G.Q.Xie,J.P.Wang2,Q.L.Zhang

Small-Scale Effect on the Static deflection of a Clamped Graphene Sheet

G.Q.Xie1,J.P.Wang2,Q.L.Zhang1

Small-scale effect on the static deflection of a clamped graphene sheet and influence of the helical angle of the clamped graphene sheet on its static deflection are investigated.Static equilibrium equations of the graphene sheet are formulated based on the concept of nonlocal elastic theory.Galerkin method is used to obtain the classical and the nonlocal static deflection from Static equilibrium equations,respectively.The numerical results show that the static deflection and small-scale effect of a clamped graphene sheet is affected by its small size and helical angle.Small-scale effect will decrease with increase of the length and width of the graphene sheet,and small-scale effect will disappear when the length and the width of graphene sheet are both larger than 200 um.

Nonlocal theory;Graphene sheet;Small-scale effect;Static deflection;Helical angle.

1 Introduction

Since carbon nanotube was discovered by Iijima(1991),it has shown a broad application prospect in various fields because of its high mechanical strength,strong energy storage and catalytic effect etc.Due to the surface effect and the small-scale effect of nanomaterials,classical continuum mechanics will lead to an inaccurate result when it is used to solve the mechanics problem of nanomaterials.Fortunately,the nonlocal theory given by Eringen(1972)can remove the shortcoming of classical continuum mechanics.Based on the nonlocal theory,Zhang,Liu,and Wang,(2004)studied the buckling of multi-walled carbon nanotube.Xie,Han,and Long(2006,2006,2007)investigated the small scale effect and the vibration of carbon nanotube.Wang(2011)used a modified nonlocal beam model to study vibration and stability of nanotubes conveying fluid.Hybrid nonlocal beam model[Zhang,Wang,and Challamel(2009)]was employed to study bending,buckling,and vibration of micro/nanobeams.Reddy(2010)presented nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates.Fang,Zhen,and Zhang(2013)carried out nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory.Liang and Han(2014)gave prediction of the nonlocal scaling parameter for graphene sheet.Recently,Miandoab,Yousefi-Koma,and Pishkenari(2015)used nonlocal and strain gradient based model to study the electrostatically actuated silicon nano-beams.Zhao and Shi(2011)used an improved molecular structural mechanics model to study poisson ratios of single-walled carbon nanotubes.

In this paper,a nonlocal model of nanoplate is developed for the static deflection of graphene sheet.Small-scale effect on the static deflection of a clamped graphene sheet is investigated.

2 Formulation

2.1 Nonlocal stress tensor

In Eringen nonlocal elasticity model,Eringen(1983)considered that the physics of material bodies whose behavior at a material point is influenced by the state of all points in the body.This result is in accordance with atomic theory of lattice dynamics and experimental observations on phonon dispersion.The most general form of the nonlocal constitutive equation involves an integral over the entire regionof interest.

For homogeneous and isotropic elastic solids,the nonlocal constitutive equation is

Where symbols ‘:’is the double dot product,is the elastic modulus matrix of classical isotropic material,denotes the nonlocal stress tensor atandis the strain tensor at any pointin the body.The kernel functionis the nonlocal modulus,is the Euclidean distance,andwhereis a constant appropriate to each material,a is an internal characteristic size(e.g.length of C-C bond,lattice spacing,granular distance etc.)and l is an external characteristic size(e.g.crack size,wave length etc.).The volume integral in Eq.(1)is over the regionV occupied by the body.However,e0and l of graphene sheet have not been found in theoretical or experimental literature.

Based on nonlocal elasticity model,we chose a representative element of graphene sheet shown as Fig.1.

The stress of a reference point xxx in the representative element can be expressed as

Figure 1:A representative element of graphene sheet.

Figure 2:A helical graphene sheet.

Taylor series

To take the average of σij(x1,x2)over the representative element,in terms of the symmetry of the representative element,we have

Where l is the length of C-C bond.

Inversion of eq.(3)yields

Whereσij(x1,x2)is the nonlocal stress tensor.

