In the present study, a finite element formulation for modeling crack propagation in hyperelastic thin membranes is developed. The crack formation and propagation are modeled by means of the material-sink approach stemming from the physical observation of the diffused bond breakage. Keeping in mind that loss of local bonds leads to localized material loss, the mass density can be considered as a variable, which numerically decreases in the area where damage localizes into a crack. This notion requires mathematical consideration of mass balance as an additional and active law, which regularizes the computational model. From the numerical point of view, the developed computational model has displacement and density degrees of freedom. Also, a monolithic approach was applied that ensures stable incrimination of the nonlinear problem. Numerical examples of the fracture of different geometries demonstrate the high robustness of the proposed approach in modeling crack propagation in membrane structures.