Numerical methods for brittle fracture propagation with application to hydraulic fracture
Keywords: Mechanical Engineering.

The breaking apart of brittle materials via fracture is ubiquitous in engineering applications and in nature. A hydraulic fracture is one driven by pressurized fluid, which is an important tool in the energy industry and a fundamental physical process in the study of volcanoes. Three major challenges in simulating brittle fracture propagation are (a) constructing discretizations of the problem geometry which account for crack paths that are not known a priori, (b) correctly modeling crack evolution, especially when the relative speeds of individual crack tips or points along a crack front are ambiguous in the context of Griffith's theory of fracture, and (c) accurately computing the singularities in the stress fields which can adversely affect convergence rates of standard numerical methods. Further, hydraulic fracture simulations require a suitable mesh of the crack geometry in order to account for the fluid flow within. In this thesis, three contributions to the problem of brittle (possibly hydraulic) fracture simulation are presented. The first is a method to simulate curvilinear hydraulic fractures under plane-strain or axisymmetric loading conditions. Discretizing the problem geometry is addressed through Universal Meshes, a novel technique to construct conforming triangulations by perturbing a limited number of nodes in a single background (universal) mesh. The conforming triangulation also provides a mesh of the crack surface by segments, and hence the Finite Element Method is used to simultaneously approximate the displacements within the rock and the pressure of the fracturing fluid. The method is benchmarked through a convergence study, a comparison with near-surface experiments, and the study of a hydraulic fracture propagating in a narrow channel. The method is then applied to a problem in volcanology: the growth of magma-filled dikes away from a depressurizing magma chamber. The fully-coupled simulations are used to make predictions about the growth rate of the dike as a function of time and about the pressure in the magma chamber as a function of the dike length. The second contribution is a method to compute the stress intensity factors along the front of a three-dimensional crack. In Linear Elastic Fracture Mechanics, the stress intensity factors are used to predict the onset of crack growth and direction of said growth. However, numerical schemes to calculate these factors in three dimensions often result in oscillatory values, which may not improve under mesh refinement. The presented method formulates functionals which are derived from the interaction integral, and that, when applied to the exact displacement field, yield weighted integrals of the stress intensity factors along the crack front. Through careful analysis of the functionals and the method, conditions under which the method is guaranteed to converge are derived. The method is validated on several crack configurations from the literature. The final contribution in this thesis addresses the challenge of predicting crack advancement in the absence of inertial effects, in particular the ambiguous crack velocities for cases in two dimensions with multiple crack tips or for cases in three dimensions. A model for crack evolution is proposed which extends Griffith's theory for quasi-static growth to account for supercritical propagation, and which resolves the crack speed ambiguity. Algorithms to apply the method in two- and three-dimensional cases are presented, and these algorithms are demonstrated on various examples