Weakly Nonlinear Theory of Dynamic Fracture

The common approach to crack dynamics, linear elastic fracture mechanics, 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 cannot be described in the linear elastic fracture mechanics framework.