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.
ZHANG Shan-yuan,ZHUANG Wei.The strain solitary waves in a nonlinear elastic rod[J].Acta Mechanica Cinia,1987,1(3):62—72.
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.
Whitham G B.Linear and Nonlinear Waves[M].New York: John Wiley & Sons, 1974, 96—113.
Bhatnager P L.Nonlinear Waves in One-Dimensional Dispersive System[M].Oxford: Clarendon Press, 1979,61—88.
Alexander M S.Strain Solitons in Solid and How to Construct Them[M].New York: Chapman & Hall/CRC, 2001, 1—198.