Nonlinear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model. (English) Zbl 1480.74118

Summary: This paper comprehensively studies the nonlinear vibration of functionally graded nano-beams resting on elastic foundation and subjected to uniform temperature rise. The small-size effect, playing an essential role in the dynamical behavior of nano-beams, is considered here applying the innovative stress driven nonlocal integral model due to Romano and Barretta. The governing partial differential equations are derived from the Bernoulli-Euler beam theory utilizing the von Karman strain-displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural frequency for four different boundary conditions is then established employing the Homotopy Analysis Method. The nonlinear natural frequencies, evaluated according to the stress-driven nonlocal integral model, are compared with those obtained by Eringen differential model. Finally, the effects of different parameters such as length, elastic foundation parameter, thermal loading and nonlocal characteristic parameter are investigated. The emergent results establish that when the nonlocal characteristic parameter increases, the nonlinear natural frequencies obtained by the stress-driven nonlocal integral model reveal a stiffness-hardening effect. On the other hand, Eringen’s differential law reveals a stiffness-softening effect excepting the case of cantilever nano-beam. Also, increase in temperature and the elastic foundation parameter leads to increase in the nonlinear frequency ratios in Eringen differential model but decrease in the frequency ratios in the stress-driven nonlocal integral model.


74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)


Full Text: DOI


[1] Wessel, J. K., The Handbook of Advanced Materials: Enabling New Designs (2004), John Wiley & Sons
[2] Witvrouw, A.; Mehta, A., The use of functionally graded poly-SiGe layers for MEMS applications, Materials Science Forum, 255-260 (2005), Trans Tech Publications
[3] Miyamoto, Y., Functionally Graded Materials: Design, Processing and Applications, vol. 5 (2013), Springer Science & Business Media
[4] Roth, S.; Baughman, R. H., Actuators of individual carbon nanotubes, Curr. Appl. Phys., 2, 4, 311-314 (2002)
[5] Li, C.; Thostenson, E. T.; Chou, T. W., Sensors and actuators based on carbon nanotubes and their composites: a review, Compos. Sci. Technol., 68, 6, 1227-1249 (2008)
[6] Sedighi, H. M.; Daneshmand, F.; Abadyan, M., Dynamic instability analysis of electrostatic functionally graded doubly-clamped nano-actuators, Compos. Struct., 124, 55-64 (2015)
[7] Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., 54, 9, 4703-4710 (1983)
[8] Hosseini-Hashemi, S.; Nazemnezhad, R.; Bedroud, M., Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity, Appl. Math. Modell., 38, 14, 3538-3553 (2014) · Zbl 1427.74055
[9] Karličić, D.; Kozić, P.; Pavlović, R., Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium, Appl. Math. Modell., 40, January(2), 1599-1614 (2016) · Zbl 1446.74023
[10] Reddy, J. N., nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci., 45, 2, 288-307 (2007) · Zbl 1213.74194
[11] Nazemnezhad, R.; Hosseini-Hashemi, S., nonlocal nonlinear free vibration of functionally graded nano-beams, Compos. Struct., 110, 192-199 (2014)
[12] Ghayesh, M. H.; Farokhi, H.; Amabili, M., Nonlinear dynamics of a microscale beam based on the modified couple stress theory, Composites Part B, 50, 318-324 (2013)
[13] Ghayesh, M. H.; Farokhi, H.; Amabili, M., In-plane and out-of-plane motion characteristics of microbeams with modal interactions, Composites Part B, 60, 423-439 (2014)
[14] Karličić, D.; Kozić, P.; Pavlović, R.; Nešić, N., Dynamic stability of single-walled carbon nanotube embedded in a viscoelastic medium under the influence of the axially harmonic load, Compos. Struct., 162, 227-243 (2017)
[15] Farokhi, H.; Ghayesh, M. H.; Amabili, M., Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory, Int. J. Eng. Sci., 68, 11-23 (2013) · Zbl 1423.74473
[16] Alves, M.; Ribeiro, P., Non-linear modes of vibration of Timoshenko nanobeams under electrostatic actuation, Int. J. Mech. Sci., 130, 188-202 (2017)
[17] Ribeiro, P., Nonlocal effects on the non-linear modes of vibration of carbon nanotubes under electrostatic actuation, Int. J. Nonlinear Mech., 87, 1-20 (2016)
[18] Ribeiro, P.; Thomas, O., Nonlinear modes of vibration and internal resonances in nonlocal beams, J. Comput. Nonlinear Dyn., 12, 3, Article 031017 pp. (2017)
[19] Salehipour, H.; Shahidi, A. R.; Nahvi, H., Modified nonlocal elasticity theory for functionally graded materials, Int. J. Eng. Sci., 90, 44-57 (2015) · Zbl 1423.74138