2.2 Geometric equations

A helical graphene sheet shown as Fig.2,(x1,x2)is the local coordinate system,andis the global coordinate system.andare parallel to both sides of the graphene sheet,respectively.

Geometric equations of the helical graphene sheet in the global coordinate are

Where w is the static deflection of the graphene sheet,

According to the transformation relationship of strains,we have

Where

2.3 Nonlocal constitutive equation

Based on Eq.(4),the nonlocal constitutive equations of graphene are rewritten as

Where E is the elastic modulus of graphene,and μ Poisson’s ratio.

Eq.(9)can be approximately expressed as

Substituting Eq.(6)into Eq.(10),we have

2.4 Physic equations

The normal and shear stresses can be collected into bending momentand torquerespectively.

The bending equilibrium equation of the plate on which a distribution force q is applied

Combination of Eqs.(11)-(14)yields

3 Numerical examples and discussions

A clamped rectangle graphene sheet is shown as Fig.3.

Figure 3:A clamped rectangle graphene sheet.

The boundary conditions of the clamped graphene sheet can be written as

Where a and b are,respectively,the length and the width of the graphene sheet.

The deflection expression of the graphene sheet in terms of Galerkin method is given by

Eq.(18)are consistent with the clamped boundary conditions of the graphene sheet.

Galerkin weak form of Eq.(15)is given by

Where m=1,2,3,...

The approximate solution of Eq.(19)is given as following

Where

Substituting of Eqs.(21)-(23)into Eq.(19)has

C1,C2andC3can be obtained from the solution of Eq.(24).

Classical static deflection of the graphene sheet can be obtained by substituting l=0 into Eq.(19).

To illustrate the small-scale effect on the static deflections of the graphene sheet,the small-scale effect factor η is de fined as

Where wcsand wnsare,respectively,the classical and nonlocal static deflections of the center point of the graphene sheet

For all the subsequent numerical example,the length of C-C bond l=0.142×10-9m,in-plane stiffness Eh=360 J/m2(Sanchez-Portal,D,1999),Poisson’s ratioμ=0.26,the thickness of the graphene sheet h=0.34 nm.

To investigate the effect of the small-scale on the static deflection of the graphene sheet,we calculated the small-scale effect factor η of the graphene sheets with the different helical angles and geometric sizes.

Figure 4:Small-scale effect factor of the static deflection of the center point of the graphene sheet with the different helical angles and geometric sizes(4 um≤a≤20 um,4 um≤b≤20 um).

Fig.4 shows that small-scale effect factor of the static deflection of the center point of the graphene sheet with the different helical angles and geometric sizes(4 um≤a≤20 um,4 um≤b≤20 um).It can be found From Fig.4 that the small-scale effect factor η is influence by the helical angle of the graphene sheet.The small-scale effect factor is far from 1,the small-scale effect is very obvious when the side length of the helical graphene sheet is smaller than 20 um.the small-scale effect will not always decrease with increase of the geometrical size of the helical graphene sheet.

Fig.5 shows that small-scale effect factor of the static deflection of the center point of the graphene sheet with different helical angles and geometric sizes(4 um≤a≤20 um,4 um≤b≤20 um).It can be seen from comparison of Fig.5 and Fig.4 that the small-scale effect factor η changes periodically with change of helical angle,the change cycle is π/4.

Figure 5:Small-scale effect factor of the static deflection of the center point of graphene sheet with the different helical angles and geometric sizes(4 um≤a≤20 um,4 um≤b≤20 um).

Fig.6 shows that small-scale effect factor of the static deflection of the center point of the graphene sheet with the different helical angles and geometric sizes(40 um≤a≤200 um,40 um≤b≤200 um).It can be seen from Fig.6 that the small-scale effect factor η is more and more close to 1 with increase of the side length of the graphene sheet no matter how much the helix angle is.When the length and the width of the helical graphene sheet are both larger than 200um,the small-scale effect of the static deflection of the graphene sheet almost disappears.

Figure 6:Small-scale effect factor of the static deflection of the center point of the graphene sheet with the different helical angles and geometric dimensions(40 um≤a≤200 um,40 um≤b≤200 um).

