Three-Dimensional Static Analysis of Nanoplates and Graphene Sheets by Using Eri

Chih-Ping WuWei-Chen Li

Three-Dimensional Static Analysis of Nanoplates and Graphene Sheets by Using Eringen's Nonlocal Elasticity Theory and the Perturbation Method

Chih-Ping Wu1,2,Wei-Chen Li1

A three-dimensional(3D)asymptotic theory is reformulated for the static analysis of simply-supported,isotropic and orthotropic single-layered nanoplates and graphene sheets(GSs),in which Eringen's nonlocal elasticity theory is used to capture the small length scale effect on the static behaviors of these.The perturbation method is used to expand the 3D nonlocal elasticity problems as a series of two-dimensional(2D)nonlocal plateproblems,the governing equations of which for various order problems retain the same differential operators as those of the nonlocal classical plate theory(CST),although with different nonhomogeneous terms.Expanding the primary field variables of each order as the double Fourier series functions in the in-plane directions,we can obtain the Navier solutions of the leading-order problem,and the higher-order modifications can then be determined in a hierarchic and consistent manner.Some benchmark solutions for the static analysis of isotropic and orthotropic nanoplates and GSs subjected to sinusoidally and uniformly distributed loads are given to demonstrate the performance of the 3D nonlocal asymptotic theory.

Eringen's nonlocal elasticity theory,graphene sheets,nanoplates,static,the perturbation method,three-dimensional nonlocal elasticity.

1 Introduction

The fields of nanoscience and nanotechnology,as well as their application in advanced industries,have developed significantly since carbon nanotubes(CNTs)[Iijima(1991)]and graphene sheets(GSs)[Novoselov,Geim,Morozov,Jiang,Zhang,Dubonos,Grigorieva,and Firsov(2004)]were first discovered.These two nanoscale materials,possessing excellent mechanical,chemical,thermal and electrical material properties,have thus been introduced in a variety of poten-tial applications to both micro-and nano-electro-mechanical systems(MEMs and NEMs)[Brischetto and Carrera(2012a,2012b);Wang(2004);Wang and Liew(2008)].They have also been combined with polymer materials to form CNT-and GS-reinforced composite structures,with studies showing that they are capable of optimally arranging their distributions in the interior domains of these to obtain various extreme characteristics that can be used in structural designs[Coleman,Khan,Blau,and Gun'ko(2006);Esawi and Farag(2007);Li,Thostenson,and Chou(2008);Liew,Lei,and Zhang(2015);Ramaratnam and Jalili(2006)].A comprehensive literature survey with regard to the application of nonlocal elastic models in the modeling of CNTs and GSs has been undertaken by Arash and Wang(2012).

Because GSs have being widely used in MEMs,NEMS and other forms of nanotechnology,it is important to be able to accurately analyze their structural behaviors in order to extend their lifetimes and enhance their performance.This paper thus focuses on a literature survey related to the mechanical analysis of nanoplates and GSs using Eringen's nonlocal elasticity theory(ENET)[Eringen(1972,2002);Eringen and Edelen(1972)].

