7.3 Intensity of Stresses
73
hence, if α = π/4 we obtain
|τ | max =
σ 1 − σ 3
2
.
(7.16)
Provided that (7.11), formula (7.16) defines the maximum tangential stress.
7.3 Intensity of Stresses
It is known [4] that the experimental values of tangential stresses are called
principal tangential stresses:
τ 13 =
σ 1 − σ 3
2
, τ 21 =
σ 2 − σ 1
2
, τ 23 =
σ 2 − σ 3
2
.
(7.17)
Octahedral tangential stress (τ o ) is the tangential stress on the area similarly
inclined to principal directions. For this area, u = v =
1
√
3
and formula (7.15)
gives:
τ
2
o =
2
9
(σ
2
1 + σ
2
2 + σ
2
3 − σ 1 σ 2 − σ 2 σ 3 − σ 3 σ 1 ).
The last formula can be represented in a different way:
τ o =
1
3
(σ 1 − σ 2 ) 2 + (σ 2 − σ 3 ) 2 + (σ 3 − σ 1 ) 2 ,
(7.18)
whereof one can make sure by opening brackets under the radical sign in the last
formula.
We will show that octahedral tangential stress is proportional to the mean square
of three principal tangential stresses. Indeed, when using formulas (7.17), we can
re-write formula (7.18) as follows:
τ o =
2
3
τ 2
12 + τ 2
23 + τ 2
31 .
(7.19)
Octahedral tangential stress is sometimes also referred to as the intensity
of tangential stresses. Along with this value, its proportional value σ ? is often
introduced, which is defined by the formula
σ ? =
2(τ 2
12 + τ 2
23 + τ 2
31 ).
(7.20)