On a New Method of Quasi-static and Dynamic Analysis of Viscoelastic Plate on Elastic Foundation

Year 2023, Volume: 34 Issue: 6, 81 - 98, 01.11.2023

Gülçin Tekin , Fethi Kadıoğlu

https://doi.org/10.18400/tjce.1268992

Abstract

In the present work, an alternative solution technique based on mixed finite element (MFE) formulation in the Laplace-Carson domain is proposed for quasi-static and dynamic analyses of viscoelastic plate (VEP) resting on an elastic foundation (EF). This work contributed a numerical solution to the problem of a viscoelastic Kirchhoff plate supported on a Winkler foundation. VEP-EF interaction problems are taken into account under different wave-type loadings. The viscoelastic material behavior of the plate is modeled by the Zener rheological solid model. A four-nodded linear isoparametric element containing sixteen degrees of freedom is used to model the VEP. Developed functional in the Laplace-Carson domain based on the Gâteaux differential method is transformed to the real time domain by utilizing the Dubner and Abate (D&A) inverse Laplace transform technique (ILTT). To evaluate the applicability of the results, five numerical samples are considered. Further analyzes are performed on different wave type loadings to offer a new perspective on the time-dependent behavior of VEP on EF.

Keywords

Viskoelastik Plak, Plak-Zemin Etkileşimi, Dalga Tipi Yükler, Gâteaux diferansiyel, Laplace-Carson Dönüşümü, Dubner&Abate

