate which is rested on Pasternak’s foundation.Sinusoidal shear deformation theory is used to describe displacement field.Four different distribution patterns are employed in our analysis.The analytical solution is presented for a functionally graded plate to investigate the influence of important parameters.The numerical results are presented to show the deflection and stress results of the problem for four employed patterns in terms of geometric parameters such as number of layers.weight fraction and two parameters of Pasternak’s foundation.

1.Introduction

Recently.researchers have focused on the various analyses of reinforced structures to increase stiffness of structures while is reducing the density of them.There are a vast number of studies representing efficient structural element formulations [1-4].Rabczuk et al.[5]presented some mesh free methods coupled with theoretical solution to analyze the fracture of structures.Some novel reinforcements are included carbon nanotubes and graphene platelets [6-8].Not only are carbon nanotubes and graphene platelets used to increase stiffness of structures but these materials also deduct density of polymer matrix.For example.some experiments by Rafiee et al.[9]were performed to show that addition of 0.1% weight fraction of reinforcement leads to increase of 31% in Young’s modulus of epoxy.In another experiment Liang et al.[10]indicated that addition of 0.7% weight fraction of reinforcement leads to increase of 76% in tensile strength and increase of 62% in Young’s modulus of elasticity.Although some experimental works on the structures reinforced with graphene nanoplatelets were published.the number of works on the theoretical aspects of mentioned issue is very limited.A literature review on the subject of this paper is presented to justify necessity of paper.

Yasmin and Daniel[11]presented some mechanical and thermal properties of graphite platelet/epoxy composites.Rafiee et al.[9]presented the mechanical properties of epoxy nanocomposites with graphene platelets.single-walled carbon nanotubes.and multi-walled carbon nanotube additives.They expressed that Young’s modulus of the graphene nanocomposite was 31% greater than epoxy with addition of 0.1% reinforcement.

Kvetkov′a et al.[12]presented a comprehensive study about fracture toughness of graphene platelets.They mentioned that increase of 1 wt%graphene platelet leads to significant improvement of fracture toughness.It is a well-established fact that computational methods are capable of dealing with a wide application range[13,14].Zhu et al.[15]employed first order shear deformation theory for static and vibration analysis of composite plates reinforced with carbon nanotube.The finite element approach was employed for solution of the problem.The functionalities of FG composite plate were assumed based on four known distributions.The effective material properties of FG nano composites were evaluated by using rule of mixture.The numerical results were presented to investigate influence of various boundary conditions and various distributions of FG plate on the bending and free vibration responses.Chatterjee et al.[16]presented a comprehensivestudy about the influence of reinforcements on mechanical and thermal properties of graphene nanoplatelets or epoxy composites.They presented comprehensive data about improvement of toughness and thermal conductivity of structures due to adding the defined percentage of graphene nanoplatelets.Chatterjee et al.[17]studied various types of mechanical reinforcement made from epoxy matrix with the addition of graphene nanoplatelets and various mixture ratios of carbon nanotubes.The influence of various sizes of graphene nanoplatelets was studied on the mechanical properties of composite structure.They mentioned that some ratios of graphene nanoplatelets and carbon nanotubes lead to significant improvement of material properties.

