Modeling the fracture of polymer networks

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This dissertation is devoted to modeling the fracture of highly elastic materials that consist of polymer networks, such as elastomers and hydrogels. These polymer materials are composed of long polymer chains of repeating molecular units, which are crosslinked to form a three-dimensional network structure. A polymer network fractures by breaking covalent bonds, but the experimentally measured strength of a polymer network is orders of magnitude lower than the strength of covalent bonds. In this dissertation, we develop mesoscale models to understand what are the necessary ingredients leading to a large reduction in the strength of polymer networks observed in experiments.

We hypothesize that the large reduction in strength is caused by statistical variation in lengths of polymer chains and a J-shaped stress-stretch relationship. The polymer chain carries entropic forces for most of the extension and carries covalent forces only for a narrow range of the extension. As a result, the statistical distribution of chain lengths causes only a small fraction of polymer chains to be highly stressed when the network is near fracture.

We test this hypothesis using two mesoscale models: an idealized parallel chain model and a two-dimensional network model. Both models assume a statistical distribution for the lengths of polymer chains. Polymer chains are represented by freely-jointed chains that feature a nonlinear J-shaped stress-stretch relationship. The parallel chain model allows for simple calculations and is amenable for analysis by analytical tools. The network model accounts for the effect of stress concentration and is amenable for numerical simulations.

Our models show that the combination of a J-shaped stress-stretch relationship and a distribution of chain lengths leads to a large reduction in strength, while keeping the variability in strength small from sample to sample. The large scatter in chain lengths causes a reduction in strength by up to two orders of magnitude, which explains a portion of the giant discrepancy between the experimentally measured strength of hydrogels and the strength of covalent bonds. Furthermore, our models demonstrate a power law relationship between the strength and the scatter in chain lengths. We provide an analytical derivation of the power law by taking advantage of the simplicity of the parallel chain model.

In addition to studying macroscopic fracture properties, we further investigate the microscopic characteristics and the breaking mechanism of the polymer network, using the network model. By examining the characteristics of shortest paths, we find that the links traversed by a large number of shortest paths are more likely to break. Finally, we connect the microstructure of the network to the macroscopic mechanical properties. It is observed that the strength of the network correlates with the growth of holes during deformation.