Analysis of torsional vibration in viscoelastic functionally graded nanotubes with viscoelastic constraints using doublet mechanics theory
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Abstract
Abstract In this study, the torsional vibration behavior of functionally graded viscoelastic nanotubes under viscoelastic boundary conditions is investigated in detail within the framework of Doublet Mechanics Theory. A comprehensive solution method is presented that allows the combined consideration of nanoscale effects and viscoelastic behavior; the effects of fundamental parameters such as viscous damping parameter, scale parameter and power law exponent on the system dynamics are analytically revealed. Physical interpretations of both vibration frequencies and damping effects are made from the obtained complex frequency solutions, and the effects of these parameters on the frequency spectrum are analyzed in detail with the help of tables and graphs. The results clearly indicate that classical elastic models are inadequate for the torsional vibration behavior of viscoelastic nanotubes and damping effects at the nano level should not be ignored. Furthermore, it is displayed that there are clear mathematical relationships between the real and imaginary components of the complex frequencies obtained in the system under the direct influence of the viscoelastic model used. In this context, the study makes an important contribution not only theoretically but also in terms of practical applications for the design of nano-mechanical systems.
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Research context
OpenAlex topics: Nonlocal and gradient elasticity in micro/nano structures · Composite Structure Analysis and Optimization · Thermoelastic and Magnetoelastic Phenomena
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References (61)
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1. Abdelrahman, A.A., Ashry, M., Alshorbagy, A.E., Abdallah, W.S.: On the mechanical behavior of two directional symmetrical functionally graded beams under moving load. Int. J. Mech. Mater. des. 17, 563–586 (2021). https://doi.org/10.1007/s10999-021-09547-910.1007/s10999-021-09547-9 ↗
2. Akbaş, ŞD., Dastjerdi, S., Akgöz, B., Civalek, Ö.: Dynamic analysis of functionally graded porous microbeams under moving load. Transp Porous Med. 142, 209–227 (2022). https://doi.org/10.1007/s11242-021-01686-z10.1007/s11242-021-01686-z ↗
3. Akgöz, B., Civalek, Ö.: Buckling analysis of functionally graded tapered microbeams via rayleigh-ritz method. Mathematics 10, 4429 (2022). https://doi.org/10.3390/math1023442910.3390/math10234429 ↗
4. 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. Compos. Part B Eng. 123, 105–111 (2017). https://doi.org/10.1016/j.compositesb.2017.03.05710.1016/j.compositesb.2017.03.057 ↗
5. Attia, M.A., Abdel Rahman, A.A.: On vibrations of functionally graded viscoelastic nanobeams with surface effects. Int. J. Eng. Sci. 127, 1–32 (2018). https://doi.org/10.1016/j.ijengsci.2018.02.00510.1016/j.ijengsci.2018.02.005 ↗
6. Avcar, M., Hadji, L., Civalek, Ö.: Free vibration analysis of porous functionally graded sandwich beams. In: Functionally Graded Structures: Modelling and computation of static and dynamical problems. IOP Publishing (2023)10.1088/978-0-7503-5301-4ch8 ↗
7. Bagheri, R., Tadi Beni, Y.: On the size-dependent nonlinear dynamics of viscoelastic/flexoelectric nanobeams. J. Vib. Controlvib. Control 27, 2018–2033 (2021). https://doi.org/10.1177/107754632095222510.1177/1077546320952225 ↗
