This paper addresses the investigation of axisymmetric thermally induced vibrations in Functionally Graded Material (FGM) cylindrical shells. It considers temperature-dependent (TD) properties and geometric non-linearity (the von Karman effect). The study systematically solves a transient heat conduction equation using finite differences method and the Crank–Nicolson method. During the heating stages, the evaluation of thermal forces and moments takes place. Equations of motion are derived through the application of Hamilton’s principle. Spatial dependencies are discretized using the generalized Ritz method, while temporal dependencies are approximated using the $β$ -Newmark method with Newton–Raphson linearization. A comparative analysis validates the procedure’s efficiency and precision. Parametric studies explore the influence of parameters, including the temperature-dependency material properties, geometric nonlinearity, and shell’s power-law index, providing valuable insights into FGM shell behavior under thermal shock.

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References

1. B. A. Boley, Thermally induced vibrations of beams, J. Aeronaut. Sci. 23(2) (1956) 179–182, https://doi.org/10.2514/8.3527. Web of Science, Google Scholar
2. B. A. Boley and A. D. Barber, Dynamic response of beams and plates to rapid heating, J. Appl. Mech.-Trans. ASME 24(3) (1957) 413–416. Crossref, Google Scholar
3. H. Kraus, Thermally induced vibrations of thin nonshallow spherical shells, AIAA J. 4(3) (1966) 500–505, https://doi.org/10.2514/3.3464. Crossref, Web of Science, Google Scholar
4. Y. Nakajo and K. Hayashi, Response of circular plates to thermal impact, J. Sound Vib. 95(2) (1984) 213–222, https://doi.org/10.1016/0022-460X(84)90543-1. Crossref, Web of Science, Google Scholar
5. S. Das, Vibrations of polygonal plates due to thermal shock, J. Sound Vib. 89(4) (1983) 471–476, https://doi.org/10.1016/0022-460X(83)90348-6. Crossref, Web of Science, Google Scholar
6. J. Venkataramana and M. K. Jana, Thermally forced vibrations of beams, J. Sound. Vib. 37(2) (1974) 291–295, https://doi.org/10.1016/S0022-460X(74)80338-X. Crossref, Web of Science, Google Scholar
7. R. C. Stroud and J. Mayers, Dynamic response of rapidly heated plate elements, AIAA J. 9(1) (1970) 76–83, https://doi.org/10.2514/3.6126. Crossref, Google Scholar
8. J. C. Bruch, S. Adali, I. S. Sadek and J. M. Sloss, Structural control of thermoelastic beams for vibration suppression, J. Therm. Stresses 16(3) (1993) 249–263, https://doi.org/10.1080/01495739308946229. Crossref, Web of Science, Google Scholar
9. Y. Nakajo and K. Hayashi, Response of simply supported and clamped circular plates to thermal impact, J. Sound Vib. 122(2) (1988) 347–356, https://doi.org/10.1016/S0022-460X(88)80359-6. Crossref, Web of Science, Google Scholar
10. D. L. Hill and J. Mazumdar, A study of the thermally induced large amplitude vibrations of viscoelastic plates and shallow shells, J. Sound Vib. 116(2) (1987) 323–337, https://doi.org/10.1016/S0022-460X(87)81305-6. Crossref, Web of Science, Google Scholar
11. A. Keibolahi, M. Heidari, Y. Kiani and M. R. Eslami, Nonlinear analysis of pin-ended deep arches under instantaneous heating, J. Therm. Stresses 45(11) (2022) 918–936, https://doi.org/10.1080/01495739.2022.2118649. Crossref, Web of Science, Google Scholar
12. T. R. Tauchert, Thermal shock of orthotropic rectangular plates, J. Therm. Stresses 12(2) (1989) 241–258, https://doi.org/10.1080/01495738908961964. Crossref, Web of Science, Google Scholar
13. J. S. Chang, J. H. Wang and T. Z. Tsai, Thermally induced vibrations of thin laminated plates by finite element method, Comput. Struct. 42(1) (1992) 117–128, https://doi.org/10.1016/0045-7949(92)90541-7. Crossref, Google Scholar
14. L. W. Chen and J. H. Lee, Vibration of thermal elastic orthotropic plates, Appl. Acoust. 27(4) (1989) 287–304, https://doi.org/10.1016/0003-682X(89)90035-2. Crossref, Web of Science, Google Scholar
