Abstract
The nonlinear forced vibrations of functionally graded material (FGM) sandwich cylindrical shells with porosities on an elastic substrate are studied. A step function and a porosity volume fraction are introduced to describe the porosities in FGM layers of sandwich shells. Using the Donnell’s nonlinear shallow shell theory and Hamilton’s principle, an energy approach is employed to gain the nonlinear equations of motion. Afterwards, the multi-degree-of-freedom nonlinear ordinary differential equations are carried out by using Galerkin scheme, and subsequently the pseudo-arclength continuation method is utilized to perform the bifurcation analysis. Finally, the effects of the core-to-thickness ratio, porosity volume fraction, power-law exponent, and external excitation on nonlinear forced vibration characteristics of FGM sandwich shells with porosities are investigated in detail.
Similar content being viewed by others
References
Miyamoto, Y., Kaysser, W.A., Rabin, B.H., Kawasaki, A., Ford, R.G.: Functionally Graded Materials: Design, Processing and Applications. Springer Science & Business Media (2013)
Duc, N.D.: Nonlinear static and dynamic stability of functionally graded plates and shells. Vietnam Vietnam Natl Univ Press, Google Sch (2014)
Wang, A., Chen, H., Hao, Y., Zhang, W.: Vibration and bending behavior of functionally graded nanocomposite doubly-curved shallow shells reinforced by graphene nanoplatelets. Results Phys. 9, 550–559 (2018)
Gao, W., Qin, Z., Chu, F.: Wave propagation in functionally graded porous plates reinforced with graphene platelets. Aerosp. Sci. Technol. 102, 105860 (2020)
Herrmann, A.S., Zahlen, P.C., Zuardy, I.: Sandwich structures technology in commercial aviation. In: Sandwich Structures 7: Advancing with Sandwich Structures and Materials, pp. 13–26. Springer (2005)
Park, K.-Y., Lee, S.-E., Kim, C.-G., Han, J.-H.: Fabrication and electromagnetic characteristics of electromagnetic wave absorbing sandwich structures. Compos. Sci. Technol. 66, 576–584 (2006)
Schwingel, D., Seeliger, H.-W., Vecchionacci, C., Alwes, D., Dittrich, J.: Aluminium foam sandwich structures for space applications. Acta Astronaut. 61, 326–330 (2007)
Duc, N.D., Seung-Eock, K., Khoa, N.D., Chan, D.Q.: Nonlinear buckling and post-buckling analysis of shear deformable stiffened truncated conical sandwich shells with functionally graded face sheets and a functionally graded porous core. J. Sandw. Struct. Mater. 109963622090682 (2020)
Vo, T.P., Thai, H.-T., Nguyen, T.-K., Inam, F., Lee, J.: A quasi-3D theory for vibration and buckling of functionally graded sandwich beams. Compos. Struct. 119, 1–12 (2015)
Liu, Y., Qin, Z.-Y., Chu, F.-L.: Analytical study of the impact response of shear deformable sandwich cylindrical shell with a functionally graded porous core. Mech. Adv. Mater. Struct. 1–10 (2020)
Liu, B., Guo, M., Liu, C., Xing, Y.: Free vibration of functionally graded sandwich shallow shells in thermal environments by a differential quadrature hierarchical finite element method. Compos. Struct. 225, 111173 (2019)
Punera, D., Kant, T.: An assessment of refined hierarchical kinematic models for the bending and free vibration analyses of laminated and functionally graded sandwich cylindrical panels. J. Sandw. Struct. Mater. 1099636220909826 (2020)
Wang, Y.Q., Liu, Y.F., Zu, J.W.: Analytical treatment of nonlocal vibration of multilayer functionally graded piezoelectric nanoscale shells incorporating thermal and electrical effect. Eur. Phys. J. Plus. 134, 54 (2019)
Zhu, J., Lai, Z., Yin, Z., Jeon, J., Lee, S.: Fabrication of ZrO2–NiCr functionally graded material by powder metallurgy. Mater. Chem. Phys. 68, 130–135 (2001)
Rodríguez-Castro, R., Wetherhold, R.C., Kelestemur, M.H.: Microstructure and mechanical behavior of functionally graded Al A359/SiCp composite. Mater. Sci. Eng. A 323, 445–456 (2002)
Ebrahimi, F., Jafari, A.: A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities. Mech. Adv. Mater. Struct. 25, 212–224 (2018)
Liu, Y.F., Wang, Y.Q.: Thermo-Electro-Mechanical Vibrations of Porous Functionally Graded Piezoelectric Nanoshells. Nanomaterials. 9, 301 (2019)
Kim, J., Żur, K.K., Reddy, J.N.: Bending, free vibration, and buckling of modified couples stress-based functionally graded porous micro-plates. Compos. Struct. 209, 879–888 (2019)
Ding, H., Li, Y., Chen, L.-Q.: Nonlinear vibration of a beam with asymmetric elastic supports. Nonlinear Dyn. 95, 2543–2554 (2019)
Dat, N.D., Thanh, N. Van, MinhAnh, V., Duc, N.D.: Vibration and nonlinear dynamic analysis of sandwich FG-CNTRC plate with porous core layer. Mech. Adv. Mater. Struct. 1–18 (2020)
Cong, P.H., Chien, T.M., Khoa, N.D., Duc, N.D.: Nonlinear thermomechanical buckling and post-buckling response of porous FGM plates using Reddy’s HSDT. Aerosp. Sci. Technol. 77, 419–428 (2018)
