YIELDING PROCESS - Fractal Mechanics of Polymers - page 69

FIGURE 4.14 The relation between increment of segments number D N Y in crystallites,
subjecting to partial melting, and increment of free volume microvoids number D N h , which is
necessary for yielding process realization, for HDPE [51].

Lately the mathematical apparatus of fractional integration and differen-
tiation [58, 59] was used for fractal objects description, which is amorphous
glassy polymers structure. It has been shown [60] that Kantor's set fractal
dimension coincides with an integral fractional exponent, which indicates
system states fraction, remaining during its entire evolution (in our case de-
formation). As it is known [61], Kantor's set (“dust”) is considered in one-
dimensional Euclidean space ( d = 1) and therefore, its fractal dimension
obey the condition d f < 1. This means, that for fractals, which are considered
in Euclidean spaces with d > 2 ( d = 2, 3, …) the fractional part of fractal
dimension should be taken as fractional exponent n fr [62, 63]:

n

fr = d f - ( d - 1).

(4.32)

The value n fr characterizes that states (structure) part of system (poly-
mer), which remains during its entire evolution (deformation). In Fig. 4.15 ,
the dependence of latent energy fraction dU at PC and poly(methyl methac-
rylate) (PMMA) deformation on n fr = d f - ( d - 1) [64] is shown. The value
dU was estimated as ( W - Q )/ W . In Fig. 4.15, the theoretical dependence
dU (n fr ) is adducted, plotted according to the conditions dU = 0 at n fr = 0 or d f
= 2 and dU = 1 at n fr = 1 or d f = 3 ( d = 3), that is, at n fr or d f limiting values
[64]. The experimental data correspond well to this theoretical dependence,
from which it follows [64]: