Cosserat生长弹性杆动力学的Gauss最小拘束原理

以自然界中具有生长、变形和运动特征的细长体为背景，用经典力学中的Gauss最小拘束原理研究生长弹性杆的动力学建模问题.在为生长弹性杆动力学建模提供新方法的同时，扩大了Gauss原理的应用范围.以Cosserat弹性杆为对象，分析弹性杆生长和变形的几何规则，表明生长应变和弹性应变是非线性耦合的；本构方程给出了截面的内力与弹性变形的线性关系；利用逆并矢，将经典力学中的Gauss原理和Gauss最小拘束原理用于生长弹性杆动力学，得到等价的两种表现形式，反映了时间和弧坐标在表述上的对称性，由此导出了封闭的动力学微分方程.给出了两种形式的最小拘束函数，表明生长弹性杆的实际运动使拘束函数取驻值，且为最小值.最后讨论了生长弹性杆的约束与条件极值等问题.

Gauss Principle of Least Constraint for Cosserat Growing Elastic Rod Dynamics

The dynamic modeling of growing elastic rods, with the background of a kind of growing, deforming and moving slender bodies in nature and engineering, was studied based on the Gauss principle of least constraint in the classical mechanics. This provides a new method for the dynamic modeling of growing elastic rods, and meanwhile expands the application scope of the Gauss principle of least constraint. With the Cosserat growing elastic rod as the object, the geometric rules for growth and deformation of the rod were analyzed, which show that the growing strain and elastic strain are in a nonlinear coupling relation. The constitutive equations were given as a linear relationship between the internal forces and elastic deformations of the rod’s cross section; through definition of the inverse of dyad, the Gauss principle of least constraint was used to model the growing elastic rod dynamics and get 2 equivalent forms of the Gauss variation, which reflect the symmetry between time and arc coordinates in the expression of rod dynamics. The closed form dynamic differential equations were derived. 2 forms of constraint functions were given, which indicate that the actual motion of an elastic rod made the function at a stationary value, and also the minimum value. Finally, some problems about the constraints and conditional extremums of the growing elastic rod dynamics were discussed.