Stability analysis of helical rod based on exact Cosserat model  

作　　者：刘延柱 薛纭 

机构地区：[1]Department of Engineering Mechanics,Shanghai Jiao Tong University [2]School of Mechanical and Automation Engineering,Shanghai Institute of Technology

出　　处：《Applied Mathematics and Mechanics(English Edition)》2011年第5期603-612,共10页应用数学和力学（英文版）

基　　金：supported by the National Natural Science Fundation of China(No.10972143)

摘　　要：Helical equilibrium of a thin elastic rod has practical backgrounds, such as DNA, fiber, sub-ocean cable, and oil-well drill string. Kirchhoff＇s kinetic analogy is an effective approach to the stability analysis of equilibrium of a thin elastic rod. The main hypotheses of Kirchhoff＇s theory without the extension of the centerline and the shear deformation of the cross section are not adoptable to real soft materials of biological fibers. In this paper, the dynamic equations of a rod with a circular cross section are established on the basis of the exact Cosserat model by considering the tension and the shear deformations. Euler＇s angles are applied as the attitude representation of the cross section. The deviation of the normal axis of the cross section from the tangent of the centerline is considered as the result of the shear deformation. Lyapunov＇s stability of the helical equilibrium is discussed in static category. Euler＇s critical values of axial force and torque are obtained. Lyapunov＇s and Euler＇s stability conditions in the space domain are the necessary conditions of Lyapunov＇s stability of the helical rod in the time domain.Helical equilibrium of a thin elastic rod has practical backgrounds, such as DNA, fiber, sub-ocean cable, and oil-well drill string. Kirchhoff＇s kinetic analogy is an effective approach to the stability analysis of equilibrium of a thin elastic rod. The main hypotheses of Kirchhoff＇s theory without the extension of the centerline and the shear deformation of the cross section are not adoptable to real soft materials of biological fibers. In this paper, the dynamic equations of a rod with a circular cross section are established on the basis of the exact Cosserat model by considering the tension and the shear deformations. Euler＇s angles are applied as the attitude representation of the cross section. The deviation of the normal axis of the cross section from the tangent of the centerline is considered as the result of the shear deformation. Lyapunov＇s stability of the helical equilibrium is discussed in static category. Euler＇s critical values of axial force and torque are obtained. Lyapunov＇s and Euler＇s stability conditions in the space domain are the necessary conditions of Lyapunov＇s stability of the helical rod in the time domain.

关 键 词：exact Cosserat model Kirchhoff＇s rod Lyapunov＇s stability Euler＇s load 

分 类 号：O317[理学—一般力学与力学基础] O343[理学—力学]