Danilo Karličić, Sanja Ožvat, Milan Cajić, Predrag Kozić, Ratko Pavlović

DOI Number
First page
Last page


In this study, we analyzed the bending vibration and stability of a multiple-nanobeam system (MNBS) coupled in elastic medium and influenced by temperature change and compressive axial load. The MNBS is modeled as the system consisting of a set of m identical and simply supported nanobeams mutually connected by Winkler’s type elastic layers. According to the Euler - Bernoulli beam and nonlocal thermo-elasticity theory, the system of m coupled partial differential equations is derived and solved by means of the method of separation of variables as well as the trigonometric one. Analytical solutions for natural frequencies and critical buckling loads of elastic MNBS are obtained. The effects of nonlocal parameter, temperature change and the number of nanobeams on the natural frequencies and the buckling loads are investigated through numerical examples. Thus, this work can represent a starting point to examine dynamical behavior and design of complex nanobeam structures, nanocomposites and nanodevices under the influence of various physical fields.


Nonlocal Elasticity, Vibration, Stability, Multiple-nanobeam System

Full Text:



Bezryadin, A., Verschueren, A. R. M., Tans, S. J., Dekker, C., 1998, Multiprobe transport experiments on individual single-wall carbon nanotubes, Physical Review Letters, 80(18), pp.4036- 4039.

Gogotsi, Y., Libera, J. A., Güvenç-Yazicioglu, A., Megaridis, C. M., 2001, In situ multiphase fluid experiments in hydrothermal carbon nanotubes, Applied physics letters, 79(7), pp. 1021-1023.

Lu, J. P., 1997, Elastic properties of carbon nanotubes and nanoropes, Physical Review Letters, 79 (7), pp. 1297-1306.

Werder, T., Walther, J. H., Jaffe, R. L., Halicioglu, T., Koumoutsakos, P., 2003, On the water-carbon Interaction for use in molecular dynamics simulations of graphite and carbon nanotubes, The Journal of Physical Chemistry B, 107(6), pp. 1345-1352.

Wen Xing, B., ChangChun, Z., Wan Zhao, C., 2004, Simulation of Young's modulus of single-walled carbon nanotubes by molecular dynamics, Physica B: Condensed Matter, 352(1), pp. 156-163.

Liew, K. M., He, X. Q., 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 Materialia, 52(9), pp. 2521-2527.

Neyts, E. C., Shibuta, Y., van Duin, A. C., Bogaerts, A., 2010, Catalyzed growth of carbon nanotube with definable chirality by hybrid molecular dynamics− force biased Monte Carlo simulations, ACS nano, 4(11), pp. 6665-6672.

Ansari, R., Gholami, R., Sahmani, S., 2013, Size-dependent vibration of functionally graded curved microbeams based on the modified strain gradient elasticity theory, Archive of Applied Mechanics, 83(10), pp. 1439-1449.

Chen, W. J., Li, X. P., 2013, Size-dependent free vibration analysis of composite laminated Timoshenko beam based on new modified couple stress theory, Archive of Applied Mechanics, 83(3), pp. 431-444.

Challamel, N., Zhang, Z., Wang, C.M., Reddy, J.N., Wang, Q., Michelitsch, T., Bernard, C., 2014, On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach, Archive of Applied Mechanics, DOI 10.1007/s00419-014-0862-x.

Natsuki, T., Endo, M., 2004, Stress simulation of carbon nanotubes in tension and compression, Carbon, 42(11), pp. 2147-2151.

Gao, Y., Wang, Z. L., 2007, Electrostatic potential in a bent piezoelectric nanowire. The fundamental theory of nanogenerator and nanopiezotronics, Nano letters, 7(8), pp. 2499-2505.

Chopra, N. G., Luyken, R. J., Cherrey, K., Crespi, V. H., Cohen, M. L., Louie, S. G., Zettl, A., 1995, Boron nitride nanotubes, Science, 269(5226), pp. 966-967.

Loiseau, A., Willaime, F., Demoncy, N., Schramchenko, N., Hug, G., Colliex, C., Pascard, H., 1998, Boron nitride nanotubes, Carbon, 36(5), pp. 743-752.

Eringen, A. C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science, 10(1), pp. 1-16. 16. Eringen, A. C., Edelen, D. G. B., 1972, On nonlocal elasticity, International Journal of Engineering Science, 10(3), pp. 233-248.

Eringen, A. C., 1992, Vistas of nonlocal continuum physics, International Journal of Engineering Science, 30(10), pp. 1551-1565.

Eringen, A. C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54(9), pp. 4703-4710.

Peddieson, J., Buchanan, G. R., McNitt, R. P., 2003, Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science, 41(3), pp. 305-312.

Reddy, J. N., Pang, S. D., 2008, Nonlocal continuum theories of beams for the analysis of carbon nanotubes, Journal of Applied Physics, 103(2), pp. 023511-16.

Reddy, J. N., 2007, Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science, 45(2), pp. 288-307.

Zhang, Y. Q., Liu, X., Liu, G. R., 2007, Thermal effect on transverse vibrations of double-walled carbon nanotubes, Nanotechnology, 18(44), 445701 (7pp).

Zhang, Y. Q., Liu, X., Zhao, J. H., 2008, Influence of temperature change on column buckling of multiwalled carbon nanotubes, Physics Letters A, 372(10), pp. 1676-1681.

Janghorban, M., 2012, Two different types of differential quadrature methods for static analysis of microbeams based on nonlocal thermal elasticity theory in thermal environment, Archive of Applied Mechanics, 82(5), pp. 669-675.

Murmu, T., Pradhan, S. C., 2009, Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Computational Materials Science, 46(4), pp. 854-859.

Murmu, T., Pradhan, S. C., 2010, Thermal effects on the stability of embedded carbon nanotubes, Computational Materials Science, 47(3), pp. 721-726.

Rašković, D., 1963, Small forced damping vibrations of homogeneous torsional system with special static constrints. Publications de l'Institut Mathématique, 3(17), pp. 27-34.

Stojanović, V., Kozić, P., Janevski, G., 2013, Exact closed-form solutions for the natural frequencies and stability of elastically connected multiple beam system using Timoshenko and high-order shear deformation theory. Journal of Sound Vibration, 332(3), pp. 563–576.

Karličić, D., Kozić, P., Pavlović, R., 2014, Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium, Composite Structures, 115, pp. 89–99.

Karličić, D., Adhikari, S., Murmu, T., Cajić, M., 2014, Exact closed-form solution for non-local vibration and biaxial buckling of bonded multi-nanoplate system, Composites Part B: Engineering, 66, pp. 328-339.

Ansari, R., Rouhi, H., Sahmani, S., 2011, Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics, International Journal of Mechanical Sciences, 53(9), pp. 786-792.

Frank, I. W., Deotare, P. B., McCutcheon, M. W., Lonĉar, M., 2010, Programmable photonic crystal nanobeam cavities, Optics Express, 18(8), pp. 8705-8712.

Karličić, D., Murmu, T., Cajić, M., Kozić, P., Adhikari, S., 2014, Dynamics of multiple viscoelastic carbon nanotube based nanocomposites with axial magnetic field, Journal of Applied Physics, 115(23), pp. 234303-14.

Ansari, R., Gholami, R., Rouhi, H., 2012, Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theories, Composites Part B: Engineering, 43(8), pp. 2985-2989.



  • There are currently no refbacks.

ISSN: 0354-2025 (Print)

ISSN: 2335-0164 (Online)

COBISS.SR-ID 98732551

ZDB-ID: 2766459-4