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The common approach to fracture dynamics, linear elastic fracture mechanics(LEFM), assumes infinitesimal deformation gradients, but, in the vicinity ofa crack’s tip predicts diverging r−1/2 crack tip strains, a result that appearsself-contradictory. We derive the leading nonlinear elastic corrections to theseasymptotic fields and show that the resulting theory quantitatively resolves anumber of discrepancies raised by recent near-tip measurements of the strainfield surrounding a dynamic crack, which are presented in an accompanyingpaper. 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 dynamicallength-scale, associated with a nonlinear elastic zone, appears naturally. WhereLEFM falls short, the theory provides excellent quantitative agreement withthe measured, near-tip, displacement and strain fields. The theory servesas a springboard for the development of a comprehensive theory of fracturedynamics.