论文标题
连续力学的主要捆绑结构
The Principle Bundle Structure of Continuum Mechanics
论文作者
论文摘要
在本文中，表明任何连续图的配置空间的结构是差异几何{\ it原理bunddle} \ cite {frankel2011thephysics}中所谓的。主束是一种结构，其中歧管的所有点（在这种情况下在这种情况下）都可以自然地投射到称为{\ it base歧管}的歧管上，在我们的情况下，它代表纯变形。所有投影到基本歧管上同一点（同一变形）上的配置称为纤维。然后，这些纤维中的每一个都与代表纯刚体运动的Lie组$ \ Mathfrak {Se（3）} $同构。此外，可以定义所谓的连接，这允许以刚体的亚动作和完全坐标的自由方式分开任何连续运动。因此，可以正确定义可以在其上定义弹性能的纯变形空间。这将使用螺丝理论\ cite {Ball：1900}进行显示，该{Ball：1900}大量用于分析刚体机理，但通常不用于分析Continua。除了恰当提到的结果外，螺丝理论还将用于将诸如螺旋和肠道之类的概念与螺丝理论概念相关联。
In this paper it is shown that the structure of the configuration space of any continua is what is called in differential geometry a {\it principle bundle} \cite{Frankel2011ThePhysics}. A principal bundle is a structure in which all points of the manifold (each configuration in this case) can be naturally projected to a manifold called the {\it base manifold}, which in our case represents pure deformations. All configurations projecting to the same point on the base manifold (same deformation) are called fibers. Each of these fibers is then isomorphic to the Lie group $\mathfrak{se(3)}$ representing pure rigid body motions. Furthermore, it is possible to define what is called a connection and this allows to split any continua motion in a rigid body sub-motion and a deformable one in a completely coordinate free way. As a consequence of that it is then possible to properly define a pure deformation space on which an elastic energy can be defined. This will be shown using screw theory \cite{Ball:1900}, which is vastly used in the analysis of rigid body mechanisms but is not normally used to analyse continua. Beside the just mentioned result, screw theory will also be used to relate concepts like helicity and enstrophy to screw theory concepts.
