Chemistry Reference
In-Depth Information
The correlation dimension
d
c
is connected with multifractal structure in-
ternal energy
U
[61] and it can be estimated according to the equation [82]:
(
)
d
-
d
D
U
=
-
c
l
c
2
F
,
(4.52)
where D
U
is internal energy change in deformation process,
c
2
is constant,
l
F
is macroscopic drawing ratio, which in the case of uniaxial deformation
is equal to l
1
. The value l
F
is determined as follows [82]:
l
F
= l1l2l3,
1
l
2
l
3
,
(4.53)
where l
2
and l
3
are transverse drawing rations, connected with l
1
by the
simple relationships [82]:
l
2
= 1 + e
2
,
(4.54)
l
3
= 1 + e
3
,
(4.55)
e
2
= e
3
= ne
1.
(4.56)
The temperature dependence of
d
c
can be calculated, as and earlier, as-
suming from the condition (4.44) that at yielding
d
c
=
d
f
= 2.82, estimating
D
U
as one half of product s
Y
e
Y
(with appreciation of practically triangular
form of curve s - e up to yield stress) and determining the constant c
2
by
the indicated above mode. Comparison of the temperature dependences of
multifractal three characteristic dimensions
d
c
,
d
I
and
d
f
, calculated accord-
As it follows from the plots of this figure, for PC the inequality Eq. (4.45)
is confirmed, which as a matter of fact is multifractal definition. The de-
pendences
d
c
(
T
) and
d
I
(
T
) are similar and their absolute values are close,
that is explained by the indicated above intercommunication of
f
g
and
U
change [89]. Let us note, that dimension
d
I
controls only yield strain e
Y
and
dimension
d
c
- both e
Y
and yield stress s
Y
. At approaching to glass transition
temperature, that is, at
T
®
T
g
, the values
d
c
,
d
I
and
d
f
become approximately
equal, that is, rubber is a regular fractal. Thus, with multifractal formalism
positions the glass transition can be considered as the transition of structure
from multifractal to the regular fractal. Additionally it is easy to show the
fulfillment of the structure thermodynamical stability condition [90]:
d
f
(
d
-
d
I
)
=
d
c
(
d
- d
c
).
(4.57)
