1.12 Determination of Strains on Inclined Planes 25SubstitutingfromEqs.(1.18)and(1.21)andrearranging,
2
∂^2 εx
∂y∂z=
∂
∂x(
−
∂γyz
∂x+
∂γxz
∂y+
∂γxy
∂z)
(1.24)
Similarly,
2∂^2 εy
∂x∂z=
∂
∂y(
∂γyz
∂x−
∂γxz
∂y+
∂γxy
∂z)
(1.25)
and
2∂^2 εz
∂x∂y=
∂
∂z(
∂γyz
∂x+
∂γxz
∂y−
∂γxy
∂z)
(1.26)
Equations(1.21)through(1.26)arethesixequationsofstraincompatibilitywhichmustbesatisfiedin
thesolutionofthree-dimensionalproblemsinelasticity.
1.11 PlaneStrain...........................................................................................
Although we have derived the compatibility equations and the expressions for strain for the general
three-dimensionalstateofstrain,weshallbemainlyconcernedwiththetwo-dimensionalcasedescribed
inSection1.4.Thecorrespondingstateofstrain,inwhichitisassumedthatparticlesofthebodysuffer
displacementsinoneplaneonly,isknownasplanestrain.Weshallsupposethatthisplaneis,asfor
planestress,thexyplane.Then,εz,γxz,andγyzbecomezero,andEqs.(1.18)and(1.20)reduceto
εx=∂u
∂xεy=∂v
∂y(1.27)
and
γxy=∂v
∂x+
∂u
∂y(1.28)
Further,bysubstitutingεz=γxz=γyz=0inthesixequationsofcompatibilityandnotingthatεx,εy,
andγxyarenowpurelyfunctionsofxandy,weareleftwithEq.(1.21),namely
∂^2 γxy
∂x∂y=
∂^2 εy
∂x^2+
∂^2 εx
∂y^2astheonlyequationofcompatibilityinthetwo-dimensionalorplanestraincase.
1.12 DeterminationofStrainsonInclinedPlanes......................................................
Having defined the strain at a point in a deformable body with reference to an arbitrary system of
coordinate axes, we may calculate direct strains in any given direction and the change in the angle
(shearstrain)betweenanytwooriginallyperpendiculardirectionsatthatpoint.Weshallconsiderthe
two-dimensionalcaseofplanestraindescribedinSection1.11.