is not convenient to directly apply Eq. (26) or (39) to calculate the stress. Instead, we
will start from Eq. (24) and reapply the Coleman-Noll procedure [34]. Using
Eq. (20) to replace S : _
E in Eq. (24) by Jσ : d and recognizing that J ¼ 1 due to
incompressibility, we can rewrite Eq. (24) as
σ : d À _
ψ À η _
T ! 0,
ð71Þ
where _
ψ is calculated using Eq. (70):
_
ψ ¼ À _
T C T ln
T
T 0
þ η 0
þ Δ _
ψ e :
ð72Þ
To determine Δ _
ψ e , we first plug Eq. (55) into Eq. (62) and find that the terms
associated with the constant ψ c0 are canceled, which turns Eq. (62) into
Δψ e X, t
ð Þ ¼
3k B T
2Nb
2
ϕ X, r, t
ð
ÞÀϕ 0 X, r, t
ð
Þ
ð
Þ r
2
þ p J À 1
ð
Þ:
ð73Þ
Recall that the angle bracket “hi” represents a volumetric integral over the chain
space as shown in Eq. (58). Taking material time derivative of Eq. (73) in the
continuum space and using Eqs. (60) and (62), we can get
Δ _
ψ e X, t
ð Þ ¼
3k B _
T
2Nb
2
ϕ À ϕ 0
ð
Þr
2
þ
3k B T
2Nb
2
∂ϕ
∂t
À
∂ϕ 0
∂t
r
2
(
)
þ tr d
ð Þp:
ð74Þ
Using Eq. (68), we can show that
∂ϕ
∂t
r
2
(
)
¼ k a c t À c X, t
ð Þ
½
p c r
ð Þr
2
À k d ϕr
2
À l : ∇ r ϕ r
ð
Þ r
2
:
ð75Þ
Using integration by parts and assuming ϕ approaches zero as r approaches
infinity in the chain space, we obtain the following identity:
l : ∇ r ϕ r
ð
Þ r
2
¼ À ϕr
2
tr l
ð Þ À 2l : ϕr r
h
i¼ À2l : ϕr r
h
i:
ð76Þ
Note that tr(l) ¼ 0 due to the incompressibility condition. Therefore,
∂ϕ
∂t
r
2
(
)
¼ k a c t À c X, t
ð Þ
½
p c r
ð Þr
2
À k d ϕr
2
þ 2l : ϕr r
h
i:
ð77Þ
The derivative ∂ϕ 0 /∂t in Eq. (74) deserves additional explanations. The distribution function ϕ 0 (X, r, t) represents the ground free energy state where all attached
chains are in their natural states and thus follow the Gaussian distribution specified
by p c (r). This ground state is introduced such that Δψ e ¼ 0 when the network is
Mechanics of Polymer Networks with Dynamic Bonds
155
will start from Eq. (24) and reapply the Coleman-Noll procedure [34]. Using
Eq. (20) to replace S : _
E in Eq. (24) by Jσ : d and recognizing that J ¼ 1 due to
incompressibility, we can rewrite Eq. (24) as
σ : d À _
ψ À η _
T ! 0,
ð71Þ
where _
ψ is calculated using Eq. (70):
_
ψ ¼ À _
T C T ln
T
T 0
þ η 0
þ Δ _
ψ e :
ð72Þ
To determine Δ _
ψ e , we first plug Eq. (55) into Eq. (62) and find that the terms
associated with the constant ψ c0 are canceled, which turns Eq. (62) into
Δψ e X, t
ð Þ ¼
3k B T
2Nb
2
ϕ X, r, t
ð
ÞÀϕ 0 X, r, t
ð
Þ
ð
Þ r
2
þ p J À 1
ð
Þ:
ð73Þ
Recall that the angle bracket “hi” represents a volumetric integral over the chain
space as shown in Eq. (58). Taking material time derivative of Eq. (73) in the
continuum space and using Eqs. (60) and (62), we can get
Δ _
ψ e X, t
ð Þ ¼
3k B _
T
2Nb
2
ϕ À ϕ 0
ð
Þr
2
þ
3k B T
2Nb
2
∂ϕ
∂t
À
∂ϕ 0
∂t
r
2
(
)
þ tr d
ð Þp:
ð74Þ
Using Eq. (68), we can show that
∂ϕ
∂t
r
2
(
)
¼ k a c t À c X, t
ð Þ
½
p c r
ð Þr
2
À k d ϕr
2
À l : ∇ r ϕ r
ð
Þ r
2
:
ð75Þ
Using integration by parts and assuming ϕ approaches zero as r approaches
infinity in the chain space, we obtain the following identity:
l : ∇ r ϕ r
ð
Þ r
2
¼ À ϕr
2
tr l
ð Þ À 2l : ϕr r
h
i¼ À2l : ϕr r
h
i:
ð76Þ
Note that tr(l) ¼ 0 due to the incompressibility condition. Therefore,
∂ϕ
∂t
r
2
(
)
¼ k a c t À c X, t
ð Þ
½
p c r
ð Þr
2
À k d ϕr
2
þ 2l : ϕr r
h
i:
ð77Þ
The derivative ∂ϕ 0 /∂t in Eq. (74) deserves additional explanations. The distribution function ϕ 0 (X, r, t) represents the ground free energy state where all attached
chains are in their natural states and thus follow the Gaussian distribution specified
by p c (r). This ground state is introduced such that Δψ e ¼ 0 when the network is
Mechanics of Polymer Networks with Dynamic Bonds
155