An approximate mathematical treatise is proposed to improve the accuracy of multiscale models for nonlinear mechanics of two-dimensional (2D) nanomaterials by taking into account the contribution of dihedral energy term in the nonlinear constitutive model for the generalized deformation (three nonzero components of each strain and curvature tensors) of the corresponding continuum. Twelve dihedral angles per unit cell of graphene sheet are expressed as functions of strain and curvature tensor components. The proposed model is employed to study the bending modulus of graphene sheets under finite curvature. The atomic interactions are modeled using first- and second-generation reactive empirical bond order (REBO) potentials with the modifications in the former to include dihedral energy term for accurate prediction of bending stiffness coefficients. The constitutive law is obtained by coupling the atomistic and continuum deformations through Cauchy–Born rule. The present model will facilitate the investigations on the nonlinear mechanics of graphene sheets and carbon nanotubes (CNTs) with greater accuracy as compared to those reported in the literature without considering dihedral energy term in multiscale modeling.
Issue Section:
Research Papers
References
1.
Annamalai
, M.
, Mathew
, S.
, Jamali
, M.
, Zhan
, D.
, and Palaniapan
, M.
, 2012
, “Elastic and Nonlinear Response of Nanomechanical Graphene Devices
,” J. Micromech. Microeng.
, 22
(10
), p. 105024
.2.
Bunch
, J. S.
, Van der Zande
, A. M.
, Verbridge
, S. S.
, Frank
, I. W.
, Tanenbaum
, D. M.
, Parpia
, J. M.
, Craighead
, H. G.
, and McEuen
, P. L.
, 2007
, “Electromechanical Resonators From Graphene Sheets
,” Science
, 315
(5811
), pp. 490
–493
.3.
Lee
, C.
, Wei
, X.
, Kysar
, J. W.
, and Hone
, J.
, 2008
, “Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene
,” Science
, 321
(5887
), pp. 385
–388
.4.
Ke
, C. H.
, Pugno
, N.
, Peng
, B.
, and Espinosa
, H. D.
, 2005
, “Experiments and Modeling of Carbon Nanotube-Based NEMS Devices
,” J. Mech. Phys. Solids
, 53
(6
), pp. 1314
–1333
.5.
Treacy
, M. M. J.
, Ebbesen
, T. W.
, and Gibson
, J. M.
, 1996
, “Exceptionally High Young's Modulus Observed for Individual Carbon Nanotubes
,” Nature
, 381
(6584
), pp. 678
–680
.6.
Krishnan
, A.
, Dujardin
, E.
, Ebbesen
, T. W.
, Yianilos
, P. N.
, and Treacy
, M. M. J.
, 1998
, “Young's Modulus of Single Walled Nanotubes
,” Phys. Rev. B
, 58
(20
), p. 14013
.7.
Wong
, E. W.
, Sheehan
, P. E.
, and Lieber
, C. M.
, 1997
, “Nanobeam Mechanics: Elasticity, Strength, and Toughness of Nanorods and Nanotubes
,” Science
, 277
(5334
), pp. 1971
–1975
.8.
Salvetat
, J. P.
, Briggs
, G. A. D.
, Bonard
, J. M.
, Bacsa
, R. R.
, Kulik
, A. J.
, Stockli
, T.
, Burnham
, N. A.
, and Forro
, L.
, 1999
, “Elastic and Shear Moduli of Single Walled Carbon Nanotube Ropes
,” Phys. Rev. Lett.
, 82
(5
), pp. 944
–947
.9.
Klintenberg
, M.
, Lebegue
, S.
, Ortiz
, C.
, Sanyal
, B.
, Fransson
, J.
, and Eriksson
, O.
, 2009
, “Evolving Properties of Two-Dimensional Materials: From Graphene to Graphite
,” J. Phys.: Condens. Matter
, 21
(33), p. 335502
.10.
Kudin
, K. N.
, Scuseria
, G. E.
, and Yakobson
, B. I.
, 2001
, “C2F, BN, and C Nanoshell Elasticity From Ab Initio Computations
,” Phys. Rev. B
, 64
(23
), p. 235406
.11.
Liu
, F.
, Ming
, P.
, and Li
, J.
, 2007
, “Ab Initio Calculation of Ideal Strength and Phonon Instability of Graphene Under Tension
,” Phys. Rev. B
, 76
(6
), p. 064120
.12.
