Stability of pre-stressed incompressible hyperelastic cylindrical tubes under axial compression
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Stability of pre-stressed incompressible hyperelastic cylindrical tubes under axial compression W. Zhao1,2 · W. Zhang2 · X. G. Yuan1 · X. C. Shang3 Received: 10 August 2020 / Revised: 20 September 2020 / Accepted: 27 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The problems of stability and controllability of the finite deformation are investigated for a pre-stressed hyperelastic cylindrical tube subjected to the axial compression. The tube is assumed to be composed of an incompressible neo-Hookean material. Based on the theory of small deformation superposed on large elastic deformation, the mathematical model is formulated by a system of incremental equilibrium equations and incremental boundary conditions. Moreover, the general solutions describing the finite deformation of the tube are obtained by the form of Bessel functions. Finally, the system of nonlinear equations governing the stability of the tube are given, the criteria of the stability discussed in terms of the corresponding numerical simulations. Keywords Stability · Eversion · Hyperelastic material · Cylindrical tube · Small deformation superposed on large elastic deformation
1 Introduction Hyperelastic materials, such as rubber, rubber-like and biological soft tissue, may be viewed as a class of important macromolecule materials with unique characteristics and wide applications [1–4]. Pre-stressed bodies are ubiquitous in several fields of science research and engineering practice. For instance, in biomechanics, the mechanical behaviors in service of many soft biological tissues, such as arterial walls, veins, skin, tendons, can be explained by modelling
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W. Zhang [email protected] X. G. Yuan [email protected] W. Zhao [email protected] X. C. Shang [email protected]
1
School of Science, Dalian Minzu University, Dalian 116600, China
2
Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China
3
Department of Applied Mechanics, University of Science and Technology Beijing, Beijing 100083, China
them as pre-stressed viscoelastic or hyperelastic materials. The bridge bearings or seismic shock absorbers under buildings are typical examples of devices operating in certain pre-stressed states, and sometimes subjected to dramatically large strains. Starting with the theory of finite deformation in continuum mechanics, our main purpose here is to consider small deformations superposed on axial compression deformations of a pre-stressed hyperelastic cylindrical tube and then investigates the stability of axial compression deformations. Zidi [5–7] considered the effects of the pre-stress on a hyperelastic tube subjected to a combined torsion, circular and axial shearing. The nonlinear system is solved numerically, and the effects of pre-stress on local volume change, circumferential stretch ratio and stress distribution are discussed. Saravanan [8] studied the inflation, extension and tw
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