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

Transcription

1 Copyright 05 Tech Science Press CMC, vol.8, no., pp.03-7, 05 Small-Scale Effect on the Static Deflection of a Clamped Graphene Sheet G. Q. Xie, J. P. Wang, Q. L. Zhang Abstract: 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 00 um. Keywords: Nonlocal theory; Graphene sheet; Small-scale effect; Static deflection; Helical angle. Introduction Since carbon nanotube was discovered by Iijima 99, 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 97 can remove the shortcoming of classical continuum mechanics. Based on the nonlocal theory, Zhang, Liu, and Wang, 00 studied the buckling of multi-walled carbon nanotube. Xie, Han, and Long 006, 006, 007 investigated the small scale effect and the vibration of carbon nanotube. Wang 0 used a modified nonlocal beam model to study vibration and stability of nanotubes conveying fluid. Hybrid nonlocal beam model [Zhang, Civil Engineering college, Hunan University of Science and Technology, Xiangtan 0, China Mianyang Vocational and Technical College, Mianyang 6000, China Corresponding author @hnust.edu.cn

2 0 Copyright 05 Tech Science Press CMC, vol.8, no., pp.03-7, 05 Wang, and Challamel 009] was employed to study bending, buckling, and vibration of micro/nanobeams. Reddy 00 presented nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Fang, Zhen, and Zhang 03 carried out nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory. Liang and Han 0 gave prediction of the nonlocal scaling parameter for graphene sheet. Recently, Miandoab, Yousefi-Koma, and Pishkenari 05 used nonlocal and strain gradient based model to study the electrostatically actuated silicon nano-beams. Zhao and Shi 0 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. Formulation. Nonlocal stress tensor In Eringen nonlocal elasticity model, Eringen 983 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 region of interest. For homogeneous and isotropic elastic solids, the nonlocal constitutive equation is σ x = C 0 : α x x,τ ε x dv x V Where symbols : is the double dot product, C 0 is the elastic modulus matrix of classical isotropic material, σ x denotes the nonlocal stress tensor at x, and ε x is the strain tensor at any point x in the body. The kernel function α x x,τ is the nonlocal modulus, x x is the Euclidean distance, and τ = e 0 a/l, where e 0 is 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. is over the region V occupied by the body. However, e 0 and 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.. The stress of a reference point x in the representative element can be expressed as

3 Small-Scale Effect on the Static Deflection of a Clamped Graphene Sheet 05 x x x o β x x Figure : A representative element of graphene sheet. o Figure : A helical graphene sheet. x Taylor series σ i j x,x = σ i j 0,0 + σ i j x,x x =0 x + σ i j x,x x =0 x =0 x x =0 + σ i j x,x! x =0 x +... x =0 To take the average of σ i j x,x over the representative element, in terms of the symmetry of the representative element, we have σi j x,x = l σ i j 0,0 3 Where l is the length of C-C bond. Inversion of eq. 3 yields σ i j 0,0 = 0.08l σ i j x,x Where σ i j x,x is the nonlocal stress tensor.. Geometric equations A helical graphene sheet shown as Fig., x, x is the local coordinate system, and x,x is the global coordinate system. x and x are parallel to both sides of the graphene sheet,respectively. Geometric equations of the helical graphene sheet in the global coordinate are ε i j = zw,i j 5

4 06 Copyright 05 Tech Science Press CMC, vol.8, no., pp.03-7, 05 Where w is the static deflection of the graphene sheet, w,i j = w. i j According to the transformation relationship of strains, we have ε i j = l i j ε i j l i j Where } l i j = cos x i,x j l i j = cos x i,x j ε i j = zw,i j Nonlocal constitutive equation Based on Eq., the nonlocal constitutive equations of graphene are rewritten as σ 0.08l σ = E µ [ε + µε ] σ 0.08l σ = E µ [ε + µε ] 0.08l σ = E + µ ε Where E is the elastic modulus of graphene, and µ Poisson s ratio. Eq. 9 can be approximately expressed as σ = E µ l [ε + µε ] σ = E µ l σ = E l ε + µ [ε + µε ] 9 0 Substituting Eq. 6 into Eq. 0, we have σ = Ez [ µ l cos β + sinβ cosβ + sin β ] w w l i i l j + µl i j i l j j σ = Ez [ µ l sin β sinβ cosβ + cos β ]

