Geometry of Incompatible Deformations: Differential Geometry in Continuum Mechanics

411 Pages·2019·5.277 MB·English

by  Sergey Lychev, Konstantin Koifman

SergeyLychevandKonstantinKoifman
GeometryofIncompatibleDeformations
De Gruyter Studies in
Mathematical Physics
|
Editedby
MichaelEfroimsky,Bethesda,Maryland,USA
LeonardGamberg,Reading,Pennsylvania,USA
DmitryGitman,SãoPaulo,Brazil
AlexanderLazarian,Madison,Wisconsin,USA
BorisSmirnov,Moscow,Russia
Volume 50
Sergey Lychev and Konstantin Koifman
Geometry
of Incompatible
Deformations
|
Differential Geometry in Continuum Mechanics
PhysicsandAstronomyClassificationScheme2010
Primary:46.05.+b;Secondary:02.40.Yy
Authors
Prof.SergeyLychev
IshlinskyInstituteforProblemsinMechanicsRAS(IPMechRAS)
[email protected]
KonstantinKoifman
DepartmentofAppliedMathematics
BaumanMoscowStateTechnicalUniversity
[email protected]
ISBN978-3-11-056201-9
e-ISBN(PDF)978-3-11-056321-4
e-ISBN(EPUB)978-3-11-056227-9
ISSN2194-3532
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
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detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
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Preface
Thisbookisintendedtoprovideasystematictreatmentofthosepartsofmoderndiffer-
entialgeometrythatareessentialforthemodelingofincompatiblefinitedeformations
insolids.Includedarediscussionsofgeneralizeddeformationandstressmeasureson
smoothmanifolds,geometricalformalizationforstructurallyinhomogeneousbodies,
variousdefinitionsformaterialconnection,andevolutionequationsforthem.
Theincompatibilityofdeformationsmaybecausedbyavarietyofphysicalphe-
nomena; among them aredistributed dislocationsand disclinations,point defects,
non-uniformthermalfields,shrinkage,growth,etc.Incompatibledeformationsresult
inresidualstressesanddistortionofgeometricalshape.Thesefactorsareassociated
withcriticalparametersinmodernhigh-precisiontechnologies,particularly,inaddi-
tivemanufacturing,andareconsideredtobeipsofactoessentialconstituentsinthe
correspondingmathematicalmodels.Inthiscontext,thedevelopmentofmethodsfor
theirquantitativedescriptionistheactualproblemofmodernsolidmechanics.The
methodsinquestionarebasedontherepresentationofabodyandphysicalspacein
termsofdifferentiablemanifolds,namelymaterialmanifoldandphysicalmanifold.
Thesemanifoldsareequippedwithspecificmetricsandconnections,non-Euclidian
ingeneral.
The book isdivided into 14 chapters. The first one is an introduction. The sec-
ondonebrieflyreviewsvariousrepresentationsofthegeometryofphysicalspaceand
time,includingEuclidean,Minkowski, andcurvedspacetimemodels. Thisreview,
ontheonehand,leadstoexhaustivedefinitionofphysicalspaceintermsofsmooth
manifolds.Ontheotherhand,itdrawsattentiontothespecificaspectsofgeometri-
calformalizationforspaceandspacetime,whichhavemuchincommonwiththeir
counterpartsinthemodelingofelasticbodies,suchassmooth(material)manifolds.
Thethirdchapterfocusesontheessentialsofconventionalnon-linearelasticity.
Inthischapter,those fundamentalsofcontinuummechanicsthatarerooted inthe
conceptofabsolute,Euclidean,spaceandabsolute,Newtonian,timearediscussed.
Specialattentionispaidtotheassumptionthatglobalstress-freeshapesofanybody
exists.Rejectionofthisassumptionreflectsthemainideaofthepresentbook.Com-
monnotationsforstrainandstressmeasures,variationalsymmetries,andconstitu-
tiveequations,whichwouldbegeneralizedintherestofthebook,arehighlighted.
Inthefourthchapter,weconsidertheissueofthephysicalinterpretationofthe
non-Euclideanstructureofthematerialmanifold.Itisshownthatatwo-dimensional
rigidsurface,whichformalizesthecurvedsubstrateusedinthedepositionprocess,
may serve as an example of a non-Euclideanphysical manifold. Affine connection
onthematerialmanifoldrepresentstheintrinsicproperties(innergeometry)ofthe
body and is determined by the field of local uniform configurations performing its
“assembly” of identical and uniform infinitesimal “bricks”.Uniformity means that
theresponsefunctionalgivesthesameresponseonalladmissiblesmoothdeforma-
https://doi.org/10.1515/9783110563214-201
VIII | Preface
tionsforthem.Asaresultofassembling,oneobtainsabodythatcannotbeimmersed
inanundistortedstateintothephysicalmanifold.Itisanessentialfeatureofresid-
ualstressedbodiesproducedbyadditiveprocesses.Forthisreason,itshouldbenefit
fromimmersionintoanon-Euclideanspace(materialmanifoldwithnon-Euclidean
materialconnection).Tothisend,itisconvenienttoformalizethebodyandphysical
