×

The generalized Schamel equation in nonlinear wave dynamics of cylindrical shells. (English) Zbl 1430.37087

Summary: The axisymmetric propagation of longitudinal waves in an integrally stiffened physically and geometrically nonlinear cylindrical shell is studied. The case is considered when the parameters characterizing nonlinearity, dispersion and thickness-radius ratio are the same order. Using the multiscale asymptotic method, the generalized Schamel equation is derived from the equations of motion of the shell. This equation contains an additional term with the fifth derivative with respect to the spatial coordinate which characterizes the high-frequency dispersion. It is shown that the derived equation does not pass the Painlevé test and therefore is not integrated by the inverse scattering transform. The geometric series method using Padé approximants is applied to construct exact solitary-wave solution of the equation. It has been established that for the existence of an exact bounded amplitude solution a “soft” type of physical nonlinearity is necessary. The Schamel-Kawahara equation containing the combined nonlinearity is considered; its exact soliton-like solution is given. Numerical simulation confirmed that the initial perturbation in the form of the exact solution generates a persistently propagating solitary wave.

MSC:

37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35C07 Traveling wave solutions
35C08 Soliton solutions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, Cambridge (2008) · Zbl 1154.74002 · doi:10.1017/CBO9780511619694
[2] Thompson, J.M.T.: Advances in shell buckling: theory and experiments. Int. J. Bifurc. Chaos 25(1), 1530001 (2015) · doi:10.1142/S0218127415300013
[3] Alijani, F., Amabili, M.: Non-linear vibrations of shells: a literature review from 2003 to 2013. Int. J. Non-Linear Mech. 58, 233-257 (2014) · doi:10.1016/j.ijnonlinmec.2013.09.012
[4] Avramov, K.V., Mikhlin, Y.V., Kurilov, E.: Asymptotic analysis of nonlinear dynamics of simply supported cylindrical shells. Nonlinear Dyn. 47, 331-352 (2007) · Zbl 1180.74022 · doi:10.1007/s11071-006-9032-1
[5] Bian, X., Chen, F., An, F.: Global bifurcations and chaos of a composite laminated cylindrical shell in supersonic air flow. Nonlinear Dyn. 96, 1095-1114 (2019) · Zbl 1437.76016 · doi:10.1007/s11071-019-04842-9
[6] Smirnov, V.V., Manevitch, L.I., Strozzi, M., Pellicano, F.: Nonlinear optical vibrations of single-walled carbon nanotubes. 1. Energy exchange and localization of low-frequency oscillations. Physica D Nonlinear Phenom. 325, 113-125 (2016) · Zbl 1364.82076 · doi:10.1016/j.physd.2016.03.015
[7] Erofeev, V.I., Klyueva, N.V.: Solitons and nonlinear periodic strain waves in rods, plates and shells (a review). Acoust. Phys. 48(6), 643-655 (2002) · doi:10.1134/1.1522030
[8] Conte, R., Musette, M.: The Painlevé Handbook. Springer, New-York (2008) · Zbl 1153.34002
[9] Shvartz, A., Samsonov, A., Dreiden, G., Semenova, I.: Evolution of bulk strain solitons in cylindrical inhomogeneous shells. AIP Conf. Proc. 1685, 070014 (2015) · doi:10.1063/1.4934451
[10] Samsonov, A.M., Dreiden, G.V., Porubov, A.V., Semenova, I.V.: Longitudinal-strain soliton focusing in a narrowing nonlinearly elastic rod. Phys. Rev. B. 57(10), 5778 (1998) · doi:10.1103/PhysRevB.57.5778
[11] Dong, K., Wang, X.: Wave propagation characteristics in piezoelectric cylindrical laminated shells under large deformation. Compos. Struct. 77, 171-181 (2007) · doi:10.1016/j.compstruct.2005.06.011
[12] Hu, Y.-G., Liew, K.M., Wang, Q., He, X.Q., Yakobson, B.I.: Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes. J. Mech. Phys. Solids 56, 3475-3485 (2008) · Zbl 1171.74373 · doi:10.1016/j.jmps.2008.08.010
