YIELDING PROCESS - Fractal Mechanics of Polymers - page 77

÷ 30Å [18, 24] ( L min < L < L max ) and in part III chains fragments with length
of L cl or of order of several tens of Ånströms ( L > L max ). In Euclidean space
the dependence s − e will be linear ( d s/ d e = const) and in fractal one -
curvilinear, since fractal space requires deformation deceleration with time.
The yielding process realization is possible only in fractal space. The stated
model of deformation mechanisms is correct only in the case of polymers
structure presentation as physical fractal.
In the general case polymers structure is multifractal, for behavior de-
scription of which in deformation process in principle its three dimensions
knowledge is enough: fractal (Hausdorff) dimension d f , informational one
d I and correlation one d c [82]. Each from the indicated dimension describes
multifractal definite properties change and these dimensions combined ap-
plication allows to obtain more or less complete picture of yielding process
[73].
As it is known [83], a glassy polymers behavior on cold flow plateau
(part III in Fig. 4.17 ) is well described within the frameworks of the rubber
high-elasticity theory. In Ref. [39] it has been shown that this is due to me-
chanical devitrification of an amorphous polymers loosely packed matrix.
Besides, it has been shown [82, 84] that behavior of polymers in rubber-like
state is described correctly under assumption, that their structure is a regular
fractal, for which the identity is valid:

d I = d c = d f . (4.44)

A glassy polymers structure in the general case is multifractal [85], for
which the inequality is true [82, 84]:

d c < d I < d f . (4.45)

Proceeding from the said above and also with appreciation of the known
fact, that rubbers do not have to some extent clearly expressed yielding point
the authors of Ref. [73] proposed hypothesis, that glassy polymer structural
state changed from multifractal up to regular fractal, that is, criterion (4.44)
fulfillment, was the condition of its yielding state achievement. In other
words, yielding in polymers is realized only in the case, if their structure is
multifractal, that is, if it submits to the inequality Eq. (4.45).
Let us consider now this hypothesis experimental confirmations and di-
mensions d I and d c estimation methods in reference to amorphous glassy
polymers multifractal structure. As it is known [82], the informational di-
mension d I characterizes behavior Shennone informational entropy I (e):