Some two-dimensional(2D)nonlocal plate theories have been used to investigate the mechanical behavior of nanoplates and GSs.By using the ENET,Aghababaei and Reddy(2009)reformulated Reddy's third-order shear deformation plate theory(TSDPT)for macroscale plates[Reddy(1984);Reddy and Phan(1985)]to obtain its counterpart for nanoplates and GSs.The TSDPT were applied to examine the bending and vibration behavior of nanoplates,in which the Navier solutions of the stress and displacement components and natural frequency parameters of simply-supported,nanoplates were presented.The small size effects on the static and vibration behavior of the nanoplates,and comparisons between their solutions and those obtained by using the nonlocal first-order shear deformation plate theory(FSDPT)and nonlocal classical plate theory(CPT),were undertaken.The nonlocal TSDPT was used by Hosseini-Hashemi,Kermajani,and Nazemnezhad(2015)to study the buckling and free vibration of rectangular nanoplates,in which the Levy solutions of critical loads and frequency parameters of rectangular nanoplates were presented.Based on the nonlocal CPT,Kitipornchai,He and Liew(2005)and Liew,He,and Kitipornchai(2006)investigated the free vibration behavior of multilayered GSs without and with being embedded in an elastic medium,respectively.By accounting for the van der Waals interactions between different GSs and Winkler's foundation model,the corresponding set of explicit formulas were derived to predict the natural frequency parameters and their associated vibration modes of double-and triple-layered GSs.In conjunction with the ENET and von Karman nonlinear strains,Reddy(2010)developed the nonlinear formulations of the non-local FSDPT and CPT for the static analysis of nanoplates.Based on the nonlocal TSDPT,Arani and Jalaei(2015)analytically investigated the static and dynamic responses of simply-supported,single-layered GSs embedded in an elastic medium under uniform and sinusoidal loads,in which a visco-Pasternak foundation model was used to simulate the interaction between the GSs and their surrounding medium.Pradhan and Phadikar(2009)presented the nonlocal FSDPT and nonlocal CPT for the vibration analysis of nanoplates and GSs by reformulating its local counterparts and using the ENET.The nonlocal FSDPT and CPT were also extended to the analysis of double-layered nanoplates,in which some effects on natural frequency parameters of the nanoplate were investigated,such as the nonlocal parameter,aspect ratio,Young's modulus of the nanoplate and stiffness of Winkler foundation effects.The CPT was reformulated by Pradhan and Kumar(2011)for the vibration analysis of single-layered orthotropic GSs with various boundary conditions by using the differential quadrature(DQ)method and ENET,in which the effects of small length scale,aspect ratio and boundary conditions on the vibration characteristics of these were examined.The nonlocal CPT was also used by Shen,Shen,and Zhang(2010)for the nonlinear vibration analysis of single-layered GSs in thermal environments.Duan and Wang(2007)presented the exact solutions for the axisymmetric bending analysis of nanoscale circular plates on the basis of the nonlocal CPT.It has been shown that the nonlocal solutions of deflections,moments and shear forces are larger than their counterpart local solutions.Wang and Li(2012)developed the nonlocal Mindlin and Kirchhoff plate theories to study the static behavior of nanoplates embedded in elastic matrix by accounting for the small length scale effect.Wen,Huang,and Aliabadi(2013)developed a 2D local boundary integral method for the nonlocal elasticity theory with 2D Eringen's model.Based on the nonlocal re fined sinusoidal shear deformation plate theory(SSDPT),Thai(2012);Thai and Vo(2012)and Thai,Vo,Nguyen,and Lee(2014)presented the analytical solutions for the bending,buckling and vibration problems of nanobeams and nanoplates,and concluded that the small scale and shear deformation effects are significant for the structural behaviors of these when the nanobeams and nanoplates become thicker.Zenkour and Sobhy(2013)and Alzahrani,Zenkour,and Sobhy(2013)applied the re fined SSDPT to the thermal buckling and hygro-thermo-mechanical bending analyses of nanoplates embedded in an elastic medium,respectively,in which the interaction between the nanoplate and its surrounding medium was modelled as a Winkler-Pasternak foundation.Li and Pan(2016)developed an SSDPT for the bending analysis of simply-supported,piezoelectric nanoplates considering the surface effects.

