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Articles

Investigating the free vibration of viscoelastic FGM Timoshenko nanobeams resting on viscoelastic foundations with the shear correction factor using finite element method

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Pages 1278-1303 | Received 10 Aug 2022, Accepted 23 Oct 2022, Published online: 07 Nov 2022

References

  • Al Rjoub, Y. S, and A. G. Hamad. 2017. Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method. KSCE Journal of Civil Engineering 21 (3):792–806. doi:10.1007/s12205-016-0149-6.
  • Altenbach, H, and V. A. Eremeyev. 2008. Analysis of the viscoelastic behavior of plates made of functionally graded materials. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 88 (5):332–41. doi:10.1002/zamm.200800001.
  • Avcar, M. 2016. Effects of material non-homogeneity and two parameter elastic foundation on fundamental frequency parameters of Timoshenko beams. Acta Physica Polonica A 130 (1):375–8. doi:10.12693/APhysPolA.130.375.
  • Avcar, M, and W. K. M. Mohammed. 2018. Free vibration of functionally graded beams resting on Winkler–Pasternak foundation. Arabian Journal of Geosciences 11 (10):1–8. doi:10.1007/s12517-018-3579-2.
  • Chaillat, S, and H. D. Bui. 2007. Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations. Comptes Rendus Mécanique 335 (12):746–50. doi:10.1016/j.crme.2007.10.005.
  • Chen, W. J, and X. P. Li. 2013. Size-dependent free vibration analysis of composite laminated Timoshenko beam based on new modified couple stress theory. Archive of Applied Mechanics 83 (3):431–44. doi:10.1007/s00419-012-0689-2.
  • Chen, W.-R, and H. Chang. 2018. Vibration analysis of functionally graded Timoshenko Beams. International Journal of Structural Stability and Dynamics 18 (01):1850007. doi:10.1142/S0219455418500074.
  • Chen, W.-R, and C.-S. Chen. 2014. Effects of locally distributed Kelvin–Voigt damping on parametric instability of Timoshenko beams. International Journal of Structural Stability and Dynamics 14 (06):1450014. doi:10.1142/S021945541450014X.
  • Coşkun, S., Bozkurt, M. Tarik Atay, and B. Öztürk. 2011. Transverse vibration analysis of Euler-Bernoulli beams using analytical approximate techniques. Advances in Vibration Analysis Research. doi:10.5772/15891.
  • Cowper, G. R. 1966. The Shear Coefficient in Timoshenko’s Beam Theory.
  • Eringen, A. C. 1983. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics 54 (9):4703–10. doi:10.1063/1.332803.
  • Eringen, A. C. 2004. Nonlocal continuum field theories, ed. A. Cemal Eringen. New York, NY: Springer New York. doi:10.1007/b97697.
  • Hamza Madjid, B, and B. Bouderba. 2022. Buckling analysis of FGM plate exposed to different loads conditions. Mechanics Based Design of Structures and Machines 1–16. doi:10.1080/15397734.2022.2068576.
  • Hutchinson, J. R. 2001. Shear coefficients for Timoshenko beam theory. Journal of Applied Mechanics 68 (1):87–92. doi:10.1115/1.1349417.
  • Maxwell, J. C. 1867. IV. On the dynamical theory of gases. Philosophical Transactions of the Royal Society of London 157: 49–88. doi:10.1098/rstl.1867.0004.
  • Koizumi, M. 1993. “The Concept of FGM.” Ceramic transactions. Functionally graded materials 34:3–10.
  • Li, L, and Y. Hu. 2015. Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. International Journal of Engineering Science 97:84–94. doi:10.1016/j.ijengsci.2015.08.013.
  • Madabhusi-Raman, P, and J. F. Davalos. 1996. Static shear correction factor for laminated rectangular beams. Composites Part B: Engineering 27 (3–4):285–93. doi:10.1016/1359-8368(95)00014-3.
  • Malikan, M., V. B. Nguyen, and F. Tornabene. 2018. Damped forced vibration analysis of single-walled carbon nanotubes resting on viscoelastic foundation in thermal environment using nonlocal strain gradient theory. Engineering Science and Technology, an International Journal 21 (4):778–86. doi:10.1016/j.jestch.2018.06.001.
  • Marinetti, A, and G. Oliveto. 2009. On the evaluation of the shear correction factors: A boundary element approach. In Proceedings of the Congress of the Italian Association of Theoretical and Applied Mechanics (AIMETA).
  • Menaa, R., A. Tounsi, F. Mouaici, I. Mechab, M. Zidi, and E. A. A. Bedia. 2012. Analytical solutions for static shear correction factor of functionally graded rectangular beams. Mechanics of Advanced Materials and Structures 19 (8):641–52. doi:10.1080/15376494.2011.581409.
