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Compatibility conditions of continua using Riemann–Cartan geometry

Böhmer, CG; Lee, Y; (2020) Compatibility conditions of continua using Riemann–Cartan geometry. Mathematics and Mechanics of Solids 10.1177/1081286520961453. (In press). Green open access

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Abstract

The compatibility conditions for generalised continua are studied in the framework of differential geometry, in particular Riemann–Cartan geometry. We show that Vallée’s compatibility condition in linear elasticity theory is equivalent to the vanishing of the three-dimensional Einstein tensor. Moreover, we show that the compatibility condition satisfied by Nye’s tensor also arises from the three-dimensional Einstein tensor, which appears to play a pivotal role in continuum mechanics not mentioned before. We discuss further compatibility conditions that can be obtained using our geometrical approach and apply it to the microcontinuum theories.

Type: Article
Title: Compatibility conditions of continua using Riemann–Cartan geometry
Open access status: An open access version is available from UCL Discovery
DOI: 10.1177/1081286520961453
Publisher version: https://doi.org/10.1177/1081286520961453
Language: English
Additional information: https://creativecommons.org/licenses/by/4.0/This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access page
Keywords: Compatibility conditions, Cosserat continuum, Riemann–Cartan geometry
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10109559
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