Figure 7:small-scale effect factor of the static deflection of the center point of the graphene sheet with the different helical angles and geometric sizes(40 um≤a≤200 um,40 um≤b≤200 um).

Fig.7 shows that small-scale effect factor of the static deflection of the center point of the graphene sheet with the different helical angle and geometric sizes(40 um≤a≤200 um,40 um≤b≤200 um).It can be seen from comparison of Fig.6 and Fig.7 that the small-scale effect factor η will change periodically with change of helical angle,the change cycle is π/4.

Fig.8 shows that comparison of small-scale effect factor of the static deflection of the graphene sheet with the different helical angles(40 um≤a≤200 um,40 um≤b ≤ 200 um).It can also be found from Fig.8 that the small-scale effect factor η of the same geometrical dimension graphene sheet will increase when the helical angles change from β =0 to β =4π/16.In other words,the small-scale effect of the same geometrical dimension graphene sheet will decrease with increase of its helical angle from β =0 to β =4π/16.

Figure 8:Comparison of small-scale effect factor of the static deflection of the graphene sheet with the different helical angles.

Fig.10 shows that comparison of the static deflection of the center point of the

Figure 9:Comparison of small-scale effect factor of the static deflection of the graphene sheet with the different helical angles.

Figure 10:Comparison of the static deflection of the center point of the graphene sheet with the different helical angles and geometrical dimensions.

graphene sheet with the different helical angles.It can be seen from Fig.10 that the static deflection of the center point of the graphene sheet will increase when the helical angle of the same length and width graphene sheet increases from β=0 to β =4π/16 or the length and width of the same helical angle graphene sheet increases.

Figure 11:Comparison of the static deflections of the center point of the graphene sheet with the different helical angles and geometrical dimensions.

Fig.11 shows that comparison of the static deflection of the center point of the graphene sheet with the different helical angles.It can be seen from Fig.11 that the static deflection of the center point of the graphene sheet will increase when the helical angle of the same length and width graphene sheet decreases from β =8π/16 to β =4π/16 or when the length and width of the same helical angle graphene sheet increases.The static deflection of the same geometrical dimension helical graphene sheet has a periodic change of β = π/4 with the change of the helical angle.

4 Conclusion

Taking the typical hexagonal element of the graphene sheet as the research object,based on the concept of nonlocal theory,the stress tensor of any point within the typical element is expanded into Taylor series,the nonlocal constitutive equations of the graphene sheet was established.Galerkin weak form is used to solve the equilibrium equation of the graphene sheet.The classical and the nonlocal static deflections of the graphene sheet were obtained from solution of the equilibrium equation.To illustrate the small-scale effect on the static deflections of the graphene sheet,the small-scale effect factor is de fined as the ratio of the static deflections of the center point of the classical plate to that of the nonlocal plate.Numerical results show that

1.When the length and width of the graphene sheet are less than 20 um,the small-scale effect factor of the static deflection of the graphene sheet will be very large,the small-scale effect is very obvious,and the small-scale effect will not always decrease with increase of the geometrical dimensions of the helical graphene sheet,the small-scale effect factor η is influence by the helical angle of the graphene sheet

2.For 40 um≤a≤200 um,40 um≤b≤200 um,small-scale effect factor of the static deflection of the graphene sheet is more and more close to 1with the increase of the length and width of the graphene sheet,the small-scale effect will disappear when the length and the width of the plate are both larger than 200um.When the helical angle of the graphene sheet changes from 0 to π/4,The small-scale effect will decrease.

3.For the same geometrical dimension graphene sheet,the small-scale effect factor has a periodic change of β = π/4 with the change of the helical angle.

4.The static deflection of the center point of the graphene sheet will increase when the helical angle of the same length and width graphene sheet reduces from β =0 to β =4π/16 or when the length and width of the same helical angle graphene sheet increases.The static deflection has a periodic change of β = π/4 with the change of the helical angle.

Acknowledgement:This work is supported by National Natural Science Foundation of China under the Grant Number 11372109

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1Civil Engineering college,Hunan University of Science and Technology,Xiangtan 411201,China

2Mianyang Vocational and Technical College,Mianyang 621000,China Corresponding author.E-mail:1020095@hnust.edu.cn