The 1/r singularity in weakly nonlinear fracture mechanics

Eran Bouchbinder, Ariel Livne and Jay Fineberg

Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

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 »rst 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 ` from a crack's tip, signi»cant 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 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

` 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.