262
17 Additions and Generalizations to the Strain Theory of Plasticity
dτ o =
1
2
1
3
σ
ij σ
ij
−
1
2 ·
2
3
σ
ij dσ
ij ,
for example,
τ o dτ o =
1
3
σ
ij dσ
ij .
By substituting the last expression into formula (17.28), we obtain
dσ
ij de ij =
1
2
1
G s
−
1
G
dσ
ij dσ
ij
+
3
2
1
G t
−
1
G s
(dτ o )
2 .
(17.29)
The expression (17.29) will be positive only if
G > G s > G t ,
which is always complied with for monotonously hardening materials. Thus, the
plasticity law of the strain theory does not contradict the requirement of the Drucker
postulate (17.27) for any loadings. Restrictions implied by the postulate can refer
only to the type of the loading surface dividing the area of active plastic strain and
unloading.
Now let us refer to the second requirement of the postulate (17.20). For a noncompressible material, the plastic strain and loading surface will depend on the
history of setting the stress deviator components and this requirement will look as
follows:
(σ
ij − σ
∗
ij )de
p
ij 0.
(17.30)
Let us use the nine-dimensional imaging space. For the plastic strain to satisfy the
requirement (17.30), the loading surface in the loading point must have a conic
feature.
Indeed, for a smooth loading surface, the vector direction de depends only on
the loading vector σ , but not on dσ . In the case of strain theory, in the expression
(17.26) of the plastic strain gain, the first addend depends directly on dσ
ij , so the
direction of the vector de p depends on dσ . In this manner, it must be suggested that
the loading point M (Fig. 17.11) is conical.
By considering the plane projection of the imaging space, let us use β to
designate the angle between the cone generatrix and the vector radius σ of the
loading point M, α to designate the angle between the vectors σ and dσ , and δ
to designate the angle between the plastic strain gain vector de p and the vector σ
(Fig. 17.11). From the Drucker postulate (p. 259), it follows that the vector de p
must lie within the cone of normal lines to the loading surface in the point M. This
condition is reduced to fulfilling the inequation:
17 Additions and Generalizations to the Strain Theory of Plasticity
dτ o =
1
2
1
3
σ
ij σ
ij
−
1
2 ·
2
3
σ
ij dσ
ij ,
for example,
τ o dτ o =
1
3
σ
ij dσ
ij .
By substituting the last expression into formula (17.28), we obtain
dσ
ij de ij =
1
2
1
G s
−
1
G
dσ
ij dσ
ij
+
3
2
1
G t
−
1
G s
(dτ o )
2 .
(17.29)
The expression (17.29) will be positive only if
G > G s > G t ,
which is always complied with for monotonously hardening materials. Thus, the
plasticity law of the strain theory does not contradict the requirement of the Drucker
postulate (17.27) for any loadings. Restrictions implied by the postulate can refer
only to the type of the loading surface dividing the area of active plastic strain and
unloading.
Now let us refer to the second requirement of the postulate (17.20). For a noncompressible material, the plastic strain and loading surface will depend on the
history of setting the stress deviator components and this requirement will look as
follows:
(σ
ij − σ
∗
ij )de
p
ij 0.
(17.30)
Let us use the nine-dimensional imaging space. For the plastic strain to satisfy the
requirement (17.30), the loading surface in the loading point must have a conic
feature.
Indeed, for a smooth loading surface, the vector direction de depends only on
the loading vector σ , but not on dσ . In the case of strain theory, in the expression
(17.26) of the plastic strain gain, the first addend depends directly on dσ
ij , so the
direction of the vector de p depends on dσ . In this manner, it must be suggested that
the loading point M (Fig. 17.11) is conical.
By considering the plane projection of the imaging space, let us use β to
designate the angle between the cone generatrix and the vector radius σ of the
loading point M, α to designate the angle between the vectors σ and dσ , and δ
to designate the angle between the plastic strain gain vector de p and the vector σ
(Fig. 17.11). From the Drucker postulate (p. 259), it follows that the vector de p
must lie within the cone of normal lines to the loading surface in the point M. This
condition is reduced to fulfilling the inequation: