Similarity ‘hot-spot’ solutions for a hypoplastic granular material

Abstract

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Hill J. M. 2000Similarity ‘hot-spot’ solutions for a hypoplastic granular materialProc. R. Soc. Lond. A.4562653–2671http://doi.org/10.1098/rspa.2000.0631SectionRestricted accessResearch articleSimilarity ‘hot-spot’ solutions for a hypoplastic granular material J. M. Hill J. M. Hill School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, New South Wales 2522, Australia Google Scholar Find this author on PubMed Search for more papers by this author J. M. Hill J. M. Hill School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, New South Wales 2522, Australia Google Scholar Find this author on PubMed Search for more papers by this author Published:08 November 2000https://doi.org/10.1098/rspa.2000.0631AbstractHypoplasticity is a theory which has been developed by the Institute for Soil and Rock Mechanics at the University of Karlsruhe to accurately model the behaviour of granular materials such as sand, soil and certain powders. The earlier hypoplasticity theories were formulated on the basis of the two assumptions of rate independence and homogeneous dependence of the stress rate on the stress. These two assumptions mean that the governing partial differential equations remain invariant under a general family of stretching similarity transformations, which in turn imply the existence of similarity solutions for which the partial differential equations can be reduced to ordinary differential equations. In two recent papers involving uniaxial compaction and cylindrical and spherical cavity expansion problems, these similarity solutions have been identified and the appearance of uniaxial ‘hot–spot” solutions has been noted. In this context, ‘hot–spot” refers to the material undergoing an infinite stress in a finite time. Here, for planar, cylindrical and spherical geometries, we summarize the general picture relating to these similarity ‘hot–spot” solutions and we give a number of simple analytical expressions for them not included in the recent papers. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Yuu S, Umekage T, Mitsuiki Y and Koga F (2008) Constitutive Relations Based on Distinct Element Method Results for Granular Materials and Simulation of Granular Collapse and Heap by Smoothed Particle Hydrodynamics, and Experimental Verification, Advanced Powder Technology, 10.1163/156855208X293808, 19:2, (153-182), . Yuu S and Umekage T (2008) Constitutive Relations and Computer Simulation of Granular Material, Advanced Powder Technology, 10.1163/156855208X294609, 19:3, (203-230), . Wu W (2006) On High-order Hypoplastic Models for Granular Materials, Journal of Engineering Mathematics, 10.1007/s10665-006-9040-7, 56:1, (23-34), Online publication date: 1-Sep-2006. Hill J and Selvadurai A Mathematics and mechanics of granular materials Mathematics and Mechanics of Granular Materials, 10.1007/1-4020-4183-7_1, (1-9) Hill J and Selvadurai A (2005) Mathematics and mechanics of granular materials, Journal of Engineering Mathematics, 10.1007/s10665-005-2729-1, 52:1, (1-9), Online publication date: 1-Jul-2005. Hill J and Selvadurai A (2005) Mathematics and mechanics of granular materials, Journal of Engineering Mathematics, 10.1007/BF02694027, 52:1-3, (1-9), Online publication date: 1-Jul-2005. Cox G and Hill J (2005) Hypoplasticity theory for granular materials—I: Two-dimensional plane strain exact solutions, International Journal of Non-Linear Mechanics, 10.1016/j.ijnonlinmec.2005.05.007, 40:9, (1171-1188), Online publication date: 1-Nov-2005. Cox G and Hill J (2002) Non-dilatant double-shearing theory applied to dynamical granular chute flow, Acta Mechanica, 10.1007/BF01171451, 159:1-4, (125-142), Online publication date: 1-Mar-2002. This Issue08 November 2000Volume 456Issue 2003 Article InformationDOI:https://doi.org/10.1098/rspa.2000.0631Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/11/2000Published in print08/11/2000 License: Citations and impact Keywordsgranular materialsuniaxial compactionhypoplasticityhot spotssimilarity solutionscavity expansion