King et al.[18]mentioned some excellent and novel properties of graphene nanoplatelet/epoxy composites because of high stiffness and low density.They expressed that by adding 6 wt% of graphene platelet,one can reach 23.5%increase of tensile modulus.Chandrasekaran et al.[19]summarized some mechanical,electrical and thermal properties of graphite nano-platelet and epoxy nanocomposite.They investigated the influence of addition of graphite nano-platelet on the fracture toughness and modulus of elasticity of the nano-composite.They presented some data to reflect improvement of properties of structures reinforced with graphite nano-platelets after addition of defined percentage of reinforcement.Das and Prusty[20]provided some applications of graphenebased polymer composites in order to improve material properties of structures which are made from it.The influence of various polymer matrixes on the various behavior of reinforced structures was studied.Yue et al.[21]presented usefulness of carbon nanotubes and graphene nanoplatelets and improvement of properties of materials by combination of them in specific ratios.Prolongo et al.[22]investigated the properties of materials due to thickness and lateral dimensions of graphene nanoplatelets.They concluded that the small nanoplatelets have a greater tendency to agglomerate in packages of several parallel particles.In addition.nanocomposites reinforced with larger and thicker nanoplatelets presented lower glass transition temperature,higher modulus and higher decomposition temperature.Liew et al.[23]presented a comprehensive review paper on the functionally graded carbon nanotube reinforced composites.They mentioned that functionally graded carbon nanotube reinforced composites were extensively used in various industries due to high density.low density and some other excellent properties.Wu et al.[24]studied thermal buckling and post buckling of functionally graded multilayer nanocomposite plates reinforced with a low content of graphene platelets.Effective Young’s modulus and other properties of graphene platelets were measured by using modified Halpin-Tsai micromechanics model.They used first order shear deformation theory and principle of virtual displacements to derive governing equations of thermal buckling and post buckling.Song et al.[25]studied nonlinear buckling and post buckling of functionally graded multilayer composite plates reinforced with a low content of graphene nanoplatelets subjected to biaxial compressive loads.The problem was formulated based on first order shear deformation theory and von K′arm′an-type nonlinear kinematic relations.The influence of some significant parameters such as total number of layers.weight fraction and other parameters such as geometries and sizes were studied.Arefi and Zenkour [26]studied bending behavior of three-layered nanoplate including a homogeneous elastic nano core and two piezomagnetic face-sheets subjected to magneto-electro-thermo-mechanical loads.The sinusoidal shear deformation theory was used to derive governing equations based on principle of virtual work.Other recent numerical methods on modeling the mechanical properties of polymer nanocomposites have been provided in Ref.[27].Moreover,for analyses with several studied parameters.other considerations such as statistical scrutiny should be taken into account.In Refs.[28,29].for example,simple MATLAB codes were provided for sensitivity analysis of computationally expensive models.Additionally.theoretical and implementation aspects of sensitivity analysis methods can be found in Refs.[30-32].

A comprehensive literature review on the subject of this paper was carefully presented.This review indicates that although some important works on the analysis of functionally graded polymer composites reinforced with graphene platelets were performed and effect of other reinforcements such as carbon nanotubes was studied on the behavior of structures in some studies.it is necessary to consider some new aspects of previously mentioned works such as application of higher order shear deformation theories and bending behaviors of these structures.Moreover,as suggestion for future investigations.the use of the Kirchhoff-Love shell theory is recommended.as represented in Ref.[36].

Based on above comments,in this work,we develop sinusoidal shear deformation theory for elastic analysis of functionally graded polymer composite plate reinforced with graphene platelets which is rested on Pasternak’s foundation.Four various patterns are considered for distribution of graphene platelets inside polymer composite.Halpin-Tsai model is used to calculate the effective material properties along the thickness direction.The governing equations are derived by using principle of virtual work.The stress and deformation analysis of the problem is performed for a simplysupported rectangular functionally graded polymer composite plate reinforced with graphene platelets.The influence of important parameters such as dimensionless geometric parameters,number of layers.weight fraction of graphene platelets and two parameters of Pasternak’s foundation are studied on the deformation and stress analysis of functionally graded polymer composite plate.

2.Problem formulation

A multi-layer GPL/polymer nanocomposite plate with thickness h and length a and width b subjected to uniform transverse static load q(x.y) is shown in Fig.1.It has NLlayers with the same thickness Δh=and is reinforced by graphene platelets (GPLs)uniformly dispersed in the polymer matrix of each layer.The GPL weight fraction changes linearly along the thickness direction to form a functionally graded material structure.

Fig.1.A multi-layer functionally graded GPL/polymer nanocomposite plate with Pasternak foundation.

The polymer composite plate reinforced with graphene platelet has four different distribution patterns which are calculated by Eq.(1) [37,38]and also is shown in Fig.2:

As it is shown in Fig.2,pattern.1 is corresponding to an isotropic homogenous plate in which GPLs are uniformly distributed at the same w.t.% across all layers while in other patterns.GPL weight fraction changes linearly.In pattern.2.GPL weight fraction decreases from the highest in the mid-plane to the lowest on both top and bottom surfaces of the plate but it is exactly reversed with the maximum weight fraction on both top and bottom surfaces and the lowest in the mid-plane of the plate in pattern.3.