5 Small-Scale Effect on the Static Deflection of a Clamped Graphene Sheet 07 σ = l i w w i l j + µl i j i j [ l Ez + µ l j + ] l i w i j In the global coordinate x,x, the stress components are l j σ i j = σ i j = l i jσ i j l i j. Physic equations The normal and shear stresses can be collected into bending moment M, M and torque M, respectively. h M = σ zdz h M = h h M = h h σ zdz σ zdz 3 The bending equilibrium equation of the plate on which a distribution force q is applied M + M + M Combination of Eqs. yields A w 6 w 6 +A 5 w +A + q = 0 + w 6 w 5 w +A 5 +A 3 Eh 3 Where D = µ A = 0.08l sin 6 β + cos 6 β 0.36l cos β sin β = q D A = l cosβ sin 3 β l cos 3 β sinβ A 3 = l sin 6 β + cos 6 β + 0.6l 3.76cos β sin β A = sin β.8sin β cos β + cos β A 5 =.5sin β.9sin β cos β +.5cos β + 6 w 5 6

6 08 Copyright 05 Tech Science Press CMC, vol.8, no., pp.03-7, 05 3 Numerical examples and discussions A clamped rectangle graphene sheet is shown as Fig. 3. x b / b / a / a / x Figure 3: A clamped rectangle graphene sheet. The boundary conditions of the clamped graphene sheet can be written as w x = a/,a/ = 0, w x = a/,a/ = 0 w x = b/,b/ = 0, w x = b/,b/ = 0 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 w = C m w m = x a x b C +C x +C 3 x m 7 Eq. 8 are consistent with the clamped boundary conditions of the graphene sheet. Galerkin weak form of Eq. 5 is given by a b [ A a b w 6 w 56 + A + A 3 w + A 5 Where m =,,3, w ] w m dx dx = 5 + A w a b a b 6 w 5 + w q D w mdx dx The approximate solution of Eq. 9 is given as following 9

7 Small-Scale Effect on the Static Deflection of a Clamped Graphene Sheet 09 w = 3 m= Where w = C m w m = x a x b C +C x x a w = x a w 3 = x a x b x b x b +C 3 x 0 x x 3 Substituting of Eqs. 3 into Eq. 9 has a b [ A a b w 6 w 56 + A 5 6 w +A w w + A = +A 5 w a = a b a b a b b q D ] x a x a [ A 6 +A 3 +A 5 w a = a b a b a b b q D x b x b + 6 w w ] x a x a [ A 6 +A 3 +A 5 w a b a b q D 6 w 55 + w dx dx dx dx + A w + A 5 x b x b + 6 w w ] x a x a 6 w 55 + w x 5 dx dx x dx dx + A w + A 5 x b x b 6 w 55 + w x 5 dx dx x dx dx

8 0 Copyright 05 Tech Science Press CMC, vol.8, no., pp.03-7, 05 C,C and C 3 can be obtained from the solution of Eq.. Classical static deflection of the graphene sheet can be obtained by substituting l = 0 into Eq. 9. To illustrate the small-scale effect on the static deflections of the graphene sheet, the small- scale effect factor η is defined as η = w cs w ns 5 Where w cs and w ns are, 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 = m, in-plane stiffness Eh = 360 J/m Sanchez-Portal, D, 999, Poisson s ratio µ = 0.6, the thickness of the graphene sheet h = 0.3 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 : Small-scale effect factor of the static deflection of the center point of the graphene sheet with the different helical angles and geometric sizes um a 0 um, um b 0 um. Fig. 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

9 Small-Scale Effect on the Static Deflection of a Clamped Graphene Sheet um a 0 um, um b 0 um. It can be found From Fig. 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, the small-scale effect is very obvious when the side length of the helical graphene sheet is smaller than 0 um. the smallscale 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 um a 0 um, um b 0 um. It can be seen from comparison of Fig. 5 and Fig. that the small-scale effect factor η changes periodically with change of helical angle, the change cycle is π/. 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 um a 0 um, um b 0 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 0 um a 00 um, 0 um b 00 um. It can be seen from Fig. 6 that the small-scale effect factor η is more and more close to 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 00um, the small-scale effect of the static deflection of the graphene sheet almost disappears.

10 Copyright 05 Tech Science Press CMC, vol.8, no., pp.03-7, 05 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 0 um a 00 um, 0 um b 00 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 0 um a 00 um, 0 um b 00 um.

