Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

30 CHAPTER 1 Basic Elasticity

Equations(1.42)maybetransposedtoobtainexpressionsforeachstressintermsofthestrains.The
procedureadoptedmaybeanyofthestandardmathematicalapproachesandgives

σx=

νE

( 1 +ν)( 1 − 2 ν)

e+

E

( 1 +ν)

εx (1.43)

σy=

νE

( 1 +ν)( 1 − 2 ν)

e+

E

( 1 +ν)

εy (1.44)

σz=

νE

( 1 +ν)( 1 − 2 ν)

e+

E

( 1 +ν)

εz (1.45)

inwhich

e=εx+εy+εz (seeEq.(1.53))

Forthecaseofplanestressinwhichσz=0,Eqs.(1.43)and(1.44)reduceto

σx=

E

1 −ν^2

(εx+νεy) (1.46)

σy=

E

1 −ν^2

(εy+νεx) (1.47)

Suppose now that at some arbitrary point in a material, there are principal strainsεIandεIIcor-
responding to principal stressesσIandσII.Ifthesestresses(andstrains)areinthedirectionofthe
coordinate axesxandy, respectively, thenτxy=γxy=0, and from Eq. (1.34), the shear strain on an
arbitraryplaneatthepointinclinedatanangleθtotheprincipalplanesis

γ=(εI−εII)sin2θ (1.48)

UsingtherelationshipsofEqs.(1.42)andsubstitutinginEq.(1.48),wehave

γ=

1

E

[(σI−νσII)−(σII−νσI)]sin2θ

or

γ=

( 1 +ν)

E

(σI−σII)sin2θ (1.49)

UsingEq.(1.9)andnotingthatforthisparticularcaseτxy=0,σx=σI,andσy=σII,

2 τ=(σI−σII)sin2θ

fromwhichwemayrewriteEq.(1.49)intermsofτas

γ=

2 ( 1 +ν)

E

τ (1.50)

ThetermE/2(1+ν)isaconstantknownasthemodulusofrigidityG.Hence,

γ=τ/G