A novel cellular automaton (CA) for simulating biological membrane rupture is proposed. Constructed via simple rules governing deformation, tension, and fracture, the CA incorporates ideas from standard percolation models and bond-based fracture methods. The model is demonstrated by comparing simulations with experimental results of a double bilayer lipid membrane expanding on a solid substrate. Results indicate that the CA can capture non-trivial rupture morphologies such as floral patterns and the saltatory dynamics of fractal avalanches observed in experiments. Moreover, the CA provides insight into the poorly understood role of inter-layer adhesion, supporting the hypothesis that the density of adhesion sites governs rupture morphology.