Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

1.15 Stress–Strain Relationships 31

andtheshearstrainsγxy,γxz,andγyzareexpressedintermsoftheirassociatedshearstressesasfollows:

γxy=

τxy

G

γxz=

τxz

G

γyz=

τyz

G

(1.51)

Equations(1.51),togetherwithEqs.(1.42),providetheadditionalsixequationsrequiredtodetermine
the15unknownsinageneralthree-dimensionalprobleminelasticity.Theyare,however,limitedinuse
toalinearlyelasticisotropicbody.
Forthecaseofplanestress,theysimplifyto

εx=

1

E

(σx−νσy)

εy=

1

E

(σy−νσx)

εz=

−ν

E

(σx−σy)

γxy=

τxy

G

⎫

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎭

(1.52)

ItmaybeseenfromthethirdofEqs.(1.52)thattheconditionsofplanestressandplanestraindonot
necessarilydescribeidenticalsituations.
Changesinthelineardimensionsofastrainedbodymayleadtoachangeinvolume.Supposethat
asmallelementofabodyhasdimensionsδx,δy,andδz.Whensubjectedtoathree-dimensionalstress
system,theelementsustainsavolumetricstraine(changeinvolume/unitvolume)equalto

e=

( 1 +εx)δx( 1 +εy)δy( 1 +εz)δz−δxδyδz

δxδyδz

Neglectingproductsofsmallquantitiesintheexpansionoftheright-handsideoftheprecedingequation
yields

e=εx+εy+εz (1.53)

Substitutingforεx,εy,andεzfromEqs.(1.42),wefindforalinearlyelastic,isotropicbody

e=

1

E

[σx+σy+σz− 2 ν(σx+σy+σz)]

or

e=

( 1 − 2 ν)

E

(σx+σy+σz)

Inthecaseofauniformhydrostaticpressure,σx=σy=σz=−pand

e=−

3 ( 1 − 2 ν)

E

p (1.54)

TheconstantE/ 3 ( 1 − 2 ν)isknownasthebulkmodulusormodulusofvolumeexpansionandisoften
giventhesymbolK.