15. N. N. Huang and T. R. Tauchert, Thermally induced vibration of doubly curved cross-ply laminated panels, J. Sound Vib. 154(3) (1992) 485–494, https://doi.org/10.1016/0022-460X(92)90781-R. Crossref, Web of Science, Google Scholar
16. N. N. Huang and T. R. Tauchert, Large amplitude vibrations of graphite reinforced aluminum cylindrical panels subjected to rapid heating, Compos. Eng. 3(6) (1993) 557–566, https://doi.org/10.1016/0961-9526(93)90052-L. Crossref, Google Scholar
17. A. A. Khdeir, Thermally induced vibration of cross-ply laminated shallow shells, Acta Mech. 151(3–4) (2001) 135–147, https://doi.org/10.1007/BF01246913. Crossref, Web of Science, Google Scholar
18. A. A. Khdeir, Thermally induced vibration of cross-ply laminated shallow arches, J. Therm. Stresses 24(11) (2001) 1085–1096, https://doi.org/10.1080/01495730152620078. Crossref, Web of Science, Google Scholar
19. J. S. Chang and J. W. Shyong, Thermally induced vibration of laminated circular cylindrical shell panels, J. Therm. Stresses 51(3) (1994) 419–427, https://doi.org/10.1016/0266-3538(94)90110-4. Google Scholar
20. S. Raja, P. K. Sinha, G. Prathap and D. Dwarakanathan, Thermally induced vibration control of composite plates and shells with piezoelectric active damping, Smart Mater. Struct. 13(4) (2004) 939–950, https://doi.org/10.1088/0964-1726/13/4/032. Crossref, Google Scholar
21. R. Kumar, B. K. Mishra and S. C. Jain, Thermally induced vibration control of cylindrical shell using piezoelectric sensor and actuator, Int. J. Adv. Manuf. Technol. 38(5–6) (2008) 551–562, https://doi.org/10.1007/s00170-007-1076-y. Crossref, Web of Science, Google Scholar
22. G. J. Simitses, Instability of dynamically-loaded structures, Appl. Mech. Rev. 40(10) (1987) 1403–1408, https://doi.org/10.1115/1.3149542. Crossref, Google Scholar
23. G. J. Simitses, Dynamic Stability of Suddenly Loaded Structures (Springer, New York, 1990). Crossref, Google Scholar
24. B. Budiansky and R. S. Roth, Axisymmetric dynamic buckling of clamped shallow spherical shells, NASA TN D-1510 (1962), pp. 597–606. Google Scholar
25. J. W. Hutchinson and B. Budiansky, Dynamic buckling estimate, AIAA J. 4(3) (1966) 525–530, https://doi.org/10.2514/3.3468. Crossref, Google Scholar
26. A. Keibolahi, Y. Kiani and M. R. Eslami, Nonlinear rapid heating of shallow arches, J. Therm. Stresses 41(10–12) (2018) 12441258, https://doi.org/10.1080/01495739.2018.1494522. Crossref, Web of Science, Google Scholar
27. A. Keibolahi, Y. Kiani and M. R. Eslami, Dynamic snap-through of shallow arches under thermal shock, Aerosp. Sci. Technol. 77 (2018) 545–554, https://doi.org/10.1016/j.ast.2018.04.003. Crossref, Web of Science, Google Scholar
28. R. K. Bhangale and N. Ganesan, Thermoelastic buckling and vibration behavior of a functionally graded sandwich beam with constrained viscoelastic core, J. Sound Vib. 295(1–2) (2006) 294–316, https://doi.org/10.1016/j.jsv.2006.01.026. Crossref, Web of Science, Google Scholar
29. S. R. Li, J. H. Zhang and Y. G. Zhao, Thermal post-buckling of functionally graded material Timoshenko beams, Appl. Math. Mech. 27(6) (2006) 803–810, https://doi.org/10.1007/s10483-006-0611-y. Crossref, Web of Science, Google Scholar
30. F. Q. Zhao, Z. M. Wang and H. Z. Liu, Thermal post-bunkling analyses of functionally graded material rod, Appl. Math. Mech. 28(1) (2007) 59–67, https://doi.org/10.1007/s10483-007-0107-z. Crossref, Google Scholar
31. Y. Kiani and M. R. Eslami, Thermal buckling analysis of functionally graded material beams, Int. J. Mech. Mater. Des. 6(3) (2010) 229–238, https://doi.org/10.1007/s10999-010-9132-4. Crossref, Web of Science, Google Scholar
32. Y. Kiani, M. Rezaei, S. Taheri and M. R. Eslami, Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams, Int. J. Mech. Mater. Des. 7(3) (2011) 185–197, https://doi.org/10.1007/s10999-011-9158-2. Crossref, Web of Science, Google Scholar
