Nonlinear elastic behavior of graphene: Ab initio calculations to continuum description (original) (raw)
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Meccanica, 2019
In the present study, the in-plane elastic stiffness coefficients of graphene within the framework of first strain gradient theory are calculated on the basis of an accurate molecular mechanics model. To this end, a Wigner-Seitz primitive cell is adopted. Additionally, the first strain gradient theory for graphene with trigonal crystal system is formulated and the relation between elastic stiffness coefficients and molecular mechanics parameters are calculated. Thus, the ongoing research challenge on providing the accurate mechanical properties of graphene is addressed herein. Using results obtained, the in-plane free vibration of graphene is studied and a detailed numerical investigation is implemented.
Elastic in-plane properties of 2D linearized models of graphene
Mechanics of Materials, 2013
Graphene is a monolayer of carbon atoms packed into a two-dimensional honeycomb lattice. This allotrope can be considered as mother of all graphitic forms of carbon. The elastic in-plane properties of graphene are studied and various existing linearized models of its elastic deformations are critically re-examined. Problems related to modelling of graphene by nonlinear multi-body potentials of inter action are also discussed. It is shown that experimental results for small deformations can be well described by both the two-parametric molecular mechanics model developed by Gillis in 1984, while some popular models have serious flaws and often the results obtained using these models do not have physical meaning. It is argued that in order to study elastic constants of linearized models of graphene layers, it is very convenient to use the four parameter molecular mechanics model. The advantages of this approach is demonstrated by its application to the Tersoff and Brenn er nonlinear interaction potentials, and by its comparison with the Gillis two-parametric model.
Homogenized elastic properties of graphene for moderate deformations
Coupled systems mechanics, 2015
This paper presents a simple procedure of obtaining a substitute, homogenized mechanical response of single layer graphene sheet. The procedure is based on the judicious combination of molecular mechanics simulation results and homogenization method. Moreover, a series of virtual experiments are performed on the representative graphene lattice. Following these results, the constitutive model development is based on the well established continuum mechanics framework, that is, the non-linear membrane theory which includes the hyperelastic model in terms of principal stretches. A proof-of-concept and performance is shown on a simple model problem where the hyperelastic strain energy density function is chosen in polynomial form.
Nonlinear Elasticity of Monolayer Graphene
By combining continuum elasticity theory and tight-binding atomistic simulations, we work out the constitutive nonlinear stress-strain relation for graphene stretching elasticity and we calculate all the corresponding nonlinear elastic moduli. Present results represent a robust picture on elastic behavior and provide the proper interpretation of recent experiments. In particular, we discuss the physical meaning of the effective nonlinear elastic modulus there introduced and we predict its value in good agreement with available data. Finally, a hyperelastic softening behavior is observed and discussed, so determining the failure properties of graphene.
Equilibrium configuration and continuum elastic properties of finite sized graphene
Nanotechnology, 2006
This paper presents a continuum mechanics approach to modelling the elastic deformation of finite graphene sheets based on Brenner's potential. The potential energy of the graphene sheet is minimized for determining the equilibrium configuration. The four edges of the initially rectangular graphene sheet become curved at the equilibrium configuration. The curving of the sides is attributed to smaller coordination number for the atoms at the edges compared to that of the interior atoms. Considering two graphene models, with only two or all four edges constrained to be straight, the continuum Young's moduli of graphene are computed applying the Cauchy-Born rule. The computed elastic constants of the graphene sheet are found to conform to orthotropic material behaviour. The computed constants differ considerably depending on whether a minimized or unminimized configuration is used for computation.
Homogenized Elastic Properties of Graphene for Small Deformations
Materials, 2013
In this paper, we provide the quantification of the linear and non-linear elastic mechanical properties of graphene based upon the judicious combination of molecular mechanics simulation results and homogenization methods. We clarify the influence on computed results by the main model features, such as specimen size, chirality of microstructure, the effect of chosen boundary conditions (imposed displacement versus force) and the corresponding plane stress transformation. The proposed approach is capable of explaining the scatter of the results for computed stresses, energy and stiffness and provides the bounds on graphene elastic properties, which are quite important in modeling and simulation of the virtual experiments on graphene-based devices.
The Bending Behavior of Graphene: a Continuum Model Inferred from Molecular Dynamics
arXiv: Materials Science, 2017
We consider a discrete model of a graphene sheet with atomic interactions governed by a harmonic approximation of the 2nd-generation Brenner potential that depends on bond lengths, bond angles, and two types of dihedral angles. A continuum limit is then deduced that fully describes the bending behavior. In particular, we deduce for the first time an analytical expression of the Gaussian stiffness, a scarcely investigated parameter ruling the rippling of graphene, for which contradictory values have been proposed in the literature. We disclose the atomic-scale sources of both bending and Gaussian stiffnesses and provide for them quantitative evaluations.
Journal of the Mechanics and Physics of Solids
We consider a discrete model of a graphene sheet with atomic interactions governed by a harmonic approximation of the 2nd-generation Brenner potential that depends on bond lengths, bond angles, and two types of dihedral angles. A continuum limit is then deduced that fully describes the bending behavior. In particular, we deduce for the first time an analytical expression of the Gaussian stiffness, a scarcely investigated parameter ruling the rippling of graphene, for which contradictory values have been proposed in the literature. We disclose the atomic-scale sources of both bending and Gaussian stiffnesses and provide for them quantitative evaluations.
Numerical investigation of elastic mechanical properties of graphene structures
Materials and Design, 2010
The computation of the elastic mechanical properties of graphene sheets, nanoribbons and graphite flakes using spring based finite element models is the aim of this paper. Interatomic bonded interactions as well as van der Waals forces between carbon atoms are simulated via the use of appropriate spring elements expressing corresponding potential energies provided by molecular theory. Each layer is idealized as a spring-like structure with carbon atoms represented by nodes while interatomic forces are simulated by translational and torsional springs with linear behavior. The non-bonded van der Waals interactions among atoms which are responsible for keeping the graphene layers together are simulated with the Lennard-Jones potential using appropriate spring elements. Numerical results concerning the Young's modulus, shear modulus and Poisson's ratio for graphene structures are derived in terms of their chilarity, width, length and number of layers. The numerical results from finite element simulations show good agreement with existing numerical values in the open literature.
The influence of strain on the elastic constants of graphene
Carbon, 2017
Indentation experiments on graphene membranes pre-stressed by hydrostatic pressure show an increase in effective elastic modulus from 300 N/m in non pressurized membranes to 700 N/m for pre-strains above 0.5 %. This pronounced dependence of the stiffness of graphene with strain is attributed to its high anharmonicity and the great influence of out of plane corrugations of this atomic thick membrane in its mechanical properties. Our experimental findings imply that graphene´s measured stiffness is highly influenced by the presence of corrugations and that the in plane elastic modulus corresponding to atomic bond stretching is more akin to 700 N/m, instead of the commonly accepted 340 N/m. 1-5