Abstract
Compared to the first-order shear deformation theory and other classical shell theories, the higher-order shear theory is deemed more accurate due to its superior ability to capture transverse shear effects, especially vital for precision in modeling thicker, doubly curved shell panels. Additionally, the third-order shear deformation theory (TSDT) is acknowledged for its computational efficiency compared to the 3D solution striking a balance between result precision and computational efficiency. This paper explores the static bending and free vibration analysis of a porous bi-directional functionally graded doubly curved sandwich shell. For the first time, a combination of TSDT theory with the p-version finite element method is applied, demonstrated for the analysis of bi-directional functionally graded doubly curved sandwich shell. In the initial phase, the mathematical formulation has been meticulously derived. Four models of sandwich FGM distributions, taking into account the porosity effect and comprising a blend of two ceramic materials and a metallic material, have been thoroughly explored. Subsequently, the study evaluates the effectiveness and accuracy of the formulation implemented in FORTRAN CODE through benchmark results, showcasing its adaptability for different shell panel geometries by adjusting the values of the radius of curvature. The latter part of the research delves into new findings related to bi-directional functionally graded porous sandwich FGM shell panels, investigating the effects of gradient indexes and porosity distribution on their behavior.
Similar content being viewed by others
Data availability
No Data associated in the manuscript.
References
Zhang, C., et al.: Additive manufacturing of functionally graded materials: a review. Mater. Sci. Eng. A 764, 138209 (2019). https://doi.org/10.1016/J.MSEA.2019.138209
Zhang, N., Khan, T., Guo, H., Shi, S., Zhong, W., Zhang, W.: Functionally graded materials: an overview of stability, buckling, and free vibration analysis. Adv. Mater. Sci. Eng. (2019). https://doi.org/10.1155/2019/1354150
Ghatage, P.S., Kar, V.R., Sudhagar, P.E.: On the numerical modelling and analysis of multi-directional functionally graded composite structures: a review. Compos. Struct. 236, 111837 (2020). https://doi.org/10.1016/J.COMPSTRUCT.2019.111837
Alipour, M.M., Shariyat, M., Shaban, M.: A semi-analytical solution for free vibration of variable thickness two-directional-functionally graded plates on elastic foundations. Int. J. Mech. Mater. Des. 6(4), 293–304 (2010). https://doi.org/10.1007/S10999-010-9134-2
Nie, G., Zhong, Z.: Dynamic analysis of multi-directional functionally graded annular plates. Appl. Math. Model. 34(3), 608–616 (2010). https://doi.org/10.1016/J.APM.2009.06.009
Kermani, I.D., Ghayour, M., Mirdamadi, H.R.: Free vibration analysis of multi-directional functionally graded circular and annular plates. J. Mech. Sci. Technol. 26(11), 3399–3410 (2013). https://doi.org/10.1007/S12206-012-0860-2
Shariyat, M., Alipour, M.M.: A power series solution for vibration and complex modal stress analyses of variable thickness viscoelastic two-directional FGM circular plates on elastic foundations. Appl. Math. Model. 37(5), 3063–3076 (2013). https://doi.org/10.1016/J.APM.2012.07.037
Mahinzare, M., Barooti, M.M., Ghadiri, M.: Vibrational investigation of the spinning bi-dimensional functionally graded (2-FGM) micro plate subjected to thermal load in thermal environment. Microsyst. Technol. 24(3), 1695–1711 (2017). https://doi.org/10.1007/S00542-017-3544-0
Lieu, Q.X., Lee, S., Kang, J., Lee, J.: Bending and free vibration analyses of in-plane bi-directional functionally graded plates with variable thickness using isogeometric analysis. Compos. Struct. 192, 434–451 (2018). https://doi.org/10.1016/J.COMPSTRUCT.2018.03.021
Shojaeefard, M.H., Saeidi-Googarchin, H., Mahinzare, M., Ghadiri, M.: Free vibration and critical angular velocity of a rotating variable thickness two-directional FG circular microplate. Microsyst. Technol. 24(3), 1525–1543 (2017). https://doi.org/10.1007/S00542-017-3557-8
