CONTINUUM MECHANICS for ENGINEERS

Hooke’s law equations, Eq 6.4-3b, for plane stress are written
along with
ε 11
ε 22
1
= ( σ (6.5-5a)
11 − vσ22)
E
1
= ( σ (6.5-5b)
22 − vσ11)
E
1+
v σ γ
E 2G 2
12 12
ε12 = σ
(6.5-5c)
12 = =
( )= −
v
v
E
1−
v
( )
ε33 = − σ + σ ε + ε
(6.5-6)
11 22 11 22
By inverting Eq 6.5-5, we express the stress components in terms of the
strains as
E
σ = ε + vε
2
1−
v
( )
11 11 22
E
σ = ε + vε
2
1−
v
( )
22 22 11
E E
σ ε γ = Gγ
1+v 2 1+v
= = ( )
12 12 12 12
These equations may be conveniently cast into the matrix formulation
⎡σ
11⎤
⎡1
v 0 ⎤ ⎡ε11⎤
⎢ ⎥ E ⎢
⎥ ⎢ ⎥
⎢
σ 22⎥
= 1 0 ε22
1−
⎢
v 2
v
⎥ ⎢ ⎥
⎣
⎢σ
12⎦
⎥
⎣
⎢0
0 1±v⎦
⎥
⎣
⎢ε12⎦
⎥
(6.5-7a)
(6.5-7b)
(6.5-7c)
(6.5-8)
In terms of the displacement components, u i (i = 1, 2), the plane stress field
equations may be combined to develop a Navier-type equation for elastostatics,
namely,
E
+v u
E
v u +
21 21−
( )
( )
i,jj j,ji
+ ρb i = 0, (i, j = 1, 2) (6.5-9)