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18 Theories of Plastic Yield
4. The loading criterion is defined by the condition dJ
2 > 0.
Based on two first pre-requisites, we can write as follows:
dε
ij = A ij kl dσ
kl , (dJ
2 > 0),
(18.17)
where the fourth-rank tensor A ij kl depends only on the stress deviator D σ .
Similarly to formula (18.1), we represent the full strain gain as a sum of gains of
its elastic and plastic parts
dε ij = dε
y
ij + de ij , (e ij = ε
p
ij ).
(18.18)
The gain of elastic strains is defined by Hooke’s law, and we assume as follows for
the gain of the plastic strain component:
de ij = C ij kl dσ
kl ,
(18.19)
where the tensor C ij kl depends on the stress deviator.
We can write as follows from the continuity condition in the case of neutral
loading:
C ij kl dσ
kl = 0, dJ
2 = 0.
(18.20)
The loading neutrality condition gives
dJ
2 = d(σ
kl σ
kl ) = 2σ
kl dσ
kl = 0.
(18.21)
Simultaneous zeroing of two lines relative to dσ
kl having forms (18.20) and
(18.21) makes us assume that
C ij kl = G ij σ
kl ,
and represent the plasticity law as
de ij =
⎧
⎨
⎩
G ij σ
kl dσ
kl ≡
1
2
G ij dJ
2 , dJ
2 > 0;
0,
d J
2 0.
(18.22)
The tensor analysis shows [1] that the most general form of tensor G ij contained in
the law (18.22) will be
G ij = G ij [(σ
ij )] = P (J
2 , J
2
3 ) + Q(J
2 , J
2
3 )J
3 t ij .
(18.23)
The law (18.22)–(18.23) expresses the Handelman–Lin–Prager yield theory. If
we assume that the third invariant of the stress deviator poorly affects the ratios of