弹性薄壳动力学比拟的曲面论基础

A Fundamental Surface Theory for Kinetic Analogy of Thin Elastic Shells

将Kirchhoff动力学比拟从弹性细杆推广到弹性薄壳,需要相应的经典曲面论新的表达形式,即用刚体动力学的概念和方法描述曲面的基本性质,形成广义Kirchhoff动力学比拟方法.从曲面非正交网格的两个刚性正交轴系出发,用其姿态坐标和Lamé系数表达曲面偏微分方程;用弯扭度和Lamé系数表达曲面的第一和第二基本二次型,得到了法曲率的表达式,由此计算了主曲率和主方向,验证了与经典曲面论的一致性;给出算例以说明该文方法的应用,这一方法可以用来表达曲面的Rodrigues方程、Weingarten公式和Gauss公式,以及曲面论的基本方程.分析表明了这一方法对表述曲面微分几何的可行性,具有推导简洁和直观的优点.这有助于为广义Kirchhoff比拟及其后续发展奠定数学基础.

The generalization of the Kirchhoff kinetic analogy from thin elastic rods to thin elastic shells,name⁃ly the generalized Kirchhoff kinetic analogy,needs a corresponding novel expression of the classical surface theory with its fundamental properties described by means of the concept and method of the rigid body dynam⁃ics.A rigid orthogonal⁃axis system and a curvature⁃twist vector were defined for the non⁃orthogonal meshing of a surface,and the Euler angles were used to express the attitude of the system and the partial differential geo⁃metric equation of the surface.The curvature⁃twist vector and the Lamécoefficient were applied to depict the 1st and the 2nd basic quadratic forms of the surface,obtain the normal curvature and calculate the principal curvature and the principal direction.The analysis demonstrates the consistency between the new and the clas⁃sical expressions of the surface theory.The application example of the proposed method shows that,this meth⁃od can reasonably express the Rodrigues equation,the Weingarten equation,the Gauss equation and the funda⁃mental equations for the surface,and well describe the differential geometry of the surface.This method has the benefits of conciseness and directness,and lays a mathematical foundation for the generalized Kirchhoff kinetic analogy and its further developments.