超细长弹性杆的力学模型及其边界条件

Mechanical Model of Super-thin Elastic Rod and Boundary Conditions

阐明一类超细长弹性杆的静力学建模问题.在平面假设下对弹性杆离散化,用方向子表述弹性杆横截面的位形,从而将杆表示成截面的弧坐标历程.通过截面形心的应变矢量和弯扭度的定义,讨论了截面的变形几何,分析了Kirchhoff模型和Cosserat模型的异同.根据微段杆的平衡条件导出了Cosserat模型下以原始弧坐标为自变量的平衡微分方程,与关于内力主矢和主矩的本构方程联立,形成封闭的微分方程组.讨论了弹性杆的端部约束及其边界条件,显示了这类边值问题的特殊性,表明了大位移下的平衡问题本质上都是静不定问题.

A static model for super-thin elastic rod is described. The rod is treated as a process of a cross section moving along its axial line at a constant velocity, assuming a plane cross-section. The geometry of deformation of the rod section is discussed and differences between the Kirehhoff model and the Cosserat model are analyzed based on the definitions of strain vector at the center and curvature-twisting vector of the section. According to the equilibrium condition of a differential segment of the rod, a differential equilibrium equation of Cosserat model is derived. Closed form constitutive equations on principal vector and principal moment of forces acted on the section are given. The constraints subjected to the ends of rod are discussed and boundary conditions for the rod are given. It is concluded that equilibrium of the rod is a statically indeterminate problem.