186
14 On the Plasticity Conditions of an Isotropic Body
Fig. 14.5 Traces of yield
surfaces on the octahedral
plane according to
Huber–Mises (circumference)
and Tresca (hexagon)
where u f is the specific potential energy of shape changing [8]
u f =
1 + μ
6E
(σ 1 − σ 2 )
2
+ (σ 2 − σ 3 )
2
+ (σ 3 − σ 1 )
2
.
(14.16)
In the case of uniaxial elongation,
σ 1 = σ s ; σ 2 = σ 3 = 0;
then using formulas (13.34) and (13.31), we obtain
σ i = σ s ,
and the second condition (14.15) can be written as
σ
2
i =
1
2
(σ 1 − σ 2 )
2
+ (σ 2 − σ 3 )
2
+ (σ 3 − σ 1 )
2
= σ
2
s .
(14.17)
In the case of pure shear,
σ 1 = −σ 3 = τ s .
By substituting this result to formula (14.17), we obtain
τ s =
1
√
3
σ s ≈ 0, 577σ s ,
(14.18)
whereas, from the Tresca plasticity condition, formula (14.13) gives τ s = 0, 5σ s .
By comparing formulas (14.16) and (14.17), we notice that expressions in their
square brackets coincide. Therefore, the Huber–Tresca condition is sometimes
called energetic and is formulated as follows: the yield state of the material is
achieved at some constant specific potential energy of shape changing.