196
15 Plasticity Theory of Henky–Nadai–Ilyushin
Fig. 15.1 Exemplary form of
the function ? )
3. Director stress and strain tensors coincide:
D σ = D ε ,
(15.3)
1
τ 0
D σ =
2
γ 0
D ε ,
(15.4)
or else
D σ =
2σ i
3ε i
D ε .
(15.5)
When switching from the tensor form to a regular one, formula (15.5) can be
written as follows:
σ x − σ 0 =
2σ i
3ε i
(ε x − ε 0 ),
· · · · · · · · · · · · · · · · · · · · · · · · ,
· · · · · · · · · · · · · · · · · · · · · · · · ;
τ xy =
2σ i
3ε i
ε xy =
σ i
3ε i
γ xy ,
· · · · · · · · · · · · · · · · · · · · · · · · ,
· · · · · · · · · · · · · · · · · · · · · · · · .
(15.6)
Formulas (15.3)–(15.5) result in that all the main axes of stress and strain tensors,
as well as the ratios of the main tangential stresses to the main shears, coincide,
e.g.
τ 12
γ 12
=
τ 23
γ 23
=
τ 31
γ 31
=
σ i
3ε i
.
This formula is a consequence of dependencies (15.6). Ratios (15.6) contain five
independent equations expressing a link between six independent components of
the stress tensor T σ and six independent components of the strain tensor T ε . The
sixth equation gives the hardening law (15.1).
Multiple experiments conducted for decades showed that the primary provisions of the strain theory of plasticity in some cases gave rather satisfactory