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The Linearized Theory of Elasticity
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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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