[13] Vijay, P.S., Sonti, V.R.: Weakly nonlinear acoustic wave propagation in a nonlinear orthotropic circular cylindrical waveguide. J. Acoust. Soc. Am. 138, 3231 (2015) · doi:10.1121/1.4935132
[14] Tomczyk, B., Litawska, A.: Micro-vibrations and wave propagation in biperiodic cylindrical shells. Mech. Mech. Eng. 22(3), 789-807 (2018)
[15] Bochkarev, A.V., Zemlyanukhin, A.I., Mogilevich, L.I.: Solitary waves in an inhomogeneous cylindrical shell interacting with an elastic medium. Acoust. Phys. 63(2), 148-153 (2017) · doi:10.1134/S1063771017020026
[16] Zemlyanukhin, A.I., Bochkarev, A.V.: Axisymmetric nonlinear modulated waves in a cylindrical shell. Acoust. Phys. 64(4), 408-414 (2018) · doi:10.1134/S1063771018040139
[17] Tobisch, E. (ed.): New Approaches to Nonlinear Waves. Lecture Notes in Physics, vol. 908. Springer, Cham (2015)
[18] Andrianov, I.V., Danishevs’kyy, V.V., Topol, H., Weichert, D.: Homogenization of a 1D nonlinear dynamical problem for periodic composites. ZAMM Z. Angew. Math. Mech. 91(6), 523-534 (2011) · Zbl 1316.74043 · doi:10.1002/zamm.201000176
[19] Andrianov, I.V., Danishevs’kyy, V.V., Kalamkarov, A.L.: Micromechanical analysis of fibre-reinforced composites on account of influence of fibre coating. Compos. Part B Eng. 39, 874-881 (2008) · doi:10.1016/j.compositesb.2007.10.002
[20] Craster, R.V., Kaplunov, J., Nolde, E., Guenneau, S.: Bloch dispersion and high frequency homogenization for separable doubly-periodic structures. Wave Motion 49(2), 333-346 (2012) · Zbl 1360.78015 · doi:10.1016/j.wavemoti.2011.11.005
[21] Kauderer, H.: Nichtlineare Mechanik. Springer, Berlin (1958) · Zbl 0080.17409 · doi:10.1007/978-3-642-92733-1
[22] Porubov, A.V.: Localization of Nonlinear Strain Waves. Fizmatlit, Moscow (2009). (In Russian) · Zbl 1189.74073
[23] Lukash, P.A.: Fundamentals of Nonlinear Structural Mechanics. Stroyizdat, Moscow (1978). (in Russian)
[24] Zemlyanukhin, A.I., Mogilevich, L.I.: Nonlinear waves in inhomogeneous cylindrical shells: a new evolution equation. Acoust. Phys. 47(3), 303-307 (2001) · doi:10.1007/BF03353584
[25] Jones, R.M.: Deformation Theory of Plasticity. Bull Ridge Publishing, Blacksburg (2009)
[26] Volmir, A.S.: The Nonlinear Dynamics of Plates and Shells. Foreign Technology Division, Wright-Patterson AFB OH, Dayton (1974)
[27] Van Dyke, M.: Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford (1975) · Zbl 0329.76002
[28] Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, vol. 4. SIAM, Philadelphia (1981) · Zbl 0472.35002 · doi:10.1137/1.9781611970883
[29] Singh, V.K., Bansal, G., Agarwal, M., Negi, P.: Experimental determination of mechanical and physical properties of almond shell particles filled biocomposite in modified epoxy resin. J. Mater. Sci. Eng. 5(3), 246 (2016)
[30] Schamel, H.: A modified Korteweg – de Vries equation for ion acoustic waves due to resonant electrons. J. Plasma Phys. 9, 377-387 (1973) · doi:10.1017/S002237780000756X
[31] Conte, R.; Ng, TW; Wu, C.; Euler, N. (ed.), Singularity methods for meromorphic solutions of differential equations, 159-186 (2018), Boca Raton
[32] Wazwaz, A.M.: Partial Differential Equations and Solitary Waves Theory. Higher Education Press, Beijing (2009) · Zbl 1175.35001 · doi:10.1007/978-3-642-00251-9
[33] Bochkarev, A.V., Zemlyanukhin, A.I.: The geometric series method for constructing exact solutions to nonlinear evolution equations. Comput. Math. Math. Phys. 57(7), 1111-1123 (2017) · Zbl 1379.35053 · doi:10.1134/S0965542517070065
[34] Awrejcewicz, J., Andrianov, I.V., Manevitch, L.I.: Asymptotic Approaches in Nonlinear Dynamics: New Trends and Applications. Springer, Berlin (1998) · Zbl 0910.70001 · doi:10.1007/978-3-642-72079-6
[35] Andrianov, I.V., Topol, H.: Asymptotic analysis and synthesis in mechanics of solids and nonlinear dynamics. arXiv:1106.1783v2 [math-ph]. Submitted on 9 June 2011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.