Sarp Adali . VARIATIONAL PRINCIPLES FOR NONLOCAL CONTINUUM MODEL OF ORTHOTROPIC GRAPHENE SHEETS EMBEDDED IN AN ELASTIC MEDIUM[J]. Acta mathematica scientia, Series B, 2012 , 32(1) : 325 -338 . DOI: 10.1016/S0252-9602(12)60020-4

[1] Geim A K, Novoselov K S. The rise of graphene. Nature Materials, 2007, 6(3): 183–191

[2] Reddy C D, Rajendran S, Liew K M. Equilibrium configuration and continuum elastic properties of finite
sized graphene. Nanotechnology, 2006, 17(3): 864–870

[3] Poot M,van der Zant H S J. Nanomechanical properties of few-layergraphene membranes. Applied Physics
Letters, 2008, 92(6): 063111

[4] Cranford S W, Buehler M J. Mechanical properties of graphyne. Carbon, 2011, 49(13): 4111–4121

[5] Stankovich S, Dikin D A, Dommett G H B, Kohlhaas K M, Zimney E J, Stach E A, Piner R D, Nguyen S T, Ruoff R S. Graphene-based composite materials. Nature, 2006, 442: 282–286

[6] Montazeri A, Rafii-Tabar H. Multiscale modeling of graphene- and nanotube-based reinforced polymer
nanocomposites. Physics Letters A, 2011, 375(45): 4034–4040

[7] Bunch J S, van der Zande A M, Verbridge S S, Frank I W, Tanenbaum D M, Parpia J M, Craighead H G, McEuen P L. Electromechanical resonators from graphene sheets. Science, 2007, 315: 490–493

[8] Sun T, Wang Z L, Shi Z J, Ran G Z, Xu W J, Wang Z Y, Li Y Z, Dai L, Qin G G. Multilayered graphene used as anode of organic light emitting devices. Applied Physics Letters, 2010, 96(13): 133301

[9] Yuan C, Hou L, Yang L, Fan C, Li D, Li J, Shen L, Zhang F, Zhang X. Interface-hydrothermal synthesis
of Sn3S4/graphene sheet composites and their application in electrochemical capacitors. Material Letters,
2011, 65(2): 374–377

[10] Arsat R, Breedon M, Shafiei M, Spizziri P G, Gilje S, Kaner R B, Kalantar-Zadeh K, Wlodarski W.
Graphene-like nano-sheets for surface acoustic wave gas sensor applications. Chemical Physics Letters,
2009, 467(4–6): 344–347

[11] Lian P, Zhu X, Liang S, Li Z, Yang W, Wang H. Large reversible capacity of high quality graphene sheets
as an anode material for lithium-ion batteries. Electrochimica Acta, 2010, 55(12): 3909–3914

[12] Mishra A K, Ramaprabhu S. Functionalized graphene sheets for arsenic removal and desalination of sea
water. Desalination, 2011, 282: 39–45

[13] Choi S M, Seo M H, Kim H J, Kim W B. Synthesis of surface-functionalized graphene nanosheets with high Pt-loadings and their applications to methanol electrooxidation. Carbon, 2011, 49(3): 904–909

[14] Yang M, Javadi A, Gong S. Sensitive electrochemical immuno-sensor for the detection of cancer biomarker
using quantum dot functionalized graphene sheets as labels. Sensors and Actuators B: Chemical, 2011, 155(1): 357–360

[15] Feng L, Chen Y, Ren J, Qu X. A graphene functionalized electrochemical apta-sensor forselective label-free
detection of cancer cells. Biomaterials, 2011, 32(11): 2930–2937

[16] Soldano C, Mahmood A, Dujardin E. Production, properties and potential of graphene. Carbon, 2010, 48(8): 2127–2150

[17] Terrones M, Botello-M′endez A R, Campos-Delgado J, L′opez-Ur′?as F, Vega-Cant′u Y I, Rodr′?guez-Mac′?as F J, El′?as A L, Mu?noz-Sandoval E, Cano-M′arquez A G, Charlier J-C, Terrones H. Graphene and graphite nanoribbons: Morphology, properties, synthesis, defects and applications. Nano Today, 2010, 5(4): 351–372

[18] He L H, Lim C W, Wu B S. A continuum model for size-dependent deformation of elastic films of nano-scale
thickness. Int J Solids Struct, 2004, 41: 847–857

[19] Kitipornchai S, He X Q, Liew K M. Continuum model for the vibration of multilayered graphene sheets. Phys Reviews B, 2005, 72: 075443

[20] Arash B, Wang Q. Discrete homogenization in graphene sheet modeling. J Elasticity, 2006, 84(1): 33–68

[21] Hemmasizade A, Mahzoon M, Hadi E, Khandan R. A method for developing the equivalent continuum
model of a single layer graphene sheet. Thin Solid Films, 2008, 516: 7636–7640

[22] Arash B, Wang Q, Caillerie D, Mourad A, Raoult A. A review on the application of nonlocal elastic models
in modeling of carbon nanotubes and graphenes. Comp Mater Sci, 2012, 51(1): 303–313

[23] Chang T, Gao H. Size-dependent elastic properties of a single-walled carbon nanotube via a molecular
mechanics model. J Mechanics and Physics of Solids, 2003, 51: 1059–1074

[24] Sun C T, Zhang H T. Size-dependent elastic moduli of platelike nanomaterials. J Appl Phys, 2003, 93: 1212–1218

[25] Ni Z, Bu H, Zou M, Yi H, Bi K, Chen Y. Anisotropic mechanical properties of graphene sheets from
molecular dynamics. Physica B: Condensed Matter, 2010, 405(5): 1301–1306

