Research Papers

Wave Propagation in Periodically Supported Nanoribbons: A Nonlocal Elasticity Approach

[+] Author and Article Information
Giuliano Allegri

Advanced Composites Centre for Innovation and Science (ACCIS),
University of Bristol,
Bristol BS8 1TR, UK
e-mail: giuliano.allegri@bristol.ac.uk

Fabrizio Scarpa

Bristol Centre for Nanoscience and Quantum Information (NSQI),
University of Bristol,
Bristol BS8 1TR, UK
e-mail: f.scarpa@bristol.ac.uk

Rajib Chowdhury

Department of Civil Engineering,
Indian Institute of Technology,
Roorkee 247 667, India
e-mail: Rajibfce@Iitr.ernet.in

Sondipon Adhikari

Multidisciplinary Nanotechnology Centre,
Swansea University,
Swansea SA2 8PP, UK
e-mail: s.adhikari@swansea.ac.uk

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 30, 2012; final manuscript received February 15, 2013; published online June 6, 2013. Assoc. Editor: Massimo Ruzzene.

J. Vib. Acoust 135(4), 041017 (Jun 06, 2013) (8 pages) Paper No: VIB-12-1127; doi: 10.1115/1.4023953 History: Received April 30, 2012; Revised February 15, 2013

We develop an analytical formulation describing propagating flexural waves in periodically simply supported nanoribbons by means of Eringen's nonlocal elasticity. The nonlocal length scale is identified via atomistic finite element (FE) models of graphene nanoribbons with Floquet's boundary conditions. The analytical model is calibrated through the atomistic finite element approach. This is done by matching the nondimensional frequencies predicted by the analytical nonlocal model and those obtained by the atomistic FE simulations. We show that a nanoribbon with periodically supported boundary conditions does exhibit artificial pass-stop band characteristics. Moreover, the nonlocal elasticity solution proposed in this paper captures the dispersive behavior of nanoribbons when an increasing number of flexural modes are considered.

