19.3 Endochronic Plasticity Theory
291
3. α ∈ [−λ 1 ; λ 1 ]. In this case τ > 0 ∀λ ∈ (−α; α) and integration when
calculating the expression (19.9) is done within −α λ α.
Thus, in the Batdorf–Budiansky theory and for the simplest two-dimensional
model, the problem of integrating plasticity equations for any loading part is
so complex that it prevents one from making any qualitative conclusions of the
nature of loading surface changes in the case of complex loading trajectories. This
conclusion was a source of pessimistic moods of Yu.N. Rabotnov [25] as to the
possible progress of plasticity theory. In Part III, we tried to show what can be
opposed to these pessimistic conclusions.
19.3 Endochronic Plasticity Theory
Almost in all variants of plasticity theory that we discussed in previous paragraphs
of Part II, the concept of a loading surface is used. It is a priori deemed that the
loading surface can be defined experimentally. To do it, we must be able to clearly
fix the moment of plastic strain, which is impossible in principle. Apparently, any of
the existing plasticity theories can be declared unsound as contradicting the data of
thin experiments. Therefore, a doubt occurred: “. . . does the loading surface concept
have any real sense and does it have to be used as a basis when building the plasticity
theory?” [26, p. 564].
Understanding of justification of this doubt resulted in efforts to build the
plasticity theory that is not based on the loading surface concept but directly
expresses components of the stress tensor as some functions on loading trajectories.
At the brink of the 1950s, A. A. Ilyushin [9, 10] created the theory of the elastic–
plastic process of strain of continuous media in the case of complex loading that
was later developed in the scientific school created by him (V. G. Zubchaninov [35],
R. A. Vasin [34] et al). The same type of theories includes the above-mentioned
(p.157) endochronic theory of plasticity, which was proposed in 1971 by K. Valanis
[32, 33] and is intensively developing today. The evolution of the endochronic theory
of plasticity is described in the article by Yu. I.. Kadashevich, and S. P. Pomytkin
[14] with a sufficient degree of detail as well as in the book by P. V. Trusov and I. E.
Keller [30].
The theory is based on the concept of the so-called internal time. Apparently, the
theory name is related to this concept: endo—internal, chronos—time (Greek). The
determinant functions of the Valanis theory have a comparatively simple structure
of hereditary type whose form does not differ from the functions of linear viscoelasticity with the substitution of physical time with the so-called internal time. In
the initial variant, the author tried to use the following type of the determinant ratio:
σ
ij =
μ
0
L(μ − μ
)dε
ij (μ
),
(19.10)
291
3. α ∈ [−λ 1 ; λ 1 ]. In this case τ > 0 ∀λ ∈ (−α; α) and integration when
calculating the expression (19.9) is done within −α λ α.
Thus, in the Batdorf–Budiansky theory and for the simplest two-dimensional
model, the problem of integrating plasticity equations for any loading part is
so complex that it prevents one from making any qualitative conclusions of the
nature of loading surface changes in the case of complex loading trajectories. This
conclusion was a source of pessimistic moods of Yu.N. Rabotnov [25] as to the
possible progress of plasticity theory. In Part III, we tried to show what can be
opposed to these pessimistic conclusions.
19.3 Endochronic Plasticity Theory
Almost in all variants of plasticity theory that we discussed in previous paragraphs
of Part II, the concept of a loading surface is used. It is a priori deemed that the
loading surface can be defined experimentally. To do it, we must be able to clearly
fix the moment of plastic strain, which is impossible in principle. Apparently, any of
the existing plasticity theories can be declared unsound as contradicting the data of
thin experiments. Therefore, a doubt occurred: “. . . does the loading surface concept
have any real sense and does it have to be used as a basis when building the plasticity
theory?” [26, p. 564].
Understanding of justification of this doubt resulted in efforts to build the
plasticity theory that is not based on the loading surface concept but directly
expresses components of the stress tensor as some functions on loading trajectories.
At the brink of the 1950s, A. A. Ilyushin [9, 10] created the theory of the elastic–
plastic process of strain of continuous media in the case of complex loading that
was later developed in the scientific school created by him (V. G. Zubchaninov [35],
R. A. Vasin [34] et al). The same type of theories includes the above-mentioned
(p.157) endochronic theory of plasticity, which was proposed in 1971 by K. Valanis
[32, 33] and is intensively developing today. The evolution of the endochronic theory
of plasticity is described in the article by Yu. I.. Kadashevich, and S. P. Pomytkin
[14] with a sufficient degree of detail as well as in the book by P. V. Trusov and I. E.
Keller [30].
The theory is based on the concept of the so-called internal time. Apparently, the
theory name is related to this concept: endo—internal, chronos—time (Greek). The
determinant functions of the Valanis theory have a comparatively simple structure
of hereditary type whose form does not differ from the functions of linear viscoelasticity with the substitution of physical time with the so-called internal time. In
the initial variant, the author tried to use the following type of the determinant ratio:
σ
ij =
μ
0
L(μ − μ
)dε
ij (μ
),
(19.10)