Some studies have carried out three-dimensional(3D)analyses of the static behaviors and dynamic responses of simply-supported,nanoplates and GSs.For exam-ples,Wu,Chiu,and Wang(2008)classified the 3D exact approaches for simplysupported macrostructures available in the literature into four different categories,namely,the state space,series expansion,Pagano and perturbation methods.A-mong these,only the state space method was successfully extended to the 3D analysis of nanostructures.Within the framework of 3D nonlocal elasticity theory,Alibeigloo and Zanoosi(2013)and Alibeigloo(2011)analyzed the static and free vibration behavior of simply-supported,rectangular GSs,respectively,by using the state space method in the thickness direction and Fourier series in the in-plane directions.The effects of the nonlocal parameter,aspect ratio and half-wave numbers on the deformations and stresses induced in the single-layered nanoplates and their natural frequency parameters were examined.The state space method was also extensively applied to the 3D free vibration analysis of multi-layered GSs embedded in an elastic medium by Alibeigloo(2012).By using the cylindrical coordinate system,Brischetto(2014)reformulated the 3D nonlocal state space equations for the free vibration behavior of simply-supported,single-layered CNTs.The natural frequency parameters and their corresponding vibration modes were obtained by using the state space method.The 3D nonlocal solutions were compared with the one-dimensional(1D)nonlocal beam ones in order to show the limitations of 1D beam models.Brischetto,Tornabene,Fantuzzi,and Bacciocchi(2015)presented a comparative study with regard to the free vibration behavior of single-and double-walled CNTs by using the re fined 2D and exact 3D shell models,in which the van der Waals interactions between the two cylinders were included.Based on the nonlocal elasticity theory,Ghavanloo and Fazelzadeh(2013)investigated the radial vibration of nanoscale spherical shells.It has been concluded that the small length scale effect on the frequency parameters of nanoshells is significant.Ansari, Shojaei,Shahabodini,and Bazdid-Vahdati(2015)studied the 3D bending and vibration analyses of functionally graded(FG)nanoplates with various boundary conditions.In their formulation,Hamilton's principle was used to derive the motion of equations of the nanoplate,the material properties of which were assumed to vary through the thickness coordinate with power-law and exponential distributions,and then the variational differential quadrature method was used to obtain the 3D nonlocal solutions for the bending and free vibration problems of FG nanoplates.Due to the mathematical complexity of 3D nonlocal elasticity theories of nanoplates and GSs,there are relatively few papers related to the 3D mechanical behavior of these available in the literature,when compared to the number of papers that examine those of macrostructures.To the best of the authors' knowledge,the perturbation method[Nayfeh(1993)]has never been applied to the 3D mechanical analysis of nanoplates and GSs,even though it has been successfully applied to that of macrostructures,such as laminated composite structures[Wu and Chiu(2002);Wu,Tarn,and Chi(1996a,1996b);Wu,Tarn,and Chen(1997)]and FG elastic/piezoelectric ones[Wu and Syu(2007);Wu and Tsai(2004)].An asymptotic theory is thus developed in this work for the 3D static analysis of simply-supported,rectangular nanoplates and GSs by using the perturbation method combined with Eringen's nonlocal elasticity theory.Using direct elimination,we first reduce the fifteen partially differential equations(PDEs)of the 3D nonlocal elasticity theory to six PDEs in terms of three displacement components and three transverse shear and normal stress ones.Through the mathematical manipulation of nondimensionalization,asymptotic expansion and successive integration,we finally obtain recurrent sets of nonlocal governing equations for various order problems.The nonlocal CPT is derived as a first-order approximation of the 3D nonlocal elasticity theory,and the nonlocal governing equations for higher-order problems retain the same differential operators as those of the nonlocal CPT,although with different nonhomogeneous terms.Expanding the primary field variables of each order as the Fourier series functions in the in-plane directions,we can obtain the Navier solutions of the leading-order problem,and the higher-order modifications can then be determined in a systematic manner.The asymptotic approach is different from methods based on the 2D nonlocal advanced and re fined theories.In the latter,a set of kinetic and kinematic models is assumed a priori,and this has to be replaced by using a much higher-order model when the accuracy of the original model needs to be improved,and thus the formulation must be totally changed.In contrast,in the former the leading-order solutions are identical to the nonlocal CPT ones,and the accuracy can be improved by performing higher-order modifications and without changing the whole formulation.Moreover,the 3D asymptotic solutions will approach the exact 3D nonlocal elasticity solutions order by order.

Some benchmark solutions for the static analysis of simply-supported,isotropic and orthotropic macro-and nano-plates subjected to sinusoidally-and uniformlydistributed loads are given to demonstrate the performance of the 3D nonlocal asymptotic theory,in which the nonlocal parameter(µ)is taken to be zero for the macroplates and positive values,which should be less than 4 nm2[Wang and Wang(2007)],for nanoplates and GSs.The 3D results for the nanoplates and GSs may provide a reference for assessing those obtained using the 2D advanced and re fined nanoplate theories.Moreover,the effects of the nonlocal parameter,aspect ratio,and in-plane dimension-to-thickness ratio on the stress and displacement components induced in the loaded nanoplates and GSs are also examined.

2 Basic equations of 3D nonlocal elasticity theory

As shown in Fig.1,we consider a simply-supported elastic nanoplate(or GS),subjected to a sinusoidally distributed load.A set of Cartesian coordinates(x,y and z)is located at the mid-plane of the plate,the thickness and in-plane dimension of which are denoted as H and(LxxLy),respectively.

