有限形变和粘弹性阻尼驱动下的KdV应变孤波的演变

THE EVOLUTION OF KdV STRAIN SOLITARY WAVES UNDER THE ACTION OF THE FINITE DEFORMATION AND VISCOELASTIC DAMPING

本文用微扰论方法,详细地研究了非线性弹性杆中纵向应变孤波在有限形变和线性粘弹性阻尼同时作用下的演变规律。结论指出有限形变和线性粘弹性阻尼对纵向应变孤波的主要影响是:(1)孤波发生形变;(2)孤波的传播速度减小;(3)孤波的能量发生耗散;(4)孤波后面会生长出一个振荡形状的小尾迹。当忽略有限形变和线性粘弹性阻尼时,应变孤波的能量是一个常数,这种能量的聚集会导致杆的非线性结构断裂,所有这些影响都在含微扰的KdV方程中作了详细的研究。

By using perturbation theory, we have studied the evolutionary rules of strain solitary waves in the presence of the finite deformation and viscoelastic damping in a non—linear elastic rod. The results show that the finite deformation and viscoelastic damping lead to four main effects on a strain solitary wave: (i) the deformation of its shape; (ii) the slowdown of its velocity; (iii) the dissipation of its energy; (iv) the formation of its “tail” which is a small amplitude vibrating wave packet with growing length. When the deformation and damping are neglected, the energy of the solitary wave is constant. The collection of the energy may lead to nonlinear structure crippling of the rod. All these effects are investigated in detail for the perturbed kortewey—de Vries equation.