On anisotropic invariants of a symmetric tensor: crystal classes, quasi–crystal classes and others

Abstract

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Xiao H. 1998On anisotropic invariants of a symmetric tensor: crystal classes, quasi–crystal classes and othersProc. R. Soc. Lond. A.4541217–1240http://doi.org/10.1098/rspa.1998.0203SectionRestricted accessOn anisotropic invariants of a symmetric tensor: crystal classes, quasi–crystal classes and others H. Xiao H. Xiao Institute of Mechanics, Ruhr–University Bochum, D–44780 Bochum, GermanyDepartment of Mathematics, Peking University, Beijing 100871, People's Republic of China, Google Scholar Find this author on PubMed Search for more papers by this author H. Xiao H. Xiao Institute of Mechanics, Ruhr–University Bochum, D–44780 Bochum, GermanyDepartment of Mathematics, Peking University, Beijing 100871, People's Republic of China, Google Scholar Find this author on PubMed Search for more papers by this author Published:08 April 1998https://doi.org/10.1098/rspa.1998.0203AbstractThis paper is concerned with invariants of a symmetric second–order tensor relative to crystal classes and quasicrystal classes and some non–continuous infinite orthogonal subgroups. Simple irreducible functional bases in unified forms, each of which consists of eight polynomial invariants only, are presented for all kinds of finite subgroups of the maximal transverse isotropy group, D∞ h, except the subgroups of the orthotropy group D2h. Results are also provided for other kinds of orthogonal subgroups including icosahedral quasicrystal classes, etc. For the trigonal, tetragonal and hexagonal crystal classes without two–fold axis, the results given here are even more compact than the corresponding results recently derived by this author (Xiao 1996a). For all non–crystal classes except the transverse isotropy groups, the presented results are the first ones. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Itin Y (2016) Quadratic Invariants of the Elasticity Tensor, Journal of Elasticity, 10.1007/s10659-016-9569-2, 125:1, (39-62), Online publication date: 1-Oct-2016. Wright T (2014) Bootstrap Elasticity III: Minimal Nonlinear Constitutive Representation for Cubic Materials, Journal of Elasticity, 10.1007/s10659-014-9507-0, 120:1, (109-119), Online publication date: 1-Jun-2015. 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Xiao H, Bruhns O and Meyers A (2006) Elastoplasticity beyond small deformations, Acta Mechanica, 10.1007/s00707-005-0282-7, 182:1-2, (31-111), Online publication date: 1-Mar-2006. Moakher M and Norris A (2006) The Closest Elastic Tensor of Arbitrary Symmetry to an Elasticity Tensor of Lower Symmetry, Journal of Elasticity, 10.1007/s10659-006-9082-0, 85:3, (215-263), Online publication date: 26-Oct-2006. Bruhns O, Xiao H and Meyers A (2003) Some basic issues in traditional Eulerian formulations of finite elastoplasticity, International Journal of Plasticity, 10.1016/S0749-6419(03)00047-0, 19:11, (2007-2026), Online publication date: 1-Nov-2003. Xiao H, Bruhns O and Meyers A (1999) On anisotropic invariants of N symmetric second-order tensors: crystal classes and quasi–crystal classes as subgroups of the cylindrical group D, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455:1986, (1993-2020), Online publication date: 8-Jun-1999.Xiao H (1998) On scalar–, vector– and second–order tensor–valued anisotropic functions of vectors and second–order tensors relative to all kinds of subgroups of the transverse isotropy group Cinfinityh, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 356:1749, (3087-3122), Online publication date: 15-Dec-1998. This Issue08 April 1998Volume 454Issue 1972 Article InformationDOI:https://doi.org/10.1098/rspa.1998.0203Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/04/1998Published in print08/04/1998 License: Citations and impact Keywordsanisotropysymmetric tensorinvariantscrystal classesirreducible functional basesquasi–crystal classes