Skip to main content
SpringerLink
Log in

Vibration and the Buckling Response of Functionally Graded Plates According to a Refined Hyperbolic Shear Deformation Theory

  • Published:
Mechanics of Composite Materials Aims and scope

A first attempt is made to study the free vibrations and the buckling response of functionally graded plates using a refined hyperbolic shear deformation theory. This theory incorporates high-order effects of shear and normal deformation with accounting for thickness stretching. A combination of hyperbolic and polynomial functions ensures a parabolic profile of shear stresses and the enforcement of zero shear stresses at the top and bottom surfaces of the plates. The need for a shear correction factor is eliminated. The plates are made from advanced composites consisting of a functionally graded material varying from a ceramic to metallic phase across the thickness. The mechanical properties of the plates are homogenized by the Voigt rule of mixtures and the Mori- Tanaka scheme. A C0 finite-element model is developed for the present theory and is included in the MATLAB code. A convergence study is performed and the efficacy of the model is validated.

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.

Similar content being viewed by others

References

  1. G. Udupa, S. S. Rao, and K. V. Gangadharan, “Functionally graded composite materials: An overview,” Procedia Mater., Sci., 5, 1291-1299 (2014).

  2. S. S. Vel and R. C. Batra, “Three-dimensional exact solution for the vibration of functionally graded rectangular plates,” J. Sound Vib., 272, No. 3-5, 703-730 (2004).

    Article  Google Scholar 

  3. A. Alibeigloo, “Exact solution for thermo-elastic response of functionally graded rectangular plates,” Compos. Struct., 92, No. 1, 113-121 (2010).

    Article  Google Scholar 

  4. S. Hosseini-Hashemi, M. Fadaee, and S. R. Atashipour, “A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates,” Int. J. Mech. Sci., 53, No. 1, 11-22 (2011).

    Article  Google Scholar 

  5. G. Jin, Z. Su, S. Shi, T. Ye, and S. Gao, “Three-dimensional exact solution for the free vibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions,” Compos. Struct., 108, 565-577 (2014).

    Article  Google Scholar 

  6. M. M. Najafizadeh and H. R. Heydari, “An exact solution for buckling of functionally graded circular plates based on higherorder shear deformation plate theory under uniform radial compression,” Int. J. Mech. Sci., 50, No. 3, 603-612 (2008).

    Article  Google Scholar 

  7. J. N. Reddy, Theory and Analysis of Elastic Plates and Shells. CRC Press (1999).

  8. M. Touratier, “Average stress in matrix and average elastic energy of materials with misfitting inclusions,” Acta Metall., 21, No. 5, 571-571 (1973).

    Article  Google Scholar 

  9. A. M. A. Neves, A. J. M. Ferreira, E. Carrera, M. Cinefra, C.M.C. Roque, R.M.N. Jorge, and C.M. Soares, “Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique,” Compos. Part B, 44, No. 1, 657-674 (2013).

    Article  Google Scholar 

  10. A. Barut, E. Madenci, T. Anderson, and A. Tessler, “Equivalent single-layer theory for a complete stress field in sandwich panels under arbitrarily distributed loading,” Compos. Struct., 58, No. 4, 483-495 (2002).

    Article  Google Scholar 

  11. D. K. Jha, T. Kant, and R. K. Singh, “Free vibration response of functionally graded thick plates with shear and normal deformations effects,” Compos. Struct., 96, 799-823 (2013).

    Article  Google Scholar 

  12. A. A. Khdeir and J. N. Reddy, “Free vibrations of laminated composite plates using second-order shear deformation theory,” Comput. Struct., 71, No. 6, 617-626 (1999).

    Article  Google Scholar 

  13. S. Yang and F. G. Yuan, “Transient wave propagation of isotropic plates using a higher-order plate theory,” Int. J. Solids Struct., 42, No. 14, 4115-4153 (2005).

    Article  Google Scholar 

  14. L. V. Tran, C. H. Thai, H. T. Le, B. S. Gan, J. Lee, and H. Nguyen-Xuan, “Isogeometric analysis of laminated composite plates based on a four-variable refined plate theory,” Eng. Anal. Bound. Elem., 47, 68-81 (2014).

    Article  Google Scholar 

  15. T. K. Nguyen, T. T. P. Nguyen, T. P. Vo, and H. T. Thai, “Vibration and buckling analysis of functionally graded sandwich beams by a new higher-order shear deformation theory,” Compos. Part B, 76, 273-285 (2015).

    Article  Google Scholar 

  16. N. Grover, B. N. Singh, and D. K. Maiti, “A general assessment of a new inverse trigonometric shear deformation theory for laminated composite and sandwich plates using finite element method,” Proc. Inst. Mech. Eng. G, 228, No. 10, 1788-1801 (2013).

    Article  Google Scholar 

  17. K. P. Soldatos, “A transverse shear deformation theory for homogeneous monoclinic plates,” Acta Mech., 94, No. 3, 195-220 (1992).

    Article  Google Scholar 

  18. S. S. Akavci, “Buckling and free vibration analysis of symmetric and antisymmetric laminated composite plates on an elastic foundation,” J. Reinf. Plast. Compos., 26, No. 18, 1907-1919 (2007).

