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

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.

Appendix

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

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

(1)

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]

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

(2)

where

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

(3)

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

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

(4)

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

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

(5)

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

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

(6)