  Despite its critical role in the study of earthquake processes, numerical
simulation of the entire stages of fault rupture remains a formidable task. The
main challenges in simulating a fault rupture process include complex evolution
of fault geometry, frictional contact, and off-fault damage over a wide range
of spatial and temporal scales. Here, we develop a phase-field model for
quasi-dynamic fault nucleation, growth, and propagation, which features two
standout advantages: (i) it does not require any sophisticated algorithms to
represent fault geometry and its evolution; and (ii) it allows for modeling
fault nucleation, propagation, and off-fault damage processes with a single
formulation. Built on a recently developed phase-field framework for shear
fractures with frictional contact, the proposed formulation incorporates rate-
and state-dependent friction, radiation damping, and their impacts on fault
mechanics and off-fault damage. We show that the numerical results of the
phase-field model are consistent with those obtained from well-verified
approaches that model the fault as a surface of discontinuity, without
suffering from the mesh convergence issue in the existing continuous approaches
to fault rupture (e.g. the stress glut method). Further, through numerical
examples of fault propagation in various settings, we demonstrate that the
phase-field approach may open new opportunities for investigating complex
earthquake processes that have remained overly challenging for the existing
numerical methods.
