  The common approach to crack dynamics, linear elastic fracture mechanics
(LEFM), assumes infinitesimal strains and predicts a $r^{-1/2}$ strain
divergence at a crack tip. We extend this framework by deriving a weakly
nonlinear fracture mechanics theory incorporating the leading nonlinear elastic
corrections that must occur at high strains. This yields strain contributions
"more-divergent" than $r^{-1/2}$ at a finite distance from the tip and
logarithmic corrections to the parabolic crack tip opening displacement. In
addition, a dynamic length-scale, associated with the nonlinear elastic zone,
emerges naturally. The theory provides excellent agreement with recent near-tip
measurements that can not be described in the LEFM framework.
