Polymer Physics

The Maxwell modelis a series connection of the two models above, representing
the linear viscoelasticity(Maxwell 1867 ), as given by

de

dt

¼

1

E

ds

dt

þ

s



(6.17)

Upon instantaneous application of a constant strain, the stress will gradually
relax down over time. This process is called thestress-relaxationexperiment. From
the condition de/dt¼0, the Maxwell model solves the stress decaying with an
exponential function, as given by

sðtÞ¼s 0 expð

t

t

Þ (6.18)

where the characteristic relaxation time is defined as

t¼



E

(6.19)

The Kelvin model (also called as the Voigt model or the Kelvin-Voigt model) is
a parallel connection of the spring and dashpot models (Kelvin, L. (Thompson, W.)
1875 ; Voigt 1892 ), representing the anealstic behavior, as given by

s¼

de

dt

þEe (6.20)

Under the condition of a constant stress, the strain slowly increases over time,

e¼

s

E

½ 1 expð

t

t

ފ (6.21)

Similarly, the characteristic retardation time is defined ast¼/E.
A series connection of the Maxwell and Kelvin models makes the four-element
model, known as the Burger’s model (Burgers 1935 ), which can describe the
viscoelastic creep behaviors of polymers, as given by

e¼

s

E

þ

s



tþ

s

E

½ 1 expð

t

t

ފ (6.22)

As illustrated in Fig.6.10,a!b represents the instant elastic response, b!c
represents the anelastic response and permanent deformation made by the viscous
fluid, c!d represents the instant elastic recovery, d!e represents the gradual
recovery from the anelastic deformation in the viscous fluid, and the height of
e represents the permanent deformation of the viscous fluid that could not be
recovered. Here, the two springs are not necessary to be identical, and neither are
the two dashpots.

102 6 Polymer Deformation