Lattice Trapping of Fracture Cracks

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Abstract
Continuum treatments of lattice defects such as dislocations and fracture cracks do not predict the resistance to the defect mobility which is due to the Peierls energy in the case of the dislocation. The discrete character of the lattice in the case of the fracture crack leads to a stress stability range for the crack above and below the Griffith stress over which the crack is stable or ``lattice trapped''. We develop here two essentially qualitative theoretical treatments of the lattice structure of cracks. In the first, we carry out a lattice sum of the bond energies of atoms facing each other across the crack plane under certain assumptions regarding the crack shape and atomic force laws. In the second, we introduce a one-dimensional lattice model of a crack which can be solved exactly. In both of these treatments the range of stress over which the crack is lattice trapped appears to be of the order of magnitude of the Griffith stress itself. As in the case of dislocations the lattice trapping is a strong function of the ``width'' of the elastic singularity at the tip, and our prediction is that one should expect materials to exist in which lattice trapping is not only observable, but is an important effect. We also find that because of lattice trapping, the true surface energy is a lower bound to the mechanical surface energy as expressed in the Griffith-stress relation.

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