8. Barretta, R., Faghidian, S.A., Luciano, R., Medaglia, C.M., Penna, R.: Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress-driven nonlocal models. Compos. B Eng. 154, 20–32 (2018). https://doi.org/10.1016/j.compositesb.2018.07.03610.1016/j.compositesb.2018.07.036 ↗
9. Barretta, R., Luciano, R., Marotti De Sciarra, F., Vaccaro, M.S.: On torsion of FG elastic nanobeams on nonlocal foundations. Compos. Struct. 340, 118146 (2024). https://doi.org/10.1016/j.compstruct.2024.11814610.1016/j.compstruct.2024.118146 ↗
10. Belabed, Z., Tounsi, A., Bousahla, A.A., Tounsi, A., Bourada, M., Al-Osta, M.A.: Free vibration analysis of Bi-directional functionally graded beams using a simple and efficient finite element model. Struct. Eng. Mech. an Int’l J. 90, 233–252 (2024)
11. Beni, Y.T.: Size-dependent torsional wave propagation in FG flexoelectric micro/nanotubes. Waves Random Complex Med. (2022). https://doi.org/10.1080/17455030.2022.209402710.1080/17455030.2022.2094027 ↗
12. Bu, Y., Tang, Y., Ding, Q.: Novel vibration self-suppression of periodic pipes conveying fluid based on acoustic black hole effect. J. Sound Vib.vib. 567, 118077 (2023). https://doi.org/10.1016/j.jsv.2023.11807710.1016/j.jsv.2023.118077 ↗
13. Challamel, N., Wang, C.M.: The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19, 345703 (2008). https://doi.org/10.1088/0957-4484/19/34/34570310.1088/0957-4484/19/34/345703 ↗
14. Challamel, N., El-Borgi, S., Trabelssi, M., Reddy, J.N.: Free vibration response of micromorphic Timoshenko beams. J. Sound Vib.vib. 591, 118602 (2024). https://doi.org/10.1016/j.jsv.2024.11860210.1016/j.jsv.2024.118602 ↗
15. Challamel, N., El-Borgi, S., Trabelssi, M., Reddy, J.N.: Buckling of micromorphic Timoshenko columns. Eur. J. Mech. a. Solids 111, 105537 (2025). https://doi.org/10.1016/j.euromechsol.2024.10553710.1016/j.euromechsol.2024.105537 ↗
16. Civalek, Ö., Uzun, B., Yayli, M.Ö.: Buckling analysis of nanobeams with deformable boundaries via doublet mechanics. Arch. Appl. Mech. 91, 4765–4782 (2021). https://doi.org/10.1007/s00419-021-02032-x10.1007/s00419-021-02032-x ↗
17. Civalek, Ö., Uzun, B., Yayli, M.Ö.: Torsional vibrations of functionally graded restrained nanotubes. Eur. Phys. J. plus 137, 113 (2022a). https://doi.org/10.1140/epjp/s13360-021-02309-810.1140/epjp/s13360-021-02309-8 ↗
18. Civalek, Ö., Uzun, B., Yaylı, M.Ö.: An eigenvalue solution for torsional vibrations of restrained porous nanorods using doublet mechanics theory. J. Braz. Soc. Mech. Sci. Eng. 44, 116 (2022b). https://doi.org/10.1007/s40430-022-03415-z10.1007/s40430-022-03415-z ↗
19. Ding, H., Zhao, C., Fan, Y., Zhao, D., Su, J.: Vibration analysis of cracked functionally graded elliptical plates based on finite element phase-field model. Int. J. Mech. Mater. des. 18, 549–565 (2022). https://doi.org/10.1007/s10999-022-09592-y10.1007/s10999-022-09592-y ↗
20. Ebrahimian, M.R., Imam, A., Najafi, M.: Doublet mechanical analysis of bending of Euler-Bernoulli and Timoshenko nanobeams. ZAMM – J. Appl. Math. Mech. / Zeitschrift Für Angewandte Mathematik und Mechanik. 98, 1642–1665 (2018). https://doi.org/10.1002/zamm.20170036510.1002/zamm.201700365 ↗
21. El-Borgi, S., Rajendran, P., Friswell, M.I., Trabelssi, M., Reddy, J.N.: Torsional vibration of size-dependent viscoelastic rods using nonlocal strain and velocity gradient theory. Compos. Struct.struct. 186, 274–292 (2018). https://doi.org/10.1016/j.compstruct.2017.12.00210.1016/j.compstruct.2017.12.002 ↗