    Article  CAS  Google Scholar 

  19. M. Karama, K. S. Afaq, and S. Mistou, “A new theory for laminated composite plates,” Proc. Inst. Mech. Eng. L, 223, No. 2, 53-62 (2009).

    Google Scholar 

  20. M. Aydogdu, “A new shear deformation theory for laminated composite plates,” Compos. Struct., 89, No. 1, 94-101 (2009).

    Article  Google Scholar 

  21. J. L. Mantari and C. G. Soares, “Static response of advanced composite plates by a new non-polynomial higher-order shear deformation theory,” Int. J. Mech. Sci., 78, 60-71 (2014).

    Article  Google Scholar 

  22. Z. Boukhlif, “A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation,” Steel Compos. Struct., 31, No. 5, 503-516 (2019).

    Google Scholar 

  23. M. Nebab, S. Benguediab, H. A. Atmane, and F. Bernard, “A simple quasi-3D HDST for dynamic behavior of advanced composite plates with the effect of variables elastic foundations,” Geomech. Eng., 22, No. 5, 415-431 (2020).

    Google Scholar 

  24. F. Z. Zaoui, D. Ouinas, and A. Tounsi, “New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations,” Compos. Part B, 159, 231-247 (2019).

    Article  Google Scholar 

  25. A. Mahmoudi, S. Benyoucef, A. Tounsi, A. Benachour, E. A. A. Bedia, and S. R. Mahmoud, “A refined quasi-3D shear deformation theory for thermo-mechanical behavior of functionally graded sandwich plates on elastic foundations,” J. Sandw. Struct. Mater., 21, No. 6, 1906-1929 (2017).

    Article  Google Scholar 

  26. M. Abualnour, M. S. A. Houari, A. Tounsi, E. A. A. Bedia, and S. R. A. Mahmoud, “A novel quasi-3D trigonometric plate theory for free vibration analysis of advanced composite plates,” Compos. Struct., 184, 688-697 (2018).

    Article  Google Scholar 

  27. H. Hebali, A. Tounsi, M. S. A. Houari, A. Bessaim, and E. A. A. Bedia, “New Quasi-3D Hyperbolic Shear Deformation Theory for the Static and Free Vibration Analysis of Functionally Graded Plates,” J. Eng. Mech., 140, No. 2, 374-383 (2014).

    Article  Google Scholar 

  28. J. L. Mantari, E. V. Granados, M. A. Hinostroza, and C. G. Soares, “Modelling advanced composite plates resting on elastic foundation by using a quasi-3D hybrid type HSDT,” Compos. Struct., 118, 455-471 (2014).

    Article  Google Scholar 

  29. S. Bouanati, K. H. Benrahou, H. A. Atmane, S. A. Yahia, F. Bernard, A. Tounsi, and E. A. Bedia, “Investigation of wave propagation in anisotropic plates via quasi 3D HSDT,” Geomech. Eng. 18, No. 1, 85-96 (2019).

    Google Scholar 

  30. B. Adhikari and B. N. Singh, “Dynamic response of functionally graded plates resting on two-parameter-based elastic foundation model using a quasi-3D theory,” Mech. Based Des. Struct. Mach., 47, 399-429 (2019).

    Article  Google Scholar 

  31. M. K. Singha, T. Prakash, and M. Ganapathi, “Finite element analysis of functionally graded plates under transverse load,” Finite Elem. Anal. Des., 47, No. 4, 453-460 (2011).

    Article  Google Scholar 

  32. M. C. Ray and H. M. Sachade, “Finite element analysis of smart functionally graded plates,” Int. J. Solids Struct., 43, No. 18-29, 5468-5484 (2006).

    Article  Google Scholar 

  33. E. Orakdöğen, S. Küçükarslan, A. Sofiyev, and M. H. Omurtag, “Finite element analysis of functionally graded plates for coupling effect of extension and bending,” Mecc., 45, No. 1, 63-72 (2009).

    Article  Google Scholar 

  34. A. Shaker, W. Abdelrahman, M. Tawfik, and E. Sadek, “Stochastic Finite element analysis of the free vibration of functionally graded material plates,” Comput. Mech., 41, No. 5, 707-714 (2008).

    Article  Google Scholar 

  35. P. V. Vinh and L. Q. Huy, “Finite element analysis of functionally graded sandwich plates with porosity via a new hyperbolic shear deformation theory,” Def. Technol., 8, No. 3, 490-508 (2022).

    Article  Google Scholar 

  36. A. Tati, “Finite element analysis of thermal and mechanical buckling behaviour of functionally graded plates,” Arch. Appl. Mech., 91, No. 11, 1-17 (2021).

    Article  Google Scholar 

  37. R. Naghdabadi and S. A. H. Kordkheili, “A finite element formulation for analysis of functionally graded plates and shells,” Arch. Appl. Mech., 74, No. 5, 375-386 (2005).

    Article  Google Scholar 

  38. M. J. Pindera and P. Dunn, “Evaluation of the higher-order theory for functionally graded materials via the finite-element method,” Compos. Part B, 28, No. 1-2,109-119 (1997).

