The 1*/r *singularity in weakly nonlinear fracture mechanics

Eran Bouchbinder, Ariel Livne
and Jay Fineberg

*Racah** Institute of Physics, Hebrew University
of Jerusalem, *

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.