单面约束弹性生长细杆的建模、稳定性分析与数值模拟

There exist unilateral constraints in the growth of thin elastic rod-like shapes in biological systems, such as cell, vessel and plant embedded in another soft material are broad phenomena in natural world. The existence of unilateral contact between the rod and substrate in its growth process will lead to different geometry. Basing on the advantages of analytical mechanics in dealing with constraint problems and modelling methods, the project will establish the reasonable model to describe the geometry of the growing thin elastic rods and analyze their dynamics characters. Specially include: We will establish the general methods for the modelling of thin elastic rod with unilateral constraints based on the unilateral constraint anayltical mechanics theory, and expand the methods to thin elastic growth rod model with unilateral constraints. Then, we will expand the stability analysis methods of statics and dynamics of thin elastic rod and thin elastic growth rod, give the critical criteria and predict the mode selection after instability. Thirdly, we will develop numerical computation methods to solve the equations of the established models above and simulate their corresponding geometry. Finally, we will apply the above results in engineering and biological systems, such as guide lines and cell growth. The carrying out the project will enrich the content of classical analytical mechanics with unilateral constraint, and also provide effective methods for modeling deformation and stability analysis of thin elastic rod with unilateral constraints. The results will propose reference to understand and predict the mechanical phenomenon in the biological and engineering area, so have reality significance and scientific interesting.

生物系统中诸如细胞、血管、植物的根茎等细长杆状结构在生长过程中往往受到单面约束的作用。单面接触对其生长过程中几何结构的形成非常重要。鉴于分析力学处理约束及建模上的优势，本项目基于分析力学方法建立合理的力学模型来探究单面约束下他们几何构型的形成机制，并分析其动力学特性。具体包括：借鉴单面约束分析力学理论探讨单面约束弹性细杆建模与分析的一般性方法，并推广至单面约束弹性生长细杆模型。发展以上模型的静力学和动力学稳定性判定方法，给出临界判据并预测失稳后的模态选择。发展数值计算方法计算其数值解，并模拟不同单面约束对其几何结构的影响。将以上理论应用于工程和生物系统，解释油井钻柱和杆状细胞生长等的动力学行为。本项目的实施既丰富了单面约束分析力学的研究内容，又为单面约束弹性细杆及弹性生长细杆的几何结构和稳定性研究提供有效的建模和分析方法，可为解释和预测生物系统和工程领域中的生长和动力学行为提供参考。

生长是自然界复杂而又普遍存在的生物现象，生长过程中受到单面约束作用从而影响其生长位形。本项目以具有细长结构的生物体为背景，例如DNA、植物藤蔓等，基于单面约束分析力学理论及超细长弹性杆理论，建立单面约束下超细长弹性杆的静力学和动力学，形成单面约束生长弹性杆建模和计算的分析力学方法。.主要研究内容包括：1)区分从自由运动进入约束，或者脱离约束过程中守恒量的变化，研究单面约束力学系统的共形不变性与精确守恒量；2)考虑轴向拉伸变形和生长耦合下生长诱发的黏性介质中弹性形态生长杆的屈曲行为；3)研究了受圆柱面单面约束弹性细杆变形的数值计算,给出了弹性细杆从自由变形到点接触变形的过程，研究结果为DNA缠绕方式、石油钻柱运动变形提供参考；4)研究了处于磁场中平面弹性环的Michell不稳定性及其后屈曲行为;5)大脑皮质结构的一维生长弹性细杆模型与解析解; 6)力-生物电耦合作用下轴突膜动力学的Noether对称性与守恒量。.本项目目前共发表研究论文和作会议报告13篇，其中邀请报告1个。已培养毕业硕士研究生2名，尚有4名硕士研究生在读，指导大学生科创竞赛2次，指导的学生获的大学生科创奖项2项。本项目的结果丰富了单面约束分析力学的研究内容，又为单面约束生长弹性细杆的位形和稳定性研究提供有效的建模和分析方法，为解释和预测生物系统和工程领域中的生长和动力学行为提供参考，具有科学价值和实际意义。