Skip to main content
Log in

Study on nonlinear vibrations of temperature- and size-dependent FG porous arches on elastic foundation using nonlocal strain gradient theory

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

A nonlocal strain gradient theory is developed in this paper to study the large amplitude vibrations of arches made of functionally graded (FG) porous material. The case of shallow arches resting on nonlinear elastic foundation is modeled via a general higher-order shear deformation theory. The third-order model of Reddy, the first-order model of Timoshenko, and the classical model of Euler–Bernoulli are analyzed. Thermomechanical properties of the arch exposed to the uniform thermal field are assumed to be temperature dependent. The nonlinear motion equations of the arch are established by employing Hamilton’s principle and the von Kármán type of geometric nonlinearity. The two-step perturbation technique and the Galerkin method are utilized to solve the nonlinear governing equations. The size-dependent linear and nonlinear frequencies of the arch are obtained for the immovable pinned-pinned boundary conditions. The comparison studies are performed to verify the present solution method with the provided data in the literature, and a good agreement is observed. The novel parametric studies covered in this research include the effects of several parameters such as elastic foundation, nonlocal and length scale parameters, porosity, temperature field, power law index, and geometrical parameters on the frequencies of FG porous arches in detail.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Y.P. Tseng, C.S. Huang, M.S. Kao, In-plane vibration of laminated curved beams with variable curvature by dynamic stiffness analysis. Compos. Struct. 50, 103–114 (2000)

    Article  Google Scholar 

  2. H. Matsunaga, Free vibration and stability of laminated composite circular arches subjected to initial axial stress. J. Sound Vib. 271, 651–670 (2004)

    Article  ADS  Google Scholar 

  3. G. Karami, P. Malekzadeh, In-plane free vibration analysis of circular arches with varying cross sections. J. Sound Vib. 274, 777–799 (2004)

    Article  ADS  Google Scholar 

  4. E. Viola, F. Tornabene, Vibration analysis of damaged circular arches with varying cross-section. Struct. Integr. Durab. (SID-SDHM) 1, 155–169 (2005)

    Google Scholar 

  5. E. Viola, M. Dilena, F. Tornabene, Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches. J. Sound Vib. 299, 143–163 (2007)

    Article  ADS  Google Scholar 

  6. P. Malekzadeh, A.R. Setoodeh, E. Barmshouri, A hybrid layerwise and differential quadrature method for in-plane free vibration of laminated thick circular arches. J. Sound Vib. 315, 212–225 (2008)

    Article  ADS  Google Scholar 

  7. Q. Lü, C.F. Lü, Exact two-dimensional solutions for in-plane natural frequencies of laminated circular arches. J. Sound Vib. 318, 982–990 (2008)

    Article  ADS  Google Scholar 

  8. C.W. Lim, Q. Yang, C.F. Lü, Two-dimensional elasticity solutions for temperature dependent in-plane vibration of FGM circular arches. Compos. Struct. 90, 323–90 (2009)

    Article  Google Scholar 

  9. P. Malekzadeh, Two-dimensional in-plane free vibrations of functionally graded circular arches with temperature-dependent properties. Compos. Struct. 91, 38–47 (2009)

    Article  Google Scholar 

  10. P. Malekzadeh, M.M. Atashi, G. Karami, In-plane free vibration of functionally graded circular arches with temperature-dependent properties under thermal environment. J. Sound Vib. 326, 837–851 (2009)

    Article  ADS  Google Scholar 

  11. L. Jun, R. Guangwei, P. Jin, L. Xiaobin, W. Weiguo, Free vibration analysis of a laminated shallow curved beam based on Trigonometric shear deformation theory. Mech. Based Des. Struct. 42, 111–129 (2014)

    Article  Google Scholar 

  12. U. Eroglu, In-plane free vibrations of circular beams Made of functionally graded material in thermal environment: beam theory approach. Compos. Struct. 122, 217–228 (2015)

    Article  Google Scholar 

  13. F. Tornabene, N. Fantuzzi, M. Bacciocchi, Refined shear deformation theories for laminated composite arches and beams with variable thickness: natural frequency analysis. Eng. Analy. Bound. Elemen. 100, 24–47 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Babaei, Y. Kiani, M.R. Eslami, Large amplitude free vibration analysis of shear deformable FGM shallow arches on nonlinear elastic foundation. Thin-walled Struct. 144, 106237 (2019)

    Article  Google Scholar 

  15. J. Fariborz, R.C. Batra, Free vibration of bi-directional functionally graded material circular beam using shear deformation theory employing logarithmic function of radius. Compos. Struct. 210, 217–230 (2019)

