Theory of Elasticity by Aldo Maceri

Theory of Elasticity by Aldo Maceri

TheTheoryofelasticitystudiesthebehaviorofthosebodiesthatrecovertheiri- tial state when the causes which produce deformations are removed. Its results constitutethefoundationsofthe Theory of structuresandthenareofmaximum importanceforengineers. The Theory of elasticity moves freely within an uni ed mathematical fra- workthatprovidestheanalyticaltoolsforcalculatingstressesanddeformationsin astrainedelasticbody. Alltheelasticproblemscanbeexactlyanalyzedemploying theclassicalMathematicalanalysis, withtheexceptionoftheunilateralproblems forwhichtheemploymentoftheFunctionalanalysisismandatory. TheTheoryofelasticitywasfoundedbythefamousmathematicianCauchyinthe eighteenth-century. Duringitshistoricaldevelopmentthisscienti csectorproposed tothemathematiciansvariousproblemsthathavecontributedorentirelygenerated thedevelopmentofcomplexmathematicaltheories, astheVariationalcalculusand theFiniteelementmethod. Thematteranalyzedinthisbookis three-dimensional problems (Chap. 1), and particularly the problem of Saint Venant(Chap. 1), two-dimensionalproblems, aspanels, plates, shells(Chap. 3), one-dimensionalproblems, asropes, beams, arches(Chap. 4), thermalstressproblems(Chap. 5), stabilityproblems(Chap. 6), anisotropicproblems, thatconstitutethebasictoolfortheanalysisofstructuresin compositematerial(Chap. 7), nonlinearelasticproblems, as niteelasticityandunilateralproblems(Chap. 8). InthisbookIhaveconstantlykeptinmindthepracticalapplicationoftheth- reticalresults. SoIhavealwaystriedtogivetoengineers, inasimpleform, aclear indicationofthenecessaryfundamentalknowledgeoftheTheoryofelasticity. In thepastsometechniquesofcalculationweredevelopedforparticularelasticpr- lemsthatcannotbeorganizedinmathematicaltheoriesbutareextremelysimpleto apply. Suchtechnicaltheorieshavealwaysfurnishedresultsexperimentallyveri ed v vi Preface withgoodapproximationandthenamongthemIhavepresentedthosethatarestill usefultoolsofveri cationintheStructuraldesign. Throughouttheanalysisoftheelasticproblemsmyconstantfocushasbeento achievethemaximumclarityandbecauseofthisIhavesacri cedvariousbright discussions. Ihavedevelopedthetreatmentofthesubjectsinclassicalway, butto thelightofthemodernMathematicaltheoryoftheelasticityandwithmoreaccented relief to the connections with the Thermodynamics. Just for this, to give a clear justi cationofthefundamentalequationoftheThermoelasticityIhaveapplieda techniqueofanalysisproperoftheFluiddynamics. Howeverinthediscussionof theunilateralproblems, wheretheFunctionalanalysisiscompulsory, Ihaverelated indetailsthemathematicalaspectsofthetheoreticalanalysis. Roma, Italy AldoMaceri October2009 Contents 1 The Three-Dimensional Problem. . . . . . . . . . . . . . . . . . . . 1 1. 1 AnalysisofStrain. . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1. 1 ComponentsofDisplacement. . . . . . . . . . . . . . . . 1 1. 1. 2 In nitesimalDeformation. . . . . . . . . . . . . . . . . . 2 1. 1. 3 ElongationandShearingStrain . . . . . . . . . . . . . . . 4 1. 1. 4 SmallDeformations. . . . . . . . . . . . . . . . . . . . . 5 1. 1. 5 ComponentsofStrain . . . . . . . . . . . . . . . . . . . . 9 1. 1. 6 PrincipalDirectionofStrain . . . . . . . . . . . . . . . . 14 1. 1. 7 InvariantsofStrain . . . . . . . . . . . . . . . . . . . . . 21 1. 1. 8 PlaneStateofStrain. . . . . . . . . . . . . . . . . . . . . 23 1. 1. 9 EquationsofCompatibility. . . . . . . . . . . . . . . . . 24 1. 1. 10MeasurementofStrain . . . . . . . . . . . . . . . . . . . 25 1. 2 AnalysisofStress. . . . . . . . . . . . . . . . . . . . . . . . . . 27 1. 2. 1 StressVector. . . . . . . . . . . . . . . . . . . . . . . . . 27 1. 2. 2 NormalStress ShearingStress . . . . . . . . . . . . . . 29 1. 2. 3 ComponentsofStress . . . . . . . . . . . . . . . . . . . . 30 1. 2. 4 Symmetryof? DifferentialEquations ofEquilibrium Cauchy sBoundaryConditions. . . . . . 31 1. 2. 5 SymmetryofStressVector. . . . . . . . . . . . . . . . . 38 1. 2. 6 RelationsBetweenNormalorShearingStress andComponentsofStress. . . . . .

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ISBN 13
Title	Theory of Elasticity
Author	Aldo Maceri
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