Figure 1:Geometric parameters and con figuration of a nanoplate(or GS)under the sinusoidally distributed load.

The main difference between the local and nonlocal elasticity theories is in their descriptions of the constitutive relation of a Hookean solid.In the former,the stress components induced at a particular material point of the loaded elastic body depend only on the strain components induced at that point,while in the latter,these will depend on the strain components induced at all the material points of the continuum,due to the small length scale effect.According to Eringen(1972,2002)and Eringen and Edelen(1972),the nonlocal constitutive behavior of an elastic body can be written as

where the elastic modulus tensor of classical isotropic elasticity is de fined as Cijkl,and stress and strain tensors are σijand εkl,respectively.The nonlocal modulus or attenuation function,which incorporates the constitutive equations into the nonlocal effect at the reference point x produced by local strain at the source x0,is de fined as α(|x−x0|,τ),in which the symbol|x−x0|is the Euclidean distance.τ=e0a/l,in which e0is a constant appropriate to each material,a is an internal characteristic length(e.g.,length of C-C bond,lattice parameter,or granular distance),and l is an external characteristic length(e.g.,crack length or wavelength).The value of e0needs to be determined from experiments or by matching the dispersion curves of plane waves with those of atomic lattice dynamics[Wang(2005)].

The integral-partial differential equations of Eq.(1)can be further reduced to singular partial differential equations of a special class of physically admissible kernel,as follows:

whereµis the nonlocal parameter,andµ=(e0a)2,and e0alt;2.0 nm for a singlewalled CNT[Wang and Wang(2007)].∇2is the Laplacian operator,in which∇2=is used for a 3D nonlocal elastic problem,while ∇2=(∂xx+∂yy)and∇2=∂xxfor the 2D and 1D nonlocal elastic ones,and∂xx=∂2/∂x2.

According to Eringen's elasticity theory,the linear constitutive equations valid for the symmetrical class of elastic materials are given by

where(εx,εy,εz,γxz,γyz,γxy)denote the strain components.and(σx,σy,σz,τxz,τyz,τxy)are de fined as the nonlocal and local stress components,respectively,in whichτyz,τxy).cij(i,j=1–6)are the elastic coefficients relative to the geometrical axes of the plate.For an isotropic material nanoplate the coefficients cijwill be reduced to c11=c22=c33=[(1−υ)E/(1+υ)(1−2υ)],c12=c13=c23=[υE/(1+υ)(1−2υ)]and c44=c55=c66=[E/2(1+υ)],in which E and υ are the Young's modulus and Poisson's ratio,respectively.

The kinematic equations in terms of the Cartesian coordinates are

in which ux,uyand uzare the displacement components.

The stress equilibrium equations of an elastic body without accounting for body forces are given by

The boundary conditions of the problem are specified,as follows:

On the top and bottom surfaces the transverse loads are given by

where h denotes one-half of the total thickness(H),and the positive directions ofandare de fined to be downward and upward,respectively.

The edge boundary conditions of the plate are considered as simply-supported ones,and are given as follows:

At the edges(x=0 and x=Lx),the boundary conditions are

At the edges(y=0 and y=Ly),the boundary conditions are

3 Nondimensionalization

A set of dimensionless coordinates and elastic field variables is de fined,as follows:

As we previously listed in Eqs.(3)–(7),there are fifteen basic equations of 3D nonlocal elasticity theory for the static analysis of nanoplates and GSs,subjected to mechanical loads.In order to make the formulation suitable for mathematical treatment,we eliminate the in-plane stress(σx,σy,τxy)and strain(εx,εy,εz,γxz,γyz,γxy)components from Eqs.(3)_(7),introduce Eq.(10)in the resulting equations,and then express the 3D basic equations in terms of the dimensionless forms of displacement(u1,u2,u3)and transverse stress(τ13,τ23,σ3)components,as follows:

where

Following a similar derivation process,we rewrite the in-surface stresses in the dimensionless form,as follows:

where

The dimensionless forms of the boundary conditions of the problem are specified as follows:

On the top and bottom surfaces the transverse loads are given by

At the edges(x1=0 and x1=Lx/L),the boundary conditions are

At the edges(x2=0 and x2=Ly/L),the boundary conditions are

4 Asymptotic expansion

Because Eqs.(11)–(15)contain terms involving only even powers of∈,we asymptotically expand the field variables in the powers∈2,as given by

Substituting Eq.(18)into Eqs.(11)–(15)and collecting coefficients of equal powers of∈,we obtain the following sets of recurrence equations.

