Numerical studies of complex materials

This dissertation presents two inter-related studies.

Chapter 2 focuses on a study of three-dimensional numerical simulations of hydraulic fractures. Hydraulic fracturing is widely used to extract shale gas from shale formations. Pressurized water is injected into horizontal wells in shale formations to create fracture networks. If a network contains many fractures, the shale gas is easier to transport to the surface via this network. To date, the shapes of hydraulic fracture networks in shale formations are still not precisely known. This research is to numerically investigate three-dimensional hydraulic fractures with molecular dynamics simulations, which integrates the physics of linear elasticity and fracture mechanics as well as the discrete channel and fluid variables to dynamically see the propagation of hydraulic fractures. This approach allows an arbitrary number of cracks and arbitrary crack paths, and I have further developed a simulator to simulate the complex networks of hydraulic fractures in response to various states of stress in the shale formation. I have also proposed an extra term to implement the Poisson effect, which connecting deformations in perpendicular directions, for a future study for more complicated three-dimensional fracture networks such as echelon cracks.

Chapter 3 presents the research on static cracks in a natural rubber sheet, seeking a strain energy functional that can accurately describe it. The type of static crack is due to the strain-induced crystallization of natural rubber and has two unusual features: pointy crack tips and a sharp boundary between the two regions with distinct deformations. When a latex rubber sheet is stretched beyond a critical stretch ratio, a slit that is cut in the middle will not cause this sheet to break apart but forms a static crack due to the crystallization of rubber. The pointy crack tips cannot be explained by the existing theories in fracture mechanics, and the sharp change of deformation is also puzzling. I have built molecular dynamics simulations to verify proposed strain energy functionals and see whether the particular crack shape in rubber can be implemented. I have got simulation results with the two features based on a strain energy functional curve with its two energy minima corresponding to the two phases. In hyperelasticity, the strain energy functional that describes the strain-induced crystallization in an isotropic material has not been mathematically formulated. The discovery shows an insight into a new form of the strain energy functional for phase separation with spatial complexity in nonlinear elasticity.