非对称截面Kirchhoff弹性细杆模型简化方法研究

研究弹性细杆Kirchhoff模型及其相关演化系统, 是深入考察宏观、微观柔性体拓扑结构与稳定性问题的重要依据. 以DNA弹性细杆数学模型为背景, 考虑截面非对称性特征的影响, 构造新的复数形式Kirchhoff系统. 在此基础上, 结合复变量扭矩设解形式, 获得了非对称截面系统的有效抗弯刚度; 并通过相关理论在高维系统简化过程中的应用, 得到了对应于原有系统的单变量二阶常微分方程. 此外, 将DNA分子具备的抗弯刚度周期变化特征转化为针对有效抗弯刚度的周期摄动形式, 以期从总体上减少理论分析对于数值积分的依赖, 为后续定量分析工作提供新的思路.

The analytical reduction of the Kirchhoff thin elastic rod model with asymmetric cross section

The Kirchhoff thin elastic rod models and related systems are always the important basis to research the topology and stability of the flexible structures in not only the macroscopic but also microscopic scale. Firstly the initial Kirchhoff equations are rebuilt in a complex style to suit the character of obvious asymmetry embodied on the cross section by considering the mathematical background of DNA double helix. Then we introduce a complex form variable solution of the torque, and extend the knowledge of effective bending coefficients as well as its facility in the high dimensional system by using the complicated system. As the result, a simplified second order ordinary differential equation with single variable is obtained. Furthermore the periodically varying bending coefficients of the DNA molecular are considered as the appended components to the effective bending coefficients. The whole reduction process makes the numerical simulation become not solely the exclusively eligible approach, and produces adaptable channel to quantitative analysis.