{"title":"Compatible Director Fields in \\(\\mathbb{R}^{3}\\)","authors":"Luiz C. B. da Silva, Tal Bar, Efi Efrati","doi":"10.1007/s10659-023-09988-7","DOIUrl":null,"url":null,"abstract":"<div><p>The geometry and interactions between the constituents of a liquid crystal, which are responsible for inducing the partial order in the fluid, may locally favor an attempted phase that could not be realized in <span>\\(\\mathbb{R}^{3}\\)</span>. While states that are incompatible with the geometry of <span>\\(\\mathbb{R}^{3}\\)</span> were identified more than 50 years ago, the collection of compatible states remained poorly understood and not well characterized. Recently, the compatibility conditions for three-dimensional director fields were derived using the method of moving frames. These compatibility conditions take the form of six differential relations in five scalar fields locally characterizing the director field. In this work, we rederive these equations using a more transparent approach employing vector calculus. We then use these equations to characterize a wide collection of compatible phases.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2023-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-023-09988-7.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Elasticity","FirstCategoryId":"5","ListUrlMain":"https://link.springer.com/article/10.1007/s10659-023-09988-7","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
The geometry and interactions between the constituents of a liquid crystal, which are responsible for inducing the partial order in the fluid, may locally favor an attempted phase that could not be realized in \(\mathbb{R}^{3}\). While states that are incompatible with the geometry of \(\mathbb{R}^{3}\) were identified more than 50 years ago, the collection of compatible states remained poorly understood and not well characterized. Recently, the compatibility conditions for three-dimensional director fields were derived using the method of moving frames. These compatibility conditions take the form of six differential relations in five scalar fields locally characterizing the director field. In this work, we rederive these equations using a more transparent approach employing vector calculus. We then use these equations to characterize a wide collection of compatible phases.
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.