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Solitary waves and chaos in nearly compressible thermo-hyperelastic cylinder

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Abstract

Solitary waves in hyperelastic structures propagate stably in absence of external forces. However, as the external forces increase, the stability of solitary waves may lose and then chaotic behaviors may appear. In this paper, solitary waves and chaotic motions of an infinitely long cylinder with an internal heat source are investigated. Moreover, the cylinder is composed of nearly compressible thermo-hyperelastic neo-Hookean materials. Based on the variational principle, in a non-uniform steady-state temperature field and under a uniformly distributed radial periodic load, a mathematical model describing motions of the infinite cylinder is established. The governing equations are analyzed by qualitative and bifurcation theorems. The effects of structure boundary temperature on the qualitative properties of traveling waves are examined, and the types of traveling waves are identified. By using Melnikov functions, the sufficient conditions for chaos are derived. Numerical simulations of Lyapunov exponents, bifurcation diagrams, trajectories, and time travel curves are presented to illustrate the process of the associated dynamical system from order to chaotic motions in some bifurcation sets. The numerical results are consistent with those of the theoretical analysis. The thresholds of physical parameters and sufficient conditions for chaos may provide critical information for nondestructive testing of structures.

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Funding

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Nos. 12102242, 12172199, 12172086) and the Program of Shanghai Municipal Education Commission (Grant No. 2019-01-07-00-09- E00018).

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200072, China

    Ran Wang, Hu Ding & Dongsheng Zhang

  2. Department of Mathematics, Shandong University of Science and Technology, Qingdao, 266590, Shandong, China

    Lijun Zhang

  3. International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho, South Africa

    Lijun Zhang

  4. College of Science, Dalian Minzu University, Dalian, 116600, China

    Xuegang Yuan

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  1. Ran Wang
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Correspondence to Hu Ding or Lijun Zhang.

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Wang, R., Ding, H., Zhang, L. et al. Solitary waves and chaos in nearly compressible thermo-hyperelastic cylinder. Nonlinear Dyn 111, 5615–5628 (2023). https://doi.org/10.1007/s11071-022-08099-7

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