Order∈0:

The boundary conditions for various order problems are specified,as follows:On the top and bottom surfaces the transverse loads are

Order∈0,

Order∈2k(k=1,2,3,etc),

Along the edges(x1=0 and x1=Lx/L),the boundary conditions are given,as follows:

order∈2k(k=0,1,2,3,etc),

Along the edges(x2=0 and x2=Ly/L),the boundary conditions are given as follows:

order∈2k(k=0,1,2,3,etc),

5 Asymptotic integration

5.1 The leading-order problem

We examine the sets of asymptotic equations and find that the present analysis can be carried out by integrating these through the thickness direction.We thus integrate Eqs.(19)and(20)to obtain

With the lateral boundary conditions on x3=-1 given in Eq.(29a),we then proceed to integrate Eqs.(21)and(22),which yields

Imposing the remaining lateral boundary conditions on x3=1 given in Eq.(29b)in Eqs.(33)and(34),we obtain

where

In this paper,the edge boundary conditions of the plate are considered as fully simply-supported edges.After the asymptotic process,we thus obtain the edge conditions for the leading order problem,as follows:

It is noted that Eqs.(35)–(37)are exactly the same as the governing equations of the nonlocal CPT[Pradhan and Kumar(2011);Shen,Shen,and Zhang(2010)],which have thus been derived as a first-order approximation of the 3D nonlocal elasticity theory.Solutions of Eqs.(35)–(37)must be supplemented with the edge boundary conditions given in Eqs.(39a)and(39b)to constitute a well-posed boundary value problem.Once the variables ofandare determined,the leading-order solutions of displacements through the plate thickness are given by Eqs.(31)and(32),the transverse shear and normal stresses by Eqs.(33)and(34)and the insurface stresses by Eq.(23).

5.2 Higher-order problems

Proceeding to order∈2k(k=1,2,3,...)and following the same process as before,we readily obtain

where

After the asymptotic process,we obtain the edge conditions for the higher-order problems,as follows:

The higher-order modi fications of mid-surface displacement componentsandcan be obtained by solving Eqs.(44)–(46)combined with the edge conditions given in Eqs.(47a)and(47b).Once these are determined,the higher-order modifications of displacement components are given by Eqs.(40)and(41),the transverse stresses by Eqs.(42)and(43)and the in-surface stresses by Eq.(28).

Equations(35)–(37)and(44)–(46)show that the differential operators among the various order problems remain identical,and the nonhomogeneous terms of higherorder problems can be calculated from the lower-order solution.The solution process of the leading-order problem can be repeatedly applied to the higher-order problems.The present asymptotic solutions can thus be determined order-by-order in a hierarchical and consistent manner.

6 Applications

6.1 Leading-order solution

The bending problem of simply-supported, single-layered isotropic/orthotropic nanoplates and GSs is studied by using Navier's method,in which the variables of displacement and stress components are expanded as double Fourier series functions in the in-plane directions.For this problem,the governing equations of the leading-order problem can thus be solved by letting

According to Eqs.(48)–(50),the other field variables should be also the forms of double Fourier series functions,such that the simply-supported edge conditions are satisfied,and these are given as

The summation sign will not be shown in the following derivation for brevity.Substituting Eqs.(48)–(50)into Eqs.(35)–(37)gives

where the applied external loads on the lateral surfaces,are expressed asand

In this work,the nanoplates and GSs are considered to be orthotropic or isotropic materials,the material properties of which are symmetric to the mid-plane,such that the coupled extension-bending coefficientsare identical to zero(i.e.,0),which leads to k13=k23=k31=k32=0.The governing equations of in-plane and out-of-plane directions given in Eq.(55)can thus be seperated and solved,as follows:

According to Eqs(57)and(58),the through-thickness distributions of various field variables of the leading-order problem can be obtained,as follows:

6.2 Higher-order modifications

Carrying out the solution to order∈2k(k=1,2,...),we find that the nonhomogeneous terms(fik(x1,x2,x3),i=1−3)and the relevant functions(ϕik(x1,x2,x3),i=1−3)in the∈2k-order equations are

