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Foundations of Engineering Mechanics Sergey A. Ambartsumian (posthumous) Micropolar Theory of Shells and Plates Foundations of Engineering Mechanics SeriesEditors VladimirI.Babitsky,SchoolofMechanical,ElectricalandManufacturing Engineering,LoughboroughUniversity,Loughborough,Leicestershire,UK JensWittenburg,Karlsruhe,Germany Moreinformationaboutthisseriesat http://www.springer.com/series/3582 Sergey A. Ambartsumian (posthumous) Micropolar Theory of Shells and Plates SergeyA.Ambartsumian(posthumous) Yerevan,Armenia ISSN1612-1384 ISSN1860-6237 (electronic) FoundationsofEngineeringMechanics ISBN978-3-030-71325-6 ISBN978-3-030-71326-3 (eBook) https://doi.org/10.1007/978-3-030-71326-3 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland DedicatedtomylatewifeSedawhowas,and remains,aconstantsourceofsupportand inspiration inmylife,mywork,andmy writings. Preface Thisbookfocusesontheappliedtheoryofshellsandplatesbasedontheunderlying assumptionsandequationsofasymmetricalelasticitytheory. This is not a new problem, but it has taken on a different meaning in recent years due to the extensive use of polycrystalline and composite materials in the micromechanicsofthinbodies. The theories proposed in this book represent intermediate links in a long chain of various theories on shells and plates. They have been constructed, on the one hand, based on theequations of theclassicaltheory of elasticity,and on theother, on equations that have thus far remained ‘untouched’ in their application to the moleculartheoriesofsoliddeformablemedia. This book has been written in one sitting, based on the results of the research undertakenbyme.Unfortunately,Iwasunabletocollect,analyze,andsummarize alltheknownresultsinthenamedareaofthetheoryofshellsandplates.Thisgaphas beenpartiallyfilledbyabriefliteraturereviewthatisincludedwiththeintroduction. The book contains certain unspecified and insufficiently rigorous provisions. I hopethattheywillbecompletedatalaterdatebyeithermyselforbyreaderswho areinterestedintheproblemslaidoutthroughout.IfinditonlyfittingtoquoteOliver Heaviside,whosaid,‘WillIgiveupmylunchjustbecauseIdon’tfullyunderstand digestion?’ Thebookconsistsofsixchapters.Thefirstchapterisareviewoftheessentialdata onthenon-symmetrictheoryofelasticity.Thesecondandthirdchaptersaredevoted tovarioustheoriesofplatebendingandsolutionstosomebasicproblems.Chapter four refers to membrane or, so-called, momentary shell theory. Chapter five deals withthetheoryofveryshallowshells.Finally,chaptersixpresentsthegeometryof thenonlineartheoryofplatesandthetheoryofveryshallowshells. Thebookisintendedforresearchers,postgraduatestudentsandengineersinter- estedintheproblemsofmechanicsofdeformablebodies,inthetheoryofshellsas wellasplatetheoryanditsapplicationsinmicromechanics. The book was initially reviewed by Prof. M. V. Belubekyan, who made a few valuablesuggestions,whichIgladlyaccepted. vii viii Preface I consider it my duty to express my deep gratitude to I. R. Arutyunyan and L.V.Melikyan,whoprovidedgreatassistanceinpreparingthebookforpublication. Copenhagen,Denmark/ SergeyA.Ambartsumian(posthumous) Yerevan,Armenia March1999 Preface to the Second Edition The main reason for publishing a new edition of this book is that the first edition wasdedicatedtosomeoneIcaredeeplyabout,andduetoanoversightonbothmy side and that of the publishers, the first version was published with some (mainly typographical)omissions. Asinthepreviousedition,Idedicatethisbooktomylatewife,Seda,towhomI oweadeptofgratitude. I would like to add that due to the limited printing and circulation of the first edition, the elements presented in this book have been largely absent from the field of academia and global mechanics. Taking advantage of this, some uneth- ical researchers, who previously openly stated (and wrote) that the problem of constructingmodelsofmicropolarthinbeams,slabsandshellswasfirstdiscovered byS.A.Ambartsumian,stoppedreferencingthisbookandotherrelevantpublica- tionswrittenbymeandmyco-authors.Assuch,theygavethefalseimpressionthat theythemselveswerepioneersinthisfieldofscience. In the new edition of the book, we seek to correct the inaccuracies and irregu- laritiesofthefirsteditionandtoaddtheresultsofnewfindingsthatmyco-authors and I discovered through our subsequent research. We have avoided making any fundamentalchangestothetext. Followingthepublicationofthisbookandotherarticles[1–5],M.V.Belubekyan, andIwroteanadditionalbooktitled‘AppliedMicropolarTheoryofElasticShells’ [1].Thisbookismainlybasedonourpreviousworksanddoesnotinanywaylimit thepossibilityofintroducingalternativetheoriesofmicropolarplatesandshells. ProfessorM.V.Belubekyan’scontributiontothepublicationofthissecondedition wassignificantandIwouldliketoexpressmysincerethankstohim. I am also very grateful to my colleagues at the Institute of Mechanics of the NationalAcademyofSciencesofArmeniaG.Z.GevorkyanandJ.A.Avdalyanfor theirhelpinpreparingthebookforpublication. Yerevan,Armenia SergeyA.Ambartsumian(posthumous) March2013 ix x PrefacetotheSecondEdition References 1. Ambartsumian,S.A.&Belubekyan,M.V.(2010).Appliedmicropolartheoryofelasticshells. Yerevan:“Gitutyun”NASofArmenia,136p. 2. Ambartsumian, S. A. & Belubekyan, M. V. (2008). Oscillation of elastic plates with the accountofinternalrotation.YSUAS.Architectureandconstruction.Vol.3.pp.25–29. 3. Ambartsumian,S.A.&Belubekyan,M.V.(2009).Theproblemofaxisymmetricvibrations ofamicropolarcircularcylindricalshell.Izv.NASofArmenia.Mechanics.Vol.62.No.3. pp.11–16. 4. Ambartsumian,S.A.&Belubekyan,M.V.(2010).Onthemicropolartheoryofveryshallow shells.Izv.NASofArmenia.Mechanics.Vol.63.No.3.pp.23–32. 5. Ambartsumian,S.A.&Belubekyan,M.V.(2012).Vibrationsofaveryshallowspherical panelwiththeaccountofinternalrotation.BulletinoftheEngineeringAcademyofArmenia. 9(1),101–106.