Reich
, S.
, Thomsen
, C.
, and Ordejon
, P.
, 2002
, “Elastic Properties of Carbon Nanotubes Under Hydrostatic Pressure
,” Phys. Rev. B
, 65
(15
), p. 153407
.13.
Sanchez-Portal
, D.
, Emilio
, A.
, Jose
, M. S.
, Angel
, R.
, and Pablo
, O.
, 1999
, “Ab Initio Structural, Elastic, and Vibrational Properties of Carbon Nanotubes
,” Phys. Rev. B
, 59
(19), pp. 12678
–12688
.14.
Goze
, C.
, Vaccarini
, L.
, Henrard
, L.
, Bernier
, P.
, Hemandez
, E.
, and Rubio
, A.
, 1999
, “Elastic and Mechanical Properties of Carbon Nanotubes
,” Synth. Met.
, 103
(1–3
), pp. 2500
–2501
.15.
Hernandez
, E.
, Goze
, C.
, Bernier
, P.
, and Rubio
, A.
, 1998
, “Elastic Properties of C and BxCyNz Composite Nanotubes
,” Phys. Rev. Lett.
, 80
(20), pp. 4502
–4505
.16.
Molina
, J. M.
, Savinsky
, S. S.
, and Khokhriakov
, N. V.
, 1996
, “A Tight‐Binding Model for Calculations of Structures and Properties of Graphitic Nanotubes
,” J. Chem. Phys.
, 104
(12
), pp. 4652
–4656
.17.
Robertson
, D. H.
, Brenner
, D. W.
, and Mintmire
, J. W.
, 1992
, “Energetics of Nanoscale Graphitic Tubules
,” Phys. Rev. B
, 45
(21
), pp. 12592
–12595
.18.
Yakobson
, B. I.
, Brabec
, C. J.
, and Bernholc
, J.
, 1996
, “Nanomechanics of Carbon Tubes: Instabilities Beyond Linear Response
,” Phys. Rev. Lett.
, 76
(14
), pp. 2511
–2514
.19.
Cornwell
, C. F.
, and Wille
, L. T.
, 1997
, “Elastic Properties of Single Walled Carbon Nanotubes in Compression
,” Solid State Commun.
, 101
(8
), pp. 555
–558
.20.
Lu
, J. P.
, 1997
, “Elastic Properties of Single and Multilayered Nanotubes
,” J. Phys. Chem. Solids
, 58
(11
), pp. 1649
–1652
.21.
Lordi
, V.
, and Yao
, N.
, 1998
, “Radial Compression and Controlled Cutting of Carbon Nanotubes
,” J. Chem. Phys.
, 109
(6
), pp. 2509
–2512
.22.
Yao
, N.
, and Lordi
, V.
, 1998
, “Young's Modulus of Single Walled Carbon Nanotubes
,” J. Appl. Phys.
, 84
(4
), pp. 1939
–1943
.23.
Liew
, K. M.
, He
, X. Q.
, and Wong
, C. H.
, 2004
, “On the Study of Elastic and Plastic Properties of Multi Walled Carbon Nanotubes Under Axial Tension Using Molecular Dynamics Simulation
,” Acta Mater.
, 52
(9), pp. 2521
–2527
.24.
Reddy
, C. D.
, Rajendran
, S.
, and Liew
, K. M.
, 2006
, “Equilibrium Configuration and Continuum Elastic Properties of Finite Sized Graphene
,” Nanotechnol.
, 17
(3
), pp. 864
–870
.25.
Sears
, A.
, and Batra
, R. C.
, 2004
, “Macroscopic Properties of Carbon Nanotubes From Molecular Mechanics Simulations
,” Phys. Rev. B
, 69
(23), p. 235406
.26.
Gupta
, S. S.
, and Batra
, R. C.
, 2008
, “Continuum Structures Equivalent in Normal Mode Vibrations to Single Walled Carbon Nanotubes
,” Comput. Mater. Sci.
, 43
(4
), pp. 715
–723
.27.
Gupta
, S. S.
, and Batra
, R. C.
, 2010
, “Elastic Properties and Frequencies of Free Vibrations of Single Layer Graphene Sheets
,” J. Comput. Theor. Nanosci.
, 7
(10
), pp. 2151–2164.28.
Jiang
, H.
, Zhang
, P.
, Liu
, B.
, Huang
, Y.
, Geubelle
, P. H.
