{"title":"New Representations for the Curvature Tensor of a Surface with Application to Theories of Elastic Shells","authors":"Nathaniel N. Goldberg, Oliver M. O’Reilly","doi":"10.1007/s10659-022-09885-5","DOIUrl":null,"url":null,"abstract":"<div><p>Consider two points <span>\\(P\\)</span> and <span>\\(Q\\)</span> on a surface. Modulo rotations about the normal vector to the surface at <span>\\(P\\)</span> and the normal vector to the surface at <span>\\(Q\\)</span>, a rotation can be defined that maps the unit normal vector to the surface at <span>\\(Q\\)</span> to the corresponding unit normal vector at <span>\\(P\\)</span>. With the help of Weingarten’s formulae, new representations are established for the components of the curvature tensor of a surface and the associated mean and Gaussian curvatures in terms of components of a pair of vectors associated with the rotation. The formulae are shown to be helpful in demonstrating how different strain measures for Kirchhoff-Love shell theory are equivalent.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"148 2","pages":"199 - 206"},"PeriodicalIF":1.4000,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-022-09885-5.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Elasticity","FirstCategoryId":"5","ListUrlMain":"https://link.springer.com/article/10.1007/s10659-022-09885-5","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Consider two points \(P\) and \(Q\) on a surface. Modulo rotations about the normal vector to the surface at \(P\) and the normal vector to the surface at \(Q\), a rotation can be defined that maps the unit normal vector to the surface at \(Q\) to the corresponding unit normal vector at \(P\). With the help of Weingarten’s formulae, new representations are established for the components of the curvature tensor of a surface and the associated mean and Gaussian curvatures in terms of components of a pair of vectors associated with the rotation. The formulae are shown to be helpful in demonstrating how different strain measures for Kirchhoff-Love shell theory are equivalent.
期刊介绍：
The Journal of Elasticity was founded in 1971 by Marvin Stippes (1922-1979), with its main purpose being to report original and significant discoveries in elasticity. The Journal has broadened in scope over the years to include original contributions in the physical and mathematical science of solids. The areas of rational mechanics, mechanics of materials, including theories of soft materials, biomechanics, and engineering sciences that contribute to fundamental advancements in understanding and predicting the complex behavior of solids are particularly welcomed. The role of elasticity in all such behavior is well recognized and reporting significant discoveries in elasticity remains important to the Journal, as is its relation to thermal and mass transport, electromagnetism, and chemical reactions. Fundamental research that applies the concepts of physics and elements of applied mathematical science is of particular interest. Original research contributions will appear as either full research papers or research notes. Well-documented historical essays and reviews also are welcomed. Materials that will prove effective in teaching will appear as classroom notes. Computational and/or experimental investigations that emphasize relationships to the modeling of the novel physical behavior of solids at all scales are of interest. Guidance principles for content are to be found in the current interests of the Editorial Board.
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