FRACTURE - Fractal Mechanics of Polymers - page 138

L > a and κ > 1, is realized in the case, when the indicated chain part loses
its fractal properties and becomes Euclidean object, that is, when its fractal
dimension D ch becomes equal to topological dimension, in other words, at
the condition D ch = 1.0 fulfillment [9].
As it is known [14], the value D ch can be determined according to the Eq.
(2.12), from which it follows, that the condition D ch = 1.0 means R ch = L ch ,
that is, it defines chain part, stretched completely between macromolecular
entanglements. Such situation is typical for the oriented polymers [16] that
predetermines dilation concept correct application for them [3]. Hence, the
condition κ > 1 is due to chain part between macromolecular entanglements
transition from fractal behavior to Euclidean one and corresponds to pho-
nons delocalization (L > a ) [10].
Let us consider the reasons of the adduced in Table 7.1 values κ < 1
for nonoriented polymers. As it has been shown in chapter two, the value
D ch characterizes molecular mobility (deformability) level of the indicated
chain part [17]. At D ch = 1.0 this mobility is suppressed completely and at
D ch =2.0 it reaches the greatest possible level, typical for rubber-like state.
Molecular mobility intensification results to corresponding stress relaxation
intensification, applied to chain part between entanglements and, as conse-
quence, to its reduction lower than macroscopic sample fracture stress sf f
[18]. Such treatment assumes availability of the correlation between param-
eters κ and D ch . This assumption is confirmed by the plot of Fig. 7.1 , where
the dependence κ 2 ( D ch ) for 10 polymers, pointed out in table 7.1, is adduced
and, besides, two points of the data for HDPE samples with a sharp notch,
accepted according to [19], were shown. As it follows from the plot of Fig.
7.1, the dependence κ( D ch ) is stronger than the linear one, that is, the value
κ 2 grows faster than D ch reduces, pointing out to strong dependence of stress
relaxation on molecular level on molecular mobility degree [10].
In insertion of Fig. 7.1 the dependence of overloading coefficient mean
values κ m (κ m = 0.5(κ 1 + κ 2 )) on value

D - is adduced. As one can see, a good
linear correlation between the indicated parameters is observed, with the
exception of only two polymers (PTFE and PSF) which have considerable,
although symmetrical, scatter. This gives the possibility to express κ m as D ch
function as follows [9]:

1
ch

- 1

κ ≈ D

. (7.4)

m ch