Scaling and instability of dynamic fracture
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This dissertation presents three inter-related studies. Chapter 2 presents a study of scaling of crack propagation in rubber sheets. Two different scaling laws for supersonic and subsonic cracks were discovered. Experiments and numerical simulations have been conducted to investigate subsonic and supersonic cracks. The experiments are performed at 85 °C to suppress strain-induced crystallites that complicate experiments at lower temperature. Calibration experiments were performed to obtain the parameters needed to compare with a theory including viscous dissipation. Both experiments and numerical simulations support supersonic cracks, and a transition from subsonic to supersonic is discovered in the plot of experimental crack speed curves versus extension ratio for different sized samples. Both experiments and simulations show two different scaling regimes: the speed of subsonic cracks scales with the elastic energy density while the speed of supersonic cracks scales with the extension ratio. Crack openings have qualitatively different shapes in the two scaling regimes. Chapter 3 describes a theory of oscillating cracks. Oscillating cracks are not seen very widely, but observed in rubber and gels. A theory has been proposed for the onset of oscillation in gels, but the oscillation of cracks in rubber has not been explained. This study provides a theory able to describe both rubber and gels and recover the experimental phase diagram for oscillating cracks in rubber. The main new idea is that the oscillations of cracks follow from basic features of fracture mechanics and are independent of details of the crack equation of motion. From the fact that oscillations exist, one can deduce some conditions on forms that equations of motion can take. A discrete model of hydraulic fracture is mentioned in Chapter 4. Hydraulic fracturing is a stimulation treatment wherein fluids are injected into reservoirs under high pressure to generate fractures in reservoirs. In this study, a lattice-based pseduo-3D model is developed to simulate hydraulic fracturing. This mode has been validated via a comparison with the KGD model. A series of pilot simulations was systematically tested for complex geometries under more realistic operation conditions, including flexible boundary conditions, randomness in elastic properties of shales and perforations. The simulation results confirm that perforation is likely to increase the complexity of fracture networks; the results also suggest that the interference between neighboring fractures is key to fracture network formation.