33. L. S. Ma and D. W. Lee, Exact solutions for nonlinear static response of a shear deformable FGM beam under an in-plane thermal loading, Eur. J. Mech. — A/Solids 31(1) (2012) 13–20, https://doi.org/10.1016/j.euromechsol.2011.06.016. Crossref, Web of Science, Google Scholar
34. Y. Kiani and M. R. Eslami, Geometrically nonlinear rapid heating of temperature-dependent circular FGM plates, J. Therm. Stresses 37(12) (2014) 1495–1518, https://doi.org/10.1080/01495739.2014.937259. Crossref, Web of Science, Google Scholar
35. S. M. Alipour, Y. Kiani and M. R. Eslami, Rapid heating of FGM rectangular plates, Acta Mech. 227(2) (2016) 421–436, https://doi.org/10.1007/s00707-015-1461-9. Crossref, Web of Science, Google Scholar
36. S. E. Ghiasian, Y. Kiani and M. R. Eslami, Nonlinear rapid heating of FGM beams, Int. J. Non-Linear Mech. 67(1) (2014) 74–84, https://doi.org/10.1016/j.ijnonlinmec.2014.08.006. Crossref, Google Scholar
37. S. Pandey and S. Pradyumna, A finite element formulation for thermally induced vibrations of functionally graded material sandwich plates and shell panels, Compos. Struct. 160(2) (2017) 877–886, https://doi.org/10.1016/j.compstruct.2016.10.040. Crossref, Google Scholar
38. M. Javani, Y. Kiani and M. R. Eslami, Geometrically nonlinear rapid surface heating of temperature-dependent FGM arches, Aerosp. Sci. Technol. 90 (2019) 264–274, https://doi.org/10.1016/j.ast.2019.04.049. Crossref, Web of Science, Google Scholar
39. M. Javani, Y. Kiani, M. Sadighi and M. R. Eslami, Nonlinear vibration behavior of rapidly heated temperature-dependent FGM shallow spherical shells, AIAA J. 57(9) (2019) 4071–4084, https://doi.org/10.2514/1.J058240. Crossref, Web of Science, Google Scholar
40. M. M. Khalili, A. Keibolahi, Y. Kinai and M. R. Eslami, Application of Ritz method to large amplitude rapid surface heating of FGM shallow arches, Arch. Appl. Mech. 92(4) (2022) 1287–1301, https://doi.org/10.1007/s00419-022-02106-4. Crossref, Web of Science, Google Scholar
41. M. M. Khalili, A. Keibolahi, Y. Kinai and M. R. Eslami, Dynamic snap-through of functionally graded shallow arches under rapid surface heating, Thin-Walled Struct. 178 (2022) 109541, https://doi.org/10.1016/j.tws.2022.109541. Crossref, Web of Science, Google Scholar
42. R. Ansari, M. Zargar Ershadi, H. Akbardoost Laskoukalayeh, M. Faraji Oskouie and H. Rouhi, Hygrothermally-induced vibration analysis of porous FGM rectangular mindlin plates resting on elastic foundation, Int. J. Struct. Stab. Dyn. 48 (2023) 1620–1633, https://doi.org/10.1142/S0219455424501001. Web of Science, Google Scholar
43. R. Ansari, S. Nesarhosseini, M. Faraji Oskouie and H. Rouhi, Dynamic response of rapidly heated rectangular plates made of porous functionally graded material, Int. J. Struct. Stab. Dyn. 22 (2022) 2250090, https://doi.org/10.1142/S0219455422500900. Link, Web of Science, Google Scholar
44. M. Faraji Oskouie, M. Zargar and R. Ansari, Dynamic snap-through instability of hygro-thermally excited functionally graded porous arches, Int. J. Struct. Stab. Dyn. 23 (2023) 2350030, https://doi.org/10.1142/S021945542350030X. Link, Web of Science, Google Scholar
45. B. Mirzavand, M. G. Javadi and M. A. Amiri Atashgah, On thermally induced vibration control of hybrid magnetostrictive beams and plates, Int. J. Struct. Stab. Dyn. 23 (2023) 2350008, https://doi.org/10.1142/S0219455423500086. Link, Web of Science, Google Scholar
46. A. Keibolahi, Y. Kinai and M. R. Eslami, Nonlinear dynamic snap-through and vibrations of temperature-dependent FGM deep spherical shells under sudden thermal shock, Thin-Walled Struct. 185 (2023) 110561, https://doi.org/10.1016/j.tws.2023.110561. Crossref, Web of Science, Google Scholar
