Polymer Physics

If the number density of the network chains,

N 0 ¼

N

V

¼

N

A 0 l 0

(3.26)

we have

s¼CN 0 kT eþ 1 

1

ðeþ 1 Þ^2

"

(3.27)

When a small deformatione<<1,

1

ðeþ 1 Þ^2

 1  2 e (3.28)

we obtain

s¼ 3 CN 0 kTe (3.29)

Thus, small deformations of a cross-linked network follow the Hooke’s law,
with the elastic modulus

E¼ 3 CN 0 kT (3.30)

Under large deformations, the following equation is more applicable,

s¼CN 0 kT l

1

l^2



(3.31)

Equation (3.31) is called the equation of state of the rubber. This equation was
firstly derived by Guth and James in 1941 (Guth and James 1941 ). We convention-
ally make an ideal-chain approximation withC¼1.
The measured results of a stretching experiment are normally treated with the
empirical Mooney-Rivlin relation (Mooney 1940 ; Rivlin 1949 ), as given by

s¼ 2 C 1 eþ 1 

1

ðeþ 1 Þ^2

"

þ 2 C 2 1 

1

ðeþ 1 Þ^3

"

(3.32)

whereC 1 andC 2 are two fitting parameters. Commonly one can say that, at the
right-hand side of the equation above, the first term represents the contribution from
the entropic elasticity of an ideal network, while the second term represents those
non-ideal contributions during the deformation, such as energetic elasticity, strain-
induced crystallization, limited extensibility of chains, and various network defects,

3.2 Statistical Mechanics of Rubber Elasticity 39