The governing equations of the higher-order problems can be solved by letting

According to Eqs.(67)–(69),the other field variables of the kth-order problems should be also in the form of double Fourier series functions,such that the simply supported edge conditions are satisfied,and these are given as

The summation sign will not be shown in the following derivation for brevity.Substituting Eqs.(67)–(69)and Eqs.(64)–(66)into Eqs.(44)–(46)gives

Because k13=k23=k31=k32=0,the mid-plane displacement modifications of the kth-order can be obtained by a matrix partition process,as follows:

According to Eqs(75)and(76),the through-thickness distributions of various field variables of the kth-order modifications can be obtained,as follows:

where

The contributions of the nonlocal parameter(µ)in the in-plane and thickness directions were separated in this dimensionless formulation asand,respectively,because the thickness dimension is much less than the in-plane one for a nanoplate.In the following numerical examples,we let,the corresponding relations of which in a dimensional form areµ1=µandµ2=µ∈4,to reasonably equalize their contributions to the mechanical behaviors of a nanoplate and GSs.In addition,µ1= µ andµ2=0 are commonly used in the advanced and re fined 2D nonlocal plate theories.

7 Illustrative examples

7.1 Single-layered isotropic/orthotropic nanoplates

A benchmark problem with regard to the static analysis of simply-supported,isotropic and orthotropic macroplates under sinusoidally and uniformly distributed loads has been presented by Reddy(2002)using the TSDPT. Reddy's solutions can be used to validate the accuracy and convergence of the 3D nonlocal asymptotic theory in the case ofµ =0.The deviation between the 3D local(µ =0)and nonlocal(µ6=0)asymptotic solutions can also be examined when the dimensions of the plate are reduced to the nanoscale.

Table 1 shows the dimensionless center displacement componentinduced in single-layered isotropic and orthotropic nanoplates under sinusoidally and uniformly distributed loads on the top surface of the plate,in whichis taken for the single-term sinusoidally distributed load andˆm=ˆn=1−19 for the uniformly distributed one.The material properties used in the isotropic and orthotropic cases are E=30 GPa,υ=0.3,andnd υLT=υTT=0.25,respectively,in which the subscripts L and T denote the directions parallel and perpendicular to the fiber direction.The geometric parameters of the plate are Lx=Ly,S=Lx/H=10 and H=1 nm.The uniform load with intensity q0is expressed as the double Fourier series functions,as follows:

For comparison purposes,a set of dimensionless forms of nonlocal field variables are de fined,as follows:

It can be seen in Table 1 that the local(µ =0)and nonlocalasymptotic solutions converge rapidly.The convergent solutions are obtained at the∈4-order level in the cases of moderately thick nanoplates(Lx/H=10),and the∈2-orderlevel in the cases of thin nanoplates(Lx/Hgt;20).The convergent local asymptotic solutions closely agree with the ones obtained by using the local TSDPT,and the deviation between these decreases when the nanoplate becomes thinner.The center displacement in the z direction increases when the value of the nonlocal parameter(µ)becomes greater,which means the small length scale effect will soften the gross stiffness of the nanoplate.The increases in the cases of the moderately thick

nanoplates(Lx/H=10)are up to about 20%and 40%forµ=1nm2andµ=2nm2,respectively,and these reduce to about 5%,10%and 0.2%and 0.4%in the cases of thin(Lx/H=20)and very thin(Lx/H=100)nanoplates.The small length scale effect on the static behaviors of the nanoplate is significant when the nanoplate becomes thicker.

Table 1:The dimensionless center displacement components induced in the singlelayer isotropic and orthotropic nanoplates under the sinusoidally and uniformly distributed loads(Lx=Ly,S=Lx/H=10,H=1 nm,

Table 1:The dimensionless center displacement components induced in the singlelayer isotropic and orthotropic nanoplates under the sinusoidally and uniformly distributed loads(Lx=Ly,S=Lx/H=10,H=1 nm,