47. H. Bagheri, Y. Kinai and M. R. Eslami, Geometrically nonlinear response of FGM joined conical-conical shells subjected to thermal shock, Thin-Walled Struct. 182 (2023) 110171, https://doi.org/10.1016/j.tws.2022.110171. Crossref, Web of Science, Google Scholar
48. R. Ansari, M. Zargar Ershadi, H. A. Laskoukalayeh and H. Rouhi, Nonlinear large-amplitude vibration analysis of annular sector plates made of FGMs subjected to cooling shock, Thin-Walled Struct. 193 (2023) 111223, https://doi.org/10.1016/j.tws.2023.111233. Crossref, Web of Science, Google Scholar
49. R. Ansari, M. Zargar Ershadi, M. Faraji Oskouei and H. Rouhi, A VDQ approach to nonlinear vibration analysis of functionally graded porous circular plates resting on elastic foundation under hygrothermal shock, Acta Mech. 234 (2023) 5115–5129, https://doi.org/10.1007/s00707-023-03649-5. Crossref, Web of Science, Google Scholar
50. M. Zargar Ershadi, M. Faraji Oskouei and R. Ansari, Nonlinear vibration analysis of functionally graded porous circular plates under hygro-thermal loading, Mech. Based Des. Struct. Mach. 52 (2024) 1042–1059, https://doi.org/0.1080/15397734.2022.21341475. Crossref, Google Scholar
51. N. T. Phuong, V. H. Nam, N. T. Trung, V. M. Duc and P. V. Phong, Nonlinear stability of sandwich functionally graded cylindrical shells with stiffeners under axial compression in thermal environment, Int. J. Struct. Stab. Dyn. 19 (2019) 2350008, https://doi.org/10.1142/S0219455419500731. Link, Web of Science, Google Scholar
52. V. H. Nam, N. T. Phuong, K. V. Minh and P. T. Hieu, Nonlinear thermo-mechanical buckling and post-buckling of multilayer FGM cylindrical shell reinforced by spiral stiffeners surrounded by elastic foundation subjected to torsional loads, Eur. J. Mech. A Solids 72 (2018) 393–406, https://doi.org/10.1016/j.euromechsol.2018.06.005. Crossref, Web of Science, Google Scholar
53. V. H. Nam, N. T. Phuong, K. V. Minh and P. T. Hieu, Nonlinear static and dynamic buckling analysis of imperfect eccentrically stiffened functionally graded circular cylindrical thin shells under axial compression, Int. J. Mech. Sci. 74 (2013) 190–200, https://doi.org/10.1016/j.ijmecsci.2013.06.002. Crossref, Web of Science, Google Scholar
54. V. H. Nam, N. T. Phuong, C. V. Doan and N. T. Trung, Nonlinear thermo-mechanical stability analysis of eccentrically spiral stiffened sandwich functionally graded cylindrical shells subjected to external pressure, Int. J. Appl. Mech. 11 (2019) 1950045, https://doi.org/10.1142/S1758825119500455. Link, Web of Science, Google Scholar
55. V. H. Nam, N. T. Phuong and N. T. Trung, Nonlinear buckling and postbuckling of sandwich FGM cylindrical shells reinforced by spiral stiffeners under torsion loads in thermal environment, Acta Mech. 230 (2019) 3183–3204, https://doi.org/10.1007/s00707-019-02452-5. Crossref, Web of Science, Google Scholar
56. N. T. Phuong, D. T. Luan, V. H. Nam and P. T. Hieu, Nonlinear approach on torsional buckling and postbuckling of functionally graded cylindrical shells reinforced by orthogonal and spiral stiffeners in thermal environment, Proc. Inst. Mech. Eng. C, J. Mech. Eng. Sci. 233 (2019) 2091–2106, https://doi.org/10.1177/0954406218780523. Crossref, Web of Science, Google Scholar
57. M. R. Eslami, Buckling and Postbuckling of Beams, Plates and Shells (Springer, Switzerland, 2018). Crossref, Google Scholar
58. J. N. Reddy, An Introduction to Nonlinear Finite Element Analysis (Oxford University Press, Oxford, 2004). Crossref, Google Scholar
59. J. N. Reddy, Mechanics of Laminated Composite Plates and Shells, Theory and Application (CRC Press, Boca Raton, 2003). Crossref, Google Scholar
60. R. B. Hetnarski and M. R. Eslami, Thermal Stresses, Advanced Theory and Applications (Springer, Amsterdam, 2009). Google Scholar
61. N. J. Hoff and V. G. Bruce, Dynamic analysis of the buckling of laterally loaded flat arches, J. Math. Phys. 32(1–4) (1953) 276–288, https://doi.org/10.1002/sapm1953321276. Crossref, Google Scholar