Continuum Mechanics using Mathematica®: Fundamentals, Methods, and Applications PDF
Preview Continuum Mechanics using Mathematica®: Fundamentals, Methods, and Applications
Modeling and Simulation in Science, Engineering and Technology Antonio Romano Addolorata Marasco Continuum Mechanics using ® Mathematica Fundamentals, Methods, and Applications Second Edition ModelingandSimulationinScience,EngineeringandTechnology SeriesEditor NicolaBellomo PolitecnicodiTorino Torino,Italy EditorialAdvisoryBoard K.J.Bathe P.Koumoutsakos DepartmentofMechanicalEngineering ComputationalScience&Engineering MassachusettsInstituteofTechnology Laboratory Cambridge,MA,USA ETHZürich Zürich,Switzerland M.Chaplain DivisionofMathematics H.G.Othmer UniversityofDundee DepartmentofMathematics Dundee,Scotland,UK UniversityofMinnesota Minneapolis,MN,USA P.Degond DepartmentofMathematics, K.R.Rajagopal ImperialCollegeLondon, DepartmentofMechanicalEngineering London,UnitedKingdom TexasA&MUniversity CollegeStation,TX,USA A.Deutsch CenterforInformationServices T.E.Tezduyar andHigh-PerformanceComputing DepartmentofMechanicalEngineering& TechnischeUniversitätDresden MaterialsScience Dresden,Germany RiceUniversity Houston,TX,USA M.A.Herrero DepartamentodeMatematicaAplicada A.Tosin UniversidadComplutensedeMadrid IstitutoperleApplicazionidelCalcolo Madrid,Spain “M.Picone” ConsiglioNazionaledelleRicerche Roma,Italy Moreinformationaboutthisseriesathttp://www.springer.com/series/4960 Antonio Romano • Addolorata Marasco Continuum Mechanics using (cid:2) Mathematica R Fundamentals, Methods, and Applications Second Edition AntonioRomano AddolorataMarasco DepartmentofMathematics DepartmentofMathematics andApplications“R.Caccioppoli” andApplications“R.Caccioppoli” UniversityofNaplesFedericoII UniversityofNaplesFedericoII Naples,Italy Naples,Italy Additionalmaterialtothisbookcanbedownloadedfromhttp://extras.springer.com ISSN2164-3679 ISSN2164-3725(electronic) ISBN978-1-4939-1603-0 ISBN978-1-4939-1604-7(eBook) DOI10.1007/978-1-4939-1604-7 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2014948090 MathematicsSubjectClassification:74-00,74-01,74AXX,74BXX,74EXX,74GXX,74JXX ©SpringerScience+BusinessMediaNewYork2006, 2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.birkhauser-science.com) Preface Themotionofanybodydependsbothonitscharacteristicsandontheforcesacting onit.Althoughtakingintoaccountallpossiblepropertiesmakestheequationstoo complextosolve,sometimesitispossibletoconsideronlythepropertiesthathave the greatest influence on the motion. Models of ideals bodies, which contain only themostrelevantproperties,canbestudiedusingthetoolsofmathematicalphysics. Addingmorepropertiesintoamodelmakesitmorerealistic,butitalsomakesthe motionproblemhardertosolve. In order to highlight the above statements, let us first suppose that a system S of N unconstrained bodies C , i D 1;:::;N, is sufficiently described by the i modelofN materialpointswheneverthebodieshavenegligibledimensionswith respecttothedimensionsoftheregioncontainingthetrajectories.Thismeansthat allthephysicalpropertiesofC thatinfluencethemotionareexpressedbyapositive i number,themassm ,whereasthepositionofC withrespecttoaframeI isgiven i i bythepositionvectorr .t/versustime.Todeterminethefunctionsr .t/,onehasto i i integratethefollowingsystemofNewtonianequations: m rR DF (cid:3)f .r ;:::;r ;rP ;:::;rP ;t/; i i i i 1 N 1 N i D 1;:::;N,wheretheforcesF ,duebothtotheexternalworldandtotheother i pointsofS,areassignedfunctionsf ofthepositionsandvelocitiesofallthepoints i ofS,aswellasoftime.Undersuitableregularityassumptionsaboutthefunctionsf , i theprevious(vector)systemofsecond-orderordinarydifferentialequationsinthe unknownsr .t/hasoneandonlyonesolutionsatisfyingthegiveninitialconditions i r .t /Dr0; rP .t /DrP0; i D1;:::;N: i 0 i i 0 i A second model that more closely matches physical reality is represented by a system S of constrained rigid bodies C , i D 1;:::;N. In this scheme, the i v vi Preface extensionofC andthepresenceofconstraintsaretakenintoaccount.Theposition i ofC isrepresentedbythethree-dimensionalregionoccupiedbyC intheframeI. i i OwingtothesupposedrigidityofbothbodiesC andconstraints,theconfigurations i ofS aredescribedbyn(cid:4)6N