Phase-field approaches to discontinuities and fractures in geologic materials

Abstract

Geologic materials contain a wide range of discontinuities and fractures, which are central to many engineering applications and geologic hazards. The fracturing process of geologic materials is characterized by its progressive softening, termed quasi-brittleness. Also, the fractured surfaces exhibit a number of features including frictional contact and roughness effects. To model the discontinuities and fracturing processes in geologic materials, the phase-field method has been increasingly applied, as it has an outstanding ability to handle complex crack geometries without using tracking algorithms. However, few phase-field studies considered the quasi-brittleness of geologic materials. More importantly, all existing phase-field approaches dismissed the frictional contact, let alone the roughness effects.

To fill the above research gaps, this thesis develops a suite of phase-field approaches to enable more reliable and systematic modeling of discontinuities and fractures in geologic materials. These approaches focus on four different but interconnected aspects of geologic discontinuities and fractures, namely frictional contact, shear fracture incorporating friction dissipation, mixed-mode rock fracture, and roughness effects of rock fractures. In the first approach, we incorporate the pressure-dependent friction into the phase-field formulation by employing a crack-oriented decomposition of the stress tensor. Each stress component is calculated by identifying the contact condition at the material point of interest. We show that the proposed method can well reproduce the results from the standard and extended finite element method without applying any algorithms to impose contact constraints.

Building on the formulation in the first approach, we develop a phase-field approach to model the shear fracture propagation that involves friction dissipation during the fracturing process. The proposed formulation is demonstrably consistent with a fracture-mechanics-based theory. We also devise a new degradation function to avoid the sensitivity of material strengths to the phase-field length parameters, allowing the proposed method to model quasi-brittle materials with prescribed strengths.

Next, we introduce a double-phase-field approach to the mixed-mode rock fracture by combining the formulations of cohesive tensile cracks and frictional shear cracks. The proposed formulation is essentially based on three steps: (i) stress decomposition in a crack-oriented coordinate system; (ii) calculation of the total potential energy according to the contact condition; (iii) determination of the dominant fracture mode following an energy-based criterion. We validate the double-phase-field approach through qualitative and quantitative comparisons between the modeling results and the experimental results.

Lastly, we introduce a phase-field modeling framework for rock fractures by incorporating roughness effects. The proposed framework aims at transforming a displacement-based constitutive law of rock fractures into a strain-based version without introducing new parameters. In doing so, the continuous phase-field method can accommodate the rough fracture models originally designed for discrete discontinuities. Numerical examples show that the phase-field results have an excellent agreement with the results obtained from the extended finite element method.