  A semi-infinite crack in infinite square lattice is subjected to a wave
coming from infinity, thereby leading to its scattering by the crack surfaces.
A partially damaged zone ahead of the crack-tip is modeled by an arbitrarily
distributed stiffness of the damaged links. While the open crack, with an
atomically sharp crack-tip, in the lattice has been solved in closed form with
help of {the} scalar Wiener-Hopf formulation (SIAM Journal on Applied
Mathematics, 75, 1171--1192; 1915--1940), the problem considered here becomes
very intricate depending on the nature of damaged links. For instance, in the
case of partially bridged finite zone it involves a $2\times2$ matrix kernel of
formidable class. But using an original technique, the problem, including the
general case of arbitrarily damaged links, is reduced to a scalar one with the
exception that it involves solving an auxiliary linear system of $N \times N$
equations where $N$ defines the length of the damage zone. The proposed method
does allow, effectively, the construction of an exact solution. Numerical
examples and the asymptotic approximation of the scattered field far away from
the crack-tip are also presented.
