Vibration Analysis of Material Size-Dependent CNTs Using Energy Equivalent Model

Document Type : Research Paper


1 Mechanical Engineering Dept., Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia

2 Mechanical Design & Production Dept., Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt


This study presents a modified continuum model to investigate the vibration behavior of single and multi-carbon nanotubes (CNTs). Two parameters are exploited to consider size dependence; one derived from the energy equivalent model and the other from the modified couple stress theory. The energy equivalent model, derived from the basis of molecular mechanics, is exploited to describe size-dependent material properties such as Young and shear moduli for both zigzag and armchair CNT structures. A modified couple stress theory is proposed to capture the microstructure size effect by assisting material length scale. A modified kinematic Timoshenko nano-beam including shear deformation and rotary inertia effects is developed. The analytical solution is shown and verified with previously published works. Moreover, parametric studies are performed to illustrate the influence of the length scale parameter, translation indices of the chiral vector, and orientation of CNTs on the vibration behaviors. The effect of the number of tube layers on the fundamental frequency of CNTs is also presented. These findings are helpful in mechanical design of high-precision measurement nano-devices manufactured from CNTs.


Main Subjects

[1] Iijima, S., Helical microtubules of graphitic carbon, Nature, 354(6348), 1991, pp. 56-58.
[2] Tu, Z.C., Ou-Yang, Z.C., Single-walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young’s moduli dependent on layer number, Physical Review B, 65(23), 2002, pp. 233407.
[3] Haile, J.M., Molecular dynamics simulation: Elementary methods, Computers in Physics, 7(6), 1993, pp. 625-625.
[4] Chen Y., Lee J.D., Eskandarian. A., Atomistic viewpoint of the applicability of microcontinuum theories, International Journal of Solids and Structures, 41, 2004, pp. 2085-2097.
[5] Eltaher, M.A., Emam, S.A., Mahmoud, F.F., Free vibration analysis of functionally graded size-dependent nanobeams, Applied Mathematics and Computation, 218(14), 2012, pp. 7406-7420.
[6] Eringen, A.C., Nonlocal polar elastic continua, International Journal of Engineering Science, 10, 1972, pp. 1–16.
[7] Eringen A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54, 1983, pp. 4703–4710.
[8] Eringen, A.C., Nonlocal continuum field theories, New York: Springer-Verlag, 2002.
[9] Mindlin, R.D., Influence of couple-stresses on stress concentrations, Experimental Mechanics, 3(1), 1963, pp. 1-7.
[10] Toupin, R.A., Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis, 11(1), 1962, pp. 385-414.
[11] Mindlin, R.D., Eshel, N.N., On first strain-gradient theories in linear elasticity, International Journal of Solids and Structures, 4(1), 1968, pp. 109-124.
[12] Yang, F.A., Chong, A.C.M., Lam, D.C.C., Tong, P., Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39(10), 2002, pp. 2731-2743.
[13] Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P., Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, 51(8), 2003, pp. 1477-1508.
[14] Wu, Y., Zhang, X., Leung, A.Y.T., Zhong, W., An energy-equivalent model on studying the mechanical properties of single-walled carbon nanotubes, Thin-Walled Structures, 44(6), 2006, pp. 667-676.
[15] Alizada, A.N., Sofiyev, A.H., Modified Young’s moduli of nano-materials taking into account the scale effects and vacancies, Meccanica, 46, 2011, pp. 915–920.
[16] Rafiee, R., Moghadam, R.M., On the modeling of carbon nanotubes: a critical review, Composites Part B: Engineering, 56, 2014, pp. 435-449.
[17] Sakharova, N.A., Pereira, A.F.G., Antunes, J.M., Brett, C.M.A., Fernandes, J.V., Mechanical characterization of single-walled carbon nanotubes: Numerical simulation study, Composites Part B: Engineering, 75, 2015, pp. 73-85.
