Geometrical Nonlinear Waves in Finite Deformation Elastic Rods

GUO Jian-gang; ZHOU Li-jun; ZHANG Shan-yuan

Volume 26 Issue 5

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 2005 > 26(5): 614-620

GUO Jian-gang, ZHOU Li-jun, ZHANG Shan-yuan. Geometrical Nonlinear Waves in Finite Deformation Elastic Rods[J]. Applied Mathematics and Mechanics, 2005, 26(5): 614-620.

Citation:

GUO Jian-gang, ZHOU Li-jun, ZHANG Shan-yuan. Geometrical Nonlinear Waves in Finite Deformation Elastic Rods[J]. Applied Mathematics and Mechanics, 2005, 26(5): 614-620.

Citation:

GUO Jian-gang, ZHOU Li-jun, ZHANG Shan-yuan. Geometrical Nonlinear Waves in Finite Deformation Elastic Rods[J]. Applied Mathematics and Mechanics, 2005, 26(5): 614-620.

PDF( 309 KB)

Geometrical Nonlinear Waves in Finite Deformation Elastic Rods

1.
Institute of Applied Mechanics, Taiyuan University of Technology, Taiyuan 030024, P. R. China;

Received Date: 2004-06-04
Rev Recd Date: 2004-11-30
Publish Date: 2005-05-15

Abstract

Abstract

By usinge Hamilton-type variation principle in non-conservation system, the nonlinear equation of wave motion of a elastic thin rod was derived according to Lagrange description of finite deformation theory. The dissipation caused due to viscous effect and the dispersion introduced by transverse inertia were taken into consideration so that steady traveling wave solution can be obtained. Using multi-scale method the nonlinear equation is reduced to a KdV-Burgers equation which corresponds with saddle-spiral heteroclinic orbit on phase plane. Its solution is called the oscillating-solitary wave or saddle-spiral shock wave. If viscous effect or transverse inertia is neglected, the equation is degraded to classical KdV or Burgers equation. The former implies a propagating solitary wave with homoclinic on phase plane, the latter means shock wave and heteroclinic orbit.
- nonlinear wave,
- finite deformation,
- viscous effect,
- transverse inertia effect,
- multi-scale method

FullText(HTML)

References(8)

References

[1]	朱位秋.弹性杆中的非线性波[J].固体力学学报,1980,16（2）:247—253.
[2]	张善元,杨绍瑞.非线性弹性细杆中的定常波[J].太原理工大学学报,1985,1（4）:33—43.
[3]	ZHANG Shan-yuan,ZHUANG Wei.The strain solitary waves in a nonlinear elastic rod[J].Acta Mechanica Cinia,1987,1(3):62—72.
[4]	ZHANG Shan-yuan,GUO Jian-gang,ZHANG Nian-mei.The dynamics behaviors and wave properties of finite deformation elastic rods with viscous or geometrical-disporsive effects[A].In:ICNM-IV[C],Shanghai,Aug 2002,728—732.
[5]	Whitham G B.Linear and Nonlinear Waves[M].New York: John Wiley & Sons, 1974, 96—113.
[6]	Bhatnager P L.Nonlinear Waves in One-Dimensional Dispersive System[M].Oxford: Clarendon Press, 1979,61—88.
[7]	Alexander M S.Strain Solitons in Solid and How to Construct Them[M].New York: Chapman & Hall/CRC, 2001, 1—198.
[8]	刘式适,刘式达.物理学中的非线性方程[M].北京: 北京大学出版社, 2002, 127—201.

Relative Articles

[1]	WANG Daohang, SUN Bo, LIU Chunxia, ZHOU Ziyi, LIU Yu. Vertical Vibration Control of Nonlinear Viscoelastic Isolation Systems With Time Delay Feedback[J]. Applied Mathematics and Mechanics, 2025, 46(2): 199-207. doi: 10.21656/1000-0887.450037
[2]	WANG Xi-gang, FU Ming-fu. Fuzzy Elasto-Visco-Plasticity Analysis of Finite Deformation Based on the L-D Plastic Flow Rule[J]. Applied Mathematics and Mechanics, 2015, 36(2): 128-139. doi: 10.3879/j.issn.1000-0887.2015.02.002
[3]	TANG You-qi, CHEN Li-qun. Internal-External Combination Resonance of Nonlinear Vibration of in-Plane Translating Viscoelastic Plates[J]. Applied Mathematics and Mechanics, 2013, 34(5): 480-487. doi: 10.3879/j.issn.1000-0887.2013.05.006
[4]	YI Zhuang-peng, ZHANG Yong, WANG Lian-hua. Nonlinear Dynamic Response and Bifurcation Analysis of the Elastically Constrained Shallow Arch[J]. Applied Mathematics and Mechanics, 2013, 34(11): 1182-1196. doi: 10.3879/j.issn.1000-0887.2013.11.008
[5]	LI Feng-ming, LIU Chun-chuan. Parametric Vibration Stability and Active Control of Nonlinear Beams[J]. Applied Mathematics and Mechanics, 2012, 33(11): 1284-1293. doi: 10.3879/j.issn.1000-0887.2012.11.004
[6]	ZHAN Jie-min, LI Yok-sheung, DONG Zhi. Chebyshev Finite Spectral Method With Extended Moving Grids[J]. Applied Mathematics and Mechanics, 2011, 32(3): 365-374. doi: 10.3879/j.issn.1000-0887.2011.03.012
[7]	ZHANG Shan-yuan, LIU Zhi-fang. Three Kinds of Nonlinear-Dispersive Waves in Finite Deformation Elastic Rods[J]. Applied Mathematics and Mechanics, 2008, 29(7): 825-832.
[8]	HE Yuan-jun, MA Xing-rui, WANG Ben-li. Stable Response of the Low-Gravity Liquid Non-Linear Sloshing in a Circle Cylindrical Tank[J]. Applied Mathematics and Mechanics, 2007, 28(10): 1135-1145.
[9]	ZHAN Jie-min, LI Yok-sheng. Generalized Finite Spectral Method for 1D Burgers and KdV Equations[J]. Applied Mathematics and Mechanics, 2006, 27(12): 1431-1438.
[10]	LIU Zhi-fang, ZHANG Shan-yuan. Solitary Waves in Finite Deformation Elastic Circular Rod[J]. Applied Mathematics and Mechanics, 2006, 27(10): 1255-1260.
[11]	WU Jun, CHEN Li-qun. Steady-State Responses and Their Stability of Nonlinear Vibration of an Axially Accelerating String[J]. Applied Mathematics and Mechanics, 2004, 25(9): 917-926.
[12]	CHEN Shu-yu, SHEN Cheng-kang, JIN Wu-gen. Study on Dynamic Constitutive Relations for Concrete With Finite Deformation[J]. Applied Mathematics and Mechanics, 2004, 25(12): 1257-1263.
[13]	CAI Jian-ping, YANG Cui-hong, LI Yi-ping. Pendulum With Linear Damping and Variable Length[J]. Applied Mathematics and Mechanics, 2004, 25(11): 1163-1168.
[14]	LI Jing-jing, CHENG Chang-jun. Differential Quadrature Method for Bending of Orthotropic Plates With Finite Deformation and Transverse Shear Effects[J]. Applied Mathematics and Mechanics, 2004, 25(8): 801-808.
[15]	ZHOU Zhe, QIN Ling-li, HUANG Wen-bin, WANG Hong-wei. Effective Stress and Strain in Finite Deformation[J]. Applied Mathematics and Mechanics, 2004, 25(5): 542-550.
[16]	Song Yanqi, Chen Zhida. Variational Principles of Asymmetric Elasticity Theory of Finite Deformation[J]. Applied Mathematics and Mechanics, 1999, 20(11): 1115-1120.
[17]	Jiang Yaodong, An Liqian, Liu Yanhua, Chen Zhida. Deformation Analysis of Laser Spectra Based on S-R Decomposition Theorem[J]. Applied Mathematics and Mechanics, 1999, 20(11): 1107-1114.
[18]	Chen Zhida. Inverse Asymptotic Solution Method for Finite Deformation Elasto-Plasticity[J]. Applied Mathematics and Mechanics, 1997, 18(11): 959-966.
[19]	Gao Hang-shan, Zhang Ru-qing. The Random Variational Principle in Finite Deformation of Elasticity and Finite Element Method[J]. Applied Mathematics and Mechanics, 1994, 15(10): 855-862.
[20]	Dai Shi-qiang. Generalization of the Method of Full Approximation and Its Applications[J]. Applied Mathematics and Mechanics, 1991, 12(3): 237-244.