13.4 Other Invariants in Plasticity Theory
173
Normal stress (σ ) on the octahedral area will be
σ = Sn =
1
3
(σ 1 + σ 2 + σ 3 ) = σ 0 .
(13.27)
The result (13.27) expresses that normal stress on octahedral areas will be equal to
the mean arithmetic from principal stresses. As said before (p. 15), this stress is also
called hydrostatic.
Let us calculate the tangential stress (τ 0 ) on the octahedral area. To do it, let us
use the formula
τ 0 =
S 2 − σ 2
0 ,
(13.28)
where we assume that
S
2
=
1
3
[σ
2
1 + σ
2
2 + σ
2
3 ],
σ
2
0 =
1
9
[σ + 1
2
+ σ
2
2 + σ
2
3 + 2σ 1 σ 2 + 2σ 2 σ 3 + 2σ 3 σ 1 ].
By substituting the last expression into formula (13.28), we obtain
τ
2
0 =
2
9
[σ
2
1 + σ
2
2 + σ
2
3 − σ 1 σ 2 − σ 2 σ 3 − σ 3 σ 1 ]
=
1
9
(σ 1 − σ 2 )
2
+ (σ 2 − σ 3 )
2
+ (σ 3 − σ 1 )
2
,
(13.29)
or otherwise
τ
2
0 =
1
3
J
2 .
(13.30)
The tangential stress on the octahedral area is referred to as octahedral tangential
stress. Based on the results (13.29) and (13.30) it is an invariant of the stress tensor.
The octahedral tangential stress modulus is defined by the formula
τ 0 =
1
3
(σ 1 − σ 2 ) 2 + (σ 2 − σ 3 ) 2 + (σ 3 − σ 1 ) 2 ,
(13.31)
and its tangential cosines can be calculated as ratios of the vector components S × n
to its module. In this manner, let us designate the unit vector of octahedral tangential
stress as
#»
β (β 1 , β 2 , β 3 ):