    Article  Google Scholar 

  39. I. Ramu and S. C. Mohanty, “Modal Analysis of Functionally Graded Material Plates Using Finite Element Method,” Procedia Mater. Sci., 6, 460-467 (2014).

    Article  CAS  Google Scholar 

  40. M. N. A. G. Taj and A. Chakrabarti, “Buckling analysis of functionally graded skew plates: An efficient finite element approach,” Int. J. Appl. Mech., 5, No. 04, (2013).

  41. A. Mahi and A. Tounsi, “A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates,” Appl. Math. Model., 39, No. 9, 2489-2508 (2015).

    Article  Google Scholar 

  42. A. Benahmed, M. S. A. Houari, S. Benyoucef, K. Belakhdar, and A. Tounsi, “A novel quasi-3D hyperbolic shear deformation theory for functionally graded thick rectangular plates on elastic foundation,” Geomech. Eng., 12, No. 1, 9-34 (2017).

    Article  Google Scholar 

  43. A. Gupta, M. Talha, and V. K. Chaudhari, “Natural frequency of functionally graded plates resting on elastic foundation using finite element method,” Procedia Technol., 23, 163-170 (2016).

    Article  Google Scholar 

  44. T. Mori and K. Tanaka, “Average stress in matrix and average elastic energy of materials with misfitting inclusions,” Acta Metall., 21, No. 5, 571-574 (1973).

    Article  Google Scholar 

  45. P. V. Vinh, N. T. Dung, N. C. Tho, D. V. Thom, and L. K. Hoa, “Modified single variable shear deformation plate theory for free vibration analysis of rectangular FGM plates,” Structures, 29, 1435-1444 (2021).

    Article  Google Scholar 

  46. S. Yin, J. S. Hale, T. Yu, T. Q. Bui, and S. P. A. Bordas, “Isogeometric locking-free plate element: A simple first-order shear deformation theory for functionally graded plates,” Compos. Struct., 118, 121-138 (2014).

    Article  Google Scholar 

  47. T. V. Vu, N. H. Nguyen, A. Khosravifard, M. R. Hematiyan, S. Tanaka, and T. Q. Bui, “A simple FSDT-based meshfree method for analysis of functionally graded plates,” Eng. Anal. Bound. Elem., 79, 1-12 (2017).

    Article  Google Scholar 

  48. H. T. Thai and D. H. Choi, “Finite element formulation of various four unknown shear deformation theories for functionally graded plates,” Finite Elem. Anal. Des., 75, 50-61 (2013).

    Article  Google Scholar 

  49. M. Talha and B. N. Singh, “Static response and free vibration analysis of FGM plates using higher order shear deformation theory,” Appl. Math. Model., 34, No. 12, 3991-4011 (2010).

    Article  Google Scholar 

  50. H. Nguyen-Xuan, C. H. Thai, and T. Nguyen-Thoi, “Isogeometric finite element analysis of composite sandwich plates using a higher order shear deformation theory,” Compos., Part B, 55, 558-574 (2013).

    Article  Google Scholar 

  51. C. H. Thai, A. J. M. Ferreira, S. P. A. Bordas, T. Rabczuk, and H. Nguyen-Xuan, “Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory,” Eur. J. Mech. - A/Solids, 43, 89-108 (2014).

    Article  Google Scholar 

  52. H. Nguyen-Xuan, L. V. Tran, C. H. Thai, S. Kulasegaram, and S. P. A. Bordas, “Isogeometric analysis of functionally graded plates using a refined plate theory,” Compos., Part B, 64, 222-234 (2014).

    Article  Google Scholar 

  53. Z. Liu, C. Wang, G. Duan, and J. Tan, “A new refined plate theory with isogeometric approach for the static and buckling analysis of functionally graded plates,” Int. J. Mech. Sci., 161, 105036 (2019).

    Article  Google Scholar 

  54. J. Rouzegar and R. A. Sharifpoor, “Finite element formulations for buckling analysis of isotropic and orthotropic plates using two-variable refined plate theory,” Iran. J. Sci. Technol. Trans. Mech. Eng., 41, No. 3, 177-187 (2016).

    Article  Google Scholar 

  55. S. E. Kim, H. T. Thai, and J. Lee, “Buckling analysis of plates using the two variable refined plate theory,” Thin-Walled Struct., 47, No. 4, 455-462 (2009).

    Article  Google Scholar 

  56. H. T. Thai and D. H. Choi, “An efficient and simple refined theory for buckling analysis of functionally graded plates,” Appl. Math. Model., 36, No. 3, 1008-1022 (2012).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, National Institute of Technology Patna, Patna, India

    J. Singh & A. Kumar

Authors
  1. J. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Kumar.

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

Singh, J., Kumar, A. Vibration and the Buckling Response of Functionally Graded Plates According to a Refined Hyperbolic Shear Deformation Theory. Mech Compos Mater 59, 725–742 (2023). https://doi.org/10.1007/s11029-023-10127-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11029-023-10127-5

Keywords

Search

Navigation