46 CHAPTER 2 Two-Dimensional Problems in Elasticity
byEqs.(1.6):
∂σx
∂x+
∂τxy
∂y+X= 0
∂σy
∂y+
∂τyx
∂y+Y= 0
andtherequiredstress–strainrelationshipsobtainedfromEq.(1.47),namely,
εx=1
E
(σx−νσy)εy=1
E
(σy−νσx)γxy=2 ( 1 +ν)
EτxyWefindthatalthoughεzexists,Eqs.(1.22)through(1.26)areidenticallysatisfied,leavingEq.(1.21)
astherequiredcompatibilitycondition.SubstitutioninEq.(1.21)oftheprecedingstrainsgives
2 ( 1 +ν)∂^2 τxy
∂x∂y=
∂^2
∂x^2(σy−νσx)+∂^2
∂y^2(σx−νσy) (2.1)FromEqs.(1.6)
∂^2 τxy
∂y∂x=−
∂^2 σx
∂x^2−
∂X
∂x(2.2)
and
∂^2 τxy
∂x∂y=−
∂^2 σy
∂y^2−
∂Y
∂y(τyx=τxy) (2.3)AddingEqs.(2.2)and(2.3),thensubstitutinginEq.(2.1)for2∂^2 τxy/∂x∂y,wehave−( 1 +ν)(
∂X
∂x+
∂Y
∂y)
=
∂^2 σx
∂x^2+
∂^2 σy
∂y^2+
∂^2 σy
∂x^2+
∂^2 σx
∂y^2or
(
∂^2
∂x^2
+
∂^2
∂y^2)
(σx+σy)=−( 1 +ν)(
∂X
∂x+
∂Y
∂y)
(2.4)
Thealternativetwo-dimensionalproblemofplanestrainmayalsobeformulatedinthesamemanner.
WehaveseeninSection1.11thatthesixequationsofcompatibilityreducetothesingleequation(1.21)
fortheplanestraincondition.Further,fromthethirdofEqs.(1.42)
σz=ν(σx+σy)(sinceεz=0forplanestrain)