Radially and axially symmetric motions of a class of transversely isotropic compressible hyperelastic cylindrical tubes

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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.

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  1. 1.

    Fu, Y.B., Ogden, R.W.: Nonlinear Elasticity: Theory and Applications. Cambridge University Press, Cambridge (2001)

  2. 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)

  3. 3.

    Knowles, J.K.: Large amplitude oscillations of a tube of incompressible elastic material. Q. Appl. Math. 18, 71–77 (1960)

  4. 4.

    Chou-Wang, M.S., Horgan, C.O.: Cavitation in nonlinear elastodynamics for neo-Hookean materials. Int. J. Eng. Sci. 27, 967–973 (1989)

  5. 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)

  6. 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)

  7. 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)

  8. 8.

    Lafortune, S., Goriely, A., Tabor, M.: The dynamics of stretchable rods in the inertial case. Nonlinear Dyn. 43, 173–195 (2006)

  9. 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)

  10. 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)

  11. 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)

  12. 12.

    Wright, T.W.: Nonlinear waves in a rod: results for incompressible elastic materials. Stud. Appl. Math. 72, 149–160 (1985)

  13. 13.

    Coleman, B.D., Newman, D.C.: On waves in slender elastic rods. Arch. Ration. Mech. Anal. 109, 39–61 (1990)

  14. 14.

    Cohen, H., Dai, H.H.: Nonlinear axisymmetric waves in compressible hyperelastic rods: long finite amplitude waves. Acta Mech. 100, 223–239 (1993)

  15. 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)

  16. 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)

  17. 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)

  18. 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)

  19. 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)

  20. 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)

  21. 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)

  22. 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)

  23. 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)

  24. 24.

    Polignone, D.A., Horgan, C.O.: Cavitation for incompressible anisotropic nonlinearly elastic spheres. J. Elast. 33, 27–65 (1993)

  25. 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)

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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.

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Correspondence to Xue-gang Yuan.



Here we will give some basic definitions and a detailed calculation of the Lagrangian function.

According to the expressions of (2.2), the right and left Cauchy–Green strain tensors, \(\mathbf{C}\) and \(\mathbf{B}\), are, respectively, given by

$$\begin{aligned} \mathbf{C}=\mathbf{U}^{2}=\mathbf{F}^{T}{} \mathbf{F},\quad \mathbf{B}=\mathbf{V}^{2}=\mathbf{FF}^{T}. \end{aligned}$$

As is well known, the constitutive relations of hyperelastic materials can be described by their strain energy functions completely. In particular, the strain energy density per unit undeformed volume for an elastic material which is transversely isotropic about the \(x_1 \) direction is given by [24]

$$\begin{aligned} W=W(I_1 ,I_2 ,I_3 ,I_4 ,I_5 ), \end{aligned}$$


$$\begin{aligned} I_1= & {} \hbox {tr}{} \mathbf{C}=\hbox {tr}{} \mathbf{B}=\lambda _1^2 +\lambda _2^2 +\lambda _3^2 ,\nonumber \\ I_2= & {} \frac{1}{2}\left[ {\left( {\hbox {tr}{} \mathbf{C}} \right) ^{2}-\hbox {tr}\mathbf{C}^{2}} \right] =\frac{1}{2}\left[ {\left( {\hbox {tr}{} \mathbf{B}} \right) ^{2}-\hbox {tr}{} \mathbf{B}^{2}} \right] \nonumber \\= & {} \lambda _1^2 \lambda _2^2 +\lambda _2^2 \lambda _3^2 +\lambda _1^2 \lambda _3^2 , \nonumber \\ I_3= & {} \det \mathbf{C}=\det \mathbf{B}=\lambda _1^2 \lambda _2^2 \lambda _3^2 , \nonumber \\ I_4= & {} C_{12}^2 +C_{13}^2 , \nonumber \\ I_5= & {} C_{11} , \end{aligned}$$

in which \(I_1 ,I_2 ,I_3 \) are three principal invariants of \(\mathbf{C}\), the same as \(\mathbf{B}\), \(\lambda _1 ,\lambda _2 ,\lambda _3 \) are three eigenvalues of the deformation gradient tensor \(\mathbf{F}\), \(C_{ij} \;(i,j=1,2,3)\) are components in \(\mathbf{C}\).

From Eq. (2.6), we know that

$$\begin{aligned} I_1= & {} \lambda _1^2 +\lambda _2^2 +\lambda _3^2 =2f^{2}+z_Z^2 +f_Z^2 R^{2}, \nonumber \\ I_3= & {} \lambda _1^2 \lambda _2^2 \lambda _3^2 =f^{4}z_Z^2 , \nonumber \\ I_5= & {} f^{2}, \end{aligned}$$

where the subscripts of zf indicate the partial derivative of those functions. Therefore, the strain energy function becomes

$$\begin{aligned} W= & {} \frac{1}{2}\mu \left( {2f^{2}+z_Z^2 +f_Z^2 R^{2}-3} \right) \nonumber \\&+\,\alpha \left( {f^{4}z_Z^2 -1} \right) +\beta \left( {f^{4}-2f^{2}+1} \right) . \end{aligned}$$

In the absence of the potential energy of external forces, the Lagrangian function L is given by

$$\begin{aligned}&L\left( {f,f_t ,f_Z ,z_t ,z_Z } \right) \nonumber \\&\quad =\int _a^b {\int _0^{2\pi } {\frac{1}{2}\rho \left( {z_t^2 +f_t^2 R^{2}} \right) R} } \hbox {d}R\hbox {d}\varTheta \nonumber \\&\qquad -\,\int _a^b {\int _0^{2\pi } {WR} } \hbox {d}R\hbox {d}\varTheta \nonumber \\&\quad =\frac{1}{4}\pi \rho \left( {b^{2}-a^{2}} \right) \left[ {2z_t^2 +\left( {a^{2}+b^{2}} \right) f_t^2 } \right] \nonumber \\&\qquad -\,\pi \left( {b^{2}-a^{2}} \right) \left[ {\mu f^{2}+\frac{1}{2}\mu z_Z^2 } \right. \nonumber \\&\qquad \left. +\,\frac{1}{4}\mu f_Z^2 \left( {a^{2}+b^{2}} \right) -\frac{3}{2}\mu \right. \nonumber \\&\qquad \left. +\,\alpha \left( {f^{4}z_Z^2 -1} \right) +\beta \left( {f^{4}-2f^{2}+1} \right) \right] . \end{aligned}$$

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Wang, R., Zhang, W., Zhao, Z. et al. Radially and axially symmetric motions of a class of transversely isotropic compressible hyperelastic cylindrical tubes. Nonlinear Dyn 90, 2481–2494 (2017).

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  • Hyperelastic cylindrical tube
  • Transversely isotropic compressible neo-Hookean material
  • Radially and axially symmetric motions
  • Bounded travelling wave solutions