Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies

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doi:10.3934/jgm.2019017

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2019, Volume 11, Issue 3: 301-324. Doi: 10.3934/jgm.2019017

This issue Previous Article Next Article A symmetry-adapted numerical scheme for SDEs

Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies

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1.
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), c\ Nicolás Cabrera, 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain
2.
Real Academia de Ciencias Exactas, Fisicas y Naturales, c\ de Valverde, 22, 28004 Madrid, Spain
3.
Department of Mechanical and Manufacturing Engineering, University of Calgary. 2500 University Drive NW, Calgary, Alberta, T2N IN4, Canada

^* Corresponding author: Víctor Manuel Jiménez Morales
^* Corresponding author: Víctor Manuel Jiménez Morales

Received: August 31, 2016

Revised: December 31, 2018

Published: August 2019

This work has been partially supported by MINECO Grants MTM2016-76-072-P and the ICMAT Severo Ochoa projects SEV-2011-0087 and SEV-2015-0554. V.M. Jiménez wishes to thank MINECO for a FPI-PhD Position. We would like to thank the referees for their valuable suggestions that have contributed to improve this paper

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Abstract

A Lie groupoid, called material Lie groupoid, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called material algebroid, is used to characterize the uniformity and the homogeneity properties of the material. The relation to previous results in terms of $ G- $structures is discussed in detail. An illustrative example is presented as an application of the theory.

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Mathematics Subject Classification: Primary: 74A20, 20L05; Secondary: 58Z05.

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