22. Eltaher, M.A., Mohamed, N.: Nonlinear stability and vibration of imperfect CNTs by Doublet mechanics. Appl. Math. Comput.comput. 382, 125311 (2020). https://doi.org/10.1016/j.amc.2020.12531110.1016/j.amc.2020.125311 ↗
23. Eltaher, M.A., Shanab, R.A., Mohamed, N.A.: Analytical solution of free vibration of viscoelastic perforated nanobeam. Arch. Appl. Mech. 93, 221–243 (2023). https://doi.org/10.1007/s00419-022-02184-410.1007/s00419-022-02184-4 ↗
24. Esen, I., Abdelrahman, A.A., Eltaher, M.A.: On vibration of sigmoid/symmetric functionally graded nonlocal strain gradient nanobeams under moving load. Int. J. Mech. Mater. des. 17, 721–742 (2021). https://doi.org/10.1007/s10999-021-09555-910.1007/s10999-021-09555-9 ↗
25. Fatahi-Vajari, A., Azimzadeh, Z.: Axial vibration of single-walled carbon nanotubes with fractional damping using doublet mechanics. Indian J. Phys. 94, 975–986 (2020). https://doi.org/10.1007/s12648-019-01547-y10.1007/s12648-019-01547-y ↗
26. Ferrari, M., Granik, V.T., Imam, A.: Introduction to Doublet Mechanics. In: Ferrari, M., Granik, V.T., Imam, A., Nadeau, J.C. (eds.) Advances in Doublet Mechanics, pp. 1–26. Springer, Berlin, Heidelberg (1997)10.1007/978-3-540-49636-6 ↗
27. Gul, U., Aydogdu, M., Gaygusuzoglu, G.: Axial dynamics of a nanorod embedded in an elastic medium using doublet mechanics. Compos. Struct.struct. 160, 1268–1278 (2017). https://doi.org/10.1016/j.compstruct.2016.11.02310.1016/j.compstruct.2016.11.023 ↗
28. Jena, S.K., Chakraverty, S., Jena, R.M., Tornabene, F.: A novel fractional nonlocal model and its application in buckling analysis of Euler-Bernoulli nanobeam. Mater. Res. Exp. 6, 055016 (2019). https://doi.org/10.1088/2053-1591/ab016b10.1088/2053-1591/ab016b ↗
29. Jena, S.K., Chakraverty, S., Mahesh, V., Harursampath, D.: Wavelet-based techniques for Hygro-Magneto–Thermo vibration of nonlocal strain gradient nanobeam resting on Winkler–Pasternak elastic foundation. Eng. Anal. Boundary Elem. 140, 494–506 (2022). https://doi.org/10.1016/j.enganabound.2022.04.03710.1016/j.enganabound.2022.04.037 ↗
30. Kadioglu, H.G., Yayli, M.O.: Axial vibration of a viscoelastic FG nanobeam with arbitrary boundary conditions. J. Vib. Eng. Technol. 13, 96 (2025). https://doi.org/10.1007/s42417-024-01671-y10.1007/s42417-024-01671-y ↗
31. Kadioglu, H.G., Uzun, B., Yayli, M.O.: Size-dependent vibration analysis of a FG viscoelastic nanotube attached with the time-dependent boundary conditions using NLSGT. ZAMM–J. Appl. Math. Mech. / Zeitschrift Für Angewandte Mathematik und Mechanik. 105, e202400934 (2005). https://doi.org/10.1002/zamm.20240093410.1002/zamm.202400934 ↗
32. Kadıoğlu, H.G., Civalek, Ö., Uzun, B., Yaylı, M.Ö.: Size-dependent vibration and static analyses of a nanobeam made of time-dependent material attached with viscoelastic boundaries using three different beam theories. Acta Mech. Mech. (2025). https://doi.org/10.1007/s00707-025-04228-610.1007/s00707-025-04228-6 ↗
33. Kafkas, U., Unal, Y., Yayli, M.O., Uzun, B.: Buckling analysis of perforated nano/microbeams with deformable boundary conditions via nonlocal strain gradient elasticity. Adv. Nano Res. 15, 339–353 (2023). https://doi.org/10.12989/anr.2023.15.4.33910.12989/anr.2023.15.4.339 ↗
34. Karamanli, A.: Structural behaviours of zigzag and armchair nanobeams using finite element doublet mechanics. European Journal of Mechanics. (2021)10.1016/j.euromechsol.2021.104287 ↗
