Stability of a single-layer graphene sheet with various edge conditions: a non-local plate model including interatomic potentials

Abstract

In this article, the biaxial buckling of a single layer graphene sheet is investigated. The model is established through the incorporation of an interatomic potential into non-local elastic plate theory to take into account the size effects and to circumvent the use of the Young’s modulus of a single layer graphene sheet since there is no accurate value of this property available in the literature. The model links the strain energy density induced in the continuum to Eringen’s non-local constitutive relations. By using the Galerkin method, explicit formulas for the critical buckling stresses of a single layer graphene sheet with arbitrary edge supports are derived from its static deflection due to a uniformly distributed load. The influences of the small size of the system and boundary conditions on the critical buckling load of the single layer graphene sheet are studied. It is found that the critical buckling load at large side lengths is almost independent of the type of boundary condition and compressive loading and is nearly immune to size effects. The analytical expressions provide a simple and quick way to evaluate accurate values of the critical buckling load. Indeed, the lack of complexity in the formulas allows a simple prediction of the value of the scale parameter that closely matches the one obtained using the complex molecular dynamics simulation technique.

Keywords

Monolayer graphene sheet buckling interatomic potential non-local elasticity Galerkin method

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References

Drexler

. Nanosystems: molecular machinery, manufacturing and computation. New York: John Wiley & Sons, 1992.

Martel

Schmidt

Shea

. Single- and multi-wall carbon nanotube field-effect transistors. Appl Phys Lett 1998; 73: 2447–2449.

Wong

Joselevich

Woolley

. Covalently functionalized nanotubes as nanometre-sized probes in chemistry and biology. Nature 1998; 394: 52–55.

Zhang

Wen

Liu

. Synthesis of vertically aligned carbon nanotubes on metal deposited quartz plates. Carbon 2002; 40: 1981–1989.

Liu

Zhang

. Multi-wall carbon nanotube as a new infrared detected material. Sensor Actuators A,Phys 2004; 116: 394–397.

Stankovich

Dikin

Dommett

GHB

. Graphene-based composite materials. Nature 2006; 442: 282–286.

Yan

Chen

Tay

. Transparent and flexible glucose biosensor via layer-by-layer assembly of multi-wall carbon nanotubes and glucose oxidase. Electrochem Commun 2007; 9: 1269–1275.

Ramanathan

Abdala

Stankovich

. Functionalized graphene sheets for polymer nanocomposites. Nat Nanotech 2008; 3: 327–331.

Chen

Liu

. Mechanics of hydrogen storage in carbon nanotubes. J Mech Phys Solids 2008; 56: 3224–3241.

10.

Zhao

Song

Ren

. A DNA nanomachine induced by single-walled carbon nanotubes on gold surface. Biomaterials 2009; 30: 1739–1745.

11.

Qin

Shen

. Facile attachment of magnetic nanoparticles to carbon nanotubes via robust linkages and its fabrication of magnetic nanocomposites. Compos Sci Technol 2009; 69: 427–431.

12.

Eringen

. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 1983; 54: 4703–4710.

13.

Eringen

. Nonlocal continuum field theories. New York: Springer, 2002.

14.

Peddieson

Buchanan

McNitt

. Application of nonlocal continuum models to nanotechnology. Int J Engng Sci 2003; 41: 305–312.

15.

Pradhan

Phadikar

. Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models. Phys Lett A 2009; 373: 1062–1069.

16.

Pradhan

Phadikar

. Nonlocal elasticity theory for vibration of nanoplates. J Sound Vib 2009; 325: 206–223.

17.

Murmu

Pradhan

. Buckling of biaxially compressed orthotropic plates at small scales. Mech Res Commun 2009; 36: 933–938.

18.

Pradhan

Murmu

. Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics. Comput Mater Sci 2009; 47: 268–274.

19.

Ansari

Sahmani

Arash

. Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys Lett A 2010; 375: 53–62.

20.

Babaei

Shahidi

. Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method. Arch Appl Mech 2011; 81: 1051–1062.

21.

Shen

Zhang

. Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Comput Mater Sci 2010; 48: 680–685.

22.

Ansari

Rajabiehfard

Arash

. Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets. Comput Mater Sci 2010; 49: 831–838.

23.

Pradhan

Kumar

. Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method. Comput Mater Sci 2010; 50: 239–245.

24.

Samaei

Abbasionb

Mirsayara

. Buckling analysis of a single-layer graphene sheet embedded in an elastic medium based on nonlocal Mindlin plate theory. Mech Res Commun 2011; 38: 481–485.

25.

Ansari

Arash

Rouhi

. Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity. Compos Struct 2011; 93: 2419–2429.

26.

Tersoff

. New empirical approach for the structure and energy of covalent systems. Phys Rev B 1988; 37: 6991–7000.

27.

Brenner

. Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys Rev B 1990; 42: 9458–9471.

28.

Born

Huang

. Dynamical theory of crystal lattices. London: Oxford University Press, 1954.

29.

Milstein

. Theoretical elastic behaviour of crystals at large strains. J Mater Sci 1980; 15: 1071–1084.

30.

Weiner

. Statistical mechanics of elasticity. New York: John Wiley & Sons, 1983.

31.

Tadmor

Smith

Bernstein

. Mixed finite element and atomistic formulation for complex crystals. Phys Rev B 1999; 59: 235–245.

32.

Arroyo

Belytschko

. An atomistic-based finite deformation membrane for single layer crystalline films. J Mech Phys Solids 2002; 50: 1941–1977.

33.

Zhang

Huang

Geubelle

. The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials. Int J Solids Struct 2002; 39: 3893–3906.

34.

Zhang

Jianga

Huang

. An atomistic-based continuum theory for carbon nanotubes: analysis of fracture nucleation. J Mech Phys Solids 2004; 52: 977–998.

35.

Huang

. Thickness of graphene and single-wall carbon nanotubes. Phys Rev B 2006; 74: 245413–246422.

36.

Hwang

Huang

. An atomistic-based finite-deformation shell theory for single-wall carbon nanotubes. J Mech Phys Solids 2008; 56: 279–292.

37.

Jones

. An approximate expression for the fundamental frequency of vibration of elastic plates. J Sound Vib 1975; 38: 503–504.

38.

Sundararajan

. An approximate solution for the fundamental frequency of plates. J Appl Mech 1978; 45: 936–939.

39.

Radhakrishnan

Nageswara Rao

Sarma

. On the relationship between the fundamental frequency and static deflections of thin elastic plates. J Sound Vib 1996; 196, 523–528.

40.

Nanorex Inc. NanoHive-1 v.1.2.0-b1, www.nanoengineer-1.com (2005, accessed, 2005).