Both these two patterns are symmetric whereas in nonsymmetrical pattern.4.GPL weight fraction increases linearly from the top surface to the bottom surface.

2.1.Effective material properties

There are different methods to calculate the effective material properties.such as Mori-Tanaka and Halpin-Tsai.which mainly depend on the concept of homogenization In this work.similar to Ref.[38].the effective Young’s modulus of the GPL/polymer composite Ecis calculated by using Halpin-Tsai model:

Fig.2.Different GPL distribution patterns.

In which EMand EGPLare Young’s moduli of the polymer matrix and GPLs,respectively.ξLand ξWare the parameters characterizing both the geometry and size of GPL nanofillers lGPL.wGPLand hGPLare the average length.width.and thickness of the GPLs,respectively.

Poisson’s ratio υcand mass density ρcof the GPL/polymer nanocomposite are presented as:

Where ρGPL,υGPLand ρM,υMare related to GPLs and polymer matrix respectively and also VM,VGPLare the volume fraction of matrix and GPLs.The volume fraction of GPLs is represented as [39,40]:

Where gGPLis weight fraction of the GPLs in the nanocomposite.

2.2.Basic equations

To describe displacement field.sinusoidal shear-deformation plate theory is used in this study.This theory contains the classical and first or higher-order plate terms which is described as[41,42]:

Where(u0，v0，w0)are the displacement components of mid-plane and (u1，v1) are related to the terms of higher-order shear deformation plate theory.and ψ(z) is a function that shows the displacement field along the direction of thickness.In this study,based on sinusoidal shear deformation plate theory.ψ(z) is presented as:

By using the displacement field described in Eq.(10),the strain components εijare derived as:

Based on shear strain components derived above,the transverse shear strains are changed along the thickness with cosine variation that satisfies zero shear strains at top and bottom.The stress-strain relations for the Kth layer are expressed as follows [43-45]:

Where

The stress resultants and shear forces are related to strain components as:

Where the stiffness coefficients are calculated as [33]:

The strain energy of composite plate can be calculated as:

And by using Eqs.(12)-(16)and also resultant components we will get the variation of U as [46,47]:

In which the resultant components are defined as:

In addition.the variation of energy due to the external work is given by:

In which Rfis the reaction of foundation and presented as:

And also k1and k2are coefficients of winkler and Pasternak foundation respectively.Based on principle of virtual displacement,the governing equations are derived as:

Substituting Eqs.(19)-(22) into Eq.(30) yields five differential equations as follows:

3.Solution procedure

In this stage.the solution of the problem is presented for a simply-supported functionally graded composite plate.To satisfy the simply supported boundary conditions of the structure in this study.the following solution is proposed:

And the static load q is considered as:

In which λnand λmare introduced as:

Substituting Eq.(36)into Eqs.(31)-(35)will give the following format:

In which ｛S｝ = ｛U0，U1，V0，V1，W0｝Tis unknown vector.[R]is the stiffness matrix and [Q]is the force vector.The details about elements of matrix [R]are given in appendix A.

4.Numerical results and discussion

The numerical results of our problem are presented in this section.Before presentation of full numerical results.the material properties and geometric characteristics are presented.All material properties and geometric characteristics are employed as:

In which lGPL.hGPL.wGPLare related to GPL nanofillers and a.b and h are related to the composite plate.

In this stage.we can present full numerical results in terms of important parameters such as weight fraction.total number of layers.aspect ratio of side lengths a/b and two parameters of Pasternak’s foundation.To present numerical results.some dimensionless parameters are employed.

4.1.Comparison and validation

To check the accuracy and correctness of our formulations and corresponding numerical results.a comparison with a valid reference is performed in this section.Our study is validated with an available research (Song et al..2017a).As it is shown in Fig.3.the dimensionless deflection of FG-X pattern in terms of total number of layers NLis compared with the same one in the research of Song et al.(2017a) and it has a good agreement.

4.2.Deflection analysis

Fig.4 shows variation of dimensionless deflectionof FG plate(Pattern I)in terms of total number of layers NLfor various weight fractions.This figure shows that the dimensionless deflection of FG plate(Pattern I:Uniform Distribution)is decreased with increase of weight fractions.One can conclude that with increase of weight fraction,the stiffness of FG plate is increased and consequently the deflections are increased.In addition.it is observed that the dimensionless deflection is insensitive to total number of layers because of uniform distribution of functionalities.