    Article  Google Scholar 

  16. O. Poit, B. Pradyumna, M. Ganapathi, Large amplitude free flexural vibration of functionally graded graphene platelets reinforced porous composite curved beams using finite element based on trigonometric shear deformation theory. Int. J. Non-linear Mech. 116, 302–317 (2019)

    Article  Google Scholar 

  17. H. Babaei, Y. Kiani, M.R. Eslami, Large amplitude free vibrations of FGM shallow curved tubes in thermal environment. Smart Struct. Syst. 25, 693–705 (2020)

    Google Scholar 

  18. H. Babaei, M.R. Eslami, On nonlinear vibration and snap-through stability of porous FG curved micro-tubes using two-step perturbation technique. Compos. Struct. 247, 112447 (2020)

    Article  Google Scholar 

  19. N.D. Duc, Nonlinear dynamic response of imperfect eccentrically stiffened FGM double curved shallow shells on elastic foundation. Compos. Struct. 102, 306–314 (2013)

    Article  Google Scholar 

  20. N.D. Duc, Nonlinear Static and Dynamic Stability of Functionally Graded Plates and Shells (Vietnam National University Press, Hanoi, 2014)

    Google Scholar 

  21. N.D. Duc, T.Q. Quan, Nonlinear dynamic analysis of imperfect FGM double curved thin shallow shells with temperature-dependent properties on elastic foundation. J. Vib. Control. 21, 1340–1362 (2015)

    Article  MathSciNet  Google Scholar 

  22. N.D. Duc, Nonlinear thermal dynamic analysis of eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations using the Reddy’s third-order shear deformation shell theory. Eur. J. Mech. A/Solid 58, 10–30 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. A.H. Sofiyev, Nonlinear free vibration of shear deformable orthotropic functionally graded cylindrical shells. Compos. Struct. 142, 35–44 (2016)

    Article  Google Scholar 

  24. A.H. Sofiyev, Large amplitude vibration of FGM orthotropic cylindrical shells interacting with the nonlinear Winkler elastic foundation. Compos. Part B 98, 141–150 (2016)

    Article  Google Scholar 

  25. A.H. Sofiyev, Z. Karaca, Z. Zerin, Non-linear vibration of composite orthotropic cylindrical shells on the non-linear elastic foundations within the shear deformation theory. Compos. Struct. 159, 53–62 (2017)

    Article  Google Scholar 

  26. N.D. Duc, P.D. Nguyen, N.D. Khoa, Nonlinear dynamic analysis and vibration of eccentrically stiffened S-FGM elliptical cylindrical shells surrounded on elastic foundations in thermal environments. Thin-walled Struct. 117, 178–189 (2017)

    Article  Google Scholar 

  27. N.D. Duc, Nonlinear thermo-electro-mechanical dynamic response of shear deformable piezoelectric Sigmoid functionally graded sandwich circular cylindrical shells on elastic foundations. J. Sandw. Struct. Mater. 3, 351–378 (2018)

    Google Scholar 

  28. A.H. Sofiyev, Review of research on the vibration and buckling of the FGM conical shells. Compos. Struct. 211, 301–317 (2019)

    Article  Google Scholar 

  29. D.Q. Chan, N.V. Thanh, N.D. Khoa, N.D. Duc, Nonlinear dynamic analysis of piezoelectric functionally graded porous truncated conical panel in thermal environments. Thin-walled Struct. 154, 106837 (2020)

    Article  Google Scholar 

  30. N.D. Duc, S.E. Kim, N.D. Khoa, D.Q. Chan, Nonlinear buckling and post-buckling analysis of shear deformable stiffened truncated conical sandwich shells with FG face sheets and a FG porous core. J. Sandw. Struct. Mater. (2020). https://doi.org/10.1177/1099636220906821

    Article  Google Scholar 

  31. N.D. Dat, N.V. Thanh, V.M. Anh, N.D. Duc, Vibration and nonlinear dynamic analysis of sandwich FG-CNTRC plate with porous core layer. Mech. Advanc. Mater. Struct. (2020). https://doi.org/10.1080/15376494.2020.1822476

    Article  Google Scholar 

  32. S.A.H. Hosseini, O. Rahmani, Free vibration of shallow and deep curved FG nanobeam via nonlocal Timoshenko curved beam model. Appl. Phys. A 122, 169 (2016)

    Article  ADS  Google Scholar 

  33. S.A.H. Hosseini, O. Rahmani, Thermomechanical vibration of curved functionally graded nanobeam based on nonlocal elasticity. J. Therm. Stress. 39, 1252–1267 (2016)

