5 Summary and Discussions
In this chapter, two approaches for modeling the mechanics of polymer networks
with dynamic bonds are described: macroscopic deformation theory (MDT) and
transient network theory (TNT). Both approaches follow the framework of continuum mechanics and thermodynamics and are capable of connecting the timedependent constitutive equation for stress to the molecular kinetics of dynamic
bonds. The MDT is based on a continuum description of the polymer network and
accounts for the effects of dynamic bonds through phenomenological kinematic
assumptions, whereas the TNT links the macroscopic mechanics to single chains
through the chain distribution function and thus has a stronger physical underpinning. Nevertheless, either MDT or TNT can be used to calculate the multiaxial stress
tensor under a prescribed deformation history, and therefore are useful for simulating
the responses of polymers with dynamic bonds under complex loading conditions.
For example, the MDT has been applied to study the fracture mechanics, i.e.,
deformation and growth of cracks, in a dual crosslink PVA hydrogel [22, 37–
39]. The TNT, together with a computational formulation based on Eulerian kinematics, has been applied to simulate large deformation in dynamic networks under
cavitation and indentation [40]. From the perspective of computational
implementations, the MDT can be readily implemented in commercial finite element
software such as ABAQUS (Dassault Systèmes Simulia, Providence, RI, USA) since
it is based on the continuum description. However, the need to track the populations
of reattached chains may pose challenges to the computation [41], because a new
internal variable needs to be added at each time increment to represent the newly
reattached chains. In contrast, the TNT tracks the chain distribution function in a
fixed domain, i.e., the chain space, and thus can avoid the need to increase the
number of internal variables with time.
There is ample room for extension in both the MDT and TNT. For example, the
neo-Hookean model adopted in the MDT can be replaced by other hyperelastic
models to account for strain stiffening [30]. Similarly, the freely jointed chain model
with Gaussian statistics adopted in the TNT can be replaced by that with Langevin
statistics to account for strain stiffening [36]. More broadly, the MDT falls into a
class of constitutive models with continuous microstructural change [42]. Although
discussions in this chapter revolve around the model system shown in Fig. 1, the
theoretical picture of chain detachment and reattachment can be generalized to other
molecular mechanisms. Here we highlight two examples: chain scission and bond
exchange. Chain scission, i.e., breaking of polymer chains under force, can be
considered as a chain detachment process, and its kinetics is highly dependent on
the chain force or stretch. Also, in the case of chain scission, detached chain cannot
reattach to the network. The stiffness degradation and loading-unloading hysteresis
caused by chain scission can be captured either using the MDT [43, 44] or the TNT
[45]. Bond exchange refers to the mechanism in which a detached chain reacts with
an attached chain and the two chains exchange connectivity [17]. Through bond
exchange, the network topology is rearranged, and the free energy stored in the
162
Q. Guo and R. Long
In this chapter, two approaches for modeling the mechanics of polymer networks
with dynamic bonds are described: macroscopic deformation theory (MDT) and
transient network theory (TNT). Both approaches follow the framework of continuum mechanics and thermodynamics and are capable of connecting the timedependent constitutive equation for stress to the molecular kinetics of dynamic
bonds. The MDT is based on a continuum description of the polymer network and
accounts for the effects of dynamic bonds through phenomenological kinematic
assumptions, whereas the TNT links the macroscopic mechanics to single chains
through the chain distribution function and thus has a stronger physical underpinning. Nevertheless, either MDT or TNT can be used to calculate the multiaxial stress
tensor under a prescribed deformation history, and therefore are useful for simulating
the responses of polymers with dynamic bonds under complex loading conditions.
For example, the MDT has been applied to study the fracture mechanics, i.e.,
deformation and growth of cracks, in a dual crosslink PVA hydrogel [22, 37–
39]. The TNT, together with a computational formulation based on Eulerian kinematics, has been applied to simulate large deformation in dynamic networks under
cavitation and indentation [40]. From the perspective of computational
implementations, the MDT can be readily implemented in commercial finite element
software such as ABAQUS (Dassault Systèmes Simulia, Providence, RI, USA) since
it is based on the continuum description. However, the need to track the populations
of reattached chains may pose challenges to the computation [41], because a new
internal variable needs to be added at each time increment to represent the newly
reattached chains. In contrast, the TNT tracks the chain distribution function in a
fixed domain, i.e., the chain space, and thus can avoid the need to increase the
number of internal variables with time.
There is ample room for extension in both the MDT and TNT. For example, the
neo-Hookean model adopted in the MDT can be replaced by other hyperelastic
models to account for strain stiffening [30]. Similarly, the freely jointed chain model
with Gaussian statistics adopted in the TNT can be replaced by that with Langevin
statistics to account for strain stiffening [36]. More broadly, the MDT falls into a
class of constitutive models with continuous microstructural change [42]. Although
discussions in this chapter revolve around the model system shown in Fig. 1, the
theoretical picture of chain detachment and reattachment can be generalized to other
molecular mechanisms. Here we highlight two examples: chain scission and bond
exchange. Chain scission, i.e., breaking of polymer chains under force, can be
considered as a chain detachment process, and its kinetics is highly dependent on
the chain force or stretch. Also, in the case of chain scission, detached chain cannot
reattach to the network. The stiffness degradation and loading-unloading hysteresis
caused by chain scission can be captured either using the MDT [43, 44] or the TNT
[45]. Bond exchange refers to the mechanism in which a detached chain reacts with
an attached chain and the two chains exchange connectivity [17]. Through bond
exchange, the network topology is rearranged, and the free energy stored in the
162
Q. Guo and R. Long