Skip to main content
SpringerLink
Log in

Fundamental frequency analysis of functionally graded plates with temperature-dependent properties based on improved exponential-trigonometric two-dimensional higher shear deformation theory

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

The objective of this paper is to provide a computational method to analyze free vibrations of advanced composite plates in thermal environments according to a recently developed higher-order shear deformation theory. This method is based upon the assumptions that displacements field include just four unknowns and considers a combination of trigonometric and exponential shear shape functions which satisfy shear stress free boundary conditions on the plate surfaces. The FG plates are simply supported and subjected to uniform, linear, nonlinear and sinusoidal temperature rise. The temperature field considered is assumed to vary in the thickness direction and constant in the axial directions of plates. It is supposed that the constituent materials possess temperature-dependent properties changing across the thickness with a simple power law function. The equations of motion are obtained by employing Hamilton’s principle and solved based on Navier’s method to determine natural frequencies of the FG plate. A parametric study for FGM plates with different values of power law index and under different sets of thermal environmental conditions has been carried out. The obtained results are compared for temperature-dependent and temperature-independent FG Plates and validated with available results in the literature.

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

References

  1. Şimşek, M.: Buckling of Timoshenko beams composed of two-dimensional functionally graded material (2D-FGM) having different boundary conditions. Compos. Struct. 149, 304–314 (2016). https://doi.org/10.1016/j.compstruct.2016.04.034

    Article  Google Scholar 

  2. Ebrahimi, M.J., Najafizadeh, M.M.: Free vibration analysis of two-dimensional functionally graded cylindrical shells. Appl. Math. Model. 38, 308–324 (2014). https://doi.org/10.1016/j.apm.2013.06.015

    Article  MathSciNet  MATH  Google Scholar 

  3. Lei, Z.X., Zhang, L.W., Liew, K.M.: Buckling analysis of CNT reinforced functionally graded laminated composite plates. Compos. Struct. 152, 62–73 (2016). https://doi.org/10.1016/j.compstruct.2016.05.047

    Article  Google Scholar 

  4. Mehditabar, A., Rahimi, G.H., Vahdat, S.E.: Integrity assessment of functionally graded pipe produced by centrifugal casting subjected to internal pressure: experimental investigation. Arch. Appl. Mech. 90, 1723–1736 (2020). https://doi.org/10.1007/s00419-020-01692-5

    Article  Google Scholar 

  5. Belkhodja, Y., Ouinas, D., Zaoui, F.Z., Fekirini, H.: A higher order exponential-trigonometric shear deformation theory for bending, vibration, and buckling analysis of functionally graded material (FGM) plates: Part I. Advanced Compos Letters 28, 1–19 (2019). https://doi.org/10.1177/0963693519875739

    Article  Google Scholar 

  6. Mantari, J.L., Soares, C.G.: A quasi-3D tangential shear deformation theory with four unknowns for functionally graded plates. Acta Mech. 226, 625–642 (2015). https://doi.org/10.1007/s00707-014-1192-3

    Article  MathSciNet  MATH  Google Scholar 

  7. Sayyad, A.S., Ghugal, Y.M.: A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich beams and plates. Int. J. Appl. Mech. 9, 1750007 (2017)

    Article  Google Scholar 

  8. Li, S., Ma, H.: Analysis of free vibration of functionally graded material micro-plates with thermoelastic damping. Arch. Appl. Mech. 90, 1285–1304 (2020). https://doi.org/10.1007/s00419-020-01664-9

    Article  Google Scholar 

  9. Li, Q., Iu, V., Kou, K.: Three-dimensional vibration analysis of functionally graded material plates in thermal environment. J. Sound Vibr. 324(3–5), 733–750 (2009). https://doi.org/10.1016/j.jsv.2009.02.036

    Article  Google Scholar 

  10. Zaoui, F.Z., Tounsi, A., Ouinas, D.: Free vibration of functionally graded plates resting on elastic foundations based on quasi-3D hybrid-type higher order shear deformation theory. Smart Struct. Syst. Int. J. 20(4), 509–524 (2017). https://doi.org/10.12989/sss.2017.20.4.509

    Article  Google Scholar 

  11. Zenkour, A.M., Radwan, A.F.: Hygrothermo-mechanical buckling of FGM plates resting on elastic foundations using a quasi-3D model. Int. J. Comput. Methods Eng. Sci. Mech. 20(2), 85–98 (2019). https://doi.org/10.1080/15502287.2019.1568618

