Radially and axially symmetric motions of a class of transversely isotropic compressible hyperelastic cylindrical tubes
- 187 Downloads
- 3 Citations
Abstract
In this paper, the radially and axially symmetric motions are examined for a hyperelastic cylindrical tube composed of a class of transversely isotropic compressible neo-Hookean materials about the radial direction. Firstly, a system of coupled nonlinear evolution equations describing the motions of the cylindrical tube is derived by Hamilton’s principle. Then the system is reduced to a system of nonlinear ordinary differential equations by the travelling wave transformations. According to the theory of planar dynamical systems, qualitative analyses on the solutions of the system are given in different parameter spaces. Specially, the influences of the material parameters on the qualitative and quantitative properties of the solutions are discussed. Two types of travelling wave solutions of the radially symmetric motion are obtained, including classical periodic travelling wave solutions and solitary wave solutions with the peak form. So does the axially symmetric motion, but solitary wave solutions with the valley form. Correspondingly, some numerical examples are shown.
Keywords
Hyperelastic cylindrical tube Transversely isotropic compressible neo-Hookean material Radially and axially symmetric motions Bounded travelling wave solutionsNotes
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Nos. 11672069, 11702059, 11232003, 11672062); the Ph.D. Programs Foundation of Ministry of Education of China (No. 20130041110050); the Research Start-up Project Plan for Liaoning Doctors (No. 20141119); the Fundamental Research Funds for the Central Universities (No. DC201502050407, DC201502050203); 111 Project (B08014). The authors also appreciate the editor’s earnest work and three anonymous reviewers for their helpful comments on an earlier draft of this paper.
References
- 1.Fu, Y.B., Ogden, R.W.: Nonlinear Elasticity: Theory and Applications. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
- 2.Ben, A.M., Bianca, C.: Towards a unified approach in the modeling of fibrosis: a review with research perspectives. Phys. Life Rev. 17, 61–85 (2016)CrossRefGoogle Scholar
- 3.Knowles, J.K.: Large amplitude oscillations of a tube of incompressible elastic material. Q. Appl. Math. 18, 71–77 (1960)CrossRefzbMATHMathSciNetGoogle Scholar
- 4.Chou-Wang, M.S., Horgan, C.O.: Cavitation in nonlinear elastodynamics for neo-Hookean materials. Int. J. Eng. Sci. 27, 967–973 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
- 5.Yuan, X.G., Zhu, Z.Y., Zhang, R.J.: Cavity formation and singular periodic oscillations in isotropic incompressible hyperelastic materials. Int. J. Non-Linear Mech. 41, 294–303 (2006)CrossRefzbMATHGoogle Scholar
- 6.Yuan, X.G., Zhu, Z.Y., Cheng, C.J.: Dynamical analysis of cavitation for a transversely isotropic incompressible hyper-elastic medium: periodic motion of a pre-existing micro-void. Int. J. Non-Linear Mech. 42, 442–449 (2007)CrossRefGoogle Scholar
- 7.Roussos, N., Mason, D.P.: Non-linear radial oscillations of a thin-walled double-layer hyperelastic cylindrical tube. Int. J. Non-Linear Mech. 33, 507–530 (1998)CrossRefzbMATHGoogle Scholar
- 8.Lafortune, S., Goriely, A., Tabor, M.: The dynamics of stretchable rods in the inertial case. Nonlinear Dyn. 43, 173–195 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
- 9.Mason, D.P., Maluleke, G.H.: Non-linear radial oscillations of a transversely isotropic hyperelastic incompressible tube. J. Math. Anal. Appl. 333, 365–380 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
- 10.Beatty, M.F.: On the radial oscillations of incompressible, isotropic, elastic and limited elastic thick-walled tubes. Int. J. Non-Linear Mech. 42, 283–297 (2007)CrossRefGoogle Scholar
- 11.Yuan, X.G., Zhang, R.J., Zhang, H.W.: Controllability conditions of finite oscillations of hyper-elastic cylindrical tubes composed of a class of Ogden material models. Comput. Mater. Contin. 7, 155–156 (2008)Google Scholar
- 12.Wright, T.W.: Nonlinear waves in a rod: results for incompressible elastic materials. Stud. Appl. Math. 72, 149–160 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
- 13.Coleman, B.D., Newman, D.C.: On waves in slender elastic rods. Arch. Ration. Mech. Anal. 109, 39–61 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
- 14.Cohen, H., Dai, H.H.: Nonlinear axisymmetric waves in compressible hyperelastic rods: long finite amplitude waves. Acta Mech. 100, 223–239 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
- 15.Dai, H.H., Huo, Y.: Solitary shock waves and other travelling waves in a general compressible hyperelastic rod. Proc. R. Soc. A Math. Phys. Eng. Sci. 456, 331–363 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
- 16.Dai, H.H., Li, J.B.: Nonlinear travelling waves in a hyperelastic rod composed of a compressible Mooney–Rivlin material. Int. J. Non-Linear Mech. 44, 499–510 (2009)CrossRefGoogle Scholar
- 17.Dai, H.H., Peng, X.: Weakly nonlinear long waves in a prestretched Blatz–Ko cylinder: solitary, kink and periodic waves. Wave Motion 48, 761–772 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
- 18.Vallikivi, M., Salupere, A., Dai, H.H.: Numerical simulation of propagation of solitary deformation waves in a compressible hyperelastic rod. Math. Comput. Simul. 82, 1348–1362 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
- 19.Zhu, M., Liu, Y., Qu, C.: On the model of the compressible hyperelastic rods and Euler equations on the circle. J. Differ. Equ. 254, 648–659 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
- 20.Shearer, T., Abrahams, I.D., Parnell, W.J., Daros, C.H.: Torsional wave propagation in a pre-stressed hyperelastic annular circular cylinder. Q. J. Mech. Appl. Math. 66, 465–487 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
- 21.Fu, Y.B., Il’Ichev, A.T.: Localized standing waves in a hyperelastic membrane tube and their stabilization by a mean flow. Math. Mech. Solids 20, 1198–2014 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
- 22.Jiang, H.J., Xiang, J.J., Dai, C.Q.: Nonautonomous bright soliton solutions on continuous wave and cnoidal wave backgrounds in blood vessels. Nonlinear Dyn. 75, 201–207 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
- 23.Clayton, J.D., Bliss, K.M.: Analysis of intrinsic stability criteria for isotropic third-order Green elastic and compressible neo-Hookean solids. Mech. Mater. 68, 104–119 (2014)Google Scholar
- 24.Polignone, D.A., Horgan, C.O.: Cavitation for incompressible anisotropic nonlinearly elastic spheres. J. Elast. 33, 27–65 (1993)CrossRefzbMATHGoogle Scholar
- 25.Yu, L.Q., Tian, L.X.: Loop solutions, breaking kink (or anti-kink) wave solutions, solitary wave solutions and periodic wave solutions for the two-component Degasperis–Procesi equation. Nonlinear Anal. Real World Appl. 15, 140–148 (2014)CrossRefzbMATHMathSciNetGoogle Scholar