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Free nonlinear vibration analysis of a functionally graded microbeam resting on a three-layer elastic foundation using the continuous piecewise linearization method

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Abstract

In this study, the free vibration analysis of a functionally graded microbeam (FGM) resting on a foundation with Winkler, Pasternak and nonlinear stiffness was undertaken. The free vibration of the FGM was modeled using the classical rule of mixture and Mori–Tanaka homogenization schemes for the material properties, the modified coupled stress theory to account for the size-dependent effects and the von Karman strain–displacement relation to account for the geometric nonlinearity. The resultant partial differential equation was transformed to an ordinary differential equation (ODE) by the Galerkin method. The governing ODE is in the form of a Helmholtz–Duffing oscillator that is applicable to isotropic microbeam as well as the FGM. The Helmholtz–Duffing model was solved using the continuous piecewise linearization method, which compared well with numerical solutions and was shown to be more accurate than other published approximate solutions. Detailed analysis of various parameters affecting the dynamic response of the FGM was conducted and revealed many interesting results. The Young modulus ratio, foundation stiffness parameters, power law index and type of boundary condition were seen to influence the vibration response of the FG microbeam. Specifically, a stronger nonlinear response was observed when (a) the length scale parameter, which represents the size effect, is significantly smaller than the thickness of the microbeam, (b) the nonlinear stiffness parameter increases, (c) the power index value is unity and (d) the boundary conditions are less restrictive. Lastly, the effect of thermal load on the buckling of the microbeam was investigated and showed that the thermal buckling load depends on the aspect ratio, length scale ratio and boundary conditions of the FG microbeam.

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References

  1. Akbas, S.D.: Forced vibration responses of axially functionally graded beams by using Ritz method. J. Appl. Comput. Mech. 7(1), 109–115 (2021)

    MathSciNet  Google Scholar 

  2. Akbaş, ŞD.: Forced vibration analysis of functionally graded porous deep beams. Compos. Struct. (2018). https://doi.org/10.1016/j.compstruct.2017.12.013

    Article  Google Scholar 

  3. Akgöz, B., Civalek, Ö.: Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory. Compos. Struct. (2013). https://doi.org/10.1016/j.compstruct.2012.11.020

    Article  Google Scholar 

  4. Alimoradzadeh, M., Salehi, M., Esfarjani, S.M.: (2020): Nonlinear vibration analysis of axially functionally graded microbeams based on nonlinear elastic foundation using modified couple stress theory. Periodica Polytechnica Mech. Eng. 64(2), 97–108 (2020). https://doi.org/10.3311/PPme.11684

    Article  Google Scholar 

  5. Ansari, R., Gholami, R., Sahmani, S.: Size-dependent vibration of functionally graded curved microbeams based on the modified strain gradient elasticity theory. Arch. Appl. Mech. (2013). https://doi.org/10.1007/s00419-013-0756-3

    Article  Google Scholar 

  6. Barretta, R., Marotti de Sciarra, F.: Variational nonlocal gradient elasticity for nano-beams. Int. J. Eng. Sci. 143, 73–91 (2019)

    Article  MathSciNet  Google Scholar 

  7. Big-Alabo, A.: Periodic solutions of duffing-type oscillators using continuous piecewise linearization method. Mech. Eng. Res. 8(1), 615–1927 (2018). https://doi.org/10.5539/mer.v8n1p41

    Article  Google Scholar 

  8. Big-Alabo, A., Ossia, C.V.: Periodic oscillation and bifurcation analysis of pendulum with spinning support using a modified continuous piecewise linearization method. Int. J. Appl. Comput. Math. (2019). https://doi.org/10.1007/s40819-019-0697-9

    Article  MathSciNet  Google Scholar 

  9. Big-Alabo, A., Ossia, C.V.: Periodic solution of nonlinear conservative systems. Progr. Relat. Chapter 14, 235 (2020). https://doi.org/10.5772/intechopen.90282

    Article  Google Scholar 

  10. Big-Alabo, A., Ezekwem, C.: Accurate solution and analysis of the transient temperature and stability of combustible micron-sized iron particle in gaseous oxidizing environment. Int. J. Appl. Comput. Math. 7, 1–18 (2021). https://doi.org/10.1007/s40819-021-00998-4

    Article  MathSciNet  Google Scholar 

  11. Dang, V.H.: Postbuckling and free nonlinear vibration of microbeams based on nonlinear elastic foundation. Math. Probl. Eng. 2018, 1–17 (2018). https://doi.org/10.1155/2018/1031237

