Finite element forced vibration analysis of refined shear deformable nanocomposite graphene platelet-reinforced beams

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In this paper, forced vibration investigation of a nanocomposite beam reinforced with different distributions of graphene platelets (GPLs) in thermal environments is performed by the development of a higher-order refined beam element. The developed beam element contains ten degrees of freedom related to axial, bending and transverse shear displacements. A uniformly distributed dynamic load with harmonic oscillations is exerted to the nanocomposite beam. Different GPL distributions such as uniform, linear and nonlinear are considered. Convergence studies of the obtained results from the finite element method are also provided. Also, the obtained results based on the presented solution are verified with a previously published paper on vibration of GPL-reinforced beams. This research shows that the resonance behavior of a nanocomposite beam can be controlled by the GPL content and dispersions. Therefore, it is showed that the dynamical deflection is significantly affected by GPL weight fraction, type of GPL distribution, temperature change, elastic substrate and excitation frequency of applied dynamic load.

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  1. 1.

    Zhang LW (2017) On the study of the effect of in-plane forces on the frequency parameters of CNT-reinforced composite skew plates. Compos Struct 160:824–837

  2. 2.

    Shenas AG, Malekzadeh P, Ziaee S (2017) Vibration analysis of pre-twisted functionally graded carbon nanotube reinforced composite beams in thermal environment. Compos Structres 162:325–340

  3. 3.

    Malikan M, Nguyen VB, Tornabene F (2018) Damped forced vibration analysis of single-walled carbon nanotubes resting on viscoelastic foundation in thermal environment using nonlocal strain gradient theory. Eng Sci Technol Int J 21(4):778–786

  4. 4.

    Kiani Y, Dimitri R, Tornabene F (2018) Free vibration of FG-CNT reinforced composite skew cylindrical shells using the Chebyshev–Ritz formulation. Compos Part B Eng 147:169–177

  5. 5.

    Keleshteri MM, Asadi H, Wang Q (2017) Large amplitude vibration of FG-CNT reinforced composite annular plates with integrated piezoelectric layers on elastic foundation. Thin Walled Struct 120:203–214

  6. 6.

    Ebrahimi F, Farazmandnia N (2017) Thermo-mechanical vibration analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets based on a higher-order shear deformation beam theory. Mech Adv Mater Struct 24(10):820–829

  7. 7.

    Yang B, Yang J, Kitipornchai S (2017) Thermoelastic analysis of functionally graded graphene reinforced rectangular plates based on 3D elasticity. Meccanica 52(10):2275–2292

  8. 8.

    Barati MR, Zenkour AM (2019) Analysis of postbuckling of graded porous GPL-reinforced beams with geometrical imperfection. Mech Adv Mater Struct 26(6):503–511

  9. 9.

    Rafiee MA, Rafiee J, Wang Z, Song H, Yu ZZ, Koratkar N (2009) Enhanced mechanical properties of nanocomposites at low graphene content. ACS Nano 3(12):3884–3890

  10. 10.

    King JA, Klimek DR, Miskioglu I, Odegard GM (2013) Mechanical properties of graphene nanoplatelet/epoxy composites. J Appl Polym Sci 128(6):4217–4223

  11. 11.

    Fang M, Wang K, Lu H, Yang Y, Nutt S (2009) Covalent polymer functionalization of graphene nanosheets and mechanical properties of composites. J Mater Chem 19(38):7098–7105

  12. 12.

    Zhao X, Zhang Q, Chen D, Lu P (2010) Enhanced mechanical properties of graphene-based poly (vinyl alcohol) composites. Macromolecules 43(5):2357–2363

  13. 13.

    Song M, Kitipornchai S, Yang J (2017) Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos Struct 159:579–588

  14. 14.

    Shen HS, Xiang Y, Lin F, Hui D (2017) Buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates in thermal environments. Compos Part B Eng 119:67–78

  15. 15.

    Shen HS, Xiang Y, Lin F (2017) Nonlinear bending of functionally graded graphene-reinforced composite laminated plates resting on elastic foundations in thermal environments. Compos Struct 170:80–90

  16. 16.

