Chemistry Reference
In-Depth Information
D
n
replacement on Young's modulus
E
the Eq.
(14.14) transforms into a classical Hooke law. The appearance in this equa-
tion of operator
It is easy to see, that at
fr
D
n
is due the indicated above polymers structure fractality.
Proceeding from the said above, it can be supposed that the operator
fr
D
n
fr
E
n
.
Hence, a solid-phase polymers deformation process is realized in fractal
space with the dimension, which is equal to structure dimension
d
f
. In such
space the deformation process can be presented schematically as the “dev-
il's staircase” [39]. Its horizontal sections correspond to temporal intervals,
where deformation is absent. In this case deformation process is described
with using of fractal time
t
, which belongs to the points of Cantor's set [30].
If Euclidean object deformation is considered then time belongs to real num-
bers set.
For processes evolution with fractal time the mathematics of fractional
integration and differentiation is used [39]. As it has been shown in Ref. [30],
in this case the fractional exponent in
fr
coincides with Cantor's set fractal di-
mension and indicates system states fraction, preserved during all evolution
time
t
. Let us remind, that Cantor's set is considered in one-dimensional Eu-
clidean space (
d
= 1) and therefore, its fractal dimension
d
f
< 1 in virtue of
fractal definition [39]. For fractal objects in Euclidean spaces with higher di-
mensions (d > 1) the fractional part of
d
f
should be accepted as n
fr
according
to the Eq. (4.32) [40]. Then the n
fr
value characterizes the fractal (polymer
structure) fraction, which is invariable in deformation process [31].
For
d
= 3 and 2 <
d
f
< 3 (the fractal object in three-dimensional Euclidean
space) the following modification of the Eq. (14.14) can be written:
should be written as
fr
d
-2
se
=
Y
E
f
,
(14.15)
Y
where e
Y
is yield strain.
to the Eq. (14.15)
s
yield stress values for extrudates of UHMPE, com-
ponors UHMPE-Al and UHMPE-bauxite, prepared with various extrusion
draw ratio, is accepted, from which the good conformity of theory and ex-
periment follows.
T
