254
17 Additions and Generalizations to the Strain Theory of Plasticity
Fig. 17.4 Angular point
Binomial representation of the dependency (17.13) takes place in the case of a
plane strain trajectory. In a general case of a 5-dimensional vector Э, the strain law
can be written as
S =
5
i=1
i p i ,
(17.15)
where i are functionals from the internal parameters of the strain trajectory, p i are
single vectors of the natural benchmark:
i = i (¸, χ 1 , . . . , χ 4 ),
χ i = χ i (¸).
It has been proved (Ilyushin, [6]) that the law (17.15) in the class of tensor–linear
ratios is general. The study of the applicability conditions of the law (17.15) in
loading different from simple one and, in particular, in the case of an angular point
at the loading trajectory (Fig. 17.4) had [6] the following results.
In an infinitely small vicinity of the angular point, gains of strain and stress
vectors are related by the dependency
dЭ =
1
N
dS +
N − P
NP
SdS
S 2
S,
(17.16)
where N does not depend and P significantly depends on the angle β (Fig. 17.4) of
the trajectory break. These dependencies must be found experimentally.
17.6 Delay Law
Assume that we know the strain trajectory in a process. In an arbitrary point K, a
single vector of the tangential line p can be built, as well as the stress vector S under
the law (17.11) (Fig. 17.5).
17 Additions and Generalizations to the Strain Theory of Plasticity
Fig. 17.4 Angular point
Binomial representation of the dependency (17.13) takes place in the case of a
plane strain trajectory. In a general case of a 5-dimensional vector Э, the strain law
can be written as
S =
5
i=1
i p i ,
(17.15)
where i are functionals from the internal parameters of the strain trajectory, p i are
single vectors of the natural benchmark:
i = i (¸, χ 1 , . . . , χ 4 ),
χ i = χ i (¸).
It has been proved (Ilyushin, [6]) that the law (17.15) in the class of tensor–linear
ratios is general. The study of the applicability conditions of the law (17.15) in
loading different from simple one and, in particular, in the case of an angular point
at the loading trajectory (Fig. 17.4) had [6] the following results.
In an infinitely small vicinity of the angular point, gains of strain and stress
vectors are related by the dependency
dЭ =
1
N
dS +
N − P
NP
SdS
S 2
S,
(17.16)
where N does not depend and P significantly depends on the angle β (Fig. 17.4) of
the trajectory break. These dependencies must be found experimentally.
17.6 Delay Law
Assume that we know the strain trajectory in a process. In an arbitrary point K, a
single vector of the tangential line p can be built, as well as the stress vector S under
the law (17.11) (Fig. 17.5).