Chemistry Reference
In-Depth Information
4 and all said above can be repeated in reference to this space: at
d
= 4 and
d
f
= 3 the value in
fr
= 0 and
dU
= 0. Let us note in conclusion that exactly the
exponent n
fr
controls the value of deformation (fracture) energy of fractal
objects as a function of process length scale. Let us note that the equality
dU
= j
l.m.
was shown, from which structural sense of fractional exponent in
polymers inelastic deformation process follows: n
fr
= j
l.m.
[64].
Mittag-Lefelvre function [59] usage is one more method of a diagrams
s − e description within the frameworks of the fractional derivatives math-
ematical calculus. A nonlinear dependences, similar to a diagrams s − e for
polymers, are described with the aid of the following equation [65]:
(
)
( )
€
‚
se s
=
1
-
E
-
e
n
‡
,
(4.34)
fr
†
0
n
,1
fr
where s
o
is the greatest stress for polymer in case of linear dependence s(e)
(of ideal plasticity), is the Mittag-Lefelvre function [65]:
∞
n
k
e
(
)
fr
∑
n
E
-=
e
fr
, n
fr
> 0, b > 0,
(4.35)
(
)
n
fr
,1
Ãk
nb
+
k
=
0
fr
where Г is Eiler gamma-function.
As it follows from the Eq. (4.34), in the considered case b = 1 and gam-
ma-function is calculated as follows [40]:
n ne
(
)
(
)
-
-
--
n
n
ne
ne
k
fr
(
)
k
fr
.
(4.36)
n
k
Ãk
+= -
1
k
fr
fr
fr
fr
fr
2
FIGURE 4.16
The experimental (1 ÷ 3) and calculated according to the Eqs. (4.34) ÷ (4.36)
(4 ÷ 6) diagrams s - e for PAr at
T
= 293 (1, 4), 353 (2, 5) and 433 K (3, 6). The shaded lines
indicate calculated diagrams s - e for forced high-elasticity part without n
fr
change [66].
