zbMATH — the first resource for mathematics

Phonon spectra prediction in carbon nanotubes using a manifold-based continuum finite element approach. (English) Zbl 1227.74079
Summary: This work develops a tensor-based, reduced-order shell (two-manifold) finite element formulation for predicting phonon spectra in finite-length cylindrical and toroidal carbon nanotubes (CNTs). The formulation does not require an assumed tube thickness. Displacements referencing two covariant basis vectors lying in the tangent space, and one basis vector orthogonal to the tangent space, capture the systems’ kinematics. These basis vectors compose a curvilinear coordinate system useful for capturing cylindrical, toroidal, and generically-curved nanotube configurations. The finite element procedure originates from a variational statement (Hamilton’s Principle) governing virtual work from internal, external (not considered), and inertial forces. Internal virtual work is related to changes in atomistic potential energy accounted for by an interatomic potential computed at reference area elements. Small virtual changes in the displacements allow a global mass and stiffness matrix to be computed, and these matrices then allow phonon spectra and energies to be predicted via a general eigenvalue problem. Results are generated for example cylindrical and toroidal CNTs documenting accurate prediction of phonon spectra, to include the expected longitudinal, torsional, bending, and breathing-like phonons.
74S05 Finite element methods applied to problems in solid mechanics
74K25 Shells
74M25 Micromechanics of solids
Full Text: DOI
[1] Lou, Liang-fu, Introduction to phonons and electrons, (2003), World Scientific Publishing Company
[2] Liu, J.; Hongjie, D.; Hafner, J.H.; Colbert, D.T.; Smalley, R.E.; Tans, S.J.; Dekker, C., Fullerene ‘crop circles’, Nature, 385, 780-781, (1997)
[3] Shyu, F.-L., Magneto-optical properties of carbon toroids: influence of geometry and magnetic field, Phys. rev. B, 72, 045424, (2005)
[4] Pozrikidis, C., Structure of carbon nanorings, Comput. mater. sci., 43, 943-950, (2008)
[5] Saito, R.; Dresselhaus, G.; Dresselhaus, M., Physical properties of carbon nanotubes, (1998), Imperial College Press
[6] MGill, D.J.; Lenzen, K.H., Circumferential axisymmetric free oscillations of thick hollowed tori, Int. J. solids struct., 3, 771-780, (1967) · Zbl 0152.43901
[7] MGill, D.J.; Lenzen, K.H., Polar axisymmetric free oscillations of thick hollowed tori, SIAM J. appl. math., 15, 678-692, (1967) · Zbl 0152.43901
[8] Zhou, D.; Au, F.T.K.; Lo, S.H.; Cheung, Y.K., Three-dimensional vibration analysis of a torus with circular cross-section, J. acoust. soc. am., 112, 2831-2839, (2002)
[9] Jiang, W.; Redekop, D., Polar axisymmetric vibration of a hollow toroid using the differential quadrature method, J. sound vib., 251, 4, 761-765, (2002)
[10] Jiang, W.; Redekop, D., Static and vibration analysis of orthotropic toroidal shells of variable thickness by differential quadrature, Thin-walled struct., 41, 461-478, (2003)
[11] Buchanan, G.R.; Liu, Y.J., An analysis of the free vibration of thick-walled isotropic toroidal shells, Int. J. mech. sci., 47, 277-292, (2005) · Zbl 1192.74161
[12] Wang, X.H.; Xu, B.; Redekop, D., FEM free vibration and buckling analysis of stiffened toroidal shells, Thin-walled struct., 44, 2-9, (2006)
[13] Ming, R.S.; Pan, J.; Norton, M.P., Free vibrations of elastic circular toroidal shells, Appl. acoust., 63, 513-528, (2002)
[14] A.K. Jha, Vibration Analysis and Control of an Inflatable Toroidal Satellite Component Using Piezoelectric Actuators and Sensors, Ph.D. Dissertation, Virginia Polytechnic University, Blacksburg, VA, 2002.
[15] Tadmor, E.B.; Ortiz, M.; Phillips, R., Quasicontinuum analysis of defects in solids, Philos. mag. A, 73, 1529-1563, (1996)
[16] Tadmor, E.B.; Smith, G.S.; Bernstein, N.; Kaxiras, E., Mixed finite element and atomistic formulation for complex crystals, Phys. rev. B, 59, 1, 235-245, (1999)
[17] Rudd, R.E.; Broughton, J.Q., Coarse-grained molecular dynamics and the atomic limit of finite elements, Phys. rev. B, 58, 10, (1998)
[18] Abraham, F.; Broughton, J.; Bernstein, N.; Kaxiras, E., Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture, Europhys. lett., 44, 783-787, (1998)
[19] Belytschko, T.; Xiao, S.P., Coupling methods for continuum model with molecular model, J. mult. comput. engrg., 1, 115-126, (2003)
[20] Chung, P.W.; Namburu, R.R., On a formulation for a multiscale atomistic-continuum homogenization method, Int. J. solids struct., 40, 2563-2588, (2003) · Zbl 1145.74303
[21] Liu, B.; Huang, Y.; Jiang, H.; Qu, S.; Hwang, K.C., The atomic-scale finite element method, Comput. meth. appl. mech. engrg., 193, 1849-1864, (2004) · Zbl 1079.74645
[22] Arroyo, M.; Belytschko, T., An atomistic-based finite deformation membrane for single layer crystalline films, J. mech. phys. solids, 55, 19941-19977, (2003)
[23] Pantano, A.; Parks, D.M.; Boyce, M.C., Mechanics of deformation of single- and multi-wall carbon nanotubes, J. mech. phys. solids, 52, 789-821, (2004) · Zbl 1106.74376
[24] Zhang, P.; Jiang, H.; Huang, Y.; Geubelle, P.H.; Hwang, K.C., An atomistic-based continuum theory for carbon nanotubes: analysis of fracture nucleation, J. mech. phys. solids, 53, 977-998, (2004) · Zbl 1112.74311
[25] Leamy, M.J., Bulk dynamic response modeling of carbon nanotubes using an intrinsic finite element formulation incorporating interatomic potentials, Int. J. solids struct., 44, 3-4, 874-894, (2007) · Zbl 1176.74014
[26] Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1983), Prentice-Hall · Zbl 0545.73031
[27] Harris, Peter J.F., Carbon nanotubes and related structures, (1999), Cambridge University Press Cambridge, UK
[28] Lee, P.S.; Bathe, K.J., Insight into finite element shell discretizations by use of the ‘basic shell mathematical model’, Comput. struct., 83, 69-90, (2005)
[29] Belytschko, T.; Xiao, S.P.; Schatz, G.C.; Ruogg, R.S., Atomistic simulations of nanotube fracture, Phys. rev. B, 65, 235430, (2002)
[30] M.J. Leamy, P.W. Chung, R. Namburu, On an Exact Mapping and a Higher Order Born Rule for Use in Analyzing Graphene Carbon Nanotubes, US Army Research Laboratory Technical Report, ARL-TR-3117, 2003.
[31] Kittel, C., Introduction to solid state physics, (2005), John Wiley & Sons, Inc. USA
[32] Srivastava, D.; Wei, C.; Kyeongjae, C., Nanomechanics of carbon nanotubes and composites, Appl. mech. rev., 56, 2, 215-230, (2003)
[33] Sokhan, V.P.; Nicholson, D.; Quirke, N., Phonon spectra in model carbon nanotubes, J. chem. phys., 113, 5, 2007-2015, (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.