Fractional calculus approach for bending of viscoelastic plate using two-variable refined plate theory

Document Type : Research Article

Authors

1 Department of Mechanical and Aerospace Engineering, Shiraz University of Technology

2 Shiraz University of Technology, Mechanical Engineering Department

3 Department of Mathematics, Shiraz university of Technology, Shiraz, Iran

Abstract

This paper deals with the time-dependent bending behavior of a rectangular viscoelastic plate based on the two-variable refined plate theory using the fractional calculus approach. The plate is fully simply-supported and is subjected to uniformly-distributed loading and the three-parameter merchant model is used for simulation of viscoelastic behavior. The time-domain governing equations are converted into frequency-domain ones using the Laplace transform and then, these equations are solved by the Navier method. The viscoelastic plate response is obtained using the elastic-viscoelastic correspondence principle so that the response of an elastic equivalent problem is extended into the viscoelastic problem. The results of this study, including plate deflection, and in-plane and transverse strains are compared with the results of the elastic model and the standard merchant model where the comparison of obtained results with the reference ones shows that the proposed approach has good accuracy. Also, the variation of deflection through the plate thickness and the effect of aspect ratio on the results are studied. This study shows that the proposed fractional model has the ability to simulation of both elastic and viscose effects simultaneously which is more compatible with the nature of viscoelastic materials.

