Skip to main content
SpringerLink
Log in

Effects of porosity and nonlocality on the low- and high-frequency vibration characteristics of Al/Si3N4 functionally graded nanoplates using quasi-3D theory

  • Original Article
  • Published:
Archives of Civil and Mechanical Engineering Aims and scope Submit manuscript

Abstract

This study explores the effects of porosity as well as nonlocal parameter on both low- and high-frequency free vibration behaviors of functionally graded nanoplates for the first times. Three distinct models are used to describe porosity distribution. A combination of the six-unknown quasi-3D theory is utilized to define the displacement field of the nanoplates. The Hamilton’s principle is exploited in combination with the nonlocal elasticity theory to establish the governing equations of the motion of the functionally graded nanoplates. The influence of some parameters such as geometric, power-law index, porosity, and nonlocal parameter on the free vibration behaviors of the functionally graded porous nanoplates is explored in detail. This pioneering study unveils a wealth of numerical findings that, for the first time, shed light on the intricate high-frequency behaviors exhibited by these nanoplates. These insights not only expand the scientific understanding of functionally graded porous nanoplates but also offer valuable knowledge that can empower scientists and engineers in their pursuit of designing and optimizing nanostructures in this domain. This research marks a significant milestone in the study of nanoplates, bridging the gap between low- and high-frequency vibration analyses and opening new avenues for future research and practical applications.

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

Access this article

Log in via an institution

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
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Data availability

The raw data used to plot the figures of this study are available from corresponding author Pham Van Vinh upon reasonable request. Other data were included in the paper.

References

  1. Chandel VS, Wang G, Talha M. Advances in modelling and analysis of nano structures: a review. Nanotechnol Rev. 2020;9:230–58. https://doi.org/10.1515/ntrev-2020-0020.

    Article  CAS  Google Scholar 

  2. Sobhy M. Piezoelectric bending of GPL-reinforced annular and circular sandwich nanoplates with FG porous core integrated with sensor and actuator using DQM. Arch Civ Mech Eng. 2021. https://doi.org/10.1007/s43452-021-00231-5.

    Article  Google Scholar 

  3. Alipour MM, Shariyat M. Nonlocal zigzag analytical solution for Laplacian hygrothermal stress analysis of annular sandwich macro/nanoplates with poor adhesions and 2D-FGM porous cores. Arch Civ Mech Eng. 2019;19:1211–34. https://doi.org/10.1016/j.acme.2019.06.008.

    Article  Google Scholar 

  4. Das M, Bhushan A. Investigation of the effects of residual stress on static and dynamic behaviour of an imperfect MEMS circular microplate. Iran J Sci Technol Trans Mech Eng. 2023. https://doi.org/10.1007/s40997-023-00627-z.

    Article  Google Scholar 

  5. Norouzzadeh A, Ansari R, Rouhi H. An analytical study on wave propagation in functionally graded nano-beams/tubes based on the integral formulation of nonlocal elasticity. Waves Random Complex Media. 2020;30:562–80. https://doi.org/10.1080/17455030.2018.1543979.

    Article  MathSciNet  ADS  Google Scholar 

  6. Koizumi M. FGM activities in Japan. Compos Part B Eng. 1997;28:1–4. https://doi.org/10.1016/s1359-8368(96)00016-9.

    Article  Google Scholar 

  7. Reddy JN. Analysis of functionally graded plates. Int J Numer Methods Eng. 2000;47:663–84. https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3%3c663::AID-NME787%3e3.0.CO;2-8.

    Article  Google Scholar 

  8. Reddy JN. A general nonlinear third-order theory of functionally graded plates. Int J Aerosp Light Struct. 2011;01:01. https://doi.org/10.3850/s201042861100002x.

    Article  Google Scholar 

  9. Su Z, Jin G, Wang Y, Ye X. A general Fourier formulation for vibration analysis of functionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations. Acta Mech. 2016;227:1493–514. https://doi.org/10.1007/s00707-016-1575-8.

    Article  MathSciNet  Google Scholar 

  10. Sheikholeslami SA, Saidi AR. Vibration analysis of functionally graded rectangular plates resting on elastic foundation using higher-order shear and normal deformable plate theory. Compos Struct. 2013;106:350–61. https://doi.org/10.1016/j.compstruct.2013.06.016.

