The 1/r singularity in weakly nonlinear fracture mechanics

TL;DR

This paper develops a weakly nonlinear fracture mechanics theory that incorporates a 1/r singularity at the crack tip, providing better quantitative agreement with experiments and maintaining core principles like path independence of the J-integral.

Contribution

It introduces a second-order displacement-gradient expansion in fracture mechanics, showing the necessity of the 1/r singularity and preserving the fundamental properties of the linear theory.

Findings

The 1/r singularity is necessary and consistent in the weakly nonlinear theory.

The weakly nonlinear J-integral remains path-independent and equals the linear elastic value.

The theory aligns well with experimental measurements near crack tips.

Abstract

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.