A theoretical description of the structure of the nonlinear zone in the vicinity of the tip of a dynamic mode I crack

The common approach to fracture dynamics, linear elastic fracture mechanics (LEFM), assumes infinitesimal deformation gradients, but, in the vicinity of a crack's tip predicts diverging r^-1/² crack tip strains, a result that appears self-contradictory. We derive the leading nonlinear elastic corrections to these asymptotic fields and show that the resulting theory quantitatively resolves a number of discrepancies raised by recent near-tip measurements of the strain field surrounding a dynamic crack, which are presented in an accompanying paper. We show that no region of r ^-1/² dominance exists and "more-divergent" strain terms occur at a finite distance from the tip. In addition, a dynamical length-scale, associated with a nonlinear elastic zone, appears naturally. Where LEFM falls short, the theory provides excellent quantitative agreement with the measured, near-tip, displacement and strain fields. The theory serves as a springboard for the development of a comprehensive theory of fracture dynamics.