Fig.3.The comparison of dimensionless deflection of plate between present and literature results (Song et al..2017a).

Fig.4.The dimensionless deflection of FG plate(Pattern I)in terms of total number of layers for various weight fractions.

Fig.5.The dimensionless deflection of FG plate(Pattern III)in terms of total number of layers for various weight fractions.

Shown in Fig.5 are variations of dimensionless deflections of FG plate in terms of total number of layers NLand various weight fractions for patterns III.The numerical results show that with increase of total number of layers of FG plate.the dimensionless deflection is tending to an asymptotic value for large values of NL.Through comparison between Figs.4-5 for various patterns.it is concluded that minimum dimensionless deflection of FG plate is observed for FG-X because of high stiffness of this pattern rather than the others.One can conclude that for FG-X more reinforcements are located at top and bottom of plates that leads to increase of stiffness of composite plate.

Figs.6-9 show variation of the dimensionless deflection of FG plate in terms of two parameters of Pasternak’s foundation(k1，k2)for patterns I.II.III and IV respectively.It is observed that with increase of both parameters of foundation.the dimensionless deflection of FG plate is decreased significantly.The numerical results show that the minimum deflection of FG plate is occurred for pattern FG-X and maximum deflection for FG-O.One can conclude that this is due to this fact that minimum bending stiffness is for FGO while maximum bending stiffness is for FG-X.

Fig.6.The dimensionless deflection of FG plate(Pattern I)in terms of two parameters of Pasternak’s foundation.

Fig.7.The dimensionless deflection of FG plate(Pattern II)in terms of two parameters of Pasternak’s foundation.

Fig.8.The dimensionless deflection of FG plate (Pattern III) in terms of two parameters of Pasternak’s foundation.

Fig.9.The dimensionless deflection of FG plate (Pattern IV) in terms of two parameters of Pasternak’s foundation.

Fig.10 shows variation of dimensionless deflection of FG plate in terms of aspect ratio λ=a/b for various distributions of FG plate.In this case.the area of plate is assumed constant through change of ratio λ.It is observed that with increase of aspect ratio λ.the dimensionless deflection of FG plate is decreased significantly.The maximum deflection of FG plate is occurred for a=b(square plate).One can conclude that with assumption of constant area.the minimum stiffness is occurred for square plate and stiffness of rectangular plate is increased with increase of aspect ratio of a/b.

Fig.10.The dimensionless deflection of FG plate in terms of λ for various patterns.

4.3.Stress analysis

In this section,the stress analysis is performed.The influence of important parameters is investigated on the stress distribution of functionally graded plate.

4.3.1.The influence of ratio λ on the stress distribution

Figs.11 and 12 show variation of dimensionless normal stresses σxx,in terms of dimensionless aspect ratio λ for various patterns(UD,FG-O,FG-X and FG-A).Fig.13 shows variation of in-plane shear stress in terms of dimensionless aspect ratio λ for various patterns(UD.FG-O.FG-X and FG-A).The numerical results show that with increase of dimensionless aspect ratio λ.the shear stress is decreased significantly because of increase of stiffness and decrease of deflections and strains.In addition,it is observed that maximum stresses are provided for FG-O distribution and minimum stress for FG-X distribution.It is concluded that the minimum stiffness is occurred for FG-O that leads to maximum deflection and then maximum strain.One can conclude that multiplication of stiffness and strain leads to maximum stress for FG-O.unlike FG-O,FG-X has maximum stiffness that leads to minimum deflection and then minimum strain that leads to minimum stress.

Fig.11.The dimensionless normal stress of FG plate in terms of λ for various patterns.

Fig.12.The dimensionless normal stress of FG plate in terms of λ for various patterns.

Fig.13.The dimensionless in-plane shear stress of FG plate in terms of λ for various patterns.

4.3.2.Stress at middle surface

In continuation.the influence of total number of layers and weight fraction is studied on the stress distribution of middle surface(z=0)of FG plate.Figs.14-17 show variation of dimensionless normal stressin terms of total number of layers NLfor various weight fractions.Investigation on the effect of weight fractions of graphene nanoplatelets on the absolute values of axial stress distribution indicates that with increase of weight fractions of reinforcement.the axial stresses are increased for patterns II.IV and decreased for patterns I.III.