parametersq ;:::;q ,whicharecalledLagrangian 1 n coordinates.Moreover,themassm ofC isnolongersufficientfordescribingthe i i physicalpropertiesofC sincewehavetoknowbothitsdensityandgeometry.To i determine the motion of S, that is, the functions q .t/;:::;q .t/, the Lagrangian 1 n expressionsofthekineticenergyT.q;qP/andactiveforcesQ .q;qP/arenecessary. h ThenapossiblemotionofS isasolutionoftheLagrangeequations d @T @T (cid:5) DQ .q;qP/; hD1;:::;n; h dt @qP @q h h satisfyingthegiveninitialconditions q .t /Dq0; qP .t /DqP0; hD1;:::;n; h 0 h h 0 h whichonceagainfixtheinitialconfigurationandthevelocityfieldofS. We face a completely different situation when, to improve the description, we adopt the model of continuum mechanics. In fact, in this model the bodies are deformable and, at the same time, the matter is supposed to be continuously distributed over the volume they occupy, so that their molecular structure is completelyerased.Inthisbookwewillshowthatthesubstitutionofrigiditywiththe deformabilityleadsustodeterminethreescalarfunctionsofthreespatialvariables andtime,inordertofindthemotionofS.Consequently,thefundamentalevolution lawsbecomepartialdifferentialequations.Thisconsequenceofdeformabilityisthe rootofthemathematicaldifficultiesofcontinuummechanics. This model must include other characteristics which allow us to describe the different macroscopic material behaviors. In fact, bodies undergo different deformations under the influence of the same applied loads. The mathematical descriptionofdifferentmaterialsistheobjectoftheconstitutiveequations.These equations, although they have to describe a wide variety of real bodies, must in any case satisfy some general principles. These principles are called constitutive axioms and they reflect general rules of behavior. These rules, although they imply severe restrictions on the form of the constitutive equations, permit us to describedifferentmaterials.Theconstitutiveequationscanbedividedintoclasses describing the behavior of material categories: elastic bodies, fluids, etc. The choice of a particular constitutive relation cannot be done a priori but instead relies on experiments, due to the fact that the macroscopic behavior of a body is strictly related to its molecular structure. Since the continuum model erases this structure,theconstitutiveequationofaparticularmaterialhastobedeterminedby experimentalprocedures.However,theintroductionofdeformabilityintothemodel doesnotpermitustodescribeallthephenomenaaccompanyingthemotion.Infact, theviscosityofS aswellasthefrictionbetweenS andanyexternalbodiesproduce heating, which in turn causes heat exchanges among parts of S or between S and Preface vii itssurroundings.Mechanicsisnotabletodescribethesephenomena,andwemust resort to the thermomechanics of continuous systems. This theory combines the lawsofmechanicsandthermodynamics,providedthattheyaresuitablygeneralized toadeformablecontinuumatanonuniformtemperature. The situation is much more complex when the continuum carries charges and currents. In such a case, we must take into account Maxwell’s equations, which describe the electromagnetic fields accompanying the motion, together with the thermomechanic equations. The coexistence of all these equations gives rise to a compatibility problem: in fact, Maxwell’s equations are covariant under Lorentz transformations,whereasthermomechanicslawsarecovariantunderGalileantrans- formations. This book is devoted to those readers interested in understanding the basis of continuum mechanics and its fundamental applications: balance laws, constitutive axioms,linearelasticity,fluiddynamics,waves,etc.Itisself-contained,asitillus- tratesalltherequiredmathematicaltools,startingfromanelementaryknowledgeof algebraandanalysis. Itisdividedinto13chapters.Inthefirsttwochapterstheelementsoflinearalge- braarepresented(vectors,tensors,eigenvalues,eigenvectors,etc.),togetherwiththe foundationsofvectoranalysis(curvilinearcoordinates,covariantderivative,Gauss andStokestheorems).Intheremainingtenchaptersthefoundationsofcontinuum mechanics and some fundamental applications of simple continuous media are introduced. More precisely, the finite deformation theory is discussed in Chap. 