A computational model for arbitrary brittle crack propagation, in a fault-like layer within a 3-d elastic domain, and its associated quasi-static and dynamic fields is developed and analyzed. It uses a FFT-based solver for the balance of linear momentum and a Godunov-type projection-evolution method for the crack evolution equation. As applications, we explore the questions of equilibria and irreversibility for crack propagation with and without surface energy, existence of strength and toughness criteria, crack propagation under quasi-static and dynamic conditions, including Modes I, II and III, as well as multiaxial compressive loadings.