Polymer Physics

viscous response with permanent deformation, exhibiting the feature of a liquid. In
ancient Greek, Heraclitus (540–475 B.C.) held the philosophical view that “every-
thing flows”. In an Old Testament scripture (the Book of Judges Chap. 5 , verse 5),
the prophetess Deborah sang: “The mountains flow before the Lord”, provided that
God’s observation time could be infinitely long.
When the solid feature dominates the mechanical response of a shear deforma-
tion, the shear stresssis proportional to the shear straing, and the proportionality
coefficient is the shear modulusE. On the other hand, when the liquid feature
dominates the response, the shear stresssis proportional to the shear rateg^0 , the
proportionality coefficient is the shear viscosity. Maxwell equation of linear
viscoelasticity can be applied to describe the continuous switching between the
solid and the liquid (Maxwell 1867 ),

g^0 ¼

s



þ

s^0

E

(6.24)

When a constant stress is imposed (its time derivatives^0 ¼0), this equation
describes the ideal Newtonian fluid under steady shear flow. When!1, this
equation describes the ideal elastic solid. The instantaneous response of the solid to
an imposed stress is elastic, and the shear modulusEcorresponds to the modulus
E 1 at high frequency. Consequently, the shear stress will relax down to zero
exponentially. Under the condition ofg^0 ¼0, the exponential function (6.18) can
be solved from (6.24), which defines the characteristic relaxation time as

t¼



E 1

ort¼J 1 (6.25)

WhenE 1 holds in constant, for instance, the glass modulus, or the rubbery
modulus exhibiting linear viscoelasticity,

t/ (6.26)

we obtain the shift factor

a 1 ¼

t 2

t 1

¼

 2

 1

(6.27)

Heret 1 andt 2 represent different time instants, anda 1 follows the Williams-
Landel-Ferry (WLF) empirical equation for polymers,

loga 1 ¼

C 1 ðTTsÞ

C 2 þTTs

(6.28)

whereTsis the reference temperature, andC 1 ,C 2 are two constants (Williams
et al. 1955 ).

104 6 Polymer Deformation