    Article  Google Scholar 

  11. Bourada F, Bousahla AA, Tounsi A, Tounsi A, Tahir SI, Al-Osta MA, Do-Van T. An integral quasi-3D computational model for the hygro-thermal wave propagation of imperfect FGM sandwich plates. Comput Concr. 2023;32:61–74. https://doi.org/10.12989/cac.2023.32.1.061.

    Article  Google Scholar 

  12. Bellifa H, Selim MM, Chikh A, Bousahla AA, Bourada F, Tounsi A, Benrahou KH, Al-Zahrani MM, Tounsi A. Influence of porosity on thermal buckling behavior of functionally graded beams. Smart Struct Syst. 2021;27:719–28. https://doi.org/10.12989/sss.2021.27.4.719.

    Article  Google Scholar 

  13. Ait Atmane H, Tounsi A, Bernard F. Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations. Int J Mech Mater Des. 2017;13:71–84. https://doi.org/10.1007/s10999-015-9318-x.

    Article  Google Scholar 

  14. Trinh MC, Kim SE. A three variable refined shear deformation theory for porous functionally graded doubly curved shell analysis. Aerosp Sci Technol. 2019;94: 105356. https://doi.org/10.1016/j.ast.2019.105356.

    Article  Google Scholar 

  15. Saleh B, Jiang J, Fathi R, Al-hababi T, Xu Q, Wang L, Song D, Ma A. 30 Years of functionally graded materials: an overview of manufacturing methods, applications and future challenges. Compos Part B Eng. 2020;201: 108376. https://doi.org/10.1016/j.compositesb.2020.108376.

    Article  Google Scholar 

  16. Fan F, Xu Y, Sahmani S, Safaei B. Modified couple stress-based geometrically nonlinear oscillations of porous functionally graded microplates using NURBS-based isogeometric approach. Comput Methods Appl Mech Eng. 2020;372: 113400. https://doi.org/10.1016/j.cma.2020.113400.

    Article  MathSciNet  ADS  Google Scholar 

  17. Zare Jouneghani F, Mohammadi Dashtaki P, Dimitri R, Bacciocchi M, Tornabene F. First-order shear deformation theory for orthotropic doubly-curved shells based on a modified couple stress elasticity. Aerosp Sci Technol. 2018;73:129–47. https://doi.org/10.1016/j.ast.2017.11.045.

    Article  Google Scholar 

  18. Farzam A, Hassani B. Isogeometric analysis of in-plane functionally graded porous microplates using modified couple stress theory. Aerosp Sci Technol. 2019;91:508–24. https://doi.org/10.1016/j.ast.2019.05.012.

    Article  Google Scholar 

  19. Razavi H, Babadi AF, Tadi Beni Y. Free vibration analysis of functionally graded piezoelectric cylindrical nanoshell based on consistent couple stress theory. Compos Struct. 2017;160:1299–309. https://doi.org/10.1016/j.compstruct.2016.10.056.

    Article  Google Scholar 

  20. Eringen AC, Edelen DGB. On nonlocal elasticity. Int J Eng Sci. 1972;10:233–48. https://doi.org/10.1016/0020-7225(72)90039-0.

    Article  MathSciNet  Google Scholar 

  21. Narendar S. Buckling analysis of micro-/nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects. Compos Struct. 2011;93:3093–103. https://doi.org/10.1016/j.compstruct.2011.06.028.

    Article  Google Scholar 

  22. Li M, Cai Y, Bao L, Fan R, Zhang H, Wang H, Borjalilou V. Analytical and parametric analysis of thermoelastic damping in circular cylindrical nanoshells by capturing small-scale effect on both structure and heat conduction. Arch Civ Mech Eng. 2022;22:14. https://doi.org/10.1007/s43452-021-00330-3.

    Article  Google Scholar 

  23. Glabisz W, Jarczewska K, Hołubowski R. Stability of Timoshenko beams with frequency and initial stress dependent nonlocal parameters. Arch Civ Mech Eng. 2019;19:1116–26. https://doi.org/10.1016/j.acme.2019.06.003.

    Article  Google Scholar 

  24. Berghouti H, Bedia EAA, Benkhedda A, Tounsi A. Vibration analysis of nonlocal porous nanobeams made of functionally graded material. Adv Nano Res. 2019;7:351–64. https://doi.org/10.12989/anr.2019.7.5.351.

    Article  Google Scholar 

  25. Fard KM, Kavanroodi MK, Malek-Mohammadi H, Pourmoayed AR. Buckling and vibration analysis of a double-layer graphene sheet coupled with a piezoelectric nanoplate. J Appl Comput Mech. 2022;8:129–43. https://doi.org/10.22055/jacm.2020.32145.1976.

    Article  Google Scholar 

  26. Shariati M, Shishehsaz M, Mosalmani R. Stress-driven approach to vibrational analysis of FGM annular nano-plate based on first-order shear deformation plate theory. J Appl Comput Mech. 2023;9:637–55. https://doi.org/10.22055/jacm.2022.41125.3704.

    Article  Google Scholar 

  27. Vaccaro MS, Sedighi HM. Two-phase elastic axisymmetric nanoplates. Eng Comput. 2023;39:827–34. https://doi.org/10.1007/s00366-022-01680-z.

    Article  Google Scholar 

  28. Soleiman A, Abouelregal AE, Fahmy MA, Sedighi HM. Thermomechanical behavior of functionally graded nanoscale beams under fractional heat transfer model with a two-parameter Mittag–Leffler function. Iran J Sci Technol Trans Mech Eng. 2023. https://doi.org/10.1007/s40997-023-00698-y.

    Article  Google Scholar 

  29. Sobhy M. Levy-type solution for bending of single-layered graphene sheets in thermal environment using the two-variable plate theory. Int J Mech Sci. 2015;90:171–8. https://doi.org/10.1016/j.ijmecsci.2014.11.014.

    Article  Google Scholar 

  30. Ansari R, Sahmani S. Prediction of biaxial buckling behavior of single-layered graphene sheets based on nonlocal plate models and molecular dynamics simulations. Appl Math Model. 2013;37:7338–51. https://doi.org/10.1016/j.apm.2013.03.004.

    Article  MathSciNet  Google Scholar 

  31. Daneshmehr A, Rajabpoor A, Hadi A. Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories. Int J Eng Sci. 2015;95:23–35. https://doi.org/10.1016/j.ijengsci.2015.05.011.

    Article  MathSciNet  Google Scholar 

  32. Aria AI, Friswell MI. A nonlocal finite element model for buckling and vibration of functionally graded nanobeams. Compos Part B Eng. 2019;166:233–46. https://doi.org/10.1016/j.compositesb.2018.11.071.

    Article  Google Scholar 

  33. Numanoğlu HM, Ersoy H, Akgöz B, Civalek Ö. A new eigenvalue problem solver for thermo-mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method. Math Methods Appl Sci. 2022;45:2592–614. https://doi.org/10.1002/mma.7942.

    Article  MathSciNet  Google Scholar 

  34. Lam DCC, Yang F, Chong ACM, Wang J, Tong P. Experiments and theory in strain gradient elasticity. J Mech Phys Solids. 2003;51:1477–508. https://doi.org/10.1016/S0022-5096(03)00053-X.

    Article  ADS  Google Scholar 

  35. Sahmani S, Aghdam MM, Rabczuk T. Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory. Compos Struct. 2018;186:68–78. https://doi.org/10.1016/j.compstruct.2017.11.082.

    Article  Google Scholar 

  36. Ghannadpour SAM, Khajeh S. Nonlinear bending and post-buckling behaviors of FG small-scaled plates based on modified strain gradient theory using Ritz technique. Adv Nano Res. 2022;13:393–406. https://doi.org/10.12989/anr.2022.13.4.393.

    Article  Google Scholar 

  37. Akgöz B, Civalek Ö. A size-dependent shear deformation beam model based on the strain gradient elasticity theory. Int J Eng Sci. 2013;70:1–14. https://doi.org/10.1016/j.ijengsci.2013.04.004.

    Article  MathSciNet  Google Scholar 

  38. Karamanli A, Vo TP. A quasi-3D theory for functionally graded porous microbeams based on the modified strain gradient theory. Compos Struct. 2021;257: 113066. https://doi.org/10.1016/j.compstruct.2020.113066.

    Article  Google Scholar 

  39. Thang PT, Nguyen-Thoi T, Lee J. Modeling and analysis of bi-directional functionally graded nanobeams based on nonlocal strain gradient theory. Appl Math Comput. 2021;407: 126303. https://doi.org/10.1016/j.amc.2021.126303.

