  Material failure by crack propagation essentially involves a concentration of
large displacement-gradients near a crack's tip, even at scales where no
irreversible deformation and energy dissipation occurs. This physical situation
provides the motivation for a systematic gradient expansion of general
nonlinear elastic constitutive laws that goes beyond the first order
displacement-gradient expansion that is the basis for linear elastic fracture
mechanics (LEFM). A weakly nonlinear fracture mechanics theory was recently
developed by considering displacement-gradients up to second order. The theory
predicts that, at scales within a dynamic lengthscale $\ell$ from a crack's
tip, significant $\log{r}$ displacements and $1/r$ displacement-gradient
contributions arise. Whereas in LEFM the $1/r$ singularity generates an
unbalanced force and must be discarded, we show that this singularity not only
exists but is {\em necessary} in the weakly nonlinear theory. The theory
generates no spurious forces and is consistent with the notion of the autonomy
of the near-tip nonlinear region. The J-integral in the weakly nonlinear theory
is also shown to be path-independent, taking the same value as the linear
elastic J-integral. Thus, the weakly nonlinear theory retains the key tenets of
fracture mechanics, while providing excellent quantitative agreement with
measurements near the tip of single propagating cracks. As $\ell$ is consistent
with lengthscales that appear in crack tip instabilities, we suggest that this
theory may serve as a promising starting point for resolving open questions in
fracture dynamics.
