IJE TRANSACTIONS A: Basics Vol. 28, No. 10, (October 2015) 1533-1542 Please cite this article as: S. Dastjerdi, M. Jabbarzadeh, Symmetrical A Non-linear Static Equivalent Model for Multi-layer Annular/Circular Graphene Sheet Based on Non-local Elasticity Theory Considering Third Order Shear Deformation Theory (TSDT) in Thermal Environment, International Journal of Engineering (IJE), TRANSACTIONS A: Basics Vol. 28, No. 10, (October 2015) 1533-1542 International Journal of Engineering Journal Homepage: www.ije.ir A Non-linear Static Equivalent Model for Multi-layer Annular/Circular Graphene Sheet Based on Non-local Elasticity Theory Considering Third Order Shear Deformation Theory in Thermal Environment S. Dastjerdi *, M. Jabbarzadeh Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran PAPER INFO Paper history: Received 15 August 2015 Received in revised form 29 September 2015 Accepted 16 October 2015 Keywords: Single and Multi-layer Graphene Sheet Non-local Elasticity Theory of Eringen Differential Quadrature Method (DQM) Semi-analytical Polynomial Method (SAPM) Winkler-pasternak Elastic Foundation Thermal Environment ABSTRACT In this paper, it is tried to find an approximate single layer equivalent for multi-layer graphene sheets based on third order non-local elasticity theory. The plates are embedded in two parameter Winkler- Pasternak elastic foundation, and also the thermal effects are considered. A uniform transverse load is imposed on the plates. Applying the non-local theory of Eringen based on third order shear deformation theory and considering the van der Waals interaction between the layers, the governing equations are derived for a multi-layer graphene sheet. The governing equations for single layer graphene sheet are obtained by eliminating the van der Waals interaction. In this study, two different methods are applied to solve the governing equations. First, the results are obtained applying the differential quadrature method (DQM), which is a numerical method, and then a new semi-analytical polynomial method (SAPM) is presented. The results from DQM and SAPM are compared and it is concluded that the SAPM results are satisfactorily accurate in comparison with DQM. Since analyzing a multi-layer graphene sheet needs a time-consuming computational process, it is investigated to find an appropriate thickness for a single layer sheet to equalize the maximum deflections of multi-layer and single layer sheets. It is concluded that by considering a constant value of the van der Waal interaction between the layers, the maximum deflections of multi and single layer sheets are equal in a specific thickness of the single layer sheet. doi: 10.5829/idosi.ije.2015.28.10a.18 1. INTRODUCTION 1 Nowadays, nano structures are applied widely in areas such as nanotubes, nanobeams and nanoplates. The graphene sheets are kind of nanomaterials which are formed in hexagonal shape by covalent bonds between carbon atoms. Special properties of graphene sheets such as high strength, low weight to area ratio, unique and extraordinary electrical properties attracted many researchers to consider this topic as their major activities [1-3]. The bending strength of graphene sheet is low, so using multi-layers of graphene sheets improves this weakness. In order to make multi-layers of graphene sheet, several single layers of graphene are 1 *Corresponding Author’s Email dastjerdi_shahriar@yahoo.com (Sh. Dastjerdi) set on each other by weak van der Waals bond between the surface atoms [4]. There are different methods to analyze the nanostructures [5]. In addition to experimental methods, there are atomic modelling [6], combination of atomic modelling and continuum mechanics [7] and continuum mechanics [8]. Since controlling experimental and atomic modelling is difficult and computations are expensive, consequently, the continuum mechanics method is used by many researchers because of convenience in formulations and acceptable results in comparison with two other methods [9]. The continuum mechanics method is categorized in three different methods: 1- couple stress theory [10], 2-modified strain gradient theory [11], and 3-The Eringen non-local elasticity theory [12]. The Eringen non-local elasticity theory is widely used to analyze the mechanical