3, andthekineticprinciples,thesingularsurfaces,andthegeneraldifferentialformulae for surfaces and volumes are presented in Chap. 4. Chapter 5 contains the general integral balance laws of mechanics, as well as their local Eulerian or Lagrangian forms.InChaps.6and7theconstitutiveaxioms,thethermo-viscoelasticmaterials, and their symmetries are discussed. In Chap. 8, starting from the characteristic surfaces,theclassificationofaquasi-linearpartialdifferentialsystemisdiscussed, together with ordinary waves and shock waves. The following two chapters cover theapplicationofthegeneralprinciplespresentedinthepreviouschapterstoperfect orviscousfluids(Chap.9)andtolinearlyelasticsystems(Chap.10).InChap.11, a comparison of some proposed thermodynamic theories is presented. The great importanceoffluiddynamicsinmeteorologyisshowedinChap.12.Inparticular, in this chapter the arduous path from the equation of fluid dynamics to the chaos is sketched. In the last chapter, we present a brief introduction to the navigation since the analysis of this problem is an interesting example of the interaction betweentheequationsoffluiddynamicsandthedynamicsofrigidbodies.Finally, inAppendixAtheconceptofaweaksolutionisintroduced. This volume has many programs written with Mathematicar [69]. These programs, whose files will be available at Extras.Springer.com., apply to topics discussedinthebooksuchastheequivalenceofappliedvectorsystems,differential operators in curvilinear coordinates, kinematic fields, deformation theory, classifi- cationofsystemsofpartialdifferentialequations,motionrepresentationofperfect viii Preface fluidsbycomplexfunctions,wavesinsolids,andsoon.Thisapproachhasalready beenadoptedbytwooftheauthorsinotherbooks(see[1,33]).1 Manyotherimportanttopicsofcontinuummechanicsarenotconsideredinthis volume, which is essentially devoted to the foundations of the theory. In a second volume[56],alreadyedited,continuawithdirectors,nonlinearelasticity,mixtures, phase changes, electrodynamics in matter, ferromagnetic bodies, and relativistic continuaarediscussed. Naples,Italy AntonioRomano AddolorataMarasco 1The reader interested in other fundamental books in continuum mechanics can consult, for example,thereferences[26,30,36,66]. Contents 1 ElementsofLinearAlgebra............................................... 1 1.1 MotivationtoStudyLinearAlgebra ............................... 1 1.2 VectorSpacesandBases............................................ 2 Examples............................................................ 3 1.3 EuclideanVectorSpace............................................. 5 1.4 BaseChanges ....................................................... 10 1.5 VectorProduct ...................................................... 12 1.6 MixedProduct ...................................................... 13 1.7 ElementsofTensorAlgebra........................................ 14 1.8 Eigenvalues and Eigenvectors of a Euclidean Second-OrderTensor ............................................... 20 1.9 OrthogonalTensors................................................. 25 1.10 Cauchy’sPolarDecompositionTheorem.......................... 28 1.11 HigherOrderTensors............................................... 29 1.12 EuclideanPointSpace.............................................. 31 1.13 Exercises ............................................................ 32 1.14 TheProgramVectorSys............................................. 38 AimoftheProgramVectorSys..................................... 38 DescriptionoftheProblemandRelativeAlgorithm.............. 38 CommandLineoftheProgramVectorSys ........................ 39 ParameterList....................................................... 39 WorkedExamples................................................... 40 Exercises ............................................................ 42 1.15 TheProgramEigenSystemAG ..................................... 42 AimoftheProgramEigenSystemAG.............................. 42 DescriptionoftheAlgorithm....................................... 43 CommandLineoftheProgramEigenSystemAG................. 43 ParameterList....................................................... 43 ix
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