[18] Tserpes, K.I., Papanikos, P., Finite element modeling of single-walled carbon nanotubes, Composites Part B: Engineering, 36(5), 2005, pp. 468-477.
[19] Shokrieh, M.M., Rafiee, R., Prediction of Young’s modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach, Materials & Design, 31(2), 2010, pp. 790-795.
[20] Shodja, H.M., Delfani, M.R., A novel nonlinear constitutive relation for graphene and its consequence for developing closed-form expressions for Young’s modulus and critical buckling strain of single-walled carbon nanotubes, Acta mechanica, 222(1-2), 2011, pp. 91-101.
[21] Bogacz, R., Noga, S., Free transverse vibration analysis of a toothed gear, Archive of Applied Mechanics, 82(9), 2012, pp. 1159-1168.
[22] Ghavamian, A., Rahmandoust, M., Öchsner, A., On the determination of the shear modulus of carbon nanotubes, Composites Part B: Engineering, 44(1), 2013, pp. 52-59.
[23] Akgoz, B., Civalek, O. Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity, Structural Engineering and Mechanics, 48(2), 2013, pp. 195-205.
[24] Feng, C., Liew, K. M., He, P., Wu, A., Predicting mechanical properties of carbon nanosprings based on molecular mechanics simulation, Composite Structures, 114, 2014, pp. 41-50.
[25] Pine, P., Yaish, Y.E., Adler, J., Vibrational analysis of thermal oscillations of single-walled carbon nanotubes under axial strain, Physical Review B, 89(11), 2014, pp. 115405.
[26] Rappé, A.K., Casewit, C.J., Colwell, K.S., Goddard Iii, W.A., Skiff, W.M., A full periodic table force field for molecular mechanics and molecular dynamics simulations, Journal of the American Chemical Society, 114(25), 1992, pp. 10024-10035..
[27] Farrokhabadi, A., Abadian, N., Rach, R., Abadyan, M., Theoretical modeling of the Casimir force-induced instability in freestanding nanowires with circular cross-section, Physica E: Low-dimensional Systems and Nanostructures, 63, 2014, pp. 67-80.
[28] Brischetto, S., A continuum elastic three-dimensional model for natural frequencies of single-walled carbon nanotubes, Composites Part B: Engineering, 61, 2014, pp. 222-228.
[29] Kiani, K., Vibration analysis of two orthogonal slender single-walled carbon nanotubes with a new insight into continuum-based modeling of van der Waals forces, Composites Part B: Engineering, 73, 2015, pp. 72-81.
[30] Stölken, J.S., Evans, A.G., A microbend test method for measuring the plasticity length scale, Acta Materialia, 46(14), 1998, pp. 5109-5115.
[31] Mindlin, R.D., Tiersten, H.F., Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11(1), 1962, pp. 415-448.
[32] Ma, H.M., Gao, X.L., Reddy, J.N., A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids, 56(12), 2008, pp. 3379-3391.
[33] Fu, Y., Zhang, J., Modeling and analysis of microtubules based on a modified couple stress theory, Physica E: Low-dimensional Systems and Nanostructures, 42(5), 2010, pp. 1741-1745.
[34] Ke, L.L., Wang, Y.S., Flow-induced vibration and instability of embedded double-walled carbon nanotubes based on a modified couple stress theory, Physica E: Low-dimensional Systems and Nanostructures, 43(5), 2011, pp. 1031-1039.
[35] Ghayesh, M.H., Farokhi, H., Amabili, M., Nonlinear dynamics of a microscale beam based on the modified couple stress theory, Composites Part B: Engineering, 50, 2013, pp. 318-324.
[36] Tounsi, A., Benguediab, S., Adda, B., Semmah, A., Zidour, M., Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes, Advances in Nano Research, 1(1), 2013, pp. 1-11.
[37] Benguediab, S., Tounsi, A., Zidour, M., Semmah, A., Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes, Composites Part B: Engineering, 57, 2014, pp. 21-24.
[38] Semmah, A., Tounsi, A., Zidour, M., Heireche, H., Naceri, M., Effect of the chirality on critical buckling temperature of zigzag single-walled carbon nanotubes using the nonlocal continuum theory, Fullerenes, Nanotubes and Carbon Nanostructures, 23(6), 2015, pp. 518-522.