Sinusoidally distributed load(ˆm=ˆn=1)Uniform load(ˆm,ˆn=1−19)Materials µ Theories Lx/H=10 Lx/H=12.5 Lx/H=20 Lx/H=100 Lx/H=10 Lx/H=12.5 Lx/H=20 Lx/H=100 Isotropic 0 Present∈0 2.8026 2.8026 2.8026 2.8026 4.4361 4.4361 4.4361 4.4361 Present∈2 2.9429 2.8924 2.8377 2.804 4.64 4.5666 4.4871 4.4381 Present∈4 2.9425 2.8922 2.8377 2.804 4.6396 4.5665 4.487 4.4381 Present∈6 2.9425 2.8922 2.8377 2.804 4.6396 4.5665 4.487 4.4381 TSDT(Reddy,2002)2.9607 2.9038 2.8422 2.8042 4.666 4.5832 4.4936 4.4438 2.8027 2.8027 2.8027 2.8027 4.4436 4.4436 4.4436 4.4436 1 Present∈0 3.3558 3.1567 2.9409 2.8081 5.2404 4.9508 4.6372 4.4441 Present∈2 3.5238 3.2578 2.9777 2.8096 5.4703 5.092 4.6898 4.4462 Present∈4 3.5233 3.2576 2.9777 2.8096 5.4709 5.0921 4.6898 4.4462 Present∈6 3.5233 3.2576 2.9777 2.8096 5.4711 5.0921 4.6898 4.4462 2 Present∈0 3.909 3.5107 3.0792 2.8137 6.0447 5.4656 4.8382 4.4522 Present∈2 4.1047 3.6232 3.1178 2.8151 6.3006 5.6174 4.8925 4.4542 Present∈4 4.1041 3.623 3.1177 2.8151 6.3022 5.6178 4.8925 4.4542 Present∈6 4.1041 3.623 3.1177 2.8151 6.3025 5.6178 4.8925 4.4542 Orthotropic 0 Present∈0 0.4312 0.4312 0.4312 0.4312 0.6497 0.6497 0.6497 0.6497 Present∈2 0.6395 0.5646 0.4833 0.4333 0.9546 0.8448 0.7259 0.6528 Present∈4 0.6344 0.5624 0.483 0.4333 0.9474 0.8419 0.7255 0.6528 Present∈6 0.6348 0.5626 0.483 0.4333 0.9467 0.8417 0.7255 0.6528 CPT(Reddy,2002)TSDT(Reddy,2002)0.6383 0.5644 0.4836 0.4334 0.9519 0.8442 0.7262 0.6528 0.4313 0.4313 0.4313 0.4313 0.6497 0.6497 0.6497 0.6497 1 Present∈0 0.5164 0.4857 0.4525 0.4321 0.7453 0.7109 0.6736 0.6507 Present∈2 0.7658 0.6359 0.5072 0.4342 1.0902 0.9224 0.7523 0.6537 Present∈4 0.7596 0.6335 0.5068 0.4342 1.085 0.92 0.7519 0.6537 Present∈6 0.7601 0.6336 0.5068 0.4342 1.0588 0.9155 0.7518 0.6537 2 Present∈0 0.6015 0.5402 0.4738 0.4329 0.8408 0.772 0.6975 0.6516 Present∈2 0.892 0.7072 0.531 0.435 1.2258 0.9999 0.7787 0.6547 Present∈4 0.8848 0.7045 0.5307 0.435 1.2227 0.9981 0.7783 0.6547 Present∈6 0.8854 0.7047 0.5307 0.435 1.1708 0.9893 0.7781 0.6547 CPT(Reddy,2002)

Figure 2 shows the convergence of the center displacement in the z direction of the isotropic nanoplate with respect to the total number of terms of the double Fourier series functions used to expand the external uniform loads,in which∈6-order solutions are presented withˆm=ˆn=1−13,1−15,1−17 and 1−19;Lx=Ly,S=Lx/H=10,H=1 nm andµ=2nm2.As expected,usingˆm=ˆn=1−19 in the numerical examples will lead to the convergent solutions,and this approach has been commonly adopted in the published literature[Reddy(2002)],and was used to obtain the convergent solutions shown in Table 1.

Figure2:∈6-order solutions of the out-of-plane displacement induced in an isotropic nanoplate under the uniform load,which were expanded with respect to the different total number of terms of double Fourier series functions.