[26] Edelen D G B, Laws N. On the thermodynamics of systems with nonlocality. Arch Rational Mech Anal, 1971, 43: 24–35

[27] Eringen A C. Linear theory of nonlocal elasticity and dispersion of plane waves. Int J Engineering Science,
1972, 10: 425–435

[28] Eringen AC. Nonlocal polar elastic continua. Int J Engineering Science, 1972, 10: 1–16

[29] Eringen A C. Nonlocal Continuum Field Theories. New York: Springer, 2002

[30] Murmu T, Pradhan S C. Vibration analysis of nano-single-layered graphene sheets embedded in elastic
medium based on nonlocal elasticity theory. J Appl Phys, 2009, 105: 064319

[31] Shen L, Shen H-S, Zhang C-L. Nonlocal plate model for nonlinear vibration of single layer graphene sheets
in thermal environments. Comp Mater Sci, 2010, 48: 680

[32] Narendar S, Gopalakrishnan S. Strong nonlocalization induced by small scale parameter on terahertz
flexural wave dispersion characteristics of a monolayer graphene. Physica E, 2010, 43: 423–430

[33] He X Q, Kitipornchai S, Liew K M. Resonance analysis of multi-layered graphene sheets used as nanoscale
resonators. Nanotechnology, 2005, 16: 2086–2091

[34] Behfar K, Naghdabadi R. Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in
an elastic medium. Composites Science and Technology, 2005, 65: 1159–1164

[35] Liew K M, He X Q, Kitipornchai S. Predicting nano vibration of multi-layered graphene sheets embedded
in an elastic matrix. Acta Mater, 2006, 54: 4229–4236

[36] Jomehzadeh E, Saidi A R. A study on large amplitude vibration of multilayered graphene sheets. Com-
putational Materials Science, 2011, 50: 1043–1051

[37] Shi J-X, Ni Q-Q, Lei X-W, Natsuki T. Nonlocal elasticity theory for the buckling of double-layer graphene
nanoribbons based on a continuum model. Computational Materials Science, 2011, 50(11): 3085–3090

[38] Pradhan S C, Phadikar J K. Scale e?ect and buckling analysis of multilayered graphene sheets based on
nonlocal continuum mechanics. J Computational Theoretical Nanoscience, 2010, 7(10): 1948–1954

[39] Arash B, Wang Q. Vibration of single- and double-layered graphene sheets. J Nanotechnology in Engi-
neering and Medicine, 2011, 2(1): 011012

[40] Pradhan S C, Phadikar J K. Small scale e?ect on vibration of embedded multilayered graphene sheets
based on nonlocal continuum models. Phys Lett A, 2009, 373: 1062–1069

[41] Pradhan S C, Kumar A. Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic
medium using nonlocal elasticity theory and di?erential quadrature method. Computational Materials Science, 2010, 50: 239–245

[42] Ansari R,Rajabiehfard R, ArashB. Nonlocal finite element model forvibrations of embedded multi-layered
graphene sheets. Computational Materials Science, 2010, 49: 831–838

[43] Pradhan S C, Kumar A. Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory
and di?erential quadrature method. Composite Structures, 2011, 93: 774–779

[44] Wang L, He X. Vibration of a multilayered graphene sheet with initial stress. J Nanotechnology in
Engineering and Medicine, 2010, 1(4): 041004

[45] Adali S. Variational principles and natural boundary conditions for multilayered orthotropic graphene
sheets undergoing vibrations and based on nonlocal elastic theory. J Theoretical Applied Mechanics, 2011,
49(3): 621–639

[46] Adali S. Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal
elasticity theory. Physics Letters A, 2008, 372: 5701–5705

[47] Adali S. Variational principles for transversely vibrating multi-walled carbon nanotubes based on nonlocal
Euler-Bernoulli beam model. Nano Letters, 2009, 9(5): 1737–1741

[48] Adali S. Variational principles for multi-walled carbon nanotubes undergoing nonlinear vibrations by semi-
inverse method. Micro and Nano Letters, 2009, 4: 198–203

[49] Kucuk I, Sadek IS, Adali S.Variational principles formulti-walledcarbon nanotubes undergoing vibrations
based on nonlocal Timoshenko beam theory. J Nanomaterials, 2010, 2010: 1–7

[50] Adali S. Variational formulation for buckling of multi-walled carbon nanotubes modelled as nonlocal Tim-
oshenko beams. J Theoretical Applied Mechanics, 2012, to appear

[51] He J-H. Semi-inverse method of establishing generalized variational principles for fluid mechanics with
emphasis on turbomachinery aerodynamics. Int J Turbo Jet-Engines, 1997, 14: 23–28

[52] He J-H. Variational principles for some nonlinear partial di?erential equations with variable coe?cients.
Chaos, Solitons and Fractals, 2004, 19: 847–851

[53] He J-H. Variational approach to (2 +1)-dimensional dispersive long water equations. Phys Lett A, 2005,
335: 182–184

[54] He J-H. Variational theory for one-dimensional longitudinal beam dynamics. Phys Lett A, 2006, 352: 276–277

[55] He J-H. Variational principle for two-dimensional incompressible inviscid flow. Phys Lett A, 2007, 371:
39–40

[56] Liu H-M. Generalized variational principles for ion acoustic plasma waves by He’s semi-inverse method.
Chaos, Solitons and Fractals, 2005, 23: 573–576

[57] Zhou W X. Variational approach to the Broer-Kaup-Kupershmidt equation. Phys Lett A, 2006, 363: 108–109

[58] Eringen A C. On di?erential equations of nonlocal elasticity and solutions of screw dislocation and surface
waves. J Appl Phys, 1983, 54(9): 4703–4710