    Article  MathSciNet  Google Scholar 

  40. Lim CW, Zhang G, Reddy JN. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids. 2015;78:298–313. https://doi.org/10.1016/j.jmps.2015.02.001.

    Article  MathSciNet  ADS  Google Scholar 

  41. Nematollahi MS, Mohammadi H. Geometrically nonlinear vibration analysis of sandwich nanoplates based on higher-order nonlocal strain gradient theory. Int J Mech Sci. 2019;156:31–45. https://doi.org/10.1016/j.ijmecsci.2019.03.022.

    Article  Google Scholar 

  42. Merzouki T, Ahmed HMS, Bessaim A, Haboussi M, Dimitri R, Tornabene F. Bending analysis of functionally graded porous nanocomposite beams based on a non-local strain gradient theory. Math Mech Solids. 2022;27:66–92. https://doi.org/10.1177/10812865211011759.

    Article  MathSciNet  Google Scholar 

  43. Daikh AA, Belarbi MO, Khechai A, Li L, Ahmed HM, Eltaher MA. Buckling of bi-coated functionally graded porous nanoplates via a nonlocal strain gradient quasi-3D theory. Acta Mech. 2023. https://doi.org/10.1007/s00707-023-03548-9.

    Article  MathSciNet  Google Scholar 

  44. Panahi R, Asghari M, Borjalilou V. Nonlinear forced vibration analysis of micro-rotating shaft–disk systems through a formulation based on the nonlocal strain gradient theory. Arch Civ Mech Eng. 2023;23:85. https://doi.org/10.1007/s43452-023-00617-7.

    Article  Google Scholar 

  45. Yue XG, Sahmani S, Luo H, Safaei B. Nonlocal strain gradient-based quasi-3D nonlinear dynamical stability behavior of agglomerated nanocomposite microbeams. Arch Civ Mech Eng. 2023;23:21. https://doi.org/10.1007/s43452-022-00548-9.

    Article  Google Scholar 

  46. Eyvazian A, Zhang C, Civalek Ö, Khan A, Sebaey TA, Farouk N. Wave propagation analysis of sandwich FGM nanoplate surrounded by viscoelastic foundation. Arch Civ Mech Eng. 2022. https://doi.org/10.1007/s43452-022-00474-w.

    Article  Google Scholar 

  47. Chandel VS, Talha M. Vibration analysis of functionally graded porous nano-beams: a comparison study. Mater Today Proc. 2023. https://doi.org/10.1016/j.matpr.2023.03.703.

    Article  Google Scholar 

  48. Coskun S, Kim J, Toutanji H. Bending, free vibration, and buckling analysis of functionally graded porous micro-plates using a general third-order plate theory. J Compos Sci. 2019. https://doi.org/10.3390/jcs3010015.

    Article  Google Scholar 

  49. Turan M, Uzun Yaylacı E, Yaylacı M. Free vibration and buckling of functionally graded porous beams using analytical, finite element, and artificial neural network methods. Arch Appl Mech. 2023;93:1351–72. https://doi.org/10.1007/s00419-022-02332-w.

    Article  Google Scholar 

  50. Karami B, Shahsavari D, Janghorban M. On the dynamics of porous doubly-curved nanoshells. Int J Eng Sci. 2019;143:39–55. https://doi.org/10.1016/j.ijengsci.2019.06.014.

    Article  MathSciNet  Google Scholar 

  51. Shahsavari D, Karami B, Fahham HR, Li L. On the shear buckling of porous nanoplates using a new size-dependent quasi-3D shear deformation theory. Acta Mech. 2018;229:4549–73. https://doi.org/10.1007/s00707-018-2247-7.

    Article  MathSciNet  Google Scholar 

  52. Kumar Y, Gupta A, Tounsi A. Size-dependent vibration response of porous graded nanostructure with FEM and nonlocal continuum model. Adv Nano Res. 2021;11:1–17. https://doi.org/10.12989/anr.2021.11.1.001.

    Article  Google Scholar 

  53. Cuong-Le T, Nguyen KD, Le-Minh H, Phan-Vu P, Nguyen-Trong P, Tounsi A. Nonlinear bending analysis of porous sigmoid FGM nanoplate via IGA and nonlocal strain gradient theory. Adv Nano Res. 2022;12:441–55. https://doi.org/10.12989/anr.2022.12.5.441.

    Article  Google Scholar 

  54. Khorasani M, Lampani L, Tounsi A. A refined vibrational analysis of the FGM porous type beams resting on the silica aerogel substrate. Steel Compos Struct. 2023;47:633–44. https://doi.org/10.12989/scs.2023.47.5.633.

    Article  Google Scholar 

  55. Addou FY, Bourada F, Meradjah M, Bousahla AA, Tounsi A, Ghazwani MH, Alnujaie A. Impact of porosity distribution on static behavior of functionally graded plates using a simple quasi-3D HSDT. Comput Concr. 2023;32:87–97. https://doi.org/10.12989/cac.2023.32.1.087.

    Article  Google Scholar 

  56. Wang Q, Wang CM. The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology. 2007;18:75702. https://doi.org/10.1088/0957-4484/18/7/075702.

    Article  CAS  Google Scholar 

  57. Li C, Lai SK, Yang X. On the nano-structural dependence of nonlocal dynamics and its relationship to the upper limit of nonlocal scale parameter. Appl Math Model. 2019;69:127–41. https://doi.org/10.1016/j.apm.2018.12.010.

    Article  MathSciNet  Google Scholar 

  58. Rezaei AS, Saidi AR, Abrishamdari M, Mohammadi MHP. Natural frequencies of functionally graded plates with porosities via a simple four variable plate theory: an analytical approach. Thin-Walled Struct. 2017;120:366–77. https://doi.org/10.1016/j.tws.2017.08.003.

    Article  Google Scholar 

  59. Sharma N, Tiwari P, Maiti DK, Maity D. Free vibration analysis of functionally graded porous plate using 3-D degenerated shell element. Compos Part C Open Access. 2021;6: 100208. https://doi.org/10.1016/j.jcomc.2021.100208.

    Article  Google Scholar 

  60. Van Vinh P. Nonlocal free vibration characteristics of power-law and sigmoid functionally graded nanoplates considering variable nonlocal parameter. Phys E Low Dimens Syst Nanostruct. 2022;135: 114951. https://doi.org/10.1016/j.physe.2021.114951.

    Article  CAS  Google Scholar 

Download references

Acknowledgements

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number ISP23-69.

Funding

The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number ISP23-69.

Author information

Authors and Affiliations

  1. Mechanical Engineering Department, Faculty of Engineering, Jazan University, P. O. Box 45142, Jazan, Kingdom of Saudi Arabia

    Mofareh Hassan Ghazwani & Ali Alnujaie

  2. Department of Solid Mechanics, Le Quy Don Technical University, Hoang Quoc Viet, Ha Noi, Vietnam

    Pham Van Vinh

  3. Mechanical Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

    Hamid M. Sedighi

  4. Drilling Center of Excellence and Research Center, Shahid Chamran University of Ahvaz, Ahvaz, Iran

    Hamid M. Sedighi

Authors
  1. Mofareh Hassan Ghazwani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali Alnujaie
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pham Van Vinh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hamid M. Sedighi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

MHG: resources, software, writing—review and editing, and funding acquisition. AA: data curation, visualization, investigation, writing—review and editing. PVV: conceptualization, methodology, formal analysis, investigation, resources, validation, writing—original draft, writing—review and editing, supervision, and project administration. HMS: methodology, software, validation, writing—review and editing.

Corresponding author

Correspondence to Pham Van Vinh.

Ethics declarations

Conflict of interest

All the authors declare no conflict of interest. The authors have no competing interests to declare that are relevant to the content of this article.

Compliance with ethical standards

The authors have no relevant financial or non-financial interests to disclose.

Ethics approval statement

All authors agreed with the content and that all gave explicit consent to submit and that they obtained consent from the responsible authorities at the institute/organization where the work has been carried out. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. This research did not contain any studies involving animal or human participants, nor did it take place on any private or protected areas. The manuscript in part or in full has not been submitted or published anywhere.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghazwani, M.H., Alnujaie, A., Van Vinh, P. et al. Effects of porosity and nonlocality on the low- and high-frequency vibration characteristics of Al/Si3N4 functionally graded nanoplates using quasi-3D theory. Archiv.Civ.Mech.Eng 24, 49 (2024). https://doi.org/10.1007/s43452-023-00858-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s43452-023-00858-6

Keywords

Search

Navigation