Damaging and Cracks in Thin Mud Layers

We present a detailed study of a two-dimensional minimal lattice model for the description of mud cracking in the limit of extremely thin layers. In this model each bond of the lattice is assigned to a (quenched) breaking threshold. Fractures proceed through the selection of the part of the material with the smallest breaking threshold. A local damaging rule is also implemented, by using two different types of weakening of the neighboring sites, corresponding to different physical situations. Some analytical results are derived through a probabilistic approach known as Run Time Statistics. In particular, we find that the total time to break down the sample grows with the dimension $L$ of the lattice as $L^2$ even though the percolating cluster has a non trivial fractal dimension. Furthermore, a formula for the mean weakening in time of the whole sample is obtained.