NDLI: On advantages of the Kelvin mapping in finite element implementations of deformation processes

Content Provider	Springer Nature Link
Author	Nagel, Thomas Görke, Uwe Jens Moerman, Kevin M. Kolditz, Olaf
Copyright Year	2016
Abstract	Classical continuum mechanical theories operate on three-dimensional Euclidian space using scalar, vector, and tensor-valued quantities usually up to the order of four. For their numerical treatment, it is common practice to transform the relations into a matrix–vector format. This transformation is usually performed using the so-called Voigt mapping. This mapping does not preserve tensor character leaving significant room for error as stress and strain quantities follow from different mappings and thus have to be treated differently in certain mathematical operations. Despite its conceptual and notational difficulties having been pointed out, the Voigt mapping remains the foundation of most current finite element programmes. An alternative is the so-called Kelvin mapping which has recently gained recognition in studies of theoretical mechanics. This article is concerned with benefits of the Kelvin mapping in numerical modelling tools such as finite element software. The decisive difference to the Voigt mapping is that Kelvin’s method preserves tensor character, and thus the numerical matrix notation directly corresponds to the original tensor notation. Further benefits in numerical implementations are that tensor norms are calculated identically without distinguishing stress- or strain-type quantities, and tensor equations can be directly transformed into matrix equations without additional considerations. The only implementational changes are related to a scalar factor in certain finite element matrices, and hence, harvesting the mentioned benefits comes at very little cost.
Starting Page	1
Ending Page	11
Page Count	11
File Format	PDF
ISSN	18666280
Journal	Environmental Earth Sciences
Volume Number	75
Issue Number	11
e-ISSN	18666299
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-05-26
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Kelvin mapping Voigt mapping Finite elements Numerical algorithms OpenGeoSys Geology Hydrology/Water Resources Geochemistry Environmental Science and Engineering Terrestrial Pollution Biogeosciences
Content Type	Text
Resource Type	Article
Subject	Global and Planetary Change Earth-Surface Processes Soil Science Environmental Chemistry Pollution Geology Water Science and Technology

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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