In this thesis, the class of axisymmetrical problems with moving boundaries, which are related to fracture, is considered. This is achieved through investigations into three separate examples; solid particle erosion, hydraulic fracturing, and particles in the Hele-Shaw cell. In this way, the methods used to study such problems will be demonstrated. The first examination considers the case of an axisymmetric indenter, defined by a power law, impacting an elastic medium. The primary motivation of this work is to determine the cause of the threshold fracture paradox, which concerns the relationship between the indenter shape and the initial energy required to cause a fracture in the impacted medium. In this study, the formation of cracks is determined through an incubation time based approach, which accounts for the dynamic nature of the impact. The effect of incorporating inertial terms in the medium is also examined, and the implications for the study of high velocity impacts by small indenters are discussed. Next, the problem of a radial (penny-shaped) hydraulic fracture is considered, with a 3D axisymmetric crack forming around a point source in the center. The aim of this effort is to provide a high accuracy numerical solver, based on an explicit level set algorithm. This is achieved through application of the proper Stefan-type condition, which describes the moving boundary, alongside extensive use of the known relationships between the parameters crack-tip asymptotics. Two cases are considered; the first is the classical formulation, while the second incorporates the effect of tangential stress induced by the fluid on the fracture walls. The level of computational error is determined against newly constructed analytical benchmarks. The obtained solutions are used to determine the accuracy of other results available in the literature. Finally, the case of multiple moving particles and stationary obstacles inside the Hele-Shaw cell is examined. The boundary is assumed to be free-moving, with additional fluid entering or leaving the system through a point source at the origin. The evolution of the fluid boundary is modeled based on a Green's function approach, which is approximated asymptotically. A numerical solver is developed to provide high accuracy approximations for the systems development over time, which also accounts for collisions between the particles. A method of utilizing this model to study the apparent viscosity of fluids used in hydraulic fracturing is outlined.
|Date of Award
|Marie Skłodowska-Curie Actions
|Gennady Mishuris (Supervisor)