IOP PUBLISHING NANOTECHNOLOGY Nanotechnology 20 (2009) 065709 (11pp) doi:10.1088/0957-4484/20/6/065709 Effective elastic mechanical properties of single layer graphene sheets F Scarpa 1,4 , S Adhikari 2 and A Srikantha Phani 3 1 Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, UK 2 School of Engineering, University of Wales Swansea, UK 3 Department of Mechanical Engineering, The University of British Columbia, Vancouver, Canada E-mail: f.scarpa@bris.ac.uk and scarpa.fabrizio@gmail.com Received 28 October 2008 Published 15 January 2009 Online at stacks.iop.org/Nano/20/065709 Abstract The elastic moduli of single layer graphene sheet (SLGS) have been a subject of intensive research in recent years. Calculations of these effective properties range from molecular dynamic simulations to use of structural mechanical models. On the basis of mathematical models and calculation methods, several different results have been obtained and these are available in the literature. Existing mechanical models employ Euler–Bernoulli beams rigidly jointed to the lattice atoms. In this paper we propose truss-type analytical models and an approach based on cellular material mechanics theory to describe the in-plane linear elastic properties of the single layer graphene sheets. In the cellular material model, the C–C bonds are represented by equivalent mechanical beams having full stretching, hinging, bending and deep shear beam deformation mechanisms. Closed form expressions for Young’s modulus, the shear modulus and Poisson’s ratio for the graphene sheets are derived in terms of the equivalent mechanical C–C bond properties. The models presented provide not only quantitative information about the mechanical properties of SLGS, but also insight into the equivalent mechanical deformation mechanisms when the SLGS undergoes small strain uniaxial and pure shear loading. The analytical and numerical results from finite element simulations show good agreement with existing numerical values in the open literature. A peculiar marked auxetic behaviour for the C–C bonds is identified for single graphene sheets under pure shear loading. (Some figures in this article are in colour only in the electronic version) 1. Introduction Graphene sheets (GS) have Young’s modulus and thermal conductivity rivalling that of graphite (1.06 TPa and 3000 W m −1 K −1 respectively) [1, 2]. They may exist as single layered or multi-layer structures. It is possible to harness the multifunctional properties of graphene sheets and design novel class of advanced composites with superior mechanical and electric performance [1–3], as well as innovative strain sensors [5]. An approach to produce graphene–polymer composites by complete exfoliation of graphite and molecular- level dispersion of GS in a polymer host has been described in [4]. The latter work, from Stankovich et al, has fuelled 4 Address for correspondence: Department of Aerospace Engineering, University of Bristol, Queens Building, University Walk, Bristol BS8 1TR, UK. a growing interest into the mechanical determination and characterization of single layer graphene sheets (SLGS), although from the experimental point of view advances have been made in measuring magneto-transport properties [9], while experimental mechanical data are still confined to graphene layers only. The enhanced flexibility of GS, despite their high Young’s modulus, has been attributed to the change in curvature given by reversible elongation of sp 2 C–C bonds [6, 8, 49]. Vibrational properties of SLGS [10] or multi- layer graphene assemblies [7] have also been evaluated using analytical and finite element simulation methods. Molecular mechanistic modelling of single layer graphene sheets has been pursued by several authors. Simple lattice models with force constants derived from an assumed potential have been developed by Bacon and Nicholson [11] and Gillis [12]. Ab initio methods have been used by 0957-4484/09/065709+11$30.00 © 2009 IOP Publishing Ltd Printed in the UK 1