Polymer Physics

 0 ¼lim

e! 0

E^00 ðoÞ

o

(6.37)

and the recoverable shear compliance

J 0 ¼lim

e! 0

E^0 ðoÞ

o^2 ^20

(6.38)

remain constant, both the storage and the loss moduli exhibit the scaling
relationships with respect to the frequency as shown in Fig.6.13. The characteristic
time required for imposing stress to make a flow is

t 0 ¼ 0 J 0 (6.39)

For a typical solid that exhibits linear viscoelasticity, the Maxwell model applies as

EðtÞ¼E 1 expð

t

t

Þ (6.40)

whereE 1 represents the elastic modulus at the infinitely high frequency. From
Fourier transform, one can obtain

EðoÞ¼io

(^1) ð
0
EðtÞeiotdt¼
iot
1 iot

E 1 (6.41)

or

EðoÞ¼E^0 ðoÞiE^00 ðoÞ (6.42)

where the real part and the imaginary part are

E^0 ðoÞ¼

o^2 t^2

1 þo^2 t^2

E 1 (6.43)

E^00 ðoÞ¼

ot

1 þo^2 t^2

E 1 (6.44)

respectively. For a typical viscous liquid, the real part and the imaginary part are

^0 ¼E^00 ðoÞ=o (6.45)

^00 ¼E^0 ðoÞ=o (6.46)

108 6 Polymer Deformation