Geometrical Nonlinear Waves in Finite Deformation Elastic Rods
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摘要: 利用有限变形理论的Lagrange描述，借助非保守系统的Hamilton型变分原理，导出了描述弹性杆中几何非线性波的波动方程．为了使非线性波动方程有稳定的行波解，计及了粘性效应引入的耗散和横向惯性效应导致的几何弥散．运用多重尺度法将非线性波动方程简化为KdV-Bergers方程，这个方程在相平面上对应着异宿鞍-焦轨道，其解为振荡孤波解．如果略去粘性效应或横向惯性，方程将分别退化为KdV方程或Bergers方程，由此得到孤波解或冲击波解，它们在相平面上对应着同宿轨道或异宿轨道．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.
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Key words:
- nonlinear wave /
- finite deformation /
- viscous effect /
- transverse inertia effect /
- multi-scale method
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