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