  The dynamics of a single crack moving through a heterogeneous medium is
studied in the quasi-static approximation. Equations of motion for the crack
front are formulated and the resulting scaling behaviour analyzed. In a model
scalar system and for mode III (tearing) cracks, the crack surface is found to
be self affine with a roughness exponent of $\zeta=1/2$. But in the usual
experimental case of mode I (tensile) cracks, local mode preference causes the
crack surface to be only logarithmically rough, quite unlike those seen in
experiments. The effects of residual stresses are considered and found,
potentially, to lead to increased crack surface roughness. But it appears
likely that elastic wave propagation effects may be needed to explain the very
rough crack surfaces observed experimentally.
