2.17 Octahedral Stresses
27
axes and not with reference to an arbitrary frame of reference. Now, denoting the
direction cosines of the plane ABC by l, m, and n, Eqs. (2.24a), (2.24b) and (2.24c)
with σ x = σ 1 , τ xy = τ xz = 0, etc., reduce to
T x = σ 1 l, T y = σ 2 m and T z = σ 3 n
(2.34)
The resultant stress on the oblique plane is thus
T
2
= σ
2
1 l
2
+ σ
2
2 m
2
+ σ
2
3 n
2
= σ
2
+ τ
2
∴ T
2
= σ
2
+ τ
2
(2.35)
The normal stress on this plane is given by
σ = σ 1 l
2
+ σ 2 m
2
+ σ 3 n
2
(2.36)
and the corresponding shear stress is
τ =
(σ 1 − σ 2 )
2 l
2 m
2
+ (σ 2 − σ 3 )
2 m
2 n
2
+ (σ 3 − σ 1 )
2 n
2 l
2
1
2
(2.37)
The direction cosines of the octahedral plane are:
l = ±
1
√
3
, m = ±
1
√
3
, n = ±
1
√
3
Substituting in (2.35), (2.36), (2.37), we get
Resultant stress T =
1
3
(σ
2
1 + σ
2
2 + σ
2
3 )
(2.38)
Normal stress = σ =
1
3
(σ 1 + σ 2 + σ 3 )
(2.39)
Shear stress = τ =
1
3
(σ 1 − σ 2 ) 2 + (σ 2 − σ 3 ) 2 + (σ 3 − σ 1 ) 2
(2.40)
Also,
τ =
1
3
2(σ 1 + σ 2 + σ 3 ) 2 − 6(σ 1 σ 2 + σ 2 σ 3 + σ 1 σ 3 )
(2.41)
τ =
1
3
2I
2
1 − 6I 2
(2.42)