4.3.3.Stress at top surface

The influence of total number of layers NLand weight fraction%gpl is studied on the stress distribution of top surface(z=h/2)of FG plate.Figs.18-21 show variation of dimensionless normal stressin terms of total number of layers NLfor various weight fractions.The effect of weight fraction of reinforcement on the axial stress indicates that the axial stresses are increased for patterns II,IV and decreased for patterns I.III.

Fig.14.The dimensionless in-plane normal stress of FG plate (Pattern I) at middle surface (z = 0) in terms of total number of layers for various weight fractions.

Fig.15.The dimensionless in-plane normal stress of FG plate (Pattern II) at middle surface (z = 0) in terms of total number of layers for various weight fractions.

Fig.16.The dimensionless in-plane normal stress of FG plate (Pattern III) at middle surface (z = 0) in terms of total number of layers for various weight fractions.

Fig.17.The dimensionless in-plane normal stress of FG plate (Pattern IV) at middle surface (z = 0) in terms of total number of layers for various weight fractions.

4.3.4.Influence of parameters of foundation

The infulence of spring and shear parameters of foundation are studied on the stress distribution of FG plate for various patterns.Shown in Figs.22-25 are distributions of normal stressin terms of two parameters of foundation k1， k2.The numerical results indicate that the dimensionless stresses are decreased withincrease of both parameters of foundation k1，k2.

Fig.18.The dimensionless in-plane normal stress of FG plate(Pattern I)at top surface(z = h/2) in terms of total number of layers for various weight fractions.

Fig.19.The dimensionless in-plane normal stress of FG plate(Pattern II)at top surface(z = h/2) in terms of total number of layers for various weight fractions.

Fig.20.The dimensionless in-plane normal stress of FG plate (Pattern III) at top surface (z = h/2) in terms of total number of layers for various weight fractions.

5.Conclusions

Fig.21.The dimensionless in-plane normal stress of FG plate (Pattern IV) at top surface (z = h/2) in terms of total number of layers for various weight fractions.

Fig.22.The dimensionless in-plane normal stress of FG plate (Pattern I) in terms of two parameters of foundation.

Fig.23.The dimensionless in-plane normal stress of FG plate (Pattern II) in terms of two parameters of foundation.

Elastic analysis of functionally graded polymer composite plate reinforced with graphene platelets resting on Pasternak’s foundation was performed based on sinusoidal shear deformation plate theory.Principle of virtual displacement was used to derive governing equations of the problem.Halpin-Tsai model was used to calculate effective material properties of functionally graded polymer composite plate reinforced with graphene platelets for four various patterns of GPL distribution.The numerical results of the problem were calculated by using Navier’s method to evaluate influence of significant parameters of the problem including aspect ratio.number of layers.weight fraction of graphene platelets and two parameters of Pasternak’s foundation.The numerical results including deflection and stress analysis were presented in terms of mentioned significant parameters.Some important discussions of this analysis are presented as follows:

Fig.24.The dimensionless in-plane normal stress of FG plate (Pattern III) in terms of two parameters of foundation.

Fig.25.The dimensionless in-plane normal stress of FG plate (Pattern IV) in terms of two parameters of foundation.

Weight fraction of graphene platelets has significant influence on the variation of deflection and stresses.The numerical results show that for patterns I.III (uniform and FG-X distributions).the dimensionless deflections are decreased with increase of weight fraction of graphene platelets.In addition,with increase of weight fraction of graphene platelets,the stress components are decreased for patterns I.III and are increased for patterns II.IV.

As another important parameter,the aspect ratio of side lengths λ=a/b was employed for our numerical analysis.The obtained results indicate that with increase of dimensionless parameter λ,all in-plane stress components are decreased.One can conclude that this is due to decrease of stiffness of plate.

Two parameters of Pasternak’s foundation have significant influence on the deflection and stress components of composite plate.The numerical results show that with increase of these parameters.all stress components and deflection are decreased.

Acknowledgments

The research described in this paper was financially supported by the University of Kashan.(Grant Number: 467893/0655).The author would also like to thank the Iranian Nanotechnology Development Committee for their financial support.

Appendix A