受曲面约束弹性细杆的平衡问题

Problems on equilibrium of a thin elastic rod constrained on a surface

作为DNA等一类生物大分子的力学模型 ,弹性细杆的非线性力学再次受到关注 ,形成一个力学与分子生物学的交叉学科 .除了不受外界约束的自由弹性细杆外 ,受曲面约束的弹性细杆静力学具有重要的应用背景 .在分析约束、约束方程和约束力的基础上建立了受曲面约束的圆截面弹性细杆的平衡微分方程 ,即曲面上的Kirchhoff方程 ,它是以截面主矢和截面姿态坐标以及中心线的Descartes坐标为变量的微分 代数方程 .作为应用 ,讨论了约束是圆柱面的情形 .此时平衡的无量纲方程仅含的物理参数是截面对形心的抗扭刚度和对主轴的抗弯刚度的比值 ,与几何参数无关 .由此导出方程的螺旋杆特解 .数值计算表明 ,对弹性细杆中心线的几何形状有显著影响的是截面主矢和姿态坐标及其导数的起始值 ,而不是物理参数 .

Nonlinear mechanics of thin elastic rod, as a model of DNA, aroused extensive interest as a joint research subject of mechanics and molecular biology. The study of the equilibrium of a thin elastic rod constrained on a surface found an important application in industry, especially in molecular biology. In the present paper the constraint equations and constraint forces of the elastic rod are analyzed, and the differential/algebraic equations of equilibrium are established with the arc-coordinate of the central line as the independent variable. In a special case when the constraint surface is a cylinder the dimensionless differential equations contain only one physical parameter, the ratio of the bending and torsional stiffness of the cross section. A special solution of helical equilibrium can be derived and is corresponding to a regular precession of the Lagrange heavy rigid body about a fixed point. The numerical analysis shows that the geometrical form of the central line is dependent on the initial conditions of the rod more than its physical parameters.