3 Macroscopic Deformation Theory
This section describes the model developed by Long et al. [25, 26] and later refined
by Guo et al. [30], which is referred to as the macroscopic deformation theory
(MDT). The theory will be introduced using the model material system shown in
Fig. 1.
3.1 Continuum Model to Capture Chain Detachment
and Reattachment
The MDT is based on a continuum description of the polymer network which does
not include its detailed molecular structure. Therefore, an equivalent theoretical
picture is needed to account for the effects of dynamic crosslinks on the macroscopic
mechanics. For the model system in Fig. 1, Long et al. [25] proposed a picture with
the following assumptions (see Fig. 3).
1. Macroscopically, the polymer is a homogeneous and incompressible solid.
Microscopically, the network consists of two types of chains: permanent and
temporary chains. The permanent chains are attached to two static crosslinks and
remain connected to the network. The rest of the chains are temporary chains
since they can detach from and reattach to the network upon dissociation and
reformation of the dynamic crosslinks. The total free energy of the network is
equal to the sum of contributions from all permanent and temporary chains.
2. The initial state (t ¼ 0) is defined as the instant when mechanical loading is
applied. In the initial state, it is assumed that all temporary chains are connected
and all chains are relaxed (see Fig. 3a). This initial state is taken as the reference
configuration Ω 0 and also the ground state for the network free energy ψ.
3. Under mechanical loading, the network’s deformation is quantified by the deformation gradient F. If all temporary chains remain attached, the total free energy of
the network is ψ 0 (F), which is specified by a hyperelastic model [4]. Moreover,
the permanent and temporary chains have the same mechanical behavior, i.e.,
they carry the same free energy if they experience the same macroscopic
deformation F.
4. When a temporary chain detaches from the network (see Fig. 3b, c), it instantaneously relaxes to the free state. This assumption neglects potential viscous drag
to a detached chain, which is reasonable for hydrogels with low viscosity solvent.
Detached temporary chain may reattach to the network (see Fig. 3d). Immediately
after reattachment of a temporary chain, it is in a relaxed state. After that the
reattached chain starts to accumulate deformation according to the macroscopic
Mechanics of Polymer Networks with Dynamic Bonds
137
This section describes the model developed by Long et al. [25, 26] and later refined
by Guo et al. [30], which is referred to as the macroscopic deformation theory
(MDT). The theory will be introduced using the model material system shown in
Fig. 1.
3.1 Continuum Model to Capture Chain Detachment
and Reattachment
The MDT is based on a continuum description of the polymer network which does
not include its detailed molecular structure. Therefore, an equivalent theoretical
picture is needed to account for the effects of dynamic crosslinks on the macroscopic
mechanics. For the model system in Fig. 1, Long et al. [25] proposed a picture with
the following assumptions (see Fig. 3).
1. Macroscopically, the polymer is a homogeneous and incompressible solid.
Microscopically, the network consists of two types of chains: permanent and
temporary chains. The permanent chains are attached to two static crosslinks and
remain connected to the network. The rest of the chains are temporary chains
since they can detach from and reattach to the network upon dissociation and
reformation of the dynamic crosslinks. The total free energy of the network is
equal to the sum of contributions from all permanent and temporary chains.
2. The initial state (t ¼ 0) is defined as the instant when mechanical loading is
applied. In the initial state, it is assumed that all temporary chains are connected
and all chains are relaxed (see Fig. 3a). This initial state is taken as the reference
configuration Ω 0 and also the ground state for the network free energy ψ.
3. Under mechanical loading, the network’s deformation is quantified by the deformation gradient F. If all temporary chains remain attached, the total free energy of
the network is ψ 0 (F), which is specified by a hyperelastic model [4]. Moreover,
the permanent and temporary chains have the same mechanical behavior, i.e.,
they carry the same free energy if they experience the same macroscopic
deformation F.
4. When a temporary chain detaches from the network (see Fig. 3b, c), it instantaneously relaxes to the free state. This assumption neglects potential viscous drag
to a detached chain, which is reasonable for hydrogels with low viscosity solvent.
Detached temporary chain may reattach to the network (see Fig. 3d). Immediately
after reattachment of a temporary chain, it is in a relaxed state. After that the
reattached chain starts to accumulate deformation according to the macroscopic
Mechanics of Polymer Networks with Dynamic Bonds
137