  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.
