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Connection Structures of Topological Singularity in Micromechanics from a Viewpoint of Generalized Finsler Space
Annalen Der Physik ( IF 2.2 ) Pub Date : 2020-10-02 , DOI: 10.1002/andp.202000306
Takahiro Yajima 1 , Hiroyuki Nagahama 2
Affiliation  

Geometric structures of Cosserat or micropolar continuum are discussed based on geometric objects in a non‐Riemannian space. A microrotation is described in a microscopic level than a macroscopic displacement level. In this case, a microscopic rotation can be expressed as a nonlocal internal variable attached to each point in a generalized Finsler space. Such non‐local hierarchy is geometrically realized by using a second‐order vector bundle viewpoint. Then, two kinds of torsion tensor in the second‐order vector bundle are obtained. One is characterized by the macroscopic displacement. The other is characterized by the microscopic rotation. These torsion tensors are equivalent to nonintegrability conditions for multivalued macroscopic displacement and microscopic rotation. Especially, a path dependency of the displacement and the microscopic rotation is represented by a non‐vanishing condition of torsion tensors. Moreover, the concept of non‐locality of the Finsler geometry implies that the approach of higher‐order geometry is applicable to a finite deformation in nonlinear mechanics. The singularity given by the multivalued function is also described as a boundary value problem. An application of the generalized Finsler geometry to a gradient theory is also discussed.

中文翻译:

广义Finsler空间视角下微力学拓扑奇异性的连接结构。

基于非黎曼空间中的几何对象,讨论了Cosserat或微极连续体的几何结构。在微观层面而不是宏观位移层面描述了微旋转。在这种情况下,微观旋转可以表示为附加在广义Finsler空间中每个点的非局部内部变量。这种非局部层次结构是通过使用二阶向量束视点在几何上实现的。然后,获得了二阶矢量束中的两种扭转张量。一种以宏观位移为特征。另一个特征是微观旋转。这些扭转张量等效于多值宏观位移和微观旋转的不可积分条件。特别,位移和微观旋转的路径相关性由扭转张量的不消失状态表示。此外,Finsler几何的非局部性概念意味着,高阶几何的方法适用于非线性力学中的有限变形。多值函数给出的奇异性也被描述为边值问题。还讨论了广义Finsler几何在梯度理论中的应用。
更新日期:2020-10-02
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