Acta Mechanica Solida Sinica

, Volume 19, Issue 1, pp 1–8

# Nonlinear waves and periodic solution in finite deformation elastic rod

• Zhifang Liu
• Shanyuan Zhang
Article

## Abstract

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.

## Key words

nonlinear wave finite deformation Poisson effect Jacobi elliptic function

## References

1. [1]
Whitham, G.B., Linear and Nonlinear Waves, New York, John Wiley & Sons, 1974, 96–113.
2. [2]
Bhatnager, P.L., Nonlinear Waves in One-dimensional Dispersive System, Oxford: Clarendon Press, 1979, 61–88.Google Scholar
3. [3]
Zhu, W.Q., Nonlinear wave of elastic rod, Acta Mechnica Solida Sinica, No.2, 1980, 247–253 (in Chinese).Google Scholar
4. [4]
Yang, G.T. and Zhang, S.Y., Dynamic Elasticity, Beijing, Chinese Railway Press, 1988, 332–343 (in Chinese).Google Scholar
5. [5]
Zhang, S.Y. and Zhuang, W., The strain solitary waves in a nonlinear elastic rod, Acta Mechanica Sinica, Vol.1, No.3, 1987, 62–72.Google Scholar
6. [6]
Zhang, S.Y., Guo, J.G. and Zhang, N.M., The dynamics behaviors and wave properties of finite deformation elastic rods with viscous or geometrical-dispersive effects, ICNM-IV, Shanghai, Aug. 2002, 728–732.Google Scholar
7. [7]
Guo, J.G., Zhou, L.J. and Zhang, S.Y., The geometrical nonlinear waves in finite deformation elastic rods, Appl. Math. Mech., Vol.26, No.5, 2005, 667–674.
8. [8]
Liu, S.K. and Liu, S.D., Nonlinear Equations in Physics, Beijing, Peking University Press, 2000, 168–201 (in Chinese).Google Scholar
9. [9]
Alexei, V. Porubov and Manuel, G.VELARDE, On Nonlinear Waves in an Elastic Solid, C.R. Acad. Sci. Series II b, 2000, 165–170.Google Scholar
10. [10]
Fu, Z.T., Liu, S.K., Liu, S.D. and Zhao, Q., New Jacobi elliptic expansion and new periodic solutions of nonlinear wave equation, Physics Letters A, Vol.290, 2001, 72–76.

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2006

## Authors and Affiliations

• Zhifang Liu
• 1
• Shanyuan Zhang
• 1
1. 1.Institute of Applied MechanicsTaiyuan University of TechnologyTaiyuanChina