    Article  MathSciNet  Google Scholar 

  12. Dang, V.H., Duong, T.-H., Bui, G.-P.: Nonlinear vibration of a functionally graded nanobeam based on the nonlocal strain gradient theory considering thickness effect. Adv. Civil Eng. (2020). https://doi.org/10.1155/2020/9407673

    Article  Google Scholar 

  13. Dang, V.H., Nguyen, D.-A., Le, M.-Q., Ninh, Q.-H.: Nonlinear vibration of microbeams based on the nonlinear elastic foundation using the equivalent linearization method with a weighted averaging. Arch. Appl. Mech. (2020). https://doi.org/10.1007/s00419-019-01599-w

    Article  Google Scholar 

  14. Hieu, D.V., Hoa, N.T., Duy, L.Q., Kim Thoa, N.T.: Nonlinear vibration of an electrostatically actuated‎ functionally graded microbeam under longitudinal magnetic‎ field. J. Appl. Comput. Mech. 7(3), 1537–1549 (2021)

    Google Scholar 

  15. Dehrouyeh-Semnani, A.M., Mostafaei, H., Nikkhah-Bahrami, M.: Free flexural vibration of geometrically imperfect functionally graded microbeams. Int. J. Eng. Sci. (2016). https://doi.org/10.1016/j.ijengsci.2016.05.002

    Article  MathSciNet  Google Scholar 

  16. Dinh Duc, N., Quang, V.D., Nguyen, P.D., Chien, T.M.: Nonlinear dynamic response of functionally graded porous plates on elastic foundation subjected to thermal and mechanical loads. J. Appl. Comput. Mech. 4(4), 245–259 (2018)

    Google Scholar 

  17. Fang, J., Gu, J., Wang, H.: Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory. Int. J. Mech. Sci. (2018). https://doi.org/10.1016/j.ijmecsci.2017.12.028

    Article  Google Scholar 

  18. He, Y., Qing, H., Gao, C.-F.: Theoretical analysis of free vibration of microbeams under different boundary conditions using stress-driven nonlocal integral model. Int. J. Struct. Stabil. Dyn. (2020). https://doi.org/10.1142/S0219455420500406

    Article  MathSciNet  Google Scholar 

  19. Aria, A.I., Rabczuk, T., Friswell, M.I.: A finite element model for the thermo-elastic analysis of functionally graded porous nanobeams. Eur. J. Mech. / A Solids (2019). https://doi.org/10.1016/j.euromechsol.2019.04.002

    Article  MathSciNet  Google Scholar 

  20. Ke, L.-L., Wang, Y.-S., Yang, J., Kitipornchai, S.: Nonlinear free vibration of size-dependent functionally graded microbeams. Int. J. Eng. Sci. (2012). https://doi.org/10.1016/j.ijengsci.2010.12.008

    Article  MathSciNet  Google Scholar 

  21. Li, L., Haishan, T., Yujin, H.: Effect of thickness on the mechanics of nanobeams. Int. J. Eng. Sci. 123, 81–91 (2018). https://doi.org/10.1016/j.ijengsci.2017.11.021

    Article  MathSciNet  Google Scholar 

  22. Li, Z., Xu, Y., Huang, D.: Analytical solution for vibration of functionally graded beams with variable cross-sections resting on Pasternak elastic foundations. Int. J. Mech. Sci. 191(2021), 106084 (2021). https://doi.org/10.1016/j.ijmecsci.2020.106084

    Article  Google Scholar 

  23. Majid, A.K.: The material length scale parameter used in couple stress theories is not a material constant. Int. J. Eng. Sci. (2018). https://doi.org/10.1016/j.ijengsci.2018.08.005

    Article  Google Scholar 

  24. Pinnola, F.P., Faghidian, S.A., Barretta, R., de Sciarra, F.M.: Variationally consistent dynamics of nonlocal gradient elastic beams. Int. J. Eng. Sci. 149, 103220 (2020)

    Article  MathSciNet  Google Scholar 

  25. Penna, R., Feo, L., Lovisi, G., Fabbrocino, F.: Application of the higher-order hamilton approach to the nonlinear free vibrations analysis of porous FG nano-beams in a hygrothermal environment based on a local/nonlocal stress gradient model of elasticity. Nanomaterials 12, 2098 (2022). https://doi.org/10.3390/nano12122098

