Table Of ContentModeling 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. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
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
