LOCAL PLASTICITY - Fractal Mechanics of Polymers - page 99

It is obvious, that this circumstance creates obstacles to the value h p predic-
tion and, respectively, to prediction of polymers quasibrittle (quasiductile)
failure parameters. Therefore, the authors of Refs. [10-12] established inter-
communication between h p and polymers structural characteristics within
the frameworks of percolation theory and fractal analysis on the example
of two polymers - HDPE and polystyrene (PS). These polymers were cho-
sen because they belong to different classes (semicristalline and amorphous
glassy ones, respectively) and are deformed by different mechanisms (shear
yielding and crazing, respectively). Therefore, receiving of identical results
for them allows to assume high enough degree of the proposed below treat-
ment community.
The percolation theory [13, 14] supposes formation at percolation thresh-
old infinite cluster, joining the system. Polymers structure itself can be con-
sidered as percolation system at high enough molecular weights, since ex-
actly entanglements network formation on the entire length of sample gives
to it strength and deformability [15]. Such percolation network formation at
temperature reduction up to T g was shown in Refs. [16, 17] with the cluster
model using. In such treatment probability of particle (statistical segment)
belonging to infinite percolation network is equal to j cl [17]. At assumption,
that the value r p is correlation length of such percolation system it can be
written [14]:

( )

φφ -

n

, (5.4)

r ~ -

p

p cl 0

where j o is percolation threshold, n p is percolation index.
As it has been shown [16, 17], the glass transition (melting) tempera-
ture is percolation threshold for polymers structure on temperature scale. At
approaching to it from below the j cl value decreases and can become any
amount of small, that allows to accept in the first approximation j o = 0. In
Fig. 5.1 , the dependence r p (j cl ) in double logarithmic coordinates for HDPE
and PC is shown, which corresponds to the Eq. (5.4).
As one can see, the data for both considered polymer lie on one straight
line, that allows to determine the value n p , which is equal to 0.76. This mag-
nitude corresponds well to classical value of correlation length percolation
index n p , which is equal to ~0.80 [14].
Let us consider the critical index n p physical significance. In Ref. [18] it
has been shown that percolation cluster is a fractal object with dimension d f ,
for which the following relationship is valid: