Analytical methods in anisotropic elasticity: with symbolic computational tools

466 Pages·2005·8.464 MB·English

by  Omri Rand, Vladimir Rovenski

About the Authors
Omri Rand is a Professor of Aerospace Engineering at the
Technion – Israel Institute of Technology. He has been involved
in research on theoretical modeling and analysis in the area of
anisotropic elasticity for the last fifteen years, he is the author
of many journal papers and conference presentations in this
area. Dr. Rand has been extensively active in composite rotor
blade analysis, and established many well recognized analytical
and numerical approaches. He teaches graduate courses in the
area of anisotropic elasticity, serves as the Editor-in-Chief of
Science and Engineering of Composite Materials,as a reviewer
for leading professional journals, and as a consultant to various
research and development organizations.
Vladimir Rovenski is a Professor of Mathematics and a well
known researcher in the area of Riemannian and computational
geometry. He is a corresponding member of the Natural Science
Academy of Russia, a member of the American Mathematical
Society, and serves as a reviewer of Zentralblatt für
Mathematik. He is the author of many journal papers and books,
including Foliations on Riemannian Manifolds and
Submanifolds (Birkhäuser, 1997), and Geometry of Curves and
Surfaces with MAPLE (Birkhäuser, 2000). Since 1999,
Dr. Rovenski has been a senior scientist at the faculty of
Aerospace Engineering at the Technion – Israel Institute of
Technology, and a lecturer at Haifa University.
Omri Rand
Vladimir Rovenski
Analytical Methods in
Anisotropic Elasticity
with Symbolic Computational Tools
Birkha¨user
Boston • Basel • Berlin
OmriRand VladimirRovenski
Technion—IsraelInstituteofTechnology Technion—IsraelInstituteofTechnology
FacultyofAerospaceEngineering FacultyofAerospaceEngineering
Haifa32000 Haifa32000
Israel Israel
AMSSubjectClassifications:74E10,74Bxx,74Sxx,65C20,65Z05,68W30,74-XX,74A10,74A40,74Axx,74Fxx,
74Gxx,74H10,74Kxx,74N15,68W05,65Nxx,35J55(Primary);74-01,74-04,65-XX,68Uxx,68-XX(Secondary)
LibraryofCongressCataloging-in-PublicationData
Rand,Omri.
Analyticalmethodsinanisotropicelasticity:withsymboliccomputationaltools/Omri
Rand,VladimirRovenski.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN0-8176-4372-2(alk.paper)
1.Elasticity.2.Anisotropy.3.Anisotropy—Mathematicalmodels.4.Inhomogeneous
materials.I.Rovenskii,VladimirY,1953-II.Title
QA931.R362004
(cid:1)
531.382–dc22 2004054558
ISBN-100-8176-4272-2 Printedonacid-freepaper.
ISBN-13978-0-8176-4372-3
(cid:2)c2005Birkha¨userBoston
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Tomyfamily, Ora, Shahar, Taland Boaz,
OmriRand
Tomyteacher, ProfessorVictorToponogov,
VladimirRovenski
Preface
Priortothecomputerera,analyticalmethodsinelasticityhadalreadybeendevelopedandim-
proveduptoimpressivelevels.Relevantmathematicaltechniqueswereextensivelyexploited,
contributingsignificantlytotheunderstandingofphysicalphenomena.Inrecentdecades,nu-
merical computerized techniques have been refined and modernized, and have reached high
levelsofcapabilities,standardizationandautomation.Thistrend,accompaniedbyconvenient
andhighresolutiongraphicalvisualizationcapability,hasmadeanalyticalmethodslessattrac-
tive,andtheamountofeffortdevotedtothemhasbecomesubstantiallysmaller.Yet,withsome
tenacity,thetremendousadvancesincomputerizedtoolshaveyieldedvariousmatureprograms
forsymbolicmanipulation.Suchtoolshaverevivedmanyabandonedanalyticalmethodologies
byeasingthetediouseffortthatwaspreviouslyrequired,andbyprovidingadditionalcapabil-
itiestoperformcomplexderivationprocessesthatwereonceconsideredimpractical.
Generallyspeaking,itiswellrecognizedthatanalyticalsolutionsshouldbeappliedtorela-
tivelysimpleproblems,whilenumericaltechniquesmayhandlemorecomplexcases.However,
itisalsoagreedthatanalyticalsolutionsprovidebetterinsightandimprovedunderstandingof
theinvolvedphysicalphenomena,andenableaclearrepresentationoftheroletakenbyeachof
theproblemparameters.Nowadays,analyticalandnumericalmethodsareconsideredascom-
plementary: that is, while analytical methods provide the required understanding, numerical
solutionsprovideaccuracyandthecapabilitytodealwithcaseswherethegeometryandother
characteristicsimposerelativelycomplexsolutions.
Nevertheless,fromapracticalpointofview,analyticsolutionsarestillconsideredas“art”,
whilenumericalcodes(suchascodesthatarebasedonthefinite-elementmethod)seemtooffer
a“straightforward”solutionforanytypeandgeometryofanewproblem.Oneofthereasons
forthisviewemergesfromthevarietyoftechniquesthatareusedforanalyticalsolutions.For
example,onehastheoptiontoselecteitherthedeformationfieldorthestressfieldtoconstruct
theinitialsolutionhypothesis,or,onehastheoptiontoformulatethegoverningequationsusing
differential equilibrium, or by employing more integral energy methodologies for the same
task.Hence, themainobstacletousinganalytical approaches seemstobethefactthatmany
researchersandengineerstendtobelievethat,asfarasanalyticsolutionsareconsidered,each
problem is associated with a specific solution type and that adifferent solution methodology
hastobetailoredforeverynewproblem.
In light of the above, the objective of this book is twofold. First, it brings together and
refreshes the fundamentals of anisotropic elasticity and reviews various mathematical tools
and analytical solution trails that are encountered in this area. Then, it presents a collection
viii Preface
of classical and advanced problems in anisotropic elasticity that encompasses various two-
dimensional problems and different types of three-dimensional beam models. The book in-
cludesmodelsofvariousmathematicalcomplexityandphysicalaccuracylevels,andprovides
thetheoreticalbackgroundforcompositematerialanalysis.Oneofthemostadvancedformu-
lationspresentedisacompleteanalyticalmodelandsolutionschemeforanarbitrarilyloaded
non-homogeneousbeamstructureofgenericanisotropy.
All classic and modern analytic solutions are derived using symbolic computational tech-
niques. Emphasis is put on the basic principles of the analytic approach (problem statement,
settingofsimplifyingassumptions,satisfyingthefieldandboundaryconditions,proofofsolu-
tion,etc.),andtheirimplementationusingsymboliccomputationaltools,sothatthereaderwill
beabletoemploytherelevantapproachtonewproblemsthatfrequentlyarise.Discussionsare
devotedtothephysicalinterpretationofthepresentedmathematicalsolutions.
From a format point of view, the book provides the background and mathematical formu-
lation for each problem or topic. The main steps of the analytical solution and the graphical
resultsarediscussedaswell,whilethecompletesystemofsymboliccodes(writteninMaple)
areavailableontheencloseddisc.
A unique characteristic of this book is the fact that the entire analytical derivation and all
solution expressions are symbolically proved by suitable (computerized) codes. Hence, the
chancefor(human)errorortypographicalmistakeiseliminated.Thesymbolicworksheetsare
therefore absolute and firm testimony to the exactness of the presented expressions. For that
reason, the specific solutions included in the text should be viewed as illustrative examples
only,whilethesolutionexactnessanditsgenericapplicabilityareprovedsymbolicallyinthe
mostgenericmanner.
Thebookisaimedatgraduateandseniorundergraduatestudents,professors,engineers,ap-
pliedmathematicians,numericalanalysisexperts,mechanicsresearchersandcompositemate-
rialsscientists.
Chapterdescription:
Thefirstpartofthebook(Chapters1–4)containsthefundamentalsofanisotropicelasticity.
Thesecondpart(Chapters5–10)isdevotedtovariousbeamanalysesandcontainsrecentand
advancedmodelsdevelopedbytheauthors.
Chapter1addressesfundamentalissuesofanisotropicelasticityandanalyticalmethodolo-
gies.Itprovidesareviewofdeformationmeasuresandstrainingenericorthogonalcurvilinear
coordinates, and reaches the complete nonlinear compatibility equations in such systems. It
then introduces fundamental stress measures and the associated equilibrium equations. Later
on,energytheoremsandvariationalanalysesarederived,followedbyageneraldiscussionof
analyticalmethodologiesandtypicalsolutiontrails.
Chapter2reviewsthemathematicalrepresentationofgeneralanisotropicmaterials,includ-
ingthespecialcasesofMonoclinic,Orthotropic,Tetragonal,TransverselyIsotropic,Cubicand
Isotropicmaterials.Later,transformationsbetweencoordinatesystemsofthecomplianceand
stiffnessmatrices(ortensors)arepresented.Thechapteralsoaddressesissuessuchasplanes
ofelasticsymmetry,principaldirectionsofanisotropyandnon-Cartesiananisotropy.
Chapter 3 defines two-dimensional homogeneous and non-homogeneous domain topolo-
gies,andpresentsvariousplanedeformationproblemsandanalyses,includingdetailedformu-
lationofplane-strain/stressandplane-shearstates.Thederivationyieldsformaldefinitionsof
generalizedNeumann/Dirichletandbiharmonicboundaryvalueproblems(BVPs).Thechapter
Preface ix
alsoaddressesCoupled-PlaneBVPformaterialsofgeneralanisotropy.Alongthesamelines,
theclassicalanisotropiclaminatedplatetheoryisthenpresented.
Chapter4presentsvarioussolutionmethodologiesfortheBVPsderivedinChapter3,and
establishes solution schemes that facilitate applications presented later on. Explicit analytic
expressionsforlow-orderexact/conditionalpolynomialsolutions,andapproximatehigh-order
polynomialsolutionsinahomogeneoussimplyconnecteddomainarederivedandillustrated.
A formulation based on complex potentials is also thoroughly derived and demonstrated by
Fourierseriessolutions.
Chapter5reviewssomebasicaspectsandgeneraldefinitionsofanisotropicbeamanalysis,
approximate analysis techniques and relevant literature. It discusses the associated coupling
characteristicsatboththematerialandstructurallevels.
Chapter6presentsananalysisofgeneralanisotropicbeamsthatmaybeviewedasalevel-
based extension of the classical Lekhnitskii formulation, and is capable of handling beams
ofgeneralanisotropyandcross-sectiongeometrythatundergogenericdistributionofsurface,
body-forceandtiploading.ThederivationisfoundedontheBVPsdeducedinChapter3,and
despite its complexity, it provides a clear insight into the associated structural behavior and
couplingmechanisms.
Chapter 7 contains a closed-form formulation for uncoupled monoclinic homogeneous
beams.Itfirstpresentssolutionsfortiploads,andthenagenericformulationforaxiallynon-
uniform distribution of surface and body loads. Later on, analysis and examples of beams of
cylindricalanisotropyarepresented.
TheentirereasoningoftheapproachinthischapterisfoundedonSt.Venant’ssemi-inverse
method of solution and may be considered as dual (though less generic) to the method pre-
sentedinChapter6.
