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
The Theory of Elasticity moves freely within a unified mathematical framework that provides the analytical tools for calculating stresses and deformations in a strained elastic body. All the elastic problems can be exactly analyzed employing the classical Mathematical analysis, with only the exception of the unilateral problems for which it is mandatory to use the Functional analysis.
This book focuses on the practical application of the theoretical results. It gives to engineers, in a simple form, a clear indication of the necessary fundamental knowledge of the Theory of elasticity. The author develops the subjects in a classical way, but in light of the modern Mathematical theory of the elasticity and with more accented relief to the connections with the Thermodynamics. To give a clear justification of the fundamental equations of thermoelasticity, he applies a technique of analysis proper of the Fluid dynamics. However in the discussion of the unilateral problems, where the Functional analysis is compulsory, he has related in detail the mathematical aspects of the theoretical analysis.
