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1、EN175:AdvancedMechanicsofSolidsDivisionofEngineeringBrownUniversity1.2Reviewofsomeconceptsfromintroductorycourses.1.2.1NotationforvectorcomponentsMostintroductorycoursesexpressvectorsascomponentsinaCartesianbasis,usingthenotation(x,y,z)tospecifythecoordinatesofthepositionvector,andtospecifyth
2、ecomponentsofavectoru.Inthiscourse,wewillbeusingamoreadvanced,andmoreconvenientnotation,whichfollowsthesameconventionasthefiniteelementpackageweareusing.AllthreedimensionalvectorswillbeexpressedinaCartesianbasis,butwewillusetodenotethethreebasisvectors.Thepositionvectorwillbeexpressedaswhered
3、enotethecomponentsofthepositionvector.Similarly,thecomponentsofavectoruwillbedenotedby.Wewillshortlybeintroducingindexnotationforvectors.Inthisnotation,weuse,wheretheLatinsubscripticantakevalues1,2,or3,todenotethecomponentsofx.Similarly,thecomponentsofanyvectoruwillbedenoted.1.2.2Strains.Yous
4、houldrecallthatstrainsareusedtodescribechangesinshapeofinfinitesimalvolumeelementswithinasolidbody.Componentsofstrainareoftenrepresentedasamatrix(oramatrixin2D)Thepicturebelowremindsyouofthephysicalsignificanceofthevariouselementsinthismatrix.Thediagonaltermsrepresentlengthchangesoftheelement
5、inthedirections,whiletheoff—diagonalterms,etc.representchangesintheanglesbetweenthexandyaxes,xandzaxes,etc.Inthiscourse,wewilluseaslightlydifferentnotationtodenotecomponentsofstrain.Asforvectors,wewillusetodenotethebasisvectorsforstrains.Thestrainmatrix(actually,nowastraintensor),willbedenote
6、dbyHere,However,justtobeconfusing,andsimilarlyforallotheroff-diagonalterms.Thereasonforintroducingthisfactorof2willbecomeapparentlater:itisdonetoensurethatthematrixofstrainsbehaveslikeatensor.Withtheoriginaldefinition(expressedintermsof)thisisnotthecase.Thecollectionofallstraincomponentsisexp
7、ressedasforshort.Here,theLatinsubscriptsiandjcantakevalues1,2,or3,sothatrepresentsasetof9numbers.Ourfiniteelementcomputationswillprintoutthetensor,notthestrainmatrixwithintheoffdiagonalterms.Observethatthestraintensorissymmetric,satisfyingthe