, Gao
, H.
, and Hwang
, K. C.
, 2003
, “The Effect of Nanotube Radius on the Constitutive Model for Carbon Nanotubes
,” Comput. Mater. Sci.
, 28
(3–4
), pp. 429
–442
.29.
Zhang
, P.
, Huang
, Y.
, Geubelle
, P. H.
, Klein
, P. A.
, and Hwang
, K. C.
, 2002
, “The Elastic Modulus of Single Wall Carbon Nanotubes: A Continuum Analysis Incorporating Interatomic Potentials
,” Int. J. Solids Struct.
, 39
(13–14
), pp. 3893
–3906
.30.
Arroyo
, M.
, and Belytschko
, T.
, 2004
, “Finite Crystal Elasticity of Carbon Nanotubes Based on the Exponential Cauchy–Born Rule
,” Phys. Rev. B
, 69
(11), p. 115415
.31.
Guo
, X.
, Wang
, J. B.
, and Zhang
, H. W.
, 2006
, “Mechanical Properties of Single Walled Carbon Nanotubes Based on Higher Order Cauchy–Born Rule
,” Int. J. Solids Struct.
, 43
(5
), pp. 1276
–1290
.32.
Wang
, J. B.
, Guo
, X.
, Zhang
, H. W.
, Wang
, L.
, and Liao
, J. B.
, 2006
, “Energy and Mechanical Properties of Single Walled Carbon Nanotubes Predicted Using the Higher Order Cauchy–Born Rule
,” Phys. Rev. B
, 73
(11), p. 115428
.33.
Huang
, Y.
, Wu
, J.
, and Hwang
, K. C.
, 2006
, “Thickness of Graphene and Single Wall Carbon Nanotubes
,” Phys. Rev. B
, 74
(24
), p. 245413
.34.
Lu
, Q.
, Arroyo
, M.
, and Huang
, R.
, 2009
, “Elastic Bending Modulus of Monolayer Graphene
,” J. Phys. D: Appl. Phys.
, 42
(10
), p. 102002
.35.
Favata
, A.
, Micheletti
, A.
, Podio-Guidugli
, P.
, and Pugno
, N. M.
, 2017
, “How Graphene Flexes and Stretches Under Concomitant Bending Couples and Tractions
,” Meccanica
, 52
(7
), pp. 1601
–1624
.36.
Favata
, A.
, Micheletti
, A.
, Podio-Guidugli
, P.
, and Pugno
, N. M.
, 2016
, “Geometry and Self-Stress of Single-Wall Carbon Nanotubes and Graphene Via a Discrete Model Based on a 2nd-Generation REBO Potential
,” J. Elasticity
, 125
(1
), pp. 1
–37
.37.
Brenner
, D. W.
, 1990
, “Empirical Potential for Hydrocarbons for Use in Simulating the Chemical Vapor Deposition of Diamond Films
,” Phys. Rev. B
, 42
(15
), pp. 9458
–9471
.38.
Brenner
, D. W.
, Shenderova
, O. A.
, Harrison
, J. A.
, Stuart
, S. J.
, Ni
, B.
, and Sinnott
, S. B.
, 2002
, “A Second-Generation Reactive Empirical Bond Order (REBO) Potential Energy Expression Hydrocarbons
,” J. Phys.: Condens. Matter
, 14
(4
), pp. 783
–802
.39.
Tersoff
, J.
, 1988
, “New Empirical Approach for the Structure and Energy of Covalent System
,” Phys. Rev. B
, 37
(12
), pp. 6991
–7000
.40.
Singh
, S.
, and Patel
, B. P.
, 2015
, “Nonlinear Elastic Properties of Graphene Sheet Under Finite Deformation
,” Compos. Struct.
, 119
, pp. 412
–421
.41.
Jiang
, H.
, Huang
, Y.
, and Hwang
, K. C.
, 2005
, “A Comparison of Different Interatomic Potentials: Radius Effect of Single Wall Carbon Nanotubes
,” Symposium on Mechanical Behaviour and Micro Mechanics of Nanostructured Materials
, Beijing, China, June 27–30, pp. 121
–133
.42.
Singh
, S.
, and Patel
, B. P.
, 2018
, “Nonlinear Elastic Properties of Graphene Sheet Using MM3 Potential Under Finite Deformation
,” Composites, Part B
, 136
, pp. 81
–91
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