Configurations and Deformations in Relativistic Elasticity

IF 0.8 Q2 MATHEMATICS Lobachevskii Journal of Mathematics Pub Date : 2024-08-28 DOI:10.1134/s1995080224602613
S. A. Lychev, K. G. Koifman, N. A. Pivovaroff
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Abstract

Configurations and deformations are the homeomorphisms that relate material manifold and shapes of a solid in conventional non-linear elasticity. They play fundamental role in kinematic description of deformable media and seem to be an essential part of non-linear continuum mechanics. These concepts nevertheless are rooted in Euclidean nature of space and time, adopted in non-relativistic physics, and their direct application in relativistic case is impossible. The paper develops the generalization for them within General Relativity. To this end the concept of relativistic shape is introduced, which is defined as an element of foliation over a congruence of worldlines that constitute the world-tube of the solid. This makes it possible to define displacements as a field of translations along elements of congruence beginning on one element of foliation and ending on another. Geometrically this is similar to Fermi–Walker transport. Each relativistic shape can be endowed with two metrics, one induced by ambient space, and another, given by pushforwarding of the metric induced on another shape (another element of foliation). These metrics are relativistic counterpart of Cauchy–Green measures of deformation. The conditions under which the foliation of congruence of worldlines exists are specified. When these conditions are violated, the natural generalization of the shape leads to a set that cannot be represented by the Riemann submanifold. This situation is similar to that which arises in the continuum theory of defects when trying to find a globally stress-free shape. To solve this problem, it is proposed to construct a material manifold whose elements represent shapes locally. The method of determination of geometry on material manifold, which turns out to be Weyl geometry, is proposed. All constructions are carried out within the framework of a variational approach to the derivation of field equations and Noether symmetries.

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相对论弹性中的配置与变形
摘要构型和变形是传统非线性弹性力学中材料流形与固体形状之间的同构关系。它们在可变形介质的运动学描述中起着基础性作用,似乎是非线性连续介质力学的重要组成部分。然而,这些概念植根于欧几里得时空性质,为非相对论物理学所采用,不可能直接应用于相对论情况。本文在广义相对论中对它们进行了概括。为此,本文引入了相对论形状的概念,将其定义为构成实体世界管的世界线全等的对折元素。这样,就可以把位移定义为沿全等元素的平移场,从一个对折元素开始,到另一个对折元素结束。在几何学上,这与费米-沃克运移类似。每个相对论形状都可以被赋予两个度量,一个是由环境空间诱导的,另一个是由另一个形状(另一个折线元素)上诱导的度量的推转给出的。这些度量是考奇-格林变形度量的相对论对应度量。世界线全同的褶皱存在的条件是明确的。当这些条件被违反时,形状的自然广义化会导致一个无法用黎曼子曲面表示的集合。这种情况类似于缺陷连续理论中试图寻找全局无应力形状时出现的情况。为解决这一问题,建议构建一个材料流形,其元素代表局部形状。提出了在材料流形上确定几何形状的方法,该方法被证明是韦尔几何。所有构造都是在推导场方程和诺特对称性的变分法框架内进行的。
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来源期刊
CiteScore
1.50
自引率
42.90%
发文量
127
期刊介绍: Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.
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