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1、Chapter1KinematicsandConservationSeveralofthephysicalprinciplesunderpinningcontinuummechanicstaketheformofcon-servationlaws,whichstatethatsomepropertyofthematerialinmotionispreservedduringthemotion,orchangesinaprescribedway.Thefundamentalpostulateofcontinuumme-chanics-tha
2、tthepropertiesofmattercanbelocalizedinarbitrarilysmallvolumes-hasitsmathematicalexpressionintheuseofdensities,i.e.amountperunitvolume,torepre-sentmass,momentum,energy,etc.Materialscompress,expand,andchangeshapeduringamotion,sothechangeinapropertydensityalsohasapurelykinem
3、aticcomponent,simplyduetothemotion.Thischapterisdevotedtotheexpressionofbasicconservationprinciplesaspartialdifferentialequations,whichaccountforbothphysicalandgeometriccausesofdensitychanges.1.1DeformationandMotionThissectionintroducessomeofthemathematicsnecessarytodescri
4、bemotionof3Dcon-tinua.Thesearemostlystandardideasfromvectorcalculus,expressedinperhapsunfamiliarnotationsuitedtotheapplication.3A“materialbody”isthevolumeoccupiedbyaspecificpieceofmaterial,sayΩ0⊂R-callthisthereferenceconfigurationofthebody.Adeformationisamapx:Ω→R3,0continuo
5、uslydifferentiablewithacontinuouslydifferentiableinverseq.Remark:Thephysicalsignificanceofcontinuousdifferentiabilityisthatnotearsorholesdevelopinthematerialasaresultofitsdeformation,nordounlimitedsqueezingorexpansionofsmallvolumes.Theserequirements,motivatedbyexperiencewithe
6、verydaymaterials,donotinthemselvesimplydifferentiabiltiy,Howeverdifferentiabilitysuffices,andistradition-1allyassumed,oftenwithoutnotice.Thederivativeofx,i.e.itsJacobianmatrix,isconventionallydenotedby∇xinthissubject:∂xi(∇x)i,j=,i,j=1,2,3∂qjI’llusethesamenotationforarbitraryv
7、ector-valuedfunctionsofvectorarguments,witharbitrarynumbersofcomponentsindomainandrange:thatis,∇denotesthederivative,expressedasamatrixrelativetosomecoordinatesystem,andtheithrowcontainsthepartialderivativesoftheithcomponent,whereasthejthcolumncontainsthejthpartialderivat
8、ivesofallofthecomponents.Withthisnotation,thechainruletakesasimpleform:ifu:Rm→Rp(p-vectorfunctio
