运动-大变形-生长非线性耦合动力学的分析力学方法与应用

The project will extend Kirchhoff kinetic analogy of elastic rod to the two-dimensional case with background of biomembrane and engineering component, so as to describe the nonlinear coupling geometrical configuration of motion-large deformation-growth of an elastic body with the unified concept, and construct the modeling system of analytical mechanics as well as their analytical and numerical caculation methods, which can be applied to specific objects. Starting with geometric analysis, Cosserat directors are devolped in a new form that their time course describes motion of the body and the space course describes configuration of the body,then the geometrical equations characterized the deformation and growth rules, as well as the constitutive equations are derived respectively, so the theory of analytical mechanics with time and space coordinates as independent variables can be fomed. The description method is completely different from applications of the analytical mechanics in continuum mechanics and field theory, and is also different from the classic manifestation of shells. The new theory is particularly suitable for coupling dynamics of constrained body with large deformation and growth. Basing on this study, the problems about nonholonomic constraint with motion- deformation - growth, conservation laws and the stability on time or space, are studied. Taking several typical Biomembrane and engineering components as examples, the analytic analysis and numerical methods are developed. The significance of this project is to present new research methods and means for the nonlinear coupling dynamics of motion - large deformation - growth, and to explore new areas of research for the classical analytical mechanics.

以生物膜和工程构件为背景，推广Kirchhoff动力学比拟思想，用统一的概念描述运动-大变形-生长非线性耦合弹性体的几何形态，形成分析力学的建模体系及其分析和数值计算方法，并应用于实际对象。从几何分析入手，改进Cosserat方向子，使其时间历程表达弹性体的运动，空间历程表达位形，导出刻划变形和生长规则的几何方程和本构方程，形成以时间和空间为自变量的分析力学理论，其描述方法完全有别于分析力学在连续介质力学和场论中的应用，也不同于经典的板壳力学的表现形式，尤其适合受约束的大变形和生长的弹性体。在此基础上研究运动-大变形-生长耦合的非完整约束问题、关于时间或空间的守恒律以及稳定性问题。以若干典型的生物膜和工程中的构件为例子，进行解析分析和发展数值计算方法。本项目的意义在于为运动-大变形-生长非线性耦合动力学提供新的有效的研究方法和手段，为经典分析力学拓展新的研究领域。

发生卷揉变形的超薄大幅弹性板壳在生物和工程领域有着广泛的应用背景，其变形特征是小应变在大尺度方向上累积成超大的位移，在形成异常复杂的卷揉变形的同时产生接触等力学现象。将处理超细长弹性杆大变形问题的有效方法——Kirchhoff动力学比拟方法运用到超薄大幅弹性板壳，旨在为后者建立一种有效建模方法，能统一表达变形和运动，即用刚体动力学的方法研究弹性薄壳的变形和运动，并且方便处理具有初始弯扭度的弹性薄壳。. 基于直法线假设，用中面上的Cosserat方向子表达两个轴系，此轴系姿态随曲线坐标线的变化率为表征中面形态的弯扭度，随时间的变化率为表征中面运动的角速度。用Euler角等表达轴系的姿态、弯扭度和角速度，以及挠曲面方程。把大幅弹性薄壳的变形和运动表达成广义坐标的弧坐标和时间历程。讨论了变形和运动的协调问题，即弯扭度之间，以及弯扭度和角速度之间的相容性问题，得到了若干非完整约束方程。. 用弯扭度表达线弹性本构方程，建立了用分布内力表达的平衡微分方程，形成超细长弹性杆平衡的Kirchhoff方程对大幅超薄弹性壳的推广。由此可以化作分析力学形式。后者这部分内容将形成待出版专著的第八章。.进一步研究了弹性细杆的矢量力学或分析力学建模问题。包括存在介质或存在约束的建模问题、以及弹性杆的生长建模问题，讨论了稳定性等问题。