Figure3shows the asymptotic solutions of assorted nonlocal field variables induced in the single-layered,orthotropic material nanoplates subjected to a sinusoidally distributed load,in which Lx=Ly,S=Lx/H=10,H=1 nm and µ=2 nm2.It can be seen that the convergence rates of the nonlocal stress and in-plane displacement components are faster than that of the out-of-plane displacement.The tendency of the through-thickness distribution functions of various field variables induced in these nanoplates(µ=2 nm2)are similar to those in the macroplates,which are well-known to be approximate linear functions for the in-plane stress and displacement components,parabolic functions for the transverse shear stress components,and higher-order polynomial functions for the transverse normal stress ones.

Figure 3:The through-thickness distributions of assorted nonlocal field variables for various orders induced in an orthotropic nanoplate under the sinusoidally distributed loads.

Figure4:∈6-order solutions for the through-thickness distributions of assorted nonlocal field variables induced in an isotropic GS under the sinusoidally distributed loads and with different values of the nonlocal parameter.

7.2 Single-layered isotropic GSs

In this section,we examine the static behaviors of simply-supported,single-layered isotropic square and rectangular GSs,subjected to the mechanical loads.The material properties and thickness of the single-layered GSs were given as E=1.02 TPa,υ =0.16 and H=0.34 nm.Table 2 shows the convergent solutions(i.e.∈6-order ones)of nonlocal displacement and stress components at the crucial positions of the square and rectangular GSs with different aspect ratios and nonlocal parameters(µ),in which a one-term double Fourier series function load(ˆm=ˆn=1)is applied.It can be seen in Table 2 that the nonlocal displacement and stress components increase when the nonlocal parameter becomes greater.By converting the dimensionless field variables to their corresponding dimensional forms,we found that the magnitude ratios between the out-of-and in-plane displacements inducedin the square GSs are about 6.6,12.9 and 63.8 in the cases of moderately thickthinand very thinplates,respectively,and these ratios between the in-plane stress and transverse shear stress components are about 7.4,14.8 and 73.8.The magnitude ratios between the out-of-plane and in plane displacements,as well as the in-plane stress and transverse shear stresses,are significant for very thin GSs,while the ratios will decrease when the GS becomes thicker.In addition,these ratios remain almost unchanged for the rectangular GSswith different values of nonlocal parameters.

Table 2:The convergent solutions(i.e.,∈6-order ones)of nonlocal displacement and stress components at the crucial positions of the single-layered graphene sheets with different aspect ratios and nonlocal parameters,and subjected to a sinusoidally distributed load(H=0.34 nm).

Figure 4 shows the through-thickness distributions of assorted local and nonlocal field variables induced in the rectangular GSs,in whichandµ=0,1 and 2 nm2.It can be seen in Fig.4 that the nonlocal displacement and stress components are always larger than their corresponding local ones,and that these will increase when the value of the nonlocal parameter becomes greater.

8 Conclusions

In this work we first reformulate the 3D elasticity theory for the static analysis of simply-supported,single-layered isotropic and orthotropic nanoplates and isotropic GSs by using the ENET and perturbation method.A 3D nonlocal elastic problem is asymptotically expanded as a series of 2D nonlocal plate problems,in which the governing equations for various order problems remain the same as those of the nonlocal CPT,although with different nonhomogeneous terms,which can be determined by the lower-order solutions.The 3D nonlocal asymptotic theory can be reduced to its local counterpart by vanishing the nonlocal parameter(i.e.,µ=0).The implementation of this 3D nonlocal asymptotic theory in the numerical examples shows the present asymptotic solutions converge rapidly.The convergent solutions are obtained at the∈4-order level in the cases of moderately thick nanoplates(Lx/H=10),and the∈2-order level in the cases of thin nanoplates(Lx/Hgt;20).The convergent asymptotic solutions agree closely with the corresponding accurate solutions in the cases of macroplates available in the literature,and those of nanoplates and GSs can serve as a standard for assessing the performance of various 2D advanced and re fined theories for nanoplates and GSs.In addition,the results show the nonlocal displacement and stress components will increase when the value of the nonlocal parameters become greater.

Acknowledgement:This work was supported by the Ministry of Science and Technology of the Republic of China through Grant MOST 103-2221-E-006-064-MY3.

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1Department of Civil Engineering,National Cheng Kung University,Tainan 70101,Taiwan,ROC

2Corresponding author.Fax:+886-6-2370804 E-mail address:cpwu@mail.ncku.edu.tw