Chapter 8 is focused on problems in various non-homogeneous domains. It first reviews
generic formulations of plane BVPs, and then extends the analysis of Chapter 7 to the case
ofmonoclinicnon-homogeneousbeamsundertiploading,whichisfoundedonextendingthe
classicaldefinitionoftheauxiliaryproblemsofplanedeformationtotheanisotropiccase.The
discussionencompassesthedeterminationoftheprincipalaxisofextension,principalplanes
of bending and shear center. The chapter also presents a generalization of the derivation in
Chapter7tothecaseofuncouplednon-homogeneousbeamsthatundergogenericdistributed
loading.
Chapter9discussescoupledsolidmonoclinicbeams.Theanalysispresentsanapproximate
modelthatprovidesinsightintoandfundamentalunderstandingofthecouplingmechanisms
withinanisotropicbeamsatthestructurallevel.Themodelalsosuppliesasimplifiedbutrela-
tivelyaccuratetoolforquantitativeestimationofcoupledbeambehavior.
Inaddition, the chapter presents anexact, level-based solutionscheme forcoupled beams.
The derivation employs a series of properly interconnected solution levels and reaches the
exactsolutioninaniterativemanner.
Chapter10 handlescoupledthin-wallmonoclinicbeamsinasimilar(approximate)manner
to Chapter 9. The analysis encompasses beams having either multiply connected domain
(“closed”)orsimplyconnecteddomain(“open”)cross-sections.
Chapter11presentsinstructionsforthesymbolicandillustrativeprogramsincludedinthis
book(implementedinMaple).
x Preface
GeneralStyleClarificationNotes:
(cid:1)
(1) As a general rule, we use a “tilde” (e.g. A), for “temporary” variables that have no
meaningfulrole,andareintroducedforthesakeofclarificationandanalyticconvenience.Such
variablesarevalid“locally”withintheimmediateparagraphsinwhichtheyappearin.Hence,
ifsuchanotationisrepeatedelsewhere,itstandsforadifferent“local”meaning;likewise,the
superscript()(cid:1)mayhavedifferentmeaningsinvariouscontexts.
(2) Duetothedependencyofmostinvolvedfunctionsonmanyparameters,bothordinary
andpartialderivatives,sayd()/dαor∂()/∂α,areabbreviatedas(),α.Similarly,d2()/dα2
or∂2()/∂α2,areabbreviatedas(),αα.
(3) Integralsappearinashortnotationbyomittingtheexplicitindicationoftheintegration
variables.Twoexamplesareanintegrationalongaclosedloopwithacircumferentialcoordi-
(cid:2) (cid:2)
(cid:1) (cid:1)
nate,s,i.e., Fds,whichiswrittensimplyas F,andtheareaintegrationinthexy-plane,
(cid:3)(cid:3) ∂Ω (cid:3)(cid:3) ∂Ω
(cid:1) (cid:1)
i.e., Fdxdy,whichiswrittenas F.
Ω Ω
(4) Within the equation notation, e.g. (1.3), the first digits stand for the chapter in which
it appears, while (1.15a) is an example for an equation in a group of (sub-)equations. By a
notationlike(1.9a:b)werefertothesecondequationofagroupofequations thatappearsin
onelinethatiscollectivelydenoted(1.9a).
(5) WithintheSection,Program,Remark,Example, FigureandTablenotation,e.g.S.1.2,
P.1.2, Remark 1.2, Example 1.2, Fig. 1.2, Table 1.2, the first digit stands for the chapter in
whichitappears.
Acknowledgements
We wish to acknowledge the great help of Dr. Michael Kazar (Kezerashvili) who had a
unique role in exposing us to some great contributions to this science made by the Eastern
academiadiscussedinChapters7,8.
WearealsothankfultothePh.D.studentMichaelGrebshteinwhomadeatremendouscon-
tributiontotherigor,theanalyticaluniformityandthesymbolicverificationofthederivation
inChapters4,7,8.
WewarmlythankAnnKostant,ExecutiveEditorofMathematicsandPhysicsatBirkha¨user
Boston,forhersupportduringthepublishingprocess.
OmriRand
VladimirRovenski Haifa,Israel