[39] Bazehhour, B.G., Mousavi, S.M., Farshidianfar, A., Free vibration of high-speed rotating Timoshenko shaft with various boundary conditions: effect of centrifugally induced axial force, Archive of Applied Mechanics, 84(12), 2014, pp. 1691-1700.
[40] Besseghier, A., Heireche, H., Bousahla, A.A., Tounsi, A., Benzair, A., Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix, Advances in Nano Research, 3(1), 2015, pp. 29-37.
[41] Akgöz, B., Civalek, Ö., Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity, Composite Structures, 134, 2015, pp. 294-301.
[42] Eltaher, M.A., Agwa, M.A., Mahmoud, F.F., Nanobeam sensor for measuring a zeptogram mass, International Journal of Mechanics and Materials in Design, 12(2), 2016, pp. 211-221.
[43] Fakhrabadi, M.M.S., Prediction of small-scale effects on nonlinear dynamic behaviors of carbon nanotube-based nano-resonators using consistent couple stress theory, Composites Part B: Engineering, 88, 2016, pp. 26-35.
[44] Akgöz, B., Civalek, Ö., Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory, Acta Astronautica, 119, 2016, pp. 1-12..
[45] Agwa, M.A., & Eltaher, M.A., Vibration of a carbyne nanomechanical mass sensor with surface effect, Applied Physics A, 122(4), 2016, pp. 1-8.
[46] Eltaher, M.A., Khater, M.E., Emam, S.A., A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams, Applied Mathematical Modelling, 40, 2016, pp. 4109–4128.
[47] Eltaher, M.A., El-Borgi S., Reddy J.N., Nonlinear Analysis of Size-dependent and Material-Dependent Nonlocal CNTs, Composite Structure, 153, 2016, pp. 902-913.
[48] Civalek, Ö., Demir, C., A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method, Applied Mathematics and Computation, 289, 2016, pp. 335-352.
[49] Hamed M.A., Eltaher M.A., Sadoun, A.M., Almitani K.H., Free vibration of Symmetric and Sigmoid Functionally Graded Nanobeams, Applied Physics A: Materials Science and Processing, 122(9), 2016, pp. 625-625.
[50] Hosseini, S.A.H., Rahmani, O., Thermomechanical vibration of curved functionally graded nanobeam based on nonlocal elasticity, Journal of Thermal Stresses, 39(10), 2016, pp. 1252-1267.
[51] Keivani, M., Mardaneh, M., Koochi, A., Rezaei, M., Abadyan, M., On the dynamic instability of nanowire-fabricated electromechanical actuators in the Casimir regime: Coupled effects of surface energy and size dependency, Physica E: Low-dimensional Systems and Nanostructures, 76, 2016, pp. 60-69.
[52] Sedighi, H.M., Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory, Acta Astronautica, 95, 2014, pp. 111-123.
[53] Keivani, M., Khorsandi, J., Mokhtari, J., Kanani, A., Abadian, N., Abadyan, M., Pull-in instability of paddle-type and double-sided NEMS sensors under the accelerating force, Acta Astronautica, 119, 2016, pp. 196-206.
[54] Keivani, M., Mokhtari, J., Abadian, N., Abbasi, M., Koochi, A., Abadyan, M., Analysis of U-shaped NEMS in the Presence of Electrostatic, Casimir, and Centrifugal Forces Using Consistent Couple Stress Theory, Iranian Journal of Science and Technology, Transactions A: Science, 2017, doi: 10.1007/s40995-017-0151-y.
[55] Wang, K.F., Zeng, S., Wang, B.L., Large amplitude free vibration of electrically actuated nanobeams with surface energy and thermal effects, International Journal of Mechanical Sciences, 131–132, 2017, pp. 227-233.
[56] Yamabe, T., Recent development of carbon nanotube, Synthetic Metals, 70(1), 1995, pp. 1511-1518
[57] Baghdadi, H., Tounsi, A., Zidour, M., Benzair, A., Thermal Effect on Vibration Characteristics of Armchair and Zigzag Single-Walled Carbon Nanotubes Using Nonlocal Parabolic Beam Theory, Fullerenes, Nanotubes and Carbon Nanostructures, 23(3), 2015, pp. 266-272.
[58] Eltaher, M.A., Agwa, M.A., Analysis of Size-dependent Mechanical Properties of CNTs Mass Sensor Using Energy Equivalent Model, Sensor and Actuator A: Physical, 246, 2016, pp. 9-17.
[59] Park, S.K., Gao, X.L., Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, 16(11), 2006, pp. 2355.