292
19 Other Variants of Plasticity Theories
where σ
ij and ε
ij are the deviators of stress and strain tensors and μ is the internal
time defined by the ratio
dμ =
dλ
f (λ)
, dλ = |dε
ij |, f(λ) > 0,
(19.11)
where f (λ) is some positive function called the hardening function. The material
is hardening if df/dλ > 0 and softening if df/dλ < 0. The requirement of fading
memory implies specific requirements to the class of functions that L belongs to,
and there must be dL(μ)/dμ < 0. In particular, this requirement is met by the
function
L(μ) =
N
i=1
E i e
−α i μ .
(19.12)
Here E i , α i are constant values of the material defined from experiments for
complex loading.
The analysis of the ratios (19.10)–(19.12) leads to the conclusion that sources
of Valanis’s approach are found in the papers by A. A Ilyushin (see, for example,
[8]), whereof we already spoke earlier (Chap. 12). We shall also note that before
Valanis, in 1969, A. A. Vakulenko [31] introduced the concept of thermodynamic
time, which allowed efficiently studying the process of building determinant ratios
of non-elastic strains.
Despite the obvious simplicity of the determinant ratios of the Valanis theory,
they allowed for a qualitative description of many interesting effects observed
in deformation beyond the yield stress. In particular, the model (19.10)–(19.12)
qualitatively describes the effects of linear and non-linear hardening, hysteresis and
stabilization hysteresis loops in cyclic deformation, “a dive” into stress intensity in
the vicinity of the fracture point of the strain trajectory, and some others. Along with
that, the quantitative correspondence of experimental data and calculation results in
most cases cannot be deemed satisfactory.
A reason for such non-compliance is probably a tempting re-simplification of the
model. Striving for simplicity usually turns into the loss of a link with the model. An
excessive complication of the model leads to computational and sometimes principal
complications. A reasonable compromise here, as advised by R. Bellman, is that
“. . . a scientist, like a pilgrim, must take a straight and narrow path between the
Traps of Re-simplification and Swamp of Over-complication” [4].
Critics of the initial variant of the Valanis endochronic theory in multiple
publications induced the author and his adherents to create “improved or corrected
variants” of the theory. They tried to correct the theory [12, 21, 33] by complicating
the type of functionality. However, the implementation of such an approach resulted
in the “Swamp of Over-complication” as said above.
19 Other Variants of Plasticity Theories
where σ
ij and ε
ij are the deviators of stress and strain tensors and μ is the internal
time defined by the ratio
dμ =
dλ
f (λ)
, dλ = |dε
ij |, f(λ) > 0,
(19.11)
where f (λ) is some positive function called the hardening function. The material
is hardening if df/dλ > 0 and softening if df/dλ < 0. The requirement of fading
memory implies specific requirements to the class of functions that L belongs to,
and there must be dL(μ)/dμ < 0. In particular, this requirement is met by the
function
L(μ) =
N
i=1
E i e
−α i μ .
(19.12)
Here E i , α i are constant values of the material defined from experiments for
complex loading.
The analysis of the ratios (19.10)–(19.12) leads to the conclusion that sources
of Valanis’s approach are found in the papers by A. A Ilyushin (see, for example,
[8]), whereof we already spoke earlier (Chap. 12). We shall also note that before
Valanis, in 1969, A. A. Vakulenko [31] introduced the concept of thermodynamic
time, which allowed efficiently studying the process of building determinant ratios
of non-elastic strains.
Despite the obvious simplicity of the determinant ratios of the Valanis theory,
they allowed for a qualitative description of many interesting effects observed
in deformation beyond the yield stress. In particular, the model (19.10)–(19.12)
qualitatively describes the effects of linear and non-linear hardening, hysteresis and
stabilization hysteresis loops in cyclic deformation, “a dive” into stress intensity in
the vicinity of the fracture point of the strain trajectory, and some others. Along with
that, the quantitative correspondence of experimental data and calculation results in
most cases cannot be deemed satisfactory.
A reason for such non-compliance is probably a tempting re-simplification of the
model. Striving for simplicity usually turns into the loss of a link with the model. An
excessive complication of the model leads to computational and sometimes principal
complications. A reasonable compromise here, as advised by R. Bellman, is that
“. . . a scientist, like a pilgrim, must take a straight and narrow path between the
Traps of Re-simplification and Swamp of Over-complication” [4].
Critics of the initial variant of the Valanis endochronic theory in multiple
publications induced the author and his adherents to create “improved or corrected
variants” of the theory. They tried to correct the theory [12, 21, 33] by complicating
the type of functionality. However, the implementation of such an approach resulted
in the “Swamp of Over-complication” as said above.