Kobayashi Shunsuke, Takemasa Katsumi, Tarumi Ryuichi
Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan.
R Soc Open Sci. 2025 Jul 16;12(7):242213. doi: 10.1098/rsos.242213. eCollection 2025 Jul.
This study presents a comprehensive mathematical model for Volterra defects and explores their relations using differential geometry on Riemann-Cartan manifolds. Following the standard Volterra process, we derived the Cartan moving frame, a geometric representation of plastic fields, and the associated Riemannian metric using exterior algebra. Although the analysis naturally defines the geometry of three types of dislocations and the wedge disclination, it fails to classify twist disclinations owing to the persistent torsion component, suggesting the need for modifications to the Volterra process. By leveraging the interchangeability of the Weitzenböck and Levi-Civita connections and applying an analytical solution for plasticity derived from the Biot-Savart law, we provide a rigorous mathematical proof of the long-standing phenomenological relationship between edge dislocations and wedge disclinations. Additionally, we showcase the effectiveness of novel mathematical tools, including Riemannian holonomy for analysing the Frank vector and complex potentials that encapsulate the topological properties of wedge disclinations as jump discontinuities. Furthermore, we derive analytical expressions for the linearized stress fields of wedge disclinations and confirm their consistency with existing results. These findings demonstrate that the present geometrical framework extends and generalizes the classical theory of Volterra defects.
本研究提出了一个关于沃尔泰拉缺陷的综合数学模型，并利用黎曼 - 嘉当流形上的微分几何探索它们之间的关系。按照标准的沃尔泰拉过程，我们使用外代数推导了卡尔丹活动标架（一种塑性场的几何表示）以及相关的黎曼度量。尽管该分析自然地定义了三种类型位错和楔形位错线的几何结构，但由于持续存在的挠率分量，它未能对位错扭转进行分类，这表明需要对沃尔泰拉过程进行修正。通过利用魏岑伯克联络和列维 - 奇维塔联络的互换性，并应用从毕奥 - 萨伐尔定律导出的塑性解析解，我们为刃型位错和楔形位错线之间长期存在的唯象关系提供了严格的数学证明。此外，我们展示了新颖数学工具的有效性，包括用于分析弗兰克矢量的黎曼全纯性以及将楔形位错线的拓扑性质封装为跳跃间断的复势。此外，我们推导了楔形位错线线性化应力场的解析表达式，并证实了它们与现有结果的一致性。这些发现表明，当前的几何框架扩展并推广了经典的沃尔泰拉缺陷理论。
