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[资源] The Linearized Theory of Elasticity

Preface
Thisbookis derivedfrom notesusedin teachingafirst-year graduate-level
course in elasticityin the DepartmentofMechanicalEngineeringatthe
UniversityofPittsburgh. Thisis a moderntreatment ofthe linearized
theory ofelasticity, whichis presentedas aspecialization ofthe general
theory ofcontinuummechanics.Itincludes acomprehensiveintroduction
to tensor analysis,arigorous developmentofthe governingfield equations
withanemphasisonrecognizing the assumptionsandapproximationsinherentin the linearized theory, specificationofboundaryconditions,and
asurveyofsolutionmethodsfor important classesofproblems. Two-and
three-dimensional problems, torsion ofnoncircularcylinders, variational
methods,andcomplexvariablemethodsarecovered.
Thisbookis intended asthe text for afirst-year graduatecoursein mechanicalorcivilengineering. Sufficientdepthis providedsuchthatthetext
canbeusedwithoutaprerequisitecoursein continuummechanics,andthe
materialis presentedin suchawayasto preparestudentsfor subsequent
coursesin nonlinearelasticity, inelasticity,andfracture mechanics.Alternatively,for acoursethatis precededbyacoursein continuummechanics,
thereis enoughadditionalcontentfor afullsemesteroflinearizedelasticity.
Itis anticipatedthat studentswillmostlyhaveundergraduatemechanicalorcivilengineeringbackgrounds,withthe mathematicaltraining that
entails. Suchstudentshaveusuallynotbeenexposedto modernreal analysisorto abstractvectorspaces, for instance. Thishasnecessarilyhad
animpact onthe mannerin whichthe materialin this bookis presented.
Anattempthasbeenmadenotto introduce asurfeitofunfamiliarmathematicalnotation. Forexample,the reader willnotfind anymathematical
expressionslike
(IR x1R) 3 (x, y)t---> x
2+y2 EIR .
Additionally,it is deemedworthwhileto spendalittle extratime onindicialnotationandtensors-students whodonotmasterthese conceptswill
increasingly find it impossible to follow the rest ofthe material.
Whenis the besttime to introduce the linearizing assumptions? This
is an important questionwhenteaching linear elasticity. Traditionally,
xv
XVI Preface
the linearization hasbeenintroduced assoonaspossible[e.g., Sokolnikoff
(1956) andTimoshenkoandGoodier(1970)]. Thisapproachhasthevirtue
ofallowingonetomoveontosolutionmethodsveryquickly. Analternative
is todevelopcompletelythe nonlineartheory ofelasticitypriorto linearizing [e.g., AtkinandFox(1980) andSpencer(1980)]. Thisgivesstudents
abroadframework that willservethem wellwhenthey take othercourses
that address related topics such as fluid dynamics and inelasticity, but
scarcely leaves time to learn howto solve the important linear elasticity
problemsthat arisein engineering. Perhapsthe bestofallworldsis onein
whichstudentsfirst take anintroductory coursein continuummechanics,
followed byspecializedclassesin elasticity,fluid dynamics,inelasticity, and
soforth. Unfortunately,therealitiesofmanpowerandteachingloads mean
that addinganadditionalintroductory coursein continuummechanicsis
oftennota practicaloption. Consequently,anattempt hasbeenmade
hereto strikeahappymiddleground. Theintroduction oflinearizing assumptionsin this bookis delayedlong enoughto providestudentswitha
contextfrom whichthey canseethe relationshipsthatexistbetweenlinear
elasticityandotherrelated subjectsandstillhavetime in aone-semester
courseto exploresomeofthe important classesofproblemsandsolution
methods.
In the analysis ofkinematics and measuresofstress, referential (Lagrangian)andspatial (Eulerian) formulations havebeenpresentedseparately. Theviewpointtaken is that linear elasticityis mostnaturallyseen
asalinearizationofthereferentialformulation, withfields in the linearized
theory viewedas beingoverthe reference configurationofthe body. If
desired,thesectionsin whichthespatialformulationsarepresentedcanbe
omittedwithminimaldisruption.
Theso-called "Gibbsnotation"for tensor analysishasbeenusedinstead
ofthe "Riccinotation"favored bymanyauthorsin continuummechanics
[e.g., TruesdellandNoll(1992)]. Forexample,thebilinearformofasecondordertensor T withrespect to the vectors u and v (in that order) is
givenasu·T·vratherthan u·(Tv). Itis the author'sopinionthat the
Gibbsnotationmakesit easierfor studentswhoarenewto the subjectto
graspthe conceptsthat aremostimportant atthis level, eventhough it
mayobscuresomeofthe moresubtle issues involving the compositionof
linear operators,Cartesianproducts,abstractvectorspaces, andthe like.
Similarly,the dyad(or tensor product)formed bytwo vectorsu andv
is givenasuvrather that u\51 v,sothat the dyadicrepresentation ofthe
second-ordertensor Tinanorthonormalvectorbasisis T=Tijeiejrather
than T=Tijei\51 ej.
Asmuchasis practical,resultsarepresentedinbothabasis-independent
tensorialform andabasis-dependentscalarcomponentform. Forinstance,
Preface
the traction-stress relation derivedin Chapter4is givenas
XVll
Indoingthis, anorthonormalvectorbasisand,whennecessary,aCartesian
coordinatesystemarepresumed.Itis felt thatstudentsareoverwhelmedby
atoo earlyintroduction to generalcurvilinearcoordinatesand,sincethey
arenotrequired for the applicationscoveredin this book,they havebeen
relegated to an appendix. Cylindricalandspherical coordinatesystems
are treated explicitly, rather than as special cases ofgeneralcurvilinear
coordinates. Thetensor notationreinforces the fact that the underlying
physicalprinciplesarevalidin anycoordinatesystem.
Themechanicsofmaterials,aspresentedto sophomoreengineeringmajors in a typical undergraduateprogramin the UnitedStates,is briefly
reviewed in Chapter1. Thismaterialsets the stage, in some sense, for
whatfollows, butmaybeomitted. Chapter2acquaintsthe studentwith
the notationandconventionsthat areto beused,introduces the concept
ofindicial notation, anddevelops the tensor analysis. Thefoundations
for the linearized theory ofelasticity are developedin Chapters3to 6.
Theremaining chapterscoversolutionmethodsfor avarietyofclassesof
problemsrangingfrom two-dimensional antiplanestrainproblemstothreedimensionalproblemsinvolving dissimilarinclusions. Theorderin which
they arecoveredis somewhatarbitrary,exceptthatChapter11oncomplex
variablemethodsassumesthatChapter7ontwo-dimensional problemshas
beencovered.
Pittsburgh,Pennsylvania WilliamS. Slaughter

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