35. Lei, Y., Adhikari, S., Friswell, M.I.: Vibration of nonlocal Kelvin–Voigt viscoelastic damped Timoshenko beams. Int. J. Eng. Sci. 66–67, 1–13 (2013a). https://doi.org/10.1016/j.ijengsci.2013.02.00410.1016/j.ijengsci.2013.02.004 ↗
36. Lei, Y., Murmu, T., Adhikari, S., Friswell, M.I.: Dynamic characteristics of damped viscoelastic nonlocal Euler-Bernoulli beams. European J. Mech.–A/Solids. 42, 125–136 (2013b). https://doi.org/10.1016/j.euromechsol.2013.04.00610.1016/j.euromechsol.2013.04.006 ↗
37. Lezgy-Nazargah, M., Trinh, L.C., Wattanasakulpong, N., Vo, T.P.: Finite element model for stability and vibration analyses of bi-directional FG curved sandwich beams. Int. J. Mech. Mater. des. 20, 849–875 (2024). https://doi.org/10.1007/s10999-023-09700-610.1007/s10999-023-09700-6 ↗
38. Li, C.: Torsional vibration of carbon nanotubes: comparison of two nonlocal models and a semi-continuum model. Int. J. Mech. Sci. 82, 25–31 (2014). https://doi.org/10.1016/j.ijmecsci.2014.02.02310.1016/j.ijmecsci.2014.02.023 ↗
39. Li, Y., Tang, Y.: Application of Galerkin iterative technique to nonlinear bending response of three-directional functionally graded slender beams subjected to hygro-thermal loads. Compos. Struct.struct. 290, 115481 (2022). https://doi.org/10.1016/j.compstruct.2022.11548110.1016/j.compstruct.2022.115481 ↗
40. Li, C., Liu, J.J., Cheng, M., Fan, X.L.: Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces. Compos. B Eng. 116, 153–169 (2017). https://doi.org/10.1016/j.compositesb.2017.01.07110.1016/j.compositesb.2017.01.071 ↗
41. Li, C., Lai, S.K., Yang, X.: On the nano-structural dependence of nonlocal dynamics and its relationship to the upper limit of nonlocal scale parameter. Appl. Math. Model. 69, 127–141 (2019). https://doi.org/10.1016/j.apm.2018.12.01010.1016/j.apm.2018.12.010 ↗
42. Li, C., Zhu, C., Lim, C.W., Li, S.: Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading. Appl. Math. Mech.-Engl. Ed. 43, 1821–1840 (2022). https://doi.org/10.1007/s10483-022-2917-710.1007/s10483-022-2917-7 ↗
43. Li, H.N., Wang, W., Lai, S.K., Yao, L.Q., Li, C.: Nonlinear Vibration and Stability Analysis of Rotating Functionally Graded Piezoelectric Nanobeams. Int. J. Str. Stab. Dyn. 24, 2450103 (2024). https://doi.org/10.1142/S021945542450103710.1142/S0219455424501037 ↗
44. Lim, C.W., Li, C., Yu, J.L.: Free torsional vibration of nanotubes based on nonlocal stress theory. J. Sound Vib.vib. 331, 2798–2808 (2012). https://doi.org/10.1016/j.jsv.2012.01.01610.1016/j.jsv.2012.01.016 ↗
45. Luo, Q., Li, C., Li, S.: Transverse free vibration of axisymmetric functionally graded circular nanoplates with radial loads. J. Vib. Eng. Technol. 9, 1253–1268 (2021). https://doi.org/10.1007/s42417-021-00294-x10.1007/s42417-021-00294-x ↗
46. Nguyen, L.B., Nguyen-Xuan, H., Thai, C.H., Phung-Van, P.: A size-dependent effect of smart functionally graded piezoelectric porous nanoscale plates. Int. J. Mech. Mater. des. 19, 817–830 (2023). https://doi.org/10.1007/s10999-023-09660-x10.1007/s10999-023-09660-x ↗
47. Nguyen, N.-D., Bui, V.-T., Trinh, L.C., Le, Q.-C.: Elastic foundation effect on the small-scale analysis of functionally graded porous microbeams using a modified strain gradient theory. Int. J. Mech. Mater. des. (2024). https://doi.org/10.1007/s10999-024-09735-310.1007/s10999-024-09735-3 ↗
48. Özmen, R., Esen, I.: Dynamic response of embedded Timoshenko CNTs exposed to magnetic and thermal fields subjected to moving load based on doublet mechanics. J. Braz. Soc. Mech. Sci. Eng. 45, 590 (2023). https://doi.org/10.1007/s40430-023-04506-110.1007/s40430-023-04506-1 ↗
49. Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007). https://doi.org/10.1016/j.ijengsci.2007.04.00410.1016/j.ijengsci.2007.04.004 ↗
50. Tang, Y., Ding, Q.: Nonlinear vibration analysis of a bi-directional functionally graded beam under hygro–thermal loads. Compos. Struct.struct. 225, 111076 (2019). https://doi.org/10.1016/j.compstruct.2019.11107610.1016/j.compstruct.2019.111076 ↗
51. Tang, Y., Wang, G., Ren, T., Ding, Q., Yang, T.: Nonlinear mechanics of a slender beam composited by three-directional functionally graded materials. Compos. Struct.struct. 270, 114088 (2021). https://doi.org/10.1016/j.compstruct.2021.11408810.1016/j.compstruct.2021.114088 ↗
52. Tang, Y., Gao, C., Li, M., Ding, Q.: Novel active-passive hybrid piezoelectric network for vibration suppression in fluid-conveying pipes. Appl. Math. Modell. 117, 378–398 (2023a). https://doi.org/10.1016/j.apm.2022.12.02810.1016/j.apm.2022.12.028 ↗
53. Tang, Y., Li, C.-L., Yang, T.: Application of the generalized differential quadrature method to study vibration and dynamic stability of tri-directional functionally graded beam under magneto–electro–elastic fields. Eng. Anal. Bound. Elements. 146, 808–823 (2023b). https://doi.org/10.1016/j.enganabound.2022.11.01610.1016/j.enganabound.2022.11.016 ↗
54. Tang, Y., Wang, G., Yang, T., Ding, Q.: Nonlinear dynamics of three-directional functional graded pipes conveying fluid with the integration of piezoelectric attachment and nonlinear energy sink. Nonlinear Dyn. 111, 2415–2442 (2023c). https://doi.org/10.1007/s11071-022-07971-w10.1007/s11071-022-07971-w ↗
55. Tariq, A., Kadıoğlu, H.G., Uzun, B., Deliktaş, B., Yaylı, M.Ö.: Modeling the viscoelastic behavior of a FG nonlocal beam with deformable boundaries based on hybrid machine learning and semi-analytical approaches. Arch. Appl. Mech. 95, 79 (2025). https://doi.org/10.1007/s00419-025-02776-w10.1007/s00419-025-02776-w ↗
56. Tounsi, A., Benguediab, S., Bedia, E.A.A., Semmah, A., Zidour, M.: Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes. Adv. Nano Res. 1, 1–11 (2013a)10.12989/anr.2013.1.1.001 ↗
57. Tounsi, A., Semmah, A., Bousahla, A.A.: Thermal buckling behavior of nanobeams using an efficient higher-order nonlocal beam theory. J. Nanomech. Micromech. 3, 37–42 (2013b). https://doi.org/10.1061/(ASCE)NM.2153-5477.000005710.1061/(ASCE)NM.2153-5477.0000057 ↗
58. Wang, S., Ding, W., Li, Z., Xu, B., Zhai, C., Kang, W., Yang, W., Li, Y.: A size-dependent quasi-3D model for bending and buckling of porous functionally graded curved nanobeam. Int. J. Eng. Sci. 193, 103962 (2023). https://doi.org/10.1016/j.ijengsci.2023.10396210.1016/j.ijengsci.2023.103962 ↗
59. Yao, L.Q., Ji, C.J., Shen, J.P., Li, C.: Free vibration and wave propagation of axially moving functionally graded Timoshenko microbeams. J. Braz. Soc. Mech. Sci. Eng. 42, 137 (2020). https://doi.org/10.1007/s40430-020-2206-910.1007/s40430-020-2206-9 ↗
60. Zeighampour, H., Tadi Beni, Y., Karimipour, I.: Material length scale and nonlocal effects on the wave propagation of composite laminated cylindrical micro/nanoshells. Eur. Phys. J. plus. 132, 503 (2017). https://doi.org/10.1140/epjp/i2017-11770-710.1140/epjp/i2017-11770-7 ↗
61. Zhen, Y.-X., Wen, S.-L., Tang, Y.: Free vibration analysis of viscoelastic nanotubes under longitudinal magnetic field based on nonlocal strain gradient Timoshenko beam model. Physica E E 105, 116–124 (2019). https://doi.org/10.1016/j.physe.2018.09.00510.1016/j.physe.2018.09.005 ↗