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Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

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Abstract

Some nonlinear dynamic properties of axisymmetric deformation are examined for a spherical membrane composed of a transversely isotropic incompressible Rivlin-Saunders material. The membrane is subjected to periodic step loads at its inner and outer surfaces. A second-order nonlinear ordinary differential equation approximately describing radially symmetric motion of the membrane is obtained by setting the thickness of the spherical structure close to one. The qualitative properties of the solutions are discussed in detail. In particular, the conditions that control the nonlinear periodic oscillation of the spherical membrane are proposed. In certain cases, it is proved that the oscillating form of the spherical membrane would present a homoclinic orbit of type “∞”, and the amplitude growth of the periodic oscillation is discontinuous. Numerical results are provided.

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Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116024, P. R. China

    Xue-gang Yuan  (袁学刚) & Hong-wu Zhang  (张洪武)

  2. School of Science, Dalian Nationalities University, Dalian, 116600, P. R. China

    Xue-gang Yuan  (袁学刚)

  3. Shanghai Institute of Applied Mathematics and Mechanics, Department of Mechanics, Shanghai University, Shanghai, 200072, P. R. China

    Jiu-sheng Ren  (任九生) & Zheng-you Zhu  (朱正佑)

Authors
  1. Xue-gang Yuan  (袁学刚)
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  2. Hong-wu Zhang  (张洪武)
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  3. Jiu-sheng Ren  (任九生)
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  4. Zheng-you Zhu  (朱正佑)
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Correspondence to Xue-gang Yuan  (袁学刚).

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Contributed by Hong-wu ZHANG

Project supported by the National Natural Science Foundation of China (Nos. 10872045, 10721062, and 10772104), the Program for New Century Excellent Talents in University (No. NCET-09-0096), the Post-Doctoral Science Foundation of China (No. 20070421049), and the Fundamental Research Funds for the Central Universities (No. DC10030104)

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Yuan, Xg., Zhang, Hw., Ren, Js. et al. Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads. Appl. Math. Mech.-Engl. Ed. 31, 903–910 (2010). https://doi.org/10.1007/s10483-010-1324-6

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  • DOI: https://doi.org/10.1007/s10483-010-1324-6

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