Thermally Induced Asymmetric Buckling of Circular Monolayer Graphene

Abstract

The asymmetric buckling behaviors of circular monolayer graphene with clamped boundary condition subjected to temperature change are analytically studied based on the nonlocal elasticity theory, including the small length effect. The axisymmetrical and asymmetric critical buckling temperatures and mode shape of different order modes are obtained. According to the analysis, the asymmetric critical buckling temperature of monolayer graphene is larger than the axisymmetric one. The axisymmetrical and asymmetric critical buckling temperatures decrease with increasing nonlocal parameter. In addition, nodal diametrical lines and nodal circles can be found from the modal shapes. In order to avoid destruction of the sensors due to buckling of the structure, they can be placed at the nodal diametrical lines or nodal circles.

1. Introduction

Graphene has recently attracted significant research interest due to its potential applications in nanobiological devices and nanoelectromechanical systems such as single-electron transistor [1], actuators [2], and photodetectors [3]. Because graphene is only one atom layer thick, it is easily amenable to external influences, including thermal loading and mechanical deformation [4–8]. The external mechanical loading can affect its electronic structure and physical properties. For example, there were significant changes in graphene’s band structure for uniaxial strains of up to 15% [5]. The magnetic behavior of multilayer graphene can be affected by mechanical transverse deformation [7].

In the last years, continuum theory has received the attention of many researchers for analysis of nanostructures. This is because available experiments on nanoscale materials are difficult to perform, and molecular dynamic simulations are highly time-consuming. Accordingly, various size-dependent continuum theories such as strain gradient theory [9], couple stress theory [10], and nonlocal elasticity theory [11] have been proposed. Among these theories, nonlocal elasticity theory including the small-scale effect has often been used to study the buckling and vibration responses of nanostructures [12–16]. The theory was initiated by Eringen [17] in the early 1970s and stated that the stresses at a point depend on strains at all the points of the body.

This paper focuses on the buckling behavior of graphene. In recent years, some researchers have studied the buckling of graphene by using different methods. For example, Frank et al. [18] experimentally studied the compression buckling strain of a graphene sheet. Neek-Amal and Peeters [19] investigated the stability of circular monolayer graphene subjected to a radial load using molecular dynamics simulations. Natsuki et al. [20] studied the buckling properties of circular double-layered graphene sheets using plate theory and found that buckling stability is significantly affected by the buckling mode shapes.

In this paper, thermally induced asymmetric buckling of circular monolayer graphene is studied based on the nonlocal elasticity theory by using the analytical method. According to the analysis, the axisymmetrical and asymmetric buckling temperatures and mode shape for the graphene with various modes are obtained. In addition, nodal diametrical lines and nodal circles in the buckling mode shape are also investigated.

2. Physical Model and Mathematical Formulation

2.1. Governing Equation and Boundary Conditions

Consider that a circular graphene sheet is clamped on its edge, which is modeled as a clamped circular plate with the radius 𝑅 and thickness ℎ subjected to temperature change 𝑇 as depicted in Figure 1. Based on the nonlocal elasticity theory, the nonlocal constitutive relation of circular plate under the thermal load is given by [21] 𝐷∇⁴𝑤(𝑟,𝜃)+1−𝑒₀𝑎²∇²𝑁_𝑇∇²𝑤(𝑟,𝜃)=0, (1) where 𝑤 is the displacement along the thickness of the graphene. 𝑟 and 𝜃 are the radial and circumferential coordinates. ∇²=((𝜕²/𝜕𝑟²)+(1/𝑟)(𝜕/𝜕𝑟)+(1/𝑟²)(𝜕²/𝜕𝜃²)) is the Laplacian operator in polar coordinates. 𝑒₀𝑎 is the nonlocal parameter revealing the nanoscale size effect. 𝑁_𝑇 and 𝐷 are the thermal buckling load and flexural bending rigidity of graphene, respectively. They are 𝑁_𝑇=𝐸𝛼𝑇1−𝜇ℎ,𝐷=𝐸ℎ³121−𝜇², (2) where 𝐸, 𝜇, and 𝛼 are Young’s modulus, Poisson’s ratio, and thermal expansion coefficient of graphene, respectively.

Figure 1 

A clamped circular graphene sheet with the radius 𝑅 under temperature change 𝑇.

The corresponding boundary conditions are 𝑤(𝑅)(𝑅)=0,𝜕𝑤𝜕𝑟=0. (3) The nondimensional variables are introduced and defined as follows: 𝑟𝜉=𝑅,𝑤𝑤=𝑅,𝑒𝜀=⁰𝑎𝑅,𝑇^∗=12𝛼𝑇𝑅²1+𝜇ℎ², (4) where 𝜉, 𝑤, 𝜀, and 𝑇^∗ denote the nondimensional radius, outplane displacement, nonlocal parameter, and buckling temperature.

Using the dimensionless variables given by (4), the governing equation and associated boundary conditions can be simplified to the following dimensionless form: ∇²∇²𝑤+𝜆²𝑤=0,𝑤(1)𝜕=0,𝑤(1)𝜕𝜉=0, (5)(6) where ∇²=𝜕²𝜕𝜉²+𝜕+𝜕𝜉𝜕𝜉𝜉²𝜕²𝜃²,𝜆²=𝑇^∗1−𝜀²𝑇^∗. (7)(8)

2.2. Asymmetric Buckling Mode

It is well known that the normal vibration of an elastic linear system is harmonic. The displacement of monolayer graphene for a harmonic vibration can therefore be separated as follows: 𝑤(𝜉,𝜃)=𝑓(𝜉)cos𝑛𝜃,𝑛=0,1,2,3,…, (9) where 𝑛 is the number of nodal diametrical lines.

Substituting the relation of (9) into (5), we have 𝑑²𝑑𝜉²+𝑑−𝑛𝜉𝑑𝜉𝜉²²𝑑²𝑑𝜉²+𝑑+𝜆𝜉𝑑𝜉²−𝑛²𝜉²𝑓(𝜉)=0. (10) The general solution of the earlier equation should be expressed as 𝑓(𝜉)=𝐴₁𝜉^𝑛+𝐴₂𝜉^−𝑛+𝐴₃𝐽_𝑛(𝜆𝜉)+𝐴₄𝑌_𝑛(𝜆𝜉),𝑛=1,2,3, (11) where 𝐽_𝑛 and 𝑌_𝑛 are the Bessel functions of the first and second kinds of order 𝑛. 𝐴₁,…,𝐴₄ are arbitrary constants depending on the boundary conditions.

Substituting the boundary conditions given by (6) into (11), we can obtain the characteristic functions as 𝐽_𝑛+1𝜆_𝑚𝑛=0, (12) where 𝜆_𝑚𝑛 are the roots of the Bessel function of the first kind of order 𝑛+1 for mode 𝑚.

Once the parameter 𝜆_𝑚𝑛 is given, the following nondimensional buckling temperature can be obtained from (8): 𝑇^∗_𝑚𝑛=𝜆²_𝑚𝑛1+𝜀²𝜆²_𝑚𝑛. (13) In addition, the nondimensional buckling mode shape can be expressed as 𝑤_𝑚𝑛(𝜉,𝜃)=𝐴_𝑚𝑛𝐽_𝑛𝜆_𝑚𝑛𝜉cos𝑛𝜃. (14) There are 𝑛 nodal diametrical lines, which are straight lines with no curve and no displacement. In addition, there are 𝑚 nodal circles, which are no displacement. The radius of nodal circle 𝜌 can be solved by 𝐽_𝑛𝜆_𝑚𝑛𝜌=0. (15)

2.3. Axisymmetric Buckling Mode

For the axisymmetric buckling mode (i.e., 𝑛=0), using (10), the differential equation can be expressed as 𝑑²𝑑𝜉²+𝑑𝑑𝜉𝑑𝜉𝑑𝜉²²+𝑑𝜉𝑑𝜉+𝜆²𝑓(𝜉)=0. (16) The general solution can be written as 𝑓_𝑠(𝜉)=𝐵₁+𝐵₂ln𝜉+𝐵₃𝐽₀(𝜆𝜉)+𝐵₄𝑌₀(𝜆𝜉), (17) where 𝐽₀ and 𝑌₀ are the Bessel functions of the first and second kinds of order zero and 𝐵₁,…,𝐵₄ are the arbitrary constants.

Applying the boundary conditions, the characteristic function, 𝐽₁(𝜆_𝑚0)=0, can be obtained. The result is the same as that derived by Farajpour et al. [21].

Using (14)-(15) and setting 𝑛=0, the nondimensional buckling temperature, buckling shape, and radii of nodal circles of axisymmetric monolayer graphene can also be determined. It is noted that there are no nodal diametrical lines, but there are 𝑚 nodal circles for axisymmetric buckling mode.

3. Results and Discussion

To study the effect of small length scale on the buckling behaviors of circular monolayer graphene, critical buckling temperature and mode shape are analyzed for different mode numbers and nonlocal parameters. Figure 2 shows the nondimensional critical buckling temperature as a function of nonlocal parameters for asymmetric monolayer graphene with clamped boundary conditions with 𝑛=1 for the first five modes. The parameter value of 𝑒₀𝑎/𝑅=0 implies that the nonlocal effect is neglected. It can be seen that the effect of nonlocal parameter on the critical buckling temperature is significant, especially at higher-order modes. The critical buckling temperature decreases with increasing nonlocal parameter. This is because the internal interaction force increases as the nonlocal parameter increases.