spaceintermsofthetheoryofsmoothmanifolds.Thedeformationisformalizedas
embedding(or,inaspecialcase,asimmersion)oftheformermanifoldintothelatter
one.
The fifth chapter is dedicated to generalization of relations for Cauchy–Green
strainmeasures.TheyaregeneratedbyembeddingofaRiemannianmanifold,repre-
sentingthebody,intoaRiemannianmanifold,representingthespace.Weconsider
suchissues asthe transposeof the deformationgradientand generalizationof the
Cauchydecompositiontheoremonsmoothmanifolds.
Thesixthchaptercoversthedefinitionofmotion,velocity,andaccelerationfields
intheframeworkofthetheoryofsmoothmanifolds.Themotionisrepresentedasa
time-dependentflow.
Theseventhchapteraddressestheissuesoftheformalizationofstressandpower
measuresbyfieldsdefinedonsmoothmanifoldsthatrepresent thebodyandphys-
ical space. It contains the systematic construction of a theory for the general non-
Euclideancase.Forcesareinterpretedascovectors,i.e.,asalinearfunctionals,whose
actiononthevelocityvectorsofmaterialpointsresultsinmechanicalpower.Theab-
stracttheory ofintegrationbased onexterior form formalismisadapted tothe ele-
mentsofthisstructure, whichallowsonetoformulatethepower balanceequation
ofthe materialmanifold (similarly to the reference descriptioninthe classicalme-
chanicsofcompatibledeformation)andofthephysicalone(similarlytothespatial
description).
In the eighth chapter the response of hyperelastic solids on smooth manifolds
isgeneralized.Onlysimplematerialsareconsidered:theirresponsedependsonthe
local configuration and material points. For such materials, the notion of material
isomorphismisintroduced.Themainassumptionthatleadstothenotionofastruc-
turallyinhomogeneoussolidisthefollowing:foreachbodyofsimplematerial,there
exists a family of configurations, which index set is identicalto the set of material
pointsconstitutingthewholebody.Eachconfigurationmapsitsindexpointtotheuni-
formstate.Thisassumptionisusedforsynthesizingthematerialmetric.Amethodis
proposedfordescribingadeformablebodyofvariablecompositionasafamilyofRie-
mannianmanifoldsover whichpartitionandjoiningoperationsaredefined. These
operationscharacterizethe structuralfeatures ofthe inhomogeneities given by the
additiveprocessscenario.
Intheninthchapter,thevariouswaystospecifythegeneralformofaffinecon-
nections on the materialand physicalmanifoldsare considered. Affine connection
endowsmanifoldswithgeometricproperties,inparticular,withparalleltranslation
rulesforvectorfields.Forsimplematerialstheparalleltranslationisanelegantmath-
Preface | IX
ematicalformalizationof the concept of a materially uniform (inparticular,stress-
free)non-Euclideanreferenceshape.Infact,onecanobtainaconnectiononphysical
spacebydeterminingtheparalleltransportruleasatransformationofthetangent
vector,whichcorrespondstothestructureofthephysicalspacecontainingshapesof
thebody.Inturn,onecanobtainaffineconnectiononmaterialmanifoldbydefining
aparalleltransportruleasthetransformationofthetangentvector,inwhichitsin-
verseimagewithrespecttolocallyuniformembeddingsdoesnotchange.Utilizingthe
conceptionofmaterialconnectionsandthecorrespondingmethodsofnon-Euclidean
geometrymaysignificantlysimplifyformulationoftheinitialboundaryvalueprob-
lemsofthetheoryofincompatibledeformations,sothechoiceofconnectionsisim-
portant.Connectiononthephysicalmanifoldiscompatiblewiththemetric,andthe
Levi–Civitarelationholdsforit.Connectionofthematerialmanifoldisconsideredin
twoalternativevariants.ThefirstleadstotheWeitzenböckspace(thespaceofabso-
luteparallelismorteleparallelism,i.e.,spacewithzerocurvatureandnon-metricity
butwithnon-zerotorsion)andgivesaclearinterpretationofthematerialconnection
intermsofthelocallineartransformationsthatreturnanelementaryvolumeofsimple
materialtotheuniformstate.ThesecondoneleadstotheRiemannianspace(space
withzeronon-metricityandtorsionbutwithnon-zerocurvature).Examplesforsuch
generalizationsareconsideredintherestofthechapter.
Thetenthchapterreferstothebalanceequations,whichareobtainedintermsof
Cartan’sexteriorcovariantderivativefromthegeneralprincipleofcovariance.
Theeleventhchaptercontainstheexamplesofinhomogeneoussolids,whichin-
homogeneitywasinducedbysomeadditivetechnologicalprocess.Varioustypesof
evolutionaryproblems,owningtodifferenttechnologicalregimes,areconsidered.The
calculationsareillustratedbynumericalcomputationsandgraphs.
In the last chapters, the required mathematical preliminaries adapted to the
presentbookareconsidered.
May2018,Moscow SergeyLychev
KonstantinKoifman