[20] Li, L.; Hu, Y., Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material, Int. J. Eng. Sci., 107, 77-97 (2016) · Zbl 1423.74496
[21] Faghidian, S. A., On non-linear flexure of beams based on non-local elasticity theory, Int. J. Eng. Sci., 124, 49-63 (2018) · Zbl 1423.74471
[22] Barretta, R.; Feo, L.; Luciano, R.; Marotti de Sciarra, F., A gradient Eringen model for functionally graded nanorods, Compos. Struct., 131, 1124-1131 (2015)
[23] Barretta, R.; Brčić, M.; Čanađija, M.; Luciano, R.; Marotti de Sciarra, F., Application of gradient elasticity to armchair carbon nanotubes: size effects and constitutive parameters assessment, Eur. J. Mech. A Solids, 65, 1-13 (2017) · Zbl 1406.74373
[24] Marotti de Sciarra, F.; Barretta, R., A new nonlocal bending model for Euler-Bernoulli nanobeams, Mech. Res. Commun., 62, 25-30 (2014)
[25] Čanađija, M.; Barretta, R.; Marotti de Sciarra, F., On functionally graded Timoshenko nonisothermal nanobeams, Compos. Struct., 135, 286-296 (2016)
[26] Barretta, R.; Feo, L.; Luciano, R.; Marotti de Sciarra, F.; Penna, R., Functionally graded Timoshenko nanobeams: a novel nonlocal gradient formulation, Composites Part B, 100, 208-219 (2016)
[27] Romano, G.; Barretta, R., Comment on the paper “Exact solution of Eringen”s nonlocal integral model for bending of Euler-Bernoulli and Timoshenko beams” by Meral Tuna and Mesut Kirca, Int. J. Eng. Sci., 109, 240-242 (2016)
[28] Romano, G.; Barretta, R.; Diaco, M., On nonlocal integral models for elastic nano-beams, Int. J. Mech. Sci., 131, 490-499 (2017)
[29] Romano, G.; Barretta, R., Stress-driven versus strain-driven nonlocal integral model for elastic nano-beams, Composites Part B, 114, 184-188 (2017)
[30] Romano, G.; Barretta, R., Nonlocal elasticity in nanobeams: the stress-driven integral model, Int. J. Eng. Sci., 115, 14-27 (2017) · Zbl 1423.74512
[31] Romano, G.; Barretta, R.; Diaco, M.; Marotti de Sciarra, F., Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, Int. J. Mech. Sci., 121, 151-156 (2017)
[32] Apuzzo, A.; Barretta, R.; Luciano, R.; Marotti de Sciarra, F.; Penna, R., Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model, Composites Part B, 123, 105-111 (2017)
[33] Liao, S., Beyond Perturbation: Introduction to the Homotopy Analysis Method (2003), CRC Press
[34] Liao, S., Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method, Nonlinear Anal. Real World Appl., 10, 4, 2455-2470 (2009) · Zbl 1163.35450
[35] Liao, S., Homotopy Analysis Method in Nonlinear Differential Equations (2012), Springer · Zbl 1253.35001
[36] Liao, S., Advances in the Homotopy Analysis Method (2013), World Scientific
[37] Jia, W.; He, X.; Guo, L., The optimal homotopy analysis method for solving linear optimal control problems, Appl. Math. Modell., 45, 865-880 (2017) · Zbl 1446.49026
[38] Nave, O.; Hareli, S.; Gol’dshtein, V., Singularly perturbed homotopy analysis method, Appl. Math. Modell., 38, 19-20, 4614-4624 (2014) · Zbl 1428.76171
[39] Kargarnovin, M. H.; Jafari-Talookolaei, R. A., Application of the homotopy method for the analytic approach of the nonlinear free vibration analysis of the simple end beams using four engineering theories, Acta Mech., 212, 3, 199-213 (2010) · Zbl 1397.74089
[40] Faghidian, S. A.; Moghimi Zand, M.; Farjami, Y.; Farrahi, G. H., Application of homotopy Padé technique in finding analytic solutions to the Volterra’s prey and predator problem, Int. J. Appl. Comput. Math., 10, 2, 262-270 (2011) · Zbl 1220.65091
[41] Kargarnovin, M. H.; Faghidian, S. A.; Farjami, Y.; Farrahi, G. H., Application of homotopy Padé technique in limit analysis of circular plates under arbitrary rotational symmetric loading using von Mises yield criterion, Commun. Nonlinear Sci. Numer. Simul., 15, 4, 1080-1091 (2010) · Zbl 1221.74052
[42] Haghani, A.; Mondali, M.; Faghidian, S. A., Linear and nonlinear flexural analysis of higher-order shear deformation laminated plates with circular delamination, Acta Mech. (2017), accepted for publication · Zbl 1390.74021
[43] Esfahani, S. E.; Kiani, Y.; Komijani, M.; Eslami, M. R., Vibration of a temperature-dependent thermally pre/postbuckled FGM beam over a nonlinear hardening elastic foundation, J. Appl. Mech., 81, 1, Article 011004 pp. (2014)
[44] Fallah, A.; Aghdam, M. M., Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation, Eur. J. Mech. A Solids, 30, 4, 571-583 (2011) · Zbl 1278.74074
[45] Nayfeh, A. H., Introduction to Perturbation Methods (1981), Wiley
[46] Mathai, A. M.; Haubold, H. J., Special Functions for Applied Scientists (2008), Springer · Zbl 1151.33001
[47] Singh, G.; Sharma, A. K.; Venkateswara Rao, G., Large-amplitude free vibrations of beams-a discussion on various formulations and assumptions, J. Sound Vib., 142, 1, 77-85 (1990)
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.