    Article  Google Scholar 

  34. F. Ebrahimi, M.R. Barati, On nonlocal characteristics of curved inhomogeneous Euler–Bernoulli nanobeams under different temperature distributions. Appl. Phys. A 122, 880 (2016)

    Article  ADS  Google Scholar 

  35. F. Ebrahimi, M.R. Barat, Size-dependent dynamic modeling of inhomogeneous curved nanobeams embedded in elastic medium based on nonlocal strain gradient theory. Proc. Inst. Mech. Eng. Part C J. Mech. 231, 4457–4469 (2017)

    Article  Google Scholar 

  36. F. Ebrahimi, M. Daman, A. Jafari, Nonlocal strain gradient-based vibration analysis of embedded curved porous piezoelectric nano-beams in thermal environment. Smart Struct. Syst. 20, 709–728 (2017)

    Google Scholar 

  37. M. Ganapathi, O. Polit, Dynamic characteristics of curved nanobeams using nonlocal higher-order curved beam theory. Physica E 91, 190–202 (2017)

    Article  ADS  Google Scholar 

  38. M. Ganapathi, T. Merzouki, O. Polit, Vibration study of curved nanobeams based on nonlocal higher-order shear deformation theory using finite element approach. Compos. Struct. 184, 821–838 (2018)

    Article  Google Scholar 

  39. L. Li, H. Tang, Y. Hu, Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature. Compos. Struct. 184, 1177–1188 (2018)

    Article  Google Scholar 

  40. H. Liu, Z. Lv, H. Wu, Nonlinear free vibration of geometrically imperfect functionally graded sandwich nanobeams based on nonlocal strain gradient theory. Compos. Struct. 214, 47–61 (2019)

    Article  Google Scholar 

  41. F. Ebrahimi, M. Daman, V. Mahesh, Thermo-mechanical vibration analysis of curved imperfect nano-beams based on nonlocal strain gradient theory. Adv. Nano Res. 7, 249–263 (2019)

    Google Scholar 

  42. X. Yang, H. Liu, J. Ma, Thermo-mechanical vibration of FG curved nanobeam containing porosities and reinforced by graphene platelets. Microsyst. Technol. 26, 2535–2551 (2020)

    Article  Google Scholar 

  43. A.N. Alizada, A.H. Sofiyev, Modified Young’s moduli of nano-materials taking into account the scale effects and vacancies. Meccanica 46, 915–920 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. A.N. Alizada, A.H. Sofiyev, The stress analysis of the substrate coated by nanomaterials with vacancies subjected to the uniform extension load. Acta Mech. 223, 1371–1383 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Gurses, B. Akgoz, O. Civalek, Mathematical modelling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation. Appl. Math. Comput. 219, 3226–3240 (2012)

    MathSciNet  MATH  Google Scholar 

  46. C. Demir, O. Civalek, Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models. Appl. Math. Model. 37, 9355–9367 (2013)

    Article  MATH  Google Scholar 

  47. B. Akgöz, O. Civalek, Longitudinal vibration analysis for microbars based on strain gradient elasticity theory. J. Vib. Cont. 20, 606–616 (2014)

    Article  MathSciNet  Google Scholar 

  48. C. Demir, O. Civalek, A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Compos. Struct. 168, 872–884 (2017)

    Article  Google Scholar 

  49. H.M. Numanoqlu, O. Civalek, On the dynamics of small-sized structures. Int. J. Eng. Sci. 145, 103164 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  50. H.M. Numanoqlu, O. Civalek, On the torsional vibration of nanorods surrounded by elastic matrix via nonlocal FEM. Int. J. Mech. Sci. 161–162, 105076 (2019)

  51. F. Ebrahimi, M.R. Barati, O. Civalek, Application of Chebyshev–Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures. Eng. Comput. 36, 953–964 (2020)

    Article  Google Scholar 

  52. O. Civalek, B. Uzun, M.O. Yaylı, B. Akgöz, Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method. Eur. Phys. J. Plus 135, 381 (2020)

    Article  Google Scholar 

  53. C.W. Lim, G. Zhang, J.N. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Physic. Solid. 78, 298–313 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Y. Gao, W.S. Xiao, H. Zhu, Snap-buckling of functionally graded multilayer graphene platelet-reinforced composite curved nanobeams with geometrical imperfections. Eur. J. Mech. A/Solid. 82, 103993 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. G.L. She, K.M. Yan, Y.L. Zhang, H.B. Liu, Y.R. Ren, Wave propagation of functionally graded porous nano-beams based on non-local strain gradient theory. Eur. Phys. J. Plus 133, 368–376 (2018)

