  To obtain the probability distribution of 2D crack patterns in mesoscopic
regions of a disordered solid, the formalism of Paper I requires that a
functional form associating the crack patterns (or states) to their formation
energy be developed. The crack states are here defined by an order parameter
field representing both the presence and orientation of cracks at each site on
a discrete square network. The associated Hamiltonian represents the total work
required to lead an uncracked mesovolume into that state as averaged over the
initial quenched disorder. The effect of cracks is to create mesovolumes having
internal heterogeneity in their elastic moduli. To model the Hamiltonian, the
effective elastic moduli corresponding to a given crack distribution are
determined that includes crack-to-crack interactions. The interaction terms are
entirely responsible for the localization transition analyzed in Paper III. The
crack-opening energies are related to these effective moduli via Griffith's
criterion as established in Paper I.
