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Applied Mathematics and Mechanics (English Edition) ›› 2010, Vol. 31 ›› Issue (7): 903-910.doi: https://doi.org/10.1007/s10483-010-1324-6

• Articles • 上一篇    下一篇

Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

袁学刚1,2 张洪武1 任九生3 朱正佑3   

  1. 1. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, P. R. China;
    2. School of Science, Dalian Nationalities University, Dalian 116600, P. R. China;
    3. Shanghai Institute of Applied Mathematics and Mechanics, Department of Mechanics, Shanghai University, Shanghai 200072, P. R. China
  • 收稿日期:2009-11-26 修回日期:2010-05-28 出版日期:2010-07-01 发布日期:2010-07-01

Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

YUAN Xua-Gang1,2, ZHANG Hong-Wu1, REN Jiu-Sheng3, ZHU Zheng-You3   

  1. 1. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, P. R. China;
    2. School of Science, Dalian Nationalities University, Dalian 116600, P. R. China;
    3. Shanghai Institute of Applied Mathematics and Mechanics, Department of Mechanics, Shanghai University, Shanghai 200072, P. R. China
  • Received:2009-11-26 Revised:2010-05-28 Online:2010-07-01 Published:2010-07-01

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摘要： 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.

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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