CONTINUUM MECHANICS for ENGINEERS

proportional to the probability per unit volume, P(r), and using Boltzmann’s
equation, the entropy may be written as
η= η + ()= η + ( πρ)+
ρ
⎛
2
⎡
r ⎞ ⎤
o kln P r o k⎢3ln
⎜ ⎟ ⎥
⎣⎢
⎝ ⎠
⎦⎥
(8.1-10)
where k is Boltzmann’s constant. Use of this in Eq 8.1-9 for an ideal rubber
gives a single chain retractive force given by
2kθ F = r 2
ρ
(8.1-11)
Consider a polymer having forces applied resulting in stretch ratios λ 1, λ 2,
and λ 3. The work done on each chain of the material is the sum of the work
done in each coordinate direction x i
() i
() i
λ xi
λ x
k
i
() i
2 θ
kθ
2
() i
W = fidxi = xdx i i =
⎡(
) −
⎤ 2
λ 1 x
∫ 2 i
x ρ ∫
2
x ρ ⎣⎢ ⎦⎥
i
(no sum)
Taking into consideration the work done on all of the chains gives total work
in each coordinate direction
(no sum) (8.1-12)
where the last summed term is the number of chains, n, times the initial
mean end-to-end distance in the x i direction. Assuming the rubber is initially
isotropic yields
(no sum)
Substituting this and ρ from Eq 8.1-6 into Eq 8.1-12 and adding all three
coordinate work terms gives
For convenience, this equation may be written as
i
() i kθ
2
() i
∑W= ⎡(
) −
⎤ 2
λ 1 ∑x
2
i
ρ ⎣⎢ ⎦⎥
n
∑
n
n
xi= n x = r
3
2 2 2
i i i
W nk 2
θ r
i 2 2 2
= λ + λ + λ −3
2 1 2 3
2 r
o
2
n
[ ]
[ ]
VG 2 2 2
W = λ1+ λ2+ λ3−3
2
(8.1-13)
(8.1-14)