    Article  MathSciNet  Google Scholar 

  12. Hieu, P.T., Van Tung, H.: Thermal and thermomechanical buckling of shear deformable FG-CNTRC cylindrical shells and toroidal shell segments with tangentially restrained edges. Arch. Appl. Mech. 90, 1529–1546 (2020). https://doi.org/10.1007/s00419-020-01682-7

    Article  Google Scholar 

  13. Woodward, B., Kashtalyan, M.: Three-dimensional elasticity analysis of sandwich panels with functionally graded transversely isotropic core. Arch. Appl. Mech. 89, 2463–2484 (2019). https://doi.org/10.1007/s00419-019-01589-y

    Article  Google Scholar 

  14. Boroujerdy, M.S., Eslami, M.R.: Nonlinear axisymmetric thermomechanical response of piezo-FGM shallow spherical shells. Arch. Appl. Mech. 83, 1681–1693 (2013). https://doi.org/10.1007/s00419-013-0769-y

    Article  MATH  Google Scholar 

  15. Guerroudj, H.Z., Yeghnem, R., Kaci, A., Zaoui, F.Z., Benyoucef, S., Tounsi, A.: Eigenfrequencies of advanced composite plates using an efficient hybrid quasi-3D shear deformation theory. Smart Struct. Syst. Int. J. 22(1), 121–132 (2018). https://doi.org/10.12989/sss.2018.22.1.121

    Article  Google Scholar 

  16. Simsek, M., Cansiz, S.: Dynamics of elastically connected double-functionally graded beam systems with different boundary conditions under action of a moving harmonic load. Compos. Struct. 94, 2861–2878 (2012). https://doi.org/10.1016/j.compstruct.2012.03.016

    Article  Google Scholar 

  17. Amirani, M.C., Khalili, S.M.R., Nemati, N.: Free vibration analysis of sandwich beam with FG core using the element free Galerkin method. Compos. Struct. 90, 373–379 (2009). https://doi.org/10.1016/j.compstruct.2009.03.023

    Article  Google Scholar 

  18. Mahmoudi A, Benyoucef S, Tounsi A, Benachour A, Adda Bedia EA (2018) On the effect of the micromechanical models on the free vibration of rectangular FGM plate resting on elastic foundation.Earthquakes Struct Int J 14(2):117-128.https://doi.org/10.12989/eas.2018.14.2.117

  19. Duc, N.D., Tran, Q.Q., Nguyen, D.K.: New approach to investigate nonlinear dynamic response and vibration of imperfect functionally graded carbon nanotube reinforced composite double curved shallow shells subjected to blast load and temperature. Aerosp. Sci. Technol. 71, 360–372 (2017). https://doi.org/10.1016/j.ast.2017.09.031

    Article  Google Scholar 

  20. Shen, H.S.: Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments. Int. J. Mech. Sci. 44(3), 561–584 (2002). https://doi.org/10.1016/S0020-7403(01)00103-500103-5

    Article  MATH  Google Scholar 

  21. Yang, J., Shen, H.S.: Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments. J. Sound Vibr. 255(3), 579–602 (2002). https://doi.org/10.1006/jsvi.2001.4161

    Article  Google Scholar 

  22. Huang, X., Shen, H.: Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. Int. J. Solids Struct. 41(9–10), 2403–2427 (2004). https://doi.org/10.1016/j.ijsolstr.2003.11.012

    Article  MATH  Google Scholar 

  23. Kim, Y.: Temperature dependent vibration analysis of functionally graded rectangular plates. J. Sound Vib. 28(3–5), 531–549 (2005). https://doi.org/10.1016/j.jsv.2004.06.043

    Article  Google Scholar 

  24. Chen, C., Chen, T., Chien, R.: Nonlinear vibration of initially stressed functionally graded plates. Thin-Walled Struct. 44(8), 844–851 (2006). https://doi.org/10.1016/j.tws.2006.08.007

    Article  Google Scholar 

  25. Zenkour, A.M., Alghamdi, N.A.: Thermoelastic bending analysis of functionally graded sandwich plates. J. Mater. Sci. 43, 2574–89 (2008). https://doi.org/10.1007/s10853-008-2476-6

    Article  Google Scholar 

  26. Li, S.R., Su, H.D., Cheng, C.J.: Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment. Appl. Math. Mech. 30(8), 969–982 (2009). https://doi.org/10.1007/s10483-009-0803-7

    Article  MATH  Google Scholar 

  27. Shariyat, M.: A generalized high-order global-local plate theory for nonlinear bending and buckling analyses of imperfect sandwich plates subjected to thermo-mechanical loads. Compos. Struct. 92, 130–143 (2010). https://doi.org/10.1016/j.compstruct.2009.07.007