  • Nguyen, T.-K., K. Sab, and G. Bonnet. 2008. First-order shear deformation plate models for functionally graded materials. Composite Structures 83 (1):25–36. doi:10.1016/j.compstruct.2007.03.004.
  • Pasternak, P. L. 1954. On a new method of analysis of an elastic foundation by means of two foundation constants. Cosudarstrennoe Izdatelstvo Literaturi Po Stroitelstvu i Arkhitekture, Moscow, USSR 1–56.
  • Pradhan, K. K, and S. Chakraverty. 2013. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Composites Part B: Engineering 51 (August):175–84. doi:10.1016/j.compositesb.2013.02.027.
  • Pradhan, K. K, and S. Chakraverty. 2015. Generalized power-law exponent based shear deformation theory for free vibration of functionally graded beams. Applied Mathematics and Computation 268:1240–58. doi:10.1016/j.amc.2015.07.032.
  • Reddy, J. N, and S. D. Pang. 2008. Nonlocal continuum theories of beams for the analysis of carbon nanotubes. Journal of Applied Physics 103 (2):23511. doi:10.1063/1.2833431.
  • Reddy, J. N. 2007. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science 45 (2–8):288–307. doi:10.1016/j.ijengsci.2007.04.004.
  • Salen, J. 2009. Viscoélasticité Pour Le Calcul Des Structures. l’École polytechnique et Presses de l’École nationale des ponts et chaussées.
  • Sarparast, H., A. Ebrahimi-Mamaghani, M. Safarpour, H. M. Ouakad, R. Dimitri, and F. Tornabene. 2020. Nonlocal study of the vibration and stability response of small-scale axially moving supported beams on viscoelastic-pasternak foundation in a hygro-thermal environment. Mathematical Methods in the Applied Sciences. doi:10.1002/mma.6859.
  • Sharma, P, and R. Singh. 2021. A numerical study on free vibration analysis of axial FGM beam. Materials Today: Proceedings 44:1664–8. doi:10.1016/j.matpr.2020.11.827.
  • Shinohara, Y. 2013. Functionally graded materials. In Handbook of advanced ceramics: Materials, applications, processing, and properties: Second Edition, ed. Y. Miyamoto, W. A. Kaysser, B. H. Rabin, A. Kawasaki, and Reneé G. Ford, Vol. 5, 1179–87. Boston, MA: Springer US. doi:10.1016/B978-0-12-385469-8.00061-7.
  • Sofiyev, A. H. 2019. About an approach to the determination of the critical time of viscoelastic functionally graded cylindrical shells. Composites Part B: Engineering 156 (June 2018):156–65. doi:10.1016/j.compositesb.2018.08.073.
  • Stephen, N. G. 1980. Timoshenko’s shear coefficient from a beam subjected to gravity loading. Journal of Applied Mechanics 47 (1):121–7. doi:10.1115/1.3153589.
  • Tabatabaei, S., J. Shahidzadeh, and A. M. Fattahi. 2022. A finite element method for modal analysis of FGM plates. Mechanics Based Design of Structures and Machines 50 (4):1111–22. doi:10.1080/15397734.2020.1744004.
  • Timoshenko, S. P. 1921. LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 41 (245):744–6. doi:10.1080/14786442108636264.
  • Wang, C. M., S. Kitipornchai, C. W. Lim, and M. Eisenberger. 2008. Beam bending solutions based on nonlocal Timoshenko beam theory. Journal of Engineering Mechanics 134 (6):475–81. doi:10.1061/(ASCE)0733-9399(2008)134:6(475).
  • Wattanasakulpong, N, and V. Ungbhakorn. 2012. Free vibration analysis of functionally graded beams with general elastically end constraints by DTM. World Journal of Mechanics 02 (06):297–310. doi:10.4236/wjm.2012.26036.
  • Winkler, E. 1867. Die Lehre von Der Elasticitaet Und Festigkeit. Dominicus. Prag.
  • Xu, M. 2006. Free transverse vibrations of nano-to-micron scale beams. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462 (2074):2977–95. doi:10.1098/rspa.2006.1712.
  • Zhang, N. H, and M. L. Wang. 2006. A mathematical model of thermoviscoelastic FGM thin plates and Ritz approximate solutions. Acta Mechanica 181 (3–4):153–67. doi:10.1007/s00707-005-0300-9.
  • Zhang, Y. Y., C. M. Wang, and N. Challamel. 2010. Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model. Journal of Engineering Mechanics 136 (5):562–74. doi:10.1061/(ASCE)EM.1943-7889.0000107.
  • Zhen, Y. X., S. L. Wen, and Y. Tang. 2019. Free vibration analysis of viscoelastic nanotubes under longitudinal magnetic field based on nonlocal strain gradient Timoshenko beam model. Physica E: Low-Dimensional Systems and Nanostructures 105:116–24. doi:10.1016/j.physe.2018.09.005.
  • Zou, L., J. Yuan, X. Liu, J. Li, P. Zhang, and Z. Niu. 2021. Burgers viscoelastic model-based variable stiffness design of compliant clamping mechanism for leafy greens harvesting. Biosystems Engineering 208:1–15. doi:10.1016/j.biosystemseng.2021.05.007.

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