  To design increasingly tough, resilient, and fatigue-resistant elastomers and
hydrogels, the relationship between controllable network parameters at the
molecular level to macroscopic quantities that govern damage and failure must
be established. Constitutive models based upon statistical mechanics have used
variants of the freely jointed chain (FJC) model with rigid links. However,
since the free energy state of a polymer chain is dominated by enthalpic bond
distortion effects as the chain approaches its rupture point, bond
extensibility ought to be accounted for if the model is intended to capture
chain rupture. To that end, a new bond potential is supplemented to the FJC
model (as derived in the uFJC framework of Buche and colleagues), which we have
extended to yield a tractable, closed-form model that is amenable to
constitutive model development. Inspired by the asymptotically matched uFJC
model response, a simple, quasi-polynomial, and anharmonic bond potential
energy function is derived. Using this bond potential, approximate yet
highly-accurate analytical functions for bond stretch and chain force dependent
upon chain stretch are established. Then, using this polymer chain model, a
stochastic thermal fluctuation-driven chain rupture framework is developed.
This framework is based upon a force-modified tilted bond potential that
accounts for distortional bond potential energy, allowing for the calculation
of dissipated chain scission energy. The model is fit to single-chain
mechanical response data collected from atomic force microscopy tensile tests
for validation and to glean deeper insight into the molecular physics taking
place. Due to their analytical nature, this polymer chain model and the
associated rupture framework can be straightforwardly implemented in finite
element models accounting for fracture and fatigue in polydisperse elastomer
networks.
