202
15 Plasticity Theory of Henky–Nadai–Ilyushin
and full strain intensity can be represented as a sum of intensities of its elastic and
plastic components
ε i = ε
y
i + ε
p
i .
15.5 Unloading Laws
We have already said (p. 195) that the unloading condition in the strain theory of
plasticity is defined by the inequation
dσ i
dt
< 0,
(15.16)
where t is the time or any other monotonously growing parameter [1, 2]. Assume
that active loading was done to some point M(ε i ; σ i ) (Fig. 15.5), then the condition
(15.16) is fulfilled, and unloading is done via the trajectory MM OA.
For an arbitrary ˜
σ i < σ i the following first unloading law is observed:
˜
σ i = 3G( ˜
ε i − ε i ).
(15.17)
Apart from the law (15.17), we should also write the second unloading law—the
law of proportionality of the stress deviator and the elastic strain deviator
D ˜
σ = 2GD ˜
ε ,
(15.18)
as well as the third law—the law of elasticity of volumetric strain
˜
σ 0 = 3K ˜
ε 0 = 3K(˜ ε 0 − ε 0 ) = 3K ˜
ε 0 .
(15.19)
Fig. 15.5 To the formulation
of unloading laws