5.2 Relationship Between Plane Stress and Plane Strain
129
If the equations for stress σ x for plane strain and plane stress are compared, it can
be observed that they are identical except for the comparison of coefficients of the
term
ε x + ε y
.
i.e.
σ x =
λ
ε x + ε y
+ 2Gε x
plane strain
2Gλ
λ+2G
ε x + ε y
+ 2Gε x plane stress
Since all the equations for stresses in plane stress and plane strain solutions
are identical, the results from plane strain can be transformed into plane stress by
replacing λ in plane strain case by
2Gλ
λ+2G
in plane stress case. This is equivalent to
replacing
ν
1−ν
in plane strain case by λ in plane stress case.
Similarly, a plane stress solution can be transformed into a plane strain solution
by replacing
2Gλ
λ+2G
in plane stress case by λ in plane strain case. This is equivalent to
replacing λ in plane stress case by
ν
1−ν
in plane strain case.
5.3 Transformation of Compatibility Equation from Strain
Components to Stress Components
5.3.1 Plane Stress Case
For two-dimensional problems, we have compatibility equation (from Eq. 3.38) as
∂
2
ε x
∂ y 2 +
∂
2
ε y
∂ x 2 =
∂
2
γ xy
∂ x ∂ y
(5.8)
Further, stress–strain relations are given by (from Eq. 5.2)
ε x =
1
E
σ x − νσ y
ε y =
1
E
σ y − νσ x
γ xy =
τ xy
G
(5.9)
Substituting for G =
E
2(1+ν)
and Eq. (5.9) in Eq. (5.8), we get
∂
2
∂ 2 y
(σ x − νσ y ) +
∂
2
∂ x 2 (σ y − νσ x ) = 2(1 + ν)
∂
2
τ xy
∂ x ∂ y
(5.10)
Now, recalling equations of equilibrium (from Eqs. 2.31a and 2.31b)