Large deformation and brittle faulting in folding rock strata

Mechanical models that follow the evolution of stresses and deformation are essential for understanding the origin of folding and the sequence of faulting in geologic strata. Although much progress has been made to understand how small-scale discontinuities (joints and deformation bands) develop during folding, we are not in a position to predict their location, orientation, or spatial density from the large-scale geometry of folded strata. Furthermore, the standard finite element method, which has been the cornerstone of large deformation analysis of complex boundary-value problems, requires expensive adaptive re-meshing to accommodate the intense deformation and displacement discontinuities in the neighborhood of a propagating crack. Our approach in this paper is to focus on the growth of a single crack in an initially intact rock. We model the quasi-static 2D crack growth using the extended finite element method, or XFEM. Within the framework of XFEM, the geometry of the crack is represented by a level set function that is interpolated with a finite element discretization, and the discontinuity in the displacement field is introduced into the model by a partition of unity method (PUM). The advantage of this proposed modeling scheme is that growth of crack can be analyzed using the finite element method without re-meshing. Finite deformation formulation is adopted to simulate fracture propagation in folding rock strata. Under a linear elastic fracture mechanics (LEFM) framework, the mode I and mode II stress intensity factors are determined with the help of path independent integrals such as the J-integral, and the crack growth direction is given by the maximum hoop stress criterion. This approach is extended to handle geometrically nonlinear fracture mechanics problems, where the J-integral is now evaluated with the Lagrangian coordinates. Similarly, other fracture mechanics parameters such as the stress intensity factors can also be related to their finite deformation counterparts. Several model problems are simulated with this new approach.