Differential Geometry and Continuum Mechanics PDF
Preview Differential Geometry and Continuum Mechanics
Springer Proceedings in Mathematics & Statistics Gui-Qiang G. Chen Michael Grinfeld R.J. Knops Editors Differential Geometry and Continuum Mechanics Springer Proceedings in Mathematics & Statistics Volume 137 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at http://www.springer.com/series/10533 Gui-Qiang G. Chen Michael Grinfeld (cid:129) R.J. Knops Editors Differential Geometry and Continuum Mechanics 123 Editors Gui-Qiang G.Chen R.J.Knops Mathematical Institute, Radcliffe Schoolof Mathematical andComputer ObservatoryQuarter Sciences University of Oxford Heriot-Watt University Oxford Edinburgh UK UK Michael Grinfeld Department ofMathematics andStatistics University of Strathclyde Glasgow UK ISSN 2194-1009 ISSN 2194-1017 (electronic) SpringerProceedings in Mathematics& Statistics ISBN978-3-319-18572-9 ISBN978-3-319-18573-6 (eBook) DOI 10.1007/978-3-319-18573-6 LibraryofCongressControlNumber:2015938751 MathematicsSubjectClassification(2010):35-06,35A30,57Q35,58-06,58D10,58D17,58J32,58J60, 58Z99,74A60,74B20,74P20,76A15 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia (www.springer.com) Preface The peer reviewed chapters in these Proceedings are mainly written versions of invited lectures delivered by internationally acknowledged specialists at the ICMS Workshop on “Differential Geometry and Continuum Mechanics” held in Edinburgh from 17 to 21 June 2013. TheaimoftheWorkshopwasinparttoencourageandfosterthestudyofrecent developments intheconceptualfoundationsandtheoreticalstructureofcontinuum mechanics. Modern demands of nanotechnology, special materials, biology and similar applied fields require that continuum mechanics no longer engages solely with predictive numerical solutions and the associated mathematical analysis of classical theories. Identification of basic principles common to all continuum the- oriesandthesubsequentderivationofgeneralmathematicalpropertiesnecessitates arigorouscriticalre-evaluationoftheaxiomaticfoundations,evocativeofHilbert’s sixth problem. Differential geometry is of obvious importance to these investigations. For example,conservationlawsarecloselyrelatedtotheGauss–Codazzi–Riccisystem. Defects can be discussed in a geometric context. The analysis of microstructure involves manifolds and conditions for their isometric embedding into Euclidean (physical)space.Geometricnotionscanbesuccessfullyemployedtomodelsurface energies. These are just some of the topics considered by the 26 speakers at the ICMS Workshop, and discussed in the following chapters. The talks confirmed that the formalism and results of differential geometry crucially underpin recent funda- mental progress in continuum mechanics, while advances in analysis (including C-convergence and compensated compactness), the calculus of variations, and partial differential equations have revealed deep connexions with long-standing problems in differential geometry. The interrelated chapters of the present Proceedings correspond to the Workshop’sprincipalthemes,andfurtheremphasizethecross-fertilisationbetween differential geometry, partial differential equations and continuum mechanics apparent even in the last century. Not included in these Proceedings are the mini-courses presented by Professors M. Epstein and T. Otway who introduced v vi Preface respectivelyappropriatenotionsofdifferentialgeometryandofequationsofmixed type. Both courses are being separately published in the Springer-Brief Series. The ICMS Workshop would have been impossible without generous financial assistance gratefully received from The Centre for Analysis and Nonlinear PDEs (CANPDE), The Oxford Centre for Nonlinear PDE (OxPDE), The London Mathematical Society (LMS), Bridging the Gap-University of Strathclyde (BTG), The Glasgow Mathematical Journal Trust (GMJT), The International Centre for Mathematical Sciences (ICMS). Thanksarealsosincerelyextendedtotheauthorsfortheirwillingcooperationin the timely preparation of contributions, to the referees for their valuable reviews, and to Joerg Sixt and Catherine Waite of Springer for encouragement and advice. It is also an immense pleasure to acknowledge the highly efficient administrative supportfromJaneWalkerandherICMScolleagues.Theoutstandingsuccessofthe Workshopwasinnosmallpartduetotheirconsistentcheerful,patientandfriendly commitment that significantly eased the organisational responsibilities. Oxford Gui-Qiang G. Chen Glasgow Michael Grinfeld Edinburgh R.J. Knops July 2014 Contents Part I General 1 Compensated Compactness with More Geometry. . . . . . . . . . . . . 3 Luc Tartar Part II Differential Geometry 2 Global Isometric Embedding of Surfaces in R3 . . . . . . . . . . . . . . 29 Qing Han 3 Singular Perturbation Problems Involving Curvature. . . . . . . . . . 49 Roger Moser 4 Lectures on the Isometric Embedding Problem ðMn;gÞ!IRm; m¼nðnþ1Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2 Marshall Slemrod Part III Defects and Microstructure 5 Continuum Mechanics of the Interaction of Phase Boundaries and Dislocations in Solids . . . . . . . . . . . . . . . . . . . . . 123 Amit Acharya and Claude Fressengeas 6 Manifolds in a Theory of Microstructures . . . . . . . . . . . . . . . . . . 167 G. Capriz and R.J. Knops vii viii Contents 7 On the Geometry and Kinematics of Smoothly Distributed and Singular Defects. . . . . . . . . . . . . . . . . . . . . . . . . 203 Marcelo Epstein and Reuven Segev 8 Non-metricity and the Nonlinear Mechanics of Distributed Point Defects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Arash Yavari and Alain Goriely Part IV Solids 9 Are Microcontinuum Field Theories of Elasticity Amenable to Experiments? A Review of Some Recent Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Christian Liebold and Wolfgang H. Müller 10 On the Variational Limits of Lattice Energies on Prestrained Elastic Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Marta Lewicka and Pablo Ochoa 11 Static Elasticity in a Riemannian Manifold . . . . . . . . . . . . . . . . . 307 Cristinel Mardare Part V Fluids and Liquid Crystals 12 Calculating the Bending Moduli of the Canham–Helfrich Free-Energy Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Brian Seguin and Eliot Fried 13 Elasticity of Twist-Bend Nematic Phases . . . . . . . . . . . . . . . . . . . 363 Epifanio G. Virga Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Part I General
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