28 CHAPTER 1 Basic Elasticity
Therefore,atapointinadeformablebody,therearetwomutuallyperpendicularplanesonwhich
theshearstrainγiszeroandnormaltowhichthedirectstrainisamaximumorminimum.Thesestrains
aretheprincipalstrainsatthatpointandaregiven(fromcomparisonwithEqs.(1.11)and(1.12))by
εI=εx+εy
2+
1
2
√
(εx−εy)^2 +γxy^2 (1.35)and
εII=εx+εy
2−
1
2
√
(εx−εy)^2 +γxy^2 (1.36)Iftheshearstrainiszeroontheseplanes,itfollowsthattheshearstressmustalsobezero,andwe
deduce,fromSection1.7,thatthedirectionsoftheprincipalstrainsandprincipalstressescoincide.The
relatedplanesarethendeterminedfromEq.(1.10)orfrom
tan2θ=γxy
εx−εy(1.37)
Inaddition,themaximumshearstrainatthepointis
(γ
2)
max=^12
√
(εx−εy)^2 +γxy^2 (1.38)or
(γ
2
)
max=
εI−εII
2(1.39)
(cf.Eqs.(1.14)and(1.15)).
1.14 Mohr’sCircleofStrain..............................................................................
We now apply the arguments of Section 1.13 to the Mohr’s circle of stress described in Section 1.8.
Acircleofstrain,analogoustothatshowninFig.1.12(b),maybedrawnwhenσx,σy,andsoonare
replaced byεx,εy, and so on, as specified in Section 1.13. The horizontal extremities of the circle
representtheprincipalstrains,theradiusofthecircle,halfthemaximumshearstrain,andsoon.
1.15 Stress–StrainRelationships.........................................................................
Intheprecedingsections,wehavedeveloped,forathree-dimensionaldeformablebody,threeequations
of equilibrium (Eqs. (1.5)) and six strain–displacement relationships (Eqs. (1.18) and (1.20)). From
thelatter,weeliminateddisplacements,therebyderivingsixauxiliaryequationsrelatingstrains.These
compatibility equations are an expression of the continuity of displacement which we have assumed
asaprerequisiteoftheanalysis.Atthisstage,therefore,wehaveobtainednineindependentequations
towardthesolutionofthethree-dimensionalstressproblem.However,thenumberofunknownstotals
15,comprisingsixstresses,sixstrains,andthreedisplacements.Anadditionalsixequationsaretherefore
necessarytoobtainasolution.