References

• Pasternak, P.L.: On a New Method of Analysis of an Elastic Foundation by Means of Two Foundation Constants. Gosstroyizdat, Moscow (1954).
• Vlasov, V.Z., Leontiev, U.N.: Beams, Plates, and Shells on Elastic Foundations. Israel Program for Scientific Translations, Jerusalem (translated from Russian) (1966).
• Wang, Y.H., Tham, L.G., Cheung, Y.K.: Beams and plates on elastic foundations: a review. Prog. Struct. Eng. Mater. 7:174-182 (2005) https://doi.org/10.1002/pse.202.
• Kerr, A.D.: Elastic and viscoelastic foundation models. J. Appl. Mech. 31:491-498 (1964) https://doi.org/10.1115/1.3629667.
• Cheung, Y.K., Zienkiewicz, O.C.: Plates and tanks on elastic foundations-an application of finite element method. Int. J. Solids Struct. 1:451- 461 (1965) https://doi.org/10.1016/0020-7683(65)90008-9.
• Dowell, E.H.: Dynamic analysis of an elastic plate on a thin, elastic foundation. J. Sound Vib. 35:343-360 (1974) https://doi.org/10.1016/0022-460X(74)90065-0.
• Katsikadelis, J.T., Armenakas, A.E.: Plates on elastic foundation by BIE method. J. Eng. Mech. 110:1086-1105 (1984) https://doi.org/10.1061/(ASCE)0733-9399(1984)110:7(1086).
• El-Zafrany, A., Fadhil, S., Al-Hosani K.: A new fundamental solution for boundary element analysis of thin plates on Winkler foundation. Int. J. Numer. Methods Eng. 38:887-903(1995). https://doi.org/10.1002/nme.1620380602.
• Sladek, J., Sladek, V., Mang, H.A.: Meshless local boundary integral equation method for simply supported and clamped plates resting on elastic foundation. Comput. Methods Appl. Mech. Eng. 191:5943-5959 (2002) https://doi.org/10.1016/S0045-7825(02)00505-4.
• Zhong, Y., Zhang, Y-S.: Theoretic solution of rectangular thin plate on foundation with four edges free by symplectic geometry method. Appl. Math. Mech. (English Edition). 27: 833-839 (2006) https://doi.org/10.1007/s10483-006-0614-y.
• Yang, Z., Yuan-Yuan, G., Fangin, B-B.: Solution for a rectangular plate on elastic foundation with free edges using reciprocal theorem method. Math. AEterna. 2:335-343 (2012).
• Rahbar-Ranji, A., Bahmyari, E.: Bending analysis of thin plates with variable thickness resting on elastic foundation by element free Galerkin method. J. Mech. 28:479-488 (2012) https://doi.org/10.1017/jmech.2012.57.
• Kägo, E., Lellep, J.: Free Vibrations of Plates on Elastic Foundation, Procedia Eng. 57:489-496 (2013) https://doi.org/10.1016/j.proeng.2013.04.063.
• Li, R., Zhong, Y., Li, M.: Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method. Proc. Royal Society A 469:20120681 (2013) https://doi.org/10.1098/rspa.2012.0681.
• Flügge, W.: Viscoelasticity (Second Ed.). Springer, Berlin (1975).
• Christensen, R.M.: Theory of Viscoelasticity (Second Ed.). Academic Press, New York (1982).
• White, J.L.: Finite elements in linear viscoelastic analysis. In: Proceedings of the 2nd Conference on Matrix Method in Structural Mechanics. AFFDL-TR-68-150: 489-516 (1986).
• Wang, Y.Z., Tsai, T.J.: Static and dynamic analysis of a viscoelastic plate by the finite element method. Appl. Acoust. 25:77-94 (1988) https://doi.org/10.1016/0003-682X(88)90017-5.
• Chen, T-M.: The hybrid Laplace transform / finite element method applied to the quasi-static and dynamic analysis of viscoelastic Timoshenko beams. Int. J. Numer. Methods Eng. 38:509-522 (1995) https://doi.org/10.1002/nme.1620380310.
• Sorvari, J., Hämäläinen, J.: Time integration in linear viscoelasticity-a comparative study. Mech. Time-Depend. Mater. 14:307-328 (2010) https://doi.org/10.1007/s11043-010-9108-7.
• Zhou, X., Yu, D., Shao, X., Zhang, S., Wang, S.: Research and applications of viscoelastic vibration damping materials: A review. Compos. Struct. 136:460-480 (2016) https://doi.org/10.1016/j.compstruct.2015.10.014.
• Zhang, C., Zhu, H., Shi, B., Liu, L.: Theoretical investigation of interaction between a rectangular plate and fractional viscoelastic foundation. J. Rock Mech. Geotech. Eng. 6:373-379 (2014) https://doi.org/10.1016/j.jrmge.2014.04.007.
• Van, H.L, Thoi, T.N, Liu, G.R., Van, P.P.: A cell-based smoothed finite element method using three-node shear-locking free Mindlin plate element (CS-FEM-MIN3) for dynamic response of laminated composite plates on viscoelastic foundation. Eng. Anal. Bound. Elem. 42:8-19 (2014) https://doi.org/10.1016/j.enganabound.2013.11.008.