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References

[1] G.W. Leibniz, G.F.A. L’Hopital, Letter from Hanover, Germany, to G.F.A. L’Hopital, September 30, reprinted 1962, Olms verlag, Hildesheim, Germany, Mathematische Schriften, 2 (1962) 301-302.
[2] P. Nabonnand, L. Rollet, Les Nouvelles annales de mathématiques: journal des candidats aux Écoles polytechnique et normale, Conferenze e Seminari dell’Associazione Subalpina Mathesis,  (2012) 1-10.
[3] B. Ross, The development of fractional calculus 1695–1900, Historia Mathematica, 4(1) (1977) 75-89.
[4] I. Podlubny, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, 1999.
[5] A. Loverro, Fractional calculus: history, definitions and applications for the engineer, Rapport technique, Univeristy of Notre Dame: Department of Aerospace and Mechanical Engineering,  (2004) 1-28.
[6] G. Catania, S. Sorrentino, Fractional derivative linear models for describing the viscoelastic dynamic behaviour of polymeric beams, in:  24th Conference and Exposition on Structural Dynamics 2006, IMAC-XXIV, Society for Experimental Mechanics (SEM), 2006.
[7] M.Q. Tang, Y.P. Li, Equilibrium paths of a fractional order viscoelastic two-member truss, in:  Advanced Materials Research, Trans Tech Publ, 2012, pp. 963-968.
[8] K. Lazopoulos, A. Lazopoulos, On fractional bending of beams, Archive of Applied Mechanics, 86(6) (2016) 1133-1145.
[9] M.F. Oskouie, R. Ansari, H. Rouhi, Bending analysis of functionally graded nanobeams based on the fractional nonlocal continuum theory by the variational Legendre spectral collocation method, Meccanica, 53(4-5) (2018) 1115-1130.
[10] Y. Wang, T. Tsai, Static and dynamic analysis of a viscoelastic plate by the finite element method, Applied Acoustics, 25(2) (1988) 77-94.
[11] S. Subramanian, Dynamic Stability of Viscoelastic Plates Subjected to Ramdomly Varying In-Plane Loads, in:  Engineering Mechanics, ASCE, 1995, pp. 191-194.
[12] Y. Sun, H. Ma, Z. Gao, On the stability of anisotropic viscoelastic thin plates, Chinese Journal of Aeronautics, 10(1) (1997) 18-21.
[13] H. Hu, Y.-m. Fu, Nonlinear dynamics analysis of cracked rectangular viscoelastic plates, Journal of Central South University of Technology, 14(1) (2007) 336-341.
[14] J. Soukup, F. Valeš, J. Volek, J. Skočilas, Transient vibration of thin viscoelastic orthotropic plates, Acta Mechanica Sinica, 27(1) (2011) 98-107.
[15] A. Zenkour, H. El-Mekawy, Bending of inhomogeneous sandwich plates with viscoelastic cores, Journal of Vibroengineering, 16(7) (2014) 3260-3272.
[16] R. Kolahchi, A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods, Aerospace Science and Technology, 66 (2017) 235-248.
[17] Y.A. Rossikhin, M.V. Shitikova, A.I. Krusser, To the question on the correctness of fractional derivative models in dynamic problems of viscoelastic bodies, Mechanics Research Communications, 77 (2016) 44-49.
[18] N. Jafari, M. Azhari, Time-dependent static analysis of viscoelastic Mindlin plates by defining a time function, Mechanics of Time-Dependent Materials,  (2019) 1-18.
[19] K. Adolfsson, M. Enelund, Fractional derivative viscoelasticity at large deformations, Nonlinear dynamics, 33(3) (2003) 301-321.
[20] Z. Xu, W. Chen, A fractional-order model on new experiments of linear viscoelastic creep of Hami Melon, Computers & Mathematics with Applications, 66(5) (2013) 677-681.
[21] R.L. Bagley, P.J. Torvik, Fractional calculus-a different approach to the analysis of viscoelastically damped structures, AIAA journal, 21(5) (1983) 741-748.
[22] L. Eldred, W. Baker, A. Palazotto, Numerical application of fractional derivative model constitutive relations for viscoelastic materials, Computers & structures, 60(6) (1996) 875-882.
[23] S. Park, Rheological modeling of viscoelastic passive dampers, in:  Smart Structures and Materials 2001: Damping and Isolation, International Society for Optics and Photonics, 2001, pp. 343-354.
[24] M. Sasso, G. Palmieri, D. Amodio, Application of fractional derivatives models to time-dependent materials, in:  Time Dependent Constitutive Behavior and Fracture/Failure Processes, Volume 3, Springer, 2011, pp. 213-221.
[25] M. Di Paola, R. Heuer, Fractional visco-elastic Euler–Bernoulli beam, International Journal of Solids and Structures, 50 (2013) 3505-3510.
[26] C.-C. Zhang, H.-H. Zhu, B. Shi, G.-X. Mei, Bending of a rectangular plate resting on a fractionalized Zener foundation, Structural Engineering and Mechanics, 52(6) (2014) 1069-1084.
[27] C. Zhang, H. Zhu, B. Shi, L. Liu, Theoretical investigation of interaction between a rectangular plate and fractional viscoelastic foundation, Journal of Rock Mechanics and Geotechnical Engineering, 6(4) (2014) 373-379.
[28] K. Lazopoulos, D. Karaoulanis, Α. Lazopoulos, On fractional modelling of viscoelastic mechanical systems, Mechanics Research Communications, 78 (2016) 1-5.
[29] G. Tekin, F. Kadıoğlu, Viscoelastic behavior of shear-deformable plates, International Journal of Applied Mechanics, 9(06) (2017) 1750085.
[30] W. Cai, W. Chen, W. Xu, Fractional modeling of Pasternak-type viscoelastic foundation, Mechanics of Time-Dependent Materials, 21(1) (2017) 119-131.
[31] A. Zbiciak, W. Grzesikiewicz, Characteristics of fractional rheological models of asphalt-aggregate mixtures, Logistyka, (6) (2011).
[32] R.P. Shimpi, Refined plate theory and its variants, AIAA journal, 40(1) (2002) 137-146.
[33] H.F. Brinson, L.C. Brinson, Polymer engineering science and viscoelasticity, 2008.
[34] J. Rouzegar, R.A. Sharifpoor, Flexure of thick plates resting on elastic foundation using two-variable refined plate theory, Archive of Mechanical Engineering, 62(2) (2015) 181-203.
Amirkabir Journal of Mechanical Engineering
Volume 53, Issue 4
July 2021
Pages 2287-2308

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