Some Qualitative Properties of Incompressible Hyperelastic Spherical Membranes Under Dynamic Loads

YUAN Xue-gang; ZHANG Hong-wu; REN Jiu-sheng; ZHU Zheng-you

doi:10.3879/j.issn.1000-0887.2010.07.011

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YUAN Xue-gang, ZHANG Hong-wu, REN Jiu-sheng, ZHU Zheng-you. Some Qualitative Properties of Incompressible Hyperelastic Spherical Membranes Under Dynamic Loads[J]. Applied Mathematics and Mechanics, 2010, 31(7): 860-867. doi: 10.3879/j.issn.1000-0887.2010.07.011

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Some Qualitative Properties of Incompressible Hyperelastic Spherical Membranes Under Dynamic Loads

doi: 10.3879/j.issn.1000-0887.2010.07.011

State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian, Liaoning 116024, P. R. China;
School of Science, Dalian Nationalities University, Dalian, Liaoning 116600, P. R. China;

Received Date: 1900-01-01
Rev Recd Date: 2010-05-12
Publish Date: 2010-07-15

Abstract

Abstract

The nonlinear dynamic properties of axisymm etric deformation were examined for a sphericalm embrane composed of a transversely isotropic incompressible Rivlin-Saundersmaterial, where the membrane was subjected to periodic step loads at its inner and outer surfaces. A second order nonlinear ordinary differential equation that approxmiately describes the radially symmetric motion of the membrane was obtained by setting the thickness of the spherical structure close to 1 and the qualitative properties of the solutions were discussed in detail. In particular, the conditions that control the nonlinear periodic oscillation of the spherical membrane were proposed. Under certain cases, it was proved that the oscillating form of the spherical membrane would present a homoclinic orbit of type "∞" and that the growth of the amplitude of the periodic oscillation was discontinuous, and numerical results were also provided.
- nonlinear dynamic property,
- hyperelastic spherical membrane,
- periodic step loads,
- nonlinear periodic oscillation

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References(15)

References

[1]	Beatty M F. Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues—with examples [J]. Applied Mechanics Review, 1987, 40(12): 1699-1733. doi: 10.1115/1.3149545
[2]	Fu Y B, Ogden R W. Nonlinear Elasticity: Theory and Applications[M].Cambridge: Cambridge University Press, 2001.
[3]	Knowles J K. Large amplitude oscillations of a tube of incompressible elastic material [J]. Q Appl Math, 1960, 18(1): 71-77.
[4]	Guo Z H, Solecki R. Free and forced finite amplitude oscillations of an elastic thick-walled hollow sphere made of incompressible material [J]. Arch Math Stos, 1963, 15(8): 427-433.
[5]	Calderer C. The dynamical behavior of nonlinear elastic spherical shells [J]. J Elasticity, 1983, 13(1): 17-47. doi: 10.1007/BF00041312
[6]	袁学刚，朱正佑，程昌钧. 具有缺陷的不可压缩neo-Hookean 球壳的动力学行为的定性分析 [J]. 应用数学和力学, 2005, 26(8): 892-898.
[7]	Verron E, Khayat R E, Derdouri A, Peseux B. Dynamic inflation of hyperelastic spherical membranes [J]. Inc J Rheol, 1999, 43(5): 1083-1097. doi: 10.1122/1.551017
[8]	Chou-Wang M-S, Horgan C O. Cavitation in nonlinearly elastodynamics for neo-Hookean materials [J]. Int J Engin Sci, 1989, 27(8): 967-973. doi: 10.1016/0020-7225(89)90037-2
[9]	REN Jiu-sheng, CHENG Chang-jun. Dynamical formation of cavity in transversely hyperelastic spheres [J]. Acta Mechanica Sinica, 2003, 19(4): 320-323. doi: 10.1007/BF02487808
[10]	YUAN Xue-gang, ZHU Zheng-you, ZHANG Ruo-jing. Cavity formation and singular periodic oscillations in isotropic incompressible hyperelastic materials [J]. Int J Nonlinear Mech, 2006, 41(2): 294-303. doi: 10.1016/j.ijnonlinmec.2005.08.001
[11]	YUAN Xue-gang, ZHANG Hong-wu. Nonlinear dynamical analysis of cavitation in anisotropic incompressible hyperelastic spheres under periodic step loads [J]. Computer Modeling in Engineering & Sciences, 2008, 32(3): 175-184.
[12]	任九生. 周期荷载下超弹性圆柱壳的动力响应 [J]. 应用数学和力学, 2008, 29(10): 1199-1207.
[13]	Polignone D A, Horgan C O. Cavitation for incompressible anisotropic nonlinearly elastic spheres [J]. J Elasticity, 1993, 33(1): 27-65. doi: 10.1007/BF00042634
[14]	Rivlin R S D, Saunders W. Large elastic deformations for isotropic materials—Ⅶ experiments on the deformation of rubbers [J]. Philos Trans Roy Soc Lond Ser A, 1951, 243(865): 251-288. doi: 10.1098/rsta.1951.0004
[15]	Gent A N, Thomas J. Forms for the stored energy function for vulcanized rubber [J]. J Polym Sci, 1958, 28(118): 625-628. doi: 10.1002/pol.1958.1202811814