274
18 Theories of Plastic Yield
some plastic strain de p occurs. Based on the Drucker postulate, the vector de p is
directed to the normal line , so df ∼ n · dσ > 0 and
˙
e
p
ij = λ
∂f
∂σ ij
.
(18.27)
Let us specify the form of the multiplier λ included in formula (18.27). Since the
plastic strain rate ˙
e p
ij must be proportional to the normal line to the surface of the
additional loading vector component dσ , the gains of the plastic strain components
de p
ij must be
∂f
∂σ kl
dσ kl . Taking this into account, formula (18.27) can be represented
as follows:
de
p
ij = H
∂f
∂σ kl
dσ kl ·
∂f
∂σ ij
,
or else
˙
e
p
ij = H
∂f
∂σ ij
∂f
∂σ kl
· ˙
σ kl
,
(18.28)
where H is the positive hardening function that, as results from the Drucker
postulate, can depend on the loading history, strain history but does not depend
on the gains dσ kl and de p
kl . The latter means that the ratio (18.28) is linearly relative
to the gains dσ kl and de p
kl .
The indicated linearity of the law (18.28) relative to gains de ?
kl and dσ kl shows, in
particular, that the link de kl ∼ dσ kl does not depend on the angle (β) of the loading
trajectory fracture (Fig. 17.4)
tg β =
dS 3
dS 1
,
which is incorrect in a general case. Therefore the law (18.28) must be refined. For
example, we can assume that H is a homogeneous function of the zero degree from
stress gain.
Let us consider a partial case. Assume that H and f are functions of the second
variant of the stress deviator J
2 (or σ ? or τ 0 , which is the same). By assuming, for
example, f = J
2 , we have
∂f
∂σ ij
=
∂
∂σ ij
(σ
ij σ
ij ) = 2σ
ij ,
as well as
∂f
∂σ kl
˙
σ kl =
∂J
2
dt
.