    Article  Google Scholar 

  26. Reddy, J.N.: Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates. Int. J. Eng. Sci. 48, 1507–1518 (2010)

    Article  MathSciNet  Google Scholar 

  27. Rezaiee-Pajand, M., Kamali, F.: Exact solution for thermal–mechanical post-buckling of functionally graded micro-beams. CEAS Aeronaut. J. (2021). https://doi.org/10.1007/s13272-020-00480-9

    Article  Google Scholar 

  28. Romano, G., Barretta, R., Diaco, M., Marotti de Sciarra, F.: Constitutive boundary conditions and paradoxes in nonlocal elastic nano-beams. Int. J. Mech. Sci. 121, 151–156 (2017)

    Article  Google Scholar 

  29. Salas, A.H.: Exact solution to Duffing equation and the pendulum equation. Appl. Math. Sci. 8(176), 8781–8789 (2014)

    Google Scholar 

  30. Salas, A. H., El-Tantawy S. A. (2021). Analytical solution to some strong nonlinear oscillator. Book Chapter. Optimization Problem in Engineering, https://doi.org/10.5772/intechopen.97677

  31. Shenas, A.G., Ziaee, S., Malekzadeh, P.: Nonlinear vibration analysis of pre-twisted functionally graded microbeams in thermal environment. Thin-Walled Struct. 118(2017), 87–104 (2017). https://doi.org/10.1016/j.tws.2017.05.003

    Article  Google Scholar 

  32. Sheng, G.G., Wang, X.: Nonlinear forced vibration of size-dependent functionally graded microbeams with damping effects. Appl. Math. Model. 71(2019), 421–437 (2019). https://doi.org/10.1016/j.apm.2019.02.027

    Article  MathSciNet  Google Scholar 

  33. Simsek, M.: Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He’ s variational method. Compos. Struct. 112, 264–272 (2014). https://doi.org/10.1016/j.compstruct.2014.02.010

    Article  Google Scholar 

  34. Şimşek, M., Reddy, J.N.: Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. Int. J. Eng. Sci. (2013). https://doi.org/10.1016/j.ijengsci.2012.12.002

    Article  MathSciNet  Google Scholar 

  35. Şimşek, M., Reddy, J.N.: A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory. Compos. Struct. (2013). https://doi.org/10.1016/j.compstruct.2013.01.017

    Article  Google Scholar 

  36. Sobamowo, M.G., Yinusa, A.A., Adesina, O.A., Oyekeye, O.M.: Nonlinear vibration analysis of an electrostatically actuated microbeam using differential transformation method. Semiconduct. Sci. Inf. Dev. 2(2), 1–4 (2020)

    Google Scholar 

  37. Togun, N., Bağdatlı, S.: Nonlinear vibration of a nanobeam on a pasternak elastic foundation based on non-local euler-bernoulli beam theory. Math. Comput. Appl. (2016). https://doi.org/10.3390/mca21010003

    Article  MathSciNet  Google Scholar 

  38. Trabelssi, M., El-Borgi, S., Ke, L.-L., Reddy, J.N.: Nonlocal free vibration of graded nanobeams resting on a nonlinear elastic foundation using DQM and LaDQM. Compos. Struct. (2018). https://doi.org/10.1016/j.compstruct.2017.06.010

    Article  Google Scholar 

  39. Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress-based strain gradient theory for elasticity. Int. J. Solids Struct. 39(2002), 2731–2743 (2002)

    Article  Google Scholar 

  40. Zaera, R., Serrano, Ó., Fernández-Sáez, J.: On the consistency of the nonlocal strain gradient elasticity. Int. J. Eng. Sci. 138, 65–81 (2019)

    Article  MathSciNet  Google Scholar 

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  1. Applied Mechanics and Design Research Group, Department of Mechanical Engineering, Faculty of Engineering, University of Port Harcourt, Port Harcourt, Nigeria

    Peter Brownson Alfred, Chinwuba Victor Ossia & Akuro Big-Alabo

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  1. Peter Brownson Alfred
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  2. Chinwuba Victor Ossia
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Alfred, P.B., Ossia, C.V. & Big-Alabo, A. Free nonlinear vibration analysis of a functionally graded microbeam resting on a three-layer elastic foundation using the continuous piecewise linearization method. Arch Appl Mech 94, 57–80 (2024). https://doi.org/10.1007/s00419-023-02505-1

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