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arXiv 1306.1624cond-mat.mtrl-scimath.DG
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非晶材料缺陷的度规描述

Metric Description of Defects in Amorphous Materials

Raz Kupferman, Michael Moshe, Jake P. Solomon

摘要:

经典弹性力学关注的是可以被建模为光滑流形的物体,该流形具有表示相邻物质元素之间的局部平衡距离的参考度量。在经典弹性力学中,与物体的构型相关的弹性能量是由实际度量和参考度量之间的差异引起的局部贡献的总和。相比之下,固体中缺陷的建模传统上涉及材料流形上的额外结构,特别是用于量化位错密度的扭转和用于表示点缺陷密度的非度量。我们表明,所有的经典缺陷可以描述的经典弹性的框架内使用张量场,只假设一个度量结构。具体来说,具有奇异缺陷的物体可以被视为仿射流形;向错和位错都被单值映射捕获,该单值映射将缺陷轨迹周围的曲线映射到仿射变换中。最后,我们证明了二维缺陷平凡monodromy是纯粹的本地意义上说,如果我们删除从流形的一个紧凑的集合,包含的轨迹的缺陷,穿孔流形可以等距嵌入在欧几里得空间。

英文摘要:

Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with a configuration of a body in classical elasticity is the sum of local contributions that arise from a discrepancy between the actual metric and the reference metric. In contrast, the modeling of defects in solids has traditionally involved extra structure on the material manifold, notably torsion to quantify the density of dislocations and non-metricity to represent the density of point defects. We show that all the classical defects can be described within the framework of classical elasticity using tensor fields that only assume a metric structure. Specifically, bodies with singular defects can be viewed as affine manifolds; both disclinations and dislocations are captured by the monodromy that maps curves that surround the loci of the defects into affine transformations. Finally, we show that two dimensional defects with trivial monodromy are purely local in the sense that if we remove from the manifold a compact set that contains the locus of the defect, the punctured manifold can be isometrically embedded in Euclidean space.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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