Wu, C.P., Yu, L.T.: Free vibration analysis of bi-directional functionally graded annular plates using finite annular prism methods. J. Mech. Sci. Technol. 33(5), 2267–2279 (2019). https://doi.org/10.1007/S12206-019-0428-5
Thai, S., Nguyen, V.X., Lieu, Q.X.: Bending and free vibration analyses of multi-directional functionally graded plates in thermal environment: a three-dimensional Isogeometric analysis approach. Compos. Struct. 295, 115797 (2022). https://doi.org/10.1016/J.COMPSTRUCT.2022.115797
Wang, C., Koh, J.M., Yu, T., Xie, N.G., Cheong, K.H.: Material and shape optimization of bi-directional functionally graded plates by GIGA and an improved multi-objective particle swarm optimization algorithm. Comput. Methods Appl. Mech. Eng. 366, 113017 (2020). https://doi.org/10.1016/J.CMA.2020.113017
Hashemi, S., et al.: Nonlinear free vibration analysis of in-plane bi-directional functionally graded plate with porosities resting on elastic foundations. Int. J. Appl. Mech. (2022). https://doi.org/10.1142/S1758825121501313
Dehshahri, K.M.Z.S.A.A.: Free vibrations analysis of arbitrary three-dimensionally FGM nanoplates. Adv. Nano Res. 8(2), 115–134 (2020). https://doi.org/10.12989/ANR.2020.8.2.115
Ahlawat, N., Lal, R.: Effect of Winkler foundation on radially symmetric vibrations of bi-directional FGM non-uniform Mindlin’s circular plate subjected to in-plane peripheral loading. J. Solid Mech. 12(2), 455–475 (2020). https://doi.org/10.22034/JSM.2019.1873720.1466
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
Ahlawat, N.: Numerical solution for buckling and vibration of bi-directional FGM circular plates. AIP Conf. Proc. 2061(1), 020020 (2019). https://doi.org/10.1063/1.5086642
Sharma, P., Khinchi, A.: On frequency investigation of bi-directional FGM beam under thermal effect. Mater. Today Proc. 47, 6089–6092 (2021). https://doi.org/10.1016/J.MATPR.2021.05.022
Tang, Y., Lv, X., Yang, T.: Bi-directional functionally graded beams: asymmetric modes and nonlinear free vibration. Compos. Part B Eng. 156, 319–331 (2019). https://doi.org/10.1016/J.COMPOSITESB.2018.08.140
Barati, A., Hadi, A., Nejad, M.Z., Noroozi, R.: On vibration of bi-directional functionally graded nanobeams under magnetic field. Mech. Based Des. Struct. Mach. 50(2), 468–485 (2020). https://doi.org/10.1080/15397734.2020.1719507
Zhao, L., Zhu, J., Wen, X.D.: Exact analysis of bi-directional functionally graded beams with arbitrary boundary conditions via the symplectic approach. Struct. Eng. Mech. 59(1), 101–122 (2016). https://doi.org/10.12989/SEM.2016.59.1.101
Şimşek, M.: Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions. Compos. Struct. 133, 968–978 (2015). https://doi.org/10.1016/J.COMPSTRUCT.2015.08.021
Lezgy-Nazargah, M.: Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach. Aerosp. Sci. Technol. 45, 154–164 (2015). https://doi.org/10.1016/J.AST.2015.05.006
Tang, Y., Ding, Q.: Nonlinear vibration analysis of a bi-directional functionally graded beam under hygro-thermal loads. Compos. Struct. 225, 111076 (2019). https://doi.org/10.1016/J.COMPSTRUCT.2019.111076
Truong, T.T., Nguyen-Thoi, T., Lee, J.: Isogeometric size optimization of bi-directional functionally graded beams under static loads. Compos. Struct. 227, 111259 (2019). https://doi.org/10.1016/J.COMPSTRUCT.2019.111259
Ohab-Yazdi, S.M.K., Kadkhodayan, M.: Free vibration of bi-directional functionally graded imperfect nanobeams under rotational velocity. Aerosp. Sci. Technol. 119, 107210 (2021). https://doi.org/10.1016/J.AST.2021.107210
Nejad, M.Z., Hadi, A.: Non-local analysis of free vibration of bi-directional functionally graded Euler–Bernoulli nano-beams. Int. J. Eng. Sci. 105, 1–11 (2016). https://doi.org/10.1016/J.IJENGSCI.2016.04.011
Zhao, L., Chen, W.Q., Lü, C.F.: Symplectic elasticity for bi-directional functionally graded materials. Mech. Mater. 54, 32–42 (2012). https://doi.org/10.1016/J.MECHMAT.2012.06.001
