From Chain Statistics to Damage and Fracture in Elastomers
Elastomers are materials composed of flexible polymer chains that are oriented randomly, connected together via cross-links, and arranged in a network structure. The leading cause for the mechanical failure of elastomer networks is fracture, which can limit their viability in real-world applications. Under increasing deformation, elastically-active polymer chains begin to elongate, and bonds composing the backbone of these chains become stretched. Eventually, some of these bonds or the bonds of the cross-links rupture. These discrete chain rupture events in the elastomer network collectively build-up, which ultimately contributes to macroscale fracture and failure. In light of the hierarchical nature of elastomer fracture, a detailed understanding of the relationship between controllable network features at the molecular level and the fracture response at the macroscopic specimen level is imperative in order to design next-generation elastomer networks with enhanced toughness, resilience, and fracture-resistance. The paradigm of micromechanical constitutive modeling provides the physics-based tools necessary to model elastomer network mechanics, damage, and fracture. This dissertation presents several micromechanical modeling endeavors that employ the principles of chain statistics (and, at times, numerical analysis considerations) to capture damage and fracture in elastomer networks. This dissertation begins by extending the arbitrarily-extensible freely-jointed chain (uFJC) model (developed by Buche and Silberstein (2021) and Buche et al. (2022)) to the “composite” uFJC (cuFJC) model. To do so, the principles of asymptotic matching are utilized to derive a simple, quasipolynomial “composite” bond potential. This potential is then supplemented to a slightly-amended version of the uFJC model. The use of approximate yet highly-accurate analytical functions help cast the resulting cuFJC model in a numerically tractable fashion. Principles of mechanochemistry are then used to derive a stochastic chain rupture framework, where the energy dissipated upon chain scission is calculated in a statistical fashion. The cuFJC model is fitted to single chain mechanical response data collected from atomic force microscopy tensile tests for validation. The associated chain scission framework is also applied here to glean deeper insight into the molecular physics taking place in the experiments. Next, the cuFJC scission model is incorporated within a modified Lake-Thomas theory of polymer fracture. The resulting Lake-Thomas fracture toughness is now richly statistical, thanks to its grounding in the cuFJC scission model. We analyze the sensitivity of this statistical Lake-Thomas fracture toughness to several different parameters, including the number of segments composing the chains, the shape of the potential energy landscape governing segment extensibility, and the rate of applied loading. The behavior of this statistical Lake-Thomas fracture toughness is also compared with the original Lake and Thomas (1967) theory and a recently-proposed adjusted theory from Wang et al. (2019). Following this, a statistical formulation is developed that models dynamically cross-linked polymer networks where reversible bond exchange and irreversible chain rupture can occur simultaneously. This formulation carefully distinguishes reversible bond exchange from irreversible chain scission in order to track the progression of overall damage in dynamic polymer networks. To accomplish this, the transient network theory of Vernerey et al. (2017) is amended to account for chains that rupture after being pulled past a critical stretch. Using the model, the observable material timescales of relaxation and self-healing are related to the kinetic rates of attachment and detachment. Chain recovery in self-healing experiments is also related to the level of irreversibly ruptured chains in the network. The efficacy of the model is demonstrated by closely matching experimental data of cyclic loading and self-healing experiments. In the next contribution from this dissertation, a series of affine and non-affine microsphere models accounting for directionally-dependent chain scission in polydisperse elastomer networks is developed. Two affine microsphere model variants are considered, assuming equal force and equal strain load sharing. A non-affine microsphere model is then subsequently developed. Implications from the domains of numerical analysis and thermodynamics were carefully taken into consideration for the derivation of the non-affine microsphere model and its computational implementation. The constitutive behavior of these microsphere models was validated via computational uniaxial tension tests. Finally, a localizing gradient-enhanced damage model for elastomers, grounded in polymer chain statistical mechanics and the generalized micromorphic framework, is developed. Polydisperse elastomer chains are conveniently described by the cuFJC model. Irreversible elastomer network damage is calculated in a statistical manner. The non-local damage and fracture response is dependent upon a non-local chain stretch, which interacts, through the micromorphic framework, with the local chain stretch (homogenized from the Arruda-Boyce eight-chain model). All of this allows for a modified J-integral to be derived, which accounts for how the non-local damage evolution influences the available crack energy release rate. From this modified J-integral, the fracture toughness is able to be recovered from the model (instead of being supplied as an a priori model input, as is the case in phase field fracture). The sensitivity of this model to the material-specific micro-micro non-local interaction length and to flaw size is investigated in single-edge notch test specimen loaded under uniaxial tension to failure.