YIELDING PROCESS - Fractal Mechanics of Polymers - page 49

density of entanglements cluster network, n cl , then the density of linear de-
fects, r d , per volume unit of the polymer can be expressed as follows [1]:
r

d = n cl × l st .

(4.1)

The offered treatment allows application of well developed mathematical
apparatus of the dislocation theory for the description of amorphous poly-
mers properties. Its confirmation by the X-raying methods was stated in Ref.
[18].
Further on, the rightfulness of application of the structural defect con-
cept to polymers yielding process description will be considered. As a rule,
previously assumed concepts of defects in polymers were primarily used for
the description of this process or even exclusively for this purpose [4-11].
Theoretical shear strength of crystals was first calculated by Frenkel, basing
on a simple model of two atoms series, displaced in relation to one another
by the shear stress ( Fig. 4.1a ) [3]. According to this model, critical shear
stress t 0 is expressed as follows [3]:

G

t =

, (4.2)

0

2 p

where G is the shear modulus.
Slightly changed, this model was used in the case of polymers yielding
[6], wherefrom the following equation was obtained:

G

, (4.3)

t

Υ = 0

p 3

where t 0Y is a theoretical value of the shear stress at yielding.
Special attention should be paid to the fact that characterizes principally
different behavior of crystalline metals compared with polymers. As it is
known [3, 19], t 0Y /t Y ratio (where t Y is experimentally determined shear
stress at yielding) is much higher for metals than for polymers. For five met-
als possessing the face-centered cubic or hexagonal lattices the following
ratios were obtained: t 0Y /t Y = 37400 ¢ 22720 (according to the data of Ref.
[3]), whereas for five polymers this ratio makes 2.9 ¢ 6.3 [6]. In essence, suf-
ficient closeness of t 0Y /t Y ratio values to one may already be the proof for the
possibility of realizing of Frenkel mechanism in polymers (in contrast with
metals), but it will be shown below that for polymers a small modification of
the law of shear stress t periodic change used commonly gives t 0Y /t Y values
very close to one [20].