An atomistic-based continuum approach for calculation of elastic properties of single-layered graphene sheet Khalid I. Alzebdeh n Department of Mechanical and Industrial Engineering, College of Engineering, Sultan Qaboos University, P.O. Box 33, Al-Khod 123, Oman article info Article history: Received 4 July 2013 Received in revised form 10 September 2013 Accepted 16 September 2013 by Y.E. Lozovik Available online 25 September 2013 Keywords: A. Graphene sheet C. Nanostructure C. Atomistic-continuum model D. Elastic properties abstract The elastic deformation of a single-layer nanostructured graphene sheet is investigated using an atomistic-based continuum approach. This is achieved by equating the stored energy in a representative unit cell for a graphene sheet at atomistic scale to the strain energy of an equivalent continuum medium under prescribed boundary conditions. Proper displacement-controlled (essential) boundary conditions which generate a uniform strain field in the unit cell model are applied to calculate directly one elastic modulus at a time. Three atomistic finite element models are adopted with an assumption that the force interaction among carbon atoms can be modeled by either spring-like or beam elements. Thus, elastic moduli for graphene structure are determined based on the proposed modeling approach. Then, effective Young's modulus and Poisson's ratio are extracted from the set of calculated elastic moduli. Results of Young's modulus obtained by employing the different atomistic models show a good agreement with the published theoretical and numerical predictions. However, Poisson's ratio exhibits sensitivity to the considered atomistic model. This observation is supported by a significant variation in estimates as can be found in the literature. Furthermore, isotropic behavior of in-plane graphene sheets was validated based on current modeling. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction Recently, nanostructured graphene sheets have captured the attention of many researchers. This can be attributed to their remarkable mechanical properties and cheap method of produc- tion as presented by Stankovich et al. [1]. In addition, character- ization of behavior of graphene facilities better understanding of other fundamental nano-materials like Carbon nanotubes (CNTs) which are viewed as a deformed graphite sheets. The complexity and high expense of investigating the mechanical behavior of graphene sheets via experiments stimulated the use of numerical simulation as proven tool capable of modeling nanostruc- tures with different dimensions. In this context, equivalent continuum-structural mechanics has been widely used to character- ize the mechanical behavior of nanostructured materials. In this approach, typical elements of structural mechanics such as rods, beams and shells are used to simulate the static and dynamic behavior of monolayer graphene. The mechanical properties of such structural elements are derived from the equivalence between steric potential of the carbon–carbon (C–C) bonds and mechanical strain energies associated with tension, torsion and bending related to the mechanical elements simulating the bonds themselves. A truss model was proposed by Odegard et al. [2], wherein rods of different degrees of stiffness represent the stretching and in-plane bending capabilities of the C–C bonds. Li and Chou [3] proposed an equivalent structural beam capable of modeling interatomic forces of the carbon covalent bonds. They adopted a molecular structural mechanics approach to compute effective elastic constants of carbon nanotubes. Meo and Rossi [4] developed a finite element model based on the use of nonlinear central spring and linear torsional spring elements to represent the modified Morse potential when simulating graphene. Cho et al. [5] carried out a molecular structural analysis to predict the elastic constants of graphite. The in-plane properties of graphite were derived by considering a single-layer graphene sheet subjected to an in-plane loading. Based on atomistic finite element approach, Shakhaee-Pour [6] investigated the elastic behavior of single-layer graphene sheets. By employing an equivalent structural beam, the elastic constants of graphene were calculated. Scrape et al. [7] proposed a truss-type model in conjunction with cellular material mechanics theory to describe the in-plane elastic properties of single-layer graphene sheets. Analytically, some researchers [8] investigated Young's modulus of graphene and CNTs based on nanoscale continuum modeling. They employed frame elements to simulate C–C bonds for which they obtained a closed form solution. Several others utilized atomistic finite elements to simulate graphene sheet using linear interatomic potential functions for bonds. Atomistic-based finite element models have been used to analyze graphene sheets in many Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ssc Solid State Communications 0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.09.017 n Corresponding author. Tel.: þ968 24142556; fax: þ968 24141316. E-mail address: alzebdeh@squ.edu.om Solid State Communications 177 (2014) 25–28