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Eringen, A. C., 1983, “On Differential-Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves,” J. Appl. Phys., 54(9), pp. 4703–4710. [CrossRef]
Li, C., Lim, C. W., Yu, J. L., and Zeng, Q. C., 2011, “Analytical Solutions for Vibration of Simply Supported Nonlocal Nanobeams With an Axial Force,” Int. J. Struct. Stab. Dyn., 11(2), pp. 257–271. [CrossRef]
Pin LuH., Lee, P., Lu, C., and Zhang, P. Q., 2007, “Application of Nonlocal Beam Models for Carbon Nanotubes,” Int. J. Solid. Struct., 44(16), pp. 5289–5300. [CrossRef]
Narendar, S., and Gopalakrishnan, S., 2010, “Nonlocal Scale Effects on Ultrasonic Wave Characteristics Nanorods,” Physica E, 42(5), pp. 1601–1604. [CrossRef]
Narendar, S., and Gopalakrishnan, S., 2011, “Nonlocal Wave Propagation in Rotating Nanotube,” Result. Phys., 1(1), pp. 17–25. [CrossRef]
Wang, Q., 2005, “Wave Propagation in Carbon Nanotubes via Nonlocal Continuum Mechanics,” J. Appl. Phys., 98(12), p. 124301. [CrossRef]
Wang, L., and Hu, H., 2005, “Flexural Wave Propagation in Single-Walled Carbon Nanotubes,” Phys Rev B, 71, p. 195412. [CrossRef]
Pradhan, S. C., 2009, “Buckling of Single Layer Graphene Sheet Based on Nonlocal Elasticity and Higher Order Shear Deformation Theory,” Phys. Lett. A., 373(45), pp. 4182–4188. [CrossRef]
Samaei, A. T., Abbasion, S., and Mirsayar, M. M., 2011, “Buckling Analysis of a Single-Layer Graphene Sheet Embedded in an Elastic Medium Based on Nonlocal Mindlin Plate Theory,” Mech. Res. Commun., 38(7), pp. 481–485. [CrossRef]
Arash, B., Wang, Q., and Liew, K. M., 2012, “Wave Propagation in Graphene Sheets With Nonlocal Elastic Theory via Finite Element Formulation,” Comput. Meth. Appl. M., 223–224, pp. 1–9. [CrossRef]
Narendar, S., Roy Mahapatra, D., and Gopalakrishnan, S., 2010, “Investigation of the Effect of Nonlocal Scale on Ultrasonic Wave Dispersion Characteristics of a Monolayer Graphene,” Comput. Mater. Sci., 49(4), pp. 734–742. [CrossRef]
Kim, K. S., Zhao, Y., Jang, H., Lee, S. Y., Kim, J. M., Kim, K. S., Ahn, J.-H., Kim, P., Choi, J.-Y., and Hong, B. H., 2009, “Large-Scale Pattern Growth of Graphene Films for Stretchable Transparent Electrodes,” Nature, 457(7230), pp. 706–710. [CrossRef] [PubMed]
Zhang, Y. B., Tan, Y. W., Stormer, H. L., and Kim, P., 2005, “Experimental Observation of the Quantum Hall Effect and Berry's Phase in Graphene,” Nature, 438(7065), pp. 201–204. [CrossRef] [PubMed]
Sakhaee-Pour, A., Ahmadian, M. T., and Vafai, A., 2008, “Potential Application of Single-Layered Graphene Sheet as Strain Sensor,” Solid State Comm., 147(7–8), pp. 336–340. [CrossRef]
Scarpa, F., Chowdhury, R., Kam, K., Adhikari, S., and Ruzzene, M., 2011, “Dynamics of Mechanical Waves in Periodic Graphene Nanoribbon Assemblies,” Nanosc. Res. Lett., 6, pp. 430–439. [CrossRef]
Lin, Y.-M., Dimitrakopoulos, C., Jenkins, K. A., Farmer, D. B., Chiu, H.-Y., Grill, A., and Avouris, P. H., 2010, “100-GHz Transistors From Wafer-Scale Epitaxial Graphene,” Science, 327(5966), p. 662. [CrossRef] [PubMed]
Mead, D. J., 1970, “Free Wave Propagation in Periodically Supported, Infinite Beams,” J. Sound Vib., 11(2), pp.181–197. [CrossRef]
Mead, D. J., 1996, “Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton 1964–1995,” J. Sound Vib., 190(3), p. 495 [CrossRef]
Ruzzene, M., and Baz, A., 2000, “Attenuation and Localization of Wave Propagation in Periodic Rods Using Shape Memory Inserts,” Smart Mater. Struct., 9, p. 805. [CrossRef]
Nakada, K., Fujita, M., Dresselhaus, G., and Dresselhaus, M. S., 1996, “Edge State in Graphene Nanoribbons: Nanometer Size Effect and Edge Shape Dependence,” Phys. Rev. B, 54(24), p. 17954. [CrossRef]
Wen, J. G., Lao, J. Y., Wang, D. Z., Kyaw, T. M., Foo, Y. L., and Ren, Z. F., 2003, “Self-Assembly of Semiconducting Oxide Nanowires, Nanorods, and Nanoribbons,” Chem. Phys. Lett., 372, pp. 717–722. [CrossRef]
Barone, V., Hod, O., and Scuseria, G. V., 2006, “Electronic Structure and Stability of Semiconducting Graphene Nanoribbons,” Nano. Lett., 6(12), p. 2748. [CrossRef] [PubMed]
Han, M. Y., Özyilmaz, B., Zhang, Y., and Kim, P., 2007, “Energy Band-Gap Engineering of Graphene Nanoribbons,” Phys. Rev. Lett., 98(20), p. 206805. [CrossRef] [PubMed]
Hod, O., and Scuseria, G. E., 2009, “Electromechanical Properties of Suspended Graphene Nanoribbons,” Nano Lett., 9(7), pp. 2619–2622. [CrossRef] [PubMed]
Law, M., Sirbuly, D. J., Johnson, J. C., Goldberger, J., Saykally, R. J., and Yang, P., 2004, “Nanoribbon Waveguides for Subwavelength Photonics Integration,” Science, 305(5688), p. 269. [CrossRef]
Yosevich, Y. A., and Savin, A. V., 2009, “Reduction of Phonon Thermal Conductivity in Nanowires and Nanoribbons With Dynamically Rough Surfaces and Edges,” Eur. Phys. Lett., 88, p. 14002. [CrossRef]
Narendar, S., and Gopalakrishnan, S., 2009, “Nonlocal Scale Effects on Wave Propagation in Multi-Walled Carbon Nanotubes,” Comput. Mater. Sci., 47(2), pp. 526–538. [CrossRef]
Scarpa, F., Adhikari, S., and Phani, A. S., 2009, “Effective Elastic Mechanical Properties of Single Layer Graphene Sheets,” Nanotechnology, 20, p. 065709. [CrossRef] [PubMed]
Scarpa, F., Adhikari, S., Gil, A. J., and Remillat, C., 2010, “The Bending of Single Layer Graphene Sheets: Lattice Versus Continuum Approach,” Nanotechnology, 21(12), p. 125702. [CrossRef] [PubMed]
Chandra, Y., Chowdhury, R., Scarpa, F., and Adhikaricor, S., 2011, “Vibrational Characteristics of Bilayer Graphene Sheets,” Thin Solid Films519(18), pp. 6026–6032. [CrossRef]
Scarpa, F., Adhikari, S., and Chowdhury, R., 2010, “The Transverse Elasticity of Bilayer Graphene,” Phys. Lett. A, 374, pp. 2053–2057. [CrossRef]
Scarpa, F., Chowdhury, R., and Adhikari, S., 2011, “Thickness and In-Plane Elasticity of Graphane,” Phys. Lett. A, 375(20), pp. 2071–2074. [CrossRef]
Liu, X., Metcalf, T. H., Robinson, J. T., Houston, B. H., and Scarpa, F., 2012, “Shear Modulus of Monolayer Graphene Prepared by Chemical Vapor Deposition,” Nano Lett., 12(2), pp. 1013–1017. [CrossRef] [PubMed]
Mead, D. J., 1986, “A New Method of Analyzing Wave Propagation in Periodic Structures; Applications to Periodic Timoshenko Beams and Stiffened Plates,” J. Sound Vib., 104(1), pp. 9–27. [CrossRef]
Huang, Y., Wu, J., and Hwang, K. C., 2006, “Thickness of Graphene and Single Wall Carbon Nanotubes,” Phys Rev B, 74, p. 245413. [CrossRef]
Xin, Z., Jianjun, Z., and Zhong-can, O.-Y., 2000, “Strain Energy and Young's Modulus of Single-Wall Carbon Nanotubes Calculated From Electronic Energy-Band Theory,” Phys. Rev. B, 62, pp. 13692–13696. [CrossRef]
Huang, Y., Wu, J., and Hwang, K. C., 2006, “Thickness of Graphene and Single-Wall Carbon Nanotubes,” Phys. Rev. B, 74, p. 245413. [CrossRef]
Przemienicki, J. S., 1968, Theory of Matrix Structural Analysis, McGraw-Hill, New York.
Tee, K. F., Spadoni, A., Scarpa, F., and Ruzzene, M., 2010, “Wave Propagation in Auxetic Tetrachiral Honeycombs,” ASME J. Vib. Acoust., 132(3), p. 031007. [CrossRef]
Brillouin, L.1953, Wave Propagation in Periodic Structures, Dover Phoenix Editions, New York.
Berdichevsky, V. L., 2009, “Theory of Elastic Plates and Shells,” Variational Principles of Continuum Mechanics II: Applications, Interaction of Mechanics and Mathematics, Springer, Berlin.
Yakobson, B. I., Brabec, C. J., and Bernholc, J., 1996, “Nanomechanics of Carbon Tubes: Instabilities Beyond Linear Response,” Phys. Rev. Lett., 76, pp. 2511–2514. [CrossRef] [PubMed]
Chang, T., and Gao, H., 2003, “Size-Dependent Elastic Properties of a Single Walled Carbon Nanotube via a Molecular Mechanics Model,” J. Mech. Phys. Solids, 51, p. 1059. [CrossRef]
Tu, Z., and Ou-Yang, Z., 2002, “Single-Walled and Multiwalled Carbon Nanotubes Viewed as Elastic Tubes With the Effective Young's Moduli Dependent on Layer Number,” Phys. Rev. B, 65, p. 233407. [CrossRef]
Blakslee, O. L., Proctor, D. G., Seldin, E. J., Spence, G. B., and Weng, T., 1970, “Elastic Constants of Compression-Annealed Pyrolytic Graphite,” J. Appl. Phys., 41(8), pp. 3373–3382. [CrossRef]
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. [CrossRef] [PubMed]
Hu, Y.-G., Liew, K. M., Wang, Q., He, X. Q., and Yakobson, B. I., 2008, “Nonlocal Shell Model for Elastic Wave Propagation in Single- and Double-Walled Carbon Nanotubes,” J. Mech. Phys. Solid., 56(12), pp. 3475–3485. [CrossRef]