    Article  Google Scholar 

  56. G.L. She, H.B. Liu, B. Karami, On resonance behavior of porous FG curved nanobeams. Steel Compos. Struct. 36, 179–186 (2020)

    Google Scholar 

  57. H. Babaei, M.R. Eslami, Size-dependent vibrations of thermally pre/post-buckled FG porous micro-tubes based on modified couple stress theory. Int. J. Mech. Sci. 180, 105694 (2020)

    Article  Google Scholar 

  58. H. Babaei, Y. Kiani, M.R. Eslami, Geometrically nonlinear analysis of shear deformable FGM shallow pinned arches on nonlinear elastic foundation under mechanical and thermal loads. Acta Mech. 229, 3123–3141 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  59. H. Babaei, Y. Kiani, M.R. Eslami, Thermomechanical nonlinear in-plane analysis of fix-ended FGM shallow arches on nonlinear elastic foundation using two-step perturbation technique. Int. J. Mech. Mater. Des. 15, 225–244 (2019)

    Article  Google Scholar 

  60. H. Babaei, M.R. Eslami, Nonlinear snap-through instability of FGM shallow micro-arches with integrated surface piezoelectric layers based on modified couple stress theory. Int. J. Struct. Stabil. Dyn. 19(8), 1950088 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  61. H. Babaei, M.R. Eslami, Thermally induced large deflection of FGM shallow micro-arches with integrated surface piezoelectric layers based on modified couple stress theory. Acta Mech. 230, 2363–2384 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  62. G.L. She, Y.R. Ren, K.M. Yan, On snap-buckling of porous FG curved nanobeams. Acta Astronaut. 161, 475–484 (2019)

    Article  ADS  Google Scholar 

  63. R.B. Hetnarski, M.R. Eslami, Thermal Stresses, Advanced Theory and Applications, 2nd edn. (Springer, Switzerland, 2019)

    Book  MATH  Google Scholar 

  64. J.N. Reddy, Mechanics of Laminated Composite Plates and Shells, Theory and Application (CRC Press, Boca Raton, 2003)

    Book  Google Scholar 

  65. H. Babaei, Y. Kiani, M.R. Eslami, Large amplitude free vibrations of FGM beams on nonlinear elastic foundation in thermal field based on neutral/mid-plane formulations. Iran J. Sci. Technol. Trans. Mech. Eng. (2020). https://doi.org/10.1007/s40997-020-00389-y

    Article  Google Scholar 

  66. M.R. Eslami, Buckling and Postbuckling of Beams, Plates, and Shells (Springer, Switzerland, 2018)

    Book  MATH  Google Scholar 

  67. M. Simsek, Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int. J. Eng. Sci. 105, 12–27 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  68. H.S. Shen, A Two-Step Perturbation Method in Nonlinear Analysis of Beams, Plates and Shells (Wiley, New York, 2013)

    Book  MATH  Google Scholar 

  69. Y. Gao, W.S. Xiao, H. Zhu, Nonlinear vibration analysis of different types of functionally graded beams using nonlocal strain gradient theory and a two-step perturbation method. Eur. Phys. J. Plus 134, 23–46 (2019)

    Article  ADS  Google Scholar 

  70. H.S. Shen, Functionally Graded Materials Nonlinear Analysis of Plates and Shells (CRC Press, Boca Raton, 2009)

    Google Scholar 

  71. L. Lu, X. Guo, J. Zhao, Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. Int. J. Eng. Sci. 116, 12–24 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  72. O. Rahmani, O. Pedram, Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. Int. J. Eng. Sci. 77, 55–70 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  73. M.A. Eltaher, S.A. Emam, F.F. Mahmoud, Free vibration analysis of functionally graded size-dependent nanobeams. Appl. Math. Comput. 218, 7406–7420 (2012)

    MathSciNet  MATH  Google Scholar 

  74. H.S. Shen, Z.X. Wang, Nonlinear analysis of shear deformable FGM beams resting on elastic foundations in thermal environments. Int. J. Mech. Sci. 81, 195–206 (2014)

    Article  Google Scholar 

Download references

Funding

This study has received no funding.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hadi Babaei.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Babaei, H., Eslami, M.R. Study on nonlinear vibrations of temperature- and size-dependent FG porous arches on elastic foundation using nonlocal strain gradient theory. Eur. Phys. J. Plus 136, 24 (2021). https://doi.org/10.1140/epjp/s13360-020-00959-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-020-00959-8

Navigation