• Hasheminejad, S.M., Gheshlaghi, B.: Three-dimensional elastodynamic solution for an arbitrary thick FGM rectangular plate resting on a two parameter viscoelastic foundation. Compos. Struct. 94:2746-2755(2012) https://doi.org/10.1016/j.compstruct.2012.04.010.
• Çalık Karaköse, Ü.H.: FE analysis of FGM plates on arbitrarily orthotropic Pasternak foundations for membrane effects. Teknik Dergi. 33:11799-11822 (2022) https://doi.org/10.18400/tekderg.878982
• Gupta, A., Talha, M., Seemann, W.: Free vibration and flexural response of functionally graded plates resting on Winkler–Pasternak elastic foundations using nonpolynomial higher-order shear and normal deformation theory. Mech. Adv. Mater. Struct. 25:523-538(2018) https://doi.org/10.1080/15376494.2017.1285459.
• Özgan, K.: Modelling laminated orthotropic plate-foundation interaction subjected to moving load using Vlasov model. Teknik Dergi. 29: 8317-8338 (2018) https://doi.org/10.18400/tekderg.339219
• Gupta, A., Talha, M., Chaudhari, V.K.: Natural frequency of functionally graded plates resting on elastic foundation using finite element method. Proc. Technol. 23:163-170(2016). https://doi.org/10.1016/j.protcy.2016.03.013.
• Zenkour, A.M., Allam, M.N.M., Sobhy, M.: Bending of a fiber-reinforced viscoelastic composite plate resting on elastic foundations. Arch. Appl. Mech. 81:77-96(2011). https://doi.org/10.1007/s00419-009-0396-9.
• Daikh, A.A., Zenkour, A.M.: Bending of functionally graded sandwich nanoplates resting on ‎Pasternak foundation under different boundary conditions. J. Appl. Comput. Mech. 6:1245-1259(2020). https://doi.org/10.22055/JACM.2020.33136.2166.
• Zenkour, A.M., Sobhy, M.: Thermal buckling of functionally graded plates resting on elastic foundations using the trigonometric theory. J. Therm. Stresses. 34:1119-1138(2011). https://doi.org/10.1080/01495739.2011.606017.
• Tabasi, H.M., Jam, J.E., Fard, K.M., Beni, M.H.: Buckling and free vibration analysis of fiber metal-laminated plates resting on partial elastic foundation J. Appl. Comput. Mech. 6:37-51(2020). https://doi.org/10.22055/JACM.2019.28156.1489
• Zamani, H.A., Aghdam, M.M., Sadighi, M.: Free vibration analysis of thick viscoelastic composite plates on visco-Pasternak foundation using higher-order theory. Compos. Struct. 182:25-35 (2017) https://doi.org/10.1016/j.compstruct.2017.08.101.
• Sofiyev, A.H., Zerin, Z., Kuruoglu, N.: Dynamic behavior of FGM viscoelastic plates resting on elastic foundations. Acta Mech. 231:1-17 (2020) https://doi.org/10.1007/s00707-019-02502-y.
• Khazanovich, L., Levenberg, E.: Analytical solution for a viscoelastic plate on a Pasternak foundation. Road Mater. Pavement Des. 21:800-820 (2020) https://doi.org/10.1080/14680629.2018.1530693.
• Aköz, A.Y., Kadıoğlu, F.: The mixed finite element method for the quasi-static and dynamic analysis of viscoelastic Timoshenko beams. Int. J. Numer. Methods Eng. 44:1909-1932 (1999) https://doi.org/10.1002/(SICI)1097-0207(19990430)44:12<1909::AID-NME573>3.0.CO;2-P.
• Kadıoğlu, F., Aköz, A.Y.: The mixed finite element method for the dynamic analysis of visco-elastic circular beams. In: Proceedings of the 4 th International Conference on Vibration Problems. Jadavpur University (1999).
• Kadıoğlu, F., Aköz, A.Y.: The quasi-static and dynamic responses of viscoelastic parabolic beams. In: Proceedings of the 11 th National Applied Mechanics Meeting (in Turkish), Bolu-Turkey (2000).
• Kadıoğlu, F., Aköz, A.Y.: The mixed finite element for the quasi-static and dynamic analysis of viscoelastic circular beams. Int. J. Struct. Eng. Mech. 15: 735-752 (2003) https://doi.org/10.12989/sem.2003.15.6.735.
• Aköz, A.Y., Kadıoğlu, F., Tekin, G.: Quasi-static and dynamic analysis of viscoelastic plates. Mech. Time-Depend. Mater. 19:483-503(2015) https://doi.org/10.1007/s11043-015-9274-8.
• Tekin, G., Kadıoğlu, F.: Viscoelastic behavior of shear-deformable plates. Int. J.Appl. Mech. 9:1750085 (23 pages) (2017) https://doi.org/10.1142/S1758825117500855.
• Ilyasov, M.H., Aköz, A.Y.: The vibration and dynamic stability of viscoelastic plates. Int. J.Eng. Sci. 38: 695-714 (2000) https://doi.org/10.1016/S0020-7225(99)00060-9.
• Oden, J.T., Reddy, J.N.: Variational Methods in Theoretical Mechanics. Springer, Berlin (1976).
• Reddy, J.N.: An Introduction to the Finite Element Method (Second Ed.). McGraw-Hill, New York (1993).
• Dubner, H., Abate, J.: Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. J. ACM. 15:115-123 (1968) https://doi.org/10.1145/321439.321446.