24 CHAPTER 1 Basic Elasticity
1.10 CompatibilityEquations............................................................................
InSection1.9,weexpressedthesixcomponentsofstrainatapointinadeformablebodyintermsofthe
threecomponentsofdisplacementatthatpoint,u,v,andw.Wehavesupposedthatthebodyremains
continuousduringthedeformationsothatnovoidsareformed.Itfollowsthateachcomponent,u,v,
andw,mustbeacontinuous,single-valuedfunctionor,inquantitativeterms,
u=f 1 (x,y,z) v=f 2 (x,y,z) w=f 3 (x,y,z)Ifvoidswereformed,thendisplacementsinregionsofthebodyseparatedbythevoidswouldbe
expressed as different functions ofx,y,andz. The existence, therefore, of just three single-valued
functionsfordisplacementisanexpressionofthecontinuityorcompatibilityofdisplacementwhich
wehavepresupposed.
Sincethesixstrainsaredefinedintermsofthreedisplacementfunctions,thentheymustbearsome
relationshiptoeachotherandcannothavearbitraryvalues.Theserelationshipsarefoundasfollows.
DifferentiatingγxyfromEq.(1.20)withrespecttoxandygives
∂^2 γxy
∂x∂y=
∂^2
∂x∂y∂v
∂x+
∂^2
∂x∂y∂u
∂yorsincethefunctionsofuandvarecontinuous,
∂^2 γxy
∂x∂y=
∂^2
∂x^2∂v
∂y+
∂^2
∂y^2∂u
∂xwhichmaybewritten,usingEq.(1.18)
∂^2 γxy
∂x∂y=
∂^2 εy
∂x^2+
∂^2 εx
∂y^2(1.21)
Inasimilarmanner,
∂^2 γyz
∂y∂z=
∂^2 εy
∂z^2+
∂^2 εz
∂y^2(1.22)
∂^2 γxz
∂x∂z=
∂^2 εz
∂x^2+
∂^2 εx
∂z^2(1.23)
Ifwenowdifferentiateγxywithrespecttoxandzandaddtheresulttoγzx,differentiatedwithrespect
toyandx,weobtain
∂^2 γxy
∂x∂z+
∂^2 γxz
∂y∂x=
∂^2
∂x∂z(
∂u
∂y+
∂v
∂x)
+
∂^2
∂y∂x(
∂w
∂x+
∂u
∂z)
or
∂
∂x(
∂γxy
∂z+
∂γxz
∂y)
=
∂^2
∂z∂y∂u
∂x+
∂^2
∂x^2(
∂v
∂z+
∂w
∂y)
+
∂^2
∂y∂z∂u
∂x