摘要
弹性杆的螺旋线平衡问题在DNA、纤维、海底电缆和输油管线等方面具有应用背景.Kirchhoff动力学比拟是分析弹性细杆平衡稳定性的有效方法.Kirchhoff模型中包括中心线无拉伸变形和截面无剪切变形的基本假定与生物大分子等软物质的实际状况有较大差异.基于精确Cosserat模型,考虑中心线的拉伸压缩变形和截面剪切变形,以及剪切变形引起杆中心线转动导致切线轴相对截面法线轴的偏离,以Euler角表达截面姿态,建立圆截面弹性杆的动力学普遍方程.在静力学范畴内讨论螺旋线平衡状态的Liapunov稳定性和Euler稳定性问题,导出稳定性条件及轴向力和扭矩的Euler临界值.证明螺旋杆平衡的静态Liapunov稳定性和Euler稳定性条件是动态Liapunov稳定性的必要条件.
The helical equilibrium of a thin elastic rod has a practical background such as DNA,fiber,sub-ocean cable and oil-well dill string.The Kirchhoff＇s kinetic analogy is an effective approach in stability analysis of equilibrium of a thin elastic rod.The main hypotheses of Kirchhoff＇s theory including no extension of the centerline and no shear deformation of the cross section are not adaptable to the real soft materials of biological fibers.The dynamic equations of a rod with circular cross section were established on the basis of exact Cosserat＇s model considering tension and shear deformations.The Euler＇s angles were applied as the attitude representation of the cross section.The deviation of the normal axis of cross section from the tangent of the centerline was considered as the result of shear deformation.The Liapunov＇s stability of helical equilibrium was discussed in static category and the Euler＇s critical values of axial force and torque were obtained.The Liapunov＇s and Euler＇s stability conditions in space domain are the necessary conditions of Liapunov＇s stability of the helical rod in time domain.
出处
《应用数学和力学》
CSCD
北大核心
2011年第5期570-578,共9页
Applied Mathematics and Mechanics
基金
国家自然科学基金资助项目(10972143)