    Article  Google Scholar 

  28. Shariyat, M.: A generalized global-local high-order theory for bending and vibration analyses of sandwich plates subjected to thermo-mechanical loads. Int. J. Mech. Sci. 52, 495–514 (2010). https://doi.org/10.1016/j.ijmecsci.2009.11.010

    Article  Google Scholar 

  29. Mahi, A., Adda Bedia, E.A., Tounsi, A., Mechab, I.: An analytical method for temperature dependent free vibration analysis of functionally graded beams with general boundary conditions. Compos. Struct. 92, 1877–1887 (2010). https://doi.org/10.1016/j.compstruct.2010.01.010

    Article  Google Scholar 

  30. Shahrjerdi, A., Mustapha, F., Bayat, M., Majid, D.L.A.: Free vibration analysis of solar functionally graded plates with temperature-dependent material properties using second order shear deformation theory. J. Mech. Sci. Technol. 25(9), 2195–2209 (2011). https://doi.org/10.1007/s12206-011-0610-x

    Article  Google Scholar 

  31. Kiani, Y., Eslami, M.R.: Thermal buckling and post-buckling response of imperfect temperature-dependent sandwich FGM plates resting on elastic foundation. Arch. Appl. Mech. 82(7), 891–905 (2012). https://doi.org/10.1007/s00419-011-0599-8

    Article  MATH  Google Scholar 

  32. Malekzadeh, P., Monajjemzadeh, S.M.: Dynamic response of functionally graded plates in thermal environment under moving load. J. Compos. B 45, 1521–1533 (2013). https://doi.org/10.1016/j.compositesb.2012.09.022

    Article  Google Scholar 

  33. Zhang, D.: Nonlinear bending analysis of FGM rectangular plates with various supported boundaries resting on two-parameter elastic foundations. Arch. Appl. Mech. 84, 1–20 (2014). https://doi.org/10.1007/s00419-013-0775-0

    Article  MATH  Google Scholar 

  34. Nejati, M., Fard, K.M., Eslampanah, A.: Effects of fiber orientation and temperature on natural frequencies of a functionally graded beam reinforced with fiber. J. Mech. Sci. Technol. 29, 3363–3371 (2015). https://doi.org/10.1007/s12206-015-0734-5

    Article  Google Scholar 

  35. Fazzolari, F.A.: Natural frequencies and critical temperatures of functionally graded sandwich plates subjected to uniform and non-uniform temperature distributions. Compos. Struct. 121, 197–210 (2015). https://doi.org/10.1016/j.compstruct.2014.10.039

    Article  Google Scholar 

  36. Kar, V.R., Panda, S.K.: Free vibration responses of temperature dependent functionally graded curved panels under thermal environment. Latin Am. J. Solids Struct. 12(11), 2006–2024 (2015). https://doi.org/10.1590/1679-78251691

    Article  Google Scholar 

  37. Attia, A., Tounsi, A., Adda Bedia, E.A., Mahmoud, S.R.: Free vibration analysis of functionally graded plates with temperature-dependent properties using various four variable refined plate theories. Steel Compos. Struct. 18(1), 187–212 (2015). https://doi.org/10.12989/scs.2015.18.1.187

    Article  Google Scholar 

  38. Ibrahimi, F., Barati, M.R.: Temperature distribution effects on buckling behavior of smart heterogeneous nanosize plates based on nonlocal four-variable refined plate theory. Int. J. Smart Nano Mater. 7(3), 119–143 (2016). https://doi.org/10.1080/19475411.2016.1223203

    Article  Google Scholar 

  39. Wang, Y.Q., Zu, J.W.: Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment. Aerosp. Sci. Technol. 69, 550–562 (2017). https://doi.org/10.1016/j.ast.2017.07.023

    Article  Google Scholar 

  40. Taleb, O., Houari, M.S.A., Bessaim, A., Tounsi, A., Mahmoud, S.R.: A new plate model for vibration response of advanced composite plates in thermal environment. Struct. Eng. Mech. Int. J. 67(4), 369–383 (2018). https://doi.org/10.12989/sem.2018.67.4.369

    Article  Google Scholar 

  41. Shahsavari, D., Shahsavari, M., Li, L., Karami, B.: A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation. Aerosp. Sci. Technol. 72, 134–149 (2018). https://doi.org/10.1016/j.ast.2017.11.004

