38 CHAPTER 1 Basic Elasticity
TheprincipalstressesarenowobtainedbysubstitutionofεIandεIIinEqs.(1.52).Thus,εI=1
E
(σI−νσII) (1.65)and
εII=1
E
(σII−νσI) (1.66)SolvingEqs.(1.65)and(1.66)gives
σI=E
1 −ν^2(εI+νεII) (1.67)and
σII=E
1 −ν^2(εII+νεI) (1.68)Atypicalrosettewouldhaveα=β= 45 ◦,inwhichcasetheprincipalstrainsaremostconveniently
foundusingthegeometryofMohr’scircleofstrain.Supposethatthearmaoftherosetteisinclinedat
someunknownangleθtothemaximumprincipalstrainasinFig.1.18.Then,Mohr’scircleofstrain
isasshowninFig.1.19;theshearstrainsγa,γb,andγcdonotfeatureintheanalysisandaretherefore
ignored.FromFig.1.19,wehave
OC=^12 (εa+εc)CN=εa−OC=^12 (εa−εc)
QN=CM=εb−OC=εb−^12 (εa+εc)Fig.1.19
Experimental values of principal strain using Mohr’s circle.