A theory of local limiting speed in dynamic fracture

A nonlinear continuum analysis is developed to show that stable, steady-state crack motion is limited not only by the macroscopic Rayleigh wave speed as asserted by the established theory of dynamic fracture, bur also by a local wave speed governed by the elastic response near the crack tip. The local limiting speed ensures that a subsonic steady-state field can be established in highly nonlinear material regions prior to rupture. A two-dimensional triangular lattice with nearest-neighbor interatomic bonding is studied as a model nonlinear elastic solid that is isotropic under infinitesimal strains, but becomes anisotropic and nonlinear when the lattice is heavily stretched. The local limiting speed is determined by considering the most critical state of deformation on the verge of bond rupture. If the critical state is assumed to be under equibiaxial stressing, the local limiting speed is found to be v(1) = c(s) root sigma(max)/mu, where c(s) is the macroscopic shear wave speed, mu is the shear modulus and sigma(max) is the equibiaxial cohesive strength of the solid (i.e. the maximum equibiaxial tensile stress that a flawless solid can stand without spontaneous rupture). The generality of this result is discussed by relaxing the restrictions in the model problem. It is also shown that lattice dispersion in front of a crack tip can further reduce the speed of bond-breaking stress waves with wavelength on the order of a few atomic spacings. This study lends further support for a viewpoint previously discussed by the author that high speed dynamic fracture involves a competition between a high inertia local crack-tip field and the surrounding low inertia apparent crack field. Motivated by recent molecular dynamics simulations of crack propagation in a 6-12 Lenard-Jones lattice, a variational principle for steady-state deformation is used together with a conjugate gradient minimization algorithm to compute atomistic responses near the tip of a crack moving with constant speed in a similar Lenard-Jones lattice. The computation is performed over a block which moves with the crack and is subjected along the boundary to a low inertia displacement held based on existing solutions for cracks moving in linear elastic solids. The critical velocity at the onset of local crack branching is found to be 0.30c(s), in almost exact agreement with the earlier molecular dynamics study. In this case, the local limiting speed is calculated to be v(1) = 0.37c(s), which is 20% larger than the observed value. This difference can be attributed to the effects of local lattice dispersion. The results are fully supportive of the notion that global-local inertia competition is a key to understanding dynamic fracture instabilities. Copyright (C) 1996 Elsevier Science Ltd