11 Small-Scale Effect on the Static Deflection of a Clamped Graphene Sheet 3 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 0 um a 00 um, 0 um b 00 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/. Fig. 8 shows that comparison of small-scale effect factor of the static deflection of the graphene sheet with the different helical angles 0 um a 00 um, 0 um b 00 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 = π/6. 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 = π/6. Figure 8: Comparison of small-scale effect factor of the static deflection of the graphene sheet with the different helical angles. Fig. 9 shows that comparison of small-scale effect factor of the static deflection of the graphene sheet with the different helical angles. It can also be found from Fig. 9 that the small-scale effect factor η of the same geometrical dimension graphene sheet will increase when the helical angles change from β = π to β = 8π/6. 6 Combination of Fig. 8 with Fig. 9, it can be seen that the small-scale effect factor η of the same geometrical dimension helical graphene sheet has a periodic change of β = π/ with the change of the helical angle. Fig. 0 shows that comparison of the static deflection of the center point of the

12 Copyright 05 Tech Science Press CMC, vol.8, no., pp.03-7, 05 Figure 9: Comparison of small-scale effect factor of the static deflection of the graphene sheet with the different helical angles. Figure 0: 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. 0 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

13 Small-Scale Effect on the Static Deflection of a Clamped Graphene Sheet 5 Figure : Comparison of the static deflections of the center point of the graphene sheet with the different helical angles and geometrical dimensions. to β = π/6 or the length and width of the same helical angle graphene sheet increases. Fig. 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. 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/6 to β = π/6 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/ with the change of the helical angle. 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 defined as the ratio of the static deflections of

14 6 Copyright 05 Tech Science Press CMC, vol.8, no., pp.03-7, 05 the center point of the classical plate to that of the nonlocal plate. Numerical results show that. When the length and width of the graphene sheet are less than 0 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. For 0 um a 00 um, 0 um b 00 um, small-scale effect factor of the static deflection of the graphene sheet is more and more close to with 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 00um.When the helical angle of the graphene sheet changes from 0 to π/, 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/ with the change of the helical angle.. 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 = π/6 or when the length and width of the same helical angle graphene sheet increases. The static deflection has a periodic change of β = π/ with the change of the helical angle. Acknowledgement: This work is supported by National Natural Science Foundation of China under the Grant Number 3709 References Eringen, A. C. 983: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics. vol. 5, no. 9, pp Eringen, A. C.; Edelen, D. G. B. 97: On nonlocal elasticity. Int. J. Enging, no. 0, pp Fang, B.; Zhen, Y. X.; Zhang, C. P. 03: Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory. Appl. Math. Modeling, vol. 37, no. 3, pp

15 Small-Scale Effect on the Static Deflection of a Clamped Graphene Sheet 7 Iijima, S. 99: Helical microtubes of graphitic carbon. Nature. vol. 35, pp Liang, Y.; Han, Q. 0: Prediction of the nonlocal scaling parameter for graphene sheet. European Journal of Mechanics-A/Solids, vol. 5, pp Miandoab, E. M.; Yousefi-Koma, A.; Pishkenari, H.N. 05: Nonlocal and strain gradient based model for electrostatically actuated silicon nano-beams. Microsystem Technologies, vol., no., pp Reddy, J. N. 00: Nonlocal nonlinear formulations for bending of classicalal and shear deformation theories of beams and plates. Int. J. Eng. Sci., vol. 8, no., pp Sanchez-Portal, D.; Emilio A. 999: A study based on ab initio structural, elastic, and vibrational properties of carbon nanotubes, Phys. Rev. B, vol. 59, no. 8, pp Wang, L. 0: A modified nonlocal beam model for vibration and stability of nanotubes conveying fluid. Phys. E., vol., no., pp Xie, G. Q.; Han, X.; Long, S. Y. 006: Effect of small-scale scale on the radial Buckling pressure of a clamped multi-walled carbon nanotube. Smart Mater. and Struct., vol. 5, no., pp Xie, G. Q.; Han, X.; Long, S. Y. 007: Long. The effect of the small-scale on dispersion characteristics of the carbon nanotube. Int. J. Solid. Struct., vol., no., pp. 55. Xie, G. Q.; Long, S. Y. 006: Elastic vibration behaviors of carbon nanotubes based on micropolar mechanics. CMC: Computers, Materials & Continua, vol., no., pp. 0. Zhang, Y. Q.; Liu, G. R.; Wang, J. S. 00: Small scale effects on buckling of multi-walled carbon nanotubes under axial compression. Phys. rev. B, vol. 70, no. 0, pp Zhang, Y. Y.; Wang, C. M.; Challamel, N. 009: Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model. J. eng. Mech., vol. 36, no. 5, pp Zhao, P.; Shi, G. 0: Study of Poisson Ratios of Single-Walled Carbon Nanotubes based on an Improved Molecular Structural Mechanics Model. CMC: Computers, Materials & Continua, vol., no., pp

16