Dynamical Formation of Cavity in a Composed Hyper-Elastic Sphere

REN Jiu-sheng; CHENG Chang-jun

Volume 25 Issue 11

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Article Navigation > Applied Mathematics and Mechanics > 2004 > 25(11): 1117-1123

REN Jiu-sheng, CHENG Chang-jun. Dynamical Formation of Cavity in a Composed Hyper-Elastic Sphere[J]. Applied Mathematics and Mechanics, 2004, 25(11): 1117-1123.

Citation:

REN Jiu-sheng, CHENG Chang-jun. Dynamical Formation of Cavity in a Composed Hyper-Elastic Sphere[J]. Applied Mathematics and Mechanics, 2004, 25(11): 1117-1123.

Citation:

REN Jiu-sheng, CHENG Chang-jun. Dynamical Formation of Cavity in a Composed Hyper-Elastic Sphere[J]. Applied Mathematics and Mechanics, 2004, 25(11): 1117-1123.

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Dynamical Formation of Cavity in a Composed Hyper-Elastic Sphere

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

Received Date: 2003-02-20
Rev Recd Date: 2004-05-28
Publish Date: 2004-11-15

Abstract

Abstract

The dynamical formation of cavity in a hyper-elastic sphere composed of two materials with the incompressible strain energy function, subjected a suddenly applied uniform radial tensile boundary dead-load, was studied following the theory of finite deformation dynamics. Besides a trivial solution corresponding to the homogeneous static state, a cavity forms at the center of the sphere when the tensile load is larger than its critical value. It is proved that the evolution of cavity radius with time displays nonlinear periodic oscillations. The phase diagram for oscillation, the maximum amplitude, the approximate period and the critical load were all discussed.
- composed incompressible hyper-elastic materials,
- finite deformation dynamics,
- cavity formation,
- non-linear periodic oscillation

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References

[1]	Gent A N,Lindley P B.Internal rupture of bonded rubber cylinders in tension[J].Proc Roy Soc London,1959,A249(2):195—205.
[2]	Ball J M.Discontinuous equilibrium solutions and cavitations in nonlinear elasticity[J].Phil Trans R Soc London,Ser A,1982,306(3):557—610. doi: 10.1098/rsta.1982.0095
[3]	Horgan C O,Polignone D A.Cavitation in nonlinearly elastic solids: A review[J].Appl Mech Rev,1995,48(3): 471—485. doi: 10.1115/1.3005108
[4]	任九生，程昌钧.不可压超弹性材料中的空穴分叉[J].应用数学和力学，2002，23(8)：783—789.
[5]	任九生，程昌钧，朱正佑.可压超弹性材料组合球体中心的空穴生成[J].应用数学和力学，2003,24(9):892—898.
[6]	Chou-Wang M-S, Horgan C O. Cavitation in nonlinear elastodynamic for neo-Hookean materials[J].Internat J Engrng Sci,1989,27(8):967—973. doi: 10.1016/0020-7225(89)90037-2
[7]	Knowles J K. Large amplitude oscillations of a tube of incompressible elastic material[J].Quart Appl Math,1960,18(1):71—77.
[8]	Guo Z H, Solecki R. Free and forced finite amplitude oscillations of an elastic thick-walled hollow sphere made of incompressible material[J].Arch Mech Stos,1963,15(3):427—433.
[9]	Calderer C. The dynamical behavior of nonlinear elastic spherical shells[J].J Elasticity,1983,13(1):17—47. doi: 10.1007/BF00041312
[10]	REN Jiu-Sheng, CHENG Chang-Jun. Dynamical formation of cavity in transversely isotropic hyperelastic spheres[J].Acta Mechanica Sinica,2003,19(4):320—323. doi: 10.1007/BF02487808
[11]	REN Jiu-Sheng, CHENG Chang-Jun. Dynamical formation of cavity in hyper-elastic materials[J].Acta Mechanica Solida Sinca，2002,15(3):208—216.
[12]	Chalton D.T., Yang J. A review of methods to characterize rubber elastic behavior for use in finite element analysis[J].Rubber Chemistry and Technology,1994,67(3):481—503. doi: 10.5254/1.3538686

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