Van Thanh, N., Khoa, N.D., Duc, N.D.: Nonlinear dynamic analysis of piezoelectric functionally graded porous truncated conical panel in thermal environments. Thin-Walled Struct. 154, 106837 (2020)
Duc, N.D., Quang, V.D., Nguyen, P.D., Chien, T.M.: Nonlinear dynamic response of functionally graded porous plates on elastic foundation subjected to thermal and mechanical loads. J. Appl. Comput. Mech. 4, 245–259 (2018)
Duc, N.D.: Nonlinear thermal dynamic analysis of eccentrically stiffened S-FGM circular cylindrical shells surrounded on elastic foundations using the Reddy’s third-order shear deformation shell theory. Eur. J. Mech. A/Solids 58, 10–30 (2016)
Vuong, P.M., Duc, N.D.: Nonlinear vibration of FGM moderately thick toroidal shell segment within the framework of Reddy’s third order-shear deformation shell theory. Int. J. Mech. Mater. Des. 16, 245–264 (2020)
Duc, N.D.: Nonlinear thermo- electro-mechanical dynamic response of shear deformable piezoelectric sigmoid functionally graded sandwich circular cylindrical shells on elastic foundations. J. Sandw. Struct. Mater. 20, 351–378 (2018)
Duc, N.D., Nguyen, P.D., Khoa, N.D.: Nonlinear dynamic analysis and vibration of eccentrically stiffened S-FGM elliptical cylindrical shells surrounded on elastic foundations in thermal environments. Thin-walled Struct. 117, 178–189 (2017)
Ding, H., Chen, L.-Q.: Nonlinear vibration of a slightly curved beam with quasi-zero-stiffness isolators. Nonlinear Dyn. 95, 2367–2382 (2019)
Zhang, W., Hao, Y.X., Yang, J.: Nonlinear dynamics of FGM circular cylindrical shell with clamped–clamped edges. Compos. Struct. 94, 1075–1086 (2012)
Zhang, W., Yang, J., Hao, Y.: Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory. Nonlinear Dyn. 59, 619–660 (2010)
Hao, Y., Li, Z., Zhang, W., Li, S., Yao, M.: Vibration of functionally graded sandwich doubly curved shells using improved shear deformation theory. Sci. China Technol. Sci. 61, 1–18 (2018)
Hao, Y.X., Chen, L.H., Zhang, W., Lei, J.G.: Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate. J. Sound Vib. 312, 862–892 (2008)
Hao, Y.X., Zhang, W., Yang, J.: Nonlinear oscillation of a cantilever FGM rectangular plate based on third-order plate theory and asymptotic perturbation method. Compos. Part B Eng. 42, 402–413 (2011)
Li, C., Li, P., Zhong, B., Wen, B.: Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions. Nonlinear Dyn. 95, 1903–1921 (2019)
Alijani, F., Amabili, M.: Effect of thickness deformation on large-amplitude vibrations of functionally graded rectangular plates. Compos. Struct. 113, 89–107 (2014)
Amabili, M.: Nonlinear vibrations and stability of shells and plates. Cambridge University Press (2008)
Qin, Z., Yang, Z., Zu, J., Chu, F.: Free vibration analysis of rotating cylindrical shells coupled with moderately thick annular plates. Int. J. Mech. Sci. 142–143, 127–139 (2018)
Amabili, M., Pellicano, F., Vakakis, A.F.: Nonlinear vibrations and multiple resonances of fluid-filled, circular shells, part 1: equations of motion and numerical results. J. Vib. Acoust. 122, 346–354 (2000)
Amabili, M., Pellicano, F., Paidoussis, M.P.: Nonlinear vibrations of simply supported, circular cylindrical shells, coupled to quiescent fluid. J. Fluids Struct. 12, 883–918 (1998)
Zhang, W., Zhao, M.H.: Nonlinear vibrations of a composite laminated cantilever rectangular plate with one-to-one internal resonance. Nonlinear Dyn. 70, 295–313 (2012)
Zhang, W., Zhao, M.H., Guo, X.Y.: Nonlinear responses of a symmetric cross-ply composite laminated cantilever rectangular plate under in-plane and moment excitations. Compos. Struct. 100, 554–565 (2013)
Zhang, W., Chen, J.E., Cao, D.X., Chen, L.H.: Nonlinear dynamic responses of a truss core sandwich plate. Compos. Struct. 108, 367–386 (2014)
Zhang, J., Yang, X., Zhang, W.: Free vibrations and nonlinear responses for a cantilever honeycomb sandwich plate. Adv. Mater. Sci. Eng. 2018, (2018)
Yang, X.-D., Zhang, W.: Nonlinear dynamics of axially moving beam with coupled longitudinal–transversal vibrations. Nonlinear Dyn. 78, 2547–2556 (2014)
Liu, Y.F., Ling, X., Wang, Y.Q.: Free and forced vibration analysis of 3D graphene foam truncated conical microshells. J. Brazilian Soc. Mech. Sci. Eng. 43, 133 (2021)
Dhooge, A., Govaerts, W., Kuznetsov, Y.A., Meijer, H.G.E., Sautois, B.: New features of the software MatCont for bifurcation analysis of dynamical systems. Math. Comput. Model. Dyn. Syst. 14, 147–175 (2008)
Loy, C.T., Lam, K.Y., Reddy, J.N.: Vibration of functionally graded cylindrical shells. Int. J. Mech. Sci. 41, 309–324 (1999)
Acknowledgement
This research was supported by the National Natural Science Foundation of China (Grant No. 11972204).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Liu, Y., Qin, Z. & Chu, F. Nonlinear forced vibrations of FGM sandwich cylindrical shells with porosities on an elastic substrate. Nonlinear Dyn 104, 1007–1021 (2021). https://doi.org/10.1007/s11071-021-06358-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-021-06358-7