Figure 2 

The critical buckling temperature as a function of nonlocal parameter for asymmetric monolayer graphene with 𝑛=1.

Figure 3 depicts the critical buckling temperature ratio for asymmetric (i.e., 𝑛=1) and axisymmetric (i.e., 𝑛=0) monolayer graphene at different modes. The critical buckling temperature of asymmetric monolayer graphene is larger than that of axisymmetric ones. The discrepancy is larger for a lower-order mode. Figures 4(a) and 4(b) illustrate the axisymmetric (i.e., 𝑛=0) buckling mode shape of circular monolayer graphene with 𝑒₀𝑎/𝑅=0.05 for 𝑚=1 and 𝑚=2, respectively. In practical applications, the lower buckling modes are easier to reach than the higher-order modes. Therefore, the parameters 𝑚=1 and 𝑚=2 are selected in the analysis. The nodal circles 𝑟=const can be seen for the figures. There are 𝑚 circular nodes for different values of 𝑚. According to the calculation, the buckling temperatures are 𝑇^∗₁₀=14.16 and 𝑇^∗₂₀=43.86 for 𝑚=1 and 𝑚=2, respectively.

Figure 3 

The critical buckling temperature ratio for asymmetric and axisymmetric monolayer graphene.

(a)

(b)

Figure 4 

Axisymmetric buckling mode shape of circular monolayer graphene with 𝑒0𝑎/𝑅=0.05 for (a) 𝑚=1 and (b) 𝑚=2.

Figure 5 shows the asymmetric buckling mode shape and nodal lines of circular monolayer graphene with 𝑒₀𝑎/𝑅=0.05 for different values of 𝑚 and 𝑛. The nodal lines are straight diametrical lines 𝜃=const and circles 𝑟=const. There are 𝑚 nodal circulars and 𝑛 straight diametrical lines for the case analyzed in the figure. The sensors used to measure modal data can be placed at the nodal diametrical lines and nodal circles in order to avoid destruction of the sensors due to structure buckling. The buckling temperatures of 𝑇^∗₁₁=24.73 for 𝑚=1 and 𝑛=1, 𝑇^∗₂₁=60.19 for 𝑚=2 and 𝑛=1, 𝑇^∗₁₂=36.95 for 𝑚=1 and 𝑛=2, and 𝑇^∗₂₂=76.9 for 𝑛=2 and 𝑚=2 are obtained from this calculation.

(a)

(b)

(c)

(d)

Figure 5 

Asymmetric buckling mode shape of circular monolayer graphene with 𝑒0𝑎/𝑅=0.05 for (a) 𝑚=1, 𝑛=1, (b) 𝑚=2, 𝑛=1, (c) 𝑚=1, 𝑛=2, and (d) 𝑚=2, 𝑛=2.

Figure 6 depicts the nondimensional radius of nodal circle of mode 1 as a function of nonlocal parameter. It can be seen that the radius of nodal circle obviously increases with increasing the nonlocal parameter. The nondimensional radii of nodal circles as a function of nonlocal parameter with 𝑚=2 are listed in Table 1. There are two nodal circles for the case of 𝑚=2. The radii of the two nodal circles increase with increasing the value of 𝑛 and nonlocal parameter. The large one can easily exceed the range of the graphene when the nonlocal parameter is too large. For the case of larger 𝑛 and nonlocal parameter, both circles may be out of the range.

Table 1 

Nondimensional radii of nodal circles as a function of nonlocal parameter with 𝑚=2.

Figure 6 

Nondimensional radius of nodal circle as a function of nonlocal parameter.

4. Conclusions

The effect of the small length scale on the critical buckling temperature and mode shape of circular monolayer graphene with clamped boundary condition was analyzed using the nonlocal elasticity theory. The results showed that the critical buckling temperature decreased with increasing nonlocal parameter. The critical buckling temperature of asymmetric monolayer graphene was larger than that of axisymmetric one. It can be observed that there were 𝑚 nodal circulars and 𝑛 straight diametrical lines for the asymmetric buckling of circular monolayer graphene. In addition, the radii of nodal circles obviously increased with increasing the nonlocal parameter.

Acknowledgment

The authors wish to thank the National Science Council of the Republic of China in Taiwan for providing financial support for this study under Projects NSC 101–2221-E-168-006 and NSC 101–2221-E-168 −009.