Fragmentation of thin films bonded to solid substrate: Simulations and a mean-field theory

We perform simulations of crack growth in a brittle-elastic two-dimensional film bonded to a rigid substrate and subjected to an isotropic tensile stress. We find that the resulting fracture patterns can be classified according to the applied tensile stress. For a significant range of stresses, the crack interactions are extremely shell range. The resulting fracture patterns are independent of stress and depend only on the initial distribution of crack nuclei. In this regime, the fracture process is well described by a fragmentation model in which a random distribution of seeds nucleates pairs of oppositely directed rays which grow until encountering other rays. Using a mean-field approach, we obtain approximate analytical results for this model which compare well with both the simulations of the fragmentation model, and with the fracture simulation results. Our approach is extensible to other models of fracture, and where numerical results exist in the literature, our mean-field solutions agree well with them.