    Feng C, Kitipornchai S, Yang J (2017) Nonlinear free vibration of functionally graded polymer composite beams reinforced with graphene nanoplatelets (GPLs). Eng Struct 140:110–119

  17. 17.

    Kitipornchai S, Chen D, Yang J (2017) Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets. Mater Des 116:656–665

  18. 18.

    Kahya V, Turan M (2017) Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory. Compos Part B Eng 109:108–115

  19. 19.

    Khan AA, Naushad Alam M, Wajid M (2016) Finite element modelling for static and free vibration response of functionally graded beam. Latin Am J Solids Struct 13(4):690–714

  20. 20.

    Taeprasartsit S (2015) Nonlinear free vibration of thin functionally graded beams using the finite element method. J Vib Control 21(1):29–46

  21. 21.

    Bounouara F, Benrahou KH, Belkorissat I, Tounsi A (2016) A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation. Steel Compos Struct 20(2):227–249

  22. 22.

    Zemri A, Houari MSA, Bousahla AA, Tounsi A (2015) A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory. Struct Eng Mech 54(4):693–710

  23. 23.

    Ebrahimi F, Barati MR (2016) A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures. Int J Eng Sci 107:183–196

  24. 24.

    Ebrahimi F, Barati MR (2017) Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory. Compos Struct 159:433–444

  25. 25.

    Vo TP, Thai HT, Nguyen TK, Maheri A, Lee J (2014) Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng Struct 64:12–22

  26. 26.

    Zare Y, Rhee KY (2017) The mechanical behavior of CNT reinforced nanocomposites assuming imperfect interfacial bonding between matrix and nanoparticles and percolation of interphase regions. Compos Sci Technol 144:18–25

  27. 27.

    Liew KM, Lei ZX, Zhang LW (2015) Mechanical analysis of functionally graded carbon nanotube reinforced composites: a review. Compos Struct 120:90–97

  28. 28.

    Ebrahimi F, Barati MR (2016) A unified formulation for dynamic analysis of nonlocal heterogeneous nanobeams in hygro-thermal environment. Appl Phys A 122(9):792

  29. 29.

    Ebrahimi F, Barati MR (2016) Thermal environment effects on wave dispersion behavior of inhomogeneous strain gradient nanobeams based on higher order refined beam theory. J Therm Stress 39(12):1560–1571

  30. 30.

    Barati MR, Zenkour AM (2018) Post-buckling analysis of imperfect multi-phase nanocrystalline nanobeams considering nanograins and nanopores surface effects. Compos Struct 184:497–505

  31. 31.

    El Abbas AB, Abdelouahed T (2016) Comparison of different local unrefined beam theories for bending and nano beams buckling analysis. J Mater Eng Struct JMES 3(1):2–13

  32. 32.

    Meziane MAA, Abdelaziz HH, Tounsi A (2014) An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. J Sandw Struct Mater 16(3):293–318

  33. 33.

    Al-Basyouni KS, Tounsi A, Mahmoud SR (2015) Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position. Compos Struct 125:621–630

  34. 34.

    Jena SK, Chakraverty S, Tornabene F (2019) Dynamical behavior of nanobeam embedded in constant, linear, parabolic, and sinusoidal types of Winkler elastic foundation using first-Order nonlocal strain gradient model. Mater Res Exp 6(8):0850f2

  35. 35.

    Jena SK, Chakraverty S, Tornabene F (2019) Vibration characteristics of nanobeam with exponentially varying flexural rigidity resting on linearly varying elastic foundation using differential quadrature method. Mater Res Exp 6(8):085051

  36. 36.

    Sepahvand K (2016) Spectral stochastic finite element vibration analysis of fiber-reinforced composites with random fiber orientation. Compos Struct 145:119–128

  37. 37.

    Hutton DV (2017) Fundamentals of finite element analysis. McGraw-hill, New York

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Correspondence to Hossein Shahverdi.

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Barati, M.R., Shahverdi, H. Finite element forced vibration analysis of refined shear deformable nanocomposite graphene platelet-reinforced beams. J Braz. Soc. Mech. Sci. Eng. 42, 33 (2020) doi:10.1007/s40430-019-2118-8

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  • Forced vibration
  • Refined beam element
  • Graphene platelet
  • GPL-reinforced beam
  • Dynamic loading