Lattice and continuum predictions of crack-tip stability

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Abstract
The stability of a crack which has a nonlinear core emerging from the tip onto the plane ahead of the crack is investigated in terms of K I−K II loading. The principal focus is to address a paradox that the Griffith relation, G=2γ s , permits equilibrium cracks to exist in pure mode‐II loading, yet under such conditions there would be insufficient tensile force to pull atoms apart to maintain free surfaces. Using a Peierls‐type framework developed by Rice [J. Mech. Phys. Solids 40, 239 (1992)], a continuum analysis of a crack with a nonlinear core ahead of the crack tip is presented to demonstrate that, indeed, the crack‐core structure is stable to self‐similar translation when the Griffith condition is met. However, the portions of the Griffith curve which have a sufficient fraction of mode‐II loading—approximately ‖K II‖/K I≥0.4 for the particular bonding properties considered here, are unattainable because dislocation emission intervenes. Corresponding studies of an equilibrium crack in a 2D hexagonal lattice demonstrate that there is a band of K I−K II values within which the crack is stable. The band is approximately centered on the Griffith curve and extends to critical K II values comparable to those at which the continuum model predicts dislocation emission to intervene. The finite width of the band occurs due to lattice trapping, and that width is observed to broaden as K II is increased. In this context, the continuum model represents the case of zero trapping. Consequently, a satisfactory explanation of the Griffith crack paradox is that pure mode‐II equilibrium cracks are unattainable because dislocation emission from the crack tip intervenes before the pure mode‐II Griffith value can be reached.

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