Crack opening calculation in phase-field modeling of fluid-filled fracture: A robust and efficient strain-based method

Abstract

The phase-field method has become popular for the numerical modeling of fluid-filled fractures, thanks to its ability to represent complex fracture geometry without algorithms. However, the algorithm-free representation of fracture geometry poses a significant challenge in calculating the crack opening (aperture) of phase-field fracture, which governs the fracture permeability and hence the overall hydromechanical behavior. Although several approaches have been devised to compute the crack opening of phase-field fracture, they require a sophisticated algorithm for post-processing the phase-field values or an additional parameter sensitive to the element size and alignment. Here, we develop a novel method for calculating the crack opening of fluid-filled phase-field fracture, which enables one to obtain the crack opening without additional algorithms or parameters. We transform the displacement-jump-based kinematics of a fracture into a continuous strain-based version, insert it into a force balance equation on the fracture, and apply the phase-field approximation. Through this procedure, we obtain a simple equation for the crack opening which can be calculated with quantities at individual material points. We verify the proposed method with analytical and numerical solutions obtained based on discrete representations of fractures, demonstrating its capability to calculate the crack opening regardless of the element size or alignment.

Figures (14)

• Figure 1: Phase-field approximation of fracture. The sharp discontinuity $\Gamma$ is diffusely approximated by the phase-field variable, $d\in[0,1]$.
• Figure 2: Sneddon problem: domain geometry and boundary conditions.
• Figure 3: Sneddon problem: initial phase-field distribution with three different regularization lengths.
• Figure 4: Sneddon problem: comparison between the analytical solution and phase-field results with (a) different regularization lengths $L$ under a fixed refinement level as $L/h = 10$, and (b) different element sizes $h$ under a fixed regularization length as $L = 0.005$ m.
• Figure 5: Sneddon problem: (a) meshes with different element orientation angles $\theta$, and (b) comparison between the analytical solution and phase-field results with different element orientations. (The refinement level and the regularization length are fixed as $L/h = 10$ and $L = 0.01$ m, respectively.)