Vol.:(0123456789) 1 3 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:3 https://doi.org/10.1007/s40430-018-1499-4 TECHNICAL PAPER Determination of rigidities, stifness coefcients and elastic constants of multi‑layer graphene sheets by an asymptotic homogenization method Masoud Tahani 1  · Sobhan Safarian 2 Received: 30 May 2017 / Accepted: 15 November 2018 © The Brazilian Society of Mechanical Sciences and Engineering 2018 Abstract This work is concerned with the analysis of bending and torsional rigidities of multi-layer graphene sheets (MLGSs) based on the Kalamkarov’s general asymptotic homogenization composite shell model. Also, the efective stifness coefcients and elastic constants of MLGSs are estimated with this analytical method. The unit cell with both in-plane and out-of-plane interactions is assumed in this model. A MLGS as a homogeneous honeycomb network sheet with the periodic hexagonal unit cell is considered here in which the layers are held together by diferent densities of van der Waals interactions. The stifness coefcients, elastic constants and rigidities of MLGS are found by considering diferent densities of the van der Waals forces and diferent number of layers. The results show good agreements in comparison with other experiments and numerical solutions. It is found that the homogenization method gives the ability to create promise analytical approach that can be used for other nanostructures. Keywords Asymptotic homogenization method · Multi-layer graphene sheet · Bending and torsional rigidities · Stifness coefcients · Elastic constants 1 Introduction The evolutionary trend in novel nanomaterials can cause the new felds of research to move forward. Such materials are single-/multi-layer graphene sheets (SLGS/MLGS) which have attracted many researchers because of their outstand- ing behaviors [1]. Limitless improvement and widely use of graphene sheets in nanotechnology motivated researchers to develop new methods for the analysis of physical and mechanical properties of the new material. In modeling of nanostructures, studies on the mechanical properties have become more well-known and advanced experimental methods involved in many technological challenges in manipulating nanometer-sized objects [2]. Among diferent properties of graphene, its role in pro- ducing nanocomposites with special properties could be noted with care. High tensile strength with fexibility and low density are some of its outstanding mechanical proper- ties that make them unique [3]. Graphene could be consid- ered as a single-layer or multi-layer sheet. Experiments show that MLGS synthesis is much easier and more efcient than SLGS. Applying SLGS needs some post-processing and purifcations that make it non-industrial. So many types of this nanostructure in nature and synthesis outputs are chiefy multi-layer graphene sheet. The experimental measurement of mechanical properties of graphene-based materials is a troublesome task, and there are few matching reported works in the literature. Often, the properties of nanostructures have been determined by using atomistic simulations and numerical methods. Furthermore, because of the nonlinear behavior of both covalent bonds and van der Waals (vdWs) interactions, atomistic simula- tions, especially for MLGS, were limited [4–9]. Tsai and Tu [4] used molecular dynamics (MD) to estimate the mechani- cal properties of graphite in the forms of single graphene Technical Editor: Paulo de Tarso Rocha de Mendonça, Ph.D. * Masoud Tahani mtahani@um.ac.ir 1 Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran 2 Department of Mechanical Engineering, Faculty of Engineering, Khayyam University, Mashhad, Iran