YIELDING PROCESS - Fractal Mechanics of Polymers - page 74

where k is reaction rate, t is its duration, h is reactionary medium nonhomo-
geneity (heterogeneity) exponent (0 < h < 1), which turns into zero only for
Euclidean (homogeneous) mediums and the Eq. (4.37) becomes classical
one: k = const.
From the Eq. (4.37) it follows, that in case h ≠ 0, that is, for heteroge-
neous (fractal) mediums the reaction rate k reduces at reaction proceeding.
One should attention to qualitative analogy of curves s − e and the depen-
dences of conversion degree on reaction duration Q ( t ) for a large number
of polymers synthesis reactions [75]. Still greater interest for subsequent
theoretical developments presents complete qualitative analogy of diagrams
s - e and strange attractor trajectories, which can be have “yield tooth,”
strain hardening and so on [76].
If to consider deformation process as polymer structure reaction with
supplied, from outside mechanical energy to consider, then the modulus d s/
d e will be k analog ( Fig. 4.17 ). The said above allows to assume, that de-
formation on parts I and III (elasticity and cold flow) proceeds in Euclidean
space and on part II (yielding) - in fractal one. The comparison with the
schematic plot of Fig. 1.2 assumes also, that transitions from one part to
another were due to measurement scale change, induced by deformation.
The fractal analysis main rules usage for polymers structure and proper-
ties description [68, 77] allows to make quantitative estimation of measure-
ment scale L change at polymer deformation. There are a several methods
of such estimation and the authors of Ref. [73] use the simplest from them
as ensuring the greatest clearness. As it was noted in chapter one, the self-
similarity (fractality) range of amorphous glassy polymers structure coin-
cides with cluster structure existence range: the lower scale of self-similarity
corresponds to statistical segment length l st and the upper one - to distance
between clusters R cl . The simplest method of measurement scale L estima-
tion is the usage of well-known Richardson equation - Eq. (2.12). For PC
at testing temperature T = 293 K included in the Eq. (2.12) parameters are
equal to: L ch = L cl = 76.5 Å, R ch = R cl = 30.1 Å [78] and the value D ch can be
calculated according to the Eq. (2.5).
In the case of affine deformation the value R cl will be changed propor-
tionally to drawing ratio l [70]. This change value can be estimated from the
equation [79]:

2(1 - n )

Rl l =

. (4.38)

cl st

(1 - 2 n )