    Article  Google Scholar 

  42. Thang, P.T., Nguyen-Thoi, T., Lee, D., Kang, J., Lee, J.: Elastic buckling and free vibration analyses of porous-cellular plates with uniform and non-uniform porosity distributions. Aerosp. Sci. Technol. 79, 278–287 (2018). https://doi.org/10.1016/j.ast.2018.06.010

    Article  Google Scholar 

  43. Tu, T.M., Quoc, T.H., Van Long, N.: Vibration analysis of functionally graded plates using the eight-unknown higher order shear deformation theory in thermal environments. Aerosp. Sci. Technol. 84, 698–711 (2019). https://doi.org/10.1016/j.ast.2018.11.010

    Article  Google Scholar 

  44. Zaoui, F.Z., Ouinas, D., Tounsi, A.: New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations. Compos. Part B Eng. 159, 231–247 (2019). https://doi.org/10.1016/j.compositesb.2018.09.051

    Article  Google Scholar 

  45. Azadi, M.: Free and forced vibration analysis of FG beam considering temperature dependency of material properties. J. Mech. Sci. Technol. 25(1), 69–80 (2011). https://doi.org/10.1007/s12206-010-1015-y

    Article  Google Scholar 

  46. Touloukian, Y.S.: Thermophysical Properties of High Temperature Solid Materials. MacMillan, New York (1967)

    Google Scholar 

  47. Reddy, J.N., Chin, C.D.: Thermo-mechanical analysis of functionally graded cylinders and plates. J. Therm. Stress. 21, 593–626 (1998). https://doi.org/10.1080/01495739808956165

    Article  Google Scholar 

  48. Javaheri, R., Eslami, M.: Thermal buckling of functionally graded plates based on higher order theory. J. Therm. Stress. 25(7), 603–625 (2002). https://doi.org/10.1080/01495730290074333

    Article  Google Scholar 

  49. Mokhtar, B., Abedlouahed, T., Adda Bedia, E.A., Abdelkader, M.: Buckling analysis of functionally graded plates with simply supported edges. Leonardo J. Sci. 8, 21–32 (2009)

    Google Scholar 

  50. Esmaeilzadeh, M., Kadkhodayan, M.: Dynamic analysis of stiffened bi-directional functionally graded plates with porosities under a moving load by dynamic relaxation method with kinetic damping. Aerosp. Sci. Technol. 93, 105333 (2019). https://doi.org/10.1016/j.ast.2019.105333

    Article  Google Scholar 

  51. Li, Q., Iu, V., Kou, K.: Three-dimensional vibration analysis of functionally graded material plates in thermal environment. J. Sound Vib. 324(3–5), 733–750 (2009). https://doi.org/10.1016/j.jsv.2009.02.036

    Article  Google Scholar 

Download references

Acknowledgements

The research reported herein was funded by the Deanship of Scientific Research at the University of Hail, Saudi Arabia, through the Project Number RG- 191241. The authors would like to express their deepest gratitude to the Deanship of Scientific Research and to the College of Engineering at the University of Hail for providing necessary support to conducting this research.

Author information

Authors and Affiliations

  1. Laboratory of numerical and experimental modelling of the mechanical phenomena, Mechanical Engineering Department, Faculty of Sciences and Technology, Ibn Badis University, 27000, Mostaganem, Algeria

    Fatima Zohra Zaoui & Djamel Ouinas

  2. Material and Hydrology Laboratory, Civil Engineering Department, Faculty of Technology, Sidi Bel Abbes University, 22000, Sidi Bel Abbés, Algeria

    Abdelouahed Tounsi

  3. Department of Civil and Environmental Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Eastern Province, 31261, Kingdom of Saudi Arabia

    Abdelouahed Tounsi

  4. Department of Materials Science and Metallurgical Engineering, University of Oviedo, Gijón, Spain

    Jaime Aurelio Viña Olay

  5. Civil Engineering Department, University of Ha’il, Ha’il, Kingdom of Saudi Arabia

    Belkacem Achour & Mabrouk Touahmia

Authors
  1. Fatima Zohra Zaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Djamel Ouinas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abdelouahed Tounsi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jaime Aurelio Viña Olay
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Belkacem Achour
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Mabrouk Touahmia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fatima Zohra Zaoui.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zaoui, F.Z., Ouinas, D., Tounsi, A. et al. Fundamental frequency analysis of functionally graded plates with temperature-dependent properties based on improved exponential-trigonometric two-dimensional higher shear deformation theory. Arch Appl Mech 91, 859–881 (2021). https://doi.org/10.1007/s00419-020-01793-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-020-01793-1

Keywords

Search

Navigation