Grahic Jump Location
Fig. 1

(a) 100 GHz transistor design based on Ref. [16]. (b) Scheme of the periodically supported nanoribbon considered in this work.

Grahic Jump Location
Fig. 2

Comparison between predicted natural frequencies for (8,0) nanoribbons at different length when considering the atomistic-FE approach and molecular mechanics models (from Ref. [15])

Grahic Jump Location
Fig. 3

First six mode shapes of a (8,0) nanoribbon with 1780 atoms. (a) 4.24 GHz, (b) 16.90 GHz, (c) 38.15 GHz, (d) 67.80 GHz, (e) 105.87 GHz, and (f) 152.34 GHz.

Grahic Jump Location
Fig. 4

Dispersion curve for the (8,0) periodically supported nanoribbon with 1780 atoms. The darker lines represent flexural modes; the lighter ones are related to torsional modes.

Grahic Jump Location
Fig. 5

Actual mode shapes of the first six nondimensional frequencies Ω for the reference (8,0) nanoribbon with 1780 atoms. Case k=π. (a) Ω1, (b) Ω2, (c) Ω3, (d) Ω5, (e) Ω5, (f) Ω6.

Grahic Jump Location
Fig. 6

Effect of the nonlocality parameter e0 on the first dispersion curve

Grahic Jump Location
Fig. 7

Atomistic-FE results (dark lines) and analytical nonlocal model (light lines) for different lengths L of the periodic simply supported nanoribbon. Five modes used for the identification on the nonlocal parameter e0.




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