Fariborz, J., Batra, R.C.: Free vibration of bi-directional functionally graded material circular beams using shear deformation theory employing logarithmic function of radius. Compos. Struct. 210, 217–230 (2019). https://doi.org/10.1016/J.COMPSTRUCT.2018.11.036
Yang, T., Tang, Y., Li, Q., Yang, X.D.: Nonlinear bending, buckling and vibration of bi-directional functionally graded nanobeams. Compos. Struct. 204, 313–319 (2018). https://doi.org/10.1016/J.COMPSTRUCT.2018.07.045
Pydah, A., Batra, R.C.: Shear deformation theory using logarithmic function for thick circular beams and analytical solution for bi-directional functionally graded circular beams. Compos. Struct. 172, 45–60 (2017). https://doi.org/10.1016/J.COMPSTRUCT.2017.03.072
Ramteke, P.M., Panda, S.K.: Free vibrational behaviour of multi-directional porous functionally graded structures. Arab. J. Sci. Eng. 46(8), 7741–7756 (2021). https://doi.org/10.1007/S13369-021-05461-6/METRICS
Nguyen-Ngoc, H., Cuong-Le, T., Nguyen, K.D., Nguyen-Xuan, H., Abdel-Wahab, M.: Three-dimensional polyhedral finite element method for the analysis of multi-directional functionally graded solid shells. Compos. Struct. 305, 116538 (2023). https://doi.org/10.1016/J.COMPSTRUCT.2022.116538
Eroğlu, M., Esen, İ, Koç, M.A.: Thermal vibration and buckling analysis of magneto-electro-elastic functionally graded porous higher-order nanobeams using nonlocal strain gradient theory. Acta Mech. (2023). https://doi.org/10.1007/S00707-023-03793-Y
Al-Osta, M.A., Al-Osta, M.A.: Wave propagation investigation of a porous sandwich FG plate under hugrothermal environments via a new first-order shear deformation theory. Steel Compos. Struct. 43(1), 117 (2022). https://doi.org/10.12989/SCS.2022.43.1.117
Nabawy, A.E., et al.: Study of the dynamic behavior of porous functionally graded suspension structural systems using finite elements methods. Steel Compos. Struct. 45(5), 697 (2022). https://doi.org/10.12989/SCS.2022.45.5.697
Kumar, H.S.N., Kattimani, S., Kumar, H.S.N., Kattimani, S.: Nonlinear analysis of two-directional functionally graded doubly curved panels with porosities. Struct. Eng. Mech. 82(4), 477 (2022). https://doi.org/10.12989/SEM.2022.82.4.477
Shan, X., Huang, A., Shan, X., Huang, A.: Intelligent simulation of the thermal buckling characteristics of a tapered functionally graded porosity-dpependent rectangular small-scale beam. Adv. Nano Res. 12(3), 281 (2022). https://doi.org/10.12989/ANR.2022.12.3.281
Zhou, J., et al.: Intelligent modeling to investigate the stability of two-dimensional functionally graded porosity-dependent nanobeam. Comput. Concr. 30(2), 85 (2022). https://doi.org/10.12989/CAC.2022.30.2.085
Zanjanchi, M., Ghadiri, M., Sabouri-Ghomi, S.: Dynamic stability and bifurcation point analysis of FG porous core sandwich plate reinforced with graphene platelet. Acta Mech. 234(10), 5015–5037 (2023). https://doi.org/10.1007/S00707-023-03638-8/METRICS
Chorfi, S.M., Houmat, A.: Non-linear free vibration of a functionally graded doubly-curved shallow shell of elliptical plan-form. Compos. Struct. 92(10), 2573–2581 (2010). https://doi.org/10.1016/J.COMPSTRUCT.2010.02.001
Belalia, S.A.: A curved hierarchical finite element method for the nonlinear vibration analysis of functionally graded sandwich elliptic plates. Mech. Adv. Mater. Struct. 26(13), 1115–1129 (2018). https://doi.org/10.1080/15376494.2018.1430277
Houmat, A.: Three-dimensional free vibration analysis of variable stiffness laminated composite rectangular plates. Compos. Struct. 194, 398–412 (2018). https://doi.org/10.1016/J.COMPSTRUCT.2018.04.028
Stoykov, S., Ribeiro, P.: Vibration analysis of rotating 3D beams by the p-version finite element method. Finite Elem. Anal. Des. 65, 76–88 (2013). https://doi.org/10.1016/J.FINEL.2012.10.008
Van Vinh, P.: Analysis of bi-directional functionally graded sandwich plates via deformation theory and finite element method. J. Sandwich Struct. Mater. (2021). https://doi.org/10.1177/10996362211025811
Daikh, A.A., Zenkour, A.M.: Free vibration and buckling of porous power-law and sigmoid functionally graded sandwich plates using a simple higher-order shear deformation theory. Mater. Res. Express 6(11), 115707 (2019). https://doi.org/10.1088/2053-1591/AB48A9
Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51(4), 745–752 (1984). https://doi.org/10.1115/1.3167719
Panda, S.K., Singh, B.N.: Nonlinear free vibration of spherical shell panel using higher order shear deformation theory—a finite element approach. Int. J. Press. Vessel. Pip. 86(6), 373–383 (2009). https://doi.org/10.1016/J.IJPVP.2008.11.023
Belalia, S.A.: Investigation of the mechanical properties on the large amplitude free vibrations of the functionally graded material sandwich plates. J. Sandwich Struct. Mater. 21(3), 895–916 (2017). https://doi.org/10.1177/1099636217701299
Houmat, A.: Mapped infinite p-element for two-dimensional problems of unbounded domains. Comput. Geotech. 35(4), 608–615 (2008). https://doi.org/10.1016/J.COMPGEO.2007.09.007
Reddy, J.N.: Energy principles and variational methods in applied mechanics (2017). Accessed: Feb. 14, 2024. https://www.wiley.com/en-gb/Energy+Principles+and+Variational+Methods+in+Applied+Mechanics%2C+3rd+Edition-p-9781119087397
Singh, V.K., Panda, S.K.: Nonlinear free vibration analysis of single/doubly curved composite shallow shell panels. Thin-Walled Struct. 85, 341–349 (2014). https://doi.org/10.1016/J.TWS.2014.09.003
Hosseini-Hashemi, S., Fadaee, M.: On the free vibration of moderately thick spherical shell panel—a new exact closed-form procedure. J. Sound Vib. 330(17), 4352–4367 (2011). https://doi.org/10.1016/J.JSV.2011.04.011
Fan, S.C., Luah, M.H.: Free vibration analysis of arbitrary thin shell structures by using spline finite element. J. Sound Vib. 179(5), 763–776 (1995). https://doi.org/10.1006/JSVI.1995.0051
Chern, Y.C., Chao, C.C.: Comparison of natural frequencies of laminates by 3-D theory, part II: curved panels. J. Sound Vib. 230(5), 1009–1030 (2000). https://doi.org/10.1006/JSVI.1999.2454
Rachid, A., et al.: Mechanical behavior and free vibration analysis of FG doubly curved shells on elastic foundation via a new modified displacements field model of 2D and quasi-3D HSDTs. Thin-Walled Struct. 172, 108783 (2022). https://doi.org/10.1016/J.TWS.2021.108783
Sayyad, A.S., Ghugal, Y.M.: Static and free vibration analysis of doubly-curved functionally graded material shells. Compos. Struct. 269(May), 114045 (2021). https://doi.org/10.1016/j.compstruct.2021.114045
Zenkour, A.M.: A comprehensive analysis of functionally graded sandwich plates: part 2—buckling and free vibration. Int. J. Solids Struct. 42(18–19), 5243–5258 (2005). https://doi.org/10.1016/J.IJSOLSTR.2005.02.016
Nguyen, V.H., Nguyen, T.K., Thai, H.T., Vo, T.P.: A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates. Compos. Part B Eng. 66, 233–246 (2014). https://doi.org/10.1016/J.COMPOSITESB.2014.05.012
Zenkour, A.M.: Bending analysis of functionally graded sandwich plates using a simple four-unknown shear and normal deformations theory. J. Sandwich Struct. Mater. 15(6), 629–656 (2013). https://doi.org/10.1177/1099636213498886
Neves, A.M.A., et al.: 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 Eng. 44(1), 657–674 (2013). https://doi.org/10.1016/J.COMPOSITESB.2012.01.089
Bessaim, A., Houari, M.S.A., Tounsi, A., Mahmoud, S.R., Bedia, E.A.A.: A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets. J. Sandwich Struct. Mater. 15(6), 671–703 (2013). https://doi.org/10.1177/1099636213498888
Acknowledgements
The Authors extend their appreciation to the Deanship Scientific Research at King Khalid University for funding this work through large group Research Project under Grant Number: RGP2/422/44.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflict of interest in preparing this article.
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.
About this article
Cite this article
Lakhdar, Z., Chorfi, S.M., Belalia, S.A. et al. Free vibration and bending analysis of porous bi-directional FGM sandwich shell using a TSDT p-version finite element method. Acta Mech (2024). https://doi.org/10.1007/s00707-024-03909-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00707-024-03909-y