Three Kinds of Nonlinear-Dispersive Waves in Finite Deformation Elastic Rods
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摘要: 在一维弹性细杆拉压、扭转和弯曲波的经典线性理论基础上，分别计入有限变形和弥散效应，借助Hamilton变分原理，由统一的方法导出了3种非线性弥散波的演化方程．对3种演化方程进行了定性分析．结果表明，这些方程在相平面上存在同宿轨道或异宿轨道，分别相应于孤波解或冲击波解．根据齐次平衡原理，用Jacobi椭圆函数展开对这些演化方程进行了求解，在一定的条件下它们均可能存在孤立波解或冲击波解，这与方程的定性分析完全一致．Abstract: On the basis of classical linear theory on longitudinal,torsional and flexural waves in thin elastic rods,taking finite deformation and dispersive effects into consideration,three kinds of nonlinear evolution equations were derived.Qualitative analyses of three kinds of nonlinear equation were completed.It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane, which correspond to solitary wave or shock wave solution respectively.Based on the principle of homogeneous balance,these equations were resolved by Jacobi elliptic function expansion method.The results show that the existence of solitary wave solution and shock wave solution are possible under certain conditions.These conclusions are consistent with that of the qualitative analysis.
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Key words:
- elastic thin rod /
- finite deformation /
- dispersive effect /
- solitary wave /
- shock wave
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