Open Access

ARTICLE

Controllability Conditions of Finite Oscillations of Hyper-Elastic Cylindrical Tubes Composed of a Class of Ogden Material Models

X.G. Yuan^1,2, R.J. Zhang³, H.W. Zhang¹

1 Department of Engineering Mechanics, State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116024, China. yxg1971@163.com, zhanghw@dlut.edu.cn
2 School of Science, Dalian Nationalities University, Dalian 116600, Liaoning, P.R. China
3 School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P.R. China, zhangrj@mail.tongji.edu.cn

Computers, Materials & Continua 2008, 7(3), 155-166. https://doi.org/10.3970/cmc.2008.007.155

Abstract

In this paper, the dynamic inflation problems are examined for infinitely long cylindrical tubes composed of a class of transversely isotropic incompressible Ogden material models. The inner surface of the tube is subjected to a class of periodic step radial pressures relating to time. The influences of various parameters, namely, the material parameters, the structure parameters and the applied pressures, on dynamic behaviors of the tube are discussed in detail. Significantly, for some given material parameters, it is proved that the motion of the tube would present a class of nonlinear periodic oscillations for any given pressures and the amplitude of oscillation is discontinuous for some special values of the given pressures. For other cases of material parameters, there exists a critical pressure such that the motion of the tube would also present nonlinear periodic oscillations if the given pressure does not exceed the critical value, however, the tube would inflate infinitely with the increasing time if the pressure exceeds the critical value. Finally, the case of periodic step pressures is considered and all controllability conditions for the finitely periodic oscillations of the cylindrical tube with time are presented by using the phase diagrams of the ordinary differential equation that governs the motion of the inner surface of the tube, especially for the neo-Hookean material, i.e., a special case of the incompressible Ogden material models. Meanwhile, some numerical examples are given.

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Keywords

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